The Connectives

The Connectives The Connectives Lloyd Humberstone The MIT Press Cambridge, Massachusetts London, England ©2011 Mas...

Author: Lloyd Humberstone

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The Connectives

The Connectives Lloyd Humberstone

The MIT Press Cambridge, Massachusetts London, England

©2011 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. MIT Press books may be purchased at special quantity discounts for business or sales promotional use. For information, please email special_sales@mitpress .mit.edu or write to Special Sales Department, The MIT Press, 55 Hayward Street, Cambridge, MA 02142. This book was set in Computer Modern Roman by the author and was printed and bound in the United States. Library of Congress Cataloging-in-Publication Data Humberstone, Lloyd. The connectives / Lloyd Humberstone. p. cm. Includes bibliographical references and index. ISBN 978-0-262-01654-4 (alk. paper) 1. Logic. 2. Grammar, Comparative and general—Connectives. 3. Language and Logic. I. Title. BC57.H86 2011 160—dc22 2010053835

10 9 8 7 6 5 4 3 2 1

DEDICATION This book is dedicated to Reed’s School, Cobham, Surrey, in gratitude for the years 1963–1967, during which I first learnt, among many other things, that there was such a subject as formal logic.

Contents Preface and Navigation Guide . . . . . . . . . . . . . . . . . . . . xiii 0 Preliminaries 0.1 CONNECTIONS AND COMBINATIONS . . . . . . . . . . . . . 0.11 Relational Connections and Posets . . . . . . . . . . . . . 0.12 Galois Connections . . . . . . . . . . . . . . . . . . . . . . 0.13 Lattices and Closure Operations . . . . . . . . . . . . . . 0.14 Modes of Object Combination . . . . . . . . . . . . . . . Notes and References for §0.1 . . . . . . . . . . . . . . . . 0.2 SOME ALGEBRAIC CONCEPTS . . . . . . . . . . . . . . . . . . 0.21 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.22 Derived Operations . . . . . . . . . . . . . . . . . . . . . . 0.23 Homomorphisms, Subalgebras, Direct Products . . . . . . 0.24 Equational Classes of Algebras . . . . . . . . . . . . . . . 0.25 Horn Formulas . . . . . . . . . . . . . . . . . . . . . . . . 0.26 Fundamental and Derived Objects and Relations: Tuple Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes and References for §0.2 . . . . . . . . . . . . . . . .

1 1 1 3 7 11 16 17 17 23 26 29 32

1 Elements of Sentential Logic 1.1 TRUTH AND CONSEQUENCE . . . . . . . . . . . . . . . . . . . 1.11 Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Consequence Relations and Valuations . . . . . . . . . . . 1.13 #-Classical Consequence Relations and #-Boolean Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.14 Forcing Consistent Valuations to Be #-Boolean . . . . . . 1.15 Two Equations and the ‘Converse Proposition’ Fallacy . . 1.16 Generalized Consequence Relations . . . . . . . . . . . . . 1.17 Generalized Consequence Relations: Supplementary Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.18 Truth-Tables, and Two More Connectives: Implication and Equivalence . . . . . . . . . . . . . . . . . . . . . . . 1.19 A More Hygienic Terminology . . . . . . . . . . . . . . . Notes and References for §1.1 . . . . . . . . . . . . . . . . 1.2 RULES AND PROOF . . . . . . . . . . . . . . . . . . . . . . . . . 1.21 Sequents and Frameworks . . . . . . . . . . . . . . . . . . 1.22 Sequents and (Generalized) Consequence Relations . . . . 1.23 A Sample Natural Deduction System in Set-Fmla . . . . 1.24 A Closer Look at Rules . . . . . . . . . . . . . . . . . . .

47 47 47 54

vii

37 43

61 67 72 73 79 82 87 100 103 103 112 114 122

CONTENTS

viii 1.25 1.26 1.27

Semantic Apparatus for Sequents and for Rules . . . . . . Two Relational Connections . . . . . . . . . . . . . . . . . Sample Natural Deduction and Sequent Calculus Systems in Set-Set . . . . . . . . . . . . . . . . . . . . . . . . . . Some Other Approaches to Logic in Set-Set . . . . . . . Axiom Systems and the Deduction Theorem . . . . . . . Appendix to §1.2: What Is a Logic? . . . . . . . . . . . . Notes and References for §1.2 . . . . . . . . . . . . . . . .

127 132

2 A Survey of Sentential Logic 2.1 MANY-VALUED LOGIC AND ALGEBRAIC SEMANTICS . . . 2.11 Many-Valued Logic: Matrices . . . . . . . . . . . . . . . . 2.12 Many-Valued Logic: Classes of Matrices . . . . . . . . . . 2.13 Algebraic Semantics: Matrices with a Single Designated Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.14 -Based Algebraic Semantics . . . . . . . . . . . . . . . . 2.15 A Third Version of Algebraic Semantics: Indiscriminate Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.16 From Algebraic Semantics to Equivalent Algebraic Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes and References for §2.1 . . . . . . . . . . . . . . . . 2.2 MODAL LOGIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.21 Modal Logic in Fmla: Introduction . . . . . . . . . . . . 2.22 Modal Logic in Fmla: Kripke Frames. . . . . . . . . . . . 2.23 Other Logical Frameworks . . . . . . . . . . . . . . . . . . Notes and References for §2.2 . . . . . . . . . . . . . . . . 2.3 THREE RIVALS TO CLASSICAL LOGIC . . . . . . . . . . . . . 2.31 Quantum Logic . . . . . . . . . . . . . . . . . . . . . . . . 2.32 Intuitionistic Logic . . . . . . . . . . . . . . . . . . . . . . 2.33 Relevant and Linear Logic . . . . . . . . . . . . . . . . . . Notes and References for §2.3 . . . . . . . . . . . . . . . .

195 195 195 211

1.28 1.29

3 Connectives: Truth-Functional, Extensional, Congruential 3.1 TRUTH-FUNCTIONALITY . . . . . . . . . . . . . . . . . . . . 3.11 Truth-Functional Connectives . . . . . . . . . . . . . . . 3.12 Pathologies of Overdetermination . . . . . . . . . . . . . 3.13 Determinant-induced Conditions on a Consequence Relation . . . . . . . . . . . . . . . . . . . 3.14 Functional Completeness, Duality, and Dependence . . . 3.15 Definability of Connectives . . . . . . . . . . . . . . . . 3.16 Defining a Connective . . . . . . . . . . . . . . . . . . . 3.17 Truth-Functions, Rules, and Matrices . . . . . . . . . . 3.18 Intuitionistically Dangerous Determinants . . . . . . . . Notes and References for §3.1 . . . . . . . . . . . . . . . 3.2 EXTENSIONALITY . . . . . . . . . . . . . . . . . . . . . . . . 3.21 A Biconditional-based Introduction to Extensionality . . 3.22 A Purified Notion of Extensionality (for GCRs) . . . . . 3.23 Some Extensionality Notions for Consequence Relations 3.24 Hybrids and the Subconnective Relation . . . . . . . . . Notes and References for §3.2 . . . . . . . . . . . . . . .

140 150 156 180 188

219 246 250 257 268 275 275 282 288 296 298 298 302 326 369

375 . 375 . 375 . 382 . . . . . . . . . . . . .

385 403 418 423 428 436 442 444 444 448 453 461 483

CONTENTS 3.3 CONGRUENTIALITY . . . . . . . . . . 3.31 Congruential Connectives . . . . 3.32 Some Related Properties . . . . . 3.33 The Three Properties Compared 3.34 Operations vs. Relations . . . . . Notes and References for §3.3 . .

ix . . . . . .

484 484 490 495 497 508

Existence and Uniqueness of Connectives 4.1 PHILOSOPHICAL PROOF THEORY . . . . . . . . . . . . . . . 4.11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Normalization of Proofs in Positive Logic . . . . . . . . . 4.13 A Proof-Theoretic Notion of Validity . . . . . . . . . . . . 4.14 Further Considerations on Introduction and Elimination Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes and References for §4.1 . . . . . . . . . . . . . . . . 4.2 EXISTENCE OF CONNECTIVES . . . . . . . . . . . . . . . . . 4.21 The Conservative Extension Proposal . . . . . . . . . . . 4.22 Conditions not Corresponding to Rules: An Example . . 4.23 Other Grounds for Claiming Non-existence . . . . . . . . 4.24 Non-partisan Existence Questions . . . . . . . . . . . . . Notes and References for §4.2 . . . . . . . . . . . . . . . . 4.3 UNIQUENESS OF CONNECTIVES . . . . . . . . . . . . . . . . 4.31 Unique Characterization: The Basic Idea . . . . . . . . . 4.32 Stronger-than-Needed Collections of Rules . . . . . . . . . 4.33 Intermission: Rules Revisited . . . . . . . . . . . . . . . . 4.34 Stronger-than-Needed Collections of Rules (Continued) . 4.35 Unique Characterization by Zero-Premiss Rules: A Negative Example . . . . . . . . . . . . . . . . . . . . . . . . 4.36 Unique Characterization in Modal Logic . . . . . . . . . . 4.37 Uniqueness and ‘New Intuitionistic Connectives’ . . . . . 4.38 Postscript on ‘New Intuitionistic Connectives’ . . . . . . . Notes and References for §4.3 . . . . . . . . . . . . . . . .

511 511 511 513 516

5 “And” 5.1 CONJUNCTION IN NATURAL LANGUAGE AND IN FORMAL LOGIC . . . . . . . . . . . . . 5.11 Syntax, Semantics, Pragmatics . . . . . . . . . . . . . . . 5.12 Temporal and Dynamic Conjunction . . . . . . . . . . . . 5.13 Why Have Conjunction? . . . . . . . . . . . . . . . . . . . 5.14 Probability and ∧-Introduction . . . . . . . . . . . . . . . 5.15 ‘Intensional Conjunction’ and ∧-Elimination . . . . . . . . 5.16 Conjunction and Fusion in Relevant Logic . . . . . . . . . Notes and References for §5.1 . . . . . . . . . . . . . . . . 5.2 LOGICAL SUBTRACTION . . . . . . . . . . . . . . . . . . . . . 5.21 The Idea of Logical Subtraction . . . . . . . . . . . . . . . 5.22 Four Choices to Make . . . . . . . . . . . . . . . . . . . . 5.23 Cancellation, Independence, and a ‘Stipulated Equivalence’ Treatment . . . . . . . . . . . . . . . . . . . . . . . 5.24 Content Subtraction and Converse Implication . . . . . . 5.25 Requirement Semantics: First Pass . . . . . . . . . . . . .

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519 535 536 536 548 566 571 576 578 578 584 588 589 597 601 605 614 626

631 631 639 645 650 658 661 673 677 677 680 683 689 692

CONTENTS

x 5.26 5.27

Requirement Semantics: Second Pass . . . . . Requirement Semantics: Final Considerations Notes and References for §5.2 . . . . . . . . . 5.3 ∧-LIKE CONNECTIVES AND THE LIKE . . . . . . 5.31 #-Like and #-Representable Connectives . . 5.32 Non-creative Definitions . . . . . . . . . . . . 5.33 #-Representable Binary Relations . . . . . . 5.34 Ordered Pairs: Theme and Variations . . . . 5.35 Hybridizing the Projections . . . . . . . . . . Notes and References for §5.3 . . . . . . . . .

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698 700 707 708 708 720 729 738 750 765

6 “Or” 6.1 DISTINCTIONS AMONG DISJUNCTIONS . . . . . . . . . . . . 6.11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 Inclusive/Exclusive . . . . . . . . . . . . . . . . . . . . . . 6.13 Intensional/Extensional . . . . . . . . . . . . . . . . . . . 6.14 Conjunctive-Seeming Occurrences of Or . . . . . . . . . . Appendix to §6.1: Conjunctive-Seeming Or with Apparent Wide Scope . . . . . . . . . . . . . . . . . . . . . . . . Notes and References for §6.1 . . . . . . . . . . . . . . . . 6.2 ARGUMENT BY CASES IN THEORY AND IN PRACTICE . . 6.21 Problematic Applications of ∨-Elimination . . . . . . . . 6.22 Commentary on the Examples . . . . . . . . . . . . . . . 6.23 Supervaluations . . . . . . . . . . . . . . . . . . . . . . . . 6.24 Distributive Lattices of Theories . . . . . . . . . . . . . . Notes and References for §6.2 . . . . . . . . . . . . . . . . 6.3 COMMAS ON THE RIGHT . . . . . . . . . . . . . . . . . . . . . 6.31 GCRs Agreeing with a Given Consequence Relation . . . 6.32 The Model-Consequence Relation in Modal Logic . . . . . 6.33 Generalizing the Model-Consequence Relation . . . . . . . Notes and References for §6.3 . . . . . . . . . . . . . . . . 6.4 DISJUNCTION IN VARIOUS LOGICS . . . . . . . . . . . . . . . 6.41 The Disjunction Property and Halldén-Completeness . . . 6.42 The ‘Rule of Disjunction’ in Modal Logic – and More on the Disjunction Property . . . . . . . . . . . . . . . . . . 6.43 Disjunction in the Beth Semantics for Intuitionistic Logic 6.44 Disjunction in the ‘Possibilities’ Semantics for Modal Logic . . . . . . . . . . . . . . . . . . . . . . . . 6.45 Disjunction in Urquhart-style Semantics for Relevant Logic . . . . . . . . . . . . . . . . . . . . . . 6.46 ‘Plus’ Semantics and Valuational Semantics for Disjunction . . . . . . . . . . . . . . . . . . . . . . . . 6.47 Quantum Disjunction . . . . . . . . . . . . . . . . . . . . Notes and References for §6.4 . . . . . . . . . . . . . . . .

767 767 767 780 789 799

7

925 925 925 948 975

“If ” 7.1 CONDITIONALS . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Issues and Distinctions . . . . . . . . . . . . . . . . . 7.12 Even If, Only If, and Unless . . . . . . . . . . . . . . 7.13 Material Implication and the Indicative Conditional

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812 816 820 820 823 830 833 842 843 843 850 854 860 861 861 872 893 899 905 910 918 922

CONTENTS

xi

7.14 7.15 7.16 7.17 7.18

Positive Implication and Clear Formulas . . . . . . . . . . From Strict to Variably Strict Conditionals . . . . . . . . Interlude: Suppositional Proof Systems . . . . . . . . . . Possible Worlds Semantics for Subjunctive Conditionals . ‘Counterfactual Fallacies’ and Subjunctive/Indicative Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.19 ‘Ordinary Logic’: An Experiment by W. S. Cooper . . . . Notes and References for §7.1 . . . . . . . . . . . . . . . . 7.2 INTUITIONISTIC, RELEVANT, CONTRACTIONLESS. . . . . 7.21 Intuitionistic and Classical Implication . . . . . . . . . . . 7.22 Ternary and Binary Connectives Involving Intuitionistic Implication . . . . . . . . . . . . . . . . . . . . . . . . . . 7.23 Relevant Implication in Set-Fmla: Proofs of Results in 2.33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.24 Relevant Implication in Fmla: Various Topics . . . . . . 7.25 Contractionless Logics . . . . . . . . . . . . . . . . . . . . Notes and References for §7.2 . . . . . . . . . . . . . . . . 7.3 BICONDITIONALS AND EQUIVALENCE . . . . . . . . . . . . 7.31 Equivalence in CL and IL . . . . . . . . . . . . . . . . . . 7.32 A Connective for Propositional Identity . . . . . . . . . . 7.33 Reducibility without Tabularity: An Austere Propositional Identity Logic . . . . . . . . . . . . . . . . . . . . . . . . . Notes and References for §7.3 . . . . . . . . . . . . . . . . 8 “Not” 8.1 INTRODUCTION TO NEGATION . . . . . . . . . . . . 8.11 Contraries and Subcontraries . . . . . . . . . . . 8.12 Negative Miscellany . . . . . . . . . . . . . . . . 8.13 Negation in Quantum Logic and Relevant Logic . Notes and References for §8.1 . . . . . . . . . . . 8.2 NEGATION IN INTUITIONISTIC LOGIC . . . . . . . . 8.21 Glivenko’s Theorem and Its Corollaries . . . . . 8.22 Dual Intuitionistic Negation . . . . . . . . . . . . 8.23 Strong Negation . . . . . . . . . . . . . . . . . . 8.24 Variations on a Theme of Sheffer . . . . . . . . . Notes and References for §8.2 . . . . . . . . . . . 8.3 THE FALSUM . . . . . . . . . . . . . . . . . . . . . . . . 8.31 Negation and the Falsum Connective . . . . . . . 8.32 Minimal Logic . . . . . . . . . . . . . . . . . . . 8.33 Variations on a Theme of Johansson . . . . . . . 8.34 Extensions of Minimal Logic . . . . . . . . . . . 8.35 The Falsum: Final Remarks . . . . . . . . . . . . Notes and References for §8.3 . . . . . . . . . . . 9

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Universally Representative and Special Connectives 9.1 UNIVERSALLY REPRESENTATIVE CLASSES OF FORMULAS . . . . . . . . . . . . . . . . . . . . 9.11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 9.12 Universally Representative Connectives . . . . . . . . . 9.13 Are There Conjunctive (Disjunctive, etc.) Propositions?

983 987 998 1007 1034 1044 1053 1057 1057 1068 1088 1091 1098 1121 1127 1127 1150 1158 1161

1163 . 1163 . 1163 . 1174 . 1186 . 1211 . 1214 . 1214 . 1222 . 1228 . 1241 . 1250 . 1252 . 1252 . 1257 . 1263 . 1274 . 1282 . 1284 1287 . . . .

1287 1287 1289 1295

CONTENTS

xii Notes and References for §9.1 . . . . . . . . . . 9.2 SPECIAL CLASSES OF FORMULAS . . . . . . . . . 9.21 Introduction . . . . . . . . . . . . . . . . . . . 9.22 Special Connectives . . . . . . . . . . . . . . . 9.23 The Biconditional in R . . . . . . . . . . . . . 9.24 Connectives Neither Universally Representative Nor Special . . . . . . . . . . . . . . . . . . . . 9.25 Special Relations Between Formulas . . . . . . 9.26 Special ‘in a Given Respect’ . . . . . . . . . . . 9.27 Pahi’s Notion of Restricted Generalization . . . Notes and References for §9.2 . . . . . . . . . .

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1312 1318 1327 1329 1335

References

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Index

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xiii

Preface and Navigation Guide This book is about the semantics and pragmatics of natural language sentence connectives and about the properties of and relations between analogous devices in the formal languages of numerous systems of propositional (or sentential) logic. And the intended readership is: all who find connectives, and the conceptual issues arising in thinking about them, to be a source of fascination. Such readers are most likely to have come from one or more of the traditional breeding grounds for logicians: philosophy, mathematics, computer science, and linguistics. These different backgrounds will of course give rise to different interests and priorities. The best way – or at least one reasonable way – to read the book is to browse through to a topic of interest, or use the index to locate such topics, and then follow the references forward and backward within it on the topic in question, or those to the extensive literature outside it. In more detail: after Chapter 0, in which some minor mathematical preliminaries are sorted out (and which can be skipped for later consultation as necessary), Chapters 1–4 and Chapter 9 contain general material on sentence connectives in formal logic, such as truth-functionality, unique characterization by rules, etc., while Chapters 5–8 concern specific connectives (conjunction, disjunction, and so on), considering their pragmatic and semantic properties in natural languages as well as various attempts to simulate the latter properties in the formal languages of various systems of propositional logic. (The word “the” in the title The Connectives picks up on this aspect of the development; an equally good title, reflecting the more general concerns of Chapters 1–4 and 9, would have been simply Connectives.) Chapter 2 surveys various different logics (typically seen as extensions of or alternatives to classical propositional logic) to provide a background for later discussion. This means the treatment is selective and many popular themes not closely related to the behaviour of specific connectives are not touched on – in particular, issues of decidability, and computational complexity more generally, are completely ignored. Similarly at the more general level, while most of the connectives attended to here are what people would happily classify as logical connectives, no attention will be given to the philosophical question of what makes something an item of logical vocabulary (a ‘logical constant’, as it is put – and of course this is a status that not only sentence connectives but also quantifiers and other expressions may merit). As with other topics not gone into in detail but bearing of matters under discussion, the notes and references at the end of each section provide pointers to the literature; if a section covers numerous topics separately, the notes and references are divided into parts, each with its own topic-indicating heading. Let us turn to matters of navigational assistance. Chapters are divided into sections, with Chapter 4, for example, consisting of three sections: §4.1, §4.2, and §4.3. Each section ends with notes and references for material covered in that section. The sections themselves are divided into subsections, labelled by dropping the “§” and adding a further digit to indicate order within the section. Thus the first and fourth subsections of §4.2 are numbered 4.21 and 4.24 respectively. We can safely write “4.21”, rather than something along the lines of “4.2.1”, to denote a subsection, as no section has more than nine subsections. References such as “Humberstone [2010b]” are to the bibliography at the end. (The work just cited contains some highlights from Chapters 1–4 and 9, leaving out proofs and most of the discussion – though also

xiv touching briefly on some topics not covered here.) Within each subsection, the discussion is punctuated by numbered items of two kinds: results and non-results. Items in the result category – Lemma, Theorem, Corollary, and Observation – are set in italics and usually have proofs terminated with an end-of-proof marker: “”. This enables them to stand out from the surrounding text. (A similar symbol, , is used for a necessity-like operator in modal logic, discussed in §2.2 and elsewhere, but this will cause no confusion.) As usual, Theorems are principal results, with Lemmas leading up to them and Corollaries drawn from them. (“Observation” is used for results not naturally falling into any of these categories, this term being chosen in place of the more usual “Proposition”, simply because we already need several different notions of proposition in connection with the subject matter itself, as opposed to the presentation of results about that subject matter.) The non-result categories are Example, Exercise, and Remark, are not set in italics, so are they indented from the left margin to make it easy to see here they begin and end. The category Remark is included to make it possible to refer precisely, as opposed to merely citing a page number, to items in this category from elsewhere in the text. All numbered items, whether results or non-results, take their numbering successively within a subsection, to make them easy to locate. Thus Theorem 1.12.3 is the third numbered item – rather than the third theorem – in subsection 1.12 (both 1.12.1 and 1.12.2 being Exercises, as it happens). Similarly, Example 2.32.18 is the eighteenth numbered item in Subsection 2.32, which happens to be an Example; and so on. The presence of Exercise as a category indicates that this material can be used selectively for teaching purposes, but instructors should note that in a few cases – usually explicitly indicated – questions asked in the exercises are addressed in the subsequent main discussion. Also, while the exercises are given to allow for practice with concepts and to ask for (mostly) routine proofs, there are also a few discussion questions to be set to one side by instructors using some of this material for purely formal courses. Definitions are not numbered, and neither are Digressions; the latter can be omitted on a casual reading but do contain material referred to elsewhere. Diagrams and tables are labelled alphabetically within a subsection. Thus the first three in subsection 2.13 would be called Figure 2.13a, Figure 2.13b, and Figure 2.13c, in order of appearance. The following abbreviations have been used with sufficient frequency to merit listing together in one place at the outset; unfamiliar terms on this list will be explained as they come up: Coro. Def. esp. Fig. gcr g.l.b. iff lhs l.u.b.

for for for for for for for for for

Corollary (by) definition especially Figure generalized consequence relation greatest lower bound if and only if left-hand side least upper bound

xv Obs. resp. rhs Thm. w.r.t.

for for for for for

Observation respectively right-hand side Theorem with respect to

We use “gcr’s” as the plural of “gcr”, sacrificing apostrophic propriety to avoid the hard-to-read “gcrs”. Other abbreviations will be explained as they arise (as indeed most of the above are) and can also be found in the index. But here it may help to mention that “CL”, “IL”, and “ML” abbreviate “Classical Logic”, “Intuitionistic Logic” and “Minimal Logic”; additional similar abbreviations are also used (see the index entry for logics and consequence relations), and the abbreviations in question may also appear subscripted to a turnstile, such as, in particular, “”. (As the preceding sentence illustrates, we use what is sometimes called ‘logical’ punctuation as opposed to the traditional convention, when it comes to ordering quotation marks and commas. The comma is inside the quotation marks only if it is part of what is quoted.) Thus CL is the consequence relation associated with classical propositional logic, sometimes also called the relation of tautological or truth-functional consequence. We also use the “”notation for generalized (alias multiple-conclusion) consequence relations, so as to be able to write down in a single way formal conditions applying to consequence relations proper and to gcr’s alike, rather than having to write them down twice, using a separate notation (such as “”) in the latter case. Similarly, the neutral verbal formulation “(generalized) consequence relation” is sometimes used, meaning: consequence relation or gcr. The set of natural numbers (taken as non-negative integers here) is denoted by N or, when convenient, by ω (a lower case omega), the latter also understood as denoting the smallest infinite ordinal. This use of “N” is standard mathematical practice, but the corresponding use of “R” and “Q” for the sets of real and of rational numbers respectively is not followed here; instead “R” denotes either of a pair of structural rules introduced in 1.22 (as do “M” and “T”), while “Q” is pressed into service for a piece of semantic apparatus in 8.33. The mathematical prerequisites for following the discussion in general are minimal: a preparedness for abstract symbolically assisted thought and a passing familiarity with proof by (mathematical) induction will suffice. Some prior acquaintance, for example in an introductory course, with formal logic would be desirable. Many topics are treated here in a way suitable for those with no prior acquaintance, but perhaps rather too many for those without some such background to assimilate (or, more importantly, to enjoy). Some remarks on notation are in order. As schematic letters for formulas the following are used: A, B, C,. . . rather than A, B, C,. . . , so that A and B can be used to denote the universes (or carrier-sets) of algebras respectively denoted by A and B, without confusion. (Similarly, propositional variables have been set as p, q, r. . . rather than p, q, r,. . . ) Thus in general notation is highly font sensitive. An exception is the use of labels along the lines of “CL”, “IL”,. . . , for classical logic, intuitionistic logic,. . . , which appear in roman except when used as subscripts (“CL ” etc.), in which case they appear in italics. Otherwise the font is significant. For example, “T” is the name of the truth-value True, while T and are a sentential constants which are associated in ways explained in the text with this truth value (as also is t); on the other hand “(T)” is the name of a

xvi condition on consequence relations and “(T)” – already alluded to above – is the name of a corresponding rule, while “T” refers to a certain modal principle, as well as to an unrelated system of relevant logic. Names of logics, whether this is understood to mean proof systems or at a more abstract level (on which see the Appendix to §1.2: p. 180), mostly attempt to follow the most widespread practice in the area concerned. Thus although we use IL for intuitionistic logic, a particular natural deduction system and a particular sequent calculus for this logic are called INat and IGen respectively (“Gen” for Gentzen), while (the core of) Girard’s system of intuitionistic linear logic is called ILL; boldface is also used for normal modal logics (K, S4, etc.) and relevant (or ‘relevance’) logics: R, RM, E, T. One usage over which there is an abstract possibility of confusion, in practice resolved by context, is the double use of “V ”, in the first place for sets of valuations (bivalent truth-value assignments) and in the second to denote the component of a Kripke model (or similar) which specifies at which points the propositional variables are true; this component we call a Valuation (with a capital “V”). Also used is “BV ”, as an unstructured label for the class of boolean valuations for whatever language is under consideration. When only a single candidate conjunction, disjunction, negation, implication (conditional) or equivalence (biconditional) connective is in play, this is written as ∧, ∨, ¬, →, ↔, respectively; metalinguistically, we sometimes use & and ⇒ for conjunction and implication (though sometimes these too are used as object language connectives). For an ordered n-tuple we use the notation a1 , . . . , an . If the objects concerned (here a1 , . . . , an ) are of different types and the tuples under consideration only allow a given type in a given position, we tend to use “(” and “)” in place of angle brackets, thus instead of writing “ W, R, V ” to denote a Kripke model, considered as a certain kind of ordered triple, with W a nonempty set, R a binary relation on W , and V – a Valuation in the sense alluded to above – is a function from propositional variables to subsets of W , the alternative notation “(W, R, V )” is used. Sometimes to indicate the mention rather than the use of a formal symbol quotation marks are used, and sometimes the symbol appears without quotation marks, simply as a name for itself. To refer to natural language expressions, quotation marks are again employed, with the use of italics as a variant. I have tried to follow a policy, in using quotation marks, of using double quotes when mentioning linguistic expressions (though, as already noted, they may be dropped in favour of italics or dropped altogether when the expressions come from a formal language) and single quotes when using expressions in order to draw attention to the fact that those expressions are being used (scare quotes, shudder quotes), as well as to refer to headings and article titles – though this attempt may not have been completely successful, especially as the distinction just drawn is not always as clear in practice as it sounds, and the policy has not been imposed in citing passages from other authors (or in index entries). For assistance with LATEX (which came at a comparatively late stage in the production, with the consequence that many of its features have not been exploited), I would like to thank Gillian Russell, Rohan French and Bryn Humberstone, and especially – for detailed customization work – Guido Governatori and Toby Handfield. For help with content, corrections, and references, I am grateful to more people than I can name, but in particular to the following: Nuel Belnap, Thomas Bull, Sam Butchart, Kosta Došen, Allen Hazen, Roger Hindley, David

xvii Lewis, Bob Meyer, Jeff Pelletier, Humphrey van Polanen Petel, James Raftery, Greg Restall, Su Rogerson, Gideon Rosen, Matthew Spinks, Brian Weatherson, Tim Williamson, and Jack Woods. Several chapters were kindly proof-read by Steve Gardner, who corrected numerous errors and misformulations; the same was done by Sam Butchart in the case of some other chapters. Thanks also to Stephen Read (on behalf of a St Andrews reading group) and to Virginia Crossman (for MIT Press) for notifying me of further needed corrections. Many additional problems throughout were astutely spotted by David Makinson, who had also provided assistance on several specific topics at an earlier stage. For their encouragement and assistance in bringing this material to the attention of suitable publishers, I am very grateful to Richard Holton and also to Jean-Yves Béziau. Other individuals are thanked (as well as some of the above) as the occasion arises, mostly in the end-of-section notes and references. I benefited from an Australian Research Council (‘ARC’) Discovery Project Grant to assist with work on this material via teaching relief for the period 2002–2006, and am much indebted to Graham Oppy for his administrative skills in organizing the successful application for that grant. Work on the material began before that period and continued after it, and so occupied a more extended stretch of time than would be ideal, which has no doubt resulted in occasional inconstancies – which if our subject were not logic, I would be happy to call inconsistencies – as well as other infelicities. I hope that those not already edited away in the course of revision do not present too much of a distraction.

Chapter 0

Preliminaries Skip or skim this chapter, returning for background explanations as necessary.

§0.1 CONNECTIONS AND COMBINATIONS 0.11

Relational Connections and Posets

Given n non-empty sets S1 , . . . , Sn and a relation R ⊆ S1 × · · · × Sn , we call the structure (R, S1 , . . . , Sn ) an n-ary relational connection. When si ∈ Si (for i = 1, . . . , n) we often write Rs 1 . . . sn for s1 , . . . , sn ∈ R, using the familiar ‘infix’ notation s1 Rs 2 for the case of n = 2. We will be especially concerned in what follows with this case – the case of binary relational connections – and will then write “S” (for “source”) and “T ” (for “target”) for S1 and S2 . A binary relational connection will be said to have the cross-over property (or to satisfy the cross-over condition) just in case for all s1 , s2 ∈ S, and all t1 , t2 ∈ T : (*)

(s1 Rt 1 & s2 Rt 2 ) ⇒ (s1 Rt 2 or s 2 Rt 1 )

The label “cross-over”, for this condition, is explained pictorially. Elements of S appear on the left, and those of T on the right. An arrow going from one of the former to one of the latter indicates that the object represented at the tail of the arrow bears the relation R to that represented at the head of the arrow: s1

s2

R

/6 R R l l l R R l R R l l R lR l l R R R l l l R R l l R R l l /(

t1

t2

Figure 0.11a: The Cross-Over Condition

Read the diagram as follows: if objects are related as by the solid arrows, then they must be related as by at least one of the broken arrows. Thus, given the horizontally connected (ordered) pairs as belonging to R, we must have at least one of the crossing-over diagonal pairs also in R. Our main interest in this condition arises through Theorem 0.14.2 below, which will be appealed to more 1

2

CHAPTER 0. PRELIMINARIES

than once in later chapters (beginning with the proof of 1.14.6, p. 69). In the meantime, we include several familiarization exercises. Exercise 0.11.1 Check that if a binary connection (R, S, T ) has the cross-over property then so does the complementary connection (R, S, T ) where R = (S × T ) R, and so also does the converse connection (R−1 , T, S) where R−1 = { t, s | s, t ∈ R}. Exercise 0.11.2 Given R ⊆ S × T , put R(s) = {t ∈ T | sRt}. Show that (R, S, T ) has the cross-over property iff for all s1 , s2 ∈ S: R(s1 ) ⊆ R(s2 ) or R(s2 ) ⊆ R(s1 ). Exercise 0.11.3 (i) For S any set containing more than one element, show that the relational connection (∈, S, ℘(S)) does not have the cross-over property. (Here ℘(S) is the power set of S, i.e., the set of all subsets of S.) (ii ) Where N is the set of natural numbers and is the usual less-thanor-equal-to relation, show that the relational connection (, N, N) has the cross-over property. Inspired by 0.11.3(i), we say that (R, S, T ) is extensional on the left if for all s1 , s2 ∈ S, R(s1 ) = R(s2 ) implies s1 = s2 , and that it is extensional on the right if for all t1 , t2 ∈ T , R−1 (t1 ) = R−1 (t2 ) implies t1 = t2 . The ‘Axiom of Extensionality’ in set theory says that such connections as that exercise mentions are extensional on the right. (They are also extensional on the left.) Part (ii) of 0.11.3, on the other hand, serves as a reminder that we do not exclude the possibility, for an n-ary relational connection (R, S1 , . . . , Sn ), that the various Si are equal, in which case we call the relational connection homogeneous. A more common convention is to consider in place of (R, S1 , . . . , Sn ) the structure (S, R), where S = S1 = · · · = Sn . Such a pair is a special case of the notion of a relational structure in which there is only one relation involved. (In general one allows (S, R1 , . . . , Rm ) where the Ri are relations – not necessarily of the same arity – on the set S. Here the arity of a relation Ri is that n such that Ri is n-ary. We speak similarly, below, of the arity of a function or operation.) Various conditions on binary relational connections which make sense in the homogeneous case, and so may equivalently be considered as conditions on relational structures, do not make sense in the general case. Three famous conditions falling under this heading, for a set S and a relation R ⊆ S × S are: reflexivity: for all a ∈ S, aRa; transitivity: for all a, b, c ∈ S, aRb & bRc ⇒ aRc; and antisymmetry: for all a, b ∈ S, aRb & bRa ⇒ a = b. If the first two conditions are satisfied, R is said to be a pre-ordering of (or ‘preorder on’) S. We will often use the notation “” for R in this case; S together with such a pre-ordering is called a pre-ordered set. If all three conditions are satisfied, R is described as a partial ordering on S, and the relational structure (S, R) is called a partially ordered set, or poset for short. In this case, we will use the notation “” or else “” for R, on the understanding that when R is given by either of these symbols, the other stands for the converse of R (i.e., R−1 , as in 0.11.1). It is worth remarking that if (S, ) is a poset, then so is

0.1. CONNECTIONS AND COMBINATIONS

3

(S, ). The latter is called the dual of (S, ), any two statements about posets differing by a systematic interchange of reference to and (or notation or vocabulary defined in terms of them are) being described as each other’s duals. If a statement about lattices is true of all posets, then so is its dual (since if the dual of a statement is false of some dual, the original statement is false of the dual of that poset.) Other standard terminology for properties of binary relations in the same vein as that recalled here (such as for the properties of irreflexivity, symmetry, and asymmetry) will be assumed to be familiar. Recall also that a relation which is reflexive, symmetric, and transitive, is said to be an equivalence relation. Exercise 0.11.4 Given a pre-ordered set (S, ), define, for s, t ∈ S: s ≡ t iff s t & t s, and put [s] = {t ∈ T | s ≡ t}. Let [S] be {[s] | s ∈ S}. Finally, define [s] [t] to hold iff s t. Show (1) that this is a good definition (that it introduces no inconsistency in virtue of the possibility that [s] = [s ], [t] = [t ], even though s = s , t = t ), and (2) that the relational structure ([S], ) is a poset. Exercise 0.11.5 Show that a relation R ⊆ S × S is a pre-ordering of S iff for all s, t ∈ S: sRt ⇔ R(t) ⊆ R(s). Posets of a special sort (lattices) will occupy us in 0.13. In the meantime, we return to the (generally) non-homogeneous setting of relational connections.

0.12

Galois Connections

Given sets S and T , a pair of functions (f, g) with f : ℘(S) −→ ℘(T ) and g: ℘(T ) −→ ℘(S) is called a Galois connection between S and T if the following four conditions are fulfilled, for all subsets S0 , S1 of S, and T0 , T1 of T : (G1) S0 ⊆ g(f (S0 )) (G2) T0 ⊆ f (g(T0 )) (G3) S0 ⊆ S1 ⇒ f (S1 ) ⊆ f (S0 ) (G4) T0 ⊆ T1 ⇒ g(T1 ) ⊆ g(T0 ) Note first that the symmetrical treatment of S and f vis-à-vis T with g in these conditions has the effect that if (f, g) is a Galois connection between S and T then (g, f ) is a Galois connection between T and S, so that we are entitled to the following ‘duality’ principle: any claim that has been established to hold for an arbitrary Galois connection (f, g) between sets S and T must continue to hold when references to f and g are interchanged in the claim, along with those to S and T . We will call this: Galois duality, to contrast it with poset duality (from 0.11 above, or lattice duality, introduced in 0.13 below). (A more explicit notation would have us call (S, T, f, g) a Galois connection between S and T , but we will rely on the context to make clear what S and T are in any given case.) Given a binary relational connection (R, S, T ), if we define, for arbitrary S0 ⊆ S and T0 ⊆ T : [Def. fR ]

fR (S0 ) = {t ∈ T | sRt for all s ∈ S0 },

[Def. gR ]

gR (T0 ) = {s ∈ S | sRt for all t ∈ T0 },

4


then the pair (fR , gR ) constitutes a Galois connection between S and T : it is not hard to check that (G1)–(G4) are satisfied. To give some feel for why these four conditions are given only as ⊆-requirements, we illustrate with an example that the converse of (G1) need not hold for (fR , gR ). Let S be a set of people and T a set of cities, and R be the relation of having visited. Then fR (S0 ) consists of all the cities (in T ) that everyone in S0 —some subset of S—has visited, and gR (fR (S0 )) is the set of people (drawn from S) who have visited all the cities that everyone in S0 has visited: obviously this includes everyone in S0 , but there may also be other people who have visited all the cities that everyone in S0 has visited, in which case gR (fR (S0 )) will be a proper superset of S0 . From now on, when convenient, we will write “fS ” instead of “f (S)”, etc. Exercise 0.12.1 Show that (G1)–(G4) above are satisfied if and only if for all S0 ⊆ S, T0 ⊆ T : T0 ⊆ fS 0 ⇔ S0 ⊆ gT0 . Exercise 0.12.2 Prove that if (f, g) is a Galois connection, then fgfS 0 = fS 0 and gfgT 0 = gT 0 , for all S0 ⊆ S, T0 ⊆ T . Next, we need to consider the way in which the set-theoretic operations of union and intersection are related in ℘(S) and ℘(T ) when Galois connections are involved. Theorem 0.12.3 If (f, g) is a Galois connection between S and T then for all S0 , S1 ⊆ S, and T0 , T1 ⊆ T : (i) f (S0 ∪ S1 ) = f S0 ∩ f S1 and (ii) g(T0 ∪ T1 ) = gT0 ∩ gT1 . Proof. We address part (i); (ii) follows by Galois duality. First, thatf (S0 ∪ S1 ) ⊆ f S0 ∩ f S1 : S0 ⊆ S0 ∪ S1 , so ‘flipping’ by (G3): f (S0 ∪ S1 ) ⊆ f S0 ; likewise to show f (S0 ∪ S1 ) ⊆ f S1 . The desired result follows by combining these two. It remains to be shown that f S0 ∩ f S1 ⊆ f (S0 ∩ S1 ). First, f S0 ∩ f S1 ⊆ f S0 , so by 0.12.1, S0 ⊆ g(f S0 ∩ f S1 ). Similarly, S1 ⊆ g(f S0 ∩ f S1 ). Putting these together, we get: S0 ∪ S1 ⊆ g(f S0 ∩ f S1 ), from which the desired result follows by another appeal to 0.12.1. Theorem 0.12.3 continues to hold if the binary union and intersection are replaced by arbitrary union and intersection (of an infinite family of sets). Notice also that the proof of the first inclusion considered in establishing part (i) here does not involve the mapping g at all. This reflects the fact that if (f, g) is a Galois connection, either of f , g, determines the other uniquely: Exercise 0.12.4 Show that if (f, g) and (f , g) are Galois connections between sets S and T then f = f , and that if (f, g) and (f, g ) are Galois connections between S and T , then g = g . There are two equations missing from Thm. 0.12.3, namely those resulting from (i) and (ii) there on interchanging “∩” and “∪”. While these do not hold generally, they can be secured in an important special case. A Galois connection (f, g) between S and T is perfect on the left if for all S0 ⊆ S, gfS 0 = S0 , and perfect on the right if for all T0 ⊆ T , fgT 0 = T0 . If it is perfect on the left and perfect on the right, then we call a Galois connection perfect.


5

Observation 0.12.5 If (f, g) is a perfect Galois connection between S and T, then f (S0 ∩ S1 ) = f S0 ∪ f S1 and g(T0 ∩ T1 ) = gT0 ∪ gT1 , for all S0 , S1 ⊆ S and T0 , T1 ⊆ T . Proof. Assume the antecedent of the Observation; we will show how the first equation follows: S0 ∩ S1 = gf S0 ∩ gf S1 = g(f S0 ∪ f S1 ), as (f, g) is perfect on the left, by 0.12.3(ii). Applying f to both sides: f (S0 ∩ S1 ) = f g(f S0 ∪ f S1 ) = f S0 ∪ f S1 , as (f, g) is perfect on the right.

Example 0.12.6 For a simple example of a perfect Galois connection, consider the connection (f, g) between a set S and itself in which for S0 ⊆ S, / S0 }). f (S0 ) = g(S0 ) = S S0 (i.e., {s ∈ S | s ∈ Before proceeding, we recall that a function f from S to T is said to map S onto T if every t ∈ T is f (s) for some s ∈ S. When we have a particular source and target in mind, we just say that f is ‘onto’ or surjective, though strictly speaking surjectivity is a property of what we might call the ‘functional connection’ (f, S, T ), by analogy with relational connections above. (Some authors refer to (f, S, T ) itself as a function, with f as its graph.) Similarly, if f (s) = f (s ) only when s = s , the function f is said to be ‘one-one’ or injective. A function f : S −→ T which is both injective and surjective is described as a bijection or one-to-one correspondence between S and T. Observation 0.12.7 Suppose (f, g) is a Galois connection between S and T. Then the following claims are equivalent: (i) (f, g) is perfect on the left; (ii) g is surjective; (iii) f is injective. As also are the following three: (i) (f, g) is perfect on the right; (ii) f is surjective; (iii) g is injective. Proof. We do the proof for the case of (i)–(iii), arguing the cycle of implications (i) ⇒ (ii) ⇒ (iii) ⇒ (i). (i) ⇒ (ii): If the given connection is perfect on the left then for any S0 ⊆ S, gfS 0 = S0 , so any such S0 is the value of g applied to some T0 ⊆ T ; i.e., g is surjective. (ii) ⇒ (iii): Suppose that g is surjective and fS 0 = fS 1 . We must show that S0 = S1 . Since g is a surjection, S0 = gT 0 and S1 = gT 1 for some T0 , T1 ⊆ T . Thus, since fS 0 = fS 1 , fgT 0 = fgT 1 ; so gfgT 0 = gfgT 1 , whence by 0.12.2, gT 0 = gT 1 , i.e., S0 = S1 .

6


(iii) ⇒ (i): Suppose that fS 0 = fS 1 implies S0 = S1 , for all S0 , S1 ⊆ S. Now for any S0 ⊆ S, we have, by 0.12.2, f gf S0 = f S0 , so we may conclude that gf S0 = S0 . Since S0 was arbitrary, the connection is perfect on the left.

Corollary 0.12.8 If (f, g) is a perfect Galois connection between S and T then f is a bijection from S to T with g as inverse. Not only does every binary relational connection give rise to a Galois connection via [Def. fR gR ] above, but every Galois connection can be represented as arising in this way. We simply state this as 0.12.9 here; a proof may be found by considering the following way of defining a relation Rf g ⊆ S × T on the basis of a Galois connection (f, g) between S and T : [Def. Rf g ]

For s ∈ S, t ∈ T : sR f g t ⇔ t ∈ f ({s}).

The rhs of this definition makes no explicit mention of g – cf. 0.12.4. Theorem 0.12.9 Every Galois connection between sets S and T is of the form (fR , gR ) for some relation R ⊆ S × T . In fact, starting with a Galois connection, and passing to a relational connection via [Def. Rf g ] and back to a Galois connection via [Def. fR gR ], we end up with the same Galois connection we started with. We can also say that starting with a relational connection from which a Galois connection is extracted by [Def. fR gR ], the relational connection delivered from this by [Def. Rf g ] is the original relational connection itself. There is, then, for any sets S and T a natural one-to-one correspondence between relations R ⊆ S × T on the one hand, and Galois connections between S and T on the other. We need the following result for later discussion (in 6.24, p. 840). Exercise 0.12.10 Show that for any Galois connection (f, g) and ‘source’ subsets S0 , S1 , we have: f S0 ∩ f S1 = f g(f S0 ∩ f S1 ). (Hint: Use 0.12.2, 0.12.3.) This corresponds – in a way that will become clear in the following subsection – to the fact that the intersection of any two closed sets is closed. The following Remark (and Warning) will not be intelligible until after that is read; the material is included here as it constitutes a commentary on the development above. Remark 0.12.11 The definition, by (G1)–(G4), of a Galois connection makes sense for arbitrary posets S and T , when f : S −→ T , g: T −→ S and the “⊆” in (G1), the antecedent of (G3) and the consequent of (G4) is replaced by “S ” (denoting the partial ordering on S), with the “⊆” in (G2), the consequent of (G3) and the antecedent of (G4) being similarly replaced by “T ”. Much of the rest of the development – you may care to check how much – then goes through if these posets are lattices, in the sense of the following subsection, when unions and intersections are replaced by appropriate joins and meets (explained there). To impose this more general perspective on the formulations above, we should have


7

to describe what we have called Galois connections between sets S and T as being instead between the sets ℘(S) and ℘(T ). Warning: Particular care is needed over adapting 0.12.5 to the more general context introduced in the above Remark. If (f, g) is a perfect Galois connection between posets A1 and A2 , then (using ∧ and ∨ for meets and joins – see 0.13 – in both cases) we can argue, as in the proof of that Observation, for a, b ∈ A1 , that f (a ∧ b) = f (a) ∨ f (b). And it may happen that the elements of A1 and (in particular, for the present illustration) A2 are themselves sets, with the partial ordering being ⊆; it does not, however, follow that f (a) ∨ f (b) is f (a) ∪ f (b). The latter union may not belong to A2 at all.

0.13

Lattices and Closure Operations

If (S, ) is a poset and S0 ⊆ S then an element b ∈ S is called an upper bound of S0 if c b for each c ∈ S0 ; if in addition it happens that for any a ∈ S which is an upper bound for S0 , we have b a, then b is called a least upper bound (“l.u.b.”, for short) of the set S0 . Note that if b1 and b2 are both least upper bounds for the set S0 , then b1 = b2 (since b1 b2 and b2 b1 and is antisymmetric). The concepts of lower bound and greatest lower bound (g.l.b.) are defined dually. (The duality concerned is poset duality, as in 0.11, not Galois duality, of course.) A poset in which each pair of elements have both a least upper bound and a greatest lower bound is called a lattice. By the above observation concerning uniqueness, we can introduce unambiguously the notation a ∨ b for the least upper bound of a and b (strictly: of the set {a, b}) and a∧b for their greatest lower bound. Similarly, if there is a least upper bound (greatest lower bound) for the whole lattice, it can unambiguously be denoted by 1 (by 0) and will be called the unit (the zero) element of the lattice (or just top and bottom elements, respectively). Note that while the existence of such elements follows from, it does not in turn entail, the existence of greatest lower bounds and least upper bounds for arbitrary sets of lattice elements; lattices in which such bounds always exist are called complete. In any lattice, however, any finite set of elements {a1 , . . . , an } has both a l.u.b. and a g.l.b., namely: a1 ∨ a2 ∨ . . . ∨ an and a1 ∧ a2 ∧ . . . ∧ an , respectively. Having agreed that the operations ∧ and ∨ take elements two at a time, we should strictly insert parentheses into the terms just written, but they are omitted since the different bracketings make no difference to what the terms denote, in view of the third of the conditions listed here, all of which are satisfied by the operations ∧ and ∨ in any lattice: a∧a=a a∧b=b∧a a ∧ (b ∧ c) = (a ∧ b) ∧ c a ∧ (a ∨ b) = a

a∨a=a a∨b=b∨a a ∨ (b ∨ c) = (a ∨ b) ∨ c a ∨ (a ∧ b) = a

(Idempotence) (Commutativity) (Associativity) (Absorption)

Such equations are to be understood ‘universally’, i.e., as claiming that the equalities concerned hold for all lattice elements a, b, c. With this universal interpretation, equations are usually called identities – thus in this sense the identities of an algebra are the equations holding (universally) in that algebra. Note the need to avoid possible confusion with the use of “identity” to mean identity element (or neutral element: see 0.21).


8

The above equations introduce us to a different perspective on lattices, allowing them to be thought of, not as relational structures, but as algebras. An algebra in this sense is a non-empty set together with some operations under which the set is closed. (For more on this concept, see 0.21.) For the moment, we simply note that given (S, ∧, ∨) satisfying the eight equations above the relation , defined by: a b ⇔ a ∧ b = a, partially orders S, and ∧ and ∨ give the g.l.b.s and l.u.b.s of the pairs of elements they operate on. We shall usually think of lattices as algebras in this way, rather than of lattices as (a special kind of) posets. When so thinking, we refer to a ∧ b as the meet, and to a ∨ b as the join, of a, b. The dual of (S, ∧, ∨) is the lattice (S, ∨, ∧), and the dual of a lattice-theoretic statement is obtained by interchanging “∨” and “∧” (as well as “” and “”, if present). The reader is left to verify that this definition of duality for lattices as algebras is consilient with the definition of duality for (lattices as) posets in 0.11. Exercise 0.13.1 Show that for any elements a, b, of a lattice (S, ∧, ∨): a∧b = a iff a ∨ b = b. Thus the latter equation could equivalently have been used instead of the former as the definition of a b. (Hint: Absorption.) The two parts of Exercise 0.11.3 suggest some illustrations of these concepts: Examples 0.13.2(i) If S is any set, then the poset (℘(S), ⊆) is a lattice with ∩ (intersection) as meet and ∪ (union) as join. (ii) The poset (N, ) is also a lattice, the meet of m and n (m, n ∈ N) being min({m, n}) and their join being max ({m, n}). Note that the first example here could equally well have been given with any collection A of subsets of S, rather than specifically by insisting on taking all of them, as long as that collection is closed under union and intersection. Considering the lattice we obtain in this way as an algebra (A, ∧, ∨) we note that it will satisfy an equation which, not following from the eight given above to define the class of lattices, is not satisfied by every lattice, namely the so-called Distributive Law: a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)

(Distributivity)

The second example features a lattice which is not only distributive, but has the stronger property that its associated partial ordering is a linear (or ‘total’) ordering, in the sense that for any elements a, b we have: a b or b a

(Connectedness)

Linearly ordered posets are also called chains. Instead of just saying “connected”, we shall more often speak of a binary relation R on some set as strongly connected when for any elements a, b of that set we have either aRb or bRa; this makes a clear distinction with what is often called weak connectedness, for which we require merely that for any distinct elements a, b, either aRb or bRa. Exercise 0.13.3 (i) Show that all lattices satisfy the condition we get by putting “” for “=” in (Distributivity). (ii) Write down the dual of (Distributivity) and show that this equation is (universally) satisfied in a lattice iff the original equation is. (iii) Show that every chain is, when considered as an algebra with ∧ and ∨, a distributive lattice.


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There is another way of making lattices out of collections of sets which is commonly encountered and which, by contrast with 0.13.2(i), is not guaranteed to lead to distributive lattices, as we shall see in 0.13.6. The key ingredient is the idea of a closure operation (or ‘closure operator’) on a set S, by which is meant a function C: ℘(S) −→ ℘(S) satisfying the following three conditions, for all X, Y ⊆ S. (C1) X ⊆ C(X) (C2) X ⊆ Y ⇒ C(X) ⊆ C(Y ) (C3) C(C(X)) = C(X) Observe that in view of (C1), (C3) could be replaced by: C(C(X)) ⊆ C(X), and that the force of all three conditions could be wrapped up succinctly (somewhat in the style of 0.12.1) by the single condition that for all X, Y ⊆ U : X ⊆ C(Y ) ⇔ C(X) ⊆ C(Y ). It should also be remarked that the conditions could be written with “” in place of “⊆”, the variables “X”, “Y ”, ranging over elements of any poset; if this poset is a complete lattice, what follows can be taken over to this more general setting, though we shall continue to concentrate on the case in which the partial ordering is the inclusion relation, ⊆, on subsets of some given set. (Cf. Remark 0.12.10.) However, it is not necessary to consider all subsets of that set, so we shall allow the above definition to stand in even when the source and target of C are not ℘(S) but some proper subset thereof. The origin of the idea of a closure operation is in topology, where the closure of a set of points is that set together with its ‘boundary’ points; in this case, certain additional features are present – in particular that C(X ∪ Y ) = C(X) ∪ C(Y ) and C(∅) = ∅ – which do not follow from the general definition. These additional features of topological closure operations are not suitable for the main logical application of the idea of closure (to consequence operations: see 1.12) – as is explained in detail in Chapter 1 of Martin and Pollard [1996]. For some practice with the general concept, we include: Exercise 0.13.4 (i) Show that if C is a closure operation on S then for any X, Y ⊆ S: C(X) ∪ C(Y ) ⊆ C(X ∪ Y ); C(C(X)∪ C(Y )) = C(X ∪ Y ); and C(X ∩ Y ) ⊆ C(X) ∩ C(Y ). (ii) Show that if (f, g) is a Galois connection between S and T then g ◦f is a closure operation on S, where g ◦ f (X) = g(f (X)), and that f ◦ g is a closure operation on T . (g ◦ f is called the composition of g with f .) (iii) A subset S0 of a set S is closed under an n-ary relation R on S (i.e., R ⊆ S n ) if whenever {a1 , . . . , an−1 } ⊆ S0 and a1 , . . . , an ∈ R, then an ∈ S0 . Where R is a collection of relations of various arities on S, define CR (S0 ) to be the least superset of S0 to be closed under each R ∈ R. Verify that CR is indeed a closure operation. (iv) Show that if X = C(X), Y = C(Y ), then C(X ∩Y ) = C(X)∩C(Y ). Parts (ii) and (iii) here have been included since they illustrate the two main ways closure operations enter into discussions of logic; in Chapter 1 it will emerge that the Galois connection route of (ii) is involved in semantic specifications of

10


(what we there call) consequence operations, and the relational route of (iii) is involved in specifying them proof-theoretically. We further remark, à propos of (ii) that when we speak of a set’s being closed under an n-ary operation (or function) f we mean that it is closed under the (n+1)-ary relation R defined by: Rx 1 . . . xn+1 ⇔ f (x1 , . . . , xn ) = xn+1 . With a particular closure operation C in mind we call a set X closed (more explicitly: C-closed ) when C(X) = X. From information as to which are the closed sets, we can recover C as mapping X to the intersection of all the closed sets ⊇ X, exploiting the fact that the intersection of any family of closed sets is itself closed. (Part (iv) of the above Exercise gives this for finite families.) Exercise 0.13.5 (i) Verify that whenever we have a closure operation on some set, the partial ordering ⊆ on the collection C of closed sets C gives rise to a lattice (as algebra), (C, ∧, ∨) in which ∧ is ∩ and ∨ is the operation ˙ = C(X ∪ Y ). (In fact we get a complete lattice, the ∪˙ defined by: X ∪Y join of arbitrarily many elements being the closure of their union.) (ii) Define a set C ⊆ ℘(S) to be a closure system on S if the intersection of arbitrarily many elements of C is an element of C. As remarked in the text, the set of C-closed subsets of S is a closure system for any closure operation C on S, and C can be recovered from C by setting C(X) = {Y ∈ C | Y ⊇ X}. (Here “X”, “Y ”, range over subsets of S.) Verify that similarly if we start with a closure system C and use the definition just given to define the closure operation C, then C can be recovered as the set of C-closed sets (i.e., those X ⊆ S with C(X) = X). Thus there is a natural one-to-one correspondence between closure operations and closure systems (on a given set). Example 0.13.6 Out of the eight subsets of the three-element set {1, 2, 3}, declare the following to be closed: {1, 2, 3}, {2}, {1, 3}, {3}, ∅. (Note that the intersection of any sets listed is also listed, and that to find the closure of any set from the original eight, take its smallest closed superset: e.g., the closure of {1} is {1, 3}.) The closed sets make up a five-element lattice which is not distributive, because, for example, {1, 3} ∩ ({2} ∪˙ {3}) = {1, 3} ∩ {1, 2, 3} = {1, 3}, whereas ({1, 3} ∩ {2}) ∪˙ ({1, 3} ∩ {3}) = ∅ ∪˙ {3} = {3}. Next, we record some properties of distributive lattices; (i) gives a kind of generalized transitivity for (thought of as defined in terms of ∧ or of ∨). Exercise 0.13.7 (i) Show that a lattice (A, ∧, ∨) is distributive iff for all a, b, c ∈ A we have: a ∧ b c & a b ∨ c ⇒ a c (ii) Prove that for any a, b, c in a distributive lattice, if a ∧ b = a ∧ c and a ∨ b = a ∨ c then b = c. Note that we have used the letter “A” for the set of elements of our lattice here; the background to this notation will be explained in 0.21. What follow are some exercises on lattices in general. Exercise 0.13.8 (i) Show that in any lattice, if a ∧ c = a and b ∧ c = b then (a ∨ b) ∧ c = a ∨ b. (Hint: rephrase everything in terms of joins, using 0.13.1.)


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(ii) Suppose that (A1 , ∧1 , ∨1 ) and (A2 , ∧2 , ∨2 ) are lattices with A2 = A1 and ∧2 = ∧1 . Show that ∨2 = ∨1 . (Hint: Use (i), rewritten with ∧1 and ∨1 for ∧ and ∨, substituting a ∨2 b for c. This gives: (a ∨1 b) ∧1 (a ∨2 b) = a ∨1 b. Similarly, we obtain: (a ∨2 b) ∧2 (a ∨1 b) = a ∨2 b. Now use the facts that ∧1 = ∧2 and that this operation is commutative.) (iii) Show that for lattice elements a, b, if a ∧ b = a ∨ b then a = b. Exercise 0.13.9 (i) Call an element a ∈ A join-irreducible in a lattice (A, ∧, ∨) if for all b, c ∈ A: a = b ∨ c ⇒ a = b or a = c; call a ∈ A join-prime if for all b, c ∈ A: a b ∨ c ⇒ a b or a c. Show that in any lattice all join-prime elements are join-irreducible, and that in any distributive lattice, the converse also holds. (ii) Similarly, call a ∈ A meet-irreducible if for all b, c ∈ A, a = b ∧ c implies a = b or a = c. Show that in any lattice, an element a is meet-irreducible iff for b, c ∈ A, a = b ∧ c implies b c or c b. We close with a sort of converse to 0.13.4(ii); the characteristic function of a set (a phrase used in the proof) is the function mapping elements of that set to the truth-value True (or “T” as we shall denote this in subsequent chapters) and non-elements of the set to the value False (or “F”; in the work of some authors the numbers 1 and 0 – or even 0 and 1 – are used to play these respective roles): Observation 0.13.10 If C is a closure operation on a set S then there is a Galois connection (f, g) between S and some set T such that C = g ◦ f . Proof. Given C and S, let T comprise the characteristic functions of the closed subsets of S, and define f and g via [Def.fR gR ] from 0.12, where R ⊆ S × T is given by: sRt ⇔ t(s) = True. Since this is automatically a Galois connection, it remains only to check that for all X ⊆ S, C(X) = g(f (X)). This is left to the reader. The characteristic functions employed in the above proof will emerge again in 1.12 under the description “valuations consistent with a consequence operation”.

0.14

Modes of Object Combination

We return now to the subject of binary relational connections as in 0.11, to consider some conditions asserting the existence of objects in the source and in the target playing special roles. Since we do not want to disallow the (‘homogeneous’) possibility that source and target are one and the same set, we will actually speak in terms of left and right instead. These conditions involve conjunction (“and”) and disjunction (“or”) in their formulation, so we will use an upward pointing triangle when the characterization is conjunctive (this being suggestive of “∧”) and a downward pointing triangle when it is disjunctive (to recall “∨”). The subscripted “L” and “R” are mnemonic for “left” and “right”

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(so, in particular, this use of “R” has nothing to do with the relation R). Four conditions are to be introduced, in terms of an arbitrary binary connection (R, S, T ); quantifier shorthand “∀”, “∃” has been used to make the structure of the conditions more visible: (L ) ∀s1 , s2 ∈ S∃s3 ∈ S such that ∀t ∈ T : s3 Rt ⇔ s1 Rt & s2 Rt. (L ) ∀s1 , s2 ∈ S∃s3 ∈ S such that ∀t ∈ T : s3 Rt ⇔ s1 Rt or s 2 Rt. (R ) ∀t1 , t2 ∈ T ∃t3 ∈ T such that ∀s ∈ S: sRt 3 ⇔ sRt 1 & sRt 2 . (R ) ∀t1 , t2 ∈ T ∃t3 ∈ T such that ∀s ∈ S: sRt 3 ⇔ sRt 1 or sRt 2 . When the first (second) of these conditions is satisfied by a relational connection, we say that this connection has conjunctive (disjunctive) combinations on the left, and call any s3 with the promised properties a conjunctive (disjunctive) combination of the given s1 and s2 . Similarly with the remaining two conditions and analogous terminology with right replacing left. If either of (L ), (L ), is satisfied in a relational connection which is extensional on the left, then the element (s3 above) whose existence is claimed is the only one having the property in question, in which case we can call it the conjunctive or disjunctive combination (on the left) of s1 and s2 , and denote it unambiguously by s1 L s2 or s1 L s2 respectively. Likewise on the right. Thus, supposing (R, S, T ) is an extensional relational connection, we have, using the R(·) notation of 0.11.2 (p. 2): (i) R(s1 L s2 ) = R(s1 ) ∩ R(s2 ) (ii ) R(s1 L s2 ) = R(s1 ) ∪ R(s2 ) (iii) R−1 (t1 R t2 ) = R−1 (t1 ) ∩ R−1 (t2 ) (iv ) R−1 (t1 R t2 ) = R−1 (t1 ) ∪ R−1 (t2 ) In fact, even without extensionality, these claims make sense if we think of “s1 L s2 ” as denoting an arbitrary s3 satisfying the condition imposed on s3 for any given s1 , s2 by (L ), regardless of whether that condition is satisfied for all alternative choices of s1 , s2 , as (L ) itself requires. And similarly in the other cases. We note, without proof, the lattice-theoretic implications of our four existence conditions. Observation 0.14.1 If (R, S, T ) is extensional on the left and also has conjunctive and disjunctive combinations on the left, then the structure (S, L , L ) is a distributive lattice; likewise with (T, R , R ) if (R, S, T ) is extensional on the right and has conjunctive and disjunctive combinations on the right. What follows the “likewise” is not really a separate fact from what precedes it, since we simply apply the preceding assertion to the converse connection (R−1 , T, S). In 0.13 we noted the possibility of generalizing the meet and join operations to form the g.l.b. or l.u.b. of an arbitrary collection of poset elements. Such generalizations of conjunctive and disjunctive combinations also make sense; and in particular we will have need below of the following generalized conjunctive combination. If (R, S, T ) is a binary relational connection then a conjunctive combination of any collection S0 ⊆ S, which we may denote by (S0 ), or more explicitly, L (S0 ), is an element s ∈ S such that for all t ∈ T , sRt iff for every s ∈ S0 , s Rt. While the property of having conjunctive combination on the left


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(to stick with this example) in this more generalized sense is of special interest in requiring the existence of such an s for every infinite S 0 ⊆ S, it is worth noting the upshot of this definition for the finite case. In particular, if S0 = {s1 , s2 } then any element qualifying as (S0 ) qualifies as s1 s2 , and vice versa; if S0 = {s1 } then s1 qualifies as (S0 ); and finally, that if S0 = ∅, then (S0 ) is an element of S which (since the “for every s ∈ S0 ” quantifier is now vacuous) bears R to each t ∈ T . Corresponding sense, mutatis mutandis, can of course be made of talk of the disjunctive combination of an arbitrary collection of objects from the left or the right of a relational connection. To recall again 0.11.3(i) from p. 2, for any non-empty set S, the relational connection (∈, S, ℘(S)) not only has conjunctive and disjunctive combination on the right – with t1 R t2 being t1 ∪ t2 , and t1 R t2 being t1 ∩ t2 – it clearly also supports the generalized versions of these modes of combination (arbitrary union and intersection). Conspicuously absent in the case of (∈, S, ℘(S)), as long as S contains more than one element, are conjunctive and disjunctive combinations on the left. A disjunctive combination of S-elements s1 and s2 , for example, would be an object which belonged to precisely the sets at least one of s1 , s2 belonged to. This object, then, would be a member of {s1 }, since s1 is, and of {s2 }, since s2 is; this implies s1 = s = s2 , which of course need not be the case if |S| 2, since we can choose for s1 and s2 distinct elements. Now Exercise 0.11.3 asked for a proof that, on the assumption that |S| 2, the connection (∈, S, ℘(S)) does not have the cross-over property. And this is no coincidence. The following result implies that no relational connection with conjunctive and disjunctive combination on the right can have either conjunctive or disjunctive combination on the left without also having the cross-over property. Theorem 0.14.2 If a binary connection (R, S, T ) has conjunctive combinations on the left and disjunctive combinations on the right, then it is has the crossover property. The same conclusion follows if such a connection has disjunctive combinations on the left and conjunctive combinations on the right. Proof. We prove the first part of the Theorem, since the second will then follow by consideration of the converse connection. Suppose we have (R, S, T ) with operations L and R as in the antecedent of the claim to be proved, and that for s1 , s2 ∈ S, t1 , t2 ∈ T , (1) s1 Rt 1 and (2) s2 Rt 2 . To demonstrate the crossover property, we must show that we then have either s1 Rt 2 or else s2 Rt 1 . From (1) and (2), we get s1 R(t1 R t2 ) and s2 R(t1 R t2 ), and from these, we infer that (s1 L s2 )R(t1 R t2 ). Therefore either (s1 L s2 )Rt1 or (s1 L s2 )Rt2 . From the first, it would follow that s2 Rt1 , and from the second, that s1 Rt2 . Three comments on this theorem and its proof are worth making. First, note that neither left nor right extensionality is needed as a hypothesis of the theorem. Secondly, as already remarked, this does not illegitimize the use of the “s1 L s2 ” (etc.) notation, for some arbitrarily selected s3 satisfying for the given s1 , s2 , our condition (L ). Finally, observe that the proof fully exploits all these conditions, in that all four of the implications involved in the two biconditionally stated conditions on the combined elements in (L ) and (R ) are used; or, to put it differently, all four of the ⊆-statements implicit in the equations (i) and (iv) above are used. Simple as it is, we shall make considerable use of 0.14.2 in the sequel. (The basic idea in the above proof may be found on

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pp. 6–7 of Strawson [1974]; the origin of 0.14.3 below may similarly be traced to work of Geach mentioned in Strawson’s discussion.) We turn our attention now to a singulary (or 1-ary) rather than binary mode of object combination, based on negation rather than on conjunction or disjunction. (Many say “unary” for the n = 1 case of “n-ary”, but this is hard to do for anyone who knows any Latin, and nobody says, for example, “duary” for the n = 2 case.) Given (R, S, T ) and s1 ∈ S, we call s2 ∈ S a negative object for s1 (“on the left”) if for all t ∈ T , s2 Rt if and only if it is not the case that s1 Rt. If every s ∈ S has such a corresponding negative object, we can say that (R, S, T ) provides negative objects on the left; as with the conjunctively and disjunctively combined objects, they are uniquely determined if the connection is extensional on the left. Right-sided analogues of these pieces of terminology are to be understood in the obvious way, so that negative object formation on the right provides for t ∈ T , some element of T to which all and only those elements of S bear R that do not bear R to t. Rather than attempting to subscript a symbol reminiscent of complementation or negation (such as ¬) with “L” or “R” to indicate “left” or “right”, we will use “neg(·)” thus subscripted. Given any set S, the set-theoretic connection (∈, S, ℘(S)) provides negative objects on the right, as in the case of conjunctive and disjunctive combinations, the form of (Srelative) complements. Note that with a binary relational connection (R, S, T ) with negative objects on the left and right for s ∈ S, t ∈ T , respectively – in the notation just introduced, neg L (s) and neg R (t) – then sRneg R (t) if and only if neg L (s)Rt. Further (by ‘De Morgan’s Laws’) if a connection has negative objects on the left then it has conjunctive combination on the left iff it has disjunctive combination on the left, and similarly with right replacing left throughout. Finally, assuming extensionality on the left (right), we have neg L (negL (s)) = s and neg R (negR (t)) = t. Observation 0.14.3 (i) If (R, S, T ) has negative objects on the right, then the relation L is symmetric, where s1 L s2 ⇔ R(s1 ) ⊆ R(s2 ). (ii) If (R, S, T ) has negative objects on the left, then the relation R is symmetric, where t1 R t2 ⇔ R−1 (t1 ) ⊆ R−1 (t2 ). Proof. We prove (i). Suppose there are negative objects on the right. We must show that for s1 , s2 ∈ S, s1 L s2 ⇒ s2 L s1 . So suppose further that not s2 L s1 , i.e., that for some t ∈ T , s2 Rt but not s1 Rt. So s1 Rneg L (t). Thus if s1 L s2 , we should have s2 Rneg R (t), which would contradict the fact that s2 Rt. Therefore not s1 L s2 . Note that extensionality on the left amounts to the relation L ’s being antisymmetric, and that since any relation which is both symmetric and antisymmetric is a subrelation of the identity relation, we have: Corollary 0.14.4 If (R, S, T ) is extensional on the left and has negative objects on the right, then s1 L s2 ⇒ s1 = s2 . Exercise 0.14.5 Suppose that (R, S, T ) is a binary relational connection which has conjunctive combinations on the left and negative objects on the right. Show that for all s, s ∈ S, s L s .


15

To counter the preponderance of non-homogeneous examples in our discussion, we include: Exercise 0.14.6 Let (S, ) be a poset, and consider the relational connection (, S, S). Show that (i) this connection has conjunctive combinations on the left and on the right iff the original poset is a lattice, and that (ii ) given its having conjunctive combinations on the right and left, this connection has disjunctive combinations on the left iff it has disjunctive combinations on the right, in which case (iii) L gives meets (∧) and R , joins (∨). We close with some further points about homogeneous binary relational connections. For the first of these, the relation concerned itself has the preorder properties, though we avoid the term “pre-order” in the formulation, so as to avoid confusion with the derivative relations L and R introduced in 0.14.3. (In fact, it is not hard to see that when R is a pre-order, the former relation, as defined in 0.14.3, coincides with R−1 .) Observation 0.14.7 If (R, S, S) is a relational connection with disjunctive combinations on the left, or with disjunctive combinations on the right, and R is both reflexive and transitive, then R is strongly connected. Proof. We show the result for the case of disjunctive combinations on the left, the right-hand case being similar. Since R is reflexive, for any s, t ∈ S we have (sL t)R(sL t), so either sR(sL t) or tR(sL t), where sL t is an arbitrary disjunctive combination of s, t, on the left. By reflexivity again, we have sRs and tRt, so (1) (sL t)Rs and (2) (sL t)Rt. If sR(sL t), then by (2) and the hypothesis that R is transitive, sRt; if tR(sL t), then by (1) and the transitivity of R, tRs. So either sRt or tRs. At the start of 4.22, we shall apply this result to show the untoward consequences of supposing that for any two statements A and B there is some statement C from which follow exactly those statements that either follow from A or follow from B. This is the ‘disjunctive combinations’ version of the theorem, taking the relation R as the relation holding between one statement (or, as we shall say in that discussion, one formula) and another just in case the latter follows from the former (according, as we shall be saying, to some given consequence relation, which in what follows we assume to treat conjunction and disjunction in the expected ways). On the other hand, there is no problem about conjunctive combinations for this connection: disjunctions of formulas constitute conjunctive combinations thereof on the left, since C follows from the statement that either A or B just in case it follows from A and also follows from B; conjunctions of formulas count as conjunctive combinations on the right. Exercise 0.14.8 Show that if (R, S, S) is a relational connection with R strongly connected and transitive, then (R, S, S) has the cross-over property. Of course from 0.14.7 and 0.14.8 it follows that a relational connection in which the relation concerned is a pre-order possesses the cross-over property if it has disjunctive combinations on either the left or the right. For a variant on 0.14.8, see Exercise 7.19.9(i). We close the present discussion with a variant on 0.14.2:

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Exercise 0.14.9 Let us say that (R, S, T ) has equivalential combinations on the left if for any s1 , s2 ∈ S there exists s3 ∈ S such that for all t ∈ T we have s3 Rt just in case s1 Rt ⇔ s2 Rt. (This means that either s1 and s2 both bear R to t or else neither s1 nor s2 bears R to t.) We adopt a corresponding understanding for equivalential combination(s) on the right. Then show that if (R, S, T ) has equivalential combinations on either side (left or right) and disjunctive combinations on the other side, it has the cross-over property. (Hint: An argument in the style of the proof of 0.14.2 can be used.) Note that for any of the informal analogues of the boolean connectives we shall meet in Chapter 1, there is a corresponding notion of combination on the left or right of a relational connection: we opened with conjunction and disjunction, turning to negation in the discussion leading up to 0.14.3, and now, with 0.14.9, we have introduced the combination corresponding to the biconditional (material equivalence). Clearly there is such an analogue in any case. For example, corresponding to exclusive disjunction (to which we attend in 6.12) there is a mode of object combination which is available on the left (for example) if for any two objects drawn from S there is some object (their ‘exclusive disjunctive combination’) which bears the relation R to an arbitrary element of T just in case exactly one of those objects bears R to that element. See Humberstone [1996a] for further discussion, in the context of valuational semantics for sentential logic; the equivalential mode of combination is very much to the fore (in this same context) in Humberstone [1992], [1993b].

Notes and References for §0.1 Some remarks on the desirability of keeping track of the source and target of a binary relation, embodied in 0.11 by the idea of a relational connection, may be found in Thatcher, Wagner and Wright [1979]. Historical information about the cross-over property may be found in Humberstone [1991b]. For posets and lattices, see Birkhoff [1967], Grätzer [1978], McKenzie et al. [1987], and Davey and Priestley [2002]. The sources for the material on Galois connections are Ore [1944], Everett [1944], Birkhoff [1967], which references explain the relationship between Galois connections (sometimes called Galois correspondences) and traditional Galois theory. The Galois connections defined in 0.12 are sometimes more specifically called antitone (or ‘contravariant’) Galois connections, because of the way conditions (G3) and (G4) reverse the direction of the ordering, as opposed to the monotone (or ‘covariant’) Galois connections often encountered in the literature. Closure operators and closure systems are thoroughly treated in Cohn [1965]. Some writers now prefer to reverse the order of inclusion (or, more generally, the given partial ordering) in the consequents of (G3) and (G4); this would not be so convenient for our purposes. A survey of various treatments is Herrlich and Hušek [1986]. The topic is touched on more briefly in many texts, as in Davey and Priestley [2002], p. 155ff.; Chapter 11 of that text also gives an introduction to the work of R. Wille, which emphasizes the relationship between lattices and (what we call) relational connections. For discussions of the application of Galois connections to topics in sentential logic, see the Notes and References to §1.1 (which begin on p. 100).

0.2. SOME ALGEBRAIC CONCEPTS

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§0.2 SOME ALGEBRAIC CONCEPTS 0.21

Algebras

The general study of algebras and relations and mappings between them is called universal algebra. Several comprehensive texts are available, and are mentioned in the notes. We will need very little of this material. In the first place, we recall that an algebra A comprises a set A together with some operations under which A is closed. The set A is called the universe (or ‘carrier’) of A. Here we are following the convention that letters A, B, . . . denote the universes of algebras denoted by the corresponding bold letters A, B,. . . Only the case in which there are finitely many operations, each finitary (i.e., each n-ary for some n 0), will be considered. Since the universe of an algebra is typically closed under many additional operations, we often use the phrase fundamental operation to refer to those in terms of which the algebra is identified. To keep track of these operations in comparing algebras, we think of them as coming in some definite order; thus a lattice A = (A, ∧, ∨) has ∧ and ∨ as its first and second fundamental operations. (A common practice – not followed here – involves separating the specification of the universe from the list of operations by a semicolon rather than a comma, so that instead of “(A, ∧, ∨)” what was written would be “(A; ∧, ∨)”, or alternatively – recalling from the Preface that we treat this difference as purely stylistic – “ A; ∧, ∨ ”.) In the case just cited, the two operations are each binary. In fact a more explicit notation is called for when explicitly comparing two – e.g., lattices – A and B, and in the following subsection we introduce a notation according to which A = (A, ∧A , ∨A ) and B = (B, ∧B , ∨B ) in this case, thinking of ∧ (or ∨) as denoting an operation symbol rather than an operation: the superscript then indicates the operation this symbol is interpreted as denoting in the context of the algebra in question. A record of the arities of the successive operations is given by a k-tuple, where k operations are involved altogether, called the similarity type (or ‘signature’) of the algebra. Thus lattices are of similarity type 2, 2 ; if we had also a third fourth, and fifth operation respectively of arities 3, 1, and 0, the similarity type would be 2, 2, 3, 1, 0 . This last case calls for some comment: a 0-ary (‘zeroplace’, ‘nullary’) operation on A is simply an element of A. So if A comes with some such operations, they may be thought of as distinguished elements of A. (Since an n-ary operation has as its value an element of A when given, in some order, n elements of A as arguments, a 0-ary operation requires no such input: it already is an element of A. Being ‘closed’ under such a zero-place operation is to be understood simply as containing the element in question.) As was mentioned above, there are various functions in addition to the fundamental operations of an algebra, under which its set of elements is guaranteed to be closed. Some of these are naturally described as (in various senses) ‘derived’ operations of the algebra and we shall touch on them in 0.22. Of course, lattices are not just any old algebras of type 2, 2 ; certain equations have to be satisfied for an algebra of this type to be a lattice, and more equations still, for example, for it to be a distributive lattice. An even more specialized class is the class of chains, thought of as lattices satisfying: for all x, y either x ∧ y = x or x ∧ y = y. In this case, we have a disjunctive condition, and it is not hard to show (0.24.1) that no equation or set of equations can be used to isolate this class.


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If one algebra is the result of altering another by deleting some of the latter’s fundamental operations, and keeping everything else (including the universe) the same, it is called a reduct of the latter, and the latter is called an expansion of the former. The reduct of a lattice we get by deleting either of its two binary operations is called a semilattice. That is, a semilattice is an algebra (A, ·) of type 2 in which the operation · is idempotent, commutative, and associative. An important class of algebras is defined by deleting the first two of these conditions and retaining only the requirement of associativity. These are called semigroups and represent a generalization not only of semilattices but also of the class of groups. The latter are algebras (A, ·,−1 , e) where · is a binary associative operation on A, and e is a distinguished element of A, usually called the identity (or neutral or unit) element, satisfying the condition that for all a ∈ A: a · e = a and e · a = a.

(Identity)

Finally, the ( )−1 part of the structure is a singulary operation (i.e., a one-place function under which A is closed) with the property that (for all a ∈ A): a · a−1 = e and a−1 · a = e.

(Inverses)

The element a−1 is called the inverse of a. To retain the conception of algebras as sets together with operations under which they are closed, we view the distinguished element e here as a 0-ary fundamental operation. For any semigroup (A, ·), at most one element of A can satisfy for all a ∈ A the condition imposed upon e by the pair of equations labelled (Identity) above: for if e1 and e2 both satisfy them we have e1 · e2 = e1 by the first and e1 · e2 = e2 by the second, so that e1 = e2 . If we are dealing with a group, then the existence of at least one such element is secured by the definition of ‘group’, so that it follows that there is exactly one identity element in the group. Notice that associativity made no appearance in the semigroup part of the last observation, which therefore holds for all algebras of similarity type 2 ; the general term for all such algebras is groupoid. We have not checked that inverses, as defined above, are similarly unique, which is required for the above formulation to be felicitous: “the element a−1 (satisfying (Inverses)) is called the inverse of a” (here quoted with differently located italics. This is left for Exercise 0.21.2(ii) below. In the context of groupoids or semigroups we can consider the two parts of the condition (Identity) and distinguish between a groupoid (A, ·)’s having a right identity in the sense of containing some element eR such that for all a ∈ A, a · eR = a, and its having a left identity eL satisfying: eL · a = a for all a ∈ A. What forced the ‘two-sided’ identity element required for groups to be unique was the requirement that there be both a left and a right identity. If this happens, the above argument gives eL = eR ; but if there is no right (no left) identity, there is nothing to stop there being several distinct left (right) identity elements. (Suggestion: find an example.) A semigroup with a distinguished (two-sided) identity element is called a monoid ; that is, (A, ·, e) is a monoid if · is associative and e · a = a · e = a for all a ∈ A. We can be similarly discriminating on the subject of inverses, assuming the presence of an identity element e, and saying when a·b = e that a is a left inverse of b, and that b is a right inverse of a. The following Exercise gives some practice


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with these concepts, specialized to the case in which the semigroup elements are mappings (functions) from some set S into itself and the semigroup operation is composition of mappings (0.13.4(ii), p. 9). Every semigroup is, to use a term introduced below (0.23), isomorphic to such a semigroup. The identity element e can be taken as the identity mapping on S, with e(x) = x for all x ∈ S. Exercise 0.21.1 (i) Verify that the operation of composition (of mappings) is indeed associative. (ii ) Show that an element of the semigroup of mappings from some set to itself has a left inverse iff that mapping is injective, and that it has a right inverse iff it is surjective. (iii) Show that if g has f as a right inverse and f as a left inverse, then f = f . (iv ) Show that if every function from S to S with a left inverse has a right inverse, then every function from S to S with a right inverse has a left inverse. 0.21.1(i)–(iii) tell us that the bijections from a set to itself form a group under composition, since every such mapping has a (two-sided) inverse. In fact, historically, group theory began as the study of such transformations (“permutations”). Part (iv) is included to evoke a well known definition of finiteness for sets, due to Dedekind: a set is finite iff it cannot be put into a one-to-one correspondence with a proper subset of itself. In other words: every injection on the set is a surjection. Thus, given (ii), (iv) amounts to the observation that Dedekind could equally well have used a converse definition: a set S is finite iff no mapping from S onto itself maps distinct elements to the same element. (A rigorous proof of (ii) in axiomatic set theory requires the Axiom of Choice.) Note that since groups (at least, as considered here) and semigroups are of different similarity types – namely 2 and 2, 1, 0 – it is not quite accurate to say, as was said above, that the concept of a semigroup is a generalization of that of a group; more accurately: semigroups generalize the groupoid reducts of groups. The latter reducts have various special properties not shared by all semigroups, and in particular, they satisfy (for all elements a, b, and c): ac = bc ⇒ a = b

ca = cb ⇒ a = b

(Cancellation Laws)

In stating these principles, the usual convention has been followed with multiplicative notation, of writing “ac” for “a · c”, etc. Incidentally, in line with this notation many writers would have “1” in place of “e”; if additive notation is used, with “+” for the binary operation, “0” would be used, and, in place of “a−1 ”, “−a”. The additive notation is particularly common for commutative groups (i.e., groups in which the fundamental binary operation is commutative); by tradition these are usually referred to as Abelian groups. To see that every group satisfies (by way of example) the first of the above cancellation laws – Right Cancellation – suppose that ac = bc (for any given elements a, b, c of the group). Then (ac)c−1 = (bc)c−1 , so, by associativity, a(cc−1 ) = b(cc−1 ) and thus, by (Inverses) ae = be, whence, by (Identity) a = b, as desired. The case of the second (‘Left’) cancellation law is similar. Also encountered in the literature are algebras intermediate in similarity type between groups and semigroups, having type 2, 0 , called monoids, in


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which the binary operation is associative, and the distinguished element is an identity element for that operation. (Cancellation is not required.) The cancellation semigroups illustrate a direction of specialization of the semigroup idea which is inimical to that with which we began, the semilattices, as will be clear from part (i) of the following Exercise. A trivial algebra is one whose universe contains only one element. Part (ii) picks up the uniqueness issue raised above. Exercise 0.21.2 (i) Show any semilattice satisfying the above Cancellation Laws is trivial. (ii) Show that if (A, ·,−1 , e) is a group and so is (A, ·, *, e), which is to say that ( )* is a map from A to A satisfying the conditions called (Inverses) above, with “a−1 ” replaced by “a*”, then the operations ( )−1 and ( )* coincide. (Hint: it suffices to show that for all a ∈ A, a* = a−1 . To do this, first show that aa* = aa−1 = (where a is an arbitrary element of A), and then invoke the appropriate Cancellation Law.) Another important property of certain semigroups which is in similar tension with the cancellation property is that of ‘having a zero (element)’, in the sense of having a semigroup element – call it 0 – with the distinctive characteristic of its namesake in the semigroup consisting of the natural numbers under (ordinary arithmetical) multiplication: for any semigroup element a, a·0 = 0·a = 0. As in the case of identity elements, we can be more discerning than this, and define a right zero to be an element 0R for which a · 0R = 0R whatever element a may be, and define a left zero to be an element 0L such that for all a, 0L · a = 0L . Then a zero is an element which is both a left zero and a right zero; the uniqueness of such an element is secured, as in the case of identity elements, by noting that since 0L = 0L · 0R = 0R , if a left zero and a right zero are both present, they are identical. Exercise 0.21.3 (i) Show that any cancellation semigroup with a zero is trivial. (A cancellation semigroup is a semigroup satisfying the cancellation laws.) (ii ) Show that if a lattice contains an element which serves as an identity element for the meet operation and also for the join operation, then the lattice is trivial. (The condition means that where (A, ∧, ∨) is a lattice with e ∈ A such that for all a ∈ A, a ∧ e = a ∨ e = a. Hint: consider, for an arbitrary element a, the element (a ∧ e) ∨ e, with the absorption laws and that condition in mind.) (iii) Show that if a lattice contains an element which serves as a zero element for the meet operation and also for the join operation, then the lattice is trivial. We turn our attention to various expansions of lattices. First, if such elements exist, we can throw in the unit (top) and zero (bottom) elements of a lattice as zero-place fundamental operations. A bounded lattice is an algebra (A, ∧, ∨, 1, 0) in which (A, ∧, ∨) is a lattice and the distinguished elements 1 and 0 are members of A satisfying for all a ∈ A: a∧ 1 = a

a∨ 0 = a


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Thus 1 is an identity element for the operation ∧ and 0 is an identity element for the operation ∨. By the previous reasoning, at most one element can function as an identity element for each of these operations (and by 0.21.3(ii), on pain of triviality, no element can play the identity element role for both operations). The interrelation of the two operations in lattices produces the following interesting twist. Appealing to the absorption laws, after ‘join’-ing and ‘meet’-ing both the sides of these equations with 1 and 0 respectively, we get that: 1 = a∨ 1

0 = a∧ 0

so that 1 is a zero element for ∨ and likewise 0 for ∧. A concept with important applications in bounded lattices and certain expansions thereof is that of an atom, which we define with the aid of the relation of covering, itself applicable in the general context of lattices (posets, indeed). An element a covers an element b when b < a and for all c such that b c a, c = a or c = b. (where “b < a” abbreviates “b a & b = a )”. In a bounded lattice, an atom is an element which covers 0. For example, a bounded chain has at most one atom, and that is its -least non-zero element. In the lattice consisting of all subsets of a given set (where corresponds to inclusion) the atoms are the singletons (or ‘unit sets’) of the elements of that set. It is convenient to have a dual term, for which purpose we will say dual atom (some prefer ‘co-atom’), to denote those elements covered by the 1 in a bounded lattice. Bounded lattices provide the perfect environment for the development of various ideas of complementation. A complement of an element a in such a lattice is an element b satisfying: (i)

a∧b=0

and

(ii )

a ∨ b = 1.

If every element has a complement, and a function mapping each element to a complement of that element is promoted to the status of a fundamental operation one speaks of a complemented lattice. 0.13.7(ii) implies that if b is a complement of a in a bounded distributive lattice, b is the only complement of a. Thus in this case there is only one way of choosing a function mapping each element to a (or rather the) complement of that element; we denote this operation by ¬. Complemented distributive lattices are called boolean algebras. Thus, a boolean algebra is an algebra (A, ∧, ∨, ¬, 1, 0) of similarity type 2, 2, 1, 0, 0

in which (A, ∧, ∨, 1, 0) is a bounded distributive lattice and ¬ is a one-place operation – called complementation – satisfying, for all a ∈ A: a ∧ ¬a = 0 and a ∨ ¬a = 1. Exercise 0.21.4 Show that the following equations are satisfied in any boolean algebra: ¬1 = 0, ¬0 = 1, ¬¬a = a, ¬(a ∧ b) = ¬a∨ ¬b, ¬(a ∨ b) = ¬a ∧ ¬b. Of course for any given set S, the structure (℘(S), ∩, ∪, ¬, S, ∅) is a boolean algebra (called the power set algebra on S); what we are calling ¬a here is ¯: the set-theoretic complement of a in S, the set sometimes denoted by a or a S a( = {s ∈ S | s ∈ / a}). As is well known, propositions form according to classical logic a boolean algebra under the operations of conjunction, disjunction, and negation. Here a (more accurately “the”) tautologous and a contradictory proposition constitute, respectively, as 1 and 0. This can be understood with

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“proposition” meaning equivalence class of formulas (or sentences) under a suitable equivalence relation, as explained in 2.13 (p. 219), as well as in rather different terms (see 2.21). For some familiar non-classical logics, the propositions assume a rather different structure and for such applications we list, in particular here, several variations on the theme of boolean complementation. A complemented lattice (not, we emphasize, presumed to be distributive) whose complementation operation ¬ satisfies the equations listed in 0.21.4 is called an orthocomplemented lattice, or ortholattice for short. These play a role in the algebraic semantics of quantum logic (2.31) in which the distribution law is rejected; see 8.13 for the treatment of negation in quantum logic. Reimposing the requirement of distributivity on the lattice reduct of an algebra (A, ∧, ∨, ¬, 1, 0), we can weaken the ‘boolean’ conditions on complementation, and specifically, the requirement that the join of an element with its complement be 1; such an algebra is called a pseudocomplemented lattice or alternatively a pseudo-boolean algebra and the operation ¬ an operation of pseudocomplementation, if for all a ∈ A: a ∧ ¬a = 0 and for every lattice element b, if a ∧ b = 0 then b ¬a. Exercise 0.21.5 (i) Show that any boolean algebra is a pseudocomplemented lattice. (ii ) Show that the above conditions on pseudocomplementation imply that a bounded distributive lattice can be expanded to a pseudocomplemented lattice in at most one such way. (iii) Show that in no non-trivial ortholattice is any element its own orthocomplement, and that in no non-trivial pseudo-boolean algebra is any element its own pseudocomplement. (iv ) Show, by contrast with (ii ), that a given lattice has more than one expansion to an ortholattice. (Hint: consider a six-element ortholattice with four pairwise -incomparable elements all of them between a top (1) element and a bottom element (0).) Pseudo-boolean algebras, and certain expansions thereof called Heyting algebras play a role in the algebraic treatment of intuitionistic (propositional) logic (2.32), providing an ‘equivalent algebraic semantics’ for that logic, in the sense explained in 2.16 (as do boolean algebras for classical propositional logic). The needed expansions add a fundamental operation called relative pseudocomplementation and usually written → (being the algebraic analogue of an implicational connective) and subjected to the condition that a ∧ (a → b) b and that for any element c such that a∧c b, we have c a → b. (Some writers disregard the terminological distinction between pseudo-boolean algebras and Heyting algebras.) Exercise 0.21.6 Show that for any element a of a Heyting algebra, we have ¬a = a → 0. Playing a somewhat similar role in the analysis of relevant logic (2.33) are distributive lattices equipped with a singulary operation ¬ satisfying the last three equations in 0.21.4, called De Morgan algebras or De Morgan lattices, and bounded distributive lattices satisfying all five of those identities, called quasiboolean algebras. There is some redundancy involved here, since given the third


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identity, each of the fourth and fifth can be derived from the other. (N.B.: For these classes of algebras, the basic complementation principles (i) and (ii) above are not required. That is, we do not insist that a ∨ ¬a = 1 and a ∧ ¬a = 0, even—for the De Morgan case—when there exist top and bottom elements 1 and 0.) Finally, we mention that modal logics have been treated algebraically by means of certain expansions (A, ∧, ∨, ¬, , 1, 0) of boolean algebras, in which the arity of is 1. These are called modal algebras if the equations 1 = 1

and

(a ∧ b) = a ∧ b

are satisfied (for all a, b ∈ A). These are sometimes called, more explicitly, normal modal algebras; the logics concerned are the normal modal logics treated in §2.2. (See the Digression on p. 290 and the discussion preceding it.) Although several other collections of algebras (such as BCK-algebras, Hilbert algebras, and to a lesser extent BCI-algebras) play an important role in the algebraic treatment of purely implicational logics, we defer consideration of them until they appear in that role below (mainly in 2.13 and 7.25; the →-reducts of the Heyting algebras mentioned above are all Hilbert algebras in the sense defined in 2.13).

0.22

Derived Operations

Given an algebra A the class of all functions from An into A (for some chosen n) includes the fundamental n-ary operations of A and many more functions besides. Between the class of such n-ary fundamental operations and this whole class of arbitrary functions of n arguments on A, there are various classes of operations with some intuitive claim to be regarded as ‘derived’ (or ‘defined’) n-ary operations of the algebra. We will be especially interested in three such intermediate classes: the term functions of A and the algebraic functions of A, as well as in a slight generalization of the former class. (The word operation may be used in place of function here.) Both term functions and algebraic functions have, for different writers at different times, constituted the intended range of application of the expression “polynomial (function or operation)”, which as a result has become dangerously ambiguous and worth avoiding for that reason. In contemporary universal algebra, this means what we mean by algebraic functions. Having given their explanation, McKenzie et al. [1987], p. 145, write: We remark that our terminology for the derived operations is not universally used, although it is gaining wide acceptance. Be aware that in the history of universal algebra prior to about 1982, the term operations and polynomial operations of an algebra were often called “polynomial functions” and “algebraic functions,” respectively.

Our response, as already indicated, will instead be to avoid the potentially confusing word “polynomial” altogether, and retain the “algebraic” terminology in its original usage, alongside the “term function/operation” terminology. (Grätzer [1968] is a prime example of the “prior to 1982” literature McKenzie et al. allude to, which, however, lingers on long past the early 1980s.) One disadvantage of thus replacing “polynomial” by “term” is that the latter has no adverbial form corresponding to “polynomially”, and this lacuna we shall fill

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by using “compositionally” in the manner indicated below, which makes term functions and compositionally derived operations the same thing, allowing us to refer to our liberalized version of this class of functions as comprising the -compositionally derived operations. (Precise definitions for this terminology will be found below.) Before getting to that, however, we need to distinguish between the compositionally derived (‘term’) functions on the one hand and the algebraic functions, on the other, of a given algebra A. The intuitive idea is that the former class comprises those functions definable by composition (including repetition and reordering of variables) in terms of the fundamental operations, while the latter is more inclusive in that we allow the definition to make reference also to specific elements of A, and not just those which happen to be distinguished elements (zero-place operations) of A. To make this informal idea more precise, we need to be a little more selfconscious about the language in which our algebras are being described, as perhaps the reference to variables in the preliminary formulation just given suggests. We will take it that for any given similarity type of algebras, there is, corresponding to each fundamental operation provided by that type, an operation symbol (so that this will be a constant in the case of the zero-place operations) and that in addition we have denumerably many variables: x1 , . . . , xn ,. . . For example if the type is 2, 2, 1 , we have two two-place function (operation) symbols, f and g, say, and one one-place function symbol, h, say. Then the operations for which these symbols stand when interpreted in a particular algebra A of the type can be denoted f A , g A , and hA . (Of course in practice, one uses a more suggestive notation, to evoke conditions associated with particular classes of algebras of the type, e.g., ∧A , ∨A , ¬A ; and when it is clear which algebra is being talked about, the superscript is omitted. This has been our policy up to this point.) On this basis we define the terms for the given type to be those expressions comprising the smallest class such that (i) each variable is in the class, and (ii) if expressions t1 , . . . , tn are in the class and f is an n-place operation symbol, then the expression f (t1 , . . . , tn ) is in the class. We turn to the interpretation of terms as term functions or as we often prefer to say, as compositionally derived operations. A term t, all of whose variables are among {x1 , . . . , xm } – to indicate which t may be written as t(x1 , . . . , xm ) – induces a function from Am into A, where A = (A, f1A , . . . , fnA ) is algebra of the appropriate similarity type, in the following way: •

If t is a variable, t must be one of x1 , . . . , xm and the induced function is the corresponding projection function: for xk (where 1 k m) the function projkm defined by projkm (a1 , . . . , am ) = ak , for a1 , . . . , am ∈ A

•

If t is of the form f (t1 , . . . , tn ) for n-ary operation symbol f , then the induced function is f A (ϕ1 (x1 , . . . , xm ), . . . , ϕn (x1 , . . . xm )) where ϕi is the function induced by ti . (Note that if n = 0, t is just f and the induced function is the zero-place f A , one of the distinguished elements of the algebra A.)


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The m-ary compositionally derived operations (or term functions) of the algebra A are then precisely the m-place functions on A induced by the terms with variables amongst {x1 , . . . , xm } in this manner. The name arises from referring to the second step of the above definition as involving the composition of the function f A with the functions ϕ1 , . . . , ϕn . This is not the same has composition of 1-ary functions as defined in 0.13.4(ii) (p. 9), which is however the special case in which n = m = 1. Another special case, in which m = 0 and n = 1 is the application of a function to its argument. To avoid confusion with the earlier notion of composition, some writers speak in of the current general form as superposition of functions. It may seem still insufficiently general, as one should surely be allowed to define a function f of arity 3, say, from given functions g and h of arities 2 and 1 respectively by setting f (x, y, z) = h(x, g(y, z)) and have this be regarded as deriving f by composition from h and g, even though we are not dealing with the strict format prescribed above. By the moves described in the following paragraph, however, it turns out that the strict format can indeed be seen to subsume such cases and there is no need for further generality that it provides. Our initial characterization of the compositionally derived operations, or term functions, makes reference to the possibility of repeating or re-ordering variables, but it may not be immediately evident how the above account of which function from Am to A is induced by a term whose formation exploits these possibilities. Take the case of re-ordering. We may wish to define in a semigroup (B, ·) a binary operation ∗ taking the pair a, b to b · a (a, b ∈ B). Such an operation is among the term functions of the semigroup, being induced by the term “x2 · x1 ”. The above recipe, using the projection functions, for determining the induced mapping from A2 to A for any algebra A of the same type as our semigroup (any groupoid, that is) proceeds by applying ·A to proj22 (x, y) and proj12 (x, y) in that order – where we have taken the liberty of writing x, y (and below, z) for x1 , x2 (and x3 ) so as to provide an interpretation of the term x ∗ y ( = y · x) in the algebra B. Along similar lines, one sees how an interpretation is provided for the case where a variable is repeated, as with the term x2 ( = x·x). (This is sometimes called: identification of variables.) We can similarly ‘pad in’ fictitious variables on which the function does not depend, as when we define $(x, y, z) to be x · z. This term induces a function from B 3 to B by: ·B (proj13 (x, y, z), proj33 (x, y, z)), or, to revert to the more customary ‘infix’ notation: proj13 (x, y, z) ·B proj33 (x, y, z). (The notion of a function’s depending on a particular argument position – here put loosely in terms of a variable’s occupying that position in a term – is defined in 3.14.10, p. 411.) The informal explanation of algebraic functions spoke of allowing reference to particular elements (of the universe) in the definition of the function. It is not hard to see how to explicate this, given the above definition of a compositionally derived operation. We simply say that the algebraically derived operations (or ‘algebraic functions’) of A are the compositionally derived operations of the algebra we obtain by expanding A by the addition of all elements of A as nullary fundamental operations. Thus if A = (A, ∨A , ∧A ) is a lattice, with a, b ∈ A, then while the ternary function ϕ(x, y, z) = x ∨A (y ∧A z) is a compositionally derived operation (term function), the singulary function ψ(x) = a ∨A (x ∨A b) qualifies only as an algebraically derived operation (algebraic function). We need, finally, to draw attention to a third kind of derived operation, foreshadowed above, intermediate between the term functions and the algebraic

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functions, of an algebra A. Compositionally derivable from operations all of which are of arity > m, for m 2, are many of lower arity, as the example above of the term x2 ( = x · x) illustrates. But composition of functions (even in the current general sense) none of which are 0-ary cannot yield a 0-place operation, and we shall now liberalize the notion of compositional derivability so as to make good this lack. The idea will be to allow a reduction in arity by eliminating an argument place on which a function does not genuinely depend. For example, if we as our algebra A have the 2, 1 reduct of a group, eliminating the identity element, then we can recover this element eA , say, by A A a ‘liberalized’ compositional derivation from ·A and ( )−1 as a ·A a−1 . The apparent dependence on a is spurious since all group-reducts of the similarity type of A satisfy the equation x · x−1 = y · y −1 . More precisely then, and abbreviating “liberalized” to “”, we say that the class of -compositionally derived operations of an arbitrary algebra A is the smallest class of functions F such that: (1) Every term function of A is in F ; (2) If a function ϕ: Am −→ A is in F , and for all a1 , . . . , am , b ∈ A, ϕ(a1 , . . . , ai , . . . am ) = ϕ(a1 , . . . , ai−1 , b, ai+1 , . . . , am ), then the function ψ: Am−1 −→ A is in F , where given any a1 , . . . , ai−1 , ai+1 , . . . am ∈ A: ψ(a1 , . . . , ai−1 , ai+1 , . . . am ) = ϕ(a1 , . . . , ai−1 , b, ai+1 , . . . , am ), for an arbitrary b ∈ A. More generally, we say that a function is -compositionally derivable from those in some set when it can be obtained by them using not only composition (with the aid of the projection functions) but also the above elimination of an inessential argument place—as in the ith place of ϕ under (2). In fact, since repetition of variables is allowed under composition, the only time at which such elimination is actually needed is in descending from 1-ary to 0-ary functions (as in the group theory example): in all other cases, compositional and -compositional derivability coincide. However, this particular case does arise for the concept of functional completeness (3.14, p. 403), which needs to be defined in terms of the liberalized notion – for the source of which, see the end-of-section notes, which start on p. 43 below. (However, we shall also meet a notion of strong functional completeness in 3.14 which uses the unliberalized form of compositional derivability.)

0.23

Homomorphisms, Subalgebras, Direct Products

A homomorphism from an algebra A to an algebra B of the same similarity type is a function h: A −→ B if for each n, every n-ary operation symbol f associated (as in 0.22) with the type, satisfies: h(f A (a1 , . . . , an )) = f B (h(a1 ), . . . , h(an )) for all a1 , . . . , an ∈ A. For example, if A and B are lattices, h must map the join (in A) of two elements of A to the join (in B) of what it maps those two elements to, and similarly for meets. If A and B are bounded lattices then h


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must take the unit and zero of A to the unit and zero of B. And so on. We call A and B respectively the source and the target of h. If the function h is surjective (maps A ‘onto B’) then the target is called a homomorphic image of the source. If h is not only surjective but also injective (‘one-one’) then h is called an isomorphism between A and B, and these algebras are described as isomorphic. (Surjectivity and injectivity were introduced in the discussion following 0.12.6, p. 5 above.) In many contexts, differences between isomorphic algebras are ignored, as when one speaks of the eight-element boolean algebra or the three-element chain. When the source and target algebra are the same, homomorphisms (isomorphisms) are called endomorphisms (automorphisms). While an isomorphic image of an algebra can be regarded as a perfectly faithful copy, homomorphisms in general destroy internal structure in the sense that if, for example, h is a homomorphism from one algebra to another, h may map distinct elements of A to the same element of B. In apparent contrast with this description of homomorphisms, the above definition of them is often summarized by saying that a homomorphism is a structure-preserving map from one algebra to another. The point of such a characterization is that although distinctions between elements may be blurred in passing to a homomorphic image, they are blurred in a highly structure-sensitive way: if the distinction between two semigroup elements a and b is thus blurred, in the sense that h(a) = h(b), then no distinction between products involving them can avoid being blurred: h(a · c) must equal h(b · c) for any semigroup element c, for example. This idea of selective blurring of differences can be more directly isolated with the concept of a congruence relation (or just “congruence”, for short) on an algebra, by which is meant an equivalence relation ≡ on the universe of the algebra for which, given any n-ary fundamental operation ϕ of the algebra and elements a1 , . . . , ak , if elements b1 , . . . , bk are such that ai ≡ bi (1 i k) then we also have ϕ(a1 , . . . , ak ) ≡ ϕ(b1 , . . . , bk ). Thus if ≡ collects together certain elements, thereby blurring the distinction between them, it must also blur all distinctions consequential on that collecting together. Clearly any homomorphism h gives rise to a congruence relation ≡, defined by: a ≡ b iff h(a) = h(b), and conversely: Exercise 0.23.1 (i) Show that if ≡ is a congruence relation on an algebra A, then for the algebra B (often called the quotient algebra determined by ≡) with universe B = {[a] | a ∈ A}, where [a] = {a | a ≡ a }, and fundamental operations f B defined for n-ary operation symbol f provided by the type of A by: f B ([a1 ], . . . , [an ]) = [f A (a1 , . . . , an )] then the function h mapping a ∈ A to [a] ∈ B is a homomorphism from A onto B. (ii ) The following concept makes sense for any algebra A which is an expansion of a semilattice with ∧ (and partial order defined by x y ⇔ x ∧ y = x). A filter in A is a set F ⊆ A such that (1) for all a, b ∈ A, a ∈ F and a b imply b ∈ F and (2) a, b ∈ F implies a ∧ b ∈ F (i.e., a ∧A b ∈ F ). Suppose that A is a boolean algebra. Show that there is a one-to-one correspondence between filters in A and congruences on A defining ≡F , for F a filter in A, by: a ≡F b iff a ↔ b ∈ F (where

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CHAPTER 0. PRELIMINARIES a ↔ b is (¬a ∨ b) ∧ (a ∨ ¬b), and again all operation symbols are understood with an “A” superscript), and, defining for a congruence ≡ on A, the filter F≡ as {a ∈ A | a ≡ 1}. It is necessary to verify that these definitions of ≡F do indeed yield a congruence relation and a filter respectively, and also that when the definition of ≡F is applied to F = F≡ , we recover the original congruence ≡ and similarly that when the definition of F≡ is applied to ≡ = ≡F , the resulting F≡ is the original F .

We required an isomorphism to be a homomorphism which was both one-one and onto. If we delete the second requirement, we get the concept of an isomorphic embedding (of the source algebra into the target algebra). Another way of describing this is to say that the source algebra is isomorphic to a subalgebra of the target, in the sense of the following definition. A is a subalgebra of B if A and B are of the same similarity type, A ⊆ B, and for each n-ary fundamental operation symbol provided by that type, f B (a1 , . . . , an ) = f A (a1 , . . . , an ) for all a1 , . . . , an ∈ A. In other words the operations do the same thing in the ‘larger’ algebra as do the corresponding operations of the ‘smaller’ algebra, when restricted to the latter’s elements. Exercise 0.23.2 (i) Define the algebra A to be the subalgebra of an algebra B generated by the (set of) elements A0 ⊆ A if A is the least subset of B which contains all the elements in A0 and is closed under the operations f B . Verify that this is indeed a subalgebra of A, and that the mapping sending A0 to A here is a closure operation. (ii ) Suppose A0 is a set of generators for A and h0 : A0 −→ B where B is the universe of some algebra B of the same similarity type as A. Show that if h and h are both homomorphisms from A to B which agree with h0 on A0 , in the sense that h(a) = h (a) = h0 (a) for all a ∈ A0 , then h = h . The last of the three concepts figuring in our title, that of a ‘direct product’ of algebras is best illustrated as applying to pairs of algebras, which generalizes in an obvious way to the case of taking direct products of finitely many algebras (the case which we need for the sequel) and a slightly less obvious way to arbitrary families of algebras, which is explained in any of the universal algebra texts cited in the end-of-section notes (beginning at p. 43). So suppose we are given two algebras A and B of the same similarity type. We define their direct product to be the algebra A ⊗ B whose universe is A × B, the cartesian product of A and B, and for each n-ary operation symbol f , the operation f A⊗B is ‘computed coordinatewise’ from f A and f B , in the sense that: f A ⊗ B ( a1 , b1 , . . . , an , bn = f A (a1 , . . . , an ), f B (b1 , . . . , bn ) . We will call A and B the factor algebras of the product A ⊗ B, and take it that the extension of the above definition to direct products of finitely many factors A1 , . . . , An is clear enough. (Just use ordered n-tuples instead of ordered pairs.) Exercise 0.23.3 Show that each of A, B, is a homomorphic image of A ⊗ B. (Hint: consider the projection functions mentioned in 0.22.)


0.24

29

Equational Classes of Algebras

To this point, several classes of algebras have been introduced: lattices, distributive lattices, chains, groups, semigroups, cancellation semigroups, boolean algebras, and so on. In some cases the classes were defined to comprise precisely those algebras of a certain similarity type as satisfied some listed equations. A class of algebras which can be so defined is called an equational class of algebras. Using the notions introduced in 0.22, we can say that if the language for a given similarity type provides terms t and u, then an equation is an expression of the form t = u, such an equation being satisfied by an algebra A of the type in question when the induced term functions tA and uA are the same. (More specifically, where {x1 , . . . , xm } includes all the variables occurring in either t or u, these terms induce the same – “compositionally derived”, as we shall say – function from Am to A.) Let us call the set of all equations satisfied by every algebra in some class K the equational theory of K, or Equ(K) for short, and the class of all algebras satisfying every equation in some class E of equations, Mod (E) – for “models of E”. Clearly, understanding these notions in terms of any single similarity type, (Mod, Equ) is a Galois connection between equations for, and algebras of, that type, and equational classes are classes of the form Mod (E) for some E. We shall meet several other analogous Galois connections in the ensuing chapters. When, as here with the ‘is satisfied by’ relation, they arise from relational connections between linguistic and non-linguistic classes of objects, we adopt the convention of placing the linguistic objects on the left. The study of the connection (Mod, Equ) is called equational logic, and the main tools for its study were provided by Birkhoff [1935], in the shape of a syntactic description of the induced closure operation Equ(Mod (·)) and an algebraic description of the dual closure operation Mod (Equ(·)). The closed elements under the first operation, which are precisely the equational theories mentioned above, are sets of equations closed under the usual rules (transitivity, replacement of equals by equals, etc.) for deducing consequences of equations, while the closed elements under the second – alias the equational classes – are, Birkhoff showed, precisely those classes K which are closed under: taking a homomorphic image of any algebra in K, taking a subalgebra of any algebra in K, and taking the direct product of any family of algebras in K. If K is closed under these three procedures, K is called a variety. It is not hard to see that an equational class of algebras must be a variety, since an equation holding in an algebra must hold in any homomorphic image or subalgebra thereof, and an equation holding throughout a family of algebras must hold in their direct product. (A further remark on this ‘preservation’ behaviour will be made in the opening paragraph of 0.25.) What is less obvious is the converse claim: that any class of algebras closed under homomorphic images, subalgebras, and direct products, is an equational class. We don’t need to give Birkhoff’s proof that K is a variety if and (more particularly) only if K is an equational class, though we will need to extract one concept used in (the ‘only if’ part of) that proof, namely, the notion of a ‘free’ algebra. First, a pause to extract some information from the more obvious half of the result. Observation 0.24.1 Chains do not constitute an equational class. Proof. The direct product of two chains is not a chain. (Draw a picture of the product of the two-element chain with itself.)


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Exercise 0.24.2 Show that cancellation semigroups do not constitute an equational class. (Hint: find a homomorphic image of a cancellation semigroup which is not itself a cancellation semigroup.) The classes of algebras featuring in 0.24.1 and 0.24.2 were introduced by, respectively, a disjunction of equations, and a conditional equation. Sometimes, such definitions, not themselves having the form of a set of equations, can be traded in for an equivalent description in equational form. Pseudo-boolean algebras are a case in point, since the key notion of the pseudocomplement of an element a was introduced (0.21.4) as an element ¬a such that a ∧ ¬a = 0 (O.K. so far – an equation) and such that for any b if a ∧ b = 0 then b ¬a. then . . . ” is the problem here – not the “” (since “b ¬a” just The “if means: “b∧ ¬a = b”). Nevertheless, an equational characterization can be given, as was observed by P. Ribenboim in 1949. (For further details, consult Balbes and Dwinger [1974].) Example 0.24.3 The relevant equations, supplementing those defining bounded distributive lattices, to obtain an E for which pseudo-boolean algebras make up Mod (E) are: a ∧ ¬(a ∧ b) = a ∧ ¬b

a ∧ ¬0 = a

¬¬0 = 0.

Exercise 0.24.4 Verify that any pseudo-boolean algebra satisfies these equations, and, conversely, that the equations are satisfied only in such algebras. We turn now to the topic of free algebras. An algebra A of some similarity type is said to be an absolutely free algebra freely generated by a set G ⊆ A when (i) A is generated by G, and (ii) an arbitrary mapping h0 from G to the universe B of any algebra B (of the given type) can be extended to a homomorphism h from A to B. By 0.23.2(ii), p. 28, this h is uniquely determined by h0 . The effect of this rather abstract definition may be gleaned from the following, which we state without proof. Observation 0.24.5 A is an absolutely free algebra with G ⊆ A as set of free generators if and only if (i) A is generated by G, (ii) no a ∈ G is ϕ(b1 , . . . , bn ) for any n-ary fundamental operation ϕ of A, and (iii) whenever ϕ(b1 , . . . , bn ) = ψ(c1 , . . . , cm ) for elements b1 , . . . , bn , c1 , . . . , cm ∈ A and fundamental operations ϕ, ψ of A, then: ϕ = ψ, m = n, and bi = ci for i = 1, 2, . . . , m. We will need the concept of an absolutely free algebra in 1.11 for our description of the formal languages of sentential logic. More germane to Birkhoff’s result, mentioned above, is the following relativized version of the concept. We simply relativize the above definition to a class K of algebras of the same similarity type, saying that A ∈ K is a (relatively) free algebra in K with G ⊆ A as its set of free generators if any mapping h0 from G to the universe B of any B ∈ K can be extended to a (unique) homomorphism h from A to B. Not every class of algebras (of a given similarity type) provides free algebras in the sense of the above definition, but if K is an equational class (alias variety), then free


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algebras in K exist with any desired number of free generators and for this case the upshot of the definition is that a free algebra has all its elements generated from G as ‘freely’ as possible, compatibly with its satisfying the equations in question. For example, in a free semigroup, we never have ab = ba except when this is required by the semigroup equations – as when b = aa, and the above identity holds in consequence of the associative law: x(yz) = (xy)z. For any variety K, the free algebra in K with countably many generators (‘the’, by identifying isomorphic algebras) plays a crucial role in the proof of the unobvious half Birkhoff’s theorem: that every class of algebras closed under homomorphisms, subalgebras, and direct products is an equational class. (See the texts mentioned in the notes – which begin on p. 43 – for a convenient exposition of the proof.) All we need these algebras for is completeness proofs in ‘algebraic semantics’ for sentential logics (2.13–14), and we do not even need to stress this aspect of the Tarski–Lindenbaum algebras constructed for such purposes. The free algebras in a variety are ‘generic’ in respect of equations, in the sense of satisfying all and only those equations satisfied in every algebra in the variety; such an algebra for K is a conjunctive combination (on the right) of all algebras in K w.r.t. the relational connection between equations and algebras, whose induced Galois connection (Mod, Equ) came to our attention in 0.24. (See further the discussion following 0.25.3 below.) One technical phrase for such equationally generic algebras is “functionally free”: for any class K of algebras, an algebra A ∈ K is functionally free in K if Equ({A}) = Equ(K). (We assume that K contains only algebras of some one similarity type, but not that K is a variety. The origin of this ‘functionally free’ terminology is described at the end of the notes on this section, which begin on p. 43). Such functionally free algebras need not be free algebras; for example, all non-trivial boolean algebras satisfy the very same set of equations, so that each of them is functionally free (in the class of boolean algebras). But the boolean algebra freely generated from k generators has 2n elements where n = 2k , so that, for example, the eightelement boolean algebra is not a free boolean algebra, its functional freeness notwithstanding. Note that being functionally free is specifically a matter of being, as it was put above, ‘generic’ in respect of equations satisfied. It is perfectly possible, for example for a free (and hence functionally free) algebra in a variety to satisfy conditional equations not satisfied throughout the variety; for example, all free semigroups satisfy the cancellation laws. (For more in this vein, see Prešić [1980] and references there cited.) Since varieties of algebras within a given similarity type are classes (namely, equational classes) of algebras, the poset which orders them under ⊆ is a lattice, with K1 ∧ K2 = K1 ∩ K2 and K1 ∨ K2 = K1 ∪˙ K2 , where (as in 0.13.5, p. 10) the last symbol denotes ‘closure of the union’, the closure of a class of algebras being understood here as the smallest variety to contain them all. (This is the closure operation Mod (Equ(·)).) This lattice has bounds 1 containing all algebras of the similarity type in question, and 0 containing just trivial algebras of that type, or, as it would more commonly be put since these are all isomorphic, containing only the trivial algebra of the type. And there will also be atoms, with no proper subvarieties apart from 0. These are called equationally complete varieties, since adding any equation not already belonging to their equational theory results in a theory proving x = y and hence having only trivial algebras as models. From the comment that all non-trivial boolean algebras satisfy the same equations, there follows part (i) of the following observation; proofs of (ii)


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and (iii) may be found in the literature (Kalicki and Scott [1955], originally; see also the expository article Duncan [1977]). Observation 0.24.6 The following varieties of algebras (from among those we have mentioned) are equationally complete: (i) boolean algebras, (ii) distributive lattices, (iii) semilattices. Equationally complete varieties are sometimes called minimal varieties, since any further shrinkage leaves only trivial algebras; the corresponding equational theories are called maximal, since they are as large as can be without being trivial (i.e., proving all equations); the latter are dual atoms in the lattice of equational theories (for a fixed similarity type). The reversal of the ordering here is just that generally associated with Galois connections (by (G3) and (G4) of 0.12). Remark 0.24.7 As well as sometimes (e.g., in 0.23.1(ii), p. 27) being very casual in not distinguishing between an operation symbol and its interpretation as an operation in a particular algebra, we have not followed a convention in many of the end-of-section references of distinguishing our use of the sign “=” to make an identity claim from our use of it to refer to the symbol for identity (equality) flanked by terms in equations under discussion. The latter is often written as “≈” in order to avoid this use-mention confusion, as in notationally distinguishing the semantical claim that for some groupoid A = (A, ·A ) the equation x · y ≈ y · x is satisfied by (or is true in) A – often written as “A |= x · y ≈ y · x” from the purely mathematical claim (spelling out the content of the former claim) that for all a, b ∈ A, a ·A b = b ·A a.

0.25

Horn Formulas

It will be helpful here to place the ‘easy’ direction of Birkhoff’s theorem—that all equational classes are closed under homomorphic images, subalgebras, and direct products—into a more general context. (However, the following discussion presumes a slightly greater familiarity with first-order logic than this book generally demands.) These three closure conditions on equational classes reflect three preservation properties which equations come to possess in virtue of their form. They are (interpreted as) purely universal, even though we generally omit the universal quantifiers and simply ask whether the equation itself (e.g., the idempotent law) is satisfied by an algebra, and it is in virtue of this feature that they are satisfied by any subalgebra of an algebra satisfying them. They are positive formulas, accounting for their preservation under passage to homomorphic images (on which more below). And they are (basic) Horn formulas: disjunctions – allowing any number of disjuncts – of atomic and negated atomic formulas in which at most one disjunct is unnegated; all such formulas are preserved under the formation of direct products. (In the case of equations, there are no negated disjuncts.) Because it will occupy us later, we need to go into a little more detail on this last feature. Our applications involve the form of certain statements of the metalanguage in which sentential logics are discussed, and we will speak of ‘Horn sentences’ rather than Horn formulas, but the topic is best introduced in the latter terminology and in the context of first-order logic.


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As already remarked, by a basic Horn formula of a first-order language, is meant a formula in prenex normal form (all quantifiers ‘at the front’, that is) in which what follows the quantifiers is a disjunction of negated and unnegated atomic formulas, at most one of which is unnegated. It is not hard to see that if a Horn formula is true in each of a class of structures, it is true in their direct product, regardless of the pattern of the initial quantifiers. There is a kind of converse to this result in the case where all initial quantifiers are universal, as well as for that in which all are existential (though not in the general case of a ‘mixed’ prefix); further information is provided in the notes, which start at p. 43. Here we are interested in the direction: if it’s a Horn formula then it’s preserved by the direct product construction. In 0.23 the notion of a direct product was explained only for algebras, and in the case of the direct product of a pair of algebras A, B, what was said was that this was an algebra A ⊗ B, with universe A × B, and operations, f A⊗B given by: f A ⊗ B ( a1 , b1 , . . . , an , bn ) = f A (a1 , . . . , an ), f B (b1 , . . . , bn )

for n-ary operation symbols f . (Treating individual constants as 0-ary operation symbols means that if f is such a constant, f A ⊗ B is f A , f B .) This has the effect that (†)

f A ⊗ B ( a1 , b1 , . . . , an , bn ) = an+1 , bn+1 ⇔ f A (a1 , . . . , an ) = an+1 and f B (b1 , . . . , bn ) = bn+1 .

It is the analogue of this feature that needs to be secured in the case of a model (or ‘structure’) for a first order language in which various relation symbols (predicates) are typically present, so, still using the A, B, notation (the more suggestive “M”, “N”, will be used below), we stipulate that relation on A × B is assigned to the m-ary relation symbol R, is RA ⊗ B as defined: (††)

RA ⊗ B ( a1 , b1 , . . . , am , bm ) ⇔ RA (a1 , . . . , am ) and RB (b1 , . . . , bm ),

for all a1 , . . . , am ∈ A, and all b1 , . . . , bm ∈ B. One could of course write (e.g.) “ a1 , . . . , am ∈ RA ” in place of “RA (a1 , . . . , am )” (etc.) if preferred. It may be thought that for the equational case we forgot to mention (††) as applied to the relation of identity (=) itself; but no such special mention is required as the condition is automatically satisfied by the ordered pairs (or more generally, ordered n-tuples) construction itself. See the beginning of 5.34 for more on this. While for a full proof that whatever (closed) Horn formulas are true in each of a family of structures are also true in their direct product, the reader is referred to the literature – e.g., to the discussion in Chang and Keisler [1990], §§6.2,3 (which also gives an account of direct products of arbitrary families rather than just pairs of models, as here) – the main idea in the proof is indicated below (0.25.3(iii)). This would not be so if we allowed more than one unnegated disjunct, since then the structures could differ as to which disjunct they verified, with the direct product verifying neither. Whenever truth for all formulas of a certain form is preserved by some construction (such as formation of direct products), this preservation behaviour is automatically inherited by

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conjunctions of formulas of the given form (see 0.25.3(ii) below), so one defines a Horn formula to be any conjunction of basic Horn formulas. (We understand a conjunction in the generous sense that there may be more than – or fewer than – two conjuncts.) Universal Horn formulas – that is, Horn formulas all of whose prenexed quantifiers are universal – are especially interesting in view of the interaction between preservation to substructures conferred by their universality, and preservation to direct products conferred by their Horn form. First-order theories with universal Horn formulas as axioms have, if consistent, minimal models. This property has been exploited especially in logic programming, a program in the language Prolog consisting of basic Horn formulas (without explicit quantifiers, but understood universally); see Lloyd [1987], Hodges [1993a]. This is closely related to a result we shall give in terms of intersections of models (with overlapping domains, notated M and N here; we require M ∩N = ∅ so that the intersection model – M ∩ N in what follows – has a non-empty domain). For simplicity, suppose that we have only one non-logical symbol, a two-place predicate letter R; nothing hangs on the arity of R, however (or indeed on the fact that only one predicate symbol is present). Then given two structures M = (M , RM ) and N = (N , RN ) with M ∩ N = ∅, we define their intersection, M ∩ N, to be (M ∩ N , RM ∩ RN ). Observation 0.25.1 Any universal Horn formula (of the above language) true in each of M and N is true in M ∩ N. Proof. (Sketch:) Since it is a Horn formula, the formula in question must be true in the direct product of M and N if it is true in M and N, and therefore also, since it is a universal formula, true in the substructure of this product with domain { a, a | a ∈ M ∩ N }; but this last structure is isomorphic to M ∩ N, by (††) above. A special case of 0.25.1 is that in which M = N ; this explains why, for example, the intersection of two transitive relations on a given set is a transitive relation, since transitivity is expressible by a universal Horn formula. (Think of the Horn formula in question as true in (M , RM ) and in (N , RN ), where M = N , and therefore in the intersection of these models.) Of course one usually writes ∀x∀y∀z((Rxy ∧ Ryz ) → Rxz ) rather than: ∀x∀y∀z(¬Rxy ∨ ¬Ryz ∨ Rxz ) but the equivalence (in classical predicate logic) is clear. (In our informal usage we would have “For all x, y, z: (xRy & yRz ) ⇒ xRz ” for the implicational form.) In terms of the more usual way of writing such formulas, the definition of a basic Horn formula as a disjunction of negated and unnegated atomic formulas in which at most one disjunct is unnegated amounts to: being of the form (ϕ1 ∧ . . . ∧ ϕn ) → ψ in which the ϕi and ψ are atomic and (i) if n = 0 we understand the formula to be the consequent of this schematically represented conditional, (ii) we allow the place marked by the schematic “ψ” not to be filled, in which case the whole schema denotes ¬(ϕ1 ∧. . .∧ ϕn ). (This amounts to taking ψ as ⊥.) What is not


35

allowed is that the consequent should itself be a disjunction of atomic formulas. (If arbitrary disjunctions were allowed in the consequent of a basic Horn formula, every formula would be equivalent to a conjunction of such formulas, since every formula has an equivalent in conjunctive normal form with quantifiers prenexed.) Examples 0.25.2As already remarked, there is nothing special about the case of binary relations, 0.25.1 applying across the board. But because various conditions on binary relations are especially familiar, we continue to illustrate matters by reference to them. The prefixed universal quantifiers are omitted here, and we use the disjunctive format for (would-be) Horn formulas – as opposed to the implicational style inset above. (i ) Rxx (reflexivity); (ii ) ¬Rxx (irreflexivity); (iii) ¬Rxy ∨ Ryx (symmetry); (iv ) ¬Rxy ∨ ¬Ryx (asymmetry); (v ) ¬Rxy ∨ ¬Ryx ∨ x ≈ y (antisymmetry); (vi ) Rxy ∨ Ryx (strong connectedness). All of (i)–(v) are basic Horn formulas, while (vi) is not, as there is more than one unnegated atomic disjunct. (Reading 0.24.1 with chains as posets and R as gives a case of non-preservation under binary direct products, so (vi) isn’t even equivalent to a Horn formula – a conjunction of basic Horn formulas, that is.) Case (i) is like that of equational logic in which there are no negated disjuncts. It, along with cases (iii) and (v), is what is called a strict basic Horn formula, meaning that there is at least one (and hence exactly one) unnegated atomic disjunct, while (ii) and (iv) is non-strict. For a Horn formula to count as strict, it has to be a conjunction of basic Horn formulas all of which are strict. (In the implicational format above, this means we do not allow ψ to disappear, or be taken as ⊥.) A special property of strict Horn formulas is that the trivial model M for a given language, in which the domain has only one element a, which (as in the case of trivial algebras) serves as the unique value for all functions, and for k -ary predicate symbols R, RM = { a, . . . , a } (a appearing k times). Taking the direct product of the empty family of structures (for a given language) to be this trivial model, then we can say that the direct product of any family of models all of which verify a strict Horn formula will in turn verify that formula, whereas if the word “strict” is to be deleted, we have to insert “non-empty” before “family of models”, since the non-strict Horn formulas are all false in the trivial model (as the latter verifies all atomic formulas). Strict Horn formulas are preserved by arbitrary direct products, whereas Horn formulas in general are preserved only by non-empty direct products (i.e., direct products of non-empty families). Universal strict Horn formulas in equational logic, in which all atomic formulas have the form t = u, or better (0.24.7, p. 32), t ≈ u, for terms t and u, are sometimes called conditional equations, or quasi-equations or – most commonly – quasi-identities, and a class of algebras of some similarity type which comprise precisely those in which some set of quasi-equations are satisfied are called

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quasivarieties (or ‘quasi-equational classes’). Thus, for example, by the cancellation semigroups constitute a quasivariety. Every variety is a quasivariety (take n = 0 in the schematic implication before 0.25.2) though not conversely (0.24.1, 0.24.2). Quasivarieties which, like the class of cancellation semigroups, are not themselves varieties (as 0.24.2 shows) are sometimes called proper quasivarieties. One of the more famous proper quasivarieties encountered in connection with propositional logic is the class of BCK -algebras, discussed below in 2.13 (which begins on p. 219). Corresponding to Birkhoff’s characterization of equational classes of algebras as varieties in the sense of classes of algebras closed under taking subalgebras, homomorphic images and direct products, there is a purely mathematical or ‘structural’ characterization of quasivarieties, in which the reference to homomorphic images is replaced, but since the replacement involves concepts we have no further use for in what follows, the interested reader is referred to the literature on this (see the end of section notes, which begin at p. 43). We conclude with some general remarks about matters of preservation. Remarks 0.25.3(i) Let be a binary relation between models for some firstorder language (say) and X be a set of formulas of that language with the property that for any ϕ ∈ X, and any models M, N: if M N, then if M |= ϕ, we have N |= ϕ. Thus preserves (the truth of) formulas in X from one model to another. On this hypothesis, one can readily check that it follows that also preserves (the truth of) conjunctions as well as disjunctions of formulas in X, because more generally, if preserves ϕ and also preserves ψ, then preserves ϕ ∧ ψ, as well as preserving ϕ ∨ ψ. Thus, inductively, we have that any formula constructed by ∧ and ∨ from formulas in X is guaranteed to be preserved by any Xpreserving relation . We can think of such formulas as the X-positive formulas. Where X comprises the set of all atomic formulas of the language, one simply says (as at the start of our discussion) positive formulas. (Notes: (1) although in some contexts – cf. the label “Positive Logic” – implications count as positive, in the current context they do not; (2) since we have spoken of the formulas being true we are thinking of closed formulas, in which case this last reference to atomic formulas to makes best sense in the context of a language with a copious supply of individual constants). (ii) The model-to-model preservation relations of (i) include such relations as M’s having N as a submodel or as a homomorphic image, but for the case of direct products, we need a relation between a class of models and another model, so we use a fancier notation to signal the shift: “ ”. In this case we say that preserves ϕ when for any family of models C and model N with C N, if M |= ϕ for each M ∈ C, then N |= ϕ. By contrast with the case under (i) above, while here we have that if preserves two formulas it preserves their conjunction, is not guaranteed to preserve their disjunction. (The universal quantification “for each M ∈ C” blocks this.) That is why conjunctions of basic Horn formulas count as Horn formulas while disjunctions of basic Horn formulas do not. (Here we are thinking of C , N as obtaining when N is the direct product of the M ∈ C.) (iii) The direct product construction is so designed – cf. (†), (††), above – that for a suitable choice of X, when N is the direct product of the


37

M ∈ C, we have: (†††)

N |= ϕ iff for all M ∈ C, M |= ϕ, for all ϕ ∈ X.

For example, again exploiting the possibility of introducing individual constant to denote the elements of the domain, we may take X to comprise the closed atomic formulas. (Here we are trying to avoid the complications caused by assignments to free variables.) The present point of interest, abstracting away from the direct product construction itself, is that whenever the condition (†††) here is satisfied by C, N and X then any ‘Horn composition’ of formulas in ϕ1 , . . . , ϕn , ψ ∈ X (ϕ1 ∧ . . . ∧ ϕn ) → ψ the truth of this formula – call it χ for brevity – is preserved in the passage from the M ∈ C to N, in the sense that If M |= χ for each M ∈ C then N |= χ. (Note that, by contrast with (†††), this is just a one-way conditional, whose converse will in general fail.) The proof of this claim is straightforward and we leave it to the reader, especially as we go through the relevant reasoning explicitly for a specific application in the proof of Theorem 1.14.5 below (for the analogue of the specific but representative case in which |C| = 2, reformulated without this restriction in the ‘if’ direction of 1.14.9, on p. 70). By analogy with X-positivity in (i) above, we could call such a χ a (basic) X-Horn formula; the fact that Horn formulas are preserved in the transition from factor structures to their direct product is the special case in which X comprises the atomic formulas and the direct product construction secures (†††) for this choice of X. What (†††) says that in the relational connection between models (for a given language L) and formulas in X ⊆ L, with relation |=, N is a conjunctive combination (on the left) of the M ∈ C. (Thus when X is the set of atomic formulas, the direct product construction provides such conjunctive combinations.) The allusions to later results (in 1.14) under (iii) arise from taking instead conjunctive combinations of (what are there called) valuations for a propositional language, the lhs of the analogue of (†††) is the claim that the propositional formula A is true on a valuation v which is the conjunctive combination of all u ∈ U (U some collection of valuations) and the rhs is the claim that A is true on each u ∈ U . In the terminology of our later discussion, 0.25.3(iii) tells us, that any Set-Fmla sequent holding on each u ∈ U holds on v as well.

0.26

Fundamental and Derived Objects and Relations: Tuple Systems

Though the topic we address in this subsection does not fall under the heading ‘Algebraic Concepts’ of the present section, it is nevertheless an important preliminary to our subsequent treatment of sentential logic, and is naturally treated at the end of a discussion of Horn formulas. For the condition that a set S be

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closed under an n-ary relation R (as in 0.13.4(iii), p. 9) is itself a particularly simple type of Horn-formulable condition, to the effect that if x1 ∈ S and . . . and xn−1 ∈ S and x1 , . . . , xn ∈ R, then xn ∈ S. In 1.24, we shall treat (n − 1)-premiss rules as n-ary relations among formal objects, which at this stage it will do no harm to think of as formulas of some formal language (actually, they will be things called ‘sequents’). There is an important distinction between (what we shall call) a proof system’s being closed under such rules, sometimes put by saying that they are admissible rules for the system, and something stronger, which can be put by saying that the rules in question are derivable (or ‘derived’). The latter notion presumes that certain rules have been taken as primitive or fundamental, and it is from these that the rules we are interested in have been derived. It is this idea of deriving one rule from others that will occupy us here, except that, since the idea has application for relations in general rather than rules in particular, we will from now on mention only parenthetically the specifically logical case that motivates the following development. A relational structure (S, R1 , . . . , Rn ) was defined in 0.11 in such a way that the set S was guaranteed to be closed under the relations Ri . We now present, for temporary consideration, a somewhat different concept: that of a relational system on a set U , which again we denote by (S, R1 , . . . , Rn ), only this time, we presume only that S ⊆ U and that each Ri is a relation on U (that is, for some k, perhaps different for different choices of i, Ri ⊆ U k ). We think of S as some initially given set of objects, with the relations Ri being ways of generating new objects from those already given. All objects under consideration are drawn from the underlying set U . Given such a relational system on U , (S, R1 , . . . , Rn ), there is a clear notion of derived object available, the set of such objects being, in the notation of 0.13.4(iii), CR (S), where R = {R1 , . . . , Rn }. This set is of course closed under the relations Ri , and is in fact the least superset of S to be so closed, and it will also be closed under many other relations, only some of which are intuitively ‘derivable’ in the relational system, others being ‘merely admissible’. By way of illustration: Example 0.26.1 We take as our underlying set in the background (the U above) the set N of natural numbers, on which we construct the relational system (S, R1 ) and S, with S = {1, 2}, and R1 = { m, n | m+6 = n}. Notice the conspicuous difference between the relations R2 and R3 in respect of their claim to be regarded as derivable: R2 = { m, n | m + 18 = n} R3 = { m, n | m is odd and m + 1 = n} While the set of derived objects is closed under both of these relations, we want an account of derived relations according to which R2 is, but R3 is not, a derived relation in this case. R2 is derived by taking the relative product of the given relation R1 with itself three times. But closure under R3 is secured by a lucky accident in the choice of initial objects S. This is not to say, of course, that derived relations do not depend on the initial objects at all. For example, if we had S = {1, 2} as before, but a ternary R= { l, m, n | 5(l + m) = n}, then amongst the derived relations we should


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certainly want to number the binary relation holding between l and n just when 5(l + 10) = n. In such a case, closure (of the set of derived objects) follows in part from the presence of certain of the initial objects but in no way from the fact that other objects are not present. Now it is possible to secure these intuitive verdicts by simply defining a relational system (S, R) – where we have now taken the liberty of parcelling up the various Ri into a set R – to have a certain relation as derivable when not only CR (S) is closed under that relation, but so also is CR (S ) for any S ⊇ S (with S ⊆ U , of course). However, for two reasons, we prefer to explicate the idea of derivedness somewhat differently. The first is that this kind of account places an inappropriately sharp division between S and the Ri : S is after all just a 1-ary relation on U . The second is that it seems preferable to have a more ‘dynamic’ account in which the idea of deriving one thing from others is clearly visible. It will transpire (0.26.6) that the account to be offered here does agree with the ‘static’ account in terms of closure properties of extended relational systems; we prefer to obtain this result by proof rather than by definition, undertaking a direct analysis of the intuitive ideas. We have spoken of fundamental vs. derived objects and relations. The first thing to fix up is an asymmetry in this way of describing the task: individual objects and relations (1-ary relations included) are at different set-theoretic levels. (The contrast just made is a generalization of the contrast between theorems and derived rules, and even here we can see that the contrast is somewhat unfortunate in that a theorem is an individual formula whereas a rule licenses typically not just one, but various, transitions from formulas to formulas.) In the general situation this asymmetry could be avoided by either contrasting primitive and derived sets of objects with primitive and derived relations, or, going down a level, by contrasting primitive and derived n-tuples of objects with primitive and derived objects. It is this latter strategy we shall pursue. If we further identify an object a with the ordered 1-tuple of that object a , then we get the added advantage of being able to treat in the same way the interaction of relations with relations (as in taking relative products) but also relations and individual objects (as in the ternary relational example of the previous paragraph). The basic idea is to work with tuples (i.e., n-tuples, but for no fixed n), deriving new tuples from old by means of an operation which generalizes what is in common between the passage from (i)

a and a, b to b

and

(ii )

a, b and b, c to a, c .

Transitions of type (i) are involved when we extend an initially given set of objects to its closure under a binary relation, i.e., in the derivation of objects, while those of type (ii) are involved in the formation of relative products, ancestrals, etc., and so in the derivation of relations (or their component tuples as we now prefer to put it). We wish to unify (i) and (ii) because we want the same operation to do the ‘deriving’ work in both cases. We take, as usual, some set U as a background against which to work. As well as the identification of an element a ∈ U with the tuple a , we will require that U not otherwise contain any tuples of elements of U : no ordered pairs, triples, etc. Both conditions are imposed by the stipulation: (*)

For all a 1 , . . . , an ∈ U , for any a: a ∈ U & a = a1 , . . . , an

⇔ n = 1 & a1 = a.


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Here is a first stab at the desired unification of (i) and (ii), using “ ” for an operation on tuples: a1 , . . . , am , b1 b1 , . . . , bn , c = a1 , . . . , am , b2 , . . . , bn , c

where the variables range over elements of U . The operation takes the lefthand (m+1)-tuple and the right-hand (n + 1)-tuple to the resulting (m + n+1)tuple by ‘splicing in’ the preterminal part of the left tuple into the position occupied by its terminal element in the right-hand tuple. Think along these lines: if you can get from a1 , . . . , am by an allowed route to b1 , and you can similarly get from b1 and the remaining bi to c, then you can get from a1 , . . . , am and those remaining bi to c. Now, obviously we would like to be able to replace not only such an element as b1 but any of the other bi in this way. So we will have a more general story to tell about the splicing operation , achieving this and certain further effects → − (mentioned below). If X is some listing of the elements x1 , . . . , xn of a finite → − set X, and similarly with Y , etc., what we say is as follows. → − − → − → − → For all X , Y , W , Z , with W, X, Y, Z ⊆ U : → − − → − → → − ( ) X , b W , b, Y , c → Z , c provided X ∪ W ∪ Y ⊆ Z. → − Here we splice X into b’s place, and also allow for rearrangements and indeed for deletion of repeated elements (if desired) and addition of new elements, these being the ‘further effects’ just alluded to. As this description makes clear, splicing as we now understand it is not strictly speaking an operation (function, mapping,. . . ), since there is no unique output tuple for a given pair of tuples as input, in deference to which fact we have replaced “=” by “ →”. Though we shall continue informally to speak of splicing as an operation, officially the notation t u → v for tuples t, u, v involves an orthographically scattered ternary relation symbol and may be read “one may get v from t and u by splicing”. (Note that even on our ‘first stab’ at a definition above, we had at best a partial operation, undefined for pairs of tuples lacking an appropriately situated common term b1 .) It is this splicing procedure which encompasses (i) and (ii), above, and more in the same vein. We shall call any set of tuples of elements of U a tuple system on U , as long as the above condition (*) is satisfied. These tuples can be thought of as the fundamental or basic tuples of the system. We use splicing to obtain the derived tuples of the system. The details are as follows: if T is a tuple system on U , let: T0 = T ∪ { a, a | a ∈ U } Ti+1 = Ti ∪ {v|t u → v for tuples t, u ∈ Ti } and put: T =

i∈ω

Ti .

T is then the set of derived tuples of T . Recalling that we identify a with a, T ∩ U can be regarded as the set of derived objects of the system. Similarly,


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we define an n-ary relation R on U to be a derived relation of the tuple system T when R ⊆ T ∩ U n . Although this definition uses the operation of splicing, in an attempt to formalize the dynamics of derivation, we can show, as promised, that it coincides with the characterization in terms of closure of the set of derived objects of any extension of the tuple system under the relation. This will be the content of a second corollary (0.26.6) of Theorem 0.26.4 below. We begin with a preliminary Lemma 0.26.2 For any tuple system T on a set U with X ⊆ U , and b, d ∈ U : → − − → If X , d ∈ T , then b, X , d ∈ T . → − − → Proof. d, d ∈ T , so we ‘splice’: X , d d, d → b, X , d

Remarks 0.26.3(i) We can also ‘contract’ by splicing: a, a, b b, b → a, b

as well as ‘permute’: a1 , a2 , b b, b → a2 , a1 , b

These properties, along with 0.26.2, are reminiscent of the structural rules of Gentzen [1934], for sequents with singleton succedents (what we call sequents of the framework Seq-Fmla, in 1.21 – though see 2.33, esp. p. 353, for Contraction.) However: (ii ) Warning: when we apply this material to sentential logic the elements of the underlying set U will not be formulas but will themselves be sequents, so that a tuple a1 , . . . , an represents the transition from sequents a1 , . . . , an−1 to the sequent an . Theorem 0.26.4 For any tuple system T on U : c1 , . . . , cn ∈ T ∪ {b}) ⇒ b, c1 , . . . , cn ∈ T Proof. By induction on the rank k of c1 , . . . , cn in (T ∪ {b}) , where the rank of a tuple t in T is the least i for which t ∈ Ti . (Thus every derived tuple has some finite rank.) Basis. k = 0. Then c1 , . . . , cn ∈ T ∪ {b} ∪ { a, a | a ∈ U }. Two cases arise: (1): c1 , . . . , cn = b. Then (cf. (*)) n = 1 and c1 = cn = b. What has to be shown is that b, b ∈ T ; but b, b ∈ T0 . (2): c1 , . . . , cn ∈ T ∪ { a, a |a ∈ U } ( = T0 ). Then c1 , . . . , cn ∈ T , and by 0.26.2, b, c1 , . . . , cn ∈ T . Inductive step. Suppose c1 , . . . , cn has rank k in (T ∪ {b}) and that the Theorem holds for tuples of lower rank. This tuple is the result of splicing such → − − → → − lower-ranked tuples X , d and W , d, Y , cn with X ∪W ∪Y ⊆ {c1 , . . . , cn−1 }. → − − → − → By the inductive hypothesis, then, since X , d ∈ (T ∪{b}) and W , d, Y , cn ∈ − → − → − → (T ∪ {b}) , we have t = b, X , d ∈ T and u = b, W , d, Y , cn ∈ T . But t u → b, c1 , . . . , cn , so b, c1 , . . . , cn ∈ T , as was to be shown.

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Corollary 0.26.5 c ∈ (T ∪ {b1 , . . . , bm }) ⇒ b1 , . . . , bm , c ∈ T Proof. By induction on m, using 0.26.4 for the inductive step.

Theorem 0.26.4 and the above Corollary are both stated as implications, but could have been stated as equivalences, the converse implications following by applying . We state a second Corollary in this stronger biconditional form to make clear that we have completed our task of providing an ‘intrinsic’ or ‘dynamic’ explication of derivedness which is co-extensive with the ‘closure-ofextensions’ characterization: Corollary 0.26.6 A relation R ⊆ U n is a derived relation of T if and only if for all T ⊇ T , the set of derived objects of (T ) is closed under R. Evidently, the notion of closure enters into the above account of derivedness in terms of tuple systems in two ways. First, the constituent tuples a1 , . . . , an

can themselves be thought of as closure conditions, as we began this section by noting: if a1 , . . . , an−1 are admitted as objects, then an is to be admitted; this shows up in the fact that the set of derived objects is always closed under such relations as have all their constituent tuples in the tuple system (as primitive or derived). Secondly, the operation (·) is a closure operation in its own right on the set of tuples of elements of the underlying set U . If we think of U as a set of formal objects (e.g. formulas), these two aspects of the general situation correspond to what will later (1.26) be distinguished as two notions of consequence—horizontal and vertical—in the specifically logical case. With such applications in mind, we state here a general ‘Induction Principle for Tuple Systems’: Theorem 0.26.7 Let T be a tuple system on U, and P be a property that all elements of T ∩ U (all ‘initial objects’) possess and all tuples in T ∩ U n , for each n 2 preserve, in the sense that for any such n-tuple c1 , . . . , cn , if each of c1 , . . . , cn−1 has the property P, then so does cn . Then (1) Every c ∈ T ∩ U (every ‘derived object’) possesses P, and (2) Every c1 , . . . , cn ∈ T ∩ U n for n 2, preserves P. Proof. Since (1) gives the n = 1 form of (2), we need not divide the proof for the two cases, understanding c ( = c) to preserve P just in case c possesses P . The proof is a straightforward induction on the rank of the derived tuple c1 , . . . , cn .

Exercise 0.26.8 (i) If R is a binary relation on U , then R is itself a tuple system on U , which gives rise to another binary relation by splicing, namely R ∩ (U × U ). Verify that this relation is the ancestral of the original relation R. (The ancestral of R is the least reflexive transitive relation R ⊇ R. Note the role of the T0 stage in securing reflexivity here.) (ii) If (S, R1 , . . . , Rn ) is a relational system on U , then S ∪ R1 ∪ · · · ∪ Rn is a tuple system on U . Verify that the set (S ∪ R1 ∪ · · · ∪ Rn ) ∩ U of


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derived objects of this tuple system is the closure CR (S) of S under R = {R1 , . . . , Rn }. (iii) Verify that the relation R2 of Example 0.26.1 is a derived relation of the tuple system S ∪ R1 associated with the relational system (S, R1 ) but that the relation R3 mentioned there is not a derived relation of this tuple system. The relation R1 of 0.26.1 is a functional relation on U in the sense that for all a ∈ U there is a unique b ∈ U such that aRb. This suggests a way of obtaining a tuple system from an algebra. So as not to depart too far from the example of 0.26.1, take the algebra with universe N and, as, fundamental operations, ordinary arithmetical addition (+) and multiplication (·). Consider the further binary operation ∗ defined by: m + n if m is odd m∗n= m · n if m is even. It is clear that any subset of N closed under the operations + and · is closed under this operation ∗; but there is no way to define ∗ by composition from those operations, even by letting constant functions drop out of the composition process, or by helping ourselves to arbitrary elements of N as composition proceeds. So we are not dealing here with a compositionally derived operation, or even an -compositionally derived or algebraically derived operation (0.22). The point of contrast is that whereas for those cases, we have operations obtained from the fundamental operations in a uniform way, for cases like ∗ the manner of their application depended on the identity of the elements to which ∗ was applied. The general property of this broader class of derived operations is that they are functions under which every subalgebra of the given algebra is closed. We take no special interest in them here, however.

Notes and References for §0.2 Grätzer [1968], Burris and Sankappanavar [1981], and McKenzie et al. [1987] are good texts on universal algebra, a subject characterized in the second of these works as the model theory of equational logic; interesting surveys of results on equational logic have been provided by Taylor [1979] and McNulty [1992]. For a clear description of the logical issues that arise when we pass from full first-order languages to this restricted language, see Henkin [1977]. A pass over the material from the perspective of category theory may be found in Manes [1976], and there is also a good summary in Part 2 of Chapter 1 of Balbes and Dwinger [1974], and a good introduction to equational logic in Chapter 3 of Burris [1998]. A concise survey of the main concepts and results of universal algebra is provided by Chapter 1 of Almeida [1994]. As well as being used (especially in universal algebra) in the sense defined here (i.e., in 0.21), the term groupoid has an entirely different use – see for example Brown [1987]. The distinction (0.22) between compositionally and -compositionally derived functions is made in the older vocabulary of ‘polynomially’ and ‘-polynomially’ derived functions in Humberstone [1993a], q.v. for references to the relevant literature on universal algebra, the most important of which on this matter being Tarski [1968].

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Proofs of Birkhoff’s theorem on varieties (as equational classes) may be found in the texts mentioned above; for example see pp. 184–5 of Burris [1998]. (Occasionally the term variety is simply defined to mean the same as equational class, in which terminology this theorem would have to be formulated as saying that the equational classes (alias varieties) are exactly those classes of algebras closed under homomorphic images, subalgebras, and direct products. There is a general discussion of preservation theorems for equational logic in Lyndon [1957]. Horn formulas (0.25) are so named from Horn [1951], though they were first isolated in McKinsey [1943]. These papers give the ‘preservation under direct products’ result for universal Horn formulas, showing also that every formula so preserved is equivalent to a conjunction of such formulas. A similar result was obtained for the case in which all quantifiers are existential, in Lyndon [1959]. If a mixture of universal and existential quantifiers is admitted, however, the result no longer holds generally (Chang and Morel [1958]), though it does continue to hold for the special case of pure equational logic (or ‘identity theory’ – no function symbols, all atomic formulas equations: Appel [1959]). The Chang–Morel paper led to interest in a variation on the direct product construction for which a converse result did hold: what are called reduced direct products. (These lie outside of our concerns in the present book; a historical account of these and related matters may be found in Zygmunt [1973].) An extensive survey of Horn formulas may be found in Hodges [1993a]. A warning à propos of 0.25.1 and the surrounding discussion: in the model-theoretic literature there is another quite different notion of (closure under) intersections of models, where the class of models whose intersection is taken are required all to be substructures of some initially given model, as in Rabin [1962]. Logic programming was mentioned in connection with Horn formulas in 0.25; a helpful presentation of numerous topics in logic programming from a specifically logical point of view may be found in Fitting [1996]. For Birkhoff-style preservation theorems on quasivarieties (as opposed to varieties) of algebras, see Theorem 2.25 of Burris and Sankappanavar [1981]; Wechler [1992], esp. pp. 205, 212; and Meinke and Tucker [1992], esp. p. 345f. and p. 387. (These authors do not all use the same terminology as each other; Wechler provides the fullest discussion.) Boolean (and other styles of) complementation is often denoted by “(·) ” – with a being the complement of a, that is. Here “ ” is used instead simply for making new variables out of old, occasionally desirable for stylistic and mnemonic purposes; for example, we might say “suppose x and y bear the relation R to x and y respectively”. We could just as well say “w” and “z”, but the use of the priming (the “ ” superscripting) helps code the “respectively” – or other similar cross-referencing – into the notation. A good reference for the “pseudo-” and “quasi-” variants on complementation mentioned in 0.21 is Balbes and Dwinger [1974], Chapters 8 and 11, respectively (though quasi-boolean algebras are there called De Morgan algebras); for “ortho-”, see Kalmbach [1983]. A further generalization of the class of quasi-boolean algebras is studied (under the name Ockham algebras) in Blyth and Varlet [1994]: these algebras come equipped with a complementation operation, f say, required to satisfy the De Morgan identities f (x ∧ y) = f (x) ∨ f (y), f (x ∨ y) = f (x) ∧ f (y), and to exchange bounds – i.e., f (0) = 1, f (1) = 0 – but are not required to satisfy the double complementation (or ‘involution’) condition that f (f (x)) = x. This omission means that while f is a dual endomorphism of such an algebra, it need not be a dual automorphism. See Blyth and Varlet [1994] for full details about


45

this variety and some of its significant subvarieties. The use (0.24) of the phrase “functionally free”, for equationally generic algebras in a class, comes from Tarski [1949]; sometimes they are simply called generic – as in Taylor [1979]. The material on tuple systems in 0.26 is treated from a related but different perspective in Chapter 0 of Segerberg [1982]; closer to our own notion of a tuple system is the notion of a ‘derivation set’ in Wasserman [1974], our 0.26.6 being a version of his ‘Derivability Principle’ (the Theorem on p. 271 of that paper). A propos of Exercise 0.21.1(ii), it was remarked that a rigorous proof requires the Axiom of Choice; this was pointed out to me by Allen Hazen.

Chapter 1

Elements of Sentential Logic §1.1 TRUTH AND CONSEQUENCE 1.11

Languages

This chapter and the next present some basic material in sentential logic. The intention is to stress those topics of particular relevance to what follows later, as well as to highlight the way individual connectives fare in the general development. We begin our discussion of logical matters by introducing the idea of a formal language. Since we shall be concerned only with sentential (or ‘propositional’) logic we shall take it that such a language consists of formulas composed ultimately from unstructured formulas called propositional variables (often called sentence letters or ‘sentential variables’) of which we presume – unless otherwise stated – there are countably many: p1 , . . . , pn , . . . To cut down on the subscripting, we will write the first four in this list as p, q, r, and s, when this is convenient. From now on, quotation marks used as in the preceding sentence to refer to linguistic expressions will generally be suppressed. So much for what is common to the various languages we shall have occasion to consider. What will distinguish one language from another will be its stock of compounding devices, or connectives, where a k-ary connective may be taken provisionally (see below) to be a symbol # which attaches to k formulas, A1 , . . . , Ak to form a formula which we may denote by #(A1 , . . . , Ak ). A language, then, may be regarded as an algebra L = (L, #1 , . . . , #n ) with L as the set of formulas generated by the propositional variables when the #i are taken as fundamental operations. This involves a double usage on our part of expressions like “#1 ”, “#2 ”,. . . first to denote a particular connective-symbol, and then to denote the syntactic operation of building a formula with the aid of that symbol together with appropriately many other formulas. (The former usage is especially tied to a conception of languages which we call the ‘concrete conception’ below.) In practice we use notation for the various #i that is suggestive of the kind of logical behaviour we expect them to exhibit; thus the preferred notation for an implicational or conditional connective, used if only one such is under discussion, is →, and for an equivalential or biconditional connective, ↔, while for conjunction and disjunction we tend to write ∧ and ∨, respectively. These connectives are all binary. For negation, a singulary connective, we use ¬. Of course there may be several candidates for these various titles – for example 47

48

CHAPTER 1. ELEMENTS OF SENTENTIAL LOGIC

we might need to contend with several negation-like connectives in play at once (as in 8.22 and 8.23) – in which case more care is needed as to how they are to be notated. But the above will be the default notation, and the usual way of readings of A → B, A ↔ B, A ∧ B, A ∨ B and ¬A are: if A then B, A if and only if B, A and B, A or B, and not A (or: it is not the case that A), respectively. As is already apparent we use capital Roman letters, occasionally subscripted, to stand for formulas of whatever language is under discussion (which is often called the object language, the language in which our discussion itself is couched being called the metalanguage; accordingly these letters themselves are often described as metalinguistic variables ranging over formulas). Often, instead of speaking of this algebra as a language, we will for convenience identify the language with the set, L, of its formulas. Only when it is necessary to stress the algebra of which this is the universe will we explicitly write “L”. Capital Roman letters “A”, “B” etc., are used here to stand for formulas, so as to avoid confusion with (italic) “A”, “B”, etc., denoting the universes of algebras A, B,. . . , as in §0.2. (See the Preface.) As in our general discussion of algebras in §0.2, we do not exclude the case in which some #i is a 0-ary operation, and hence a formula in its own right. Such formulas are also called propositional (or ‘sentential’) constants, though more commonly they will be called zero-place, nullary, or 0-ary connectives. An atomic formula is a formula not constructed out of other formulas: the atomic formulas thus comprise any sentential constants (nullary connectives) there may be, together with all the propositional variables. (Some authors exclude the constants from the class of atomic formulas. This is a matter of terminological convenience.) Those formulas, by contrast, whose construction involves the application of at least one connective of arity 1, will be called complex (or ‘compound’) formulas, and the total number of such non-zeroplace connectives involved in the construction of a formula will be called the complexity of the formula. (Thus atomic formulas are of complexity 0.) We will frequently have occasion to prove things about all formulas of this or that language by induction on the complexity of formulas, a method of proof with which the reader is assumed to be comfortable. The algebraic perspective just introduced lends itself to a less concrete construal of how formulas are constructed than that presumed above. There we thought of connectives primarily as symbols, just like the propositional variables, which physically attached to formulas to make new and accordingly longer formulas of which the originals were proper parts. The corresponding (and in our discussion, not notationally distinguished) operations were the operations of attaching these symbols to those immediate proper parts. A more abstract construal is also possible, on which we do not care how a primitive (singulary, say) connective # forms the formula #A from the formula A, as long as this is effected by a well-defined 1-ary fundamental operation of a certain kind of algebra (see below) whose universe consists of formulas. In particular, we do not insist that #A should consist of some symbol stuck onto the front of the formula A. It is still possible, with this more abstract conception of how connectives work, to impose some conditions to preserve some of the distinctive formal features of the concrete conception, and foremost among such features is that sometimes called the ‘unique readability’ property, which is the following feature of the concrete construal: each formula is built up from atomic formulas by the application of connectives in only one way. The way to secure an ana-

1.1. TRUTH AND CONSEQUENCE

49

logue of this property on the abstract conception is to define languages to be absolutely free algebras freely generated by the set {pi }1i 0) can be replaced by applications in which the cut-formulas are of lower complexity—in particular, in which it is the components of the given formula which are ‘cut’. (This reduces the problem to showing that we don’t need to apply the rule in the case of atomic formulas, but this further reduction is not to the point here; also the reader should be warned that the inductive organization of Gentzen’s proof has further complexity than is exposed here. See the references given in the notes, under ‘The Framework Set-Set’, at p. 190.) We will attend to this feature of

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the rules when we talk about a pair of left and right rules, or the connective they govern, as being ‘cut-inductive’ in later discussion. (See 2.33.27, p. 365, and surrounds; the logical framework there in play for sequent calculus is based on multisets rather than sets, but that difference is not significant.) Now, it is has often been observed – see for example Hacking [1979], p. 297 – that a similar situation arises for the structural rule (R). In Gen and IGen (the latter explained in 2.32), we can restrict the application of this rule, A A, to the case in which A is of complexity 0. If the primitive rules of a sequent calculus are such that there is no loss of provable sequents when this restriction is imposed on (R) these rules are called regular in Kaminski [1988]. We will use the same term with regard to an individual pair of insertion rules for a connective, or for the connective itself, understood as governed by such rules. The rules for an n-ary connective # are regular when we can derive (R) for a compound #(A1 , . . . , An ) (i.e., prove #(A1 , . . . , An ) #(A1 , . . . , An )) from the hypothesis that we have (R) for the components, using the rules for #. Example 1.27.11 (i) To illustrate this property for the ∧ rules, we show how, instead of helping ourselves to (R) for A = B ∧ C, we can derive it, on the assumption that we already have (R) for B, C:

∧ Left 1

BB

CC

B∧CB

B∧CC

B∧CB∧C

∧ Left 2 ∧ Right

Here we have used the original Gentzen (∧ Left) rules, instead of the Ketonen-modified versions given in our official presentation of Gen; those would take an extra step, appealing to (M). (ii) We have not given sequent calculus rules for any nullary connectives but suppose that we have ⊥ as primitive, with zero-premiss Left insertion rule (providing, as (R) itself does, further initial sequents – starting points – for proofs): Γ, ⊥ Δ. We could equally well just use the case in which Γ or Δ (or both) is ∅ for that purpose, in the present setting. (There is no “⊥ Right” rule.) Then the initial sequent ⊥ ⊥ is available without appeal to (R) but as the special case in which Γ = ∅ and Δ = {⊥}. Thus this treatment of ⊥ counts as regular. In terms of the general definition earlier, requiring for regularity that we can derive (R) for #(A1 , . . . , An ) from the hypothesis that we have (R) for the Ai using the rules for #, the hypothesis is vacuously satisfied when n = 0, so we must derive the (R)-sequents using just the rules for #, as in the case of ⊥ just illustrated. Exercise 1.27.12 Show that the rules in Gen for ∨, →, and ¬ are also regular. Returning to the topic of invertibility, let us note that the invertible forms of the operational rules of the classical sequent calculus rules, with invertibility defined as above in syntactic terms (having admissible converses) are also invertible in the following semantic sense: the converse rules are guaranteed to preserve the property of holding on an arbitrary boolean valuation. (They

1.2. RULES AND PROOF

149

thus preserve tautologousness, but we will also be considering unprovable sequents and accordingly need the local preservation characteristic rather than just the global version, in the terminology of 1.25 – though 1.25.8, p. 131, tells us that this distinction is nullified for rules, such as those under consideration here, which are substitution-invariant.) We can use this feature to show the completeness of sequent calculi based on such rules, and for good measure, illustrate the redundancy of the Cut Rule by using the rule (RM): given before 1.27.9 above as our sole structural rule, with the invertible rules governing ∧ from 1.27.9 itself. (These ideas are illustrated with rules governing the Sheffer stroke in Riser [1967].) A further concept we shall employ is that of a sequent being reducible to a set of sequents, which is a hereditized version of the relation of the conclusion sequent to the premiss sequents of an application of a sequentto-sequent rule. More precisely, let us say that reducibility, here abbreviated to Red, is the least relation between sequents and sets of sequents satisfying the following conditions (which depend on a background set of sequent-to-sequent rules): (i ) (ii )

σ Red {σ} for any sequent σ. Whenever σ Red Σ and Σ is a set of premisses for the application of one of the background rules with conclusion σ ∈ Σ, then σ Red Σ {σ } ∪ Σ .

In other words, σ is reducible, against the background of a certain set of rules, to Σ, when those rules can be used to derive σ in a succession of stages from Σ as initial sequents. At each stage, working in reverse, we replace σ by a set of premisses Σ from which it can be derived by one of the rules. Exercise 1.27.13 (i) Show that taking as background rules the ∧ rules of 1.27.9, for every Set-Set sequent σ in the language with ∧ as sole connective, there is a unique set of ∧-free sequents Σ such that σ reduces to Σ (i.e., σ Red Σ). (ii) Show that an ∧-free sequent of the present language is tautologous if and only if it is of the form ΓΔ with all formulas in Γ ∪ Δ propositional variables and Γ ∩ Δ = ∅. (iii) Denoting the Σ promised in (i) for a given σ by Red (σ), list the sequents in Red (σ) for σ as the unprovable sequent p ∧ q, r q ∧ s, p ∧ s and then for σ as the provable sequent p ∧ (q ∧ r) (p ∧ q) ∧ r. We now illustrate the bearing of these considerations on completeness (following the lead of Riser [1967]). Example 1.27.14 Every tautologous sequent in the language with ∧ as sole connective is derivable in the proof system with structural rule (RM) and the operational rules of 1.27.9. The reasoning goes as follows, in which Red (σ) is understood as in 1.27.13(iii) and we abbreviate “σ is tautologous” and “every sequent in Red (σ) is tautologous” to “σ valid” and “Red (σ) valid”, respectively: σ valid ⇒ Red (σ) valid ⇒ Red (σ) provable ⇒ σ provable. (1)

(2)

(3)

(1) is justified by the already noted ‘semantic invertibility’ of the 1.27.9 ∧ rules, (2) by 1.27.13(ii) and the availability of (RM), and (3) by application of the ∧ rules in their usual (‘downward’) direction. Note that (T)

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CHAPTER 1. ELEMENTS OF SENTENTIAL LOGIC was not needed, and nor – illustrating the regularity of the operational rules involved – were any appeals to (RM) with non-atomic formulas involved.

The next time considerations of rule invertibility will be raised is in connection with intuitionistic logic in 2.32 (see p. 322 onward), where, we shall see, a sequent calculus treatment using pure and simple invertible rules is not similarly available. (A relatively recent discussion relating treatability by invertible rules to semantic considerations can be found in Avron, Ciabattoni and Zamansky [2009].)

1.28

Some Other Approaches to Logic in Set-Set

We have had a look at a sequent calculus approach and a natural deduction approach to logic in Set-Set; we now consider five other types of proof systems that could be used in this framework (together with further variations beginning with 1.28.1). In all cases except the fourth, we assume the structural rules (R), (M), and (T) are taken as primitive rules, and concentrate on the shape rules for the connectives might take. The first approach would be to turn the sequent calculus rules ‘upside down’, and consider rules which always remove the connectives they govern, either from the left, or from the right, of the “”. (We say remove rather than eliminate for the same reason as in the preceding subsection – see the discussion after 1.27.5 – insert was used in preference to introduce.) For example, for ∧ we could take the (∧E) rule from NAT as elimination rule on the right, with Γ, A ∧ B Δ Γ, A, B Δ to do the job on the left. There is no difficulty in writing a pair of rules in this style for the other connectives, or in showing that the resulting proof system suffices for the proof of all (and only) tautologous sequents by giving proofs for the critical sequent-schemata, in the manner sketched for Gen and NAT above (1.27.3). As a second possibility, consider turning the natural deduction approach ‘back to front’ and having the introduction and elimination of connectives to/from sequents occur on the left of the “” instead of on the right. The above rule for ∧ would qualify, along with its upside-down version, the Ketonenmodified Gentzen rule (∧ Left) used in formulating Gen, as elimination and introduction rules for this connective, while for ∨ we could use rules allowing passage from Γ, A ∨ B Δ to Γ, A Δ and to Γ, B Δ to do the eliminating, with (∨ Left) from Gen for the introducing. Again, we leave the reader to complete the picture for the remaining connectives. This ‘action on the left’ approach takes on a particularly simple appearance in Set-Fmla since by contrast with natural deduction there (‘action on the right’) the possibility of discharging assumptions – dropping a formula from the left, does not arise, as the conclusion-sequent of any rule attempting an analogous deletion on the right would lie outside Set-Fmla. The ‘particularly simple’ form, that proofs then take on consists in every sequent in the proof of a sequent Γ C having the same formula C on the right. This amounts to translating the various #-classicality conditions on consequence relations into ‘rule’ form.


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A third approach, illustrated in Scott [1974b], though probably originating in Popper [1948], and appearing in something close to the present notation in Kneale [1956], is to use two-way rules, given by schematic figures to be read both downward and upward (which is what the double line is intended to indicate): for ∧:

Γ, A, B Δ Γ, A ∧ B Δ

for ∨:

Γ A, B, Δ Γ A ∨ B,Δ

Such rules – which, we stress, as in the preceding section, are not interderivable with the (Gentzen-style) sequent calculus rules in weaker logics lacking some of the structural rules or working in more refined frameworks (Mset-Mset, for example – as in 2.33 and in Example 7.25.25 below, and the surrounding discussion) very neatly spell out the significance (for boolean valuations) of commas on the left and the right of the “”; the other two connectives in the language of this section involve some jumping over the “”: for →:

Γ, A B, Δ ΓA → B,Δ

for ¬:

ΓA Δ Γ ¬A, Δ

An alternative pair equivalent to those given for ¬ would have A on the right above the double line with ¬A on the left below. (Warning: there is an unrelated usage, going back to Gentzen [1934], of doubled horizontal lines in the proof-theoretic literature: not to display two-way rules, but in the depiction of sequent-to-sequent proofs, to indicate the application of certain structural rules, such as (M), or its analogue in Seq-Seq, Seq-Fmla, etc., as well as a structural rule for such frameworks allowing the permutation and contraction of formula occurrences.) This approach is related to the idea of invertibility of rules from the end of the preceding subsection. The invertible operational rules there considered are sequent calculus (complexity-increasing) rules whose inverse (complexity-reducing) forms are admissible though not derivable (at least in the absence of (T)). Here such inverse forms are actually taken as primitive (and are thus a fortiori derivable). (Two-way rules of the present kind have found favour among several subsequent authors, such as D. Scott and K. Došen – see Scott [1974b], p. 423, for example, and Došen [1989b]. The equivalence of premiss and conclusion in two-way rules like those above plays also an important role – under the not entirely satisfactory name “definitional equation” for the connective concerned – in the philosophically motivated ‘Basic Logic’ of Sambin, Battilotti and Faggian [2000], into whose details we do not enter here. In fact for their Basic Logic proper, as opposed to various liberalized extensions thereof, the side-formulas are heavily constrained, so that for example, Δ, in the above two-way rules for ∨, is required to be empty; also because the unextended system is substructural, those rules – for instance – are really the rules for the multiplicative disjunction denoted by + in 2.33 below. They envisage liberalizing the system both by adding structural rules and also by relaxing the constraints on side-formulas.) A fourth approach is presented in Kleene [1952], §80, as a variation on the Gentzen sequent calculus approach designed to minimize the application of structural rules and thereby simplify the procedure of deciding whether a sequent is provable by searching backwards (or “upwards”) through possible ways in which it might be proved. He gives two such proof systems, called classical


152

and intuitionistic G3, and shows them to yield proofs of the same sequents as the standard sequent calculus systems; we consider only the former here for illustrative purposes, touching on the latter at the end of 2.32. (In fact there is also a variation Kleene calls G3a which provides for shorter proofs. Kleene’s G1 and G2 are versions of the sequent calculus already considered by Gentzen.) The novelty of this approach is that the formula whose main connective is inserted on the left (on the right) by the application of a sequent calculus rule is present on the left (on the right) in the premiss sequent. Thus where Gentzen has (as remarked in the previous subsection, not quite what we offered in the proof system Gen we named after him):

(∧ Left 1)

Γ, A Δ

(∧ Left 2)

Γ, A ∧ B Δ

Γ, B Δ Γ, A ∧ B Δ

Kleene’s G3 has instead what in our notation would be: (∧ Left 1)

Γ, A ∧ B, A Δ

(∧ Left 2)

Γ, A ∧ B Δ

Γ, A ∧ B, B Δ Γ, A ∧ B Δ

Obviously with the latter rules in mind it would not quite be appropriate to read “(∧ Left i)” (i = 1, 2) as the “ith rule for inserting ∧ on the left”, since the ∧ is evidently already on the left before the rule is applied. We leave the reader to decide on anything more definite than “left-hand rule for dealing with ∧” if that is desired. Given the usual structural rules, the Gentzen and Kleene forms of (∧ Left i) (i = 1, 2) are interderivable. To obtain the latter form of (∧ Left 1) from the former, start with Γ, A ∧ B, A Δ and apply the former rule. In our Set-Set framework the this gives the conclusion Γ, A ∧ B Δ of the latter rule. (In Mset-Mset, we get Γ, A ∧ B, A ∧ B Δ and must apply the Contraction rule to replace the two occurrences of A ∧ B by one. This structural rule is one that Kleene wanted to purge because of the way it complicates the kind of backward proof-search strategy alluded to above. We return to Contraction, though for multisets rather than sequences of formulas, in 2.33.) Conversely, if we start with a premiss for the Gentzen rule, we can ‘thin’ in (by (M)) an occurrence of A ∧ B and apply the Kleene rule. A similar interderivability obtains in the case of (∧ Left 2), and of (∧ Right), given below, and indeed for the rules governing the other connectives. Despite this, we give the Kleene proof system separate treatment here because of the striking difference of the operational rules, taken individually: they do not insert compound formulas into sequents, but instead remove components. Turning, then, to the rule for the right-hand side, instead of a Gentzen-style (∧ Right), given here in the simplified form in which the Γ and Γ of our account of Gen are identified, as also for Δ and Δ : (∧ Right)

Γ A, Δ

Kleene offers:

Γ B, Δ

Γ A ∧ B, Δ

1.2. RULES AND PROOF (∧ Right)

Γ A, A ∧ B, Δ

153 Γ B, A ∧ B, Δ

Γ A ∧ B, Δ

As part of the effort to reduce appeals to the structural rules, all the ‘thinning’ which would in Gen by achieved by invoking (M) is done at the top of the proof by using a ‘thinned’ form of (R): Γ, A A, Δ There is no difficulty in proving on the basis of these rules sequents asserting the commutativity and associativity of conjunction, or in working out what the Kleene rules for the other connectives should be; in any case one may consult Kleene [1952] for enlightenment on this score and further information about the approach. (See also Troelstra and Schwichtenberg [1996], especially Chapter 3.) Some words from Kleene [1952], p. 481, on the motivation behind this system: The system G3 is designed to minimize the number of choices of premise(s) for a given conclusion, when we are attempting to exhaust the possibilities for proving a given endsequent, especially in showing the endsequent to be unprovable.

We return to the G3 style of sequent calculus at the end of 2.32, where a restriction to the framework Set-Fmla0 (for intuitionistic rather than classical logic) will occasion some modifications. Finally, as a fifth approach, we could simply use as zero-premiss rules along with (R) the sequent-schemata of 1.27.3 that we have needed to appeal to for showing completeness, with (M) and (T) as the only rules in the everyday sense (i.e., the only rules which have one or more premisses for their application). Completeness would then be immediate. (The availability of such an option was noted in Gentzen [1934], III §2.2.) This is very like the pursuit of the axiomatic approach in Set-Set, in that a loss in ways of deriving one thing from another is made up for by an increase in the number of possible points for derivations to start from. It is an interesting question whether in Set-Fmla a reduction of the non-zero-premiss rules to (M) and (T) is similarly possible. Naturally, this depends on how much similarity is required. One could just use all tautologous sequents, or zero-premiss sequent-schemata representing them, as ‘axioms’ (the distinction indicated by “or” here will receive attention in the Appendix to this section, p. 180 onward). This move would conspicuously fail to make proofs (all of them one line long) explanatory of how the logical powers of the individual connectives contribute to the presence of sequents involving them in the logic concerned (cf. 4.14), and so does not seem desirable even in case membership is decidable (as in classical sentential logic), let alone elsewhere (e.g., first-order logic). Details concerning these zero-premiss rule systems and of their completeness w.r.t. various classes of #-boolean valuations for various choices of # will be found in 3.11, when we discuss the conditions imposed on gcr’s by what we there call the ‘determinants’ of a truth-function – roughly: the individual lines in a truth-table specifying that function (we shall be speaking of the determinantinduced conditions on a gcr). Here we pause to note the utility of the zeropremiss rules in modulating between various different but, given (R), (M) and (T), interderivable non-zero-premiss rules.


154

Exercise 1.28.1 Let # be a one-place connective in some language, and consider the following two rules in the Set-Set framework over that language: (1) Γ, #A Δ (2) Γ, A Δ Γ A, Δ Γ #A, Δ Show that with the aid of the structural rules (R), (M) and (T), each of rules (1) and (2) can be derived from the other. (Hint: show that each is thus interderivable with the zero-premiss rule A, #A.) It is not important that # is a 1-ary connective here: the point would just as well be made for any context in which a (single) formula A might appear. (The term “context” is intended to be self-explanatory here. We treat the notion of a context more explicitly in 3.16.1 on p. 424 and the immediately preceding discussion.) For example, we could replace “#A” in (1) and (2) by “A → B”. The resulting rules preserve the property of holding on an arbitrary →-boolean valuation. (Of course a similar point could be made for (1) and (2) themselves, with # rewritten as ¬.) Exercise 1.28.2 Find rules for singulary # in the style of (1) of 1.28.1 (i.e., # in the premiss-sequent schematically indicated but not in the conclusionsequent) and in the style of (2) (i.e., the reverse) which are interderivable, given the standard structural rules, with the following zero-premiss rules (in the style of the ‘Hint’ for 1.28.1): (i) A, #A

(ii ) A #A

(iii) #A A.

For n-ary connectives with n 2 there are even more possibilities: Example 1.28.3 We extend the numbering from 1.28.1. Consider the rule (3), governing a connective #, now assumed binary: (3)

Γ, A # B Δ Γ, A B, Δ

This time the sequent-schema analogous to that suggested in the Hint to 1.28.1 would be: A B, A#B, and the following three rules are all like (2) in that the connective makes its appearance below rather than above the line separating premiss-sequent from conclusion-sequent, and all (given (R), (M) and (T)), interderivable with (3): (4)

Γ, A Δ Γ A # B, B, Δ

(6)

Γ, B Δ Γ A, Δ Γ A # B, Δ

(5)

Γ, B Δ Γ, A A # B, Δ

Although (4), (5), and (6) all insert # with their application, only (6) is a sequent calculus style right insertion rule, in that, like the rules of Gen from 1.27, explicit mention of the components of the #-compound concerned is made only in the premiss-sequents. (In fact, to be in line with the rules of 1.27, we should have to allow the sets of side formulas


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to be different for the two premiss-sequents of (6) and take unions as we pass down, but clearly this makes no difference given (R), (M) and (T).) If we take # as → and write compounds on the other sides of the “ ” from where they appear in 1.28.3, a range emerges of alternatives to the sequent calculus rule (→ Left), which is (6) with # as →. (The 0-premiss rule is then ‘horizontal’ Modus Ponens: A, A → B B.) We have, for instance, the rules: Γ, B Δ

and

Γ, A, A → B Δ

Γ A, Δ Γ, A → B B, Δ

which are the analogues of (5) and (4) respectively, once the compounds are relocated to the other side of the “”, while (3) emerges under this transformation as: Γ, A → B Δ Γ, A B, Δ Again, all these rules for → are, as we might put it, (RMT)-interderivable. The availability of the above variations on the theme of the zero-premiss (‘horizontalized Modus Ponens’) rule A, A → B B has nothing to do with the fact that the formula represented schematically before the “ ” is a compound whose components are represented, one on that side and the other on the other side. If we use subscripted occurrences of “ ϕ” as meta-schematic letters either for compounds or components with – to stick with this example for illustration – the composition being effected by a binary connective #, then we can find (RMT)-interderivable variations on the theme of ϕ1 , ϕ2 ϕ3 , with ϕ1 , ϕ2 , ϕ3 , chosen in any order from A, B, A # B: Γ ϕ1 , Δ

Γ ϕ2 , Δ

Γ ϕ3 , Δ

Γ ϕ1 , Δ

Γ ϕ2 , Δ

Γ, ϕ2 ϕ3 , Δ

Γ, ϕ1 ϕ3 , Δ

For example, taking ϕ1 , ϕ2 , ϕ3 , respectively, as A, B, A ∧ B, so that we are reformulating the zero-premiss rule A, B A ∧ B (‘horizontal ∧-Introduction’), the first of these three is the version of (∧ Right) mentioned earlier in this subsection, while the last is the rule: Γ B, Δ Γ, A A ∧ B, Δ In this example (as well as some others), the Δ’s can all be erased to give nonzero-premiss rules interderivable with the use of (R), (M) and (T) as understood for Set-Fmla rather than Set-Set. This will not be possible of course if the zero-premiss rule itself requires multiple or empty right-hand sides – for example, as it would had we run through the above variations on the theme, not of ϕ1 , ϕ2 ϕ3 , but instead, of ϕ1 ϕ2 , ϕ3 (the ϕi as above). Anyone embarking on the following exercise might do well to begin with this simple case. Exercise 1.28.4 Find rules in the style of (3) and in the style of (6), under 1.28.3 (“in the style of” meaning what it meant in 1.28.2) which are

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CHAPTER 1. ELEMENTS OF SENTENTIAL LOGIC interderivable given the structural rules with the schema indicated by (i) and then for that indicated by (ii ) for (m + n)-ary connective #: (i) B1 , . . . , Bm , #(A1 , . . . , Am+n ) C1 , . . . , Cn (ii ) B1 , . . . , Bm #(A1 , . . . , Am+n ), C1 , . . . , Cn in which, in both cases, {B1 , . . . , Bm ,C1 , . . . , Cn } = {A1 , . . . , Am+n }. (Strictly, this exercise should be formulated with the aid of ‘metaschematic’ letters, which is how we intend the Bs and Cs to be understood, as with the “ϕ” notation above. For example, one case falling under (i) would be, where m + n = 3: A1 , A3 , #(A1 , A2 , A3 ) A2 and a second would be the schema A3 , #(A1 , A2 , A3 ) A1 , A2 . In the first of these cases, m = 2, n = 1, B1 is A1 , B2 is A3 , and C1 is A2 . In the second, m = 1, n = 2, B1 is A3 , C1 is A1 and C2 is A2 .)

1.29

Axiom Systems and the Deduction Theorem

A proof system in Fmla has rules transforming sequents of that framework into sequents of that framework. The distinction between the sequent A and the formula A comes to little, and one usually speaks of the provability of A when A is provable, and calls A a theorem (or ‘thesis’ or just ‘provable formula’) when that sequent is provable. Classical sentential logic in this framework amounts proof-theoretically to the choice of a proof system which yields as theorems precisely the tautologies. The non-zero-premiss rules are called axioms, and historically the tendency has been to use only few non-zero-premiss rules—a conspicuous favourite being Modus Ponens in the form A→B

A

B and perhaps also Uniform Substitution. We could represent Modus Ponens as above but with occurrences of “” preceding the premisses and conclusion, but as we shall before long be considering the application of this rule to formulas which are not provable – “as a rule of inference” we shall say, rather than “as a rule of proof”, the representation as it stands is preferable. (In any case, while working in Fmla, we tend to identify a sequent A with the formula A itself.) Alternatively, the latter rule may be replaced by choosing axiomschemata rather than individual axioms (so that, in effect, all substitutions are made at the start of the proofs: for more detail, see the discussion following 1.29.4 below.). Such axiom systems are no doubt familiar. (Instead of taking the axioms as zero-premisses rules, one may treat the axiom-schemata as such rules; see the Appendix to the present section for further discussion: p. 180 onward.) Here, by way of example, is a version using schemata of the system of Hilbert and Bernays [1934]. The axioms are all formulas of the following forms, and the only (non-zero-premiss) rule is Modus Ponens: (HB1) (HB2)

A → (B → A) (A → (A → B)) → (A → B)

1.2. RULES AND PROOF (HB3) (HB4) (HB5) (HB6) (HB7) (HB8) (HB9) (HB10) (HB11) (HB12)

157

(A → B) → ((B → C) → (A → C)) (A ∧ B) → A (A ∧ B) → B (A → B) → ((A → C) → (A → (B ∧ C))) A → (A ∨ B) B → (A ∨ B) (A → C) → ((B → C) → ((A ∨ B) → C)) (A → B) → (¬B → ¬A) A → ¬¬A ¬¬A → A

By contrast with the natural deduction approach and the sequent calculus approach to logic, the axiomatic approach, exemplified here by Hil, presents us with, as Segerberg [1968] nicely puts it (p. 31) the “familiar and distressing fact that when it comes to the actual derivation of theorems, a set-up of this type is rather awkward”. The difficulties are especially acute with the very much more economical axiom systems than Hil which abound in the literature on propositional logic (see Prior [1962], Appendix I, for examples and references; considerable heuristic advice on the practical problem of finding a proof from given axioms is provided in Hackstaff [1966]). At least Hil is organized, for ∧ and ∨ at any rate, very much along introduction-and-elimination lines. (The axioms have been grouped in threes to bring out this feature.) But even with Hil, the pervasive role of implication, used to code occurrences of “” which in a more liberal framework would have formulas to its left, can make the task of searching for a proof more complicated than one would like. For example, the reader (especially one not accustomed to the axiomatic approach) might care to find proofs in Hil for the following formulas: p → p;

(p → (q → r)) → (q → (p → r));

p → (q → (p ∧ q))

Life is made easy if one proves first a ‘Deduction Theorem’ for the system to the effect that, for a certain consequence relation defined in terms of the axioms and rules, B is a consequence of A1 , . . . , An iff An → B is a consequence of A1 , . . . , An−1 . For Hil, we define this relation, call it Hil , thus: Γ Hil B iff there is a sequence C1 , . . . , Ck of formulas each of which is either an axiom (a formula of one of the forms (HB1–HB12)) or an element of Γ, or else follows from earlier formulas in the sequence by Modus Ponens, and whose final formula Ck is the formula B. This sequence is called a deduction of B from Γ (“in Hil ” to be explicit). We call this relation Hil the consequence relation associated with the axiom system Hil. (For the more roundabout type of formulation needed if the rule of Uniform Substitution is used, see below.) The Deduction Theorem for Hil thus states that whenever A1 , . . . , An Hil B, we have A1 , . . . , An−1 Hil An → B; repeated appeals to this result—or its analogue for a different axiom system—allow us to prove implicational theorems (or more correctly, to demonstrate their provability) in the context of the axiomatic approach by means of reasoning from assumptions (the hypotheses to the left of the turnstile), thereby partially simulating the natural deduction approach. We shall not give a proof of the Deduction Theorem for Hil, but comment that the consequence relation Hil thus defined occupies a special place amongst those which, as we shall put it in the following paragraph, ‘agree with’ the collection

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of provable formulas of the system: it coincides with the relation CL of tautological consequence. (See the notes for references on the Deduction Theorem, p. 193.) The Deduction Theorem will be proved for several systems later in this section, however, starting at 1.29.10(ii), on p. 168. Say that a consequence relation agrees with a set Θ of formulas if for any formula A, A ∈ Θ iff A is a consequence, according to that relation, of the empty set. If we think of a consequence relation as in effect a collection of Set-Fmla sequents (as we can, essentially, for a finitary consequence relation) then we are asking that this collection return the same verdict – in or out – as a set of Fmla sequents for those cases in which the Set-Fmla sequents, having empty right-hand sides happen also to be Fmla sequents. (The same idea can be applied to Set-Set and Set-Fmla, and we shall be considering the topic of a gcr which ‘agrees with’ a given consequence relation in 6.31.) Given a set of formulas, Θ, for a consequence relation to agree in the sense just defined, with Θ, it must lie somewhere between two extremes, which we shall call min and max . These are the least and most extensive consequence relations agreeing with Θ, and they may be defined thus: Γ min A iff A ∈ Γ or A ∈ Θ Γ max A iff Γ ⊆ Θ ⇒ A ∈ Θ There is no difficulty verifying that the relations so defined are indeed consequence relations, or in fact that they are not only finitary consequence relations, but are finitary to an extreme degree in the sense that Γ min A implies for some Γ0 ⊆ Γ, with |Γ0 | 1, we have Γ0 min A, and likewise for max . (A more explicit notation would acknowledge the relativity to Θ.) Exercise 1.29.1 Show that min is included in every other consequence relation that agrees with Θ, and that max includes every other such consequence relation. There is thus plenty of room for choosing a consequence relation which agrees with Hil on the set of sequents provable in that system. The only constraint, as we have just seen is that taking Θ for the set of formulas which, preceded by “”, are provable in Hil, and defining min and max in terms of Θ as above, we have: min ⊆ ⊆ max . The consequence relation Hil defined in terms of the axiomatization of Hil occupies, from a semantic point of view, a privileged position amongst all satisfying this ‘betweenness’ condition: as was mentioned above, it coincides with the semantically characterized relation CL of tautological consequence, and therefore also with the earlier proof-theoretically characterized relation Nat . (This follows from the Deduction Theorem and the fact that Hil is sound and complete in the sense of proving all and only tautologies. We are not establishing either of these results here, however.) We pause for some exercises before proceeding to notice a variation on the system Hil ; the terminology is taken from the discussion (‘Axiom-Chopping’) in §8.3 of Anderson and Belnap [1975]; the labels abbreviate “Rule of Suffixing”, “Rule of Transitivity” and “Rule of Prefixing”. Exercise 1.29.2 (i) Show that the rule (RSuff) is a derived rule of Hil :

1.2. RULES AND PROOF (RSuff)

159

A→B (B → C) → (A → C)

(ii ) Do the same for the following Rule of Transitivity: A→B B→C (RTrans) A→C (iii) And the same again – a harder exercise (see the Digression immediately following) – for the prefixing rule: A→B (RPref) (C → A) → (C → B) (iv ) Using, as needed, any of the above derived rules (rather than just Modus Ponens) give a proof in Hil of (p → q) → ((p → (q → r)) → (p → r)). Digression. We include some indications here by way of assistance with parts (iii) and (iv) of the above Exercise. Let us begin with a proof in Hil of the formula version of the schema sometimes called ‘Assertion’ (mentioned in 2.33 below): p → ((p → q) → q), helping ourselves to the results of 1.29.2(i), (ii). (1) (2) (3) (4) (5)

p → ((p → q) → p) ((p → q) → p) → ((p → q) → ((p → q) → q)) p → ((p → q) → ((p → q) → q)) ((p → q) → ((p → q) → q)) → ((p → q) → q) p → ((p → q) → q)

HB1 HB3 1, 2 RTrans HB2 3, 4 RTrans

Using the formula here proved, we can obtain the principle for permuting antecedents, sometimes called (Perm), (p → (q → r)) → (q → (p → r)), which begins with a substitution instance of the theorem just proved (cf. 1.29.4): (1) (2) (3) (4)

q → ((q → r) → r) (((q → r) → r) → (p → r)) → (q → (p → r)) (p → (q → r)) → (((q → r) → r) → (p → r) (p → (q → r)) → (q → (p → r))

See above. 1, RSuff HB3 2, 3 RTrans

Clearly by using any formulas A, B, C, in place of p, q, r, we obtain a proof of the corresponding instance of the Permutation schema (called C below) (A → (B → C)) → (B → (A → C)), and so by Modus Ponens we shall be able to pass from any theorem of the form A → (B → C) to the corresponding permuted form B → (A → C). This derived rule takes us from (HB3) to a corresponding prefixing schema (A → B) → ((C → A) → (C → B)) called B later in our discussion ((HB3) itself being known as B). The derivability of (RPref) then follows by Modus Ponens. We treat the formula mentioned under 1.29.2(iv), more sketchily. Start with the prefixing formula (q → r) → ((p → q) → (p → r)) and prefix a “p”:

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CHAPTER 1. ELEMENTS OF SENTENTIAL LOGIC (p → (q → r)) → (p → ((p → q) → (p → r)))

We leave the reader to extend this by permuting the internal antecedents p and p → q and appealing to the Contraction axiom (HB2) to contract the double “p” antecedent at the end, giving a proof of ((p → (q → r)) → ((p → q) → (p → r)) sometimes called Self-Distribution, or just S – as mentioned after 1.29.10 below (p. 169). We will later use this last label for the formula (or the corresponding schema, with schematic letters in place of propositional variables). ((HB2) is similarly known as W , and (HB1) as K. The provability of S and K is particularly important for one way of showing that the Deduction Theorem holds for Hil, as we explain below, where some explanation will also be found of the provenance of these labels B, S, K etc.) The desired formula from 1.29.2(iv) is just a version of this with the antecedents permuted. End of Digression. Exercise 1.29.3 Where denotes conjunctive combination of valuations in the relational connection between formulas (of some language) and (arbitrary) valuations, it is immediate that if a sequent of Fmla holds on v1 v2 then it holds on v1 and it holds on v2 . (This is in any case a consequence of 1.14.5 since sequents of Fmla are sequents of Set-Fmla.) No such result obtains for Set-Set; but verify that if “and” is replaced by “or”, then the modified claim is correct for sequents of Set-Set (and hence also for sequents of Set-Fmla). Exercise 1.29.4 Show that the rule of Uniform Substitution is a ‘merely admissible’ (admissible but not derivable) rule of the proof system Hil. (Hint: consider the local/global contrast from 1.25 between preservation characteristics.) Exercise 1.29.4 leads us to an instructive comparison of Hil with a variant proof system, axiomatized by selecting particular instances of the schemata (HB1)–(HB12) together with the rules Modus Ponens and Uniform Substitution (or U.S. as we shall sometimes say for brevity). We need to make sure that distinct propositional variables replace the distinct schematic letters – making the formula chosen what we call a representative instance of the schema: for definiteness, say, replacing A, B, and C by p, q, and r respectively. Call the instances of these schemata (HB1)◦ –(HB12)◦ , and call the proof system thus axiomatized USHil. In any proof in this new system all applications of U.S. can be made to precede applications of Modus Ponens, for suppose we have a sequence consisting an application of Modus Ponens: A→B

A

B followed by one of U.S.: B s(B) s being some substitution. Then we can replace it by a sequence of applications of U.S. first and Modus Ponens second, as follows, ending in a proof of the same


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formula (i.e., s(B)). First we apply the substitution s just mentioned, appealing to Uniform Substitution, to the premisses of the above Modus Ponens. We then have premisses s(A → B) and s(A) for a different application of Modus Ponens, since the formula s(A → B) is one and the same as the formula s(A) → s(B) (“substitutions are endomorphisms”): s(A) → s(B)

s(A)

s(B) By repeating this procedure, any appeal to U.S. following an application of Modus Ponens can be replaced by an appeal to U.S. which precedes any such application. Any proof from (HB1)◦ –(HB12)◦ can therefore be replaced a proof of the same formula in which all applications of U.S. are made to the axioms themselves, rather than to theorems derivable therefrom. It can accordingly be thought of as a proof in Hil since any substitution instance of the axioms of USHil is an instance of the axiom-schemata (an axiom, then) of Hil. From this and 1.29.4, we conclude that the same formulas are provable in the two systems. (Our formulations above assume that the presentation of a proof system by axiom-schemata has the instances of those schemata as axioms, ignoring certain subtleties – not to the point here – raised in connection with independence in the Appendix to this section.) If, however, we wish to define a relation analogous to Hil for the system USHil which will coincide with CL and for which the Deduction Theorem will hold, we need to be careful about which rules are allowed to apply to the ‘hypotheses’ A1 , . . . , An in the derivation of B therefrom. In particular, the rule of Uniform Substitution must not be allowed for such derivations, since the substitution instances of a formula are not generally amongst its tautological consequences – we do not in general have A CL s(A), that is, for an arbitrary formula A and substitution s. The simplest way to modify the definition of Hil given above for the present case would be to say that the relation we are interested in, USHil holds between Γ and B just in case there is a sequence C1 ,. . . ,Ck of formulas each of which is either a theorem of USHil or an element of Γ, or else follows from earlier formulas in the sequence by Modus Ponens, and whose final formula Ck is the formula B. The point at which this differs from the earlier definition is that a reference to theorems now replaces a reference to axioms. Proofs of the theorems from the axioms are of course permitted to employ not only Modus Ponens but also Uniform Substitution. This is a special case of the distinction drawn in the terminology of Smiley [1963a] between rules of inference – intended in Fmla to be used for inference from assumptions/hypotheses/premisses – and rules of proof – intended only for the deduction of theorems from axioms (or already-proven theorems). Modus Ponens is not just a rule of proof, but a rule of inference, whereas U.S. is, on this terminology, a rule of proof which is not a rule of inference. (One often says “rule of proof” to mean “rule of proof which is not a rule of inference”.) Gabbay [1981] (p. 9) uses the phrases provability rule and consequence rule to mark the same distinction, though this is not ideal since rules of both sorts give rise to consequence relations, rules of inference giving the horizontal and rules of proof giving the vertical consequence relations distinguished in 1.26 (as applied to Fmla); the distinction will be important again in our discussion of modal logic (§2.2) where the rule of necessitation is a rule of proof which is not a rule

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of inference—and this time, unlike U.S., a substitution-invariant rule. Further discussion of Smiley’s distinction can be found in Humberstone [2008a] and at greater length in Humberstone [2010a]. An extended conception of the axiomatic approach might, in the light of the above distinction, take an axiomatization to consist in a set of axioms, and two sets of formula-to-formula rules, the rules of proof and the rules of inference, defining a consequence relation (such as USHil ) by allowing only the latter to develop the consequences of a given set of formulas. Even this refinement falls far short of what is available in Set-Fmla, however, since although the consequence relation just alluded to, like any other finitary consequence relation, can be though of as (the set of ‘Thinnings’ of) a collection of sequents in the latter framework, we are only able to discuss sequent-to-sequent rules of SetFmla to the extent that the Set in question is empty. Although we have been thinking about passing from a set of formulas (the set of theorems of an axiom system) to a consequence relation in the case in which the set in question is given syntactically, some aspects of the discussion do not depend on this. Just as, earlier, we described two consequence relations min and max on the basis of a given set Θ of formulas, so we can envisage the following route from such a set to a consequence relation mp ; this time it will be useful to make the dependence on Θ visible, so let us write, more explicitly, mp(Θ) : Γ mp(Θ) B iff B belongs to every superset of Γ ∪ Θ which is closed under Modus Ponens. Exercise 1.29.5 (i) Show that if Θ is the set of classical tautologies in ∧, ∨, ¬, →, then mp(Θ) is the relation of tautological consequence (alias CL ) on the language with those connectives. (ii ) Give an equivalent definition of mp(Θ) by completing the following: “Γ mp(Θ) B if and only if there exist formulas A1 ,. . . ,Ak such that Ak = B and . . . ” Naturally the mp idea works only given an appropriately behaving →; in various settings, one might give analogous definitions appealing to other connectives. What about a recipe for moving from sets of formulas to consequence relations which, like min and max does not depend on any particular connectives, but which, unlike them, delivers the relation of tautological consequence when the set with which we begin is the set of tautologies? That such a recipe exists is implicit in 1.25 above: given Θ we define the structurally complete consequence relation determined by Θ, which we shall denote by sc(Θ) , thus: Γ sc(Θ) B iff for all substitutions s such that s(Γ) ⊆ Θ, we have s(B) ∈ Θ. (Here, s(Γ) is {s(C) | C ∈ Γ}). Now if a consequence relation is substitutioninvariant and Θ is the -closure of ∅, then it follows that is included in sc(Θ) . In this case, we say further that is structurally complete if the converse inclusion also holds; that is, if = sc(Θ) (where, as before, Θ = Cn (∅)). In other words, is structurally complete if Γ B whenever for every substitution s with s(C) for each C ∈ Γ, we also have s(B). Exercise 1.29.6 (i) Show that the consequence relation CL (on a language with the connectives of, say, Nat) is structurally complete in the above sense. (Suggestion: use 1.25.8 from p. 131.)


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(ii ) It is convenient to have a less long-winded way of speaking than that provided by the immediately preceding discussion, so let us abbreviate “sc(Θ) ”, for Θ = Cn (∅), to “”. In this notation (which derives, inter alia, from Dummett (1973)), the claim that CL is structurally complete appears as: CL = CL . Show that for any substitution-invariant consequence relation , we have ⊆ and that these consequence relations agree on the consequences of the empty set; i.e., for all formulas A, we have A if and only if A. (More refined considerations in this vein appear in Makinson [1976]; see also Humberstone [2006a].) Thus the set of tautologies determines the relation of tautological consequence not only via the mp route, but also via the route (in the notation of (i) of the above Exercise); for the case of intuitionistic logic, whose consequence relation we shall introduce as IL in 2.32 below, the former relationship holds but not the latter, as we shall have occasion to observe in 6.42. We shall exploit this in our discussion of the Deduction Theorem below (1.29.25, p. 179). How is the concept, introduced here, of a structurally complete consequence relation, related to that of a structurally complete proof system, introduced in 1.25? To answer this (along the lines given in Humberstone [2006a]), we need to recall that although Pogorzelski [1971] defined a proof system to be structurally complete when all of its substitution-invariant admissible rules were derivable, he was thinking of proof systems for Fmla, which accordingly could be identified as ordered pairs R, Θ , with R a set of rules and Θ a set of formulas (the axioms). This gives rise to a consequence operation Cn R,Θ or the associated consequence relation by the same definition as yields the relation Hil from the above basis for Hil (we could have defined the operation Cn Hil instead), and then the structural completeness of the proof system coincides with the structural completeness of this consequence relation. (However, since rules are required to be finitary while consequence relations are not, there is room for a discrepancy here, noted in Makinson [1976]; the idea of structural completeness for arbitrary consequence relations and not just those of the form Cn R,A appeared in Tokarz [1973]. Note that for a description more in keeping with our own general practices – see the Appendix to this section (p. 180) – we would use just R to present a proof system, subsuming axioms as zero-premiss primitive rules. Finally we add that where we speak of substitution-invariant rules, Pogorzelski speaks of structural rules, and where we speak of admissibility, he speaks of permissibility.) This ignores the distinction between rules of proof and rules of inference, taking into account which distinction would lead us to identify axiomatizations with triples rather than pairs, say as R, R0 , Θ in which R0 ⊆ R. R would comprise the rules of proof, and R0 those among them which are to be used as rules of inference. This is a somewhat cumbersome attempt to attain the distinction easily formulated for sequent-to-sequent rules in Set-Fmla between zero-premiss rules of the form A B (corresponding to rules of inference) and rules allowing the passage from A to B (corresponding to rules of proof). In the latter setting, however, this simple twofold distinction, as well as the horizontal/vertical characterization just given of it, does not cover every possible formula-to-formula transition, and should only be regarded as a first step in characterizing such transitions. (See §6 of Humberstone [2000b], as well as §3 of Humberstone [2008a].) In the present discussion, we shall mostly ignore the case in which there are rules of proof which are not rules of inference.


164

As has been suggested already, the conceptual apparatus just summarised is not adequate to discussing logic in frameworks richer than Fmla, for all its introduction of consequence relations. The ordered pairs in a consequence relation are themselves sequents of Set-Fmla (at least, when the first element is a finite set) and the above apparatus leaves out of account the status of sequentto-sequent transitions. The rule whose applications are pairs Γ A ∧ B; Γ A , for example, is admissible though not derivable in the system Gen − of 1.27 ( = “Cut-free Gen”), for the same reasons as were indicated in the last two paragraphs of that subsection for the rule (T), though both that rule and (T) are substitution-invariant. That system is accordingly structurally incomplete, while Nat, for example, is structurally complete (as we saw in 1.25.9); since the associated consequence relations are one and the same, any account which pins structural completeness down as a property of consequence relations is inadequate to the general case. (Since Gen is a proof system for Set-Set rather than Set-Fmla, we are exploiting 1.22.3 in the above formulation.) We turn now to a proof of a variation on the Deduction Theorem for a weak logic called BCI logic, which extends (1.29.10: p. 168) to a proof of the Deduction Theorem proper for intuitionistic and also classical logic, and which will also lead us back to the theme of structural completeness. In our discussion of the axiom system we called Hil above, we defined a relation Hil , which definition can be generalized to any set Θ of axioms in a language with the binary connective → in the following way. Γ Θ B if and only iff there is a sequence C1 ,. . . ,Ck of formulas each of which is (i) an axiom – an element of Θ, that is – or (ii) an element of Γ, or (iii) else follows from earlier formulas in the sequence by Modus Ponens, and whose final formula Ck is the formula B. More explicitly, alternative (ii) means that for the formula Cj in the sequence C1 ,. . . ,Ck , there exist h, i < j with Ch = Ci → Cj . Note that this is just what we previously notated more explicitly as Γ mp(Θ) B. If, as with the examples that will occupy us for a while, Modus Ponens is the only rule – or the only “non-zero-premiss” or “non-axiomatic” rule, to be more precise – of the axiom system, this relation Θ is what we have called the consequence relation associated with that axiom system. (We use the less explicit notation here so that it can be generalized to the case in which rules other than Modus Ponens are permitted in the deductions, as in 1.29.12 below.) Let us introduce the labels “B”, “C”, and “I” for the schemata: B: (A → B) → ((C → A) → (C → B)) C:

(A → (B → C)) → (B → (A → C))

I:

A→A

The logic axiomatized by taking all instances of these schemata as axioms and Modus Ponens as the sole rule is called BCI logic and will be discussed at somewhat greater length in 7.25, along with some logics similarly based on schemata K and W : K: A → (B → A)

W : (A → (A → B)) → (A → B)

(As it happens in our later discussion we use an axiomatization with representative instances – going by the same names – of these schemata, along with a rule of Uniform Substitution, but the present formulation is more convenient here.) We will note the significance of these potential further axioms below (Coro.


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1.29.10.), for the moment pausing for practice with BCI logic; we will make use of (i) and (ii) of Exercise 1.29.7 which follows a Digression on the etymology of the labels employed here, while (iii)–(v) are for practice only. Digression. The lambda calculus employs terms constructed from individual variables x, y, z, . . . in accordance with the principle that any such variable is a term, as are M (N ) for any terms M and N , and λv M for any term M and variable v. (Intuitively, where M stands for a function and N for a candidate argument – using “argument” for what a function is applied to (its ‘input’) – M (N ) represents the application of M to N ; and for a term M typically including the variable v, the lambda term λv M represents that function mapping a value of the variable v to the corresponding value of the term M .) A term M is closed when every variable v occurring in M occurs within a part of the form λv N . These closed terms are intimately related to the combinators of Combinatory Logic, and we shall oversimplify by identifying combinators with certain such closed terms. (For a fuller account without the oversimplification, see Hindley and Seldin [1986], or the revised version thereof, Hindley and Seldin [2008].) In particular, we make the following identifications: B = λxλyλz x(y(z)) K = λxλy x

C = λxλyλz [x(z)](y) W = λxλy [x(y)](y)

I = λx x S = λxλyλz [x(z)](y(z))

(The axiom corresponding to S will appear below, after 1.29.10.) The square brackets here are a visual aid to the internal structure of the terms. Thus for S, “[x(z)](y(z))” means the application of the function x(z) to argument y(z). Usually such punctuation is not employed and parentheses are omitted in accordance with an “associate to the left” convention. With this convention in force, S, for example, would appear as λxλyλz xz(yz). The above examples are unrepresentative in two respects: first, in that the number of variables in the lambda calculus terms matches that in the corresponding propositional formula, and secondly in that all the “λ”s come at the start of the terms. To illustrate the latter point, we note that corresponding to (the most obvious natural deduction proof of) the formula ((p → p) → q) → q is the term λx x(λy y), while both points at once are illustrated by the case of (((p → q) → q) → q) → (p → q) and the corresponding term λxλy x(x(λz z(y))). But what is the correspondence we have in mind here? A type-assignment is an expression M : A in which M is a lambda calculus term and A is a formula of the propositional language with → as its only connective. Informal reading: “M : A” says that the term M is of the type A; for example “x: p → (p → q)” says that x is a function from some domain represented by “p” to functions from that set to some set (represented by) q. In what follows we use , ⊥, . . . to stand for type-assignments and Γ, Δ, . . . for sets thereof. A variation on the implicational subsystem of the natural deduction system Nat of 1.23 starts with assumptions having the form of type-assignments with variables as terms and arbitrary (implicational) formulas as types. Here is a sequent-to-sequent formulation in which the structural principles (R) and (M) have been combined into one rule:


166 (RM)

Γ, α α

subject to the proviso that:

each type-assignment in Γ ∪ {α} is of the form v : A, with v a variable not occurring in any other such type-assignment (within the same proof). (→I)

Γ, v : A M : B Γ λv M : A → B

(→E)

ΓM :A→B

ΔN :A

Γ, Δ M (N ) : B

For a sample proof in this type-assignment system, we offer the following, which explains why the schema B (a representative instance of which appears in line (8) below) is so called; the reader is invited to construct similar proofs for the other combinator-labelled principles: (1) (2) (3) (4) (5) (6) (7) (8)

x: p → q x: p → q y: r → p y: r → p z: r z: r y: r → p, z: r y(z): p x: p → q, y: r → p, z: r x(y(z)):q x: p → q, y: r → p λz x(y(z)):r → q x: p → q λyλz x(y(z)): (r → p) → (r → q) λxλyλz x(y(z)): (p → q) → ((r → p) → (r → q))

(RM) (RM) (RM) 2, 3 →E 1, 4 →E 5 →I 6 →I 7 →I

It turns out that the theorems of the implicational fragment of intuitionistic logic are then precisely the types A for which there exists a term M with M : A provable. It may happen that there are several terms – even once ‘bound alphabetic variants’ (such as λxλy [x(y)](y) and λyλz [y(z)](z)) are identified – to which this system assigns the same type. These terms correspond (roughly) to different proofs of the same formula (the formula appearing as the type). This connection between lambda calculus and natural deduction (or a similar connection between combinatory logic and the axiomatic approach) is the beginning of what is usually called the Curry–Howard isomorphism, for details of which see Hindley [1997], especially Chapter 6. See also Barendregt and Ghilezan [2000] and Bunder [2002b], and, for a textbook exposition, Sørensen and Urzyczyn [2006]. The idea of labelling axioms or axiom-schemata and the logics they axiomatize by means of combinators goes back to work of C. A. Meredith in the 1950s: see the notes to §2.1 for references (p. 274). Extending the type-assignment perspective to classical logic has been somewhat problematic; a review of some alternatives and the presentation of a new proposal may be found in Parigot [1992]. See also Chapter 2 of Pym and Ritter [2004]. End of Digression. Exercise 1.29.7 Show that on the basis of the above axiomatization of BCI logic, the following rules are derivable: (i) From E → (F → G) to E → ((B → F) → (B → G)) (ii) From E → (F → G) to (A → E) → (F → (A → G)) (Hint: it may help first to show the derivability of the Prefixing rule (RPref) of 1.29.2(iii).) Show, further, that the following three rules are respectively underivable, derivable, and underivable: (iii) From A → (A → B) to A → B


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(iv) From A → (A → B) and A to B (v) From A to (A → (A → B)) → B on the basis of the suggested axiomatization of BCI logic, taking as given the fact that W is not provable in this logic. (Hint: for (iii) and (v), first check that p → [p → ((p → (p → q)) → q] is BCI -provable.) Finally, show, given that K is not provable in BCI logic, that the following rule is not admissible in this logic: (vi) From A to B → A. Lemma 1.29.8 For all formulas A1 , . . . , Am , B1 , . . . , Bn , C, D, we have BCI [A1 → (A2 → . . . → (Am → (C → D)) . . .)] → [[B1 → (B2 → · · · → (Bn → C) . . .)] → [A1 → (A2 → . . . → (Am → (B1 → (B2 → . . . → (Bn → D) . . .))) . . .)]]. Proof. By I, we have BCI (C → D) → (C → D), so by n appeals to the rule in 1.29.7(i), with the “B” of that formulation instantiated successively to B1 ,. . . ,Bn , and with E, F, and G being respectively C → D, C and D, we get BCI (C → D) → [[B1 → (B2 → . . . . → (Bn → C). . . )] → [(B1 → (B2 → . . . → (Bn → D). . . ))]] Now by m appeals to the rule in 1.29.7(ii), inserting A1 ,. . . ,Am as in the statement of this lemma, we have the result claimed. (Here E is C → D, F is the formula B1 → (B2 → . . . . → (Bn → C). . . ) and G is (B1 → (B2 → . . . → (Bn → D). . . )).) In our earlier discussion, the Deduction Theorem was said to hold for (an axiomatization with axiom-set) Θ if whenever Γ, A Θ B, we had Γ Θ A → B. A weaker form of the theorem is what we shall show here in the case of BCI (as we shall denote {B,C,I} ). Define A →m B thus: A →0 B = B A →m+1 B = A → (A →m B) m So A → B is an iterated implication with B as final consequent and m antecedents of the form A. We are now ready for a statement and proof of the BCI Deduction Theorem, which in the terminology of Czelakowski [1986] is a ‘local’ deduction theorem, meaning that the formula whose deducibility from Γ records the deducibility of B from Γ ∪ {A} varies from case to case: sometimes it will be A →1 B, sometimes A →2 B, and so on. (In fact because Czelakowski understands the Deduction Theorem as a metalinguistic biconditional – as in the formulation (→) following 1.29.22 below – he takes the local versions biconditionally also.) Theorem 1.29.9 For any Γ, A, B: if Γ, A BCI B, then for some m: Γ BCI A →m B. Proof. Suppose Γ, A BCI B, i.e. that there is a deduction C1 ,. . . ,Ck , as above, of B from Γ ∪ {A}. We argue by induction on the number of applications of Modus Ponens in this deduction that given such a deduction, there is also, for some n, a BCI -deduction of A →m B from Γ. The basis case for this induction

168


is 0. If no applications of Modus Ponens have been made, this means that B (= Ck ) is either an axiom or an element of Γ. In the former case, the desired conclusion, that for some m, we have Γ BCI A →m B, is obtained by taking m = 0. The latter case divides into two subcases, since the “Γ” of our earlier definition is now instantiated as Γ ∪ {A}: (a) Ck ∈ Γ, and (b) Ck = A. In subcase a, the previous strategy works: putting m = 0 gives Γ BCI A →m B. For subcase b, we appeal to the schema I, setting m = 1, since A →1 B = A → B = Ck → Ck . For the inductive step, suppose that Modus Ponens has been applied in the deduction, with Ck derived from Ch = Ci → Ck and Ci (where h, i < k). These deductions involve fewer applications of Modus Ponens, so by the inductive hypothesis, we have Γ BCI A →m (Ci → Ck ) and Γ BCI A →n Ci for some m, n. By Lemma 1.29.8, putting A1 = . . . = Am = B1 = . . . = Bn = A, Ci = C, Ck = D, we get BCI (A →m (Ci → Ck )) → [(A →n Ci ) → (A →m+n Ck )], and therefore extending the deductions establishing Γ BCI A →m (Ci → Ck ) and Γ BCI A →n Ci by two appeals to Modus Ponens, we have that for some m or other (the sum of the previous m and n): Γ BCI A →m Ck .

Corollary 1.29.10 (i) For any Γ, A, B, if Γ, A BCIW B, then: either Γ BCIW A → B or Γ BCIW B. (ii) For any Γ, A, B, if Γ, A BCKW B, then Γ BCKW A → B. (iii) For any Γ, A, B, if Γ, A BCK B, then for some m, we have for all n > m, Γ BCK A →n B. Proof. (i): The same proof given for 1.29.9 works in the presence of additional axioms, such as, here, W , so if Γ, A BCIW B, then we have Γ BCIW A →m B for some m. If m = 0 then we have Γ BCIW B, while if m 1 we can reduce this to m = 1 by appeal to W , getting Γ BCIW A → B. (ii): The two alternatives just distinguished for (i) need not be distinguished here since, by the instance B → (A → B) of the schema K, we have Γ BCKW A → B whenever Γ BCKW B. (iii): left as an exercise.

Part (i) of the above Corollary is well known in relevant logic (BCIW giving the implicational fragment of the relevant logic R): it is Theorem 2.7 on p.21 of Diaz [1981], for instance. (For the → and ¬, → fragments of RM, a similar result appears as Theorem 30 on p. 34 of Blok and Raftery [2004].) Part (iii) has the interesting consequence, not available in the BCI case, that if Γ, A BCK B1 , and Γ, A BCK B2 , then for some m, we have Γ BCK A →m B1 and Γ BCK A →m B2 : something of considerable interest if we were to add a conjunction connective to the language. The many-valued logics of Łukasiewicz are extensions of BCK logic without W (contractionless logics, in the sense of 7.25 below), satisfying various additional principles. The three-valued logic from amongst this range will occupy us at some length in 2.11, and for this we have the following especially simple variant on 1.29.9 (noted in Church [1956], Exercise 19.8): if for the consequence relation concerned, we have Γ, A B,


169

then Γ A →2 B. (If a table for →2 is calculated using the table for → in Figure 2.11a, p. 198, the only difference is that the entry for 2 → 3 changes from 2 to 1. Alternatively: change the 2 → 3 entry in the → table of the matrix of Figure 1.29a, below, from 3 to 1. For a consideration of the Deduction Theorem in the context of Łukasiewicz’s many-valued logics more generally, see Pogorzelski [1964].) Part (i) of Corollary 1.29.10 is obtained in Church [1950]; our presentation with 1.22.9, 1.29.10, is similar to that in Curry [1954]. Note that BCIW logic is the implicational fragment of the relevant logic known as R (2.33). Similarly for part (ii), we have in BCKW logic the implicational fragment of intuitionistic logic (also called the logic of positive implication). A more commonly encountered axiomatization uses K alongside the single further schema called S in 2.13 and 7.25; see the preceding Digression for the reason. (Mnemonically, as already mentioned before 1.29.3 on p. 160: “S” for Self-Distribution – i.e., of →.) Actually, we use this label for the specific formula with propositional variables rather than schematic letters in that discussion. For present purposes, we give the schematic version: S:

(A → (B → C)) → ((A → B) → (A → C))

SK logic, understanding this as for the terms “BCI logic” etc. is the same as BCKW logic, the implicational fragment of intuitionistic logic (in Fmla). While S was originally used as an axiom(-schema) in the nineteenth century by Frege (and indeed is called Frege in Prior [1956]), it is hard to think of it, anachronistically, as anything other than devised specifically for the purpose of pushing through the inductive step of a proof of the Deduction Theorem for intuitionistic logic, given above as Coro. 1.29.10(ii). Instead of deriving that result, as we have done, from the BCI version of the Deduction Theorem, a direct proof is very straightforward with the axiomatization suggested by the labelling “SK logic”. The basis of the induction (on the number of applications of Modus Ponens in the deduction of B from Γ ∪ {A}) is provided by K and the SK -provable I, while for the inductive step, we consider an application of Modus Ponens to formulas C and C → D in a deduction from Γ ∪ {A}. We need to show that there is a deduction of A → D from Γ. The inductive hypothesis gives us deductions of A → C and A → (C → D) from Γ, which can be pooled to give a deduction of A → D by inserting the relevant instance of the schema S, namely (A → (C → D)) → ((A → C) → (A → D)) and then applying Modus Ponens twice to obtain the desired formula. As for the axiom system Hil with which we began this subsection, one can prove the deduction theorem by showing (for the basis case) that I and K are provable, and (for the inductive step) that for the associated consequence relation, A → D is a consequence of A → (C → D) and A → C. All of this can be done just using the implicational axioms (HB1), (HB2) and (HB3), which, with Modus Ponens as the sole rule, are in the combinator-derived nomenclature, K, W and B , this last being a suffixing rather than (like B itself) prefixing form of transitivity: B

(A → B) → ((B → C) → (A → C))

We can easily derive I from K and W , and for the above “rule form” of S, given (i) A → C and (ii) A → (C → D), we get (C → D) → (A → D) from (i) by B and Modus Ponens (this being an application of the derived rule RSuff from

170


1.29.2(i), p. 158), from which, together with (ii), we get by two applications of Modus Ponens and B (this being an application of RTrans from 1.29.2(ii) the conclusion A → (A → D) from which our final conclusion A → D follows by Modus Ponens and W . Since we have Modus Ponens and the Deduction Theorem for this implicational proof system, B KW logic is none other than the logic of intuitionistic implication, and so coincides – not just in its stock of provable theorems but in the associated consequence relation – with BCKW logic and SK logic. The BCI Deduction Theorem has further applications beyond those cited in Coro. 1.29.10, one of which, from Surma [1973d], we give here. 7.31.3(iii) at p. 1132 below, approaches this result from a different angle. Note that the word equivalential here is just an adjective meaning “involving or pertaining to the connective ↔”, analogously to the use of implicational in connection with →: in particular there is no intention to call to mind the notion of an equivalential consequence relation in the sense introduced in the discussion between 1.19.9 (p. 97) and 1.19.10 above. Corollary 1.29.11 Where Θ is the equivalential fragment of CL in Fmla, but with ↔ written as →, if Γ, A mp(Θ) B then either Γ mp(Θ) A → B or else Γ mp(Θ) B. Proof. The hypothesis that Γ, A mp(Θ) B amounts to asserting the existence of a deduction of B from Γ ∪ {A} in which not only B, C, and I but any other formula which, like them, is tautologous when → is interpreted as ↔, is allowed as an axiom. Since no new rules have been included, 1.29.9 remains intact and we have Γ mp(Θ) A →n B for some n. But if n is odd then a deduction of A →n B can be extended to one of A → B (from the same set of formulas) and if n is even, then a deduction of A →n B can be extended to one of B, by interposing suitable equivalential tautologies. Because we can axiomatize classical logic in Fmla by the addition of further axioms without having to make use of further (one or more premiss) rules, the same argument gives the Deduction Theorem for any such axiomatization of classical propositional logic, with Modus Ponens as sole rule. If further – or more generally, other – rules are added, and the notion of a deduction is modified so that these may be used in obtaining a deduction of B from Γ ∪ {A} as well as in the deduction promised by the Deduction Theorem when such a deduction of B exists, of A → B from Γ, then we have to check the inductive part of the proof again. The induction will then be over the number of applications of any of the rules permitted, rather than simply the number of applications of Modus Ponens, as above. (This means that we are considering the new rules as rules of inference rather than as mere rules of proof. If rules of the latter type are present then the notion of a deduction must be modified as indicated above, in our introduction of this distinction between types of rules, so that formulas are not derived from those in Γ by means of rules of proof which are not rules of inference.) We illustrate the kind of difficulty that can arise, using the following matrix, for a language with two binary connectives → and ∧. (This discussion presumes some familiarity with matrix methodology, as explained in 2.11.)

1.2. RULES AND PROOF → *1 2 3

1 1 1 1

171 2 2 1 1

3 3 3 1

∧ 1 2 3

1 1 3 3

2 3 3 3

3 3 3 3

Figure 1.29a

Example 1.29.12 Consider the axiom system with the Positive Implication axiom-schemata S and K as well as the schemata (i) (A ∧ B) → A and (ii) (A ∧ B) → B, with the rules Modus Ponens and Adjunction, the latter being the two-premiss rule that takes us from A and B to A ∧ B. To see that the Deduction Theorem does not hold for this logic note that while, by Adjunction, p ∧ q is deducible from {p, q}, q → (p ∧ q) is not deducible from {p}. We use the matrix of Figure 1.29a to show that this is so. The axiom system described is sound w.r.t. the above matrix, and indeed more generally that this holds for the consequence relation which holds between Γ and A when there is a deduction using Modus Ponens and Adjunction from formulas drawn from Γ and the set of axioms, and terminating in A. However, we do not have p q → (p∧q), for consider the assignment of values 1 and 2 to p and q respectively. This gives the left-hand formula a designated value (the sole designated value, 1) and the right-hand formula the value 2 → (1 ∧ 2) = 2 → 3 = 3. (For a variation on the example, introduce a 1-ary connective with the table it would have if A had been an abbreviation of A ∧ A; then note that p p, though p → p.) Remark 1.29.13 The matrix in Fig. 1.29a is the result of changing the 3element Gödel matrix of 2.11.3 (p. 202) by replacing all “2”s in the body of the ∧-table by “3”s. The latter matrix in turn represents the various models on a two-element Kripke frame (see the discussion in 2.11.3). Preservation of the value 1 amounts to preservation of the property of being true throughout a model on this frame. For the language of intuitionistic logic, as conventionally understood, this property coincides with the property of preserving truth at an arbitrarily given element of such a model (2.32.4, p. 309), but this is not so if ∧ is interpreted as in Figure 1.29a, which amounts to saying that at a point (in a Kripke model) A ∧ B is true if and only if at every point in the model, each of A, B, is true. Exercise 1.29.14 Consider a variation on the axiom system of 1.29.12 for a logic with connectives ∧ and → like that described there except that the rule of Adjunction is replaced by a ‘conditional’ form of Adjunction: from C → A and C → B to C → (A ∧ B). Does the Deduction Theorem hold for this system? Prove or refute, as appropriate. (Compare the logic described parenthetically at the end of 1.29.12: if we think of the rule “from A to A” replaced by a conditional form “from C → A to C → A”, then it is clear that we have the Deduction Theorem in this case.) See the discussion following 1.29.18, if difficulties arise.

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While we have concentrated on consequence relations of the form mp(Θ) , though lately suppressing the “mp” part of this notation, as well as allowing other rules to join (or replace) Modus Ponens in deductions, we can more generally say that a consequence relation , however defined, satisfies the Deduction Theorem whenever Γ, A B implies Γ A → B for all Γ, A, B in the language of , presumed to have → as one of its connectives. Such a claim will be trivial if is the consequence relation associated (in the sense introduced just before 1.22.3) with a conventional natural deduction or sequent calculus proof system because of the employment of the rule (→I), or (→ Right), as it would be called in the latter context. The issue becomes non-trivial when what we are dealing with is the consequence relation associated with a proof system in Fmla – an axiom system, that is. One could also say, though we will not need this generalization here, that satisfies the Deduction Theorem for a binary connective #, primitive or derived, in the language of , when Γ, A B always implies Γ A # B, whether or not → itself is present: see Porte [1960b], [1981], for example. Czelakowski [1985] generalizes even further and considers the replacement of the single formula here represented as A # B by a set of formulas. In fact, Czelakowski (amongst several other writers) understands by the Deduction Theorem not just the implication just described but the conjunction of this with its converse (a conjunction given as (→) below: cf. further the reference to Rautenberg [1986] in 1.29.23(ii). With regard to this general notion, a striking observation was made, perhaps first in Prucnal [1972a], with repercussions for a vast range of consequence relations including all those which can be thought of as representing the implicational fragments of ‘intermediate’ logics in Set-Fmla. Intermediate here means ‘intermediate between intuitionistic and classical logic’. (See 2.32 for further details, including information on the consequence relation, IL , of intuitionistic logic.) What matters really is just that we are considering substitution-invariant ⊇ IL , or more accurately, since we suppose that the only primitive connective present is →, all ⊇ BCKW (or more simply, SK ). The key ingredient in Prucnal’s Lemma (1.29.15 below) is a collection of substitutions which we shall call sA , for each formula A. For a given formula A, this is the unique substitution behaving as follows on the propositional variables: sA (pi ) = A → pi . (When we have a subscripted turnstile, as with “SK ” in the statement of the following Lemma we write the subscript only once when using the “ ” notation, so “C SK D” means “C SK D and D SK C”.) Lemma 1.29.15 For any formulas A, B in the language of SK , we have: sA (B) SK A → B. Proof. By induction on the complexity of B. The basis case (B = pi ) is settled by the definition of sA .There is only one inductive case, that in which B = C → D. Given, by the inductive hypothesis, that sA (C) SK A → C and sA (D) SK A → D, we have only to observe that A → (C → D) SK (A → C) → (A → D) to conclude, since sA (C → D) is the formula sA (C) → sA (D), that


173

sA (C → D) SK A → (C → D),

as required.

We then argue, following Prucnal [1972a], Wroński [1986], Wojtylak [1991], Slaney and Meyer [1992], though some of these authors are more interested in the topic of structural completeness than the Deduction Theorem itself (on which more below): Theorem 1.29.16 Every substitution-invariant ⊇ SK (with → the sole primitive connective of the language of ) satisfies the Deduction Theorem. Proof. Suppose for as described, we have Γ, A B. By substitutioninvariance, it follows that sA (Γ), sA (A) sA (B). By Lemma 1.29.15 this means that {A → C | C ∈ Γ}, A → A A → B. Since ⊇ SK , we have A → A, so {A → C | C ∈ Γ} A → B. But also since ⊇ SK , for each C ∈ Γ, we have C A → C, so we conclude that Γ A → B. Clearly the crucial feature of implication that is exploited in this line of thought is the fact, used in the proof of Lemma 1.29.15 that A → (C → D) SK (A → C) → (A → D). Let us say that an n-ary connective # is consequent-distributive according to , presumed to have → amongst its connectives, when for all A, B1 ,. . . ,Bn in the language of : A → #(B1 , . . . , Bn ) #((A → B1 ), . . . , (A → Bn )) We can summarise the crucial equivalence exploited in the proof of Lemma 1.29.15 roughly by saying that → itself is consequent-distributive according to SK (or more informally, according to intuitionistic logic). For a more accurate statement we need to introduce a strengthening of this notion. If the left and right side of the schematic -condition inset above are not only equivalent but synonymous (according to ) in the sense that the results of replacing either by the other in any longer formula give equivalent formulas, then we will say that # is strongly consequent-distributive according to . Note that just such replacements are made in the case of # = → in the proof of 1.29.16 above. (The “synonymy” terminology, which derives from Smiley [1962a], will be introduced more carefully for various logical frameworks in 2.13, 3.15, and 3.16.) This is what is needed for the inductive step for the connective # in the proof of any analogue of that Lemma, leaving us with the following distillation: Observation 1.29.17 Let ⊇ SK be any substitution-invariant consequence relation on a language with the connective →, with the property that every primitive connective of the language of is strongly consequent-distributive according to . Then the Deduction Theorem holds for .


174

Corollary 1.29.18 Every substitution-invariant ⊇ IL with → and either or both of ∧, ↔, satisfies the Deduction Theorem. Proof. By “IL ” here is meant the fragment in whichever of ∧, ↔ is/are present alongside →. The justification for the claim is Observation 1.29.17 and the fact that each of ∧, ↔, is strongly consequent-distributive according to IL . This discussion bears on the question of Exercise 1.29.14, incidentally, about an axiom system extending that for positive implication axioms with (A ∧ B) → A and (A ∧ B) → B, and rule (alongside Modus Ponens) for passing from C → A and C → B to C → (A ∧ B), since one can check that according to the consequence relation associated with this axiom system, ∧ is consequentdistributive. It is not, however, strongly consequent-distributive – so the proof via a mimicking (with the new connective ∧) 1.29.15 breaks down. Consider the following tables for → and ∧, on whose provenance see the Notes under “Deduction Theorem”. The latter table gives the direct product of the standard two-valued table for ∧ with itself (see the discussion surrounding Figure 2.12a at p. 213 in the following chapter) while that for → departs from the product of two-valued (boolean) → with itself at two positions: the entries for 2 → 4 and for 3 → 4, which in the product should be 3 and 2 respectively but here are both = 4. → *1 2 3 4

1 1 1 1 1

2 2 1 2 1

3 3 3 1 1

4 4 4 4 1

∧ 1 2 3 4

1 1 2 3 4

2 2 2 4 4

3 3 4 3 4

4 4 4 4 4

Figure 1.29b

You may check that for the consequence relation associated with the axiom system (involving → and ∧) of 1.29.14, call it 0 , and for the consequence relation determined by the matrix of Figure 1.29b – here helping ourselves to terminology from §2.1 below – call it 1 , we have 0 ⊆ 1 . (This amounts to saying that the axioms mentioned in 1.29.14 all assume the designated value 1 – as indicated by the asterisk – and that the rules preserve, for each assignment of values, the property of taking the value 1.) For = 1 one may check that although (p → q) → (p ∧ q) (p → q. → p) ∧ (p → q. → q) as a special case of consequent-distributivity (of ∧), holding therefore also for = 0 , we do not have strong consequent-distributivity, the left and right hand sides here not yielding equivalent results when spliced into the context ((p → q) → p) → as we see by considering the assignment of values 3 and 2 to p and q respectively. Since 0 ⊆ 1 , these longer formulas are not equivalent according to 0 either. In the terminology (already used in passing in 1.19) of §3.3, where a consequence relation for which any equivalent formulas are synonymous is called


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congruential, this makes 0 ⊆ 1 non-congruential. In a refined version of that terminology, in which a connective is called congruential according to a consequence relation if it makes equivalent compounds from equivalent components – a consequence relation being then congruential just when each connective in its language is congruential according to it – the above example shows that → is not congruential according to our current pair of consequence relations.) For a simple counterexample to the Deduction Theorem for each of 0 , 1 taken as , note that p q → (p ∧ q) while p → (q → (p ∧ q)) the latter point being established by the same matrix assignment as above (assign 3 to p, 2 to q). Note that instead of saying “strongly consequentdistributive” in 1.29.17, we could have left it at “consequent-distributive” if we had inserted the further condition that the consequence relation be congruential. Digression. For a given rule ρ whose applications are, for some formulas A1 ,. . . ,An , exactly the substitution instances of A1 , . . ., An , B let us mean by the prefixed form of ρ the rule whose applications are those substitution instances of pi → A1 , . . ., pi → An , pi → B where pi is a variable not occurring in A1 , . . . , An , B. Thus the rule, just mentioned, taking us from C → A and C → B to C → (A ∧ B) is the prefixed form of Adjunction from 1.29.12, described as a “conditional form” of Adjunction in 1.29.14. It is an observation frequently made (from at least Porte [1960b], Theorem 1, to Czelakowski [1985], Theorem 1.3) that while the provability of I and K suffices, in the proof of the Deduction Theorem, for the basis case of the induction on number of rule applications, for the inductive step it suffices that for each primitive rule ρ whose applications are substitution instances, as above, of A1 , . . . , An , B , that the consequence relation associated with the axiom system in question, should satisfy: pi → A1 ,. . . ,pi → An pi → B, with pi as above. (Note that the condition this imposes on in the case of the rule Modus Ponens is a consequence-relational version of the schema S, to the effect that for all formulas A, B, C, we have A → (B → C), A → B A → C.) One should not think, however, that extending a proof system by adding prefixed forms of all of its rules automatically results in a system for which the Deduction Theorem holds, since in adding new rules we have added further conditions for the relation to satisfy. End of Digression. A somewhat different use of Lemma 1.29.15, other than in proving the Deduction Theorem, appears in the sources cited above, namely in proving the structural completeness of various consequence relations. Recall that the structural completeness of an axiom system (in which there is no rule of proof/rule of inference distinction) coincides with the structural completeness of the consequence relation associated with that axiom system. Theorem 1.29.19 Let be a consequence relation satisfying the conditions of 1.29.17. Then is structurally complete. Proof. Suppose that for every substitution s with s(A1 ),. . . , s(An ), we have s(B), with a view to showing that A1 , . . . , An B. We denote sAi (as defined


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for 1.29.15) by si , and consider the substitution s = s1 ◦ s2 ◦ . . . ◦ sn (◦ indicating composition). By Lemma 1.29.15, extended (1.29.17) to accommodate any other consequent-distributive connectives in addition to →, we have s(C) A1 → (A2 → . . . (An → C). . . ), and instantiating C to A1 ,. . . ,An gives s(C) in each case, so putting C = B the supposition gives s(C) for this case too. (See the Example below for elaboration.) But since is an extension of SK , and we have D, D → E E, for all formulas, which implies that A1 ,. . . ,An B. Although for simplicity the above result hypothesizes that the consequence relation concerned satisfies all the conditions imposed in 1.29.17, in fact we do not need one of those conditions: substitution-invariance. Example 1.29.20 A numerically definite but still schematic illustration of Observation 1.29.19: Suppose n = 3. Then the supposition is that every substitution s for which s(A1 ), s(A2 ) and s(A3 ) are provable (i.e., are -consequences of ∅) gives s(B) provable. Take s = s1 ◦ s2 ◦ s3 . By Lemma 1.29.15, s3 (A1 ) A3 → A1 and then, again, s2 (s3 (A1 )) A2 → (A3 → A1 ), and thus s1 (s2 (s3 (A1 ))) A1 → (A2 → (A3 → A1 )). Recall that s1 (s2 (s3 (A1 ))) = s1 ◦ s2 ◦ s3 (A1 ) = s(A1 ), so s(A1 ) A1 → (A2 → (A3 → A1 )). Similarly, we have: s(A2 ) A1 → (A2 → (A3 → A2 )) and s(A3 ) A1 → (A2 → (A3 → A3 )). In each case (and clearly also for n > 3) the final consequent of the implicational formula is identical with one of the several antecedents and so is SK -provable, meaning that each of the s(Ai ) is provable. Thus s(B) is provable, for which by the same considerations we have s(B) A1 → (A2 → (A3 → B)). But then since A1 , A1 → (A2 → (A3 → B)) A2 → (A3 → B) – substituting in p, p → q q – and A2 , A2 → (A3 → B) A3 → B and A3 , A3 → B B, we have (appealing to (T)) A1 , A2 , A3 B. Many structurally incomplete consequence relations are known, and of these many share their Fmla logics (Cn (∅)) with much loved structurally complete consequence relations. In keeping with much of our recent discussion, we give a purely implicational example, based on a modification of the SK - or BCKW axiomatization of the implicational fragment of intuitionistic logic. We shall need to make reference to the following matrix:


177 → *1 *2 3

1 1 1 1

2 3 1 1

3 3 1 1

Figure 1.29c

Example 1.29.21 Consider the following ‘subrules’ (special cases) of Modus Ponens: (i) From A1 → A2 and (A1 → A2 ) → B to B (ii) From A and A → (B1 → B2 ) to B1 → B2 (iii) From A1 → A2 and (A1 → A2 ) → (B1 → B2 ) to B1 → B2 . It is clear, since no propositional variable is provable by Modus Ponens from S and K, that any premiss or conclusion of an application of Modus Ponens in a proof (note: a proof, not a deduction from assumptions), that for purposes of proving theorems, Modus Ponens could be replaced by any one of rules (i)–(iii) without resulting in the loss of any theorems. For definiteness here, we will work with rule (i). Let be the consequence relation holding between Γ and B when there is a deduction of B from Γ in the sense of the Deduction Theorem, i.e., we can obtain B from formulas in Γ and instances of the schemata S and K by appeal to rule (i). Since, by the previous considerations A if and only if SK A, we have that for any substitution s, if s(p) and s(p → q), then s(q). (Modus Ponens is an admissible rule.) But p, p → q q, as we can see from Figure 1.29c, which depicts a matrix in which we may assign both formulas on the left designated values: by giving p and q the values 2 and 3 respectively, p → q takes on the value 2 → 3 = 1, so both left hand formulas have designated values, 2 and 1, while the formula on the right has the undesignated value 3. On the other hand, rule (iii) is designation-preserving on every assignment (essentially because no implicational formula can receive the value 2) and all instances of S and K are valid in this matrix, from which we draw the conclusion that p, p → q q, so Modus Ponens, though a substitution-invariant admissible rule, is not derivable in our weakening of the usual axiomatization of SK logic. Note also that ‘Peirce’s Law’, the schema ((A → B) → A) → A, which added to S and K with Modus Ponens yields the implicational fragment of classical logic, also has all instances valid in the matrix of Figure 1.29c, so here again by weakening Modus Ponens to its special case, rule (i), we obtain a structurally incomplete consequence relation for which the consequences of the empty set are a familiar logic in Fmla (the set of classical tautologies in →). Rules such as (i)–(iii) of 1.29.21 – special subcases of Modus Ponens – have been considered in Pogorzelski [1994] and Nowak [1992]. The latter paper discusses a famous system from Hiż [1959] which provides a weaker consequence relation than CL (in the language with connectives → and ¬) but agrees with CL when there is nothing on the left. In fact the matrix of Figure 1.29c can be found on p. 195 of that paper (alongside a table for ¬); even earlier, it appeared

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in the service of a similar point in Lyndon [1951], p. 459. This phenomenon was also illustrated in Porte [1960a], whose example we recapitulate here: Example 1.29.22 The effect of the Hilbert–Bernays axioms (HB10)–(HB12) governing ¬ can be wrapped up in the single schema of “Decontraposition”: (the converse of the contraposition schema) (¬ A → ¬B) → (B → A); a complete axiomatization of the classical tautologies in negation and implication can be derived by adding this schema to S and K, with Modus Ponens as the sole rule. (Peirce’s Law is then derivable.) A minor variation on Porte [1960a] then consists in the following observation. Take as axioms all instances of the double negations of these three schemata and as rules (i) from ¬¬(A → B) and ¬¬A to ¬¬B and (ii) from ¬¬A to A. Every formula obtainable by Modus Ponens from S, K, and Decontraposition is then provable (why?) and no other formula is provable, but Modus Ponens, though admissible – since admissibility just depends on the set of theorems – is not derivable (again: why?). Answers – albeit somewhat informally expressed – to our parenthetical why-questions may be found in Porte [1960a]. We return to the topic of the Deduction Theorem, recalling from the discussion preceding 1.18.1 (p. 84), the condition (→)

Γ A → B ⇔ Γ, A B

which summarises the logical behaviour of → according to as IL . (For this reason, in 2.33 – p. 329 – we call consequence relations satisfying this condition →-intuitionistic.) One might expect a stronger description, to the effect that (→) will be satisfied by all ⊇ IL , at least if this inclusion is understood to require the language of to be that of IL . The example at the end of 1.29.12 (p. 170) shows the need for such a restriction, since the ⇐ direction of (→) is just the claim that the Deduction Theorem holds for , when this is understood in the more general sense in which it is not just for defined as the consequence relation associated with an axiom system (though certainly IL could be so introduced). Keeping this “same language” restriction intact for the consideration of ⊇ IL , we find that the two halves of (→) fare very differently. For any consequence relation 0 satisfying the forward (i.e., ⇒) direction of (→), any extension ⊇ 0 will also satisfy this condition, since satisfaction of the condition by any is equivalent satisfying to the ‘unconditional’ Modus Ponens style condition that for all A, B, we have A, A → B B. Thus its satisfaction is always inherited by extensions. For the half of (→) of current interest, its backward direction, this is not so, however, as we shall see. Exercise 1.29.23 (i) Suppose that we were considering not consequence relations but generalized consequence relations. Show that any extension of a gcr satisfying the following variant (with side formulas on the right) of the Deduction Theorem condition, must also satisfy the condition: Γ, A B, Δ ⇒ Γ A → B, Δ.


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(Suggestion. Find a pair of unconditional -statements satisfying both of which is equivalent to satisfying the above conditional -statement, if necessary by consulting 1.18.4, p. 85.) (ii ) Returning to the subject of consequence relations, the two-way form (→) is, as noted earlier, sometimes itself referred to as the Deduction Theorem (for or for an axiom system for which is the associated consequence relation), or less confusingly, as the Deduction-Detachment Theorem (as in Font and Jansana [1996], p. 48, Modus Ponens often being referred to as the Rule of Detachment). This exercise, including the ‘hint’, is taken from Rautenberg [1986], and presumes familiarity with intuitionistic logic – the consequence relation IL – on which see 2.32. Show that for no binary connective # definable in the {∧, ∨, ¬}fragment of intuitionistic logic do we have, for all formulas A, B of that fragment: Γ A # B ⇔ Γ, A B (Hint: Note that any # satisfying this condition satisfies (a) A #A and (b) A # B, A B, for all A, B, and show no connective definable using the available primitives satisfies (a) and (b).) We can use the fact that IL is structurally incomplete, illustrated in the discussion preceding 6.42.14 on p. 882 with examples from Harrop and Mints, to show that not only are there extensions of IL that do not satisfy the Deduction Theorem, but such cases can be found amongst the intermediate logics. Here “intermediate” has the special sense (on which see further 2.32) of logics intermediate between classical and intuitionistic logic. Since we are thinking of Set-Fmla here, we identify such logics with intermediate consequence relations, i.e., substitution-invariant consequence relations for which we have IL ⊆ ⊆ CL . There is still some ambiguity here, since we have not specified what the logical vocabulary is, but we have in mind the usual primitive connectives →, ∧, ∨, and ¬ or ⊥. In fact for convenience let us take it that ⊥ is amongst our primitive connectives, since this will allow us formulas constructed without the aid of propositional variables; for good measure, let us help ourselves to as well. We know already that we have to step outside of the {→, ∧} fragment in order to obtain an extension of IL not satisfying the Deduction Theorem, by 1.29.17. Mints’s example – see our discussion in 6.42 for substantiation of the following claim – of structural incompleteness for IL itself can be put as follows, using the “” notation of 1.29.6(ii) (p. 162): Example 1.29.24 (p → r) → (p ∨ q) IL ((p → r) → p) ∨ ((p → r) → q) although (p → r) → (p ∨ q) IL ((p → r) → p) ∨ ((p → r) → q). We now turn this fact to our present advantage (as in Humberstone [2006a]): Observation 1.29.25 The consequence relation IL is an intermediate consequence relation not satisfying the Deduction Theorem. Proof. We have to show first that IL is an intermediate consequence relation, i.e., that IL is substitution-invariant and IL ⊆ IL ⊆ CL . Substitutioninvariance follows from the definition of IL given by: Γ IL A iff for all

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substitutions s with s(Γ) all intuitionistically provable, we have s(A) intuitionistically provable, since a failure of substitution invariance would mean that for some Γ, A with Γ IL A there was substitution s for which s (Γ) IL s (A), i.e. for some substitution s , say, s (s (Γ)) were intuitionistically provable but s (s (A)) was not: contradicting the hypothesis that Γ IL A, since the composition s ◦s is then an s sending all of Γ, but not A, to theorems of intuitionistic logic. Secondly, that we have IL ⊆ IL is an application of 1.29.6(ii). Finally, it remains to check that IL ⊆ CL . Suppose that Γ CL A; let v be a boolean valuation verifying all of Γ but not A, and let sv be the substitution defined in 1.25.7 (p. 130), except with ⊥ and in place of pi ∧ ¬pi and pi → pi , respectively. By 1.25.7(ii), sv (Γ) CL sv (A). But it is not hard to check that CL and IL are alike as far as formulas constructed from no propositional variables are concerned, so sv (Γ) IL sv (A), from which we conclude that Γ IL A, completing our proof that IL ⊆ CL . For the failure of the Deduction Theorem, let A and B be the formulas on the left and right of the “IL ” in 1.29.24. By 1.29.6(ii), IL and IL agree in the Γ = ∅ case, so since, by the second claim under 1.29.24, IL A → B, even though, as the first claim there says, A IL B. Thus IL is an intermediate consequence relation not satisfying the Deduction Theorem. A consequence relation + is said to be an axiomatic extension of a consequence relation when for some set of formulas Θ, we have for all Γ, A: Γ + A if and only if Γ, Θ A. The terminology arises from the case in which is the consequence relation associated with an axiom system, in which case + is the consequence relation associated with an axiom system with additional axioms (i.e. those in the set Θ). Many familiar intermediate consequence relations are axiomatic extensions of IL in this sense; for example – one mentioned in 2.32 – we obtain LC by taking Θ to consist of all formulas of the form (A → B) ∨ (B → A). Exercise 1.29.26 Show, using 1.29.25, that the consequence relation IL there treated is not an axiomatic extension of IL . (Hint: consider what is required for the proof of the Deduction Theorem for IL .)

Appendix to §1.2: What Is a Logic? But there is no generally accepted account of what a logic is. Perhaps this is as it should be. We need imprecision in our vocabulary to mirror the flexible imprecision in our thinking. – Aczel [1994], p. 261.

The question in the title above is to be distinguished from the question – on which, predictably, there has been considerable philosophical debate – “What is logic?”. That debate usually focuses on the question of what entitles an expression in a natural language, or its analogue in a formal language, to be counted as part of the logical rather than the non-logical vocabulary of that language. (See notes, under ‘Logical Constants’.) Suppose you have answered that question to your satisfaction, or are prepared to set the question aside in the meantime. (We are certainly not going to consider it further, here.) Candidate codifications of the area delimited by such an answer it is natural


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to call logics. To emphasize this count-noun use of “logic”, we will sometimes speak of a system (of logic) rather than a logic. The term “system” here is not to be conflated in advance of discussion with our precisely defined “proof system” (as in 1.23). What exactly it should be taken to mean depends on the answer returned to the question forming the title of this Appendix. The general tenor of our discussion will be to discourage thinking of anything as the answer; rather, there are several different degrees of refinement for the individuation of logics, answering well to different needs – as the above quotation from Aczel is perhaps suggesting. Our initial pass over the terrain will distinguish three, in order of decreasing discriminatory fineness. We will then have a closer look and find that the three-tier structure needs really to be replaced by a fourfold classification. The usefulness of distinctions between logics individuated at the various levels will be illustrated. We restrict our attention to asking when systems in the same language should be identified, without considering a level of analysis which abstracts from the particular language in which a system is formulated and acknowledges the possibility of a system in another language being ‘essentially the same’ as a given system. (Allowing the language to change would raise many interesting issues about translation which are accordingly not treated here. See Caleiro and Gonçalves [2005], for discussion and references; for further information on translational embeddings consult the references cited after the ‘Hint’ to Exercise 6.42.3, p. 873f.) Suppose we want to talk about Lemmon’s system of propositional logic, or at any rate that version of it called Nat in 1.23. What should we mean by “Lemmon’s system”? That depends on what we want to say about it. Three candidates come to mind: (1) The proof system Nat. (2) The set of derived tuples (Nat ) of that proof system. (3) The set of all sequents provable in Nat. (The “ ” notation in (2) is from 0.26, as deployed in 1.24.) Correspondingly, we could return any one of three answers to the question of when proof systems S and S are proof systems for the same logic. The answer corresponding to (1) would be: when S = S ; to (2): when S = (S ) ; and to (3) when, for L the language of S and S , S ∩ seq(L) = (S ) ∩ seq(L). Clearly, the third policy makes the fewest distinctions amongst logics, and the first the most, while the second is intermediate. If we are interested in the (semantic) completeness of what we want to think of as Lemmon’s system, then there is no harm in adopting the third – coarsest – policy. It does not distinguish between systems with the same stock of provable sequents, but then, this is not a distinction to which the question about completeness is sensitive. All we want to know is whether in amongst that stock are to be found all the tautologous sequents. By contrast, if it is some notion of rule-completeness (1.25) we are interested in, then this third level is not going to be of much use, since such questions are sensitive to the distinction between derivable and admissible (sequent-tosequent) rules, a distinction which cannot be drawn on the basis of the stock of provable sequents. In general there is a loss of information as we pass down the list from (1) to (3), which might prompt the reaction that we always identify logics in accordance with the most finely discriminating policy, at level (1). But this would be tedious, since one often wants to speak of a reformulation of such

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and such a logic, at the end of which we want the same logic to have been formulated anew, while the original tuple-system may not survive this process. We illustrate this, and the general three-tier structure, by turning now from the example of Nat to that of Hil, from 1.29. We consider propositional logic in Fmla (with, for definiteness, connectives ∧, ∨, →, ¬). One could take a system of logic to be—the narrowest conception— a particular axiomatization. Or, more liberally, one could ignore differences in axiomatization—differences over what the axioms and primitive rules were—as long as the same set of theorems and derivable rules are forthcoming on the basis of the differing axiomatizations. (It is not really necessary to say “theorems and derivable rules”, since the reference to theorems is really to provable sequents and hence can be subsumed as applications of zero-premiss rules.) On this policy, the system USHil presented in 1.29 via the axioms (HB1)◦ –(HB13)◦ would be the same as that, mentioned in connection with the Deduction Theorem axiomatized by taking the formula (p → (q → r)) → ((p → q) → (p → r)) in place of (HB2) and (HB3), keeping the other axioms and rules the same. Finally, and least discriminatingly, we might ignore differences of axiomatization which did not result in differences between theorems provable. On this policy, not only would the two axiomatizations just mentioned determine the same logic, but so also would the axiomatization by schemata without Uniform Substitution as a rule (Hil as given in 1.29). The underivability of that rule on the basis of the resulting axiomatization makes for a difference on the earlier ‘intermediate’ policy, of course. Again, such questions as those of soundness and completeness can be satisfactorily addressed on this level, while matters such as rule soundness, rule completeness, and – to balance the examples with a non-semantic property – structural completeness, need the finer intermediate level as their habitat. On the other hand, such a question as “Are the axioms of Hilbert and Bernays’s system independent?”, which asks whether any axiom could be deleted and still be forthcoming as a theorem on the basis of the remaining axioms (and rules), obviously requires interpreting system in the first way, as simply meaning axiomatization. In the last two paragraphs, we have considered for Lemmon’s system in SetFmla and Hilbert and Bernays’s system in Fmla, three increasingly stringent ways of specifying what we have in mind as the system in question. It might appear superficially that it was the same threefold classification that was articulated for each of the two frameworks. This was not quite so, however. At the most discriminating level in the former case we placed the policy of individuating logics as proof systems, while in the latter case, occupying this role was not the proof system but a particular axiomatization. The remaining two levels did correspond. But clearly we can apply the proof system/axiomatization distinction in any logical framework, and once we have spelt this out, the original suggestion of a three-tiered structure can be replaced with one recognizing all four levels as potentially useful places for agreement at which to suffice for identity of logic. Since the term axiomatization is especially associated with Fmla, it seems preferable to offer either of the more neutral terms basis or presentation in our general account. Given our broad understanding of the concept of a rule, we can say that a collection R of rules is a basis for, or is a presentation of, a proof system S just in case S is the union of the rules in R. Thus the axiomatic presentation in 1.29 of the proof system Hil consisted of twelve zero-premiss


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rules (the axioms), one two-premiss rule (Modus Ponens) and a one-premiss rule (Uniform Substitution). The union of these various rules, which are relations of various arities on the set of sequents (according to Fmla) of the language with connectives ∧, ∨, →, ¬, is then a tuple system, which we take to be the proof system Hil. Similarly, the collection of rules with which we defined Nat in 1.23 is a presentation of that system in the sense that it is the tuple system arrived at by union from the collection. In fact, in our discussion of Hil and Nat (and Gen and NAT ) we sometimes spoke of these proof systems when the context made it clear that it was the particular presentation offered that was at issue. This involves an element of intensionality which it would have complicated the discussion excessively to have weeded out. For example, we felt happy to speak of the rules of Nat as for the most part introducing or eliminating a connective from the right of the “”. Now of course, strictly speaking, Nat is just a collection of tuples of sequents, not bearing any special subdivision into rules. Naturally, one understands such a reference to the rules of Nat as being to those rules Nat was defined as being the union of, and aside from the present discussion, in which we are being especially careful, we shall quite happily make remarks about proof systems that strictly apply to presentations of proof systems, when the remark in question is obviously to be understood with reference to a particular presentation (by which the proof system was defined) or else is not sensitive to the choice of proof system. While we are being careful, though, the general point is that though such a presentation uniquely determines a proof system, there is no unique path back from a proof system to any particular collection of rules of which it is the union. Sometimes, the choice amongst alternative collections is not worth attending to: witness the impatience in 1.24 over the question of whether to consider Nat as presented by (inter alia) two rules of ∨-Introduction, one from the left disjunct, and one from the right disjunct, or rather a single rule which was the union of these two. On the other hand, such matters as that of independence, raised above, are very sensitive to the particular choice of a basis, as we now illustrate. The general notion of independence is that of the independence of a rule in a collection of rules, since this subsumes the type of example (axioms) considered above given our recognition of zero-premiss rules. It should be understood as follows, where we write R for the union of the rules in the collection R of rules. (Thus R is the proof system presented by R; recall that an individual rule is itself a set of tuples of sequents.) A rule ρ ∈ R is independent in R when ρ is not derivable in (R {ρ}). Recall that what, more fully spelt out, this amounts to is that not every tuple in ρ belongs to (R {ρ}) . Thus for the case of axioms such as those of Hilbert and Bernays, we are concerned with ρ of the form { A} for each of twelve particular formulas A, and our definition counts such a ρ independent when if the axiom in question is deleted, that same formula is not forthcoming as a theorem on the basis of the remaining axioms. Similarly, a primitive non-zero-premiss rule fails to be independent in the basis for a proof system if deleting it from that basis results in a collection of rules on the basis of which it is a derived rule. For example, the presentation of Nat in Lemmon [1965a] included (for pedagogical reasons) the rule (¬¬I) which we saw in 1.23.5 (p. 121) to be a derived rule of the proof system whose basis did not include it: this rule therefore fails to count as independent. The notion of independence just characterized for rules coincides with the usual one, though a somewhat less usual notion does have some currency. Notice

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that a rule which fails to be independent in our sense can be deleted from the basis in which it figures without thereby affecting the stock of sequents (or formulas, as would more commonly be said by those working in Fmla) provable on the basis of the reduced basis: for the purposes of proving sequents, the rule is simply redundant. The converse does not hold for our notion of independence, while the other notion would just identify independence with redundancy of the sort described. (Porte [1960a], working in Fmla, distinguishes our sense as D-independence, with “D” for “deducibility”, and the weaker sense as Tindependence, with “T” for “theorem”. Or see Porte [1965], p. 36 and pp. 73–75.) The rule (T) in Gen, for example, is redundant in the collection of rules with which we presented that system, since Gen − , presented without its aid, proves the same sequents, but it is still (in our sense) an independent rule since, as noted in 1.27 (or Wang [1965]), it is not derivable in Gen − . We defined independence in terms of the derivability of the rule in question in the proof system from whose basis it was removed, while the weaker ‘redundancy’ concept would be got by speaking instead of admissibility. Notice that this distinction does not arise for zero-premiss rules, since for them derivability and admissibility coincide (and amount simply to the presence of a class of sequents amongst the derived objects of the tuple system concerned). Digression. We are concerned here with the independence of logical principles (in this or that logical framework), but it should be noted that there is also another notion of independence which applies to arbitrary sets of formulas, such as might be thought of as potential sets of axioms for non-logical theories, relative to a given consequence relation . (Thus the theory in question consists of the consequences by of the set chosen.) In this setting one describes a set Γ of formulas of the language of as independent when for all A ∈ Γ, Γ{A} A. There are several interesting results in the literature for this notion, addressing the question, for various choices of , of which sets Δ of formulas admit of an independent axiomatization (or ‘basis’): i.e., a set Γ with Cn (Γ) = Cn (Δ) with Γ independent in the sense just defined. For this topic the interested reader is referred to Wojtylak [1988], mainly with an eye to IL , but also surveying literature on the sometimes contrasting case of = CL . End of Digression. Returning to our preferred notion of independence, it would be as well here to point out why the notion cannot be characterized at the level of proof system. The obvious thought would be to define, for a proof system S and a rule ρ ⊆ S, ρ as independent when ρ is not derivable in the proof system S ρ. In other words, we take out from S all of the tuples belonging to ρ, giving rise to S ρ, and then see if we get them all back again in (S ρ) . The reason this does not work in general to characterize the independence of ρ (in the preferred—and customary—sense) is that in taking out the tuples belonging to ρ we may be removing tuples belonging to other rules used to present S. Thus, while what we were interested in was the derivability of ρ on the basis of the other rules, we have here potentially weakened those other rules to the extent that they overlap (as sets of applications) with ρ. We can illustrate the point simply with an artificial example. Take a language with one singulary connective, O, and one binary connective, . The logical framework for the example is Fmla, and to avoid clutter we omit the prefatory “”. Let ρ1 and ρ2 be the rules schematically indicated thus:

1.2. RULES AND PROOF (ρ1 )

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AB A

(ρ2 )

A OB A

Thus ρ1 is the set of pairs AB, A (A, B any formulas), and analogously for ρ2 . It is very clear that ρ2 is not independent in the collection R = {ρ1 , ρ2 }, since ρ2 is included in ρ1 : a rather extreme example of overlap. Our definition of independence directs us to ask: do we have ρ2 ⊆ (R {ρ2 }) , which to say: do we have ρ2 ⊆ ρ 1 ? As just noted, we have something much stronger, namely: ρ2 ⊆ ρ1 . The alternative suggested definition would direct our attention instead to the question: do we have ρ2 ⊆ (ρ1 ρ2 ) ? The tuple system ρ1 ρ2 consists of those pairs A B, A in which A is any formula, and B any formula not of the form OC. One may verify by induction on n that for no n does any pair A OB, A belong to (ρ1 ρ2 ) ; so this alternative definition returns the wrong verdict in this case. Although the complete inclusion of one rule in another is seldom encountered in any collection of rules by means of which a proof system is presented, the general feature of overlap is rather common. For example, suppose we counted the two forms of ∨-Introduction distinguished above as two rules. Then applications in which the inferred disjunction was of the form A ∨ A would be applications of both rules. Another example arises with the ‘axiom-schema’ variation on the presentation of Hil in 1.29, at least if we take this (and see the next paragraph for another option) to consist of the two-premiss rule Modus Ponens and twelve zero-premiss rules, one for each axiom-schema. For then the formula (p → p) → ((p → p) → (p → p)) or more accurately the corresponding sequent, is an application of the rule given by (HB1) in schematic form: A → (B → A) as well as of that given by (HB3): (A → B) → ((B → C) → (A → C)). The need for discussions of independence to locate themselves at the level of presentation of a proof system rather than at that of the proof system itself, in view of the fact that the former is not determined uniquely by the latter, arise even if the rules in the alternative presentations one considers are disjoint. To see this, consider the example just mentioned of the schematic version of our presentation of Hil. An alternative basis to that in terms of the thirteen rules just described would be to treat each instance of the twelve schemata as a separate zero-premiss rule. (The talk of an infinite set of axioms does not represent a commitment to this presentation, since we may group together all axioms of a given form under a single zero-premiss rule.) The sensitivity of questions of independence to this distinction over the choice of basis has been well illustrated by Dale [1983], where a presentation of classical truth-functional logic in → and ¬ in terms of schemata is discussed. The three schemata, which suffice (with Modus Ponens) for all tautologies in these connectives and which are due to Łukasiewicz, are those mentioned in 1.29.22 above, namely: A → (B → A) (A → (B → C)) → ((A → B) → (A → C)) (¬A → ¬B) → (B → A) no two of which share a common instance. What Dale shows is that if we think of these as three zero-premiss rules then each of them is independent, while if we think of them as representing infinitely many such rules (one for each axiom), then, rather strikingly, none of the rules in this infinite collection is independent.

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The example of independence indicates that there are some questions about systems of logic which can only sensibly be raised if system of logic is understood as meaning collection of rules. There are numerous other such examples, including the question of whether (e.g.) intuitionistic logic uniquely characterizes the connective ¬. A consideration of questions of this kind is deferred until §4.3, since it will require a slightly more abstract conception of rules than has been in play up to this point. In the meantime, it will be taken that we have justified adding a level (0) to the three levels of fineness in individuating logics with which we began: The logic presented by the collection of rules R for a language L is the same logic as that presented by rules R just in case: (0) R = R (individuation by presentation) (1) R = R (individuation by proof system) (2) ( R) = ( R ) (individuation by derivable rules) (3) ( R) ∩ seq(L) = ( R ) ∩ seq(L) (individuation by provable sequents) If there is here any level standing in need of special justification, it is level (1). We include it as a convenient intermediary between (0) and the later levels, the theory of tuple systems (0.26) providing a simple account of primitive and derived objects and relations. Since for each of the four understandings of what constitutes a logic provided by the above scheme, we identify a logic with a set of some kind, the scheme automatically provides us with four appropriate ways to understand “is a subsystem of” (or “is a sublogic of”): namely in terms of the relation ⊆; conversely for talk of one system’s being an extension of another. At level (0), one axiomatization (or more generally, presentation) is a subsystem of another if its rules (subsuming the axioms) are included in that other’s. At level (3), we trace the subsystem relation in terms of inclusion of provable sequents. And similarly for the two intermediate levels. Often, in reaction to the impoverished perspective afforded by identifying a logic in Fmla with its set of theorems, it is suggested that a logic should be taken as a consequence relation. (See the quotations from Dummett and Czelakowski in the notes, which begin on p. 188.) Here, a consequence relation is understood in the received sense as a closure operation on the language in which the logic is formulated, and not in the analogically extended sense as anything so exotic as a closure operation on a set of Set-Fmla sequents. There are two ways of seeing this reaction in the light of the four-level hierarchy of this section: (i)

We are to work in Set-Fmla rather than Fmla; logics are individuated in accordance with policy (3).

(ii)

We continue in Fmla but move from level (3) to level (2) for purposes of individuating logics.

Which construal is preferable depends on what else the proponent of the identification of logics with consequence relations does. For example, if explicitly formulated rules are all of the formula-to-formula type, then they are rules for the framework Fmla and construal (ii) is perhaps preferable. Note that


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how our terminology applies will be sensitive to which construal is made; for example, what is described in terms of rule-completeness on construal (ii) becomes completeness simpliciter on construal (i). The only point to be made here is that whichever construal is adopted, one ends up with a perspective insufficiently rich to do justice to the full range of phenomena that would figure in a full discussion of sentential logic. This is because (a) both construals leave us with only one consequence relation, while our treatment of Set-Fmla in 1.26 called for two (vertical and horizontal) and (b) the consequence relation it leaves us with relates only formulas (or sequents in Fmla) and not sequents in Set-Fmla (let alone Set-Set). What (a) means is that the distinction between rules of proof and rules of inference cannot be captured. (This distinction was mentioned at the end of 1.29, where we suggested that it does not suffice to resurrect within Fmla all the Set-Fmla distinctions one would like to see.) And (b) means that the distinction between the derivability and the mere admissibility of a rule like (→I) or Conditional Proof – from Γ, A B to Γ A → B – is completely lost. All we can say (in this instance) is that for any Γ, A, B, if Γ, A B then Γ A → B, where is the consequence relation concerned: which comes down to the admissibility of the rule. There is no way of indicating that Γ A → B is actually derivable from Γ, A B. Yet some record to the latter effect will be what we want if what we want of a logic is that it should codify the inferential dispositions of a population: one such disposition might be that of passing from the acceptance of an argument with A (inter alia) as premiss and B as conclusion to accepting the argument based on such other premisses as remain with a conditional conclusion having A as antecedent and B as consequent. For more in this vein, see the discussion following 4.34.2 (p. 594). None of this is to say that no good can come of identifying (à la construal (i) or (ii)) logics with consequence relations; it is just that not everything of significance for logical theory which remains invariant under changes in presentation can be captured by such an identification. If you move from Fmla to Set-Fmla (or even Set-Set) because, to put it in semantic terms, you are dissatisfied with just ‘churning out the logical truths’, you should note that you are now just ‘churning out’ the valid sequents. We close by noting that instead of explicitly alluding to sequent-to-sequent rules in speaking of logics, it is possible to obtain a similar effect by allowing logics (perhaps of some specific kind) to be collections of sequents, without allowing that every such collection counts as a logic (of the given kind). Rather, only such as are closed under the rules to which one wishes to give a special status, are to be considered. For example, in the framework Fmla, a normal modal logic (as in 2.21) might be defined to be a collection of formulas closed under Modus Ponens, Uniform Substitution, and Necessitation, and containing certain initially given formulas. The status of closure under Uniform Substitution in such lists of conditions for being a logic deserves a moment’s explicit attention. Such a condition may seem reasonable given that we are doing formal logic, or given that we want to treat the propositional variables as variables—as representatives of arbitrary formulas about which therefore, a logic should say nothing special. (For on the notion of formulas’ being treated in a “special” way by a logic, see 9.21 below.) The former sentiment may be found (e.g.) in Czelakowski [1984]; the latter is expressed in Stalnaker [1977]. On the other hand, the fact remains that many

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logicians have had occasion to explore systems which are not closed under U.S., and these explorations give every appearance of being perfectly continuous with the more usual (and more convenient) investigation of systems which are so closed. A list of some examples may be found in the notes on this section (p. 191). However, while in the cases listed it is true that the propositional variables receive a special treatment, in that provable sequents cease to be provable on replacing such variables uniformly by complex formulas (or constants like and ⊥), they are at least all treated in the same special way. This suggests a weaker condition than closure under Uniform Substitution, namely closure under uniform substitution of propositional variables (rather than arbitrary formulas) for propositional variables. We may notice that a restricted condition along these lines would be enough to retain the distinction, in connection with which the unrestricted closure condition is sometimes invoked, between propositional logics in Fmla and theories in a propositional language (Meyer [1976a]). Again, see the notes, under ‘Uniform Substitution’, p. 191.

Notes and References for §1.2 Logical Frameworks. We have contrasted (in 1.21) what we call logical frameworks ( = conceptions as to what kind of sequents the rules of a proof system are to manipulate) with approaches to logic ( = choices as to which such manipulations are to be employed). Logical frameworks departing more extensively than any we consider in the present work are on display in Gabbay [1996]. Warning: the phrase “logical framework” has acquired, in certain computer science circles, a rather different sense (as in Huet and Plotkin [1991]). The Framework Set-Fmla. I am grateful to Aubrey Townsend for the use made in this section (in the discussion following Example 1.23.1) of the distinction between arguments in the premisses-&-conclusion sense and arguments in the course-of-reasoning sense. This distinction can, incidentally, be used to deflect a criticism made by Shoesmith and Smiley, who say, at p. 106 of their [1978]: “Textbooks say that an argument is valid if the conclusion follows from the premisses, but this cannot be right”, going on to cite examples of arguments which draw their conclusions by means, inter alia, of clearly fallacious intermediate steps, which happen not to count against the final conclusion’s following from the initial premisses. The natural response is that the ‘textbooks’ were speaking of arguments in the premisses-&-conclusion sense (to which correspond formally sequents) and not in the course-of-reasoning sense (to which correspond formally proofs). The practice of using “” as a separator in sequents is inspired by Blamey [1986] or [2002]; cf. Blamey and Humberstone [1991]. Reasons for preferring it over “” and “→” (or the increasingly common “⇒”) are given, in effect, in the body of the section: one needs to avoid confusion with a symbol for a relation and also with a connective symbol. Dummett [1977] uses a colon (“:”) for this purpose, which is not bad, though perhaps lacking a suitable ‘directedness’; better in this respect is the symbol “” from such papers as Raftery [2006b]. Burgess and Humberstone [1987] follow Dummett’s example, though unaccountably referring to this (p. 198) as ‘semicolon notation’. (The use of the colon notation in Andreoli [1992] is quite different, separating two parts of the right-hand side of a one-sided sequent. Thus although the two parts are both


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multisets of formulas, one would not call the logical framework here in play Mset-Mset.) The terminology of a sequent’s holding on a valuation, first used here in 1.25, is taken from Bendall [1978]; the uncouth practice (on the part of some authors) of calling a sequent holding on a valuation true on that valuation encourages the confusion of sequents with formulas, with serious effects outside of the logical framework Fmla. (“What is the negation of p q ∨ r?”, etc.) However, we have no hesitation in saying that an equation holds in an algebra when it is true in that algebra for all values of the variables involved. The framework Fmla-Fmla or the equivalent (treating logics as binary relations between formulas instead of as consequence relations or gcr’s) figures prominently in, for example, Johansson [1953], Makinson [1973a], Goldblatt [1974a], Vakarelov [1989], Ardeshir and Ruitenburg [1998], Crolard [2001], and we shall employ it more than once in what follows (e.g. in 2.31). The style of natural deduction (namely Nat) which we have associated especially with Lemmon’s name had been used before Lemmon [1965a] by Suppes [1957] (though the treatment of quantifiers, not relevant to our concerns here, is very different in the two books); cf. also Mates [1965]. The style associated with Prawitz was the original style of Gentzen [1934]; Prawitz’s techniques for setting out proofs and for reasoning about such proofs are also employed in Tennant [1978]. This ‘tree presentation’ of natural deduction proofs (and likewise for sequent calculus proofs such as Example 1.27.5) should not be confused with the so-called tree (or the closely related, older, tableau) method of proof, as in Jeffrey [1967], (also Smullyan [1968]: see below; these proof techniques were originated by E. W. Beth), where the branching accommodates multiplicity of conclusions rather than of premisses. Such systems of proof continue to enjoy a certain popularity in textbooks – e.g., Priest [2008] – but are not discussed here. (In the author’s opinion they are not conducive to a clear-headed separation of proof-theoretic from semantic considerations.) Nor will attention be given to the ‘graph’ proofs of Shoesmith and Smiley [1978] or the ‘proof-nets’ of Girard [1987a] – for a survey of options surrounding which see de Oliveira and de Queiroz [2003]. Other graph-theoretic proof systems can be found in Hughes [2006], Restall [2007]; further use of diagrams in related connections appears in Urquhart [1974b]. Assumption-rigging in Nat is discussed in Coburn and Miller [1977]. As evidence for the remark in the text about Lemmon’s attitude to the proofs of intuitionistically unacceptable sequents, consider his comment at note 1 (p. 596) of Lemmon [1965b] on the availability in a certain natural deduction for predicate logic of a four line proof of ∃y(∃x(Fx) → Fy): “this (intuitionistically unacceptable) theorem ought to be hard to prove in predicate calculus”. A sequent-to-sequent formulation of natural deduction rules, with some useful asides, may be found in Dummett [1977], from p. 168 of which we quote the following expression of the sentiment mentioned in the text about the axiomatic approach in Fmla: The use, before Gentzen, of axiomatic formalizations of logic had other, unfortunate, effects. It was, in the first place, conceptually misleading. It diverted attention from the obvious fact that, whereas in a theory our concern is to establish true statements, and the derivation of statements from others is only a means to this end, in logic the process of derivation is itself the object of study. Axiomatic formalizations, unlike natural

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CHAPTER 1. ELEMENTS OF SENTENTIAL LOGIC deduction, tempt us to regard logic as a search for logical truths.

Similar sentiments are expressed on p. 434 of Dummett [1973], and the opening passages of Czelakowski [1984] defend a viewpoint according to which “logic is a set of valid inferences, not just valid formulas. The difference is essential: the notion of valid formula can be defined in terms of valid inferences but, in general, not vice versa.” According to the view of the present work, however, while the move from formulas to sequents is one step in the right direction, it does not go far enough: for many purposes, one needs to attend to precisely which sequent-to-sequent transitions are licensed by a collection of logical principles. (That is, to the ‘vertical’ rather than just the ‘horizontal’ notion of consequence, as this distinction is drawn in 1.26.) The comments in the text about how Lemmon’s system is to be adapted for minimal and intuitionistic logic appear in the preface to Lemmon [1965a]. Lemmon says nothing about the relation of his system to the topic of relevant logic, or about the role of assumption-rigging in his proofs. It is actually these features of the system, which we will draw on in subsequent discussion (2.33), that prompted its choice here over that of Prawitz [1965] as an exemplar of the natural deduction approach. It is customary to remark that the idea of natural deduction was also worked out independently of Gentzen by S. Jaśkowski, developing some ideas expounded in class by Łukasiewicz in the 1920’s. A later descendant of Jaśkowski’s approach is to be found in the natural deduction systems of F. B. Fitch; see Anderson and Belnap [1975] for further details, references, and applications. The Framework Set-Set. The fact that one has to move from Set-Fmla to Set-Set in order to secure sequent-definability for the various classes of #boolean valuations was noticed by Carnap, though he did not put it in this terminology. According to Carnap [1943], a ‘full formalization’ of truth-functional logic should exclude ‘non-normal interpretations’; see further the notes to §1.1 (Strong vs. Weak Claim: p. 101). In summary, whereas Gentzen [1934] was led to consider Set-Set for the provision of an elegant and proof-theoretically illuminating treatment of classical logic, and subsequent work (such as that of Scott) has shown that this framework, in view of the work it allows the purely structural rules to do and of its symmetry between left and right lends itself ideally to proofs of semantic completeness, it was the above considerations of definability that drove Carnap into the framework. Sequent Calculus. The term “sequent calculus” is sometimes (mis)used to cover any approach to logic in which sequents are manipulated by rules – e.g. in Kremer [1988], Bostock [1997], §7.2. It would be less confusing to use “sequentlogic” for such purposes (as in the title of Blamey and Humberstone [1991]), so that the former term is reserved – this being the standard usage – to cover proof systems whose non-structural rules all insert connectives (on the left or the right). Anderson and Belnap [1975] call sequent calculi “consecution-calculuses”; strictly speaking the plural of “calculus” should not be “calculi” – they are correct (it’s the wrong declension, in Latin) – but it seems pedantic to press the point and be lumbered with this mouthful. Gentzen (in translation) uses the terms antecedent and succedent for the material on the left and on the right, respectively, of the “”. To avoid any risk of confusion in the former case with


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the use of “antecedent” (vs. “consequent”) we speak instead of the left-hand side and the right-hand side of the sequent (with infrequent occasional use of “succedent” in the latter role). The relation between natural deduction and sequent calculus has been discussed by many authors, of whom we mention here only Negri and von Plato [2000], esp. Chapters 1, 8 and ‘Conclusion’; see also Negri and von Plato [2001]. Cut Elimination. There are good discussions of the Cut Elimination Theorem in Kleene [1952], Dummett [1977] and Chapter 5 of Girard et al. [1989], Chapters 4 and 5 of Troelstra and Schwichtenberg [1996], and Chapter 6 of Restall [2000]; see also Sanchis [1971], Ungar [1992]. A category-theoretic treatment of the topic is provided by Došen [1999]. Although we do not include a proof of the Cut Elimination theorem here for any of the various sequent calculus proof systems under consideration, one pertinent aspect of the situation is introduced for 2.33.27 at p. 365 below: what we call cut-inductive rules (or connectives, as governed by such rules). On different styles of rule in Set-Set, there is a good discussion in Segerberg [1982], §3.6; see especially the tables on pages 68 and 69, which display the main alternatives in a conveniently surveyable manner. (Segerberg’s discussion is not conducted along quite the lines of our own since he considers generalized consequence relations rather than logics in Set-Set, so that what are displayed are actually metalinguistic implicational statements about the -relation; however because these statements, not having disjunctive consequents, are strict Horn sentences of the metalanguage, they all ‘correspond to’ sequent-to-sequent rules in our sense.) See further, on this same topic, §0.3.5 of Došen [1999]. For a comparison of different sequent calculus systems, see Chapter 3 of Troelstra and Schwichtenberg [1996], Chapter 5 of Negri and von Plato [2000], and for a survey of various developments in proof theory, Buss [1998b]. Recent treatments inspired by developments in substructural logics can be found in Ciabattoni [2004] and Ciabattoni and Terui [2006]. Structural Completeness. Since Pogorzelski [1971], mentioned in 1.25, there have been many published discussions of this topic. We mention the following, noting that further references to the early literature may be found in the paper by Latocha: Prucnal [1972a,b], [1976], [1979], [1985]; Makinson [1976]; Latocha [1983]; Wojtylak [1983], [1991], §16.7 of Chagrov and Zakharyaschev [1997]. Note, however, that when we consider structural completeness in connection with a framework such as Set-Fmla, we have in mind the derivability of all substitution-invariant sequent-to-sequent rules for the framework, which is rather different from what the above references are concerned with. (The issue of structural completeness as a property of proof systems and as a property of consequence relations was touched on after 1.29.6; see further (2) in the Digression after 6.42.14 below, on p. 882.) The phrase “structural completeness” is used in an unrelated sense in Buszkowski [1988], p. 80, a usage taken up in several later discussions of logically oriented categorial grammar. Uniform Substitution. Let us repeat the reference, from the Appendix to §1.2, to Stalnaker [1977] for a defence of the requirement that logics be closed under Uniform Substitution. As was mentioned, many proposed logics do not in fact satisfy this condition, though they typically satisfy the weaker condition

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mentioned in the Appendix of being closed under variable-for-variable substitution. A stronger condition on substitutions – for which the condition on logics of closure under substitution would be correspondingly weaker – would restrict attention to invertible substitutions s, satisfying s(pi ) = s(pj ) ⇒ i = j, as is urged in Schurz [2005] and somewhat less explicitly, though for broadly similar reasons, in Smiley [1982].) A list of some examples of such non-substitutioninvariant logics was promised. They may be found in Prior [1967a], Chapter 7, §4; Cooper [1968]: we discuss this case in 7.19 below; Thomason [1970]; Åqvist [1973]; Cocchiarella [1974], especially p. 56 and the discussion in §6 of David Kaplan’s system S13 (though these are essentially variations on the theme of non-substitution-invariant modal logic à la Carnap – for discussion of which, along with numerous further references, see Schurz [2001] and [2005]; Davies and Humberstone [1980] (see further 9.11.3 below); Bunder [1979] and Diaz [1981], p. 72 and Chapters 7–8; Veltman [1981]; Burgess and Humberstone [1987]; Buss [1990]; Halpern [1996]; 8.35 below; Makinson [2005], esp. pp. 14–16; Silvestre [2006];; van Benthem [2006], p. 151. See also the reference on p. 99 of Humberstone [2002b] to a paper (not in our bibliography) by von Kutschera. There is an interesting philosophical discussion of Uniform Substitution in Smiley [1982], the gist of which is that the inclusion of such a rule as primitive (or derived) marks a misunderstanding of the nature of logic – the ‘schematic fallacy’, Smiley calls it – according to which symbols intended for use in talking about interpreted languages, come to have a life of their own as symbols in (formal) object languages. Thus the terminology is somewhat unfortunate since schematic presentations in the sense of the present work (and indeed generally) are precisely those in which, eschewing appeals to Uniform Substitution, traces of Smiley’s ‘schematic fallacy’ are least evident. (Some aspects of Smiley [1982] are further clarified in Oliver [2010].) Whatever the pros and cons of doing logic in the way Smiley opposes, the fact remains that no complete account of how logic has actually been done could omit discussion of Uniform Substitution. Further, the use of schematic symbols (rather than propositional variables, which Smiley calls schematic symbols) itself requires attention (in the metalanguage) to substitution, as when one says that any formula of the form A → (A → A) is a formula of the form A → (B → A). No doubt a discussion of these matters more careful than that attempted here would, still setting propositional variables to one side, distinguish between schematic letters which hold a place for object language formulas and metalinguistic variables which range over such formulas. Although in the Appendix we mention Meyer [1976a] as citing closure under Uniform Substitution as a useful point for distinguishing logics (in Fmla) from theories, in fact our suggested replacement for playing this role was introduced in another paper (and à propos of an unrelated topic) by Meyer published in the same year: Meyer [1976c] calls variable-for-variable substitutions strict substitutions; in Homič [2008], they are called simple substitutions. A less felicitous term was introduced in Belnap [1976], namely “variable identifications” – less felicitous, since as Belnap notes, such a substitution as applied to a given formula may not result in the identification of variables within that formula. We have applied the term “substitution-invariant” both to rules and to (generalized) consequence relations. It should be noted that consequence relations are sometimes called invariant when any substitution instance of a formula is a consequence of that formula (e.g., Suszko [1971c]), which is of course an entirely different notion from what we are calling substitution-invariance (which Suszko


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and others would call structurality – terminology we avoid for reasons given just prior to 1.25.5 on p. 129). Deduction Theorem. See for example Kleene [1952], and for additional reading, Pogorzelski [1968] and Porte [1982a]. The latter paper has an extensive bibliography. Many variations on the theme of the Deduction Theorem are treated in Anderson and Belnap [1975]. For a treatment from the perspective of category-theoretic approach to logic in the style of J. Lambek (as well as a general introduction to that approach), see Došen [1996]. Exercise 1.29.14 with its request to “prove or refute” the claim that the Deduction Theorem held for the system there under consideration was originally written in the expectation that the correct reply consisted in a proof, which expectation was corrected by Darius Lane (a student in the class to which this material was presented) who noted that the matrix in Figure 1.29b supplied instead a refutation of the claim (see the discussion following 1.29.18). Thanks also to Andy Smith, who used John Slaney’s MaGIC program to obtain this matrix for Lane (though we present it with the values differently labelled and arranged). The mistake the author had made was to overlook the need to insert the “strongly” before “consequent-distributive” in the formulation of 1.29.17. In the discussion (two paragraphs before 1.29.11) of B KW logic as intuitionistic implicational logic under another name, I am much indebted both to Darius Lane and to Allen Hazen. The fact that intermediate consequence relations (intermediate between IL and CL , that is) cannot be counted on to satisfy the Deduction Theorem is stressed in Rautenberg [1986]. See further Humberstone [2006a]. Logical Constants. In the Appendix to §1.2 we distinguished the question of what logic is from the question of what a logic is, and proceeded to address the latter question. (A similar passage, coincidentally, appears in the opening paragraph of Feferman [1989].) The former question is usually addressed in terms of the project of determining what makes something a piece of logical vocabulary – a ‘logical constant’ as it is often put, as opposed to belonging to the non-logical vocabulary (of an interpreted language; note that there is no connection with sentential or propositional constants, alias zero-place connectives, intended by this terminology). For those who wish to pursue this project, we cite some references from the mid 1970s onward: Peacocke [1976], Westerståhl [1976], Hacking [1979], McCarthy [1981], [1998], Schroeder-Heister [1984], Tarski [1986], Došen [1989b], Sher [1996], McGee [1996], Feferman [1999], Warmbr¯ od [1999], Hodes [2004], van Benthem [1989], [2002]. The last of these papers is particularly thoughtful, as is the very well-organized critical survey article on this topic, MacFarlane [2005]. The closest we come to touching on the subject of what makes something an item of logical vocabulary, aside from the discussion of unique characterization by rules (§4.3) which has sometimes been thought to be peculiarly appropriate in this connection, is the allusion to the ‘topic-neutral’ account of such vocabulary which lies behind the joke from Woody Allen in the quotation at the beginning of Chapter 5 below (p. 631).

Chapter 2

A Survey of Sentential Logic §2.1 MANY-VALUED LOGIC AND ALGEBRAIC SEMANTICS 2.11

Many-Valued Logic: Matrices

There are two related respects in which our treatment of semantic matters up to this point might be generalized. As has been remarked, our boolean valuations are homomorphisms from the language on which we concentrated in §1.1 (with connectives ∧, ∨, ¬, , ⊥) to the two-element boolean algebra. This suggests that we might consider a notion of validity tailored to an arbitrary boolean algebra, or indeed to an algebra of any other sort of the same similarity type as the language under consideration (we need this if homomorphisms into the algebra are to play the role of valuations). This project we call algebraic semantics. In fact, several projects are involved under this general rubric, and they will be considered in 2.13–2.16. The other respect in which what we have done may appear as a special case which cries out for generalization is that our valuations (boolean or otherwise) are functions mapping formulas to the twoelement set {T, F}: but what is so special about the number two? Why not allow more than two truth-values, and liberalize the idea of truth-functionality accordingly? This project is usually called many-valued logic, and it will be our topic in the present subsection. The two projects are intimately related because in specifying the truth-functions, in the newly liberalized sense, with which our connectives are to be associated, we are specifying operations in an algebra whose universe is the set of (truth-)values. Indeed, as we shall note in 2.13, what was just characterized in very rough terms as algebraic semantics is sometimes regarded as a special case of the style of many-valued semantics with which we are now concerned. (We shall also see, in 2.14 and 2.15, kinds of algebraic semantics which are not a special cases in this way.) In the 1920s and 1930s, interest in many-valued logic arose from two sources, one technical and one philosophical. On the technical front, the use of manyvalued truth-tables provided a useful device in proofs of the independence of axioms in axiomatic (or ‘Hilbert’) systems such as Hil in 1.29. Such a use, which also applies outside Fmla, in no way represents a repudiation of the idea that there are two and only two genuine truth-values – the principle of bivalence – and 195

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CHAPTER 2. A SURVEY OF SENTENTIAL LOGIC

there is something therefore misleading in speaking in this connection of manyvalued semantics at all. We shall illustrate this method after we have at our disposal the key notion of a ‘matrix’, showing (2.11.3, p. 202) that axiom-schema (HB12) of Hil is independent of the other schemata, in the sense of having some instances that cannot be proved from the axioms instantiating (HB1)–(HB11). (This use of matrices to establish unprovability was foreshadowed with several examples in 1.29.) On the philosophical front, a variety of qualms were aired with the presumption, apparently needed for the application of formal logic to logical questions arising in connection with interpreted languages, that the statements of such languages admitted of a two-fold partition into the true and the false. So motivated, the use of many-valued logic is to be taken as (at least toying with) a repudiation of the principle of bivalence. Łukasiewicz, the pioneer here, was in particular concerned that this principle committed one to an unacceptable fatalism or ‘logical determinism’ if one took seriously the thought that it was (for example) either true that one would be in Dublin on January 1, 2030, or false that one would be there then. (See Łukasiewicz [1922].) He thought that, if a is an agent capable of freely deciding such matters as those involved in travel plans, then the statement that a is/will be in Dublin on the date in question is not yet either true or false, but has some third status – indeterminate, say. With the passage of time, such indeterminate statements typically resolve themselves into the true and the false, depending on how the agents concerned act, but (thought Łukasiewicz) to assign them to either of these two ‘determinate’ categories prematurely is to deny that these agents can have any freedom in respect of the actions on which the truth-values of the statements depend. Let us not concern ourselves here with the merits of the claim that bivalence threatens freedom in this way (see further 6.22, 6.23) and just have a look at how Łukasiewicz,who assumed this was so, reacted. Toward the end of this subsection, we will indicate (with some quotations from M. Dummett) the possibility of a philosophically motivated use of many-valued logic which need not involve a repudiation of bivalence; because of considerations like this, the phrase many-valued logic is felt by some to be an embarrassment, better replaced by talk of ‘matrix methodology’ (cf. the works of Wójcicki in our bibliography). When later, in the discussion preceding 2.12.3, we come to define ‘many-valued’ – or k-valued for some k – the case of k = 2 is not excluded (and nor is the case of k infinite, the inappropriateness of the “k” notation for this case notwithstanding). The truth-tables from, e.g., Łukasiewicz [1920], [1930], in the three values (we shall call) 1, 2, and 3, for “(determinately) true”, “indeterminate”, and “(determinately) false”, respectively, appear here as Figure 2.11a; the name Ł3 is given both to the matrix there depicted and to the logic determined by that matrix. (This terminology is explained presently.) Historically, these have sometimes been called by their initials T, I, and F, but we want to avoid the use of “T” and “F” for a reason which will become clear after 2.11.10 below. These values have also been called 1, 1/2, and 0, prompting Łukasiewicz, his collaborators and successors to investigate, in an experimental spirit, the possibility of taking as truth-values all the rational numbers, and even all the real numbers, between 0 and 1 as truth-values, with truth-functions in a certain sense analogous to those given for the three-valued case here. To make the analogy explicit, we illustrate with another finite case, the six-element Łukasiewicz matrix, with equi-distant

2.1. MANY-VALUED LOGIC AND ALGEBRAIC SEMANTICS

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rational numbers 1 (the ‘designated’ value – a notion explained below) alongside 4/5,3/5,2/5,1/5, and 0 (the five ‘undesignated’ values). Conjunction and disjunction are computed using the standard arithmetical minimum and maximum operations while for formulas ¬A and A → B the values are 1−a and min(1, 1−a+b), where a and b are the values of A and B. (Here min is again just the usual minimum operation.) These same instructions are used for calculating values in all the finite cases as well as in the infinite cases. In what follows, we discuss only the three-valued case. For references to the full range of logics, see the end-of-section notes. Digression. All of Łukasiewicz’s logics are extensions of BCK logic – see 2.13 and 7.25 below – and their implicational fragments also contain the formula called Q in 2.13 (p. 237) and the interesting formula (Ł): (Ł)

((q → p) → (p → q)) → (p → q).

traditionally used in axiomatizing the weakest Łukasiewicz logic, now usually called Łω , but shown to be redundant in Meredith [1958] and Chang [1958], (see also Slaney [2002]) given the axioms B, C, K, and Q and an axiom involving negation (¬q → ¬p) → (p → q). (In fact the original axiomatization, due to Wajsberg, used B , K and Q, in place of B, C, K, and Q. See the list following 2.13.5 (p. 228) for the formulas to which these labels refer, except Q, mentioned after 2.13.16: p. 237. See also the Digression on p. 335 below.) Wajsberg axiomatized the logic with which are currently concerned, Ł3 , by taking B , K, the above axiom for negation, and the following special case of the Fmla form of Peirce’s Law: ((p → ¬p) → p) → p. In the axiomatizations mentioned here the rules were Uniform Substitution and Modus Ponens. The absence of any mention of ∨ and ∧, appearing in Figure 2.11a below, is accounted for by the fact that Łukasiewicz took these as defined rather than primitive. In the case of ∨, A ∨ B was defined to be (A → B) → B, from which ∧ can be defined De Morgan style. A detailed description of the other finite matrices, yielding the logics Łn , may be found in Urquhart [2001], as well as of the infinite matrix yielding Łω , and the fact that the latter logic, conceived as a set of formulas, is the intersection of all the different Łn for n ∈ ω. For a result to the effect that every consistent extension of Łω is one of these Łn , see Komori [1978a], at least if we interpret these labels as denoting the respective implicational fragments of the logics concerned.) The fact that Łω is weaker than all the Łn should not invite the conclusion that Łn is weaker than Łm whenever m > n: something that is far from being the case, as is made clear in the discussion of these logics in any of the references mentioned above or in the end-of-section notes, under ‘Łukasiewicz’s Logics’ (p. 271). End of Digression. More recently (‘fuzzy logic’), such values have been urged as representing ‘degrees of truth’ intermediate between definite falsity and definite truth for the sentences of a language containing vague expressions: see the notes to this section (p. 268). The connectives with which we are concerned in our presentation of three-valued logic are conjunction, disjunction, negation, and implication, for which we use the symbols with which we are familiar. The first three tables of Figure 2.11a are to be understood like this: a compound with one of the binary connectives here as the main connective is to have the value marked in the cell at which a row and column intersect, the row


198 ∧ *1 2 3

1 1 2 3

2 2 2 3

3 3 3 3

∨ 1 2 3

1 1 1 1

2 1 2 2

3 1 2 3

→ 1 2 3

1 1 1 1

2 2 1 1

3 3 2 1

¬ 1 2 3

3 2 1

Figure 2.11a: Łukasiewicz’s 3-Valued Tables

being chosen in accordance with the first component formula’s truth-value and the column in accordance with that of the second. In the case of the singulary connective ¬ here (and in similar tables elsewhere) we have kept the format of putting the ‘input’ value in the first column, under the connective, with the ‘output’ value to its right. Admittedly, there is a clash of expectations caused by such truth-table presentations as appear in Figures 1.18a,b. Following the latter pattern, one would expect instead – imagining the second column in the ¬ table headed with a formula – that the value written under ¬ would be that of the negation of that formula. We either have this clash or else the clash with the remaining tables of Figure 2.11a itself. The present choice has been to avoid the latter. In some cases, the reasons for the entries in these tables are very clear. For example, if A has value 2 and B has value 1, then as we see from the table for ∧, second row, first column, A ∧ B takes the value 2. This is reasonable since if we are thinking of B as something whose truth is settled and A as something which, depending on how free agents influence the course of events, could go either way, then the conjunction of A with B ought to be classified as something which could also go either way, since which way it goes depends on which way A goes. But other cases, in particular the central boxes in all the tables for the binary connectives, are problematic, and it does not take much to see that there is actually no way of filling out these positions which both respects the intended meanings of the connectives (∧ behaving like and, etc.) and the motivating account, in terms of determinacy and indeterminacy, of the truth-values. These connectives just aren’t functional in those values. For this reason, further elaborated in the notes to this section (‘Philosophy’), Łukasiewicz’s three-valued logic cannot do the job it was designed to do, which is not to say that the tables offered are without interest for other purposes. Note that those for ∧ and ∨ can be more economically described by saying that the functions in question are the meet and join operations in the three element chain with 1 > 2 > 3. And what more natural first step in considering variations on the two-element chain can be envisaged than looking to the three-element chain? (Here we are just getting into the mood for three-valued logic; rather than endorsing the sentiment expressed. The discussion below of the valuations vh will reveal this enthusiastic attempt to generalize from the two-valued case to be somewhat superficial.) While it is in keeping with ordinary usage to refer to any one of the tables displayed in Fig. 2.11a as a matrix, in fact the convention has grown up in work on many-valued logic to reserve this term for something very like the totality of the tables, or the functions there represented, for all the connectives under consideration together. Something ‘very like’, but not quite. For what we have


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so far specified is an algebra (in the sense of 0.21) with universe {1, 2, 3} and operations of arities 2, 2, 2, and 1 corresponding to the four connectives. We need to specify something more to turn this algebra into a matrix, which will then be a complete piece of apparatus for the semantic description, in many-valued terms, of formulas and sequents. The missing ingredient is the information as to which of the values is/are to be regarded as playing the role played by the value T in the bivalent account of a notion of validity. We need to know, that is, which set of values is such that regardless of how values are assigned to the propositional variables in a formula, that formula takes on a value in the set before it counts as a ‘(three-valued) Łukasiewicz-tautology’ – or Ł3 -tautology, for short (see the above Digression) – and is such that a sequent of Set-Fmla is to be regarded as valid if it always preserves possession of such a value from the left to the right of the “”, and so on. (Normally we reserve all talk of tautologousness for the bivalent boolean case: this “Ł” prefixed usage will be the sole exception.) These will be called the designated values of the matrix. In Łukasiewicz’s case, the only designated value was 1; thus for example, while p → p emerges as an Ł3 -tautology, p∨¬p does not, since this formula takes on an undesignated value (namely 2) when p has that value. (This was in accordance with Łukasiewicz’s wishes, since it is in the form of the law of excluded middle that classical logic most directly exposes its fatalistic presumptions, he thought; see further 6.22, 6.23.) A common practice, which we have followed for Fig. 2.11a, in setting out many-valued tables is to indicate by an asterisk at their first appearance in the display those values which are to be taken as designated. We can wrap all of this up into a neat package, by giving the following general definitions, which abstract from the specific features of Łukasiewicz’s three-valued logic. A matrix for a language L is a pair (A, D) in which A is an algebra (see 0.21) of the same similarity type as the language L, whose universe A we call the set of values, and D is a subset of A, whose elements are called the designated values. If M = (A, D) is a matrix then by an M-evaluation for L is meant a homomorphism from L to A. (Reminder : when discussing languages we generally suppress the distinction between what would otherwise appear as L—the algebra of formulas—and L, the universe of this algebra, writing simply “L” for both.) If h is such a homomorphism and Γ ⊆ L we will write h(Γ) for {h(B) | B ∈ Γ}. A Set-Set sequent Γ Δ of L holds on an M-evaluation h just in case: (1)

h(Γ) ⊆ D ⇒ h(Δ) ∩ D = ∅.

In the context of discussing a particular matrix M we will usually just say “evaluation” rather than “M-evaluation”; we avoid the word “valuation” – reserved for the bivalent case as in 1.12 – so that this term is free for use below in some comparative remarks. (We will be introducing the concept of the valuation vh induced by a matrix evaluation h.) Since Set-Fmla and Fmla are specializations of Set-Set, we take the above definition to dictate what it is for a sequent of either of those frameworks to hold on an evaluation. Thus a sequent of Set-Fmla, for example, holds on an evaluation unless under that evaluation all formulas on the left of the “” receive designated values, while the formula on the right receives an undesignated value; this we record as (2) below, for the case of the sequent Γ C, whose consequent gives the situation for a sequent ( C) of Fmla to hold on h (in M): (2) h(Γ) ⊆ D ⇒ h(C) ∈ D.

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Finally, a sequent is valid in the matrix M if it holds on every M-evaluation. This is our desired notion of validity (though in 2.12 we will consider validity over a class of matrices). In a matrix whose algebra has a natural linear ordering, such as those considered by Łukasiewicz there is another notion of validity based on preserving the ‘height’ of elements w.r.t. that ordering; we consider this notion under the heading “-based semantics” in 2.14. The Łukasiewicz manyvalued matrices (from amongst which we are concentrating on the 3-valued matrix) are treated from this point of view in Scott [1974b]; for much information on this topic, see Font, Gil, Torrens an Verdú [2006]. The matrix consequences of ∅ coincide on either conception of validity, and since Łuksiewicz worked in Fmla, each approach qualifies as a generalization of his practice to the more general frameworks. Instead of defining a tautologous sequent to be one holding (in the sense of 1.25) on every boolean valuation, we could equivalently offer a characterization in terms of the matrix based on the two-element boolean algebra, with its 1 as the only designated value: a tautologous sequent is one which holds (in the current sense) on every evaluation in this matrix. Remarks 2.11.1(i) There is a slight oversimplification here: we cannot insist that the algebra is in this narrow sense a boolean algebra (A, ∧, ∨, ¬, 1, 0), since the requirement on being a matrix for a language of having an algebra of the same similarity type as that language would then exclude most of the cases we wish to discuss. For the language with connectives ∧, ∨, →, and ¬ we mean by a boolean matrix a matrix with algebra (A, ∧, ∨, →, ¬) of type 2, 2, 2, 1 in which (A, ∧, ∨, ¬) is the reduct to the listed operations of a boolean algebra in the official sense – introduced for 0.21.4 – and further, for all a, b ∈ A, a → b = ¬a ∨ b. (More generally, we can use any algebra which is, in the sense suggested by the discussion of 0.22, -compositionally equivalent to a boolean algebra.) The two-element boolean matrix for a language for all of whose connectives # §1.1 defined a notion of #-booleanness (pp. 65 and 83) has operations defined on the reduct to those operations of the two-element boolean algebra. (ii ) A matrix validating all the sequents of a logic is said to be a matrix for that logic. Taking logics as consequence relations or gcr’s this amounts to saying, in terms of a notation introduced below, that M is a matrix for when ⊆ |=M . (See the discussion following 2.11.4 for |=M .) The following terminology, introduced in Rasiowa [1974] has considerable currency: given an algebra A of the same similarity type as the language of a set D ⊆ A is a -filter when (A, D) is a matrix for . The explanation for this terminology is that in many cases in which A is the expansion of a lattice by additional operations, for some famous choices of e.g., = CL , the latter understood as denoting the consequence relation of classical logic, these coincide with filters on A in the sense of 0.23.1(ii) (see p. 27). The Ł3 -tautologies are formulas A whose corresponding sequents A are valid in the Łukasiewicz matrix described above. (As usual, we will not fuss excessively, in what follows, to distinguish sets or properties of sequents in Fmla with sets or properties of formulas.) Interestingly, Łukasiewicz worked only in


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Fmla and did not consider Set-Fmla. This may help account for a respect in which the table above for → differs from the others: it describes an operation under which the subset {2} is not closed. In fact the tables of Figure 2.11a differ from some offered in 1938 by Kleene (see Kleene [1952], §64) for the semantics of discourse involving partial functions only in that, if we use the same symbol for the connective and the associated truth-function, in the (so-called ‘strong’) Kleene tables we have 2 → 2 = 2. If Łukasiewicz had made this choice, then he would have no three-valued logic to speak of: there are no sequents in Fmla valid in the resulting matrix, since one can always assign 2 to every formula (because of the above-noted closedness of {2} under all the operations) and hence get an evaluation on which the sequent A does not hold. Thus there are no Ł3 -tautologies all of whose connectives are drawn from the set {∧, ∨, ¬}, for the same reason as there are no classical two-valued tautologies whose only connectives are in {∧, ∨}. Had Łukasiewicz worked in, for example, Set-Fmla instead, then the Kleene choice on → would not have left him without any logic, since plenty of sequents in this framework are valid in the Kleene three-valued matrix. In terminology introduced in 2.11.6 below, we would say that such a choice would have led to an atheorematic logic. (Other considerations may also then. . . may seem less fatalistic be relevant: p → p, read with “→” as if in its upshot than p ∨ ¬p.) In fact, we have yet to specify which values were to be taken as designated, but the point just made is correct for each of the two choices {1}, and {1, 2} we might make. Let us call the Kleene matrices resulting from making these two choices K1 and K1,2 , respectively. (As we have said, the tables describing the algebra of these matrices are sometimes called the strong Kleene tables, and of the two matrices it is usually only K1 that is associated with Kleene’s name; we have no cause in this chapter to discuss the ‘weak’ Kleene tables, though they will receive brief mention in 7.19.10. The latter tables, with 1 designated, constitute a matrix more commonly associated with D. Bochvar, discussed for example in Rescher [1969] and Urquhart [2001]. K1,2 has attracted the attention of paraconsistent logicians – see 8.13 – with the idea that the values 1, 2, 3 represent “true and not false”, “true and also false”, and “false and not true” respectively. See further 7.19.10, p. 1050.) Exercise 2.11.2 (i) Show that the sequent p ∧¬p q is valid in K1 but not in K1,2 whereas for the sequent p q ∨¬q it is the other way around. Check that the sequent p ∧ ¬p q ∨ ¬q is valid in both matrices. (ii ) Give an example of a sequent (of Set-Fmla) which is valid in Łukasiewicz’s three-element matrix but not in K1 , and an example of a sequent valid in K1 but not in Łukasiewicz’s three-element matrix. (iii) Call a matrix (A, D) degenerate if either D = ∅ or D = A. In Fmla the sequents valid in a degenerate matrix with no designated values is empty. What happens if there are instead no undesignated values? Give a syntactic description of the sequents of Set-Fmla and Set-Set valid in degenerate matrices of these two kinds. Include a reference to the gcr’s baptized No and Yes in 1.19 (p. 91). (iv ) Is the contraction formula (p → (p → q)) → (p → q) (= W in the list after 2.13.5 on p. 228 below) valid in Łukasiewicz’s three-element matrix? What about the validity of the two sequents: p → (p → q) p → q

p → (p → q), p q

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CHAPTER 2. A SURVEY OF SENTENTIAL LOGIC in this same matrix? What do your answers reveal about the natural deduction rule (→I), or the sequent calculus rule (→ Right) for Łukasiewicz’s three-valued logic? Now check the validity of the formula (p → (p → (p → q))) → (p → (p → q)) in this matrix. (See the discussion after 1.29.10, p. 168.) (v) Devise a suitable notion of submatrix for which the two-valued boolean matrix is isomorphic (identifying T and F with 1 and 3) with a submatrix of K1 and also of K1,2 and from which it follows that any Set-Fmla (or indeed Set-Set) sequent which is classically tautologous is K1 -valid and also K1,2 -valid. (A solution appears in 4.35.4, p. 598, and preceding discussion.) (vi ) Show that, by contrast with the case of Set-Fmla (cf. part (i) of this exercise), in Fmla the two-element boolean matrix and the matrix K1,2 yield the same logic. (Suggestion: the preceding exercise gives us that if A is valid in K1,2 , A is a classical tautology. For the converse, show by induction on the complexity of A, where h is the unique evaluation differing on the propositional variables pi from a K1,2 -evaluation h by having h (pi ) = 1 whenever h(pi ) = 2, that whenever h(A) ∈ {1, 3}, h (A) = h(A). We obtain the requested result because if A is not K1,2 valid, we have h(A) = 3 for some h, which can be adjusted to h with h (A) = 3 which is a renotated version of a bivalent valuation assigning F to A.) (vii) Suppose we consider a matrix like Łukasiewicz’s three-element matrix except that the values 1 and 2 are designated (as in K1,2 ). Are the classical two-valued tautologies exactly the formulas valid in this matrix, and if not, in which direction does the inclusion fail? (Assistance and some interesting historical information on this question may be found in Rescher [1969], p. 27, mid-page; the same example – due to Turquette – can also be found on p. 90 of Kijania-Placek [2002], along with a reference to a generalization by Michael Byrd of the point at issue.) (viii) Show that for any two matrices M1 = (A, D1 ), M2 = (A, D2 ) on the same algebra, if a sequent σ of Set-Fmla (and a fortiori of Fmla) is valid in M1 and also in M2 , then σ is valid in the matrix (A, D1 ∩ D2 ). (ix ) Would the claim with which (viii) is concerned be correct for sequents of Set-Set?

Example 2.11.3 For the independence proof mentioned in the second paragraph of this subsection, we introduce a fourth three-valued matrix, from Gödel [1932], with the following tables for → and ¬: → *1 2 3

1 1 1 1

2 2 1 1

3 3 3 1

¬ 1 2 3

3 3 1

and for ∧ and ∨, the same tables as Łukasiewicz and Kleene provide. (This matrix was considered earlier by Heyting and later by Jaśkowski;


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in each case the interest was in connection with intuitionistic logic. See Heyting [1930] – esp. p. 326 of the translation in Mancosu [1998].) Rather laboriously, one may check that all formulas instantiating (HB1)–(HB11) are valid in this matrix, and that the rule Modus Ponens preserves this property, whereas the instance ¬¬p → p of (HB12) fails on an evaluation h with h(p) = 2. (The algebra of this matrix, with values 1 and 3 as unit and zero, is a Heyting algebra in the sense of 0.21.6 and surrounding discussion (p. 22). For an interesting application, see the Digression on p. 313, as well as in the discussion following 2.32.11 (p. 319), where it is pointed out that the above matrix is a description of the two-element Kripke frame for intuitionistic logic in the connectives ∧ and ¬. The values 1, 2, and 3 represent the set comprising both points, just the second, and the empty set, respectively; there is no representation of the set containing just the first element because this is not closed under the accessibility relation and so cannot be thought of as a ‘persistent’ proposition. See 8.21.12 (p. 1221) for the fact that the {∧, ¬}-fragment of IL in Set-Fmla is in fact determined by this single matrix.) We turn to a more general observation concerning validity in a matrix; as with several of the results stated in this chapter, for reasons of space, we omit the proof. Observation 2.11.4 For any matrix M, the set of sequents (of Set-Set, SetFmla, or Fmla) valid in M is closed under Uniform Substitution. Sometimes a formulation in terms of consequence relations (or gcr’s) is more convenient than one in terms of classes of sequents of Set-Fmla (or Set-Set): cf. 1.21.3 on p. 109. We define the generalized consequence relation determined by M, denoted by |=M , to be that relation standing between (possibly infinite) Γ and Δ when for every M-evaluation h, the condition (1) above, repeated here for convenience, is satisfied: (1) h(Γ) ⊆ D ⇒ h(Δ) ∩ D = ∅ We use the same notation (i.e., “|=M ”) for the consequence relation determined by M, understanding this in the same way except that attention is restricted to the case in which Δ is {B} for some formula B. This terminology gives us a convenient way of speaking of logics in Set-Set and Set-Fmla which does not prejudge questions of compactness and also builds in closure of the corresponding set of sequents under the framework-appropriate forms of (R), (M) and (T). We want to build these in because the set of sequents valid in any given matrix (indeed holding on any given matrix evaluation) is always so closed. For Fmla these structural rules have no application, and nor do considerations of compactness (finitariness), for which reason there is here no distinction analogous to that between a consequence relation and a collection of Set-Fmla sequents: all we have is a set of formulas (or: set of sequents in Fmla). A slight extension of the claim of 2.11.4 (since infinite sets of formulas are allowed), which is no less obviously correct, would be the following: Observation 2.11.4 (Second Version) For any matrix M, the (generalized) consequence relation |=M is substitution-invariant.

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In Fmla, closure under Uniform Substitution is not only necessary but sufficient for a set of sequents (= formulas) to comprise precisely those valid in some matrix, or, as we shall put it, to be determined by some matrix, as was first observed by A. Lindenbaum: Theorem 2.11.5 Let S be any set of formulas in a language L. Then a necessary and sufficient condition for it to be the case that for some matrix M for L, S consists of precisely the formulas valid in M, is that S is closed under Uniform Substitution. Proof. Necessity: by 2.11.4. Sufficiency: Given S closed under Uniform Substitution, we supply a special matrix M, called the Lindenbaum matrix for S, with the desired properties. The first component of M is the algebra L of formulas of L, the second component, specifying the designated values, consists of S itself. We now claim that S is determined by M, as required, a claim which has a ‘soundness’ half: (1) that every formula in S is valid in M; and a ‘completeness’ half: (2) that every formula valid in M belongs to S. To show (1), suppose B ∈ S and suppose for some M-evaluation h, we have h(B) undesignated, i.e., h(B) ∈ / S. Here h is a homomorphism with the same source and target, which is to say, an endomorphism L, i.e., a substitution. Thus h(B) is actually a substitution instance of B, so we cannot have h(B) ∈ / S if S is to be closed under Uniform Substitution. To show (2), suppose B valid in M; then for every M-evaluation h, h(B) is designated, i.e., h(B) ∈ S. Now the identity mapping is one such h; therefore B ∈ S. As a first guess, one might suppose that an analogous result can be obtained for Set-Fmla and Set-Set. That is, one might expect that for any substitution-invariant consequence relation on L, there is some matrix M with = |=M . That this first guess would be wrong may most straightforwardly be shown with an example from Smiley [1962a], note 7, which uses a certain ‘logic without tautologies’ developed by Suszko: Example 2.11.6 Let be the relation of tautological consequence (alias CL ) on the language with (for definiteness: the choice does not affect the point) ∧, ∨, ¬, →. Define Suszko by: Γ Suszko A iff Γ A and Γ = ∅, for all Γ, A. So defined, Suszko is a substitution-invariant consequence relation. Note that we do not have Suszko q ∨ ¬q, for this relation since (by the nonemptiness condition) we do not have Suszko A for any formula A. Thus if Suszko = |=M for M = (A, D) there is some M-evaluation h with h(q ∨ ¬q) ∈ / D. Now consider the fact that there exists also an Mevaluation h with h (p) ∈ D: for otherwise we should have, as we do not, p Suszko B for every formula B. Then, since the formulas q ∨ ¬q and p do not have any propositional variables in common, we can, by 2.11.7(i) below, ‘amalgamate’ h and h to give an evaluation which agrees with them on each of their respective verdicts on these two formulas. On


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this evaluation (on any such evaluation, that is), h call it, we therefore have h (p) ∈ D and h (q ∨ ¬q) ∈ / D, so p M q ∨ ¬q. Yet from the way Suszko was defined, we do have p Suszko q ∨ ¬q. So Suszko = |=M after all. Disregarding the syntactical connotations of the word theorem, we describe a consequence relation (or any proof system of which it is the associated consequence relation) as atheorematic when for no formula A do we have A. (These are called ‘purely inferential’ consequence relations in Wójcicki [1988].) There is no harm in extending this terminology to the case of gcr’s. Suszko is, then, an example of an atheorematic consequence relation. Exercise 2.11.7 (i) Show that if h1 and h2 are M-evaluations, for some matrix M, and no propositional variable occurs in each of two formulas A1 and A2 , then there exists an M-evaluation h3 with h3 (A1 ) = h1 (A1 ) and h3 (A2 ) = h2 (A2 ). Suppressing the distinction between matrix evaluations and the induced bivalent valuations (h vs. vh ) introduced below (just before 2.11.9), this has the consequence – exploited in 6.41.2 below (p. 862) à propos of Halldén-completeness – that for any boolean valuations u and v and formulas A, B, composed using connectives # for which the notation of a #-boolean valuation is defined, if A and B have no propositional variables in, common then there is a boolean valuation w for which w(A) = u(A) and w(B) = v(B). (ii ) Show that Suszko = Log(BV ∪ {vF }) (we use the terminology of §1.1 here). (iii) Let be a substitution-invariant consequence relation, let S = Cn (∅), and let Lind(S) be the consequence relation determined by the Lindenbaum matrix for S. Show that is structurally complete if and only if = Lind(S) . (Just unpack the definitions of the terms involved. Structural completeness was defined for consequence relations just before 1.29.6, which appeared on p. 162. The present result is Theorem 3 of Tokarz [1973].) A second approximation toward characterizing the consequence relations determined by matrices would be one taking into account 2.11.7(i). Again this suggestion (from Łoś and Suszko [1958]) is not quite right, but it will require only a minor emendation. If is a consequence relation (on a language L), call a set Γ( ⊆ L) -inconsistent if Γ A for every formula A (of L); otherwise Γ is -consistent. Our second stab runs: a consequence relation is |=M for some matrix M iff (a) is invariant and (b) for all Γ, Θ, B: (**)

If Γ, Θ B and Θ is -consistent and no propositional variable occurring in any formula in Θ occurs in any formula in Γ ∪ {B}, then Γ B.

While this can be shown to be equivalent to the corrected form in terms of (***) below, taken from Shoesmith and Smiley [1971], in the special case in which is a finitary consequence relation, for the general case one needs to allow for the possibility of various different sets playing the role of Θ in (**). We state here, without giving a proof (using 2.11.7(i)), the suitably adjusted version of this characterization.

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Theorem 2.11.8 Let be a consequence relation on a language L. Then a necessary and sufficient condition for it to be the case that for some matrix M for L, = |=M that both (a) is substitution-invariant and (b) for all Γ, Θi ⊆ L, B ∈ L: (***)

If Γ, Θ1 , Θ2 , . . . , Θn , . . . B and each Θi is -consistent and no propositional variable occurring in any formula in Θi occurs in any formula in Θj for j = i or in any formula in Γ ∪ {B}, then Γ B.

The natural next step would be to consider the analogue of 2.11.5 for sets of formulas and 2.11.8 for consequence relations, for the case of generalized consequence relations. Shoesmith and Smiley ([1971], [1978]) call conditions such as (**) and (***) cancellation conditions since under the indicated hypotheses about consistency and lack of shared variables, they say that Θ (in (**)) or the union of the Θi (in (***)) can be cancelled out in the claim that Γ, Θ B or that Γ, i∈I Θi B, enabling us to conclude that Γ just by itself bears the relation to B without any need for assistance from these Θ-formulas. (Such cancellation conditions go under the name of uniformity in Łoś and Suszko [1958], where the corresponding condition is somewhat more obscurely – and not quite correctly (see Wójcicki [1974] and references) – presented. This terminology is generally followed, e.g., in Wójcicki [1988], Urquhart [2001], both of which give the proof that this suffices for determination by a single matrix – but it is too reminiscent of the phrase ‘Uniform Substitution’ for us to follow suit here. Indeed, disregarding the potential for confusion entirely, Avron and Lev [2001] call use the phrase uniform consequence relation to mean substitution-invariant consequence relation.) In Shoesmith and Smiley [1978], a cancellation condition is proposed for generalized consequence relations which looks like this: If Γ1 , . . . , Γn , . . . Δ1 , . . . , Δn , . . . and no Γi ∪ Δi shares any variables with Γj ∪ Δj for j = i, then for some i we must have Γi Δi . It is clear from 2.11.7(i) that for a generalized consequence relation of the form |=M this condition must be satisfied, and Shoesmith and Smiley are able to show that in the finitary case, invariance and the satisfaction of this condition, which can for this case be simplified by setting n = 2, guarantee the existence of a determining matrix for a generalized consequence relation. The result has not, however, been extended to the non-finitary case ([1978], p. 281). We turn to some methodological remarks prompted by 2.11.7(ii), in which good old-fashioned bivalent valuations make an appearance, apparently conflicting with the opening theme of the present discussion: idea that many-valued matrices would give us something more general than the treatment of semantic matters in Chapter 1 in terms of exclusively bivalent valuations. The latter then came to appear as a special case of the concept of a matrix evaluation, namely that in which the matrix in question was a two-element matrix with one designated and one undesignated element (T, F, respectively). This is a useful perspective to take for many purposes. For example, we can treat the fact that the relation of tautological consequence is substitution-invariant as a special case of the fact (2.11.4) that any matrix-determined consequence relation is substitution-invariant. But there is another perspective which is no less useful, since we know from 1.12.3 (on p. 58) that whenever we are dealing with a consequence relation not relating Γ to C, there is some valuation (consistent


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with that consequence relation) verifying Γ but not C. However many values we have, they still divide into the designated and the undesignated, yielding a binary division; we can accordingly regard an evaluation h as giving rise to a valuation vh thus: for any formula A, vh (A) = T if h(A) ∈ D, and vh (A) = F if h(A) ∈ / D where D is the set of designated elements of the matrix in question. Then a sequent ‘holds’ in the sense of the present subsection on h iff it ‘holds’ in the sense of Chapter 1 on the valuation vh . According to this second perspective, then, (many-valued) evaluations are not a generalization of (two-valued) valuations at all, by contrast with what the descriptions in parentheses here might suggest: rather the two-valued divisions effected by valuations are always present in the background of the many-valued setting, in the form of the ubiquitous designated/undesignated distinction. To connect the matrix-based vocabulary with that of Chapter 1 note that to say a consequence relation or gcr is determined by a matrix M amounts to saying that is Log({vh | h is an M-evaluation}. To illustrate its utility, we include an exercise on proof systems. The methodological moral of the above discussion is simply: beware of the idea that there is a uniquely optimal all-purpose perspective to take on the relation between two things. Exercise 2.11.9 Call the proof system in Set-Set for the language with connectives ∧, ∨, and ¬, presented by the following rules Kle: (R), (M), (T), the sequent-schemata involving ∧ and ∨ in 1.27.3, together with the ‘simultaneous contrapositions’ (see below) of these schemata, and the two schemata (1) ¬¬A A, (2) A ¬¬A. Let Kle 1 and Kle 1,2 be the systems with presentations extending this by, respectively A, ¬A and A, ¬A. By the simultaneous contraposition of a sequent Γ Δ is meant the sequent ¬Δ ¬Γ (with ¬Δ = {¬D | D ∈ Δ}, and similarly for ¬Γ). Note that we are using simultaneous contraposition here to characterize the initial sequents of Kle rather than as a sequent-to-sequent rule; the only non-zero-premiss rules in play are (M) and (T). (i) Show that a sequent is provable in Kle 1 iff it is valid in K1 . (ii ) Show that a sequent is provable in Kle 1,2 iff it is valid in K1,2 . Suggestion: for the completeness (‘if’) halves of (i) and (ii), use 1.16.3 (p. 75) to get Γ+ , Δ+ maximal-consistently extending Γ, Δ when Γ Δ, for as the gcr associated with the proof system in question. For (i), specify a matrix evaluation h by h(A) = 1 if A ∈ Γ+

h(A) = 2 if A ∈ Δ+ and ¬A ∈ Δ+

h(A) = 3 if A ∈ Δ+ and ¬A ∈ / Δ+ (i.e., A ∈ Δ+ and ¬A ∈ Γ+ ). It must be verified that this h is indeed a K1 -evaluation; note that all formulas in Γ, and no formulas in Δ, receive the designated value 1 under h. (For (ii): put instead h(A) = 1 if A ∈ Γ+ and ¬A ∈ / Γ+ , h(A) + + + = 2, if A ∈ Γ and ¬A ∈ Γ ; h(A) = 3 if A ∈ Δ .) Completeness proofs of this style may be found in Cleave [1974a], Martin [1974], and – for related logics, including Ł3 – Brady [1982]. A somewhat different approach to proving completeness w.r.t. K1 appears in Kearns [1974]. For a matrix characterization of the basic system Kle itself, see 3.17.5 (p. 430); for the close connection between this system and relevant logic, see 2.33.6(iii) (p. 340) and the discussion that follows it.

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Note that Kle 1 , or the associated gcr, is atheorematic in the sense introduced in 2.11.6, as is the consequence relation determined by K1 . Unlike our earlier example of this phenomenon, Suszko , the present example shows that such consequence relations are to be found amongst those determined by a single matrix. We pause to note a general feature of these cases, involving an interaction between structural completeness (1.29.6 on p. 162, cf. 2.11.7(iii)) and triviality (1.19.2: p. 92). Observation 2.11.10 Every non-trivial atheorematic consequence relation is structurally incomplete. Proof. Suppose is non-trivial and atheorematic. By non-triviality, for some A, B, in the language of , we have A B. If were structurally complete, there would accordingly be some substitution s for which (i) s(A) and (ii) s(B), but (i) here is impossible, since is atheorematic. The second perspective we have introduced on the relation between evaluations and valuations suggests a question. Given the definition of vh as above in terms of h and the correlation between holding on h and holding on vh , we may ask: what is the point of introducing matrices and matrix-evaluations in the first place? Why not just make do with the machinery of Chapter 1? There is a one-word answer to this question: truth-functionality (or more generally: compositionality – see 2.11.11 below). Although we have yet to treat this topic in detail (§3.1), it is clear that even if all we are ultimately interested in is vh (A) for various formulas A and some given evaluation h, we need not be able to compute this for compound A knowing only vh (B) for each component (or immediate subformula) B of A. For example, Kleene–Łukasiewicz negation takes 1 to 3, 2 to 2, and 3 to 1, exhibiting the kind of functionality just isolated; but, taking the value 1 alone as designated (as in the Łukasiewicz matrix, and in the Kleene matrix K1 ) given that vh (A) = F, we cannot say whether vh (¬A) = T or vh (¬A) = F. If h(A) = 2, then vh (¬A) = F, since h(¬A) is the undesignated value 2, but if h(A) = 3, which is the other way to have vh (A) = F, then vh (¬A) = T, since in this case h(¬A) is the designated value 1. (Here, incidentally, we see some of the advantages of not using a notation such as “T”, “I”, “F” for 1, 2 and 3: it is convenient to keep the ranges of our matrix evaluations and of their induced bivalent valuations clearly distinguished.) Since the term “truth-functional” is well entrenched for functionality in the matrix values, we could record the failure of functionality in T and F just observed by saying that the connective ¬ fails to be ‘designation-functional’ in the matrix concerned. The raison d’être of many-valued logic is such failures of designation-functionality. Faced with such failures, the device of many-valued tables allows us to retain a ‘compositional’ treatment, by working at the level of the evaluations h instead of at the level of the induced valuations vh . Remark 2.11.11 A semantic account of some language which proceeds by assigning semantic values of some kind to its expressions is said to be compositional if the semantic values assigned to a complex expression is always uniquely determined by those assigned to the components. Many-valued logic offers a compositional treatment with semantic values


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taken as matrix elements, though not when the associated designationstatus (T or F) is itself taken as a semantic value. (See the notes for bibliographical information.) This perspective on many-valued logic has been emphasized especially and over many years by Dummett, beginning with Dummett [1959a], who notes that while we are accustomed to hearing people – such as Łukasiewicz – say for various reasons that some sentences deserve to be classified as neither true nor false, such claims become problematic when we consider (what Dummett calls) the point of the true/false dichotomy: as applied to whole sentences which might be used to make assertions—as opposed to subsentences embedded within them—there seems no reason to hold any particular sentence to fall on neither side of the dichotomy. A famous passage from Dummett [1959a], p. 11f.: We need to distinguish those states of affairs such that if the speaker envisaged them as possibilities he would be held to be either misusing the statement or misleading his hearers, and those of which this is not the case: and one way of using the words ‘true’ and ‘false’ would be to call states of affairs of the former kind those in which the statement was false, and the others those in which the statement was true.

The formulation in terms of misusing the statement or misleading one’s hearers is not quite right, perhaps, since there are certain ways of doing these things – such as by inappropriate conversational implicature (a notion explained in 5.11 below) – which are not to the point; but let us set such subtleties aside for this discussion. See further the notes to the present section. Dummett addresses, with the above criterion in mind, several particular areas in which statements have been described as ‘neither true nor false’ should be classified as true or else, depending on the area at issue, as false. The quotation given below provides an example of subdividing the true—and using more than one designated value—while Łukasiewicz’s three-valued matrix provides an example of subdividing the false: not ‘false’ in the sense of having true negations (this covers only the value 3 case) but false on the understanding conveyed by the passage above: roughly, unfit for assertion. The ‘values’ of many-valued matrices are thus to be thought of as different ways of being true (in the case of the designated values) or false (in the case of the undesignated values): we register the differences so that, in effect, by assigning these more finely differentiated semantic values (as opposed to mere truth and falsity as construed in the passage quoted above) so that a compositional treatment can be provided. One such area mentioned in Dummett [1959a] is that of conditionals, though the following retrospective remarks are from Dummett [1963], p. 155: I once imagined a case in which a language contained a negation operator ‘−’ which functioned much like our negation save that it made ‘−(A → B)’ equivalent to ‘A → −B’, where ‘→’ is the ordinary twovalued implication. In this case, the truth or falsity of ‘−(A → B)’ would not depend solely on the truth or falsity of ‘A → B’, but on the particular way in which ‘A → B’ was true (whether by the truth of both constituents or by the falsity of the antecedent). This would involve the use of threevalued truth-tables, distinguishing two kinds of truth. In the same way, it might be necessary to distinguish two kinds of falsity.

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This example is typical of what happens when the word “false” is used by those claiming a particular sentence to be neither true nor false. On the usage in question, a preferred negation operator is singled out and the non-true sentences whose negations are true are those called false. (Cf. the suggestion for Exercise 2.11.9.) This usage contrasts conspicuously with that emphasized in the quotation from Dummett [1959a] above, of course. The idea about conditionals presented in the Dummett [1963] quotation occurred to several people independently, and we devote 7.19 to the development given to it in Cooper [1968]. A perspective somewhat similar to Dummett’s may be found in Scott [1973], p. 266. (Scott [1974b] gives additional technical details, and [1976] some philosophical reflections: see further the reply, Smiley [1976]). An approach similar to Scott’s can be found in Urquhart [1974a]. Perhaps the author most famously advocating an emphasis on bivalent valuations including in the treatment of n-valued logics with n > 2 was Roman Suszko, who gives a treatment of Łukasiewiz’s three-valued logic from this vantage point in Suszko [1975]. (See further Malinowski [1977], [1985], [1994], [1997], Béziau [1999b], and the discussion and further references in Caleiro, Carnielli, Coniglio and Marcos [2005].) The question of how logical considerations dictate a need for a proliferation of more than one designated or more than one undesignated value cannot be dealt with here, since it requires the extensionality notions introduced in §3.2. See the discussion from 3.23.9 (p. 458) to the end of 3.23. Dummett’s own version of the logical considerations involves instead the concept we call congruentiality, in §3.3; there is a comparative discussion of these two reactions, and of further aspects of Dummett’s perspective on many-valued logic, in Humberstone [1998c]. The above-noted ubiquity of the designated/undesignated distinction, whether or not one chooses to take Dummett’s view of its significance, is associated with the (usual) pursuit of many-valued logic in ‘binary’ logical frameworks (Set-Fmla, Set-Set, etc.). However, since to say that a sequent holds on a valuation is to say that it does not verify all the left-hand formulas while falsifying all the right-hand formulas, a more direct analogue in n-valued logic would be to have n-place sequents, Γ1 Γ2 . . . Γn−1 Γn , say, such a sequent being deemed to hold on a matrix-evaluation h iff it is not the case that h(C) = 1 for all C ∈ Γ1 , h(C) = 2 for all C ∈ Γ2 , and . . . , and h(C) = n for all C ∈ Γn , and to be valid when holding on all evaluations. This idea is due to G. F. Rousseau ([1967], [1970a]), and we supply no more than the briefest discussion here; a similar idea was developed by at the same time by M. Takahasi – see Chapter 3 of Bolc and Borowik [2003]. (An alternative suggestion as to how to avoid the underlying bivalence produced by the valuations vh may be found in Malinowski [1994], §§4,5). For the case of n = 3, the appropriate version of the structural rule (R) consists of all sequents of the forms: A A ∅

A

∅ A

∅ A

A

Any such sequent holds on an arbitrary evaluation because an evaluation cannot assign more than one value to a given formula. Exercise 2.11.12 Give appropriate “3-Set” versions of the rules (M) and (T). The idea of translating what in the terminology of 3.11 below are called determinants into determinant-induced conditions on a gcr applies here also:


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for example if the binary connective # is interpreted by a three-valued function with 1, 2, 1 as one of its elements (as in the case of ∨ for Łukasiewicz and Kleene), then the appropriate condition takes the following two-part form: A

B

A#B

A, A # B

B ∅

The validity of all sequents of the first form amounts to the impossibility of an evaluation’s assigning 1 and 2 to A and B respectively, while assigning 3 to A # B; the second form reflects the impossibility of an evaluation’s assigning 1 and 2 to A and B respectively while assigning the value 1 to A # B. Thus together, they force (‘consistent’) evaluations to assigning the value 2 to a compound whose components have the values 1 and 2 (in that order). Such determinant-induced conditions – to borrow ahead from terminology introduced in 3.11 and 3.13 – are in effect involved in the ‘Suggestion’ for Exercise 2.11.9 above, though we were there working in a traditional binary framework: SetSet. To modulate to the latter framework from 3-Set, for example, with values 1 and 2 designated, one defines the relation thus: Γ Δ iff ∃Γ1 , Γ2 such that Γ = Γ1 ∪ Γ2 and the sequent Γ1 Γ2 Δ is provable; if only the value 1 is to be designated, one would instead take: Γ Δ iff ∃Δ1 , Δ2 such that Δ = Δ1 ∪ Δ2 and the sequent Γ Δ1 Δ2 is provable. Such a modulation is analogous, for sequents, to the modulation, for value assignments, from evaluations to the induced bivalent valuations. In both cases, a general version of Dummett’s query – presented above – remains unanswered, as to what possible basis, other than instrumental utility for compositional semantics or more convenient proof system presentation (in the n-Set framework), there might be for such an n-fold distinction. In the case of the matrices we have the question of how the various values correspond to what we are prepared to tolerate, given in the facts, by way of an assertion, and in the case of the sequents we have the question of what connection there might be between such sequents and the task of representing arguments in the familiar premisses-and-conclusions sense.

2.12

Many-Valued Logic: Classes of Matrices

We have considered matrices singly as determining logics in the frameworks Fmla, Set-Fmla, and Set-Set. We could call a logic a many-valued logic in the narrow sense if there exists a determining matrix for it (or if it “has a finite strictly characteristic matrix” to use a common form of words – see the notes, under ‘Terminology’: p. 272). Of course, a still narrower sense could be devised, requiring a finite determining matrix: such logics are called tabular. In a broader sense, one might consider as many-valued a logic which was determined by a class of matrices, in the sense of proving precisely those sequents valid in every matrix of the class. Here we subsume formulas as sequents of Fmla. Again, one might restrict attention to the case in which the matrices in the determining class were finite, though we shall pay no special attention to this case. (Gödel [1932] proved that intuitionistic logic in Fmla is not tabular – has no determining finite matrix, that is; see our 6.42.21 below. Jaśkowski [1936] showed, by contrast, that there does exist a determining class of finite matrices, which, even though the class is infinite, can be used to provide a decision procedure for theoremhood in intuitionistic logic; see further Surma [1973c] and Surma, Wroński and Zachorowski [1975]. The general property manifested here – determination by a class of finite matrices – is called the

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finite model property, and we shall touch on it briefly in 2.13.5 – p. 228 – and the end-of-section notes, at p. 273.) Similarly, we say that a consequence relation is determined by a class of matrices if for all Γ and B, we have Γ B iff for each M in the class Γ |=M B and likewise for gcr’s with “B” replaced by “Δ”. Note that, in terms of valuations induced by matrix evaluations, to assert that a (generalized) consequence relation is determined by a collection C of matrices is to assert that = Log({vh | h is an M-evaluation for some M ∈ C}). So far we have drawn a distinction between being many-valued in a broader and a narrower sense. It is clear that the so called broader sense includes the narrower, since if a logic is determined by the single matrix M it is determined by {M}, but it remains to be seen if any logics which are many-valued in the broader sense fail to be so in the narrower sense. If that turned out not to be the case, the distinction would be without point. In the case of logics in Fmla, this is precisely what happens, as we now show. Given a set of formulas determined by some family of matrices, the obvious plan for constructing a single matrix determining the set of formulas would be to reach for the direct product construction reviewed for algebras in 0.23. The direct product of a family of matrices Mi (for some language L) should have for its algebra the direct product (0.23) of their respective algebras Ai (which are of the same similarity type since we are considering matrices for the one language L) and for its set of designated values the cartesian product of their sets Di of designated elements. Thus, to take the simplest case of interest, if the family has only two members M1 = (A1 , D1 ) and M2 = (A2 , D2 ) then their direct product is the matrix M1 ⊗ M2 = (A1 ⊗ A2 , D1 × D2 ). Note that if h is an evaluation from L into this product matrix we can recover an M1 -evaluation h1 and an M2 -evaluation h2 such that for all B ∈ L, h(B) = h1 (B), h2 (B) . This is sometimes put, in terms of the truth tables for the Mi , by saying that the values in the product matrix are computed coordinatewise from those tables. We will call such evaluations as h1 , h2 here, the factor evaluations corresponding to h. (As well as direct products, the algebraic notion of a homomorphism – in various different forms, depending on how the designated and undesignated elements are to be treated – can be put to useful work in studying logics by matrix methods; the same goes for the concept of a submatrix, raised in 2.11.2(v), p. 201. Some discussion of these topics is provided in 4.35.) Theorem 2.12.1 A formula of a language L is valid in the direct product of a family of matrices for L iff it is valid in each of those matrices. Proof. ‘If’: Where Mi are the matrices concerned, if B ∈ L is not valid in the product matrix, then some evaluation h assigns to B an undesignated value, in which case at least one of the corresponding factor evaluations, hk say, assigns to B an element of Ak Dk , meaning that B is not valid in Mk . ‘Only if ’ : Suppose that for some hk : L −→ Ak , B does not hold on hk . Since hk (B) ∈ / Dk , any h evaluating B in the product matrix which has hk as its kth factor will assign an undesignated value to B.


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Corollary 2.12.2 If sets of formulas S1 , S2 , are determined respectively by matrices M1 , M2 , then S1 ∩ S2 is determined by M1 ⊗ M2 . Theorem 2.12.1 is not going to be of use in telling us that a logic in Fmla is determined by a single matrix rather than by a family of matrices, since a necessary condition on a set of formulas for being determined by such a family is closure under Uniform Substitution, which we already know, by 2.11.5 (p. 204), is a sufficient condition for being determined by a single matrix. Nevertheless, if we know that a logic is determined by some particular family of matrices then we know it is determined by their direct product, which may be of interest if the family contains only finitely many members each of which is finite: for in that case the direct product of those matrices is a single matrix, itself finite, which determines the logic. The converse direction of 2.12.1 is also not without interest in showing (together with the direction just mentioned) some of the multiplicity of determining matrices for a given logic. Take the set of classical tautologies, for example, and the product of the two-valued boolean matrix with itself. The latter four-element matrix then also determines this same set of formulas. Frequently its elements T, T , T, F , F, T and F, F are denoted, respectively, 1, 2, 3, and 4, and the tables for, e.g., ∨ and ¬ are depicted in Figure 2.12a; we have followed the convention mentioned in 2.11 of using an asterisk as a reminder of which elements are designated. (The explanation offered à propos of the ¬ table in 2.11a applies here also.) ∨ *1 2 3 4

1 1 1 1 1

2 1 2 1 2

3 1 1 3 3

4 1 2 3 4

¬ 1 2 3 4

4 3 2 1

Figure 2.12a: Product tables for disjunction and negation

The entry in the second row, third column (of the ∨-table in Figure 2.12a), for example, saying that 2 ∨ 3 (to use the same symbol for our connective and the operation interpreting it) is 1, is arrived at by computing T, F ∨ F, T

coordinate by coordinate = T ∨ F, F ∨ T = T, T = 1. Thus one could talk (in the Fmla framework) of classical propositional logic, or CL as we generally call it, as a four-valued logic, or indeed, since we can take any ‘power’ of the two-element matrix and get the same result, as 2k -valued logic for any k we please. However, it is usual to describe a logic which is manyvalued (in the narrow sense) as n-valued when it is determined by a n-element matrix but by no matrix with fewer than n elements; n here should also be allowed to be an infinite cardinal. (There is also a natural explication of the idea of being n-valued in the setting of the algebraic semantics described in 2.16 below, to be found in García Olmedo and Rodríguez Salas [2002].) Another variation on the theme of one set of formulas being determined by different matrices emerges if we consider what, following Kalicki [1950], may be called the direct sum of a family of matrices, defined exactly as for the direct product except that the direct sum designates a value when some (rather than every) coordinate is designated in its respective factor (or ‘summand’) matrix.

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Exercise 2.12.3 (i) Show that the gcr determined by the product of the twoelement boolean matrix with itself is not the tautological gcr (i.e., the gcr determined by that two-element matrix). (Hint: for example, show that for this gcr as , we do not have p ∨ q p, q.) (ii ) Show that the set of sequents in Set-Fmla which are valid in the sum of the two-element boolean matrix with itself is not the set of tautologous sequents. What happens if we consider the ‘dual’ framework Fmla-Set (for whose sequents Γ Δ we require |Γ| = 1)? (iii) Conclude from (ii) that the formulas valid in this sum matrix are precisely the tautologies. (Other examples of matrices agreeing in Fmla but not in Set-Fmla with CL may be found in Łoś and Suszko [1958], Church [1953].) The collapsing of the distinction between the broader and narrower senses of ‘many-valued logic’ which we have witnessed in the case of Fmla does not extend to Set-Fmla, still less to Set-Set. That is, we can have a consequence relation (or a generalized consequence relation) which is determined by a family of matrices without there being any single matrix which determines it. A well-known example concerns the consequence relation associated with ‘minimal logic’ (M L in 8.32); we could also have used the example of Suszko from 2.11.6, 2.11.7(ii). Here is another. Observation 2.12.4 The consequence relation determined by {K1 , K1,2 } is not |=M for any matrix M. Proof. Where is the consequence relation in question, we have p ∧ ¬p q ∨ ¬q (2.11.2(i), p. 201) but q ∨ ¬q (because K1 is in the determining class), even though p ∧ ¬p is -consistent (since for example p ∧¬p q, as K1,2 is in the determining class) and the formulas p ∧ ¬p and q ∨ ¬q have no propositional variables in common. Here we have a violation of the cancellation condition (**), mentioned on p. 205, taking Γ = ∅, Θ = {p ∧ ¬p} and B = q ∨ ¬q.) This example shows that Theorem 2.12.1 above on direct products fails if “formula” is replaced by “sequent of Set-Fmla”, since our findings for this example clash with the Corollary to that Theorem, reformulated in Set-Fmla, or alternatively (though finitariness is not at issue with this example) reformulated for consequence relations as: the consequence relation determined by M1 ⊗ M2 is the intersection of those determined by M1 and M2 . For, writing Log({M1 , M2 }) for the consequence relation determined by {M1 M2 }, we certainly have Log({K1,2 , K1 }) = Log(K1,2 ) ∩ Log(K1 ), and yet we have just that for no M, and therefore not for M = K1 ⊗ K1,2 , is this consequence relation determined by M. (Note that we abbreviate “Log({M})” to “Log(M)”, and that what this denotes in the official terminology of Chapter 1 is Log({vh | h is an M-evaluation}).) Since the claim that a given Set-Fmla sequent is valid in a matrix is expressed by a universal Horn sentence (2.25), validity of such sequents is preserved in passing from a family of matrices to their direct product. So it must be the converse direction that fails: the “only if” half of 2.12.1. And indeed it is. For as the consequence relation determined by {K1 , K1,2 }, consider the object


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language analogue of the (incorrect) consequence statement that p ∧ ¬p q, corresponding to the ex falso quodlibet sequent p ∧ ¬p q. This sequent is valid in the product of the Kleene matrices but it is not valid in the second factor K1,2 . The latter invalidity was noted at 2.11.2(i), p. 201; it is due to the possibility of assigning the designated value 2 to each of p and ¬p. In the product matrix by contrast, two of the nine values, namely 1, 1 and 1, 2 are designated, and negation takes each of them to an undesignated value ( 3, 3

and 3, 2 , respectively). Thus no evaluation can map each of p and ¬p to a designated element, so that p ∧ ¬p q is valid in the product matrix. (By contrast, the dual sequent p q ∨ ¬q remains invalid in the product, just as it was in the first factor.) The situation that led to the counterexample presented in 2.12.4 is representative, in the following respect. (See Czelakowski [2001], Exercise 0.1.1, p. 59, and references there cited, where this point is stated in fuller generality, with arbitrary direct products and not just binary products.) Let Cn 1 , Cn 2 and Cn 1⊗2 be the consequence operations determined by matrices M1 , M2 , and M1 ⊗ M2 . Then for any Γ, Cn 1⊗2 (Γ) = Cn 1 (Γ) ∩ Cn 2 (Γ) provided that there is some M1 evaluation which assigns designated values (of M1 ) to all formulas in Γ and also some M2 evaluation which assigns designated values (of M2 ) to all formulas in Γ. If this condition – which could equally well be put by saying that Γ is both Cn 1 - and Cn 2 -consistent – is not satisfied, then Cn 1⊗2 (Γ) is the set of all formulas of the language concerned. Even the direction (from factors to product) in which the direct product construction preserved validity for Set-Fmla sequents fails in Set-Set, so that Theorem 2.12.1 reformulated for this framework is now false in both directions, ‘if’ and ‘only if’ (pace Urquhart [1986], p. 77). This is as one would expect since we have departed from the realm of Horn sentences by allowing multiple right-hand sides in our sequents. To illustrate the point, consider: Example 2.12.5 Take a language with one connective $ (of arity 2), and two two-element matrices M1 = (A1 , D1 ) and M2 = (A2 , D2 ) with Ai = (Ai , $i ), i = 1, 2 where A1 = A2 = {T, F}, D1 = D2 = {T}, x$1 y = x, while x$2 y = y, for all x, y ∈ {T, F}. In M1 the sequent p $ q p is valid; in M2 the sequent p $ q q is valid. Therefore in each the sequent p $ q p, q is valid. But this is not valid in M1 ⊗ M2 since we may consider any evaluation h with h(p) = T, F , h(q) = F, T : neither of these is the designated value, whereas the formula p $ q on the left of the “” in the sequent last mentioned is mapped by h to the designated value T, T , so the sequent does not hold on h and is accordingly not valid in the product matrix. In terms of gcr’s this means that we do not even have Log(M1 ) ∩ Log(M2 ) ⊆ Log(M1 ⊗ M2 ). (We return to this example in Chapters 3 and 5: see the paragraph leading up to 3.24.2, as well as 5.35.) We need to take note in 3.24 of what has been established to this point, so we summarize our findings: Observation 2.12.6 (i) Validity of a sequent of Set-Fmla in each matrix in some collection implies validity of the sequent in their direct product, though in general (by contrast with the case of Fmla) validity in the product does not

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imply validity in each of the factor matrices. Alternatively put: the consequence relation determined by the product matrix includes the intersection of the factordetermined consequence relations, though not necessarily conversely. (ii) Validity of a sequent of Set-Set in each matrix in some collection neither implies nor is implied by the validity of the sequent in their direct product. Alternatively put: the gcr determined by the product matrix in general neither includes nor is included in the intersection of the factor-determined gcr’s. There is a reprieve from the negative parts of this Observation when the factor-matrices of the product are all the same (in which case the product is often called a ‘(direct) power’ of the matrix in question), here given as 2.12.7 only for the case in which the factor appears twice, for notational convenience. A more generally applicable reprieve will appear for consequence relations, allowing the converse whose absence is noted in 2.12.6(i), as 3.24.6. Observation 2.12.7 Any sequent of Set-Set valid in a matrix M⊗M is valid in M; accordingly, this is the case for Set-Fmla and Fmla, since the sequents of these frameworks are sequents of Set-Set and are assessed for validity in a matrix in the same way as they would be when considered as any other Set-Set sequents. Corollary 2.12.8 For any sequent σ of Set-Fmla, σ is valid in a matrix M iff σ is valid in M ⊗ M. Alternatively put: the consequence relations determined by M and M ⊗ M are one and the same. Proof. ‘If’: by 2.12.6(i); ‘Only if’: by 2.12.7.

As we have already seen (2.12.3(i)), this Corollary fails in the ‘only if’ direction if “Set-Fmla” is replaced by “Set-Set”. (Alternatively put, the inclusion |=M ⊇ |=M⊗M fails if the “|=” notation understood as denoting the gcr determined by the matrix concerned.) The preceding comment is closely connected with our finding (1.14.5) that the conjunctive combination of valuations consistent with a consequence relation was itself consistent with that consequence relation, reworded (1.26.2) to say that Set-Fmla sequents holding on two valuations held on their conjunctive combination, a result we emphasized did not carry over to Set-Set. To link that discussion with the present matrix material, we need the concept of the bivalent valuation vh induced by a matrix evaluation h, introduced toward the end of 2.11, and also the observation that every evaluation h into a product matrix M1 ⊗ M2 can itself be viewed as a ‘product’ of an M1 -evaluation h1 and an M2 -evaluation h2 , where for any formula A, h(A) = h1 (A), h2 (A) . In this case we write h as h1 ⊗ h2 . To avoid excessive subscripting, where g = hi , we write vg as vi , and where g = hi ⊗ hj , we write vg as vi⊗j . Notice that, in view of the way designation is defined for product matrices, we have the following simple connection with conjunctive combinations of valuations: vi⊗j = vi vj . Let σ be a Set-Fmla sequent invalid in M1 ⊗ M2 , failing accordingly on some evaluation h = h1 ⊗ h2 , and thus on the bivalent valuation v1⊗2 and therefore on v1 v2 . Then by the discussion from Chapter 1 just recalled, σ fails on v1 and so is not M1 -valid, or else σ fails on v2 and so is not M2 -valid. This reconstructs of the proof of 2.12.6(i), above, in the terms of Chapter 1, and we


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can see that the failure (as evidenced by the example mentioned in the ‘hint’ to Exercise 2.12.3(i), for instance) of the result to extend to Set-Set is due to the last step in the argument, when we inferred that σ could not hold on each of v1 and v2 while failing on their conjunctive combination: this is precisely what can happen in Set-Set. (In the case of our example, it happens that h1 and h2 are themselves already bivalent valuations, with vi = hi , and, linking up further with the discussion in §1.1, what we have here is two ∨-boolean valuations whose conjunctive combination v1 v2 , is not ∨-boolean.) The point has been made that for consequence (and generalized consequence) relations, unlike sets of formulas, being many-valued in the broad sense and being many-valued in the narrow sense do not coincide. The question then naturally arises as to exactly what it takes to be many-valued in the broad sense. When we were looking at consequence relations with a view to answering the corresponding narrow question we found that substitution-invariance was not sufficient for the existence of a determining matrix. (Recall Suszko .) By contrast, this is sufficient if all we are looking for is determination by a class of matrices: Theorem 2.12.9 Let be any (generalized) consequence relation on a language L. Then a necessary and sufficient condition for to be determined by some collection of matrices is that is substitution-invariant. A proof of this result (credited to J. Zygmunt for the gcr case) may be found at p. 336 (4.7.10(B)) of Wójcicki [1988]. We present the argument in the case of consequence relation because it leads to an interesting generalization of the notion of a matrix. Given a substitution-invariant consequence relation , consider the collection of matrices {MΘ | Θ is a -theory}, where for a given -theory Θ, the matrix MΘ has for its algebra the algebra of formulas of the language of (as in the proof of 2.11.5) and for its set of designated elements the set Θ. For the soundness of w.r.t. this class of matrices, suppose that Γ B and for some MΘ –evaluation h, h(Γ) are all designated, with a view to showing that h(B) is also. This supposition for Γ means that Θ h(C) for each C ∈ Γ. Since Γ B and is substitution-invariant, h(Γ) h(B). (As in the proof of 2.11.5, p. 204, we exploit the fact that the matrix evaluations h are just the substitutions.) Thus by the condition (T+ ) on consequence relations, Θ h(B), so since Θ is a -theory, h(B) ∈ Θ, i.e., h(B) is a designated element of the matrix MΘ . The proof of the completeness of w.r.t. this collection of matrices is much as in 2.11.5: if Γ B then the identity substitution is an MΘ -evaluation taking Θ = Cn (Γ) on which all of Γ take designated values while B does not. This proof shows that we can make a stronger claim than 2.12.9 does, since we have shown that a substitution-invariant is determined by a collection of matrices on the same algebra. Such a class of matrices is often called a bundle of matrices, and the particular class used in the preceding argument the Lindenbaum bundle for . Indeed Wójcicki [1973](see §4 thereof and earlier papers there cited) ‘bundles’ them up together to baptize a structure {(A, Di )}i∈I , by which is meant, for some index set I, {(A, Di ) | i ∈ I}, a generalized matrix (or ‘ramified matrix’ in Wójcicki [1988]). A somewhat more suggestive way of packaging the bundle comes from Smiley [1962a], namely as pairs (A, C) in which C is a closure operation on the universe A of the algebra A. These are accordingly

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called ‘Smiley models’ (abbreviated to “S-models”) in, e.g., Harrop [1968], and “Smiley matrices” in Wójcicki [1973], but a more common description of them, going back to Bloom, Brown and Suszko [1970], is as abstract logics (see Brown and Suszko [1973], Bloom and Brown [1973]). While it seems awkward to speak of a logic (as consequence relation) being determined by an ‘abstract logic’, and the present author would have preferred something like ‘Smiley structure’ or ‘closure structure’, the phrase retains considerable currency (e.g. in Font and Jansana [1996]) so we will use that terminology here. (In Czelakowski and Pigozzi [2004] they are called ‘second-order matrices’. See further the discussion following 2.12.12.) We say that a Set-Fmla sequent Γ B holds on an evaluation h in such a structure, h being as usual a homomorphism from the language concerned to the algebra A, just in case h(B) ∈ C(h(Γ)), where as usual h(Γ) = {h(A) | A ∈ Γ}. A sequent is valid in (A, C) when it holds on every evaluation. A consequence relation is sound w.r.t. the abstract logic (A, C) when every σ ∈ is valid in (A, C), complete w.r.t. (A, C) when only those σ ∈ are valid in (A, C), and determined by (A, C) when both sound and complete w.r.t. (A, C). (For this definition to be appropriate in the case of non-finitary , the notion of a Set-Fmla sequent σ should be taken as generalized to allow infinitely many formulas to the left of the “”.) Abstract logics generalize the notion of a matrix in the sense that while every matrix gives rise to a Smiley structure which is in a way we shall see (2.12.10(ii)) equivalent to the original matrix, while the converse is not the case. Given a matrix (A, D), the abstract logic induced by (A, D) is the abstract logic (A, CD ) in which the closure operation CD is defined thus, A being the universe of A: For all X ⊆ A, CD (X) = {a ∈ A | X ⊆ D ⇒ a ∈ D}. Exercise 2.12.10 (i) Check that the above definition is equivalent to the following. For all X ⊆ A: D if X ⊆ D CD (X) = A if X D (ii ) Show that for any homomorphism h from a language into an algebra A (of the same similarity type) and any Set-Fmla sequent σ ( = ΓB), σ holds on h in the matrix (A, D) if and only if σ holds on h in the induced abstract logic (A, CD ). From this we may conclude that (A, D) and (A, CD ) are equivalent in the sense that the same sequents are valid in (A, D) and (A, CD ). Abstract logics generalize not only individual matrices but also matrix bundles, in the sense that any matrix bundle {(A, Di )}i∈I gives rise to an abstract logic validating the same Set-Fmla sequents (or: determining the same consequence relation). Just define (A, C) by setting, for any X ⊆ A: C(X) = {a ∈ A | ∀i ∈ I X ⊆ Di ⇒ a ∈ Di }. It is clear enough that so defined, C is a closure operation, and a routine verification in the style of that required for 2.12.10(ii) establishes the following. Observation 2.12.11 With C defined as above, the consequence relations determined by {(A, Di )}i∈I and by the abstract logic (A, C) coincide.


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Recalling from 0.13.5(ii) (see p. 10) the one-to-one correspondence between closure operations and closure systems, we note that the matrix bundle {(A, Di )}i∈I giving rise to the abstract logic (A, C) by the definition above cannot be uniquely recovered from the induced (A, C). For the above definition of C(X) amounts to setting this equal to the intersection of all Di ⊇ C, as in the definition (in 0.13.5(ii)) of a closure operation in terms of a closure system: but we have not required the sets Di to comprise a closure system (on A), since we did not require them to be closed under arbitrary intersections. Thus adding in further intersections of the Di would give rise to the same C. There would, indeed, be no change to the consequence relation determined by such additions – as foreshadowed in the case of finite intersections by Exercise 2.11.2(viii), p. 201. (In view of the equivalence between closure operations and closure systems mentioned here, it will not come as a surprise that in fact some of the references given earlier take an abstract logic to have a closure system on A, rather than a closure operation, as the second ingredient alongside A.) Since we know – 2.12.9 and subsequent discussion – that any substitutioninvariant consequence relation is determined by not just a collection of matrices but by a bundle of matrices, 2.12.11 gives us the conclusion that every such consequence relation is determined by an abstract logic, but here we give a more direct derivation of this result: Observation 2.12.12 Every substitution-invariant consequence relation is determined by an abstract logic. Proof. Given a substitution-invariant consequence relation , consider the abstract logic (A, C) in which A is the (free) algebra of formulas (of the language of ) and C is Cn . Soundness of w.r.t. this abstract logic follows from substitution-invariance, and for completeness, we use the identity map as the evaluation h for which h(B) ∈ / C(h(Γ)) whenever Γ B. The above proof spells out how, trivially, on a certain conception of what a logic is, a (substitution-invariant) logic is sound and complete w.r.t . . . itself! The relevant conception of a logic is as a special kind of abstract logic, taken a logic as a pair consisting of the underlying language, taken as an algebra, together with a consequence operation. Because the language of a consequence relation is fixed once the consequence relations is, and the notions of consequence relation and consequence operation are interdefinable (via [Def. ] and [Def. Cn] in 1.12), this is equivalent to the identification of logics as consequence relations. Abstract logics are simply a generalization of this conception of a logic, since we abstract away from the requirement that the algebraic component be a language (an absolutely free algebra – see the discussion preceding 1.11.1, p. 49), so that we make the corresponding terminological change in the second component and allow arbitrary closure operations as opposed, specifically, to consequence operations.

2.13

Algebraic Semantics: Matrices with a Single Designated Value

The label algebraic semantics covers any use of algebraic concepts and results in semantics. A more precise definition would be inappropriate, foreclosing options

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possibly yet to emerge (though a specific meaning is given to the phrase “has an algebraic semantics” in 2.16). The matrix methods surveyed above obviously fall within its purview since the matrices involved come equipped with an algebraic structure. The label many-valued logic, however, evokes somewhat different associations, and to go part of the way towards acknowledging this difference, we reserve the label algebraic semantics in this and the following subsections for treatments in which the role of the set of designated values takes a low profile. We will look at four types of treatment falling under this rubric. In the first, matrices are employed in each of which a single value is designated. This may not appear to give matters of designation a ‘low profile’. But such a description comes to seem fitting especially when the designated value is itself a distinguished element (nullary fundamental operation) of the algebra of the matrix under consideration, such as the 1 of a boolean algebra, in which case it is natural to speak of a formula’s validity in the algebra. (See Wójcicki [1973], opening paragraph of §2, and p.227 of Wójcicki [1988] for related considerations, and Czelakowski [1981] for the use of the phrase “algebraic semantics” in the present sense.) Whether or not the designated element has this status, as long as there is exactly one such element, we shall, following Czelakowski [2001] speak of a unital matrix, and it is aspects of semantic analysis of logics in terms of such matrices that we shall be concerned in the present subsection. Note that our leading example of matrix semantics in 2.11, the Łukasiewicz three-element matrix, was in fact just such a unital. matrix, as was one of the two Kleene matrices there discussed (namely K1 ). The second style of treatment is one in which designated values do not put in an appearance at all, and instead a notion of validity is provided in terms of a partial ordering supplied by the algebras, and a sequent A B (say) is valid in an algebra if for every homomorphism h from the language to the algebra, h(A) h(B). We call this: -based semantics and it will be the subject of 2.14. Thirdly, in 2.15 we shall consider a special case of determination by a class of matrices, where the class consists of all matrices based on a single algebra; this deserves to be called algebraic semantics because in taking account all matrices on the algebra, we are abstracting from any selection of a particular set of elements as the designated elements. We consider also a variation on this idea due to W. Rautenberg. A final subsection (2.16), alluded to already above, addresses very briefly some ideas about algebraic semantics developed in Blok and Pigozzi [1989] and other works by the same authors. What is the range of application of matrix semantics with a restriction to unital matrices? Without this restriction, we answered this question for Fmla in 2.11.5. Attending to the restriction, the analogous result is given as 2.13.1 here. The added ‘semantic’ restriction turns out to correspond precisely to a ‘syntactic’ condition on the logics S (as sets of formulas) for whose formulation we need the concept of formulas B and C being, as we shall say S-synonymous, meaning by this that for any formulas A1 and A2 differing only in that one has B as a subformula in zero or more places where the other one has C, A1 ∈ S iff A2 ∈ S. (This is a Fmla-oriented version of the concept of synonymy introduced in 1.19.) In other words, S-synonymous formulas are freely interchangeable in any other formulas without making a difference to the membership of those other formulas in S. We abbreviate this to: B ≡S C. This notation is suggestive of the fact that S-synonymy is a congruence relation (0.23) on the algebra of the Lindenbaum matrix for S figuring in the proof of 2.11.5 (p. 204). The


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induced quotient algebra, sometimes called the Lindenbaum algebra or ‘Tarski– Lindenbaum algebra’ for S plays the same role in the following argument. For a more concise formulation, we call a set of formulas (which in all cases of interest will be considered as a logic in Fmla) monothetic if any two of its elements are synonymous relative to the whole set (i.e., if B ∈ S and C ∈ S, then B ≡S C). In other words, thinking of S as a logic specified by some proof system, S is monothetic when it has, to within S-synonymy, only one theorem. (Thesis is an older term for theorem; “monotheorematic” seems too much of a mouthful. The word “monothetic” has other meanings in various areas – taxonomy, statistics, group theory, and elsewhere – from which there will be no danger of confusion with the present usage.) While many commonly encountered logics are monothetic, we shall meet one that is not, at the end of the present subsection. (The notion of a monothetic logic has a natural extension to frameworks other than Fmla, understanding for example, a logic in Set-Fmla to be monothetic whenever formulas B and C are synonymous – freely interchangeable in arbitrary sequents in the logic – whenever the sequents A and B both belong to the logic. But here we need only the Fmla notion described above.) As usual, “|D|” denotes the cardinality of D: Theorem 2.13.1 Let S be any non-empty set of formulas in a language L. Then a necessary and sufficient condition for it to be the case that for some matrix M = (A, D) with |D| = 1, S is determined by M, is that (i) S is closed under Uniform Substitution, and (ii) S is monothetic. Proof. Necessity: Of (i), by Observation 2.11.4 (p. 203); as to (ii), say D = {d} and suppose B, C ∈ S. Then for all evaluations h, h(B) = d and h(C) = d, so h(B) = h(C). An induction on formula complexity shows that h(A1 ) = h(A2 ), where A1 and A2 are formulas differing in some replacements of B for C, completes the argument. (Do the induction on the complexity of A where A1 , A2 result from A by uniform substitution of B, C, respectively, for some propositional variable in A; an argument along these lines appears in the proof of 3.31.1, p. 485.) Sufficiency: We construct the desired algebra A out of the equivalence classes [B] for B ∈ L where [B] = {C ∈ L | C ≡S B} and for each k-ary connective # of L we define the corresponding operation #A in A by: #A ([B1 ],. . . ,[Bk ]) = [#B1 , . . . , Bk )]. Thus if we have the binary connective ∧ then [B1 ] ∧A [B2 ] is stipulated to be the equivalence class [B1 ∧ B2 ]. Such a stipulation could be inconsistent if B2 and some formula C were S-synonymous, so that [B2 ] = [C], but B1 ∧ B2 was not S-synonymous with B1 ∧ C. But this is impossible, since if a difference in having B1 ∧ B2 instead of B1 ∧ C as a subformula in some position makes a difference to membership in S, then that is itself an instance where a difference over B2 and C makes such a difference, contradicting the assumption that B2 ≡S C. Thus here, and in the general case of k-ary #, the algebraic operations #A are well defined.

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The set D of designated elements is defined to be {[B] | B ∈ S}. Note that this set contains at least one element since we have assumed S is non-empty, and that it contains at most one element by part (ii) of the condition on S. It remains to be verified that for all B ∈ L, (1) if B ∈ S then B is valid in this matrix, and (2) if B is valid in this matrix, then B ∈ S. (2) is proved much as in the proof of 2.11.5 (p. 204), except here the crucial evaluation is not the identity mapping but the homomorphism h defined by: h(B) = [B]. (h is a homomorphism, by 0.23.1, p. 27.) For (1), we also need a modification of the earlier proof. We should begin by noting, where “h” ranges over (A, D)-evaluations, “s” over substitutions, and “B” over formulas, and writing “sB” for “s(B)”, that: (*) ∀h∃s∀B: h(B) = [sB]. To establish this, suppose we are given h. Look at h(pi ) for the propositional variables pi , and select some formula Ai from h(pi ). (Thus h(pi ) = [Ai ].) The substitution s we want can be taken as that which maps each pi to Ai . We must check that ∀B.h(B) = [sB], by induction on the complexity of B. The way we obtained s secures this result automatically for B = pi , leaving the inductive part of the proof to do. To reduce notation, we give the case of compounding by a binary connective #. Thus we suppose we are dealing with B = B1 # B2 . The inductive hypothesis is used in the second equality: h(B1 # B2 ) = h(B1 ) #A h(B2 ) = [sB1 ] #A [sB2 ] = [sB1 # sB2 ] = [s(B1 # B2 )]. Now, to establish (1), we note that if B is not valid in the matrix we have constructed, this means that h(B) = [C] for some C ∈ / S; by (*) there exists / S, and so, since S some substitution s with h(B) = [sB]. Since sB ≡S C, sB ∈ is closed under Uniform Substitution, B ∈ / S. For a formula B, the equivalence class [B] figuring in the above argument is often called the proposition expressed by B, by analogy with the fact that one thinks of different sentences of an interpreted language as expressing the same proposition when they are in some appropriate sense equivalent. Synonymy (in the current sense—taken from Smiley [1962a] and [1963a]) is a very strong equivalence relation since it amounts to the indiscernibility of the formulas concerned by the lights of the logic employed. More generally, especially some practitioners of algebraic semantics describe the evaluations h as assigning to formulas propositions, encouraging one to think of the elements of the algebras as more analogous to meanings of sentences than to previously uncountenanced additional truth-values. (Dunn [1975a] is a good example, though the talk of propositions in this context goes back at least to Halmos [1956]. This use of proposition also appears on p. 116 of Smiley [1963a].) Thus if we are thinking of, for instance, ∧ as the operation of formula conjunction, then ∧A is to be thought of as an operation which forms from two propositions the corresponding conjunctive proposition. (Cf. the comments toward the end of 2.11.) In the case of many logics S in Fmla, there is some (primitive or derived) binary connective # of the language of the logic with the property that for all formulas B and C the relation ≡S holds between B and C iff B # C ∈ S.


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For example, in the case of classical logic (or intuitionistic logic, or minimal logic) the connective ↔ (1.18) is such a connective. (Porte [1980] calls the synonymy relation ≡S in this case a formula-definable congruence. See also Porte [1965], p. 48 and pp. 64–68. In the latter source, Porte allows ‘definability’ of a congruence by means of a finite set of formulas. Relaxing that still further to allow an arbitrary set of formulas, we have a Fmla version of the notion of an equivalential logic defined for logics as consequence relations – or logics in Set-Fmla – at the end of 1.19 above.) When working a proof like that just given for such a logic, it is usual to define [B] in terms of the connective in question – thus in the case just mentioned one might set [B] = {C | B ↔ C ∈ S}. Sometimes there will not be a single such derived connective but there may be a class of them, say #1 , . . . , #n with a necessary and sufficient condition for the S-synonymy of B and C being the membership in S of each of the formulas B #i C. (See also the discussion of equivalential logics in 2.16 below; there, however, it is logics as consequence relations rather than as sets of formulas that are at issue.) As an example, for which we can make do with n = 2, consider the pure implicational fragments of the various logics just mentioned: B #1 C = B → C, B #2 C = C → B. When our S has been proof-theoretically specified we can apply Theorem 2.13.1 to provide an algebraic completeness results for the axiom system doing the specifying. The interest of such results varies directly with the extent to which the algebras concerned are independently familiar or can in some other way be invested with informal significance, rather than coming in for special attention merely for the sake of the result. A good example of such a class of algebras is the class of boolean algebras, so that it is of no small interest to show that some axiom system (putatively) for CL (in Fmla) yields as theorems precisely the formulas valid in all unital matrices (A, {1}) where A is a boolean algebra with 1 as its unit element. In fact, downplaying (as promised) the role of value-designation, we will just say a formula is valid in the boolean algebra when it is valid in such a matrix. Some axiomatizations lend themselves more readily than others to such a treatment, and we will describe the general strategy without exhibiting any one of them. (See Dunn [1975a] for more concrete detail here.) To have as close a fit as possible with boolean algebras as defined in 0.21, let us suppose our language L to have connectives ∧, ∨, ¬, , ⊥. A formula A ∈ L holds on a homomorphism (or evaluation) h into a boolean algebra A = (A, ∧, ∨, ¬, 1, 0) when h(A) = 1 and is valid in A when it holds on every such evaluation. Note that we require h() = 1, h(⊥) = 0. Now we suppose that we have a set of formulas in L laid down as axioms, and some non-zero-premiss rules, as in 1.29. First we should check soundness w.r.t. the class of boolean algebras by showing that each of the axioms is valid in every boolean algebra and that each of the rules preserves validity. The application of Theorem 2.13.1 is to completeness. We need the Lindenbaum algebra figuring in its proof to actually be a boolean algebra, as it will be if we have so chosen our axioms that they suffice, with the rules, for the proof of every tautology. Note that we may take ≡S (where S is the set of theorems) to be defined by: A ≡S B iff each of ¬A ∨ B and ¬B∨A belongs to S. (Alternatively, we could use the characterizations given earlier if we have either or both of →, ↔, as additional primitive connectives and work with an appropriately expanded version of boolean algebras – see 2.11.1, p. 200.) Thus we have to check that the defining conditions for being a

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boolean algebra are satisfied by the algebra defined as in the proof of 2.13.1. For example, since the meet operation in such an algebra must be commutative, we need to make sure that [B] ∧A [C] = [C] ∧A [B], for all formulas B, C, which means showing that [B ∧ C] = [C ∧ B], i.e., that (for the indicated choice of ≡S ) each of ¬(B ∧ C) ∨ (C ∧ B) and ¬(C ∧ B) ∨ (B ∧ C) is provable on the basis of our axioms, for any B, C. Since, by 2.13.1, every unprovable formula is invalid in this algebra, once we have shown it to be a boolean algebra, we have our completeness proof: all those formulas valid in every boolean algebra are provable in our axiom system. (In fact we needed to show only that each unprovable formula is invalid in some boolean algebra, while what we have shown is stronger: that there is some one boolean algebra in which all the unprovable formulas are invalid.) The following formulation combines this completeness result with the corresponding soundness result: Corollary 2.13.2 The set of formulas B such that CL B, where CL is the consequence relation associated with classical logic as formulated with the primitive connectives in the above discussion, is determined by the class of boolean algebras, i.e., by the class of matrices (A, {1}) with A a boolean algebra and 1 its top element. A similar result is available for many other Fmla logics: for example, we can replace the reference to CL with one to IL , and the reference to boolean algebras by one to Heyting algebras, for which purposes → should be amongst the primitive connectives. A natural question at this point arises over the question of what can be said along the lines of Theorem 2.13.1 for logics in Set-Fmla (consequence relations) or in Set-Set (gcr’s). The latter question we shall not address, and the former will receive our attention at a suitable point in the following chapter (see 3.23.9, p. 458), so for the moment we continue to discuss unital matrix semantic for logics in Fmla. As formulated here, 2.13.2 does not look much like – what it was described to be – a soundness and completeness result, since no axiomatization has been specified: but such axiomatizations abound in the literature. (See for example the {∧, ¬} and {∨, ¬} axiomatizations in §6 of Appendix 1, Prior [1962], which can be supplemented with suitable definitions of ∨ and ∧ respectively, as well as of , ⊥, to stick to the letter of the above presentation.) In the case of any such axiom system, the Lindenbaum algebra is the free boolean algebra with countably many free generators (0.24), these being the equivalence classes [pi ] of the propositional variables. If for a moment we rescind our standing assumption that we deal only with languages generated by countably many propositional variables, we can easily depict the Lindenbaum algebra for the tautologies in a language with, say, just p and q as generators; alternatively, think of this as the subalgebra – generated by these variables – of the full Lindenbaum algebra. The elements here have been labeled arbitrarily by means of representative formulas of their equivalence classes; we could have written p∨¬p instead of for the top element, for instance, since [p ∨ ¬p] = [], and similarly we could have written ¬(p ∧ q) in place of ¬p ∨ ¬q, and so on. Figure 2.13a depicts the boolean algebra freely generated by {[p], [q]}. With it before us, three comments may help to ward off possible confusions. First, we note while {p, q} is a set of generators for the restricted language currently under


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oo ?O?O?OOO o o ?? OOO oo ?? OOO ooo ? p∨q OO ¬p ∨ ¬q p ∨¬q oooo ¬p∨q TTT oTTT joOOO ?? j o o T T j T T j OO ? o o o TTTT jjTjTjToTo jTToToTo TTTT OOOO ??? oooo j j j OOO?? o o j o TTTT TTTT oo O T T jTjjj oToTo oo j T O ? j TTTT q TTTT ooo ¬q jj ¬poooop q p ↔q ?O?OO p j j O o T T j ?? OOO TTTT ooToTjTjTjj o ?? OOO oT j T ooo ? OO ojojojojojjTjTTTTT ToToToToToT OO OOO ??¬p∧¬q p ∧q ooo ¬p ∧ q p∧¬q o OOO ??? o OOO?? ooooo OO? ooo ⊥

Figure 2.13a: The 16-Element Boolean Algebra

consideration – considered as (absolutely) free algebra of formulas – and their equivalence classes generate the Lindenbaum algebra, which is not the same fact reported twice over. These particular equivalence classes are not unique in the latter respect. For the Lindenbaum algebra is also freely generated by the (again minimal) set of generators {[p], [¬q]}, as well as by {[p], [p ↔ q]}, and so on, while the current language is not (freely or otherwise) generated by {p, p ↔ q}: there is no way, for example, of building the formula q out of these two with the help of the connectives (even though we can obtain [q] from [p] and [p ↔ q] since [q] = [p] ↔ [p ↔ q] = [p ↔ (p ↔ q)]. A second potential source of confusion calls for the following terminological warning: p and q are atomic formulas or atoms (along with and ⊥) in the sense of not being constructed from other formulas – and so are atoms in the sense of things without proper parts on the ‘concrete’ construal of languages mentioned in 1.11 – but their ≡-classes, the (candidate) generators [p] and [q] are evidently not atoms in the Lindenbaum algebra, a status possessed by only the four elements on the second to bottom row of the diagram in Figure 2.13a. Thirdly we need to comment on the partial order associated with that boolean algebra (i.e., with its lattice reduct as in 0.13) has [A] [B] just in case A → B is a tautology, or equivalently, to employ a formulation more in keeping with the following subsection, in which this order comes to the fore, just in case A CL B (“B is a tautological consequence of A”). The point to note here is in the sense in which formula is at least as strong as another if it has the latter as a consequence, and is strictly stronger when the converse does not obtain, this ‘is at least as strong as’ (resp. ‘is stronger than’) relation emerges as the relation (resp. < ) between the corresponding propositions. This is not what one might informally expect of the notation, which tempts to think of the lower or lesser as weaker rather than stronger, whereas in fact on a diagram such as Figure 2.13a, the higher one goes the weaker are the propositions concerned. In the Lindenbaum algebra for the full language of propositional logic with all countably many generators p1 ,. . . , pn ,. . . there are, by contrast, no atoms at all since for any [A] = [⊥] we can select pi not occurring in A (whether or not pi occurs in some B with [A] = [B]) and note that [⊥] < [A ∧ pi ] < [A]. Its atomlessness is an indication that the (full) Lindenbaum algebra is not isomorphic to any power set algebra, since such an isomorphism always delivers atoms corre-


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sponding to the singleton subsets, something perhaps already evident because power set algebras are always either finite or uncountable and our algebra has only countably many elements. (An additional, though not unrelated, give-away sign is that the Lindenbaum algebra is not a complete boolean algebra: there is no (infinite) join for the subset {[p1 ],. . . ,[pn ],. . . }, for example.) In the context of CL, the Lindenbaum algebras obtained from proof systems in Fmla are usually regarded as interim structures on the way to showing that if a formula is not provable on the basis of such an axiomatization as we have been presuming to be under consideration, then it is not a tautology – in other words, that it is not valid in the two-element boolean algebra. To obtain this further result, one needs the additional fact (proved in several of the references given for this material), that given any a = 1 in a boolean algebra A, there is a homomorphism f from A to the two-element boolean algebra with f (a) = 0. We defer a discussion of unital matrix semantics for Set-Fmla and Set-Set until 3.23. (Actually, in 3.23.9 – p. 458 – we address only the Set-Fmla case, in the terminology of consequence relations.) Sometimes a ‘multiplication table’ style of presentation of a Lindenbaum algebra is more convenient than a Hasse diagram, as in the case of the implicational fragment of CL. As with Figure 2.13a, we prefer to have something small and manageable to deal with, so we depict the Lindenbaum algebra of equivalence classes of formulas in this fragment built from the two propositional variables p, q. In the interests of brevity, we write p, q for [p], [q], and 1 and p ∨ q respectively for [p → p], and [(p → q) → q] ( = [(q → p) → p]). In the last case, the labelling is suggested by the CL-equivalence of (A → B) → B and A ∨ B, for any formulas A, B. Alternatively, one may regard the entries, p → q for example, as simply being representative formulas chosen from their equivalence classes, rather than names of the equivalence classes themselves (in the present instance [p → q], alias [p → (p → q)] etc.). → 1 p q p→q q→p p∨q

1 1 1 1 1 1 1

p p 1 q→p p p∨q q→p

q q p→q 1 p∨q q p→q

p→q p→q p→q 1 1 p→q p→q

q→p q→p 1 q→p q→p 1 q→p

p∨q p∨q 1 1 p∨q p∨q 1

Figure 2.13b: Two-variable formulas in classical →

There is an implicit inductive argument here, to the effect that every formula A constructed from the variables p and q by means of → is CL-equivalent to one of the six listed above, the induction being on the complexity of A. If this is 0 we are dealing with p or with q, each of which is on the list. For the inductive step, suppose A is of complexity > 0, in which case A is B → C for some B, C, and by the inductive hypothesis each of B, C, is equivalent to one of the six listed formulas, in which case the table in Figure 2.13b shows which formula – again on the list – B → C is equivalent to.


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Exercise 2.13.3 Draw the Hasse diagram whose elements are the six equivalence classes of Figure 2.13b (with the partial ordering indicated à propos of Figure 2.13a). Another exercise in this vein, but in terms of the consequence relation IL (of intuitionistic logic) described in 2.32, p. 302 rather than CL , may be found on p. 1058: Exercise 7.21.2. The following remark pertains to a contrast between CL and IL which may be of most interest to those who have already had occasion to peruse 2.32 (p. 302 onward). Remark 2.13.4 While the variety of boolean algebras is locally finite in the sense that any such algebra with finitely many generators is itself finite this is not so for the variety of Heyting algebras (see discussion preceding 0.21.6, p. 22). Even the Heyting algebra with a single free generator – the so-called Rieger–Nishimura lattice – has infinitely many elements (Nishimura [1960]). In logical terms, this means that infinitely many distinct equivalence classes, relative to IL , arise even for formulas built from a single propositional variable. (A proof of this fact, which is due to Gödel, may be found in McKinsey and Tarski [1948], where the result is given as Theorem 4.5. What Nishimura [1960] does is to provide an explicit description of the equivalence classes via representatives, and the implicational relationships between these representatives.) If we select one formula from each equivalence class, getting formulas C1 (p),. . . ,Cn (p),. . . we have in effect specified countably many definable intuitionistic 1-ary connectives, #1 , . . . , #n . . ., where #n A is the formula C(A), i.e., the result of substituting A for p in C(p). See the discussion of contexts after 3.16.1 below, p. 424. (Some discussions of IL would lead one erroneously to think there are only three such formulas C(p), to within equivalence: p, ¬p and ¬¬p. For an example of this, see Dietrich [1993], p. 139f.) If formulas with only → are considered, however, there is a return to ‘local finiteness’: for any fixed number of variables, there are only finitely many pairwise non-equivalent such formulas – Diego [1966], Thm. 18, p.36; see also Urquhart [1974c]. (A diagram of the logical relations between the purely implicational pairwise non-equivalent formulas in two fixed variables may be found in Skolem [1952] – see p. 183f. See also de Bruijn [1975].) Diego’s discussion is couched in terms of the corresponding variety of algebras: Hilbert algebras. We describe these presently. (See further 7.21.2(ii), p. 1058.) A similar result was obtained, more generally, for the case of ∨-free formulas of the language of IL in McKay [1968a] (see p. 260). For the one-variable case, analogous observations concerning various fragments of IL may be found in the Remark following Thm. 4.5 in McKinsey and Tarski [1948]. We transfer the “locally finite” terminology from (varieties of) algebras to logics, in any framework, by describing a logic as locally finite when from any finite number of propositional variables, only finitely many pairwise non-synonymous formulas can be constructed. (“Locally tabular” is another phrase for the same property.) The connection with tabularity in the sense of determination by a single finite matrix is given by part (i) of 2.13.5; we include part (ii) to give a fuller picture. It uses the concept of the finite model property which a logic (in


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Fmla) is said to possess just in case each of its non-theorems is invalid in some matrix validating all the logic’s theorems – in other words if it is determined by some class of finite matrices. The latter property is especially connected with decidability (see Harrop [1958], [1965] and, for a helpful warning in this connection, Urquhart [1981]) – a topic not on the agenda for the present work. A few further words on the finite model property will be found in the notes (under that heading, on p. 273).

Observation 2.13.5 (i) Any tabular logic is locally finite. (ii) Any locally finite logic has the finite model property. Proof. (i): Suppose a logic is determined by M = (A, D) with A finite (where A is the universe of A). Any formula in k distinct propositional variables induces in the obvious way a function from Ak to A, so since there are only |A|n such functions where n = |A|k , and any two formulas inducing the same function are synonymous according to the logic in question, there are at most finitely many pairwise non-synonymous formulas. (Namely: at most |A|n such.) A proof of (ii) is left to the reader.

Neither of the implications recorded in 2.13.5: S is tabular

⇒ S is locally finite

⇒

S has the f.m.p.

is reversible (i.e., the converse does not hold for arbitrary S). For certain purposes a more quantitatively explicit terminology in dealing with local finiteness is useful, as in Humberstone [2003a], where a logic S (in Fmla for that discussion) is said to be k-locally finite with extent n to mean that the n is the least natural number for which a set of n formulas exists such that every formula constructed from p1 , . . . , pk is synonymous in S with some formula in the set. The implicational fragment of the relevant logic R – BCIW logic in the terminology of 1.29 and of what follows in this subsection – is 1-locally finite (with extent 6) but not 2-locally finite (with any extent, that is) and thus not locally finite tout court. This was shown in Meyer [1970]; see also Anderson and Belnap [1975], p.392. There is a contrast here with the implicational fragment of IL (BCKW logic), as 2.13.4 reported: this logic is locally finite (k-locally finite for each k ∈ N). It is, incidentally, like the corresponding fragment of CL, 1locally finite with extent 2, as is BCK logic, since the one-variable implicational fragments of IL, CL and BCK logic all coincide: see 7.21.1 and 7.25.1. The remainder of this subsection addresses algebraic treatments of some purely implicational logics in Fmla, with axioms drawn from the formulas on the following list, labelled in accordance with an analogy with the combinators of combinatory logic. The same labels were used for the corresponding schemata – replacing distinct propositional variables by distinct schematic letters – in our discussion in 1.29, where some remarks on the connection with the combinators may also be found. (See the Digression starting on p. 165 and – for further historical information – the notes to the present section.)

2.1. MANY-VALUED LOGIC AND ALGEBRAIC SEMANTICS K:

p → (q → p)

S:

(p → (q → r)) → ((p → q) → (p → r))

B:

(q → r) → ((p → q) → (p → r))

B:

(p → q) → ((q → r) → (p → r))

C:

(p → (q → r)) → (q → (p → r))

I:

p→p

W:

(p → (p → q)) → (p → q)

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We string the labels together, as in 1.29, to describe the logic obtained from the labelled formulas as axioms with Uniform Substitution and Modus Ponens as rules of proof. BCIW logic is the →-fragment of the relevant logic R described in 2.33, while BCI logic is the →-fragment (in Fmla) of Girard’s Linear Logic, also treated there (from p. 345 on). BCKW logic is the →-fragment of intuitionistic logic. SK gives the same logic. We take especially these last two facts for granted here. (A proof would most conveniently be given with the aid of the Deduction Theorem for the SK axiomatization – see the discussion following 1.29.10, on p. 168.) B and B are equivalent given C, a principle which allows us to permute (or, mnemonically, to Commute) antecedents in an iterated implication, so in view of the widespread use of C, one seldom encounters both B and B in axiomatizations, though an exception is the system of ‘ticket entailment’ or T (discussed especially in Anderson and Belnap [1975]), whose implicational fragment is BB IW logic. All the above formulas, or the corresponding schemas, play the role in Fmla of certain structural rules in the sequent calculus in Seq-Fmla (or Seq-Seq); in particular K corresponds to a of “thinning” (on the left) analogous to (M) in such a framework, while C corresponds to a structural rule allowing permutation of formulas on the left of the “” (and so is built into Set-Fmla, or indeed MsetFmla: see 1.21 for all these framework distinctions), and W to the rule allowing contraction of two occurrences of a formula on the left to a single occurrence. As was mentioned in 1.21.8 (p. 111), logics lacking one or more such principles are sometimes accordingly termed substructural logics; relevant logics in particular lack K (see 2.33) while contractionless logics lack W (see 7.25), and in linear logic (2.33 again: from p. 345 on) both are absent. We have already encountered an example of a logic – as an attempt at Exercise 2.11.2(iv), p. 201, will have revealed – not containing W , in the form of (especially the Fmla version of) Łukasiewicz’s three-valued logic (2.11). Roughly speaking, the presence of K allows hypotheses not required for the derivation of a conclusion to be cited as antecedents with that conclusion as consequent, while the presence of W allows multiple appeals to a single hypothesis to be counted as though only one appeal had been made. This description is somewhat skewed in favour of linear logic, since the latter then appears as a paradigm of honest book-keeping in the matter of dependence on hypotheses. Even weaker substructural possibilities are available since we could resist the move from sequences (“Seq”) to multisets (“Mset”), insisting that adjacent formula occurrences may not be permuted (or exchanged); in the Fmla simulations this amounts to dropping the principle C above. What is sometimes called Lambek logic – or the Lambek Calculus – does indeed drop

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not only (the rules corresponding to) K and W but also C, but we do not treat it here. (A formulation of the Exchange rule may be found in the notes to §2.3, under ‘Linear Logic’, p. 371.) Finally: note that K, W , and B – or rather the corresponding axiom-schemata (distinct schematic letters supplanting distinct propositional variables) – appeared as (HB1)–(HB3) in our discussion of the axiomatic approach in 1.29. We will consider some corresponding classes of algebras of similarity type 2, 0 , where the binary operation will still be denoted by → and the distinguished element is represented by 1. (Actually “→” will be playing three roles in our discussion: to stand for a sentence connective – as in K, S, etc., to stand for an operation symbol, and to stand for the operation associated with that operation symbol in a given algebra. That is, we do not explicitly mark the distinction which in the notation of 0.22 would be “→” vs. “→A ”, where A is one of the algebras under discussion.) There is a translation of formulas of the implicational language considered above into terms in the language of such algebras, giving for each formula A a term tA in accordance with the following: (1) If A is a propositional variable pi , tA is xi ; if A is B → C, tA is tB → tC . Thus, writing x1 , x2 , x3 as x, y, z (just as we write p1 , p2 , p3 as p, q, r), we have as the term tC : (x → (y → z)) → (y → (x → z)). We have various classes of algebras corresponding to the different implicational logics under consideration about which we are concerned especially to ask whether the following relationship holds: (2)

A is provable in the logic if and only if every algebra in the class satisfies the equation t A = 1.

This places such enquiries well within the purview of unital matrix semantics since it amounts to a soundness and completeness result for the logic in question with respect to the class of all matrices on the algebras in question with 1 as the designated value. After all, to say that a particular algebra satisfies the equation tA = 1 is to say that however elements of the algebra are assigned to the variables xi in tA , the result is still 1: but such an assignment amounts to a matrix-evaluation by using it as an assignment to the pi in A, and so in particular a matrix-evaluation on which the formula A (or more pedantically the sequent A) holds. There is one slight snag in this; namely, that for matrix semantics we required the similarity type of the algebra to coincide with that of the language, and we are envisaging a language here with → as sole connective, yet the algebras we are dealing with are of type 2, 0 . However, we could equally well, for the cases of principal concern below, not require the element behaving as we shall require 1 to behave to be a fundamental nullary operation a distinguished element of the algebra, that is. We might equally well have asked only that the algebra should contain some element behaving as the various sets of conditions demand that 1 behave. In some of the published literature on (e.g.) BCK -algebras, they are treated in this way, as groupoids. It should be noted that although here we are discussing algebraic semantics for these various logics, there exist in most cases more revealing model-theoretic semantics for them, such as the Kripke semantics for IL (intuitionistic logic,


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of which SK logic is the implicational fragment), described in 2.32, and the Urquhart semantics for the relevant logic R (implicational fragment: BCIW logic), described in 2.33; adaptations of the latter to suit the contractionless BCI and BCK systems will be described in 7.25. As well as such tA = 1 equations, we consider three additional principles, two quasi-identities (conditional equations) and one identity: (AS) (x → y = 1 & y → x = 1) ⇒ x = y (MP) 1 → y = 1 ⇒ y = 1 (SMP) 1 → y = y. The labels abbreviate “antisymmetry”, “Modus Ponens”, and “Strong Modus Ponens”, respectively. The connection between the familiar rule Modus Ponens and (MP) will become clear in due course (see the proof of 2.13.8). (SMP) is a strengthening of (MP); if (SMP) is satisfied and we have the antecedent of (MP) for a given algebra element a, that is, if we have 1 → a = 1, then by (SMP), since 1 → a = a, we have a = 1, the consequent of (MP) is satisfied. (This shows that (MP) follows from (SMP) without the aid of any additional assumptions; we leave the reader to show that the converse is not so.) In much of the literature on BCK algebras, Hilbert algebras, etc.—phrases we shall explain presently—a relation is defined on such an algebra (A, →, 1) by putting, for a, b ∈ A: a b ⇔ a → b = 1, a relation which turns out to be a partial order on A, with (AS) securing its antisymmetry. We will use this notation in what follows when (AS) is being assumed. Now by a BCK AS -algebra, a BCK M P -algebra, or a BCK SM P -algebra we mean an algebra satisfying, alongside the equations tB = 1, tC = 1 and tK = 1, the conditions (AS), (MP) and (SMP) respectively. We adopt a similar usage for the case of other labels listed above. BCK AS - and BCI AS -algebras are what are respectively known in the literature as BCK-algebras and BCI -algebras respectively. (See Bunder [1981a]; the original definitions were somewhat different.) We shall frequently use this same terminology, reverting to the use of the “AS” subscript only when a contrastive emphasis is required. SK AS -algebras are the Hilbert algebras of Diego [1966], and we shall use the latter terminology in what follows, which shows that 1 is a ‘right zero’ for the operation → in any Hilbert algebra. (The condition (SMP) above similarly says that 1 is a ‘left identity’ element for →; we shall see—2.13.10—that it too is satisfied in all Hilbert algebras.) The present discussion concentrates initially on Hilbert algebras; we include some remarks on BCK -algebras below, as well as on BCI algebras. (Warning: the latter do not provide an algebraic semantics for BCI logic: see the discussion after 2.13.25 below.) Observation 2.13.6 Every Hilbert algebra satisfies the equation y → 1 = 1. Proof. By the equation tK = 1 (i.e., x → (y → x) = 1) corresponding to K, we have 1 → (y → 1) = 1 (taking x as 1). Thus “pre-multiplying” both sides by y → 1, we get: (y → 1) → (1 → (y → 1)) = (y → 1) → 1. But the lhs here = 1, by the K-equation again, which gives us: 1 = (y → 1) → 1. Since both 1 → (y → 1) = 1 and (y → 1) → 1 = 1, or to use the notation,

232


both 1 y → 1 and y → 1 1, by (AS) we have y → 1 = 1.

Note that in the notation, what we have just shown is that y 1 (for all y), so 1 is a top element with respect to the partial order , though we have yet to show that, in the case of (say) Hilbert algebras, is indeed a partial ordering. We do in 2.13.9 below. First we show that SK AS -algebras are all of them SK M P -algebras: Observation 2.13.7 Every Hilbert algebra satisfies (MP), the quasi-identity: if 1 → y = 1 then y = 1. Proof. Suppose 1 → a = 1 for a given Hilbert algebra element a. Since by 2.13.6, we also have a → 1 = 1, the condition (AS) gives the conclusion a = 1, so (MP) is satisfied. We are now in a position to prove the ‘only if’ direction of (2) above for the case of Hilbert algebras. Theorem 2.13.8 If a formula A is provable from substitution instances of S and K with the aid of Modus Ponens, then the equation tA = 1 holds in every Hilbert algebra. Proof. By induction on the length of a shortest proof of A from substitution instances of S and K. Basis case: A is itself such a substitution instance of S or K. Since the definition of Hilbert algebras as SK AS -algebras secures that tS = 1 and tK = 1 hold in every such algebra, it suffices to note that when B is a substitution instance of A (an arbitrary formula B, not the specific formula we have called B), the term tB is obtained from tA by substitution of terms for variables, so that whenever tA = 1 holds (universally) in an algebra, tB = 1 does also. For the inductive part of the proof, we are entitled to assume that we have formulas A and B for which A and A → B have the property claimed for all formulas in the statement of the Theorem, and must show that this implies that B, obtainable from them by Modus Ponens, also has this property. Thus the equations tA = 1 and tA→B = 1 hold in all Hilbert algebras. But the latter is tA → tB = 1, and so, given the former, can be written as 1 → tB = 1. By 2.13.7, then, substituting tB for y, we have tB = 1 in an arbitrary Hilbert algebra, as required. The condition on formulas given in the statement of 2.13.8—to the effect that they should be provable from substitution instances of S and K with the aid of Modus Ponens—was worded to aid the proof rather than for its familiarity. Recall from 1.29 that any proof from axioms using Modus Ponens and Uniform Substitution can be replaced by a proof of the same formula and using these same rules, but in which all applications of Uniform Substitution preceded any application of Modus Ponens. (This is the fact that makes such an axiomatization replaceable by one in which Uniform Substitution is dropped and the original axioms replaced by axiom-schemata.) Thus the condition in question amounts simply to provability in SK logic, alias the implicational fragment of IL. Since it is easy to check by semantic means (the Kripke semantics for IL


233

is given in 2.32 below) which formulas lie in this fragment, we can use this information to together with 2.13.8 to tell us a lot about Hilbert algebras. Since B and I are in the fragment (recall that, with the help of Modus Ponens, SK delivers the same theorems as BCKW ), we have in particular: Corollary 2.13.9 For any Hilbert algebra, the relation defined as above (a b ⇔ a → b = 1) is reflexive and transitive, and hence – by (AS) – partially orders the algebra. Proof. The case of reflexivity is immediate from the SK -provability of I. For transitivity, suppose that for elements a, b, c of some Hilbert algebra a b and b c, that is, we have a → b = b → c = 1. We must show that a c. By the SK -provability of B, 2.13.8 gives us that every Hilbert algebra satisfies: (y → z) → ((x → y) → (x → z)) = 1, so in the case at hand we have (b → c) → ((a → b) → (a → c)) = 1. Since b → c = 1, by 2.13.7 the condition (MP) is satisfied, and thus (a → b) → (a → c) = 1; but a → b = 1, so by 2.13.7 again, a → c = 1, i.e., a c. We can show in the same way that our strengthened version of (MP) is satisfied by every Hilbert algebra; in terms of our elaborately explicit notation, then, SK AS -algebras all turn out to be (not only SK M P -algebras, but) SK SM P algebras. Corollary 2.13.10 Every Hilbert algebra satisfies the condition (SM P ): 1 → y = y. Proof. The formula p → ((p → q) → q) is SK -provable (see below), so every Hilbert algebra satisfies: x → ((x → y) → y) = 1, by 2.13.8. Putting 1 for x and appealing to (MP) gives (1 → y) → y = 1, and (SMP) follows by an appeal to the K-equation y → (1 → y) = 1 and (AS). As to the provability of p → ((p → q) → q), we take it as having been verified that B, C, I are derivable from S and K, and note that the (simple) proof of this formula in BCI logic may be found in the course of the proof of 2.13.26 (p. 244). Now the fact that (SMP) is satisfied is particularly suggestive of an alternative and perhaps purely equational characterization of the class of Hilbert algebras. Their characterization as SK AS -algebras is not equational because of the role of the quasi-identity or conditional equation (AS), but perhaps with the help of (SMP) we can avoid the need to appeal to (AS). A first thought might be that the class of Hilbert algebras (SK AS -algebras) just is the class of SK SM P -algebras; but that is not quite correct, as is shown by the example depicted in Figure 2.13c, devised by Komori for a related application (see Idziak [1983], Komori [1984]). In the three-element algebra for which the → table is given in Fig. 2.13c the equations tS = 1 and tK = 1, corresponding to S and K, are satisfied, as is (SMP), but (AS) is not satisfied (since a → b = b → a = 1 although a = b); so this is an SK SM P algebra which is not a Hilbert algebra. (Checking that the relevant equations hold even in such a small structure is somewhat tedious. In the case of K, we have to verify that


234

→ 1 a b

1 1 1 1

a a 1 1

b b 1 1

Figure 2.13c

(x → (y → z)) → ((x → y) → (x → z)) = 1 holds when the variables range over the three elements a, b, 1, of Komori’s algebra. Doing this by brute force involves constructing a 27-line “truth-table” recording the successive combined assignments of elements to the variables, checking the equation in each case.) Accordingly we shall need to supplement S, K and (SMP) in order to obtain a purely equational basis for the class of Hilbert algebras. We need an equation (or perhaps several) satisfied by all such algebras and whose satisfaction secures that the condition (AS) is satisfied. We note that the conditional with antecedent A → B with A = (p → q) → ((q → p) → p) and B = (p → q) → ((q → p) → q) is provable from S and K with the aid of Modus Ponens, as is the converse implication B → A. (To put this in the notation of 2.32, we have A IL B for this particular choice of A, B.) Thus by 2.13.8 and (AS), the equation (3)

(x → y) → ((y → x) → x) = (x → y) → ((y → x) → y)

holds in every Hilbert algebra. This equation, along with (SMP) implies the antisymmetry condition (AS), because if x → y = y → x = 1, the two sides of (3) reduce, giving us: (4)

1 → (1 → x) = 1 → (1 → y)

and thus, after several appeals to (SMP) (5)

x = y.

We summarize what has been proved: Theorem 2.13.11 The class of Hilbert algebras is an equational class, with equational basis {S, K, (3), (SMP)}. The formula (3) above which, following Diego [1966], we used to obtain this equational characterization, could have been replaced by others. Consider for example: (6)

(((x → y) → y) → x) → x = (((y → x) → x) → y) → y.

Exercise 2.13.12 Show that an algebra (A, →, 1) is a Hilbert algebra iff it is an SK SM P -algebra satisfying (6). Remarks 2.13.13(i) A propos of (6), we remark that the equivalence, in intuitionistic logic, of (((p → q) → q) → p) → p and (((q → p) → p) → q) → q is noted in Skolem [1952], p. 183.


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¨ B, on which see 4.22.10 (ii ) If we abbreviate (A → B) → B to A ∨ (p. 555) below, then we can record the equivalence mentioned in (i) more ¨ q) ∨ ¨ p and (q ∨ ¨ p) ∨ ¨ q. A ∨ ¨ B enconveniently as obtaining between (p ∨ joys some disjunction-like properties in BCK logic, for example in being a formula that is provably implied by A and (thanks to the K axiom) by B, though not – by contrast with A → A, for example – by every formula. A more complicated definition for a disjunction-like connec... tive we shall denote by ∨ was suggested for the case of BCIW logic in Church [1951a], sometimes – e.g. in Meyer [1983] – accordingly referred to as Church disjunction: ... A ∨ B = (B → A) → ((A → B) → B). This is provably implied by B in the logic in question (i.e., without K), as well as by A, though in the latter case W is needed. (Clearly BCI logic itself provides no such “∨-introductive” connective, compounding any two formulas into one provably implied by each of them, for such implications would then survive into the equivalential fragment of classical logic, an extension of BCI logic if we think of ↔ as written “→”, rendering any two formulas provably equivalent.) In BCIW logic, alias purely ... implicational R, as Church remarks, ∨ is idempotent, commutative and associative (though not – even in IL – “∨-eliminative: that would have meant that ∨ was definable in terms of → in IL). The commutativity of the corresponding algebraic operation is what Diego’s identity (3) above asserts, as one sees on permuting antecedents on the rhs, showing that this identity holds not just in every Hilbert algebra but in arbitrary BCIWAS -algebras. (Our earlier warning about the mismatch between the class of BCI -algebras and BCI logic applies here also; see the discussion – p. 244 – after 2.13.25 below.) (iii) The fact that (AS) can be avoided by use of (6) above is noted in the context of BCK -algebras (i.e., BCK AS -algebras) in Cornish [1981], where some remarks about the equational class in question may be found. (See also Bunder [1983b].) In this context (6), rewritten here in the notation introduced in Remark 0.24.7 (see p. 32): (((x → y) → y) → x) → x ≈ (((y → x) → x) → y) → y, is an equation Cornish finds to be an identity of surprisingly many ¨ for the derived operation analogously to the BCK -algebras. Using ∨ connective in (ii) above we note that Cornish’s identity, like Diego’s above (as remarked in (ii)), asserts the commutativity of the binary op¨ y)¨∨x. We note in passing that the eration ∗ defined by: x ∗ y = (x ∨ corresponding sentence connective (“Cornish disjunction”?) again forms compounds which in BCK logic are provably implied by each of their components though not by arbitrary formulas. (For the case of the first component, only BCI resources are needed.) Thus the question of its commutativity is not without (propositional) logical interest. There is an interesting contrast to note between the class of Hilbert algebras and the class of BCK -algebras in connection with Remark 2.13.13(iii). While Hilbert algebras form an equational class, as we saw in 2.13.11, this

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is not so in the case of BCK -algebras. Wroński [1983] showed that the class of BCK -algebras is not a variety – in particular because a homomorphic image of a BCK -algebra need not itself be a BCK -algebra—and so by Birkhoff’s theorem identifying varieties and equational classes of algebras (0.24), there is no analogue of 2.13.11 for BCK -algebras. No equations can play the role of (3) or (6), in the case of Hilbert algebras, for BCK -algebras: the role of rendering (AS) redundant, that is. In particular (3) and (6) themselves do not hold in all BCK-algebras – something which follows immediately from a result (much conjectured previously) proved in Nagayama [1994]: the only equations t = u holding in all BCK -algebras are those in which the rightmost variable in t is the same as the rightmost variable in u, with the exception of the case in which t = 1 (and hence u = 1) holds in all such algebras. For instance x → (y → z) = y → (x → z) illustrates the common rightmost variable (here z) phenomenon, with the (“t = 1”) exception being illustrated by the case of x → (y → x) = y → y. We do not reproduce Nagayama’s proof here. (It relies on a sequent calculus formalism developed in Komori [1984] for a slightly different class of algebras.) We simply note that it gives us a simple criterion showing that (3) and (6) are not BCK -identities (equations holding in all BCK -algebras, that is). The following line of thought has considerable intuitive plausibility. To get the effect of (AS) using equations. we need terms t(x, y) and u(x, y) in the indicated variables with t(x, y) = u(x, y) a BCK -identity which reduces on the hypothesis that (for a particular choice of x, y) x → y = y → x = 1 to the (AS) conclusion, x = y, and that this reduction will have to proceed via appeals to (SMP), another BCK -identity, in such a way that the “same rightmost variable” condition was violated (because the last appeal to (SMP) will be in the context of 1 → x = 1 → y). This promises a ‘proof-theoretic’ proof of Wroński’s theorem, mentioned above; the details may be found in Nagayama [1994], where the result appears as Theorem 2.7. We have been taking a lot for granted about the class of BCK -algebras, and it is time to notice explicitly how little of the earlier discussion of Hilbert algebras actually depended on all the defining characteristics of that class of algebras. Recall that Hilbert algebras can be defined as BCKW AS -algebras (this being equivalent to their characterization as SK AS -algebras); so it is only the absence of the equation tW = 1 (or equivalently, x → (x → y) = x → y)) that distinguishes the theory of BCK -algebras from that of Hilbert algebras. But in fact, much of what was said about Hilbert algebras drew on much less than the ‘BCK ’ parts of the story. The proofs of 2.13.6 and 2.13.7 establish parts (i) and (ii) respectively of the following more general statement: Observation 2.13.14 Any algebra (A, →, 1) satisfying tK = 1 and (AS) satisfies (i) y → 1 = 1, and (ii) the condition (MP): if 1 → y = 1 then y = 1. Since it was only part (ii) here that was needed for the proof (2.13.8) that tA = 1 is satisfied in every Hilbert algebra for any formula A derivable by Modus Ponens from substitution instances of S and K, we have an analogous conclusion for BCK -algebras (with “B” and “C” replacing “S”): Theorem 2.13.15 If a formula A is provable from substitution instances of B, C, and K with the aid of Modus Ponens, then the equation tA = 1 holds in every BCK-algebra.


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The Hilbert algebra corollaries (2.13.9, 2.13.10) are accordingly available here also: Exercise 2.13.16 (i) Show that defining in a BCK -algebra by ab ⇔ a→b=1 makes a partial ordering of the elements. (ii ) Show (as a corollary of 2.13.15) that every BCK -algebra satisfies the condition (SMP): 1 → y = y. However, as already noted, at this point we cannot continue the parallel with Hilbert algebras to the point of using (SMP) and supplementary identities to provide an equational characterization of BCK -algebras. This has not prevented people from investigating various equational classes of BCK -algebras, however, of which we have already given several examples: the Hilbert algebras comprise one such class, those mentioned in 2.13.13(iii) another. For additional examples, we might consider the principles P:

((p → q) → p)) → p

Q:

((p → q) → q) → ((q → p) → p).

No combinatory logic pedigree is claimed for these labels (though see Hirokawa, Komori and Takeuti [1996], on this issue with regard to the first), which are simply mnemonic for “Peirce’s Law” and “Quasi-commutative”. (The latter term is from Abbott [1967]; BCK -algebras satisfying Q are frequently called ‘commutative’ in the literature, though it is not the binary operation corresponding to → which is commutative, but that corresponding to the defined connective ¨ ” in 2.13.13(ii) above.) As with the earlier examples, it suffices to we called “ ∨ show that (AS) is secured by the new principles. In the case of Q, we can argue as follows: since Q brings with it, by Uniform Substitution, its own converse, BCKQ-algebras must satisfy the equation (7) (x → y) → y = (y → x) → x. Now suppose, with a view to deriving (AS), that a → b = b → a= 1 in such an algebra, with b as x and a as y in (7), we accordingly get (8) 1 → a = 1 → b and thus, since (SMP) is satisfied in all BCK -algebras, a = b, establishing (AS). Thus BCKQ-algebras comprise an equational class. As in the cases considered earlier, we have to take, alongside the logic-derived equations, tB = 1, etc. (including now also tQ = 1), the (SMP) equation 1 → y = y. For the case of BCKP -algebras ( = SKP -algebras), which provide an algebraic semantics for the implicational fragment of classical logic (which is just BCKP or SKP logic), the situation is simpler, because these automatically satisfy tW = 1, and are thus Hilbert algebras to which an obvious modification of 2.13.11 applies. It suffices to show (for tW = 1) that all such algebras satisfy x → (x → y) = x → y. But it is not hard, with some help from 2.13.20(ii) below, to see that every BCK -algebra satisfies (9): (9) x → (x → y) = ((x → y) → y) → (x → y)

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and by tP = 1, we can replace the rhs by x → y. (Note that BCK -algebras satisfying tP = 1, satisfy (w → z) → w = w; to make the replacement indicated in (9), we put x → y for w and y for z.) Remark 2.13.17 Whilst BCKQ logic is a new logic (distinct from BCK-logic and SK logic), SKQ logic is nothing new: just the →-fragment of classical logic, alias SKP logic again). An elegant algebraic treatment of this case may be found in Abbott [1967] (or §7.4 of Abbott [1969]). For more information on Peirce’s Law, see 7.21 below, where we will see (7.21.18) that instead of saying “BCKP ” here, we could equivalently, and therefore more economically, have said “BCIP ”. Returning to the case of BCK -algebras, the fact that they do not constitute an equational class is no obstacle to their providing an algebraic semantics (in the sense of unital matrix semantics) for BCK logic. We treat the case of Hilbert algebras and BCK -algebras together to bring out the parallel; the formulation below uses “valid in an algebra” to apply to formulas A for which the equation tA = 1 is satisfied in the algebra in question. (That is, we are referring to validity in the matrix based on the given algebra, with 1 as the sole designated value. Results of this form were foreshadowed at (2) above.) Theorem 2.13.18 A formula is provable in SK logic (BCK logic) if and only if it is valid in every Hilbert (resp. BCK-) algebra. Proof. The ‘only if’ (soundness) direction is given by 2.13.8 for the SK case and 2.13.15 for the BCK case. For the ‘if’ (completeness) direction, we use the Lindenbaum algebras of the two logics. The elements are the synonymy classes [A] of the formulas A, which in either case comprise all those formulas B for which A → B and B → A are both provable. The element 1 is taken as [A] for an arbitrary provable formula A. (Any two provable formulas are synonymous, in the case of each logic.) It is routine to verify that, in the one case, tS = 1, tK = 1 and (AS) are satisfied, and in the other, that tB = 1, tC = 1, tK = 1, and (AS) are satisfied. Thus if A is an SK -unprovable formula, h(A) = 1, where h is the evaluation given by: h(C) = [C] for all formulas C, so there is either case, an algebra of the required kind in which any unprovable formula is invalid. We collect here some of the similarities and differences between the class of Hilbert algebras and the class of BCK -algebras, some of which have already been touched on, and the rest of which will be treated below. Since the contraction axiom W (or tW = 1, in equational form) is what differentiates the two logics (or classes of algebras), this is a selective chart of the difference this axiom makes. After that, we conclude with a few words on the case of BCI logic. Remarks 2.13.19(i) Hilbert algebras constitute an equational class (or variety), while BCK -algebras do not. (As already remarked, this was first shown in Wroński [1983].) (ii ) The class of Hilbert algebras is locally finite (see 2.13.4, p. 227) while the class of BCK -algebras is not; put in terms of logics, SK logic is – while BCK logic is not – locally finite. (See 2.13.20(i).)


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(iii) Hilbert algebras and BCK -algebras alike provide algebraic semantics for the respective logics (SK and BCK ): witness 2.13.18. (iv ) A binary operation · on a set A is said to be anti-commutative just in case for all a, b ∈ A, a · b = b · a only when a = b. The operation → is anti-commutative in the case of Hilbert algebras but not in the case of (arbitrary) BCK -algebras. (Some further remarks on anti-commutativity may be found below.) Exercise 2.13.20 (i) Taking for granted the fact that W is not provable in BCK logic, show that no two of the formulas in the following infinite sequence are equivalent (i.e. provably imply each other) in BCK logic: p → q, p → (p → q), p → (p → (p → q)), p → (p → (p → (p → q))), .. . Note that it follows from this that BCK logic is not locally finite. (ii ) Consider now repetition in the consequent, with the following infinite sequence of formulas: p → q, (p → q) → q, ((p → q) → q) → q, (((p → q) → q) → q) → q, .. . Show that the first and third formulas on this list are BCK -equivalent (‘the Law of Triple Consequents’), and calculate how many formulas— to within BCK -equivalence (alias BCK -synonymy: see the proof of 2.13.18)—appear on the infinite sequence of which the listed formulas comprise the initial segment. Notice, incidentally, that this Law of Triple Consequents holds even in BCI logic. (iii) Prove or refute the claim that BCI = BCK ∩ BCIW, where BCI (etc.) denotes the set of theorems of BCI logic. A propos of the formulas enumerated in 2.13.20(i) here, which we may here temporarily refer to as A1 ,. . . ,An ,. . . , we note that W is A2 → A1 , and one could equally well consider as potential new axioms to add to BCK logic, not just W (giving SK ), but any other Ai+1 → Ai instead, giving progressively weaker logics with i increasing. The corresponding infinite chain (under ⊆) of classes of BCK -algebras are studied in Cornish [1980a]; each of them is an equational class. As to the fact that BCK -algebras do not themselves form an equational class, it is not clear why the early papers such as Iséki [1966] introducing such algebras should have chosen precisely the definition—as what we have more explicitly called BCK AS -algebras—that they did choose, given only the desire to provide an algebraic analogue of BCK logic. (Note the vagueness of the title of Iséki [1966].) Suppose that, as in 2.13.19(iii), the desire is simply for an

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algebraic semantics for a logic in Fmla—and without regard to any preferred consequence relation on the set of formulas concerned—such as the version of BCK which supplied the motivation was (and has been in our discussion). In that case, since all that matters is which equations hold throughout the class of algebras in question, then the class of algebras satisfying the equations satisfied by every BCK -algebra would do just as well, and would by definition be an equational class. In the notation of 0.24, if we let K be the class of BCK algebras, then the class we are considering is Mod (Equ(K)). (Two further notes: (1) At present it is not known whether the equational theory in question, Equ(K) for the current choice of K, has a finite equational basis. (2) We emphasize the restriction of these remarks to the impoverished ‘logics as sets of formulas’ conception – cf. §1.2, Appendix (p. 180) – because if logics are individuated as consequence relations, then a kind of algebraic semantics, treated in 2.16 below, has been developed according to which one can say that the class of BCK -algebras “is the (unique) equivalent algebraic semantics” for BCK logic. The quoted words come from Blok and Raftery [1995].) We can in fact provide a much more manageable variety of algebras than that just mentioned. For all that matters to the connection with a logic of the type we are now envisaging is which equations of the specific form t = 1 (t a term) hold throughout the class of algebras in question. Thus consider in particular the class of BCK SM P -algebras, an equational class which properly includes the class of all BCK AS -algebras. (We restore the “AS” subscript here in the interests of clarity.) The “includes” is given by 2.13.16(ii); the example of Figure 2.13c shows the “properly” – and much else besides, for the general theory of BCK -algebras: see Blok and Raftery [1995], where this algebra is called B. One key point is easy enough to see: any model for the equational theory of BCK -algebras which is not itself a BCK -algebra must have a subalgebra isomorphic to B. We state the point as: Observation 2.13.21 A formula is provable in BCK logic if and only if it is valid in every BCKSM P -algebra. Proof. ‘If’: A formula valid in every BCK SM P -algebra is valid in every BCK AS algebra in view of the inclusion mentioned above, so the result follows from 2.13.18. ‘Only if’: By the ‘BCK ’ part of the definition of the notion of a BCK SM P algebra, B, C, and K and all their substitution instances are valid in every such algebra, and by (SMP), as in the proof of 2.13.8 (and 2.13.15), Modus Ponens preserves this property. We have thus shown BCK logic to be determined by an equational class of algebras, by dropping (AS) from the characterization of the class in question. It may seem undesirable to have distinct elements a, b, for which a → b = 1 and b → a = 1, such as the class of BCK SM P -algebras allows for, when the distinction makes no difference to sentential formulas evaluated in the algebras. In that case one can supplement the completeness half of 2.13.21 by the observation that although the relation holding between a and b when a → b = 1 is no longer a partial order, but only a pre-order, we can turn it into a partial ordering in accordance with 0.11.4 on the equivalence classes there described. The


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equivalence relation in question (which relates a and b when a → b = b → a = 1) is actually a congruence relation so the induced quotient algebra (0.23.1: p. 27) can be used to invalidate a BCK-unprovable formula – and here the invidious distinctions have been eliminated. We turn now to the concept of anti-commutativity, defined for the sake of stating a contrast between Hilbert algebras and BCK -algebras in 2.13.19(iv) above. Recall that according to the definition, a binary operation · (on A) is said to be anti-commutative just in case for all a, b (∈ A), we only have a · b = b · a for the case in which a = b. Thus being anti-commutative is as far from being commutative as a binary operation can get. (It cannot be that we always have a · b = b · a, since whenever a = b we must also have a · b = b · a.) Talk of a BCK -algebra, Hilbert algebra, etc., being anti-commutative, means that the operation → of the algebra is anti-commutative in the sense just defined. The anti-commutativity of (what in our terminology amount to) SKP AS -algebras is stressed by Abbott in the works cited in 2.13.17, but it is well worth noting that this holds likewise for the much broader class of Hilbert algebras (SK AS algebras). Let us abbreviate the name of the condition we want to show is satisfied by such algebras to “(AC)”, and repeat “(AS)” alongside it for the sake of comparison: (AS)

(x → y = y → x = 1) ⇒ x = y

(AC)

x → y = y → x ⇒ x = y.

It is thus clear that (AC) is a stronger condition than (AS), since it promises the same consequent on the basis of a weaker antecedent. Note, in connection with the labelling here, that while it is a binary operation (namely →) which (AC) says is anti-commutative, it is a binary relation (namely ) which (AS) is antisymmetric; the two terms would not make sense literally applied to any one thing. (For more on this theme, see 3.34, which begins on p. 497.) Observation 2.13.22 Hilbert algebras all satisfy (AC). Proof. Suppose a → b = b → a (in some Hilbert algebra); pre-multiplying by a, we get: a → (a → b) = a → (b → a). But the left-hand side is equal to a → b (since tW = 1) and the right hand side to 1 (since tK = 1); thus the conclusion that a = b follows by (AS). The appeal to W (or rather, to tW = 1) is essential here, as we see from the following example: → 1 2 3 4 Figure 2.13d

1 1 1 1 1

2 2 1 1 1

3 3 2 1 2

4 4 2 2 1

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Remarks 2.13.23(i) Figure 2.13d depicts a four-element BCK -algebra in which (AC) is not satisfied. That we are dealing with a BCK-algebra here is left to the reader to verify; the counterexample to (AC) comes from 3 → 4 = 4 → 3 although 3 = 4. (ii ) Note that the first three rows and columns of the above table describe a three-element subalgebra of the original, which is therefore a BCK -algebra, though this time one which does satisfy (AC) but does not satisfy tW = 1, showing that whereas all BCK algebras satisfying tW = 1 satisfy (AC), the converse is not the case. The three-element BCK -algebra mentioned in 2.13.23(ii) is of course none other than the →-reduct of the (algebra of) Łukasiewicz’s three-valued logic, as depicted in Figure 2.11a (p. 198). It is in fact (up to isomorphism) the only threeelement quasi-commutative BCK -algebra – where quasi-commutative means: satisfying tQ = 1. (More information about the algebraic side of Łukasiewicz’s many-valued logics, the MV -algebras of C. C. Chang, and BCK-algebras may be found in Komori [1978a], Font, Rodríguez Salas and Torrens [1984], Mundici [1986], Pałasiński and Wroński [1986]. A book-length treatment of these topics has appeared in the form of Cignoli, d’Ottaviano and Mundici [2000]; there is a concise historical survey in Cignoli [2007]. On the axiomatization of the pure implicational fragments of Łukasiewicz’s three-valued and denumerably infinitevalued logic, see respectively McCall and Meyer [1966] and Meyer [1966], and references therein.) From a logical point of view the anti-commutativity condition says that if an implication is equivalent to its converse, then its antecedent is equivalent to its consequent. We can consider this in the form of a rule:

(AC) Rule:

(A → B) → (B → A)

(B → A) → (A → B)

A→B

Here we have stated the conclusion of the rule simply as A → B, because the symmetry between the two premisses means that if for all A and B the stated conclusion follows from the premisses, then so does B → A. (Since we wish to avoid using connectives other than →, we eschew a formulation with a single premiss, (A → B) ↔ (B → A) and conclusion A ↔ B, more directly corresponding to our earlier gloss in terms of the equivalence of an implication with its converse. Note that, more explicitly, we should prefix the premisses and conclusion of the rule above with a “”.) Conclusions of the (AC) rule are certainly not derivable by application of Modus Ponens to the corresponding premisses and theorems of BCK logic, as 2.13.23(i) shows. The question remains open as to whether the rule is, nonetheless, admissible: whenever formulas of the form indicated by its premisses are provable in BCK logic, so is the corresponding conclusion. (The second premise already provably implies the conclusion in the Łukasiewicz many-valued logics as well as in IL, this being the content of the principle (Ł) mentioned in the Digression on p. 197.) We turn briefly to the case of BCI-algebras, by which is meant BCI AS algebras, and the associated logic, BCI logic. The association with the logic is in this case, far from as intimate as one might expect from the SK and BCK cases, but begin with a respect in which we do have a parallel:


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Observation 2.13.24 BCI-algebras satisfy (M P ): 1 → y = 1 ⇒ y = 1. Proof. Suppose 1 → a = 1, for some element a of a BCI -algebra; pre-multiplying by a we get: a → (1 → a) = a → 1. But the lhs here = 1 → (a → a), since we have x → (y → z) = y → (x → z) in BCI -algebras for the same reason as in BCK -algebras (i.e., the defining equation tC = 1 and the condition (AS)). Thus we have a → 1 = 1 → (a → a), and thus a → 1 = 1 → 1 = 1. From the fact that 1 → a = 1 = a → 1, we conclude by (AS) that a = 1, establishing (MP). The above proof was provided by Brian Weatherson, who observes, further, that we made two distinct appeals to the hypothesis that 1 → a = 1 in the course of the argument to the conclusion that a = 1, thus in effect exploiting the contraction principle W (not available in BCI logic) in the metalanguage. (Had we been conducting the meta-logical reasoning in a language itself governed by a contractionless logic, we should be entitled only to the claim that BCI -algebras satisfy the condition: 1 → y = 1 ⇒ (1 → y = 1 ⇒ y = 1). A similar remark appears in note 1, p. 386 of Restall [1993a].) The present case is interestingly different from the case of BCK -algebras, for which (2.13.14, p. 236) we simply noted the availability of the argument given for (MP) for Hilbert algebras (2.13.7), via the fact that all such algebras satisfy y → 1 = 1 (“y 1”). This is not a fact in the BCI case, however, and it is easy to see that the only BCI -algebras for which it holds are BCK -algebras, since it gives y → (x → x) = 1 and hence, commuting antecedents (“C”), we have x → (y → x) = 1 (that is, tK = 1). (See 2.13.26 on p. 244, however, for a proof of the ‘strong’ Modus Ponens property (SMP) in the case of BCI -algebras, which of course does not appeal to the K-equation.) We can use the fact that (MP) is satisfied by all BCI -algebras as we used the corresponding fact for SK - and BCK -algebras (2.13.8, 2.13.15): to mimic, in theorizing about the class of algebras, deductions in the sentential logic, with (MP) doing the work of Modus Ponens: Corollary 2.13.25 If a formula is provable in BCI logic, then it is valid in every BCI-algebra. This, of course, is the soundness half of what would be a soundness and completeness theorem if there were one – but it is here that the parallel with the SK and BCK -cases breaks down. (Bunder [1983a], Kabziński [1983b].) The most conspicuous source of trouble is the fact—itself most easily established with the aid of the semantics for relevant logic introduced in 2.33—that the formula (10) (10)

(p → p) → (q → q)

is unprovable in the relevant logic R described in that section. ((10) violates Belnap’s criterion of relevance: that the antecedent and consequent of a provable implication should have at least one propositional variable in common.) Since BCI logic is a subsystem of R (whose implicational fragment, as we have already mentioned, is given by BCIW ), (10) is not BCI -provable. But of course, the equation corresponding by the tA translation to (10), namely (11)

(x → x) → (y → y) = 1

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does hold in every BCI -algebra, since tI , namely x → x = 1 is satisfied by exactly the algebras that y → y = 1 is, so we have x → x = y → y = 1, and (11) is just another way of saying 1 → 1 = 1. The point here is simply that BCI logic, unlike BCK logic, is not what we have called monothetic: the two provable formulas p → p and q → q are not synonymous in the logic—with (12) here being provable and (10) above not, for instance: (12)

(p → p) → (p → p).

And as we saw at the beginning of the present subsection (2.13.1, p. 221), being monothetic is a necessary (as well as a sufficient) condition for a logic (in Fmla) to be determined by a unital matrix. By 2.12.1, determination by a class of unital matrices coincides with determination by a single such matrix, once we observe that the matrix delivered by the direct product construction from a family of unital matrices is itself a unital matrix. Thus only a monothetic logic could be determined by the class of BCI algebras, considered as matrices with 1 as the designated value. (Recall that we are discussing Fmla; this last reasoning, concerning products, fails in Set-Fmla, as we noted in 2.12.6. A discussion of determination by unital matrices in the latter framework may be found at 3.23.9, p. 458.) The point at which a Lindenbaum algebra completeness proof in the style of 2.13.18 would break down is clear: in specifying the 1 of the algebras concerned in that result, we exploit the fact this could be taken as the equivalence class of any provable formula, since the latter were all synonymous: a consequence of the now-absent K. We conclude the present subsection with a result on the subject of the ‘strong’ Modus Ponens condition (SMP) for BCI-algebras and a remark on the issue of non-monotheticity. More information on BCI and BCK logics and the corresponding classes of algebras, as well as some issues in contractionless logic, may be found in 7.25. Observation 2.13.26 Every BCI-algebra satisfies the condition (SMP): 1 → y = y. Proof. As in the case of 2.13.10 (for Hilbert algebras) we show (i) that (1 → y) → y = 1 is satisfied in all BCI -algebras, by seeing this as a special case of x → ((x → y) → y) = 1, appealing to the BCI -provability of the corresponding formula, and then (ii) that y → (1 → y) = 1 is also satisfied in all BCI -algebras; from (i) and (ii) the result follows by (AS). For (i), then, we begin by noting that BCI logic proves (p → q) → (p → q), since this is a substitution instance of I. Then substituting p → q for p, p for q and q for r in C, we get ((p → q) → (p → q)) → (p → ((p → q) → q)), so by Modus Ponens, BCI logic proves (the “Assertion” formula): p → ((p → q) → q).


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Thus by 2.13.25, every BCI -algebra satisfies x → ((x → y) → y) = 1 and thus, taking x as 1 and appealing to 2.13.24, we conclude that every BCI -algebra satisfies (1 → y) → y = 1. For (ii) note that by two appeals to the equation corresponding to I, we see that every BCI-algebra satisfies (y → y) → (y → y) = 1 and y → y = 1, and therefore 1 → (y → y) = 1, so permuting antecedents (justified by (AS) and the equation corresponding to C) we get y → (1 → y) = 1, which is (ii).

Remark 2.13.27 The fact that BCI logic is not monothetic may prompt us to explore some monothetic extension of the logic – perhaps the least monothetic extension. (We shall do so in 7.25; see also §4 of Humberstone [2006b].) Alternatively, we may consider some variation on the semantics in terms of BCI -algebras, relative to which a soundness and completeness result can be obtained for BCI logic. Meyer and Ono [1994] take the latter course, understanding by a BCI -algebra an algebra of the type we have been considering (more precisely, of type 2, 0 ) supplemented by a partial ordering , not, as in the above discussion, taken as defined, but rather as a separate primitive and subject to the condition that a b iff 1 a → b. (Thus the structures concerned are not strictly speaking algebras at all, but Meyer and Ono already have a different use for the phrase “BCI structures” in their discussion.) This allows them to state a soundness and completeness theorem w.r.t. the resulting expanded algebras on which a formula A is counted as valid just in case 1 h(A) for every evaluation h (rather than, as on our understanding, in case 1 = h(A) for each h). This falls outside of the purview of the present subsection since it is not then a case of unital matrix semantics, though it is very much within the style of algebraic semantics associated with the relevance logic tradition (cf. Dunn [1975a]). Another problem with Meyer and Ono [1994], is mentioned in note 14 of Humberstone [2002a]. The definitions of BCK - and BCI -algebras with which we have been working, using the tB = 1, tC = 1 etc. identities and the quasi-identity (AS), can be simplified considerably, and we close with an illustration of this for the latter case. (For the former, see Blok and Raftery [1995].) The definition figuring in the following exercise appeared in Dudek [1986] and Kabziński [1993], for example; note that only one of the identities uses more than a one variable, and exploits the fact already demonstrated (2.13.26) that we can use the strong (SMP) Modus Ponens condition. Exercise 2.13.28 Show that the class of BCI -algebras is precisely the class of algebras (A, →, 1) of similarity type 2, 0 satisfying the conditions (SMP), tB = 1, and (AS). (Recall that B is B with its antecedents commuted: a suffixing rather than a prefixing principle. One small hint: substituting 1 for the first two variables in tB = 1 gives something which, appealing to (SMP) will yield 1 → (z → z) = 1 and thus, by (SMP) again, (z → z) = 1, i.e., the identity corresponding to the axiom I.)


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2.14

-Based Algebraic Semantics

We turn now to what was described as -based semantics in our opening remarks in 2.13. For the first time we consider a logic in the framework Fmla-Fmla. The language we shall consider has connectives ∧ and ∨, and the sequents have he form A B for formulas A and B of this language. (We could consider instead the implicational language of the bulk of the preceding subsection, with the partial order on BCI and BCK algebras as defined there, but it seems appropriate instead to air some aspects of the logic of conjunction and disjunction.) We call the proof system we are about to define Lat (for lattice). In this framework, the rule (M) has no application, (R) applies as usual to give all sequents B B, and (T) takes us from A B and B C to A C. (Thus the analogue of a consequence relation or of a gcr here would simply be a pre-order of the set of formulas; the analogue of the abstract completeness theorems 1.12.3 (p. 58), 1.16.3 (p. 75), is given as Theorem 4.2 in Humberstone [1992], to the effect that every pre-order is determined by the class of all valuations which are in the obvious sense consistent with it. The pre-order vs. partial order distinction will come up presently.) So much for the structural rules of Lat. For the two connectives, we give the four obvious zero-premiss rules: A∧BA

A∧BB

AA∨B

B A ∨B

and two two-premiss rules, versions of (∧I) and (∨E) from Nat (1.23) with formula-variables rather than set-variables on the left, as is called for by the current framework: CA

CB

CA∧B

AC

BC

A∨BC

Now suppose that A = (A, ∧A , ∨A ) is a lattice whose associated partial ordering (§1.1) is (which we would write as “A ” if we were being fussy). Then a Fmla-Fmla sequent B C for B, C in the current language will be said to hold on a homomorphism h from that language to A when h(B) h(C), and to be valid in A when it holds on every such homomorphism. The informal idea behind these definitions is that we think of the elements of A as propositions between which represents a relation of implication or entailment, and to which a homomorphism h maps formulas thereby treating a given formula as expressing a given proposition: one formula, C, is then deemed to follow from another, B, when however this mapping is done the corresponding propositions stand in the appropriate relation of implication: this is what makes the sequent B C valid in the given lattice. Let us call formulas A and B equivalent (in Lat) when the sequents A B and B A are both provable in Lat. As above, let C be a formula containing some propositional variable which is replaced by A to give C(A) and by B to give C(B); then one may show by induction on the complexity of C that if A and B are equivalent, so are C(A) and C(B), so that equivalent formulas are synonymous for the present logic, which could accordingly, by a minor adaptation of the terminology to be given in 3.31, be termed a ‘congruential’ logic. (See further 2.14.7(ii).) We use this in our proof of the completeness of Lat in 2.14.3, but the first half of showing that Lat lives up to its name is a soundness result w.r.t. the current -based semantics: Observation 2.14.1 Every sequent provable in Lat is valid in every lattice.

2.1. MANY-VALUED LOGIC AND ALGEBRAIC SEMANTICS Proof. By induction on the length of proofs from the above basis for Lat.

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Corollary 2.14.2 Not every tautologous sequent of Fmla-Fmla in the language of Lat is provable in Lat. Proof. Consider the sequent p ∧ (q ∨ r) (p ∧ q) ∨ (p ∧ r) and note that we can find a homomorphism into a non-distributive lattice on which this sequent does not hold. For example, the five-element lattice of 0.13.6 (p. 10) can be used for this purpose.

Observation 2.14.3 Every sequent of Fmla-Fmla which is valid in every lattice is provable in Lat. Proof. We define a single lattice, like the Lindenbaum algebras in the proof of 2.13.1 (p. 221), in which every unprovable sequent is invalid. Let us call this lattice A = (A, ∧A , ∨A ); A consists of the equivalence classes [B] of formulas B where ≡ is the synonymy relation referred to in the comments preceding 2.14.1 and [B] = {C | B ≡ C}, and the operations ∧A and ∨A are defined as in the proof of 2.12.1. Verify that this is indeed a lattice and that, [A] [B] in this lattice just when the sequent A B is provable in Lat. Thus the homomorphism which maps a formula C to its synonymy class [C] is one on which any unprovable sequent A B fails to hold. To compare the present -based semantics with the unital matrix semantics of the preceding subsection, suppose we consider bounded lattices, with top and bottom elements 1 and 0; the former will be used as the designated element, and the latter does not interfere so we include it for good measure. A sequent B C of Fmla-Fmla is valid in a matrix (A,{1}), in which A is an algebra of the similarity type of bounded lattices, if for every homomorphism h from the language of Lat to the ∧, ∨-reduct of A, h(B) = 1 implies h(C) = 1. Let us call the sequents valid in this sense in every matrix whose algebra is not only of the same similarity type as a bounded lattice but actually is a bounded lattice, 1-valid, and every sequent valid in the -based sense in every lattice -valid. Note that every -valid sequent must be 1-valid, though as we shall now see, the converse does not hold: Exercise 2.14.4 Show that the Lat-unprovable ‘distribution’ sequent from the proof of 2.14.2 is 1-valid. (Note that we are still considering 1-validity in terms of arbitrary lattices, and not specifically distributive lattices.) This discrepancy between 1-validity and -validity does not mean that the latter notion is not amenable to description in matrix terms as in 2.13, with validity going by preservation of the property of having a designated value. It’s just that we have to allow for more than one designated value. For note that a sequent is valid in the -sense in a lattice A just in case it is valid in each of the matrices (A, Da ) where for a ∈ A, Da = {b ∈ A | a b}. The discrepancy between 1-validity and -validity cannot, incidentally, arise with distributive

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lattices, by certain considerations about prime filters – filters in the sense of 0.23.1(ii), p. 27, to which the join of two elements only belongs when at least one of these elements does – which §0.2 did not go into. (For those familiar with this terminology, the point is that whenever, in such a lattice not a b, then there is a prime filter containing a but not b; but the characteristic function of a prime filter in a lattice A is a homomorphism from A to the two-element chain, sending a to 1 and b to 0.) As in 2.13, each formula B of the propositional language under consideration corresponds to a term tB apt for interpretation in algebras of the same similarity type as that language, so that corresponding to the Lat formula p ∧ (q ∨ r), for example, we have the term x ∧ (y ∨ z), interpreted in (specifically) a lattice as denoting the meet of x with the join of y and z. (0.22.) Continuing to restrict our attention to lattices (amongst algebras of type 2, 2 ) we note that in the framework Fmla-Fmla over the language of Lat, we each sequent B C similarly corresponds to -statement tB tC , and thus to an equation tB = tB ∧ tC (or equivalently tC = tB ∨ tC ). As we can extend the equational theory of lattices to obtain the equational theory of distributive lattices, we can make a parallel extension of Lat to a proof system we call DLat, adding the distribution schema as a new zero-premiss rule: A ∧ (B ∨ C) (A ∧ B) ∨ (A ∧ C). Exercise 2.14.5 Prove the analogues of 2.14.1 and 2.14.3 which result from changing “Lat” to “DLat”, and references to lattices into references to distributive lattices. Having arrived at DLat, however, there is no further room for growth, unless we expand the language. This is a (propositional) logical analogue of the equational completeness of the variety of distributive lattices (0.24.6(ii), p. 32), and would normally be expressed by saying that the system DLat is Post-complete. What this can be taken to mean—several variations are possible—is that any proper extension in the same language, closed under Uniform Substitution, of its class of provable sequents, is inconsistent in the sense of containing every sequent in the given framework (Fmla-Fmla here). The general theme behind the various notions of Post-completeness in the literature is that one fixes on a language and a particular logical framework, and considers the lattice of logics in that framework, typically requiring these to be substitution-invariant (cf. Appendix to §1.2 – starting at p. 180). The Post-complete logics are then the dual atoms of this lattice (covered, that is, by the inconsistent logic, in which all sequents for the given framework, in the language concerned). We shall meet this concept from time to time in various logical frameworks; for the case of Fmla, see 7.25.3 (p. 1100) and surrounding discussion. We can extend the -based account of validity to the frameworks Set-Fmla and Set-Set. We simply define a sequent C1 ,. . . ,Cm D1 , . . . , Dn (with n = 1, for the former case) to hold on an evaluation h in the bounded lattice A = (A, ∧A , ∨A , 1A , 0A ) when h(C1 ) ∧A . . . ∧A h(Cm ) h(D1 ) ∨A . . . ∨A h(Dn ), taking the meet for m = 0 as 1A and the join for n = 0 as 0A . (The definition makes sense even if there are further fundamental operations, so we should really be speaking about algebras which are expansions of bounded lattices.) We may as well suppose here that the language also includes the zero-place connectives and ⊥, which evaluations are required to map to the top and bottom elements,


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and for which appropriate rules are provided. (Note that since we take sequents always to involve only finitely many formulas on the left or right of “”, we do not need infinite joins and meets, and so do not need to suppose we are working only with complete lattices.) A sequent of Set-Set or Set-Fmla is then valid, as before, in a lattice when it holds on every evaluation into that lattice. And, as with Lat in Fmla-Fmla, we could now ask about the shape of proof systems delivering proofs for precisely the sequents of these frameworks which are valid in every lattice, or again in every distributive lattice. We defer consideration of such questions until 2.31. Exercise 2.14.6 Show that the -based notion of validity, generalized as above to Set-Set, in the algebra of the Kleene matrices of 2.11, coincides with the property of being provable in both Kle 1 and Kle 1,2 . (Recall that we presume the ordering 3 < 2 < 1.) Hint: Check that this -based notion of validity amounts to the same as being valid in both of the matrices K1 and K1,2 ; then the result follows by 2.11.9 on p. 207. (The algebra here, as well as the fourelement algebra of the matrix described in 3.17.5, p. 430, for Kle, is a quasi-boolean algebra in the sense of 0.21.) By way of comment on the above exercise, we remark that the intersection of the consequence relations determined by K1 and K1,2 figures in Townsend [1989] under the heading (intelligible in the light of the exercise) of the ‘non-decreasing criterion of validity’, where a pleasant description is provided in terms of normal forms (in the style of Anderson and Belnap [1975], pp.157f.). The associated gcr – on a language with some additional connectives not discussed here, though see 7.19 – is the favoured ‘partial logic’ of Blamey [1986], [2002], who thinks in terms of forward truth-preservation (1-preservation) and backward falsitypreservation (3-preservation). Both authors regard the value 2 as signalling the absence of a real truth value (as arising in cases of presupposition-failure, etc.) and would be unhappy with the description of their work as ‘many-valued’. (This same gcr was urged in Cleave [1974] as providing a logic for vagueness.) Blamey thinks of the ‘absence of a value’ construal of taking (as we put it) the value 2 as motivating a special notion of expressive completeness distinct from the general notion of functional completeness. The latter, as we shall see in more detail in 3.14 – p. 403 – for the two-valued case, is a matter of providing for the expression of every n-ary function from the given set of values to that set, whereas Blamey’s suggestion is that for the construal in question the only functions whose expression is required (or indeed permitted as genuinely intelligible on the present construal) are those which are monotone (order-preserving) w.r.t. the partial order according to which 2 1 and 2 3, while 1 and 3 are incomparable. (Thus x y means that x is equal to or ‘less defined than’ y.) See Blamey [1986] or [2002] for details, including the connection with Kleene’s tables, and Kijania-Placek [2002] for similar sentiments. As it happens, monotonicity w.r.t. this same order would also seem to be required by Łukasiewicz’s motivation for his three-valued logic, whereas the function he associates with → is lacks the property, since we get a counterexample to x1 y1 and x 2 y2 imply x 1 → x2 y1 → y2 by taking x1 = x2 = 2 and y1 = 1, y2 = 3; in this case x1 → x2 = 1, which is certainly not y1 → y2 = 3. The effect of this is that the kind of changes

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in truth-value with the passage of time that Łukasiewicz envisaged, in which statements once indeterminate (having value 2) become either determinately true or determinately false (values 1, 3 resp.), have repercussions for longer statements containing them of a type Łukasiewicz did not envisage, in which a statement can change in value from 1 to 3 or vice versa. This point, and several examples of illustrating it, may be found in note 6 of Seeskin [1974]. (A notion of monotone truth-function is defined in 4.38.7 below – p. 621), – but there the order being preserved is one according to which only two values T and F are involved and we have F < T.) Remarks 2.14.7(i) A -based notion of validity in Set-Set (and thus SetFmla) can be given an which does not presume that we dealing with (expansions of) bounded lattices, but required only the weaker assumption that the elements are partially ordered by . As usual, validity will be a matter of holding on each h, but instead of saying that C1 ,. . . ,Cm D1 , . . . , Dn holds on h in the algebra A concerned when h(C1 ) ∧A . . . ∧A h(Cm ) h(D1 ) ∨A . . . ∨A h(Dn ), we can define such a sequent to hold on h when for all c, d ∈ A: If c h(Ci ) for each i = 1,. . . ,m, and d h(Dj ) for each j = 1,. . . ,n, then c d. This gives a notion of holding on h (and hence of validity) equivalent to that provided in terms of meets and joins in the circumstance that such meets and joins exist (as glb’s and lub’s w.r.t. ). (ii ) One may for certain purposes want something in the same vein as, but more general than, -based semantics even with the alternative definition of validity supplied in (i), for the following reason. As currently conceived, only congruential logics – those, as mentioned above, satisfying a ‘replacement of provable equivalents’ condition – can be treated using this semantic approach. For if B C and C B are valid in an algebra A with partial ordering , then for each evaluation h into A, h(B) h(C) and also h(C) h(B), which implies, by the antisymmetry of , that for each such h, h(B) = h(C), in which case B and C are interreplaceable in compounds without affecting the holding of sequents constructed using those compounds on h. The obvious remedy for such a defect, if that is what it is deemed to be, is simply to drop the antisymmetry condition, and recast the semantics in terms of pre-orders rather than partial orders.

2.15

A Third Version of Algebraic Semantics: Indiscriminate Validity

In the preceding subsection we had occasion (2.14.6, p. 249) to recall the logic determined by {K1 , K1,2 }, comprising the two Kleene matrices first introduced in 2.11. Let us stick with this example to illustrate a general phenomenon. Some sequents belonging to this logic (valid in both matrices) are there because of the


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restriction to the particular matrices concerned on the given algebra, and some are there regardless of any such restriction. An example of the former class was given in 2.11.2(i), p. 201: p ∧ ¬p q ∨ ¬q. Representative of the latter class would be, for example, any sequent of the form A A ∧ A. If we have a matrix on an algebra A in which the operation ∧A is idempotent, then we do not have to enquire as to which elements are designated: we know that regardless of matters of designation, a sequent of the latter form must be valid, because if h(A) is designated, so is h(A ∧ A) = h(A) ∧A h(A) = h(A). Accordingly, let us say that a sequent is indiscriminately valid in an algebra A, to mean that it is valid in every matrix based on A. Mostly in the ensuing discussion we shall simply say that the sequent in question is ‘valid in’ A, since in this subsection this is the principal notion of validity at issue. (We sometimes include the ‘indiscriminately’ for emphasis.) Given a class of algebras of the same similarity type as some sentential language, even if they do not have a particularly salient distinguished element to designate, giving unital matrix semantics, and even if they are associated with no partial ordering in terms of which a -based semantics seems especially natural, we can still consider what logic is determined by the class in the following sense: the set of all sequents (indiscriminately) valid in every algebra in the class. Despite its wide availability, this particular notion of algebraic validity does not receive much attention in the literature. (Though for a closely related notion, see the discussion leading up to 2.15.9 below.) That is because it has an unusual feature which may well be thought to deprive it of much logical interest, to which we now turn. Although we have been referring to the matrix-evaluations throughout this section, note that whether h has this status for a given matrix depends only on the algebra of the matrix, so for the present subsection, where A is such an algebra (presumed to be of the same similarity type as whatever language is under discussion) we shall instead speak of A-evaluations, to mean homomorphisms from the language to A. Further, let us call formulas B and C A-equivalent, writing “B =A C” when for every A-evaluation h, we have h(B) = h(C). For sequents of Fmla-Fmla, there is a very direct connection between this concept and that of the indiscriminate validity of a sequent in an algebra: Observation 2.15.1 B C is indiscriminately valid in A iff B =A C. Proof. ‘If’: Clear. ‘Only if’: Suppose that B is not A-equivalent to C; thus for some A-evaluation h, h(B) = h(C). Consider the matrix (A, D) with D = {h(B)}. B C does not hold on h in this matrix, and so is not valid in this matrix, and so is not indiscriminately valid in the algebra A.

Corollary 2.15.2 If B C is indiscriminately valid in an algebra A then so is C B. It is this corollary which reveals that aspect of indiscriminate validity in an algebra which may seem to render it only of marginal specifically logical interest, since we do not expect an account of what follows from what according to which when C follows from B, B automatically follows from C also. (And of course we do not have this result with validity in a matrix—unital or not—or of validity

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on the -based semantics of the preceding subsection.) We shall continue to consider the notion nonetheless; after all, it is not without technical interest as providing a sufficient condition for the arguably worthier notion of validity in a matrix (based on the algebra in question). Obs. 2.15.1 deals with sequents of Fmla-Fmla, or at least with sequents of Set-Set or Set-Fmla which do not exploit the extra possibilities of those frameworks. What happens in these richer frameworks more generally? One might expect, based on 2.15.1, that in the case of the former framework that Γ Δ would be valid in an algebra A just in case some formula in Γ was Aequivalent to some formula in Δ, and that in the case of Set-Fmla, Γ C would be valid in A just in case C was A-equivalent to some element of Γ. In fact the correct formulations are somewhat different, and though that for Set-Set entails that for Set-Fmla, we record them separately for the reader’s convenience; the “h(Γ)” notation is as explained in 2.11 (just before (1) on p. 199). Observation 2.15.3 (i) Γ Δ is indiscriminately valid in A if and only if for every A-evaluation h, h(Γ) ∩ h(Δ) = ∅. (ii) Γ C is valid in A if and only if for every A-evaluation h, h(C) ∈ h(Γ). Proof. It suffices to prove the ‘only if’ direction of (i). Suppose that for some A-evaluation h, h(Γ) ∩ h(Δ) = ∅. Let D = h(Γ) and proceed as in the proof of 2.15.1. Although the proof indicated for 2.15.3 establishes that claim without establishing what was described above as the result one might expect, namely that (in the case of Set-Set) Γ Δ is valid in A just in case some formula in Γ is A-equivalent to some formula in Δ, we have only asserted, and not actually demonstrated, that the latter claim is false. Let us note a consequence of the claim, with a view to refuting it by means of a simple counterexample: Remark 2.15.4 If the claim under discussion above were correct, then we should have the following left-primeness property for Set-Fmla: if A (indiscriminately) validates Γ C, then for some B ∈ Γ, A validates B C. (Cf. 1.21.5, p. 110.) In the more general framework of Set-Set, the corresponding consequence – ‘two-sided’ primeness – would be: if A validates Γ Δ, then for some B ∈ Γ, C ∈ Δ, A validates B C. The example which follows shows the Set-Fmla property (and a fortiori the Set-Set property) to fail. Example 2.15.5 Consider the two-element algebra of truth-values on which the familiar (boolean) matrix for CL is based. For any formulas A, B, the sequent A, B A ∧ B is indiscriminately valid in this algebra because for x, y ∈ {T, F}, x ∧ y (a candidate value of h(A ∧ B)) is always equal to x or to y, thus to h(A) or h(B) respectively – though in the context of this algebra we usually write h(A), etc., as v(A). (Here we are using “∧” not only for the connective, but for the associated operation; in a more explicit notation introduced in 3.14 below (p. 403), we denote the latter by “∧b ”, with the subscript intended to suggest boolean. Alternatively, where A is the 2-element boolean algebra, we could write ∧A . Often


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this particular algebra is simply called 2, with its distinguished elements 1 and 0 written as T and F, as we elide the now insubstantial distinction between v and hv .) Thus p, q p ∧ q is indiscriminately valid in the algebra in question, whereas neither p p ∧ q nor q p ∧ q is, thereby refuting (via 2.15.4) the suggestion that whenever Γ C is valid in an algebra A, C is A-equivalent to some B ∈ Γ. The example just given had to be chosen with some care; for example, we could not have used the sequent p, q p ↔ q. Though valid in the standard matrix, with D = {T} (i.e., tautologous), putting D = {F} would allow h(p) = h(q) = F with h(p ↔ q) = T as an evaluation on which the sequent fails. In this case, then, the element of ‘discrimination’ involved in paying attention to what the set of designated values is to be does make a difference. Exercise 2.15.6 (i) Say which of the following sequents are indiscriminately valid in the algebra described in 2.15.5: p p ∨ q; p, p → q q; p, q p ∨ q; p, p ↔ q q. (ii ) Show that if both A B and B A are tautologous, then A B is indiscriminately valid in that two-element algebra. (iii) Does it follow from the validity (in the designation-preserving sense) of each of A B and B A in any given matrix based on a three-element algebra, that A B is indiscriminately valid in that algebra? Justify your answer. The falsity of the claim queried in (iii) of this exercise notwithstanding, the symmetry of the condition cited in 2.15.3(i) suffices for an analogue of the untoward result given as 2.15.2 above: Corollary 2.15.7 If Γ Δ is indiscriminately valid in A then so is Δ Γ. Thus the gcr determined by A (to use this phrase in the obvious sense without explicitly defining it) is always a symmetric relation. This is not an unheard of property for a gcr; for example the smallest gcr on any language possesses it (since for this gcr , we have Γ Δ iff Γ ∩ Δ = ∅), and we shall encounter it again in 8.11.6 as a feature of the pure negation fragment of CL (in Set-Set). (In the notation of 1.19.3, p. 93, this means that = −1 .) However, what is perhaps special about its appearance in the present setting is that the symmetry holds for each algebra A, and not just for certain special cases. We return to the topic of symmetry below, after 2.15.11, for the present noting only that the ‘rule’ form of symmetry (from: Γ Δ to: Δ Γ) is clearly a structural rule, in the sense of not involving in its formulation reference to any particular connectives; this prompts us to think about the status of the more familiar structural rules in the present setting: Exercise 2.15.8 The Set-Set structural rules (R) and (M) obviously possess and preserve (respectively) the property of being indiscriminately valid in any given algebra, though this is less clear for the case of (T). Show that this rule preserves validity in an algebra using the characterization provided by 2.15.3(i).

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Although we introduced indiscriminate validity, at the start of this subsection, as a notion of sequent validity which competes with that provided by our earlier samples of algebraic semantics, -based validity and preservation of the designated value on unital matrix semantics, there is a natural variant on the indiscriminate notion suggested by the latter. Instead of considering validity in every matrix on the algebra in question, we could consider validity in every unital matrix on that algebra. This notion is deployed to considerable effect in Rautenberg [1991], and we shall this conception of validity Rautenberg-validity. (Rautenberg does not actually write in terms of sequent validity, but we have translated his discussion—which considers consequence relations en bloc rather than isolating the individual sequents—into our terms here. The present notion was introduced in Rautenberg [1985], p. 4, with the terminology “algebraic consequence”.) Clearly, if a sequent is indiscriminately valid in an algebra, it is Rautenberg-valid in that algebra. To see that the converse fails, consider: Example 2.15.9 Expand the sentential language of 2.11 by a constant (zeroplace connective) c, and the algebra of the Kleene matrices correspondingly by taking the value 2 as a distinguished element for which we require that in every evaluation h, h(c) = 2. In the resulting algebra, the sequent p, ¬p c is Rautenberg-valid, since the only value for which it and the result of applying ¬ to it coincide (and so for which each of the formulas on the left can assume it as a designated value when exactly one value is designated) is 2, which is the value assumed by c. But this sequent is not indiscriminately valid in the algebra in question, since we can choose {1, 3} as our set of designated elements. (Note however that as an immediate consequence of the definitions a sequent of the form A B is Rautenberg-valid in an algebra iff it is indiscriminately valid in that algebra.) Digression. Rautenberg [1991] also considers what, adapting his own terminology slightly, we will call coset-validity for sequents of Set-Fmla, which is intermediate between indiscriminate validity and what we have called Rautenbergvalidity. By cosets of an algebra A here is meant ∅ together with any equivalence class of a congruence relation on A. Γ B is coset-valid in A if it is valid (in the usual designation-preserving sense) in any matrix (A, D) with D a coset of A. (See 0.23 for the notion of a congruence.) A simple algebra is one in which the only congruence is the identity relation, so coset validity on a simple algebra is almost the same as Rautenberg-validity, differing only because of the status of ∅ as an honorary coset. End of Digression. Following the suggested terminology of 2.15.4, we call a gcr two-sidedly prime when Γ Δ implies C D for some C ∈ Γ, D ∈ Δ. Observation 2.15.10 If is the gcr determined, in the indiscriminate validity sense, by a class of algebras which is closed under taking direct products, then is two-sidedly prime. Proof. To save space we prove an illustrative case from which the general case is easily extracted. Suppose the gcr determined by a class K of algebras (of the same similarity type as the language of ) closed under direct products, but that lacks the primeness property mentioned. Then there are formulas A, B,


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C, D, with A, B C, D while (i) A C, (ii) A D, (iii) B C, (iv) B D. Thus we have four homomorphisms h1 , h2 , h3 , h4 , into algebras A1 , A2 , A3 , A4 in K, with h1 (A) = h1 (C), h2 (A) = h2 (D), h3 (B) = h3 (C), and h4 (B) = h4 (D). Then the A1 ⊗A2 ⊗A3 ⊗A4 -evaluation h1 ⊗ h2 ⊗ h3 ⊗ h4 invalidates the supposedly -provable sequent A, B C, D. The notation we use at the end of this proof here (“h1 ⊗ h2 ⊗ h3 ⊗ h4 ”) is to be understood as in the discussion after 2.12.7 (p. 216). Remark 2.15.11 We could cast the point of 2.15.10 in the context of a generalized version of equational logic rather than sentential logic by allowing, for sets of terms T , U of some fixed similarity type ‘generalized equations’ of the form (say) T // U , understood to hold in an algebra of the type in question just in case every assignment h to the variables of T ∪ U results in h(t) being the same element of that algebra as h(u), for some t ∈ T , u ∈ U . The above point is then that T // U holds over a class of algebras closed under direct products only if for some t ∈ T , u ∈ U , the generalized equation {t} // {u} (which amounts to the traditional ungeneralized equation “t = u”) holds over the class. (On the other hand, just like equational classes themselves, all generalized equational classes are closed under subalgebras and homomorphic images.) A related generalization of equational logic may be found in Furmanowski [1983]. We illustrate the case of two-sided primeness below (2.15.14). Let us return to symmetry as a property of gcr’s, this time considering the valuational semantics of Chapter 1 rather than algebraic semantics; there is a natural analogue of the idea behind indiscriminate validity, since given a class V of (bivalent) valuations, we may define a gcr ∩ V thus: (1) Γ ∩ V Δ if and only if for all v ∈ V, v(Γ) ∩ v(Δ) = ∅. For notational parity with ∩ V as defined by (1) we use the notation V for the gcr determined in the usual truth-preservation sense by V (i.e., Log(V ) in the sense of 1.16). In (1) the notation v(Γ) denotes {v(C) | C ∈ Γ}; using this notation the definition of ∩ V is (2) Γ V Δ if and only if for all v ∈ V, v(Γ) ⊆ {T} ⇒ T ∈ v(Δ) (Warning: this differs from the convention for “v(Γ)” introduced in Chapter 1, on p. 58, in that what was there expressed as v(Γ) = T would on the present convention be expressed by: v(Γ) ⊆ {T}, or equivalently, by: F ∈ / v(Γ)). Lemma 2.15.12 For any class V of valuations, ∩ V ⊆ V . Proof. If Γ V Δ then for some v ∈ V , v(Γ) ⊆ {T } while v(Δ) ⊆ {F }. For such a v, then, v(Γ) ∩ v(Δ) = ∅, so Γ ∩ V Δ.

Observation 2.15.13 For any class V of valuations, ∩ V = V if and only if V is symmetric.

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Proof. ‘If’: By 2.15.12, we always have ∩ V ⊆ V , so it suffices to show, on the ∩ hypothesis that ∩ V is symmetric, that V ⊆ V . For a contradiction, suppose ∩ instead that Γ V Δ while Γ V Δ. The latter implies that either (i) v(Γ) ⊆ {T} and v(Δ) ⊆ {F} or else (ii ) v(Γ) ⊆ {F} and v(Δ) ⊆ {T}. (i) immediately conflicts with the supposition that Γ V Δ, while (ii ) conflicts with the fact that, since Γ V Δ and V is symmetric, we have Δ V Γ. ∩ ‘Only if’: Suppose that ∩ V = V . Since V is evidently symmetric, so too is V . In some cases, such a result means that a symmetric V is determined by a two-element algebra (or class of such algebras) where “determined” is taken in the sense of the indiscriminate validity version of algebraic semantics. For instance, this is so when V is taken as the class of ¬-boolean valuations (and the only connective in the language is ¬) – the case treated in 8.16 below. But it is not hard to see that even restricting attention to substitution-invariant symmetric gcr’s, not all of them are determined (‘indiscriminate validity’ sense) by classes of algebras. An obvious further necessary condition is congruentiality, briefly mentioned in 2.14 above (§3.3 for details). Suppose we have a 1-ary connective # and formulas A and B are A-equivalent for each A in some putative determining class K. This implies that #A and #B are also A-equivalent for each such A, so we must have #A #B if we have A B (“# congruential according to ” in the terminology of 3.31, which has the upshot that -equivalent formulas are synonymous when each connective in the language of is congruential according to ); because of symmetry, we can of course reformulate this as: A B implies #A #B (or indeed: A B implies #B #A). Setting aside the question of what additional syntactic conditions (if any) might be required to guarantee the amenability of to semantic treatment via indiscriminate validity, we close instead with a simple example of such treatment for a two-sidedly prime gcr. Theorem 2.15.14 Let be the smallest congruential gcr in the language with one binary connective ◦ such that for all formulas A, B of that language, we have A ◦ B B ◦ A. Then is sound and complete, in the sense of the indiscriminate validity semantics, w.r.t. the class of all commutative groupoids. Proof. Soundness is clear, so we turn to completeness, for which we need the Tarski–Lindenbaum algebra of , with elements [A] for formulas A of this language, where [A] = {A | A A }, and fundamental operation ◦ (we use the same notation as for the connective), well-defined in view of congruentiality, by: [A] ◦ [B] = [A ◦ B] Since we required that A ◦ B B ◦ A, this algebra is a commutative groupoid, and if Γ Δ, then on the evaluation h for which h(A) = [A] for all formulas A, we have h(Γ) ∩ h(Δ) = ∅ (since otherwise we would have Γ Δ).

Exercise 2.15.15 (i) Why have we not had to stipulate that the gcr of the preceding result should be symmetric, as well as being congruential? (See below.)


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(ii ) Clearly a gcr cannot be determined by a class of algebras in the indiscriminate validity sense unless it is substitution-invariant; why then did we not have to stipulate for the of 2.13.14 (p. 236) that was not only congruential but also substitution-invariant? We said earlier that 2.15.14 would illustrate the case of a two-sidedly prime gcr, so it is perhaps appropriate to note explicitly why this is so. We could give an inductive argument based on the syntactic characterization of (cf. 1.25); but a justification of this claim from the semantic perspective lies more immediately to hand: since is sound and complete w.r.t. (is determined by) the class of commutative groupoids—understanding this terminology in the ‘indiscriminate validity’ sense—the claim follows immediately from 2.15.14 by 2.15.10, since, this being an equational class of algebras, it is clearly (0.24) closed under taking direct products. It would be interesting to know—raising again the question set aside before 2.15.14—what a general syntactic characterization might look like of those gcr’s which are determined (in the sense of this subsection) by classes of algebras, or indeed by single algebras. On the ‘classes of algebras’ question, in particular, one wonders whether the gcr’s so determined are precisely those which are symmetric, congruential, and substitution-invariant. There are many variations on the concepts under consideration here. For example we could define B1 ,. . . ,Bn C to be congruence valid in A just in case for any evaluation h into A and any congruence ≡ on A, and any equivalence class X of ≡: if {h(B1 ),. . . ,h(Bn )} ⊆ X then h(C) ∈ X. For such further variations, the interested reader is referred to Rautenberg [1991].

2.16

From Algebraic Semantics to Equivalent Algebraic Semantics

Blok and Pigozzi [1989] sets out to abstract from the close relationship between classical logic and the class of boolean algebras, or to take another example, intuitionistic logic and the class of Heyting algebras. The aim is to see what makes for the closeness of the relationship in such cases, between the logics, conceived as consequence relations, and the classes – in the present instances, varieties – of algebras. This leads them to define the notion of an algebraizable logic, or alternatively put, a logic with what they call an equivalent quasivariety semantics; the paper ushered in a mass of work in the genre that became known as abstract algebraic logic (or ‘AAL’) and we give only the most selective introductions here. A logic here is understood to be a substitution-invariant consequence relation; we will sometimes omit the qualification “substitution-invariant” in what follows, however. (In addition, Blok and Pigozzi restrict attention to finitary consequence relations, but this restriction is often dropped in subsequent work in abstract algebraic logic – see the survey Font, Jansana and Pigozzi [2003] – and we will not attend here to the resulting distinctions in the more general setting: ‘algebraizable’ vs. ‘finitely algebraizable’, etc.) The property of having an equivalent algebraic semantics will emerge as stronger than merely having – being sound and complete w.r.t. – some algebraic semantics or other (in the sense that (1) below is satisfied), and we begin with this weaker property. Propositional (or sentential) logics deal in formulas constructed from propositional variables by means of the available connectives, while the corresponding

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linguistic items in the case of algebras are equations. In 1.13 we took pains to distinguish the terms of an equational language from the corresponding propositional formulas – tB from B – but to continue to enforce this distinction would make a summary of the main idea from Blok and Pigozzi rather heavy-going so from now on we shall use the propositional language exclusively, so that terms are now constructed from the variables pi rather than xi , and the terms (for the equational language of a particular similarity type) are identified with the formulas (of a propositional language of that similarity type). Thus the transition from propositional languages to equational languages becomes the transition from terms to equations. So in the first instance we must associate with each term (thought of as a propositional formula) A an equation t = u. At this point, having relaxed the distinction between terms and (propositional) formulas, we will increase vigilance with regard to another distinction, and follow a common practice in equational logic – mentioned in 0.24.7 (p. 32) – and write equations as t ≈ u. That is, in the interests of clarity, we distinguish between the use and the mention of the identity predicate: if we want to say that one thing is equal to another (!), e.g. that the interpretation tA of the term t in the algebra A is the same as the interpretation uA of u in A, we say that tA = uA , but if we want to talk instead about the formal language of equational logic we will speak of the equation t ≈ u. Of course there is a connection between these two things, in that for terms t and u, we have A |= t ≈ u if and only if tA = uA . “A |= t ≈ u”, as remarked in 0.24.7 (p. 32), means that t ≈ u holds in (is true in, is satisfied by) the algebra A. (For other than closed terms, see the explanation in 0.22.) So the question becomes: how to translate a term into an equation, in the most general way. For this generality, we should allow the equation we end up with to depend on the particular term we started with, and this means that the “≈” in the equation should be flanked by terms which themselves depend, not necessarily in the same way, on the term t, say, we started with. Since equations always consist of terms of the similarity type in question, this means that with t we must associate a left-replacement term δ(t), say, and a right-replacement term, ε(t), say. (We follow the notation of Blok and Pigozzi at this point, but oversimplify in a respect to be clarified later: in general more than one such pair of terms δ(t), ε(t), will be allowed.) If these terms are to make sense as applied to an algebra of the same type as the propositional language we started with, then δ and ε must themselves be provided by that similarity type as (in the terminology of 0.22) fundamental or compositionally derived 1-ary operation symbols, since we need δ(t), ε(t) to be terms of the relevant equational language. In view of the identification of propositional and equational languages we write instead δ(A) and ε(A) since “A” is our more usual notation as a variable ranging over (propositional or sentential) formulas. Blok and Pigozzi then say that a class K of algebras of the given similarity type is an algebraic semantics with defining equations δ(p) ≈ ε(p), for the consequence relation (assumed to be substitution-invariant) just in case for all formulas B1 ,. . . ,Bn ,C of the language of we have the following, in which the “|=K ” on the right means that relative to any an assignment of elements of the universe of any algebra in K to the variables, if each δ(Bi ) ≈ ε(Bi ) holds relative to that assignment in the given algebra, then so does δ(C) ≈ ε(C): (1)

B1 ,. . . ,Bn C iff δ(B1 ) ≈ ε(B1 ),. . . , δ(Bn ) ≈ ε(Bn ) |=K δ(C) ≈ ε(C).


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The rather long-winded gloss on the equational consequence relation |=K amounts to saying that K satisfies the quasi-identity: (2)

(δ(B1 ) ≈ ε(B1 ) & . . . & δ(Bn ) ≈ ε(Bn )) ⇒ δ(C) ≈ ε(C),

and Blok and Pigozzi [1989] point out that the effect of this is that whenever a logic has an algebraic semantics, it has an algebraic semantics with the class K of algebras concerned being a quasivariety (which of course may be a variety – see the end of 0.26 for these concepts). We now indicate the respect in which we have been oversimplifying the story. Instead of encoding a propositional formula B into a single equation δ(B) ≈ ε(B), Blok and Pigozzi allow that a set of equations may be involved in this capacity, so that we need not a single pair of ‘defining equation’ terms δ, ε, but rather a finite system of such pairs, say δ1 , ε1 , . . ., δk , εk , with the right-hand side of (1) above having each “δ(Bi ) ≈ ε(Bi )”, i = 1,. . . ,n, replaced by “δ1 (Bi ) ≈ ε1 (Bi ),. . . , δk (Bi ) ≈ εk (Bi )”, and with “δ(C) ≈ ε(C)” replaced with “δj (C) ≈ εj (C) for each j = 1,. . . ,k”. The provision of δ and ε (or more accurately, of the various δj , εj ) allows us to translate from the propositional language with its consequence relation into the (equational) language of a quasivariety of algebras with its consequence relation |=, and when we have (1), the translation is faithful in the sense that we have not only the forward (‘only if’) direction but also the backward (‘if’) direction of (1) – or its more explicitly corrected form in terms of the various δj , εj . These two directions amount to a soundness and a completeness claim for w.r.t. the class of algebras concerned, though with a formalized semantical metalanguage (with its |=K -claims). Blok and Pigozzi describe the class as an algebraic semantics for and demand more for it to qualify as an equivalent algebraic semantics for . What more could one ask for? What Blok and Pigozzi asked for, since they saw this as a most striking aspect of the classic examples of algebraic methods in application to propositional logic (such as in relating CL to the class of boolean algebras), was that we should not only be have a faithful translation of into |=, but also a faithful translation going in the other direction (which inverts the given translation, as explained presently). Remarks 2.16.1(i) Traditional matrix methodology also has the resources to offer ‘something more’ than that CL is determined by the class of boolean algebras, namely the result that for any simple matrix (A, {1}) for the consequence relation CL , A is a boolean algebra with top element 1. (We did not go into this in the discussion in 2.13.) The word simple (also called “reduced” or “factorized”) is explained as follows. A simple matrix is one in which the only matrix congruence is the relation of identity, a matrix congruence being a congruence on the algebra of the matrix which does not identify designated with undesignated elements. (I.e., ≡ is a matrix congruence on (A, D) if ≡ is a congruence of A such that (for a, b ∈ A) a ≡ b implies a ∈ D ⇔ b ∈ D.) Thus given a suitable notion of indiscernibility, a simple (or reduced) matrix is one satisfying the Leibnizian principle of the identity of indiscernibles. Further discussion may be found in Rautenberg [1993]. Simple matrices were called ‘normal’ in Smiley [1962a], where an interesting application of the concept can be found, and ‘simple’ at p. 262 of Shoesmith and

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CHAPTER 2. A SURVEY OF SENTENTIAL LOGIC Smiley [1978]; see further pp. 258–265 of that work for related concepts. See, further, the end of 2.16.4(iii) below. (ii) Examples of consequence relations which do not have an algebraic semantics in the present sense can be found in Blok and Pigozzi [1989], p. 18, Blok and Rebagliato [2003], p. 170, and Raftery [2006a], pp. 126 and 135. (The latter is especially significant in that it concerns a logic – BB I logic – of independent interest.) But having an algebraic semantics does not itself seem to be of any logical significance. Any consequence relation lacking such a semantics can be extended to one having an algebraic semantics, by adding a new 1-ary ‘identity’ connective # to the language with the property that #A and A are synonymous according to the extended consequence relation for all formulas A; this follows from Theorem 3.3 of Blok and Rebagliato [2003]. (For example, to BB I logic we could add as axioms A → #A and its converse, for all formulas A. The consequence relation Raftery’s observation pertains to is that obtained from the axioms – consequences of ∅ – by allowing the formulato-formula Modus Ponens to yield further consequences from any set of formulas, as in 1.29.) But for all ordinary purposes, the envisaged # is an obviously redundant new primitive connective, #A amounting to “It is the case that A” – just a long-winded way of saying A.

We return to the reverse translation theme. This means that we need to translate equations t ≈ u into propositional formulas, recalling that in the nature of the case there is already a one-to-one correspondence (represented in the present notation by using the same notation, and in the preceding section by means of the tB –versus–B notation) between the terms themselves and such formulas. So what we need to translate t ≈ u, alias B ≈ C, is something that gives us a formula which is determined on the basis of the two formulas B, C: in other words, we need a binary connective or rather, a binary generalized connective in the style of 1.19: mapping each pair of formulas not to a single formula but to a set of formulas. (The generalization involved here matches the generalizing move from single δ and ε above to the collection of the various δj , εj .) Making use of the notation for a set of equivalence formulas from that subsection (after 1.19.9), we write E, or more explicitly E(p, q), with E(B, C) understood as the result of replacing p and q (uniformly) by B and C. When we use the “E” notation here we presume that E(p, q) is (relative to the propositional consequence relation under consideration) indeed a set of equivalence formulas as defined in 1.19. What we require of such an E is that it should ‘undo’ or ‘invert’ the translation from sentential to equational logic provided by δ and ε; that is we say that K is an equivalent algebraic semantics for when (1) above is satisfied, making K an algebraic semantics for ) and also (3), for all formulas B, C of the language of ( = terms in the language of |=K ): (3)

B ≈ C =||=K δ(E(B,C)) ≈ ε(E(B,C)).

Here we use the simple “δ, ε” notation, even though what is involved in general is a family of several δj , εj , as explained above. More explicitly when K is an equivalent algebraic semantics for in virtue of (1) and (3), it is called an equivalent algebraic semantics “with defining equations


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δ(p) ≈ ε(p) and equivalence formulas E(p, q)”; any for which there exists such an equivalent algebraic semantics is called algebraizable. (Since, as Blok and Pigozzi observe, K can always be taken to be a quasivariety in this case, the phrase “equivalent quasivariety semantics” is also used in place of “equivalent algebraic semantics”.) As something of a motivating example, we are reminded that the class of boolean algebras is an equivalent algebraic semantics for CL with defining equations given by δ(p) ≈ ε(p) with δ(p) = p and ε(p) = . (So here we have just one (δ, )-pair. Recall that here we use a uniform notation for the algebras and the propositional languages, and so write “” rather than “1”.) This makes the class of boolean algebras an algebraic semantics for CL ; taking E(p, q) as {p ↔ q} or {p → q, q → p} or indeed, matching the most likely choice of primitives in the equational context, {¬p ∨ q, ¬q ∨ p}, makes this same class an equivalent algebraic semantics, since (3) is satisfied. By way of further justification for this “equivalent”, we record without proof the content of Coro. 2.9 of Blok and Pigozzi [1989] as: Observation 2.16.2 If K is an equivalent algebraic semantics for with defining equation terms δ, ε, and equivalence formulas E(p, q), then: (i) A1 ≈ B1 ,. . . ,An ≈ Bn K C ≈ D if and only if E(A1 , B1 ), . . ., E(An , Bn ) E(C, D), for all A1 ,. . . ,An ,B1 ,. . . ,Bn , C, D; and, for all B: (ii) B E(δ(B), ε(B)). Parts (i) and (ii) here should be compared with (1) and (3), displaying the same behaviour for the translation from propositional formulas to equations. Blok and Pigozzi ([1989], Thm. 4.7) give a characterization of algebraizability couched in terms of propositional logic, without the explicit existential quantification over a class (quasivariety) of algebras, repeated here: Observation 2.16.3 A consequence relation is algebraizable if and only if the language of provides a set of equivalence formulas E(p, q) and a set of formulas δi (p), εi (p), where i ∈ I for I finite, for which we have, for all formulas B in the language of : B {E(δi (B), εi (B)) | i ∈ I}. Here we have used the set notation on the right in the interests of brevity: the “”-direction of the inset claim means that B C for each formula C in the set in question. Thus, by way of example, if I = {1, 2, 3} and E(p, q) is {p → q, q → p} the set in question is the union of E(δ1 (B), ε1 (B)), E(δ2 (B), ε2 (B)), and E(δ3 (B), ε3 (B)), and thus is the set: {δ1 (B) → ε1 (B), ε1 (B) → δ1 (B), δ2 (B) → ε2 (B), ε2 (B) → δ2 (B), δ3 (B) → ε3 (B), ε3 (B) → δ3 (B)}. There is some redundancy in the formulation of 2.16.3 – though not in the version in Blok and Pigozzi [1989] – in that the initial stipulation that E(p, q) is a set of equivalence formulas can, thanks to the congruence condition E(A1 , B1 ), . . . , E(An , Bn ) E(#(A1 , . . . , An ), #(B1 , . . . , Bn )),


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be weakened by omission of the detachment or Modus Ponens condition in the definition in 1.19 ( = (6) below), here formulated for simplicity for the case in which there is only one δ, ε pair rather than a collection of them, and assumed to satisfy and the ‘inversion’ condition – p E(δ(p), ε(p)). For the congruence condition implies (for all A, B) that (4) E(A, B), E(δ(A), ε(A)) E(δ(B), ε(B)). So, since A E(δ(A), ε(A)), we have – by the conditions (M) and (T) on consequence relations – (5), and then since also E(δ(B), ε(B)) B, we finally obtain (6): (5) E(A, B), A E(δ(B), ε(B)). (6) E(A, B), A B. Aside from the cases of classical and intuitionistic logic and their fragments (with equivalent algebraic semantics in the form of boolean and Heyting algebras and suitable reducts), various further examples of algebraizable and nonalgebraizable consequence relations are given in Blok and Pigozzi [1989]. Some of them which we collect here: Examples 2.16.4(i) Taking Γ BCK A to mean that A can be obtained by repeated applications of Modus Ponens from Γ and theorems of BCK logic, or equivalently by Modus Ponens from Γ substitution instances of B, C, and K (cf. 1.29), BCK is algebraizable, the equivalent algebraic semantics consisting of the quasivariety of BCK -algebras – a proper quasivariety in this case (i.e., not a variety). In this case as for CL and IL , the defining equation δ(p) ≈ ε(p) can be taken as p ≈ p → p, and the set E(p, q) of equivalence formulas as {p → q, q → p}. (In fact the claim here that BCK -algebras provide the equivalent quasivariety semantics for BCK logic is not quite correct, since the equational language has a constant 1 not corresponding to any sentential constant. Blok and Pigozzi [1989], p. 62 note the need for a more general notion of equivalent algebraic semantics to cover such cases; alternatively we could make the point in a version of BCK logic augmented by such a constant – , say, with axioms A → for all formulas A – or modify the notion of a BCK -algebra by eliminating the distinguished element from the fundamental operations, with a concomitant change to the defining quasi-identities. Thus Blok and Pigozzi’s formulation of the claim at issue here – their Thm. 5.11 – reads, more cautiously, “The class of BCK-algebras is definitionally equivalent to the equivalent quasivariety semantics for BCK”, BCK being what we are calling BCK logic.) (ii) With a corresponding understanding (to that given under (i)) of BCI and BCIW , neither of these consequence relations is algebraizable. (The fact that the latter is not algebraizable implies that the former isn’t either, since the axiomatic extension of an algebraizable consequence relation is algebraizable.) See 9.23.8(i) below (p. 1311), and the discussion leading up to it. (iii) Where Γ R A means that A can be obtained from substitution instances of the axioms of the relevant logic R (see 2.33) by means of the rules Modus Ponens and Adjunction (in the present context this can be


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thought of as the rule of ∧ Introduction), we have R algebraizable, with equivalence formulas as in (i) and with ε(p) as there (i.e., = p → p) but with δ(p) as p∧(p → p). (Plain old p won’t work here: see the Digression below.) Note that by contrast with the cases under (i), although we use the same ε as there, here this actually depends on its argument: i.e., ε(A) and ε(B) are not in general equivalent. (Thus a logic need not be monothetic, in the sense explained in 2.13 and related there to unital matrix semantics, in order to be algebraizable. See Blok and Pigozzi [1989], Thm. 2.4, for some remarks on the relation between these concepts, as well as pp. 42–45 for the relationship with matrix semantics in general. The key point applies in connection with the weaker property of having an algebraic semantics in the Blok–Pigozzi sense explained above, as opposed to that of having an equivalent algebraic semantics. Any logic with an algebraic semantics is determined by the class of matrices (A, D) where A is one of the algebras concerned and D = {a ∈ A | δ A = εA }. This is what is referred to by the phrase “equational definability of truth-predicates” in the title of Raftery [2006a], q.v., for further information.) The examples listed under (ii) and (iii) might raise an eyebrow as to the purely (propositional) logical significance of the notion of algebraizability. According to them, the {→}-fragment of R (alias BCIW logic) is not algebraizable – on which see further 9.23 below – but the {→, ∧}-fragment of R is algebraizable (since as long as we have ∧ we can construct the defining equation mentioned. It is difficult to see any deep contrast of logical interest between the two cases, however; this point prompts the suggestion at p. 276 of Dunn and Hardegree [2001] that one might look into a variant on the notion of algebraizability which replaces algebras with partially ordered algebras and the emphasis on equations with a generalization to -statements; work along these lines appears in Raftery [forthcoming]. (Actually, Dunn and Hardegree have a specific kind of “po-algebra” in mind, called by them a tonoid, in which every fundamental operation is either monotone or antitone w.r.t. the partial ordering.) Digression. In 2.16.4(iii) it was remarked that for the case of = R , we cannot simply take δ(p) as p. The reason is that we need p {δ(p) → (p → p), (p → p) → δ(p)} and we do not have p R p → (p → p). On the other hand, as can be seen from Blok and Pigozzi [1989], p. 49, choosing δ(p) as p ∧ (p → p), we do indeed manage to get the equivalence inset above. Some aspects of that equivalence may seem puzzling to anyone familiar with R through sequent calculus (e.g., as given in 2.33), for while we do have, for instance p R (p ∧ (p → p)) → (p → p), the ‘corresponding’ sequent p (p ∧ (p → p)) → (p → p) would not be provable in any sequent calculus (or in any natural deduction system) for R. Essentially the inset R claim holds because the formula to its

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right is R-provable outright, so we can automatically (in virtue of the condition (M)) on consequence relations, weaken it on the left by adding in an additional formula – here p; the traditional Deduction Theorem does not however hold for R , while the rule (→ Right), or in the natural deduction case, the rule (→I), applies in the sequent formulation for which reason the structural rule of Left Weakening (or Left Thinning) must be disallowed in the sequent calculus formulation (and be carefully prevented from being derivable in the natural deduction formulation – see the discussion of ‘assumption-rigging’ in 1.24). A simpler example of the same phenomenon is that because R p → p, we also have q R p → p, but the ‘corresponding’ sequent is not provable, since we do not have R q → (p → p); the “q” here could equally well be “p” again, to make this point. In the implicational fragment (alias BCIW logic), this is a special case of a variation on the traditional deduction theorem, given in 1.29.10 (p. 168); but this does not cover the presence of ∧ with the attendant use of the adjunction rule. These issues come up again in 9.23. End of Digression. An earlier adverse reaction to the proposals of Blok and Pigozzi [1989] was that the notion of algebraizability was too restrictive in another respect (i.e. other than that raised above à propos of the implication vs. implication– conjunction fragments of R). Font and Verdú [1991] argue that a wider range deserves consideration if what we have in mind is some kind of equivalence between a logic and a class of algebras: such a relationship should hold, for instance between the pure classical logic of conjunction and disjunction on the one hand and the class of distributive lattices on the other. But the latter variety does not constitute an equivalent algebraic semantics for the consequence in question, which is not algebraizable at all because it is not even equivalential (in the sense introduced toward the end of 1.19): there is no set of equivalence formulas E(p, q) at all, let alone one in terms of which we can mirror the quasi-identities of the variety using some suitably chosen δ, ε. This is because the consequence relation in question is atheorematic in the sense introduced in 2.11.6: Cn(∅) = ∅, to put it in terms of the associated consequence operation. The point can be strengthened even further: an atheorematic consequence relation does not even provide a set of implication formulas, let alone a set of equivalence formulas, except in a trivial case. Exercise 2.16.5 Show that if a consequence relation has A B for some A, B, then if is atheorematic, it has no set of implication formulas. (Suggestion. Suppose E(p, q) is a set of implication formulas. This set must be empty if is to be atheorematic. Show how this clashes with non-triviality: A B for some A, B.) Thus we conclude that the pure logic of conjunction and disjunction does not provide a set of implication formulas, and so is not even protoalgebraic, to use a concept from Blok and Pigozzi [1986], defined as follows: is protoalgebraic if and only whenever formulas A and B are S-synonymous for any -theory S, A and B are S-equivalent in the sense that S, A B and S, B A. (Recall that S-synonymy means that for any context C(·), we have C(A) ∈ S iff C(B) ∈ S.) It is easy to see that if provides a set of implication formulas then S-synonymy is sufficient for S equivalence. For suppose that A and B are


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S-synonymous but (without loss of generality) that S, A B, where E(p, q) is a set of implication formulas for . We work through the case in which E(p, q) has a single element, #(p, q), say – which we shall write with infix notation as p # q. Thus we are assuming that for all formulas A, B: (7) A # A

and

(8) A, A # B B.

Since S, A B, (8) implies that A # B ∈ / S; but in view of (7), A # A ∈ S. This contradicts the assumption that A and B are S-synonymous. Exercise 2.16.6 (i) Generalize the above argument to the case of arbitrary (as opposed to singleton) E(p, q). (ii) Give an example of a consequence relation which is neither protoalgebraic nor (like the {∧, ∨}-fragment of CL ) atheorematic. (If in doubt, consult 1.19.10, p. 99.) The converse, namely that being protoalgebraic implies having a set of implication formulas, is less obvious; for a proof, see Theorem 1.1.3 in Czelakowski [2001]. The three concepts in play in our discussion are related as follows; neither of the arrows (indicating implication) is reversible: algebraizable

⇒

equivalential

⇒

protoalgebraic.

A much more elaborate diagram indicating the implicational relations amongst a range of related concepts including these and variations thereon may be found (as Figure 1) on p. 49 of Font, Jansana and Pigozzi [2003]. The equivalence between having a set of implication formulas and being protoalgebraic prompts the question of whether there is a similar alternative characterization to that in terms of having a set of equivalence formulas of the concept of ’s being equivalential. Though rather more similar to the original than in the case just considered, the following alternative characterization has some prominence in the literature (e.g., Wójcicki [1988], p. 222): Observation 2.16.7 A substitution-invariant consequence relation is equivalential if and only if there is a set of formulas E(p, q) such that for every -theory S and pair of formulas A, B: A and B are S-synonymous just in case E(A, B) ⊆ S. This is closer to the original characterization in the sense that not only does it feature the existential quantification over a set E(p, q), but this set is a set of equivalence formulas for in the original sense (given in 1.19). Let us return to the problem of the narrowness of the Blok–Pigozzi notion of algebraizability, as illustrated by the fact that the logic of (classical) conjunction and disjunction, conceived as a consequence relation, is not even – in view of 2.16.5 – protoalgebraic, let alone algebraizable. The problem had been raised by the Barcelona group in AAL (i.e., Josep Font and colleagues), and from the same group came a promising solution, which is expounded, for example in Font and Jansana [1996], and which can be described as lifting all the concepts in play in Blok and Pigozzi [1989] up a level, from formulas to sequents (of SetFmla). Instead of translating an equation B ≈ C into a set of propositional formulas E(B, C), such as {B → C, C → B}, we translate it into a set of sequents

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which will secure a similar interreplaceability of the formulas concerned. Under favourable circumstances, the set {B C; C B} will, for example, have this effect. (We follow the policy from 1.23 of using a semicolon rather than a comma when listing sequents, to avoid conflicts with ‘sequent-internal’ commas.) The favourable circumstances in question we discuss in §3.3 under the heading of congruentiality; when they do not obtain, another set of sequents can often be found to play the E-role. Example 2.16.8 For instance, for the logic of (intuitionistic logic with) strong negation – see 8.23 – one would use instead {B C; C B; −B −C; −C −B} where “−” is the 1-ary ‘strong negation’ connective under discussion there. With this raising of levels from formulas to sequents, the problems posed by atheorematic logics would now become problems for logics without provable sequents. But such logics do not exist, at least in the presence of the ubiquitous (R). It might be thought that (R), along with (M) and (T), are going to be required on the (propositional) logic side, since just as with Blok and Pigozzi [1989], we are still concerned with mirroring the consequence relation |=K for a class K of algebras: but this is not quite correct, as is explained below (2.16.10). We also need to translate in the reverse direction, from sequents to (sets of) equations; one such translation mentioned by Font and Jansana [1996], Def. 4.21, works in the presence of conjunction (behaving classically): the sequent Γ B is translated into the equation B ∧ C ≈ C, where C is the conjunction of formulas in Γ; Font and Jansana call this equation t∧ (Γ B). If Γ is empty, this leaves us at as loss as to what C should be, especially for applications to atheorematic logics, such as that of ∧ and ∨, so when using this translation, let us specifically restrict attention to the logical framework Set1 -Fmla. We can then ask that these two translations do what was expected of their ‘lower level’ analogues in Blok and Pigozzi [1989], (1) and (3) above (beginning on p. 258), though the horizontal transitions from the left to the right of / need to replaced by vertical sequent-to-sequent transitions, with the original reference to a consequence relation being replaced by a proof system with set R sequent-to-sequent rules. To indicate the derivability by means of rules in R of σ from a set Σ of sequents we will write Σ R σ. Below, we use “R ” when this relation obtains in both directions. (9) σ1 , . . . , σn R σ if and only if t∧ (σ1 ), . . . , t∧ (σn ) |=K t∧ (σ). (10)

B ≈ C =||=K {B ∧ C ≈ B, C ∧ B ≈ C}.

The right-hand side of (10) results from translating the equation B ≈ C into the corresponding set of sequents, by the simple-minded translation mentioned above (suited only to congruential logics), and then translating the elements of this set, namely B C and C B, into their respective equations using t∧ . To consider the general situation we follow the lead of Font and Jansana [1996], p. 71f., and let s and t be any translations from (sets of) equations to


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(sets of) sequents and vice versa – mnemonically s delivers sequents – and point out that from the generalized versions of (9) and (10) in this notation: (9 ) (10 )

σ1 , . . . , σn R σ if and only if t(σ1 ), . . . , t(σn ) |=K t(σ). B ≈ C =||=K t(s(B ≈ C)).

the translational ‘duals’, (9 ) and (10 ), follow: (9 ) B1 ≈ C1 , . . . , Bn ≈ Cn |=K B ≈ C if and only if s(B1 ≈ C1 ), . . . , s(Bn ≈ Cn ) R s(B ≈ C) (10 )

σ R t(s(σ)).

To see that (9 ) follows from (9 ) and (10 ) let us work with the illustrative case of n = 1. We must show that B1 ≈ C1 |=K B ≈ C if and only if s(B1 ≈ C1 ) R s(B ≈ C) (for arbitrary B, C, B1 , C1 ). By (10 ) the left-hand side is equivalent to t(s(B1 ≈ C1 )) |=K t(s(B ≈ C)), which is equivalent to the right-hand side by (9 ). Since this same reasoning applies in the case of arbitrary n, we take (9 ) to have been established, and proceed to (10 ). By (9 ), it suffices to show that t(σ) =||=K t(s(t(σ))). But any B ≈ C ∈ t(σ) is |=K -equivalent by (10 ) to t(s(B ≈ C)), giving the desired result. Thus t and s satisfying (9 ) and (10 ), when they can be found, do indeed suffice in the same way (cf. 2.16.2) as Blok–Pigozzi’s E and δ, ε, to set up an invertible translation between a propositional logic and the quasi-equational theory of a class of algebra, except that now we conceive of the propositional logic as given by proof system with sequent-to-sequent rules (in fact identifying the system with this set of rules) rather than as a consequence relation. Returning to the case of the logic of conjunction and disjunction we may take R, for instance, to comprise the structural rules and operational rules governing these connectives from the rules of IGen (described in 2.32 below, and chosen over Gen because we wish to avoid ‘multiple succedents’), or indeed as instead the rules (R), (∧I), (∧E), (∨I) and (∨E) of the natural deduction system Nat, we have a proof system with ‘equivalent algebraic semantics’ in the sense of the previous remarks as the class of distributive lattices. Exercise 2.16.9 Show, for either of the choices of R just described, that (9) and (10) above hold when K is taken as the variety of distributive lattices. (Suggestion. For the ‘if’ direction of (9) – a kind of completeness result for R – consider the algebra of congruence classes of formulas under the relation ≡R defined by: A ≡R B just in case Σ R A B and Σ R A B, with operations ∧, ∨ defined as in the case of a Tarski– Lindenbaum algebra , though here when Σ = ∅ we are not dealing with the free distributive lattice on countably many generators [pi ], but on a quotient algebra thereof, induced by ≡R .)

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Remark 2.16.10 Working through 2.16.9, for either choice of R, one is confronted with the case of the structural rule (R), with t∧ mapping its instances A A to equations A ∧ A ≈ A; thus here it is specifically the idempotence of the meet operation (in distributive lattices) that corresponds to (R). With different translations, this rule might translate to an equational condition that was not satisfied. Earlier it was said that “it might be thought that (R), along with (M) and (T), are going to be required on the (propositional) logic side, since just as with Blok and Pigozzi [1989], we are still concerned with mirroring the consequence relation |=K for a class K of algebras: but this is not quite correct.” More explicitly, we note now that what corresponds to the structural conditions on the consequence relations |=K are the corresponding conditions on the ‘closure relations’ (cf. 6.47.4–6.47.5 below) R , rather than the internal structural rules of the propositional proof system under consideration. We refrain from using the term “consequence relation” here as R relates sets of sequents to individual sequents rather than sets of formulas to individual formulas. There is a bona fide consequence relation in the vicinity, though, called in Avron [1988], p. 163, the external consequence relation of the proof system (with rules) R, defined to hold between Γ and C when { B | B ∈ Γ} R C. (Avron contrasts this with the ‘internal’ consequence relation of the proof system, which is roughly our consequence relation associated with a proof system, as in 1.22.2–1.22.3, (see p. 113) though not exactly, as Avron does not – in this particular publication – understand by a consequence relation a relation between sets of formulas and formulas; he uses multisets rather than sets and does not impose the analogue of the weakening/thinning condition (M) – or its converse, the multiset structural rule of Expansion. See Avron [1991a]. An exposition more orthodox exposition in respect of the consequence relation terminology is given on p. 697 of Avron [1998].) As in the case of Blok and Pigozzi [1989] and related discussions, the Barcelona school’s AAL work extends far beyond anything covered in this discussion. In particular, some interesting results for sequent-to-sequent rules have been obtained using the semantics of abstract logics or generalized matrices we encountered at the end of 2.12. See Font and Jansana [1996] (esp. Chapters 2–4) for details and further references, as well as §5 of Font, Jansana and Pigozzi [2003]. In the initial developments after Blok and Pigozzi, many-valued semantics was absorbed into AAL only in the “designated values” preservation conception of validity, with what in 2.14 we called the -based conception of validity receiving scant attention. This imbalance was corrected in Font, Gil, Torrens and Verdú [2006]; see also Font and Jansana [2001], Font [2008], and Paoli, Spinks and Veroff [2008]. Other pertinent reading is listed under ‘Abstract Algebraic Logic’ at p. 275 of the end-of-section notes.

Notes and References for §2.1 Fuzzy Logic. The description in 2.11 of fuzzy logic (for vagueness) is oversimplified. Instead of taking linearly ordered truth-values representing degrees of truth, the original proponent of fuzzy logic now takes what he calls ‘fuzzy


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subsets’ of the real interval [0,1], rather than the reals themselves, as the truthvalues (Zadeh [1975] – see Haack [1978] Chapter 9, §4, and Urquhart [1986], §3.4, for some discussion), while another approach described as fuzzy (Goguen [1969]) uses lattices (of truth-values) which aren’t chains, thereby avoiding many of the objections (cf. Fine [1975]) to the use of multiple truth-values in this connection (e.g., p ∧ ¬p does not get a low enough truth-value if p is infected by vagueness) since join and meet no longer mean max and min. It is not clear that this approach is faithful to the ‘degrees of truth’ idea, however (cf. Burgess and Humberstone [1987], §1), and it is perhaps better thought of as an algebraic incarnation of the alternative ‘supervaluational’ semantics rather than a many-valued treatment in its own right (cf. Kamp [1975]). Supervaluations are explained in 6.23. Critical discussions of many-valued and supervaluational approaches to vagueness may be found in chapters 4 and 5, respectively, of Williamson [1994a]. The phrase fuzzy logic later came to have several different applications and connotations. See note 5 – p. 59 – and Chapter 19 of Gottwald [2001] as well as Hájek [1998a], [1998b], [2003]. The title of Cignoli, Esteva, Godo and Torrens [2000] sums up Hájek’s approach: ‘Basic Fuzzy Logic is the logic of continuous t-norms and their residua’. By a t-norm or more explicitly, ‘triangular norm’, in the context of some ordered set with a top element is meant an associative and commutative binary operation which is monotone (i.e., preserves the given order) in both positions and has that top element as an identity or neutral element. With special reference to the real interval [0, 1] and its usual ordering, a t-norm is continuous if it is continuous, in the usual sense of functions of a real variable, in each of its two positions (when the other is held fixed). The word “residua” means residuals, as explained in 4.21 below: roughly speaking if we think of a t-norm as a conjunction-like function then we can think of its residual as an implication-like function satisfying an importation/exportation equivalence. (More precisely, writing these as ◦ and →: x y → z iff x ◦ y z.) Metcalfe [2003] provides a useful survey of topics in the proof theory of fuzzy logics in this sense. A slightly different approach can be found in Cignoli [2006], and some comparative remarks in Novák [2006]. Various aspects of fuzzy logic continue to provoke controversy about the legitimacy of the enterprise; two examples (on opposite sides in the debate) illustrating this are Copeland [1997] and Pelletier [2004].

Primitive Connectives; Polish Notation. The presentation in Łukasiewicz [1920] of the three-valued matrix described in this section is somewhat different in that Łukasiewicz takes only ¬ and → as primitive connectives, defining the others with their aid: see Prior [1962], the chapter on ‘Three-Valued and Intuitionistic Logic’, esp. p. 243 on the present matter. Prior followed Łukasiewicz in using the so-called ‘Polish notation’, writing binary connectives before rather than between their arguments, and symbolizing them somewhat differently also: “K” for “∧”, “A” for “∨”, “C” for “→”, and “N” for “¬”. This enables formulas to be referred to unambiguously without the use of parentheses; e.g., KNpq and NKpq for our ¬p ∧ q and ¬(p ∧ q), respectively, KpAqr as opposed to AKpqr for p ∧ (q ∨ r) as opposed to (p ∧ q) ∨ r, and CCpqr vs. CpCqr for (p → q) → r vs. p → (q → r).

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Philosophy. The problem of the ‘middle box’ alluded to in the text can be seen very clearly with ∧. Where a is some specific individual (you, say) the statement that a is in Dublin at noon on Jan. 1, 2030, should have the value 2, as should its negation, if the value 2 is to be assigned to statements whose eventual truth or falsity depends on later free actions. Thus the conjunction of this statement with its negation, by the table for ∧, should receive the value 2 also. But this now conflicts with the intended way of interpreting the values, since whatever any agent might decide on, it cannot come out anything other than false that a both is and is not in Dublin at noon on January 1, 2030. Nor is there any way to change the table to avoid the trouble. For this case, we would like to see 2 ∧ 2 = 3; but if for the second conjunct we pick, not the negation of the first but the first conjunct itself, or indeed any other independent statement, for instance that b (where b = a) is in Dublin at noon. . . , then the original 2 ∧ 2 = 2 verdict seems right. Thus the status of a conjunction as determinately true, indeterminate, or determinately false is not fixed solely on the basis of which cells of this threefold partition its conjuncts occupied, and three-valued logic (for this application) is accordingly a mistake. The attempt to incorporate modal operators for representing necessity and contingency into this three-valued framework, as in Łukasiewicz [1930], makes its inappropriateness even more evident, since (using Łukasiewicz’s suggested symbol “M” to be read “it is possible that” and to have the associated truth-table M(1) = M(2) = 1, M(3) = 3) the formulas Mp ∧ Mq on the one hand and M(p ∧ q) on the other, receive the same truth-value on every evaluation. For a clear statement of this type of argument against similar would-be many-valued logics, see Fine [1975] and Kamp [1975], esp. pp. 131–133. It can also be found on p. 3 of Prior [1967b], which compares interestingly with the much more favourable appraisal of Łukasiewicz’s idea in Prior [1953]. Other locations for the argument include Seeskin [1971] and Smiley [1976]. The earliest published version, in English, of this style of objection to popular applications of many-valued logic appears to be that of Geach [1949], though Malinowski [1993], p. 21, reports that as early as 1938, the point had been noted by F. Gonseth. Some different (and less conclusive) objections may be found in Krolikoski [1979a]. There is scarcely any area for which the suggestion—however farfetched—has not at some time been made that three-valued logic finds there a useful application, from ethical pluralism (Cohen [1951]) to primitive thought (Cooper [1975]); it would not be practicable to go carefully into such suggestions. Even in a case of the application of many-valued logic which is not vulnerable to objections of the sort rehearsed above, we saw from the passages of Dummett quoted in 2.11 that there is no need to think of the principle of bivalence as being violated since what matters is bivalence at the level of valuations (vh ) rather than the level of evaluations (h). While claims to the effect that this or that is a counterexample to the principle of bivalence are criticized, Dummett [1959a] is by no means an endorsement of the principle of bivalence, a fact which has led to some confusion. Matters are clarified somewhat by the 1972 Postscript appended to the reproduction of his [1959a] in Dummett [1978], and clarified considerably more by the Preface to the latter work. (For more detail on Dummett’s changes of mind as well as on the connections between various themes in his work, see Green [2001], especially – for the present topic – pp. 24–30 and Chapter 3.) It should be added that the quotation from Dummett [1959a] which was given in 2.11 is taken from a discussion in which there is a qualification we


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did not mention: Dummett is excluding vague and ambiguous sentences for consideration when he says that we need to distinguish “those states of affairs such that (. . . ) and those of which this is not the case”. With regard to vagueness, in particular, the rationale for such exclusion is problematic (as is noted in Burgess [1980], p. 214); Dummett [1995] addresses this question. Further, it is not completely clear that the cases of intuitionistic logic, which Dummett regards as embodying a deep repudiation of the principle of bivalence, and the case of Łukasiewicz’s three-valued logic, in which the repudiation is merely superficial, are as different as he makes them out to be: see Humberstone [1998c]. Finally, Dummett’s account of the role of many-valued tables, as providing compositional semantics in the face of a failure of designation-functionality, works clearly only for many-valued logic in the narrow sense. Matters are less straightforward in the case of a logic which is many-valued in the broad but not the narrow sense (determined by a class of matrices but by no single matrix) – such as the Set-Fmla logic determined by {K1 , K1,2 }, – which is not many-valued in the narrow sense, by 2.12.4. In this case we cannot say that the designated values are subdivisions of the true and the undesignated values subdivisions of the false, since there is no single distinction between designated and undesignated values (the value 2 changing status as we consider now the one matrix, now the other). The contrast in our discussion between matrix evaluations h and (bivalent) valuations vh is formulated as a contrast between algebraic valuations and logical valuations, respectively, in Suszko [1977]. Łukasiewicz’s Logics. An introductory but informative treatment of the various many-valued logics of Łukasiewicz is provided by Ackermann [1967], which describes the arithmetical aspects of the inclusion relations between them. (See Karpenko [2005] for fuller details.) Ackermann also goes into the question of axiomatizing these logics, a topic we do not cover – even for the three-valued case at the centre of attention in 2.11 – except briefly in a Digression in that subsection. A representative collection of work in Poland on these logics is to be found in Wójcicki and Malinowski [1977]. For further details and references, see Urquhart [2001] and Hähnle [2001]. Compositionality. References for this topic (introduced in 2.11.11): Partee [1984], Pelletier [1994], Morrill and Carpenter [1990], Chapter 10 of Hintikka and Kulas [1983], (esp. Chapter 6 of) Hintikka [1996], Appiah [1987] – in which the word “componentiality” is used in place of “compositionality”, unwisely since it risks evoking connotations of componential analysis, an unrelated topic in the lexical semantics of natural languages – Hodges [1997], [1998], Janssen [1997], Westerståhl [1998] and references therein, and Higginbotham [2003] for an interesting case study. The references just given are a small sample of those explicitly discussing the pros and cons of compositionality in semantics; for examples simply endorsing or following the principle, see Frege [1892] – to which the idea is often credited, though Pelletier [2001] here provides a dissenting voice (on which see also Hodges [1998]) – and Lewis [1972], for a streamlined series of applications. Direct Products. Direct products of matrices probably first appeared, along with (inter alia) Theorem 2.12.1, in Wajsberg [1935]; they also appear in

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Jaśkowski [1936]. Direct sums of matrices seem to be due to Kalicki [1950]. Both constructions get some discussion from Rescher [1969]. Terminology. As an alternative to speaking of a logic as determined by a matrix, the following terminology is common for consequence relations. A matrix M is said to be ‘strongly characteristic’ (or ‘strongly adequate’) for such a relation if is determined by M, and ‘characteristic’ (‘adequate’) if the induced logic in Fmla, the set {B | ∅ B}, is determined by M. In accordance with this usage, instead of saying that a logic is tabular (or finitely many-valued in the narrow sense, as we have sometimes put it), one says that the logic has a finite characteristic matrix. As such terminology suggests, attention to consequence relations has been something of an afterthought, with the primary concern having been with logics in Fmla. An interesting criticism of the relevant literature along these lines, together with a more refined review of terminological distinctions than has just been given, may be found in Dummett [1973], pp. 435–437. A matrix in which all sequents in some logic are valid is sometimes called a ‘model’ for that logic. We prefer to reserve this term for the structures provided by model-theoretic semantics for first-order logic or for those from the Kripke (or analogous) model theory for modal logic (§2.2), intuitionistic logic (2.32) etc. The distinction, as far as Set-Fmla is concerned at any rate, between manyvalued logics in the broad and narrow sense is an on/off version of a quantitative measure called the ‘degree of complexity’ Wójcicki [1973] ( = ‘ramification number’ in Wójcicki [1988]) of a consequence relation, by which is meant the least amongst the cardinalities of classes of matrices determining the consequence relation. Thus “many-valued in the narrow sense” amounts to being of complexity 1. For information, see Hawranek and Zygmunt [1981], [1984], Hawranek [1987]. The first and third of these papers show that the consequence relation of Minimal Logic (8.32) is of complexity 2, as we saw in 2.12.4 this was also the case for the gcr determined by {K1 ,K1,2 }. Matrix Methodology. Many of the subtleties falling under this heading and not touched on here, as well as more information on those that are, may be gleaned from Harrop [1965], Wójcicki [1973], and Part III of Shoesmith and Smiley [1978]. A comprehensive treatment may be found in Wójcicki [1988]. See also van Fraassen [1971], §5 of Chapter II and §1 of Chapter V. Harrop’s paper is somewhat biased in favour of discussions of decidability, and makes no reference to the pertinent work of the Polish logicians, but contains numerous points of interest. Chagrov [1985] provides a systematic discussion in matrix terms of extensions of intuitionistic logic, possibly with additional (e.g., modal) connectives. The survey article Urquhart [1986] could also be consulted, in conjunction with Wroński [1987] for the correction of errors, which are not repeated in the successor article Urquhart [2001]; see also Hähnle [2001]. Our cursory discussion of many-valued logics has omitted all mention of Emil Post’s systems, concentrating on those of Łukasiewicz, Kleene, and the product matrices, as being of greater philosophical interest; the text and bibliographies of Rescher [1969], Urquhart [1986], Bolc and Borowik [1992], and Malinowski [1993] should be consulted to fill such lacunae. See also the bibliography (compiled by R. G. Wolf) of Dunn and Epstein [1977]. Various—mostly three-valued—logics


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are discussed in Goddard and Routley [1973]. Hodes [1989] and Avron [1991b] provide systematic discussions of the three-valued case; an older and briefer foray into roughly similar territory is Dienes [1949a]. (Dienes [1949b] presents a similarly cursory discussion of implication in early many-valued logics; see also Turquette [1966]. An extensive discussion of the matrix treatment of implication in the three-valued case was given in Maduch [1978] as well as in Matoušek and Jirků [2003].) A shorter tour through some of the material of 2.11 and 2.12, though including certain additional points cast in slightly different notation and terminology, is provided by Humberstone [1998c]. Chapter 7 of Dunn and Hardegree [2001] also provides much information on matrix methodology. In our discussion we have used “D” to denote the set of designated values of a matrix; an increasingly common practice, associated especially with abstract algebraic logic, is to use “F ” (to suggest “filter”) for this. The Finite Model Property. Although, as noted before 2.13.5, the concept of the f.m.p. was introduced by Harrop with validating matrices for a logic as its models, the corresponding notion arises in other areas also, such as with the algebras figuring in various non-matrix-based algebraic semantics (such as the equivalent quasivariety semantics of an algebraizable logic – see 2.16), as well as in the Kripke semantics for modal logic (reviewed in §2.2). In all these cases there is a notion of validity which is relativized to structures (algebras, Kripke models, Kripke frames) with a universe of elements, which can accordingly be either finite or infinite, and determination by a class of structures of the former kind is called having the finite model property. As it happens, the Kripkesemantical version of the property – which itself can be regarded as the finite model property or the finite frame property (see Segerberg [1971], Coro. 3.8) – coincides with the matrix version when the frames are turned into suitably equivalent matrices. (See the early Digression in 2.23, and note that for the normal modal logics the algebraic semantics is unital matrix semantics as in 2.13.) Logics with the finite model property are also called finitely approximable in some of the literature. When logics are identified with consequence relations, it is necessary to distinguish the traditional f.m.p., meaning that the induced Fmla logic – the set of consequences of ∅ has the f.m.p. as described above, from the strong f.m.p., meaning that the consequence relation itself is determined by a class of finite matrices (or algebras or structures of whatever kind the semantical description employs): see §10 in van Alten and Raftery [2004]. Early Algebraic Semantics. To the above references, we may add the following as useful specifically for the material on algebraic semantics: Halmos [1956], Dunn [1975a] and Chapters 6 and 7 of Rieger [1967]. The Halmos paper (like the book by Rieger) treats all aspects of logic in an algebraic way, and not just the semantics; a minor – in the sense of not mattering to Halmos’s main expository purposes – but interesting error was pointed out in Hiż [1958]. Halmos and Givant [1998] is an introductory text explaining in great detail the relationship between classical propositional logic and boolean algebra(s) but not touching on other logics. The classic books on algebraic semantics of the unital matrix sort are Rasiowa and Sikorski [1963] and Rasiowa [1974]; as with most of the other references given, there is an unfortunate preoccupation with the Fmla framework here.

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Notes on the BCK (etc.) Material in 2.13. Iséki and Tanaka [1978] is a survey of the basics of BCK -algebras, and cites as the source of the notion the paper Imai and Iséki [1966] – where, however, this class of algebras is not isolated (under the name BCK -algebra or any other name); the appropriate reference is to Iséki [1966]. In all of the earlier papers and most of the later ones in this genre a different (‘dual’) notation is used; typically, “b∗a = 0” appears in place of what we write as “a → b = 1”, and “b a” for our “a b”. One reason for this is the general algebraists’ predilection for the bottom end rather than the top end of algebras equipped with an ordering—see Dunn [1975a], p. 185, on the comparative interest taken by logicians and algebraists in filters and ideals, respectively—and another is the intention to regard a ∗ b as generalizing the complement of b relative to a (a b, with a and b being sets). The significance of the analogy between relative complementation for sets and implication in propositional logic was already noted in Kalman [1960]. What we call BCK algebras are called BCK *-algebras in Torrens [1988] to acknowledge that they are duals of the traditional BCK -algebras. A textbook treatment of BCK algebras is provided by Meng and Jun [1994]. The originator of the labels “BCK” and “BCI” for logics was C. A. Meredith; see Meredith and Prior [1963], esp. §§7, 8. (The original analogy between implicational principles and the combinators of combinatory logic may be found in Chapter 9 of Curry and Feys [1958]; see further Bunder [1981a], [1983a], Hindley [1993], and, perhaps most comprehensively, Hindley and Meredith [1990]. (Hindley’s coauthor here is David Meredith, a cousin – we learn from Prior [1956] – of C. A. Meredith.) The papers of Bunder just cited sometimes make a confusing use of the term “algebra” to mean class of algebras; the same confusion is evident in the very title of Iséki [1966].) Prior himself favoured somewhat more informal names for familiar implicational principles than the combinatorinspired labels; these, as well as further historical remarks on Meredith’s work, may be found in Prior [1956]. (See further the notes to §7.2 bearing on the material in 7.22 and 7.25.) A confusing feature of some of the subsequent literature on BCK -algebras and the like is the departure from the convention of abbreviating combinator-inspired axioms in the labels; thus for example, Komori [1984] introduces a class of “BCC -algebras”, a label which makes no sense from the point of view of that convention. (Komori [2001] revises the nomenclature accordingly.) (Kabziński [1993], fully aware of the relevant etymology, considers what he nonetheless calls BCII -algebras, where the second “I ” is a reminder of Suszko’s identity connective – see 7.32.) Aside from what we (following Diego [1966] and many other writers) call Hilbert algebras, an unrelated class of algebras also goes by the same name—as in Johnson and Lahr [1980], for instance. Hilbert algebras were first isolated in Henkin [1950], where they are called implicative models. (Cf. further Lyndon [1951], §3.) Although we gave a reason for not calling quasi-commutative (in the sense of Abbott [1967]) BCK algebras ‘commutative’, it must be conceded that the term quasi-commutative has itself been used differently in connection with such algebras: see Cornish [1980b]. The BCK -algebras corresponding to classical implicational logic are in the terminology of 2.13 BCKP -algebras (alternatively SKP -algebras). They have gone by various names in the literature, being for instance the ‘Tarski algebras’ of Monteiro [1978], though in the mainstream BCK -algebraic work they are most often called implicative BCK -algebras (compare the terminology of ‘positive implicative’ for Hilbert algebras); this seems, however, not especially

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suggestive—though in 7.25 we follow suit for the sake of continuity with the literature—since we are dealing with algebraic versions of implication throughout. (There is perhaps no corresponding objection to the use of “implication algebra” in Abbott [1967], [1969], since there only an algebraic version of classical implication is at issue—as in, perhaps the earliest venture into this field, Forder and Kalman [1962]. Rasiowa [1974], on the other hand, expects her readers to keep track of a distinction between ‘implicative algebras’ and ‘implication algebras’.) For much information, assistance and advice on matters relating to BCK -algebras (and the like) and the corresponding logics, and improvements to the relevant discussion in 2.13, I am grateful to Matthew Spinks and to Brian Weatherson. All of Remarks 2.13.23 are due to Weatherson, as is the algebra in Figure 2.13d (p. 241). Abstract Algebraic Logic. In addition to the main references cited in 2.16 – Blok and Pigozzi [1989], Font and Jansana [1996], and Font, Jansana and Pigozzi [2003] – there are many further articles of interest, some going back to earlier years, such as Blok and Pigozzi [1986] and Czelakowski [1981]. The latter discusses equivalential logics (in the sense explained in 2.16), a class of logics isolated in the 1970s by Prucnal and Wroński and treated more recently in a contemporary ‘AAL style’ in Chapter 3 of Czelakowski [2001]; see also Malinowski [1989]. The interesting question of what becomes of algebraizability à la Blok and Pigozzi when generalized consequence relations rather than consequence relations are at issue is raised on p. 275 of Dunn and Hardegree [2001]. However, the reader should be warned that the definition (purporting to summarize Blok and Pigozzi’s intent) of algebraizability on p. 275 of this work (Definition 7.13.3) fails to do so, completely omitting the crucial ingredient of invertible translations in both directions; instead the definition amounts, given Coro. 2.3 of Blok and Pigozzi [1989] to a description of what those authors call consequence relations having an algebraic semantics (as opposed to: having an equivalent algebraic semantics, which is what algebraizability is). A unified algebraic approach (despite the prominence of a partial ordering emerging from the algebras) to a range of substructural logics may be found in Galatos, Jipsen, Kowalski, and Ono [2007].

§2.2 MODAL LOGIC 2.21

Modal Logic in Fmla: Introduction

Traditionally, modal logic would be described as studying the logical properties of notions of necessity and possibility. In fact the subject has metamorphosed into something more general, since many other notions exhibit sufficient similarity to lend themselves to study alongside necessity and possibility, using the same apparatus. Think of the project of extending, in order to formalize these notions, some account of the boolean connectives we have already encountered (namely ∧, ∨, ¬, →, ↔, and ⊥), and for simplicity let us take the account in need of extension to be that given by classical logic. In this subsection, we work in Fmla, so our interest in classical logic is in a stock of formulas in an expanded language which includes all substitution instances of the (classical) tautologies. We add a new connective to this language, , to symbolize some

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unspecified notion of necessity (e.g., logical necessity); we could also add an additional primitive for possibility, but instead we regard this as defined: A = ¬¬A. and are called modal operators. (Incidentally, if we chose to pursue modal logic against the background of intuitionistic, rather than classical, logic, it would be better to take and both as primitive rather than try to define one in terms of ¬ and the other; the situation is analogous to the relation between the universal and existential quantifiers in intuitionistic predicate logic. See the notes to this section.) As it happens, C. I. Lewis’s work in modal logic, which stimulated the twentieth-century revival of interest in the topic, was inspired not so much by an interest in necessity and possibility themselves as in their usefulness for isolating the concept of strict implication, where A strictly implies B when it is necessarily the case that A materially implies B: when (A → B) (equivalently, ¬(A ∧ ¬B)) is true, that is. This is usually abbreviated to A B, (though of course one can consider as a primitive connective in its own right, and define the usual modal operators in terms of it and the boolean connectives). Lewis thought of as vehicle for formalizing the informal idea of entailment; see further the discussion surrounding 6.13.1 on p. 789, and, more generally, §7.2. Indeed it is for such later applications (of which there are many) that we include this section as a crash course in modal logic. The informal rendering of “” and “” as necessarily and possibly, sometimes called the alethic interpretation of this symbolism, is only one possible interpretation, as was intimated in our opening paragraph. Amongst the many non-alethic readings current are a deontic reading (taking A to represent “it ought to be the case that A” or “it is morally – or legally – obligatory that A”), a doxastic reading (where A is thought of as saying of some cognizing agent a: a believes that A), and an epistemic reading (“a knows that A”). There are also various temporal readings. One would be of A as “always A”; a more directional reading would be “it will always in the future be that A”. When the latter reading is used to guide the investigation, typically alongside another operator with an oppositely directed interpretation, for “it has in the past always been that A”, one speaks of tense logic—touched on below. In each case “” takes on the reading suggested by its definition as “¬¬”; for example, in the deontic case this would be “it is permissible that”, and in the (future) tense-logical case, “it will at some time be that”. In addition to these traditional areas, aimed at formalizing this or that everyday notion (perhaps somewhat idealized), two more recent applications of modal techniques have been to the treatment of provability in a first-order theory (typically Peano arithmetic) as an operator, and to the description of computer programs (‘dynamic logic’). References to the seminal literature on these various areas of modal logic ‘in the broad sense’ may be found in the notes. So as not to suggest any one interpretation of the language of modal logic, it is recommended that “” and “” simply be read respectively as “box” and “diamond”. The logical behaviour these various notions have in common can be described in terms of ‘normality’. Individuating logics in Fmla at the crudest possible level, in terms of the stock of formulas they contain, we understand by a modal logic any set of formulas in the present language, containing all classical tautologies and closed under Uniform Substitution and tautological consequence. If S is a modal logic, we will write “S A” in place of “A ∈ S”, since in practice we shall always identify named modal logics as consisting of the theorems of

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proof systems described by the name used (see below). A normal modal logic is a set of formulas containing every formula falling under one or other of the schemata: (A0) A

where A is any tautology in the non-modal connectives

(A1) (A → B) → (A → B) and closed under Uniform Substitution, Modus Ponens, and the rule of: Necessitation (‘Nec.’): From A to A. (It follows easily from this characterization that any normal modal logic is a modal logic in the more general sense introduced above.) As the appearance of “” here betrays, we are thinking of the above specification of a normal modal logic as an axiomatization of the smallest such logic, which is called K (after Kripke – no connection with the matrices K1 and K1,2 of §2.1, after Kleene). Notice that since we have used schemata (A0) and (A1), we do not need Uniform Substitution as a separate rule in the basis, and will take the only rules of the present proof system to be Modus Ponens and Necessitation. (We have been very lazy with the axiom-schema (A0); it could of course have been replaced by any collection of individual axioms, such as (HB1)–(HB12) from 1.29.) Exercise 2.21.1 (i) Show that the following is (for n 0) a derived rule of the proof system for K just described (A1 ∧ . . . ∧ An ) → B (A1 ∧ . . . ∧ An ) → B (ii ) Show that any modal logic closed under the above rule is normal. We denote by K + X the smallest normal modal logic containing all instances of the schema X; this can be thought of as axiomatized by extending the above basis for K with X as an additional axiom-schema. When the schema X has been given a special name or number, we omit the ‘+’. Thus where T = A → A B = A → A .3 = (A → B) ∨ (B → A)

4 = A → A 5 = A → A Ver = ⊥

D = A → A .2 = A → A

we denote K + A → A by KT, and the smallest normal extension of this system by A → A will be called KT4. (Or rather, would be called KT4 if we didn’t introduce the further convention below rebaptizing this as S4.) This and the nomenclature introduced below come from various sources – principally Lemmon and Scott [1977], Segerberg [1971] and Chellas [1980]; some of the etymology behind the various labels can be gleaned from these sources, as well as from Bull and Segerberg [1984]/[2001]. For example, the letter “D” stands for “deontic”, the principle so named commonly being regarded as a weakening appropriate for deontic logic, of the alethic principle T, since what is obligatory need not be the case, but is at least permissible. Such weakenings are also often considered in the passage from epistemic to doxastic logic. In these cases there obviously considerable idealization involved (“logical omniscience” is the phrase often used to convey what normality requires), and D for belief is a further idealization, imposing a requirement of consistency on the believer. Following an idea of Segerberg’s, we propose that for a further rapprochement with more

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traditional terminology, the prefix “KT” appearing before a number be abbreviated to “S”. Thus S4 and S5 are the two C. I. Lewis logics known by those names (though differently axiomatized by Lewis); Lewis’s weaker systems S1, S2, and S3 are non-normal modal logics and so fall outside the ambit of the present survey. (These are not normal extensions of KT and so we write “S1”, etc., rather than “S1”.) S4.2 and S4.3, the best known logics between S4 and S5, will receive our attention in 6.33. Warning: For conformity with the literature regarding weaker modal logics, such as K4.3, we should really formulate .3 differently, namely with the first disjunct in the form “((A ∧ A) → B)”. (The second disjunct can remain intact; see 6.33.1(iii), p. 855.) A convenient notational device from Chellas [1980] enables us to coin some new labels. If X denotes some schema with → as main connective, then we use Xc to denote the converse implicational schema, and X! for the corresponding biconditional. (Does this use of the “c” subscript involve committing the ‘converse proposition’ fallacy of 1.15? No: here we are not dealing with propositions, but with syntactic objects – formulas and schemas.) Thus Tc and T! are respectively, A → A and A ↔ A. Accordingly KT! is the smallest normal modal logic in which each formula is equivalent to its necessitation; it is not hard to see that we can delete the word “normal” here and still obtain a correct description. This apparently degenerate system – sometimes also called the Trivial system of modal logic – occupies something of a special position amongst normal modal logics, as we now explain in the following paragraph. An exception to this (·)c convention is that it is not applied in the case of .2 (which is also, as is mentioned in the notes, known as G); the converse here is referred to as M (for “McKinsey’s Axiom”). Normal modal logics form a lattice under inclusion (ordinary class inclusion ⊆, since these are sets of formulas), in which meets are intersections and the join of two normal modal logics is—not their union, but—the smallest normal modal logic including both of them. This lattice has bounds, the zero being K and the unit being the inconsistent system. The dual atoms of this lattice are the Postcomplete normal modal logics: the only normal modal logic extending them is the inconsistent logic. The general (framework- and language-independent) idea is that Post-complete logics are dual atoms of any lattice of logics with the inconsistent logic as its unit element. (See the comments on DLat after 2.14.5, p. 248.) Now what is special about the system KT! mentioned in the previous paragraph is that it is one of the two Post-complete normal modal logics (the other being KVer). Many examples of proofs of formulas and derivations of rules on the bases of axiomatizations like that given above for normal modal logics may be found in Hughes and Cresswell [1968]. (Actually, for the most part the authors do not there consider normal modal logics weaker than KT; in fact on some earlier definitions of normal – e.g., that of Kripke [1963], T’s presence was demanded; a later reworking of some of the material which makes good this lack is Hughes and Cresswell [1996].) We pass to a semantic description of normal modal logics in terms of Kripke models, which is to say triples (W, R, V ) with W = ∅, R ⊆ W × W , and V :{pi }1i 2: the ‘vacuity’ noted à propos of 3.17.1, is in a certain sense essential to the exhibiting of a substitution-invariant rule with only the weaker preservation characteristic in the two-valued case, though not, as 3.17.6 illustrates, for n > 2. We shall show, in Thm. 3.17.11 below, that as long as there are sequents of Fmla valid in a twoelement matrix, and for such a sequent, we extend our earlier specifically boolean use of the term ‘tautology’ to apply it to the formula involved, then any rule (for the framework Fmla) which preserves validity in the matrix also preserves for an arbitrary matrix-evaluation, the property of holding on that evaluation. In discussing the two-valued case, we have in mind the ‘non-degenerate’ twoelement matrices (with one designated and one undesignated element), so we shall discuss the case in terms of valuations (rather than evaluations) and the values T and F. Thus what we shall show is that any substitution-invariant rule which leads from tautologies only to tautologies (the global characteristic), also leads only from truths (on an arbitrarily selected valuation v) to truths (on v: the local characteristic), as long as there are any tautologies at all in the given matrix. In view of the generality of the result just described, we must be careful to avoid assuming the availability of any or all of the boolean connectives of we have specifically singled out for attention (by associating some pre-assigned truth-function with them in defining a notion of #-boolean valuation, as at p. 65. When we discussed preservation characteristics of rules à propos of the system Nat in 1.25, for example, we were working with a functionally complete set of connectives in the sense that over the class of valuations under consideration— the boolean valuations—every truth-function was the truth-function associated with some (primitive or derived) connective. We exploited part of this expressive richness in order to correlate with each sequent σ a formula fm(σ) true on precisely the valuations on which σ held: here we used the availability of ∧ (to link formulas to the left of ) and → (to tie the result to the formula on the right). This feature of that discussion is irrelevant to our present concerns, since it concerned rules for Set-Fmla whereas we are now considering only the simpler case of Fmla. More to the point here is the use of ¬ in providing, on the basis of any (boolean) valuation v, what we may call ‘constantizing’ substitution instance for any formula with the property of being true on no (boolean) valuation if the original formula was false on v and true on every such valuation if the original formula was true on v. The trick was to replace, using the substitution sv in 1.25.7 (p. 130), each propositional variable true on v by a tautology (p → p, we used) and each variable false on v by a formula whose negation was a tautology (we used p ∧ ¬p); of course ¬(p → p) or any number of other formulas would have done as well: but given the remaining primitive connectives some use of ¬ to obtain this effect was essential. (We are here drawing on the discussion immediately following 1.25.7 rather than the actual formulations used in that Lemma.) Since we are now trying to prove something for all languages and two-valued matrices which do provide tautologies the first part of this constantizing strategy will always be available (pick a matrix tautology to play the p → p role); but the second part need not be. It turns out to be useful to divide all the possible cases into these two types: those in which the second part of the strategy is available because there is some formula false on every valuation under consideration (the

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CHAPTER 3. TRUTH-FUNCTIONAL, EXTENSIONAL, . . .

f cases, we will call them) and those on which it is not available for want of such a formula (the ‘no f ’ cases: we will introduce t and f as temporary ad hoc notations presently). We turn to setting up the apparatus for the proof proper. Let M be a (non-degenerate) two-element matrix for some language. As announced above, we will refer to M-evaluations as valuations, to formulas true on every valuation as tautologies, an arbitrarily selected one of which we denote by t. Similarly, if there happen to be formulas true on no valuation (‘contra-valid formulas’), we let f stand for an arbitrarily selected such formula. Formulas true on exactly the same valuations will be called equivalent. For each connective # of the language we denote by # the truth-function associated with # over the class of valuations; we use the same convention when dealing with defined connectives and the correspondingly derived truth-functions. (One could perhaps make use of “#b ” rather than “#”, but the latter notation will be more convenient.) We reformulate the analogue of Lemma 1.25.7(i): Lemma 3.17.7 With M as above, if there exist formulas t, f, as described there, then for any formulas A1 , . . . , An , B, and valuation v such that v(A1 ) = . . . = v(An ) = T and v(B) = F, there exists a substitution s with s(A1 ), . . . , s(An ) tautologies and s(B) not a tautology. Proof. Take s as sv , as for 1.25.7(i), but with t and f playing the roles of pi → pi and pi ∧ ¬pi respectively. We pass now to the ‘no f ’ case. An initial observation we may make pertains to two of the four one-place truth-functions, introduced under the names I, V, F, N in 3.14: the identity function, the ‘constant true’ function, the ‘constant false’ truth-function and complementation (boolean negation). Observation 3.17.8 If, with M as above, there exists a formula t but no formula f satisfying our conditions on such formulas, then for no (compositionally) derived connective # of L do we have: # = F or # = N. Proof. If # = F then #A, for any A, could be taken as f ; if # = N then #t could be taken as f. For a result like that of 3.17.7 for the ‘no f ’ case, we employ an alternative to the ‘constantizing’ substitution of the proof of that Lemma, and follow— roughly—the lead of R. V. Setlur. (The present discussion can be regarded as rounding out Setlur [1970a].) Instead of choosing the now unavailable f to replace the propositional variables false on the valuation in terms of which the substitution is defined, we use the substitution s∗v defined in 1.25.12 on p. 132, to replace them all by an arbitrarily selected but fixed variable (p). We use t to do the work done in that exercise by p → p. That is, we set s∗v (pi ) = t if v(pi ) = T, s∗v (pi ) = p if v(pi ) = F. Because of the greater intricacy of the ‘no f ’ case, we split the reasoning corresponding to that of 3.17.7 into two parts, Lemma 3.17.9 working out the effects of our new substitution strategy (a form of 1.25.12(i) suited to the present more general setting), and Lemma 3.17.10 (corresponding to 1.25.12(ii)), drawing the desired corollary.

3.1. TRUTH-FUNCTIONALITY

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Lemma 3.17.9 With M as above, if there is a formula t in the language of M but no formula f (t, f, satisfying the conditions on such formulas imposed in the discussion leading up to 3.17.7), then for all formulas A, if v(A) = T then s∗v (A) is a tautology and if v(A) = F, then s∗v (A) is equivalent to p, for any valuation v. Proof. By induction on the complexity of A. The basis step being secured by the definition of s∗v , we pass to the inductive part of the argument, taking A = #(B1 , . . . , Bm ), with # a primitive m-ary connective. First, we suppose that v(A) = T; we must show that s∗v (A) = #(s∗v (B1 ), . . . , s∗v (Bm ) is a tautology. Without loss of generality, we may suppose that for some k (0 k m), v(B1 ) = . . . = v(Bk ) = F while v(Bk+1 ) = . . . = v(Bm ) = T. The inductive hypothesis then gives that s∗v (A) is equivalent to #(p,. . . ,p,t,. . . ,t) where p occurs k times and ‘t’, m − k times, so it is this latter formula that must be shown to be a tautology. Defining the singulary connective # by # C = #(C, . . . , C, t, . . . , t ), k times

m−k times

our task reduces to showing that # p is a tautology, i.e., that # is V. Now by 3.17.8 we know that # is either V or I, and we can rule out I as a possibility since we know that v(#(B1 . . . , Bm ) = T, so #(v(B1 ), . . . , v(Bm )) = T, which means that #(F, . . . , F, T, . . . , T) = T, where there are k “F”s and m − k “T”s, so # (F) = T. Next, suppose that v(A) = F; we must show that s∗v (A) = #(s∗v (B1 ), . . . , s∗v (Bm ) is equivalent to p. Making the same assumption about k as before, we have by inductive hypothesis that s∗v (A) is equivalent to #(p, . . . , p, t, . . . , t), with p occurring k times, and, with # defined as above, our task reduces to showing that # p is equivalent to p, i.e., that # is I. If not, by Obs. 3.17.8, # would have to be V: but this is impossible since, by reasoning as for the preceding case, we know that #(F, . . . , F, T, . . . , T) = F, so # (F) = F.

Lemma 3.17.10 With M as introduced before 3.17.7 above, if there is a formula t of L but no formula f in the language of M – with t,f, as above – then for any formulas A1 , . . . , An , B, of that language and valuation v such that v(A1 ) = . . . = v(An ) = T and v(B) = F, there exists a substitution s with s(A1 ), . . . , s(An ) tautologies and s(B) not a tautology. Proof. Choose s as s∗v and apply Lemma 3.17.9.

The upshot of all of this is, as promised, that for substitution-invariant rules for the framework Fmla, the distinction between the local and the global preservation characteristic cannot arise for two-element matrices, as long as these matrices provide at least one tautology: Theorem 3.17.11 With M a non-degenerate two-valued matrix for a language L, and ρ a substitution-invariant rule of the framework Fmla over L, if ρ preserves tautologousness (validity in M), then ρ preserves truth on a valuation (an M-evaluation), provided that at least one formula of L is valid in M.

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Proof. Suppose that ρ is a substitution-invariant rule as described, and that A1 , . . . , An , B ∈ ρ, v(A1 ) = . . . = v(An ) = T, while v(B) = F. We must show that there exist C1 , . . . , Cn , D with C1 , . . . , Cn , D ∈ ρ, with each Ci , but not D, being a tautology. If some formula of L is false on every valuation, we take it as the f of 3.17.7 and apply that Lemma to get sv (Ai ) as Ci (i = 1, . . . , n) and sv (B) as D, with the desired properties. If there is no such formula, we invoke instead 3.17.10, again taking s(Ai ) as Ci and s(B) as D, though this time with s = s∗v . In either case, C1 , . . . , Cn , D ∈ ρ, since A1 , . . . , An , B ∈ ρ and ρ is substitution-invariant. The condition in 3.17.11 is not only necessary, but also sufficient, for the local/global collapse there spoken of: Exercise 3.17.12 Show that with M and L as in 3.17.11, if no formula of L is valid in M, then a formula-to-formula substitution-invariant rule can be found with the global preservation characteristic w.r.t. M but without the local preservation characteristic. This completes our excursus on rules and truth-functions, though we note two issues which would deserve attention in a more extended treatment of the topic. The first would be looking for analogous results in Set-Fmla and SetSet. The second issue is the relationship between the local/global distinction amongst preservation characteristics of rules w.r.t. matrices and the derivable/admissible distinction between rules w.r.t. proof systems. In particular, when provability in a proof system coincides with validity in a matrix M, then clearly admissibility for rules coincides with possession of the global preservation characteristic for M (preservation of M-validity, that is). However, if the proof system is determined by some collection of matrices but by no single matrix (as discussed in 2.12) then admissibility coincides with the even more ‘global’-type characteristic of preserving the property of being valid-in-every-matrix-in-thecollection. This a rule might possess without preserving, for each matrix in the collection, validity in that matrix (our global characteristic in the above discussion) let alone preserving for an arbitrary matrix-evaluation into for some matrix in the collection, the property of holding on that evaluation (our local characteristic). An analogous multi-layered structure was seen to arise in the Kripke semantics of modal logic in 2.23.

3.18

Intuitionistically Dangerous Determinants

In this subsection we examine some phenomena brought to light by N. M. Martin and (especially) S. Pollard, though without using their closure space + “irreducible closed sets” conceptual framework to discuss them. (See Pollard and Martin [1996], Pollard [2002], [2006], and for background to their approach, Martin and Pollard [1996]. “Closure space” means closure system in the sense of 0.13.5(ii), p. 10; “irreducible” means meet-irreducible in the sense of 0.13.9(ii), p. 11, with meets as arbitrary rather than just binary intersections.) Instead, we formulate matters in terms of valuations consistent with consequence relations making use of the partial order on valuations for a given language defined (as in 1.26) by u v iff for all formulas A, if u(A) = T then v(A) = T. Note that this is the partial order associated with the lattice of valuations under the


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operations of conjunctive and disjunctive combination ( and ); in particular u v ⇔ u = u v. The top and bottom elements of this lattice are the all-verifying and all-falsifying valuations vT and vF . Theorem 3.18.1 Suppose that # is an n-ary connective (not necessarily primitive, n 0) in a language for which u, v, are valuations with u respecting the determinant F, . . . , F, T (n occurrences of “F”) for # and v respecting some determinant for # with final element F. Then u v implies u = v or v = vT . Proof. Suppose u and v respect determinants for # as described and that u v, with a view to showing that either u = v or v = vT . With n = 0 we immediately have a contradiction, since the conditions on u and v require that u(#) = T while v(#) = F, contradicting u v. So consider n 1. For a contradiction in this case, suppose that u = v and v = vT . Since u v, the first of these suppositions implies that for some formula A, u(A) = F while v(A) = T; the second means that for some formula B, v(B) = F. Note that it follows from this that u(B) = F Letting an F-final determinant respected by v be x1 , . . . , xk , y1 , . . . ym , F , with k + m = n, we can assume without loss of generality that x1 = . . . = xk = T and y1 = . . . = ym = F. (Otherwise just rearrange the positions of A and B in what follows.) Since u satisfies Fn , T , the superscript n indicating an n-termed sequence of Fs (and similarly in what follows, this notation being that used in the proof of 3.13.15, p. 395), we have u(#(Ak , Bm )) = T, while since v satisfies Tk , Fm , F , we have v(#(Ak , Bm )) = F, contradicting the hypothesis that u v. Theorem 3.18.1 is our version of (some of the) material in the papers by Pollard cited above; we add a dual form: Theorem 3.18.2 Suppose that # is an n-ary connective (not necessarily primitive, n 0) in a language for which u, v, are valuations with v respecting the determinant T, . . . , T, F (n occurrences of “T”) for # and u respecting some determinant for # with final element T. Then u v implies u = v or u = vF . Proof. As in the previous proof, we suppose the stated conditions on u, v and # are satisfied and that u v while u = v and u = vF . Thus there are formulas A, B with u(B) = F u(A) = T, v(B) = v(A) = T. Let a Tfinal determinant for # respected by u be Tk , Fm , T , with the result that u(#(Ak Bm )) = T, whence, as u v, v(#(Ak Bm )) = T, contradicting the hypothesis that v respects Tn , F . By now it will be clear that the title of this subsection does not refer to determinants which are ‘intuitionistically dangerous’ individually, but rather to pairs of such determinants which pose some kind of threat when taken together. More precisely still, it is the conditions on consequence relations induced by these determinants that pose a threat of non-conservatively extending IL into CL (or worse). (Here “IL” means: a proof system whose rules embody those conditions as well as the condition of being →-intuitionistic, as defined on p. 329 – such as INat or IGen.) Let us look at this in detail in the case of the two kinds of determinants in 3.18.1. We had first for n-ary #, Fn , T which we could call the compositional subcontrariety determinant for #, since it says that

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over a class of valuations respecting it, any #-compound and its n (not necessarily distinct) components cannot all take the value F on a valuation in the class. (The term subcontrary may not be familiar; we shall meet it several times before its gets to share in a subsection title at 8.11.) Similarly, in the case of 3.18.2, the determinant Tn , F can be thought of as a compositional contrariety determinant for an n-ary connective, since it disallows the simultaneous truth of the compound and all its components. But let us return to looking at 3.18.1. For the valuations u and v which were imagined to be such that u v, with dire ‘flattening’ consequences (u = v unless v = vT ), u was supposed to satisfy the full subcontrariety determinant for # and v just some determinant with F as final entry. To make notation easier, let us simply assume that the n pre-final entries in this determinant consist of k occurrences of T and then m occurrences of F, where m + k = n; that is, we are dealing the determinant Tk , Fm , F . Thus the two determinant-induced conditions on a gcr are: A1 , . . . , Ak , B1 , . . . , Bm , #(A1 , . . . , Ak , B1 , . . . , Bm ) and A1 , . . . , Ak , #(A1 , . . . , Ak , B1 , . . . , Bm ) B1 , . . . , Bm . Rather than converting these into the corresponding conditions on consequence relations by means of the methods of arbitrary and general consequences described in 3.13, let us help ourselves to the effects of the rules governing disjunction in a proof system for IL such as INat or IGen and simply replace the commas on the right by disjuncts, using ⊥ for the case of the empty disjunction. At the same time, we will present them as sequent-schemata rather than consequence relation statements. The result of these transformations turns the above conditions into (1) and (2), respectively: (1) A1 ∨ . . . ∨ Ak ∨ B1 ∨ . . . ∨ Bm ∨ #(A1 , . . . , Ak , B1 , . . . , Bm ). (2) A1 , . . . , Ak , #(A1 , . . . , Ak , B1 , . . . , Bm ) B1 ∨ . . . ∨ Bm . Next, noticing that the various Ai (i = 1, . . . , k) march together here, as to the various Bj (j = 1, . . . , m), let us identify the Ai with each other, just writing “A” for them, and similarly in the case of the Bj . This gives the following manageable special cases of the above schemas, in which # (defined by # (A, B) = #(Ak , Bm )) may be taken to be at most binary, though we retain the prefix – rather than switch to infix – notation: (1 )

A ∨ B ∨ # (A, B);

(2 )

A, # (A, B) B.

Informally speaking – thinking of the “∨”s on the right of (1 ) as commas – we want to apply the cut rule (T) to these two, but as they stand such an application leads nowhere. But if we are working with a language also containing → subject to its usual insertion-on-the-right or introduction rule (depending on whether we are working with IGen or INat), then we can move from (2 ) by this rule to: (2 )

# (A, B) A → B.

We can now we can proceed with the envisaged cut, or, more accurately, make use of (∨E) or (∨ Left), to obtain as an offspring of (1 ) and (2 ):


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A ∨ B ∨ (A → B), and this – i.e., a representative instance of this, with distinct propositional variables for the distinct schematic letters – is not IL-provable. Thus the schemata corresponding to the conditions induced on consequence relations by the “dangerous” (pair of) determinants result in a non-conservative extension of either of the standard proof systems – the crucial feature being (→I) or (→ Right) – for IL, since A and B can just be, for example, p and q, giving a formerly unprovable sequent derived by means of (in our formulation here, zero-premiss) rules governing #, even though that sequent does not itself feature a formula constructed by means of #. Exercise 3.18.3 A simpler form of the last inset schema, without the middle disjunct, is more familiar than the version displayed, as a principle for ‘classicizing’ intuitionistic logic. The two forms are IL-equivalent, however, and here we address the less obvious direction of that equivalence. Show that if ⊇ IL and A ∨ B ∨ (A → B) for all A, B, then A ∨ (A → B) for all A, B. (Suggestion. Make a judicious substitution for “B” which will enable the second and third disjuncts to be collapsed into one, IL-equivalent to A → B, exploiting the contraction properties of intuitionistic →.) The derived connective # above was described as ‘at most’ binary. If k = 0 (while m > 0) or m = 0 (while k > 0), the formula represented as A or B (respectively) disappears. In the former case, and we are left with (1 ) and (2 ) in the forms below, with # 1-ary: B ∨ # B

# B B,

which is bad news all round (rather than just for the intuitionist), as is especially clear if the ∨ in the first of these is written as a comma and we can see that the resulting Cut with the second yields B. Since B is any formula, we have the inconsistent consequence relation here: every Set-Fmla sequent is provable. In the latter (m = 0) case, what we are left with as (1 ) and (2 ) are: A ∨ # A

A, # A ⊥.

in which now the empty disjunction ⊥ makes its first explicit appearance. By contrast with the previous case, here, as in the general case above, the (2 )-to(2 ) move is essential; writing ¬A for A → ⊥, this move leads to # A¬A, which with the sequent on the left above gives the IL-unprovable A ∨ ¬A. The inset conditions above make #A behave like the classical negation of A, of course, and here we see them spreading via an application of (→I) or (→ Right) to infect the previously weaker-than-classical ¬. (Alternatively, we could conduct the discussion of this example entirely in terms of primitive ¬, without → or ⊥, as Pollard does in some of the publications listed in our opening paragraph, in which case the ⊥ in the second inset schema above should be replaced by a new schematic letter, “C”, say, and instantiating this to ¬A gives rise to a (2 )-to-(2 ) style transition which passes from A, # A ¬A to # A ¬A.) Exercise 3.18.4 What happens in the case that k = 0 and also m = 0?


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To see the crucial role of the (2 )-to-(2 ) transition – and thus of the → rule mentioned under its natural deduction and sequent calculus names above – let us look at these syntactical manipulations in the light of Theorem 3.18.1, note that for a class V of valuations consistent with IL for which that rule preserves V -validity (see 1.26.14, p. 139), for any u ∈ V , and any formulas A, B: (∗)

(2 ) holds on u if and only if for all v u, (2 ) holds on v.

Now the hypothesis that (1 ) holds on u for all formulas A, B, is equivalent to the hypothesis that u respects the compositional subcontrariety determinant for # (simplified down from that for #), and the hypothesis that (2 ) holds on v for all formulas A, B, is equivalent to the hypothesis that v respects the chosen F-final determinant (chosen for #, that is, and then simplified as above) for # . Thus 3.18.1 takes us from these hypotheses to the conclusion that for all v as in (∗), v = u or v = vT . So if both (1 ) and (2 ) both hold on u, u is either a boolean valuation or else vT , with the result that we end up with CL or with inconsistency (the ‘bad news all round’ case, as it was put above). To get to (2 ), of course, we needed to apply (→I)/(→ Right) before the final (fatal) cut. Digression. Since we are officially working in Set-Fmla with commas on the right encoded by disjunction, references to the cut rule (T) are really references to the following rule(s): Γ C1 ∨ . . . ∨ Ci ∨ D ∨ Ci+1 ∨ . . . ∨ Cn

Γ , D E

Γ, Γ C1 ∨ . . . ∨ Ci ∨ E ∨ Ci+1 ∨ . . . ∨ Cn Of course the order of the disjuncts is immaterial and this particular display, putting E into the same position occupied by D, is chosen simply for clarity. There is no difficulty seeing this rule as derivable with the aid of the usual ∨ rules and the (standard) structural rules. We stress the → rules in our discussion rather than the ∨ rules since the current syntactic discussion can be reformulated without reference to ∨, as we shall see immediately. End of Digression. Since it is the afforded to → by the familiar intuitionistic rules that has been stressed here, we should just check that the appeals to disjunction can be avoided. We have already seen that the ‘nullary disjunction’, ⊥ can be avoided, since it only occurs in the above discussion where it can be replaced by a new schematic letter. This was the method of ‘arbitrary consequences’ from 3.13; for ∨ proper we need a version of the method of ‘common consequences’ from the same discussion. (Alternatively, one could think of this as a version of deductive disjunction, generalized upward from the binary case mentioned described in 1.11.3: p. 50.) That is, we rephrase (1 ) to: A → C, B → C, #(A, B) → C C,

while from (2 ) we have (by ‘suffixing’): (A → B) → C # (A, B) → C. Cutting these together, we get: A → C, B → C, (A → B) → C C, which is not IL-provable, as one sees most easily by taking C as A: A → A, B → A, (A → B) → A A.


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Here the first formula is provable and so can be dropped, and the second follows (intuitionistically) from the third so it too can be dropped, leaving us with Peirce’s Law. Having seen that we do not need it, we resume the discussion with the aid of disjunction, this time to consider much more briefly than for the 3.18.1 case, the case of the conditions on consequence relation induced by the determinants mentioned in 3.18.2. Here we have the compositional contrariety determinant, leading via the earlier identifications to the schema (3 ), in place of (1 ), and T-final determinant, similarly giving (4 ). (Imagine (3) and (4) as having been provided along the lines of (1) and (2).) (3 )

A, B, # (A, B) ⊥;

(4 )

B A ∨ # (A, B).

From (3 ) we derive using the crucial (→I)/(→ Right) rule: (3 ) # (A, B) A → ¬B, which cuts with (4 ) to deliver the IL-unprovable B A ∨ (A → ¬B). So again the IL proof system (e.g., INat) is extended non-conservatively – indeed all the way to classical logic, as we see most easily by taking B as p → p – by the addition of rules corresponding to the conditions on consequence relations induced by the dangerous pair of determinants. The use of ⊥, ¬ and ∨ are all inessential and can be got rid of as in our earlier discussion. Since we built the idea of being closed under the (→I)/(→ Right) rule into the notion of an →-intuitionistic consequence relation (at p. 329), we can give summarise with a formulation that doesn’t explicitly address proof systems and rules: Theorem 3.18.5 If for a consequence relation we have ⊇IL and is →intuitionistic, then ⊇CL whenever (i) for some connective #, satisfies the conditions induced by the compositional subcontrariety determinant for # as well as some F-final determinant, or (ii) satisfies the conditions induced by the compositional contrariety determinant for # as well as some T-final determinant. As remarked at the start of this subsection, this result is essentially due to S. Pollard, though we have filled out the picture in some respects (such as bringing in (ii)). Let us note further that the two determinants for classical negation (the two elements of ¬b , or N, as we also sometimes call it), T, F and F, T , taken together, fall foul of (i) here, as noted above, and they also fall foul of (ii). (We stressed this in connection with the induced conditions – A ∨ # A and A, # A ⊥ – in the former case.) But the requirement that should be →-intuitionistic is crucial here, as elsewhere; for an example of ⊇ IL while ⊇ CL , despite satisfying the (now, not so dangerous) determinant-induced conditions, see Humberstone [2006a], especially the discussion of “¬Ω ” at p. 67.

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Notes and References for §3.1 Much of the material in 3.11–13 is treated in Segerberg [1982], [1983]; our term “fully determined” is adapted from Segerberg’s use there of “type determined”. See also Sanchis [1971]. In Humberstone [1986] the term “determinant” is used for what are here called determinant-induced conditions (and no term is used for what we call determinants here). Pollard [2002] refers to determinants as truthconditions, and denotes the determinant we write as (for example) T, T, F with the notation “T, T ⇒ F”. Partially determined connectives (and some manyvalued analogues thereof) are called quasi-truth-functional in Rescher [1962] (where it is noted that this term is also used – though with a different sense – in Goddard [1960]) and §26 of Rescher [1969]; some general theoretical results about the apparatus involved here may be found in Avron and Lev [2005]. In this paper (as well as in Avron and Lev [2001], concentrating on the 2valued case) the authors use the phrase non-deterministic matrix – abbreviated to “Nmatrix” – for some apparatus like a matrix but in which the evaluation of a compound is merely constrained rather than uniquely determined by the values of the component. (See also Béziau [2002].) The uses of “quasi-truthfunctional” just remarked on have motivated the choice of the (different) term pseudo-truth-functional for our purposes in the following section, to avoid confusion; the latter term is defined in the discussion preceding 3.22.5. For an application of the distinction between not being fully determined and being completely undetermined, see Humberstone [1997b]. Problems with certain extant explications of truth-functionality and related notions, especially as such notions apply to natural language expressions (a topic avoided in our discussion) are discussed in Martin [1970] and Sanford [1970a]; for a more recent venture into this field, see Schnieder [2008], and, for examples of mistaken claims about truth-functionality, Section 4 of Humberstone [1997b]. The main publications known to the author on the subject of if-then-else (from 3.11.3(ii)) approach it from the perspective of equational logic: Bloom and Tindell [1983], Hoare [1985], Guessarian and Meseguer [1987], and Manes [1993]. An accessible presentation of the results in Post [1941] – mentioned in 3.14 – may be found in Pelletier and Martin [1990], as well as in Gindikin [1985] and §5 of Urquhart [2009]. These include not just that on ‘Sheffer’ functions described in the Digression beginning on p.165, but the more general question of which sets of truth-functions are functionally complete and on the enumeration of the (countably many) clones of truth-functions; see also Lyndon [1951] and Rautenberg [1981], [1985], on this last topic. (For reasons of space, we have not included a discussion of a logically interesting near-relation of such Sheffer functions, the ‘universal decision elements’ described in Sobociński [1953], Pugmire and Rose [1958], Rose [1958].) As was mentioned in 3.14, Gindikin [1985] may be consulted for an accessible proof of Post’s Theorem on functional completeness (p.95f.); however the reader should be warned against a typographical error in the formulation of the result (as Exercise 6.13 on p. 89): the second occurrence of “Φ” should be replaced by “the class in question”. Pelletier and Sharp [1988] is an interesting paper, with a pedagogical emphasis, on some questions of functional (in)completeness. The case treated in Example 3.14.7 (functional incompleteness of {¬b , ↔b }), is discussed by them in considerable detail; it is also treated in Massey [1977] (as mentioned in 3.14.7). Humberstone [1993a] fills out the historical picture on functional completeness, with special emphasis


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on the distinction between compositional and -compositional (under the names “polynomial” and “-polynomial”) derivability of a truth-function from a set of truth-functions, and provides references to concepts related to functional completeness that have been studied in universal algebra; we repeat one reference from the body of 3.14 here: Tarski [1968]. The term clone (due to P. Hall, according to Cohn [1965]) from the same subsection is not etymologically related to its biological homonym, arising (it is said) instead as a portmanteau for “closed network of operations”. The definition appearing in Burris [2001] of this notion, alluded to in the discussion after 3.14.6 is as follows: Given a set A let F (A) be the set of all constants from A and all finitary functions on A. A clone on A is a subset X of F (A) such that 1 X is closed under composition; 2 X contains all projection functions; 3 if a constant c is in X then all finitary functions with constant value c are in X; 4 if a finitary function with constant value c is in X then the constant c is in X.

(A small change in the wording of 3 has been made here in the interests of grammaticality; the slight misuse of the term “constant”, common in the writings of algebraists, has been preserved.) Functionally complete n-ary truth-functions for n > 2 are discussed in Wheeler [1961], Tomaszewicz [1972], Gindikin [1985], p.99—in all cases touching on the question of how many such there are for a given n. (For n = 3, there is a detailed study in Cunningham-Green [1959].) For discussion of functional completeness in many-valued logic, see §11 of Rescher [1969], and Rosenberg [1977]. The generalized Post duality patterns in 3.15 are related to (though more general than) some material presented in Gottschalk [1953], which describes some connections between duality and the traditional ‘square of opposition’ from Aristotelian (syllogistic) logic; cf., also on this theme, Williamson [1972]. The topics of essentially n-ary functions the dependence of a function on particular arguments, which occupy us in the last part of 3.15 have been extensively discussed; see for example Breitbart [1967], Salomaa [1965], Čimev [1979], and references therein. In the paragraph following 3.14.6 bibliographical references to the work of Menger and associates on functional composition (without automatic assistance from projection functions) were promised. The following sources will give the flavour: Menger [1962a, b], Menger [1964a, b], Menger and Schultz [1963], Menger and Whitlock [1966], Calabrese [1966], Skala [1966]. What we call composition, Menger calls substitution (more exactly: ‘parenthesis substitution’). The concept of a generalized Post duality pattern introduced in 3.14 is intimately related to – indeed is a special case of – the notion of an isotopy, a generalization of the concept of isomorphism, concerning which the interested reader is referred to §1.3 of Dénes and Keedwell [1974], where further references can also be found. The discussion of definability and definition in 3.15 draws on Smiley [1962a] and the appendix to Humberstone [1993a]; the distinction between what we call the metalinguistic and object-linguistic views of definition is colourfully drawn in Meyer [1974a], in which – by contrast with 3.15 – a preference for


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the latter view is expressed. (See further Humberstone [1998a].) Kuznetsov’s 1965 result, reproduced here as 3.15.8(ii) was duplicated in Hendry [1981]; for an alternative to the example in 3.15.8(i), see Došen [1985b]. See also Rousseau [1968]. Kuznetsov completeness (as we call it) is also discussed in Ratsa [1966], [1971] and Kuznetsov and Ratsa [1979].

§3.2 EXTENSIONALITY 3.21

A Biconditional-based Introduction to Extensionality

Where is a (generalized) consequence relation with the connective ↔ in its language and such that ⊇ Log(BV↔ ), we can formulate an important ‘replacement’ condition on connectives # of that language, L, say, as they behave according to , which is satisfied by all connectives which are fully determined according to . (It will turn out to be satisfied by others as well.) Let us say that #, assumed k-ary, is ↔-extensional in the i th position (where 1 i k), according to if for all B1 , . . . , Bk , C ∈ L: (↔-Exti )

Bi ↔ C #(B1 , . . . , Bk ) ↔ #(B1 , . . . , Bi−1 , C, Bi+1 , . . . , Bk ).

In view of the restriction to ⊇ Log(BV ↔ ), this can be read as saying that formulas with the same truth-value (on any valuation consistent with ) can always be substituted, one for another, into the ith place in a #-compound without changing the truth-value of that compound. The above terminology derives from a usage of the term extension as applied to statements, to denote the truth-value of the statement concerned. In the setting of predicate logic and various enrichments thereof (modal predicate logic, predicate logic with predicate modifiers, etc.) there are various other notions of extensionality current, mostly as applied to arbitrary contexts, for which here we use the underlining convention of 3.16.1 (p. 424), though for the first of these we use this for a context for a name (or individual constant), and for the second, a context for a predicate letter. The matching notion of extension for names calls the extension of a name the object denoted by the name, and for (one-place) predicates, the set of objects of which the predicate is true. The extensionality of a context means that splicing expressions with the same extension into the context yields results with the same extension; here we are only concerned with the case in which the results in question are sentences (or formulas as we say, when dealing with one of our formal languages). Putting this in terms of a favoured (generalized) consequence relation with ↔ in its language means that the following -claims amount to claims that the contexts indicated are extensional in name position, predicate position and sentence position respectively: a=b

a

↔

b

∀x(F x ↔ Gx)

F

↔

G

A ↔ B

A

↔

B

Derivatively we can apply this terminology to the case of a connective # as supplying the contexts in question; for illustration we take # as singulary.

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Thus we could say that # is extensional in name position, when (however the names a and b are chosen) a = b #A ↔ #B, where A is an atomic formula B differing from A only in having one or more occurrences of a replaced by b. (A is taken as atomic here because we not want to deny extensionality in name position to # – for example, taken as ¬ – just because co-extensive names cannot be replaced one by another in some longer formula with # as main connective but in whose scope there is some other connective responsible for the failure of extensionality of the resulting context. Similarly in the following case.) Likewise, # would be extensional in predicate position, when – sticking with the case of monadic predicates – ∀x(F x ↔ Gx) #A ↔ #B, with B is an atomic formula differing from A in having F replaced by G. Because we do not have, for extensionality in sentence position, this difference over parts of speech (name vs. sentence, etc.), the extensionality in sentence position of # (according to ) is simply a matter of the third claim above for the case in which the context is simply provided by # (i.e., A is #A). Further, since we are concerned only with sentential logic, this qualification will not be needed. Even in this setting, though, it should be remarked that extensionality as a property of contexts is worth bearing in mind, though we shall not dwell on it. All contexts are extensional according to CL , for example. (Having not yet purified the notion of its dependence on ↔ and its behaving suitably – for example, as specified in the opening sentence of this subsection – we are not in a position to ask whether this is so for IL . In 3.23 it will become evident that, reverting to the C(·) notation for a 1-ary context mentioned after 3.16.1, although A ↔ B IL C(A) ↔ C(B) for all A, B, C, not all contexts – indeed not even all connectives – are extensional according to the consequence relation IL : see 3.23.2, p. 455, for example. But we have not even introduced the notion of ↔-extensionality which is to be thus purified. That comes next.) We will call a k-ary connective ↔-extensional (according to ) if it is ↔extensional (according to ) in the ith position for each i (1 i k). Equivalently, #, of arity k, is ↔-extensional according to if (↔-Ext)

B1 ↔ C1 , . . . , Bk ↔ Ck #(B1 , . . . , Bk ) ↔ #(C1 , . . . , Ck ).

Notice first that, by contrast with the property of being fully determined according to , these concepts apply straightforwardly not only to gcr’s but also to consequence relations , since always exactly one formula appears schematically represented on the right of the “”. Notice, secondly, that we have not included the case of k = 0 here (or in (↔-Exti )); in this case, the left-hand formulas vanish and are left with the condition, automatically satisfied given the conditions on , that # ↔ #. And, finally, observe that we could see the above definitions as special cases of a more general notion: that of the # -extensionality (according to ) of a connective #, where # is any binary connective. We have given only the above special case (# = ↔) as we regard the notion of ↔extensionality as of interest only to motivate a notion of extensionality which will be purified of reference to any such ancillary connectives. Now, if ⊇ Log(BV # ) for each connective # for which the notion of a #-boolean valuation has been defined, then # is ↔-extensional according to . This fact may tempt one to confuse the notions of truth-functionality (w.r.t. BV ) and extensionality; in fact, however, there are extensional connectives which are not fully determined (according to some one ): and in issuing this correction we have adjusted the talk of truth-functionality (a concept which


446

is relativized to classes of valuations) to talk of full determination (a concept relativized to consequence relations or gcr’s) so as to have coordinate concepts to compare. By way of illustration, consider the smallest gcr extending Log(BV ↔ ) and satisfying for a singulary connective $ in its language, both: ($1)

$A A

and ($2)

A ↔ B $A ↔ $B,

for all formulas A and B. Because of ($2), which is of course nothing other than (↔-Ext) for $ as #, this connective is extensional (according to ), but since for we have neither: (i)

$A, A , for all A

nor

(ii )

A $A, for all A,

$ is only partially determined (according to ), and so not truth-functional (by 3.11.5, p. 379) w.r.t. Val (). To see that neither (i) nor (ii) holds, note that a valuation consistent with can associate with $ the truth-function I, with v($A) = v(A) for all A, and a valuation consistent with can also associate with $ the function F, which is to say: v($A) = F, for all A. (The special V, F, I, N notation for singulary truth-functions was introduced after 3.14.5 (p. 406). ($1) and ($2) rule out neither possibility. But a valuation behaving in the former way, as long as it assigns the value F to at least one formula, is not consistent with any gcr satisfying (i), and one behaving in the latter way, as long as it assigns the value T to at least one formula, is not consistent with any gcr satisfying (ii). To tighten up this discussion, let us be more specific about the gcr under consideration, and specify a gcr $↔ as the least gcr on the language with connectives ↔ (binary) and $ (singulary) which satisfies the conditions (1)–(4) on ↔ in 3.11.2 (p. 378), as well as ($1) and ($2) above. Also let Vf be the class of valuations over which the truth-function f is associated with $. Since $ is singulary, f will be one of V, F, I, N. Then we have: Observation 3.21.1 Val($↔ ) = BV↔ ∩ (VI ∪ VF ). Proof. ⊇: Suppose v is ↔-boolean and associates with $ either I or F; then none of (1)–(4) of 3.11.2 nor ($1), ($2), constitutes a counterexample to the claim that v is consistent with $↔ , and this property is preserved as we apply (R), (M) and (T) to obtain the least gcr satisfying these conditions. ⊆: Val ($↔ ) ⊆ BV ↔ , by 3.11.2 (p. 378) and 3.11.4 (p. 378). It remains only to show that Val ($↔ ) ⊆ (VI ∪ VF ). So suppose v ∈ V al($↔ ) but v ∈ / VI ∪ VF . / VI there Since Val ($↔ ) ⊆ BV ↔ , we know that v is ↔-boolean. Since v ∈ exists a formula C with v(C) = v($C), which means, since v is consistent with / VF , for $↔ that (1) v(C) = T and (2) v($C) = F, in view of ($1). Since v ∈ some formula D such that (3) v($D) = T, and so by ($1) again, (4): v(D) = T. Now by (1) and (4) and the ↔-booleanness of v, we get v(C ↔ D) = T. But then the following instance of the condition ($2) is violated: C ↔ D $↔ $C ↔ $D since (2) and (3) tell us that v($C) = v($D). This contradicts the assumption that v ∈ V al($↔ ). Note that not even ($1) was needed for the above example of a non-truthfunctional connective which was nonetheless ↔-extensional. With only ($2), we should have had a completely undetermined extensional connective (according to $↔ so modified). Further, we may note that ($1) is a determinant-induced

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condition (for $) on , the determinant being T, T , and that by itself it does not guarantee that $ is extensional according to a gcr satisfying it. For example, the connective of §2.2 is not extensional according to the gcr KT described at the end of 2.23 – though in that subsection this notation was explicitly used for the inferential consequence relation, rather than the inferential gcr, associated with KT.) The relation between truth-functionality and ↔-extensionality is most clearly revealed after we have passed to the ↔-independent concept of extensionality simpliciter below, and then completed the proportionality fully determined is to truth-functional as extensional is to _ _ _ The blank will be filled by “pseudo-truth-functional” a term to be introduced in the following subsection, and the relationship explicitly charted in Theorem 3.22.5 below (p. 451). To describe this relationship in terms of the present ↔-involving notion of extensionality would require the intrusive imposition of restrictions to ↔-classical gcr’s and ↔-boolean valuations. In the meantime we conclude this subsection with another pair of examples of extensional but not fully determined connectives, this time specified in terms of the Kripke semantics for modal logic. To stick with gcr’s, what we need is the C-inferential gcr where C is the class of all frames; for brevity, we denote this by K . (See the final paragraph of 2.23.) Rather than sticking with the language of §2.2, we add, for each of our examples, a new singulary connective. Example 3.21.2 Add to the language of §2.2 a singulary connective O1 , for which the following clause in the definition of truth is supplied: for any model M = (W, R, V ), any x ∈ W and any formula A M |=x O1 A iff either (R(x) = ∅ & M |=x A) or (R(x) = ∅ & M |=x A). Thus at any point either every formula O1 A has the same truth-value as the formula A (which happens if that point bears R to at least one point) or else every formula O1 A has the opposite of A’s truth-value (which happens if the point bears R to nothing). Clearly (↔-Ext) is satisfied: A ↔ B K O1 A ↔ O1 B, since at any point where A and B have the same truth-value, so do O1 A and O2 B, both having the value that A and B receive, should that point bear R to something, and both having the other truth-value otherwise. It is left to the reader to check that O1 is not, however, fully determined according to K . (In fact O1 is completely undetermined according to K .) Note that we can package up the right-hand side of the above clause for O1 more concisely thus: M |=x O1 A iff (R(x) = ∅ ⇔ M |=x A). We will use this more compact style of formulation for the second example: Examples 3.21.3(i) As for 3.21.2, except that we have a new connective O2 , with clause: M |=x O2 A iff (xRx ⇔ M |=x A). We leave the reader to verify that O2 is extensional but not fully determined according to K . (ii) Extensional connectives which are not fully determined connectives, can be found in first-order logic also. Some are presented under the name of generalized connectives (and seen as a special case of generalized quantifiers) in Caicedo [1986], where the following example appears

448

CHAPTER 3. TRUTH-FUNCTIONAL, EXTENSIONAL, . . . (p. 778). Let # be interpreted according to the principle that A # B is true in a structure (for a first order language) if and only if either (a) the domain of the structure is finite and each of A, B, is true in the structure, or (b) the domain is infinite and at least one of A, B, is true. (A more precise formulation would run in terms of satisfaction rather than truth, to allow for the occurrence of free individual variables in the scope of #.) Thus # is interpreted as ∧ or as ∨, depending on the cardinality of the domain.

Exercise 3.21.4 What happens to Examples 3.21.2 and 3.21.3(i) respectively if we consider KD and KT instead of K ? (These are the C-inferential gcr’s for C = the class of all serial frames, and C = the class of all reflexive frames, respectively. See 2.23.2 for background.) We note that O1 from 3.21.2 can be taken as a compositionally derived connective of the conventional language of modal logic: O1 A = A ↔ ; this is not the case for O2 from 3.21.3(i). (With some non-conventional alternatives, the latter example does become definable, however. See the end of 8.32.4 on p. 1259 for an example.) Caicedo’s example in 3.21.3(ii) is like the second modal example in view of the non-definability of finiteness in first order terms, though we could equally give an example resembling 3.21.2 in this respect: for example having # amount to conjunction when the domain has cardinality 5 and as disjunction otherwise. Remarks 3.21.5(i) Our various examples can be presented, given suitable understandings of the key terms in the framework Fmla. Restricting attention to extensions S of CL in this framework, closed under tautological consequence, and with all the boolean connectives of §1.1 present, we say that a connective # is fully determined according to the set S if #p is synonymous in S with some boolean formula, and that # is extensional according to S if (A ↔ B) → (#A ↔ #B) ∈ S for all formulas A, B. These definitions are given for the case of # singulary; the intended application in the general case is obvious. Then it is easily seen, for example, that according to the set of formulas of the language of 3.21.2, O1 is extensional but not fully determined. (ii ) The method of determinant-induced conditions on gcr’s and consequence relations from 3.11 and 3.13 show us how to ‘read off’ a complete logic in Set-Set or Set-Fmla from a proposed truth-table for a connective can be adapted for logics in Fmla. The locus classicus for such ventures is Henkin [1949], in which a recipe is given for axiomatizing these logics as extensions of the purely implicational fragment of CL in Fmla. Supplementary literature includes l’Abbé [1951], Thomas [1960], Surma [1973a].

3.22

A Purified Notion of Extensionality (for GCRs)

As has already been mentioned, it is tedious to have to conduct the discussion of extensionality and other such notions in terms of the behaviour of particular connectives like ↔. This creates a need to insert constant provisoes to the effect that only ↔-classical gcr’s, or only ↔-boolean valuations, are to be considered,

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449

and has the attendant disadvantage that the concepts involved cannot be applied in connection with languages lacking the requisite connectives (here ↔), or even compositionally derived surrogates. For these reasons, a ‘pure’ treatment, abstracting from such details, would be preferable. In the present case, it is not hard to see how to rephrase the above definitions so as to preserve their spirit, while eliminating the reference to the material biconditional. We define a k-ary connective # to be extensional in the i th position (for i k) according to an arbitrary gcr if for all Γ, Δ, B1 , . . . , Bk ,C: (Exti ) Γ, Bi C, Δ and Γ, C Bi , Δ imply Γ, #(B1 , . . . , Bk ) #(B1 , . . . , Bi−1 , C, Bi+1 , . . . , Bk ), Δ. and to be extensional tout court (according to ), as before, when # is extensional in each of its positions (according to ). In other words, abbreviating “Γ, Bi Ci , Δ and Γ, Ci Bi , Δ” to “Γ / Bi Ci / Δ”, # is extensional according to when (Ext)

Γ / B1 C1 / Δ and . . . and Γ / Bk Ck /Δ imply Γ/#(B1 , . . . , Bk ) #(C1 , . . . , Ck ) / Δ.

For # singulary, what this comes to is that for all Γ, Δ, A, B: Γ, A B, Δ and Γ, B A, Δ imply Γ, #A #B, Δ. Note that we do not need to add to the consequent of this metalinguistic conditional: “and Γ, #B #A, Δ”, thereby giving the full effect of “Γ / A B / Δ” because its antecedent is symmetrical in A and B (so the addition would be redundant: the same simplification was made in the formulation of (Exti ) above). In terms of Val (), this formulation may be read: if, on the assumption that all formulas in Γ are true and all those in Δ false (on v ∈ Val()), then if A and B have the same truth-value, the formula #A can be true only if the formula #B is. If happens to be a ↔-classical gcr, then the displayed conditional holds of just in case the earlier requirement, that for all A, B: A ↔ B #A ↔ #B, is met. (And likewise for # of arity > 1, whence our description of the present redefinition as preserving the spirit of the original ↔-involving definition.) An unconditional formulation is also available, equivalent to the above, namely that for all A, B: (E)

A, B, #A #B and #A #B, A, B

which ‘says directly’ that any valuation consistent with satisfying it cannot assign T to each of A and B, and T to #A but F to #B, or assign F to each of A and B and T to #A but F to #B. This forces a valuation consistent with to associate some truth-function, not necessarily the same one for different valuations, with #: see the definition of pseudo-truth-functionality below. (This talk of the truth-function associated with a connective on a valuation was introduced in the discussion preceding 3.11.1, p. 376.) Exercise 3.22.1 (i) Show that a singulary connective # is extensional according to a gcr iff the above condition (E) is satisfied for all formulas A, B, in the language of . (ii) Show that a binary connective # is extensional according to a gcr iff the following fourfold (but unconditional) requirement is satisfied, for all A, A , B, B :

450

CHAPTER 3. TRUTH-FUNCTIONAL, EXTENSIONAL, . . . (E2 )

A, B, A , B , A # A B # B and A, B, A # A B # B , A , B and A , B , A # A B # B , A, B and A # A B # B , A, B, A , B .

(iii) Is the following claim correct or otherwise? (Give a proof or a counterexample.) If the binary connective # is extensional according to a gcr , then for all formulas A in the language of , we have: A, A # A A # (A # A). The reader is invited to generalize (E2 ) (from 3.22.1(ii)) to the case of # n-ary, for arbitrary n. Most of the exposition to follow concentrates on the n = 1 case. (We take up the n = 2 case in the discussion after 3.24.12, on p. 475.) Exercise 3.22.2 (i) Show that if # is a singulary connective in the language of a gcr according to which # is extensional, then for each formula A: #A ###A. (Note: this is a difficult exercise, so the following are given as preparatory steps. Try it first, before proceeding.) (ii ) Show that under the above conditions on # and , we have, for all A: A, #A ##A and also ##A A, #A. (Hint: simply make substitutions for “B” in the first and second conjuncts of (E), respectively) (iii) Under the same conditions, show that A, #A ###A. (Hint: use (M) and (T) on the first -statement in (ii) and the result of taking ##A for B in the first conjunct of (E).) (iv ) Under the same conditions, show that #A ###A, A. (Hint: again put ##A for B, but this time, in the second conjunct of (E), and put #A for A in the second -statement in (i). Apply (M) and (T), the cut formula in this case being ##A.) Final Note: from (iii) and (iv) we obtain #A ###A, by ((M) and) (T), this time with A as the cut formula. We leave the reader to show the converse -statement. We can now ‘purify’ (and simplify) the $ example of 3.21.1 (p. 446). Let $ be the least extensional gcr on any language containing the singulary connective $ and satisfying $A $ A, for all A. Observation 3.22.3 Val($ ) = VI ∪ VF . Proof. The proof can easily be reconstructed from that of 3.21.1, disregarding the features pertaining to ↔. The point is to show that for any v ∈ Val ($ ) there is some singulary truth-function fv such that v($A) = fv (A) for all A, and that fv = I or fv = F.

Corollary 3.22.4 $ = Log(VI ∪ VF ). Proof. From 3.22.3 by 1.16.3 (p. 75).

By 3.11.5 on p. 379, the semantic concept corresponding to the property of being fully determined according to a gcr is truth-functionality (w.r.t. Val ());

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our various examples (and especially the proof of 3.21.1) have no doubt suggested the analogous semantic concept for extensionality. We will call the property that needs to isolated ‘pseudo-truth-functionality’. For ease of contrast, we repeat the definition of truth-functionality from 3.11, alongside our definition of this new concept. Where V is a class of valuations (for a particular language L) we decided to call an n-ary connective # (of L) truth-functional w.r.t a class V of valuations for L, if there exists some function f : {T, F}n −→ {T, F}, such that for all v ∈ V , for all B1 , . . . , Bn ∈ L: v(#(B1 , . . . , Bn )) = f (v(B1 ), . . . , v(Bn )). And we call # pseudo-truth-functional w.r.t. V if for all v ∈ V , there exists some function f : {T, F}n −→ {T, F}, such that for all B1 , . . . , Bn ∈ L: v(#(B1 , . . . , Bn )) = f (v(B1 ), . . . , v(Bn )). Thus we weaken an ∃∀ condition to an ∀∃ condition, and it is clear that truthfunctionality w.r.t. V implies pseudo-truth-functionality w.r.t. V , with the above example of $ serving to show the failure of the converse implication: note that the function called fv mentioned in the proof of 3.22.3 depends on the choice of v. It is, in the terminology introduced before 3.11.1 on p. 376, the truth-function associated with # on the valuation v: pseudo-truth-functionality w.r.t. V allows different truth-functions to be associated with # on different v ∈ V , while truthfunctionality w.r.t. V does not: at least one must be chosen that works for all v ∈ V . What the method of proof yields is something more general than the specific case dealt with, namely (i) of: Theorem 3.22.5 (i) A k-ary connective # is pseudo-truth-functional w.r.t. Val(), for any gcr , iff # is extensional according to . (ii) # is pseudo-truth-functional w.r.t. V, for any class V of valuations, iff # is extensional according to the gcr Log(V ). (iii) If # is fully determined according to a gcr , then # is extensional according to . Part (ii) here is a reformulation of (i), in the light of 3.13.8(i) from p. 391. We return to the case of singulary # for further practice. Notice that 3.22.2 is a reflection, given the above correspondence between extensionality and pseudotruth-functionality, of the fact that each singulary truth-function f satisfies f (x) = f (f (f (x))). In fact this holds for f as an arbitrary term function of any boolean algebra. (See Lyngholm and Yourgrau [1960].) Indeed it holds for arbitrary algebraic such functions, a point which may be put into (sentential) logical terms by saying that for any context C(·), the formulas C(p) and C3 (p) = C(C(C(p))) are classically equivalent. (On this use of the term context see the discussion before and after 3.16.1, p.424.) This is noted in Ruitenburg [1984], where it is shown that no analogous result hold for intuitionistic logic: more specifically, that while in the term-function (or ‘derived connective’) case, i.e., the case in which p is the only variable to occur in C(p), we always have C2 (p) IL C4 (p), for the general (‘algebraic function’) case in which other variables may occur in C(p) – giving a 1-ary context which does not constitute a derived connective – all we can say is that for each C(·) there is some n for which Cn (p) IL Cn+2 (p) – there being no n for which this equivalence holds

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for all C(·). (These are Proposition 2.3 and Theorem 1.9 of Ruitenburg [1984]. It worth pausing to find an example of a derived connective # of IL, for which, by contrast with what the ‘law of triple negation’ – see 2.32.1(iii) at p. 304 – says of ¬, #p and ###p are not IL-equivalent.) Part (iii) of 3.22.5, which justifies the opening sentence of 3.21, can be established by using part (i) and 3.11.5 on p. 379, and noting, as above, that pseudo-truth-functionality (w.r.t. Val ()) is a weaker concept than truth-functionality. We can revisit Example 3.21.2 with the above apparatus in mind, recalling from 2.21 (p. 279) the valuations vxM for a model M = (W, R, V ) with x ∈ W . With respect to the class of all such valuations, O1 is pseudo-truth-functional but not truth-functional, O1 being associated with I on vxM when R(x) = ∅ and with N on vxM when R(x) = ∅. Exercise 3.22.6 We consider a language with the singulary connective # as its sole (primitive) connective. Let 1 and 2 be, respectively, the least gcr’s on this language to satisfy the conditions (for all formulas A, B): (1) #A #B; (2) B, #A #B. Note that each condition suffices for # to be extensional according to . We use the above “Vf ” notation for the class of valuations over which # is associated with f . (i) Show that Val (1 ) = VV ∪ VF (ii ) Show that Val (2 ) = VV ∪ VF ∪ VI (iii) Give a syntactic description, in the style of (1), (2), for with Val () = VI ∪ VN . (Note: Further practice in this vein is available at Exercise 3.32.4(ii), p. 492.) Our various examples might prompt the speculation that every gcr according to which the sole singulary connective in its language is extensional has for its class of consistent valuations the union of one (in the fully determined case) or more (otherwise) of VV , VF , VI , VN . This is not so, however: Exercise 3.22.7 With a language as in 3.22.6 (whose numbering continues here), let 3 be the least among the gcr’s on that language satisfying, for all formulas A and B: (3) A #B. (i) Check that # is extensional according to 3 . (ii ) Show that Val (3 ) = VV ∪ {vF }, where vF is the valuation assigning the value F to every formula of the language. (Compare 2.11.7(ii).) (iii) Give a description in the style of (1), (2) from 3.22.6 and (3) of the present exercise, of the gcr Log(VF ∪ {vT }), where vT is, as usual, the valuation assigning T to every formula. A different kind of example can also be used to show the incorrectness of the claim that every gcr according to which the sole singulary connective in its language is extensional has for its class of consistent valuations is the union of one or more of VV , VF , VI , VN . We take as our language that of 3.22.3 above, but instead of considering the gcr $ there treated, we consider a stronger gcr we shall call $Q . Example 3.22.8 $Q is to be the least gcr on the language just mentioned satisfying for all A:

3.2. EXTENSIONALITY $A A

453 $A q

and A, q $A.

Note that if has ∧ in its language, and is ∧-classical, these conditions amount to: $A A ∧ q, for all formulas A. Now, $ is extensional according to $Q , and indeed Val ($Q ) ⊆ VI ∪ VF by our earlier reasoning (3.21.1 from p. 446, 3.22.3 in the present subsection). But the converse fails. Any $Q -consistent valuation v will deliver as fv either I or F, but this time there is a further constraint: fv = I if v(q) = T and fv = F if v(q) = F. In what respect is this an example ‘of a different kind’ from that in 3.22.7? Well, the gcr $Q is, unlike all others that we have been considering, not substitutioninvariant. Although we have, for example $p $Q q, we do not have $p $Q r. The reader may recall a similar situation arising in our discussion of substitutioninvariance à propos of definitions with ancillary formulas on the object-linguistic approach to definition (3.16.2), where OA was introduced by definition to be A ∧ q. The latter, construed as an algebraic function of (or ‘context for’) A, first appeared as Example 3.16.1(i) on p. 424, and indeed, if we are prepared to apply the terminology of the present chapter not only to primitive, and compositionally derived, but also to algebraically derived ‘connectives’, then we can make the point of Example 3.22.8 without sacrificing substitution-invariance, since we can simply understand $A as being the formula A ∧ q, and use in place of $Q , any ∧-classical gcr. Remark 3.22.9 For $ as a primitive connective, as in 3.22.8, we note a failure of a result like 3.11.7 to transfer to Val ($Q ): it is not the case that the truth-value of a formula in the language of $Q on a valuation v ∈ Val($Q ) depends only on the propositional variables in A; for example we can have v(p) = v (p) but v($p) = v ($p). In this case A is just the formula p; of course, we can say that v(A) depends only on the variables in A together with q. In the general case of an extensional but not fully determined connective #, 3.11.7 fails more dramatically, since we need information about what truth-function is associated on v with # which will not be forthcoming even given knowledge of v(pi ) for each pi in the language.

3.23

Some Extensionality Notions for Consequence Relations

We should note here the convenience of working with gcr’s over consequence relations. While the left-hand condition imposed on # and by (E) makes sense for as a consequence relation, the right-hand condition would need to be reformulated with “∨” replacing its commas, or with any two of the formulas appearing after “” prefixed by “¬” and moved over to the other side of “”. We could then conduct the discussion only against the background of (respectively) ∨-classical consequence relations (and ∨-boolean valuations) or ¬-classical consequence relations (and ¬-boolean valuations); there would be a loss of generality analogous to that already complained of in connection with “↔”. (Likewise with the original (Exti ) condition, because of the “Δ” on the right.) However, using a conditional formulation, (RE ) below, it is possible to

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simulate for consequence relations (some of) the effect of the second conjunct of (E), using the ‘common consequences’ device from 3.13. For what follows, we need to separate the two conjuncts of (E) above, as right-extensionality and left-extensionality; that is, we define a singulary connective # to be left-extensional according to if, for all formulas A, B, in the language of , we have (LE) A, B, #A #B and to be right-extensional according to when (RE) #A #B, A, B for all A, B in the language of . (There will be no danger of confusion with the unrelated use of the “left/right-extensional” terminology in 0.14 to abbreviate “extensional on the right/left” as applied to binary relational connections – a terminology introduced in 0.11.) As noted above we may take “# is leftextensional according to ” as just defined to apply in the case of a consequence relation , rather than only for gcr’s. To obtain the promised condition (RE ), we apply the ‘method of common consequences’ of 3.13, where it was used to trade in determinant-induced conditions on gcr’s for (conditional) determinantinduced conditions on consequence relations. If is a consequence relation, we say that (singulary) # is right-extensional according to if for all Γ, A, B, C in the language of : (RE ) Γ, #B C and Γ, A C and Γ, B C imply Γ, #A C Whereas any valuation consistent with a left-extensional gcr must ‘treat truths alike’ in the sense of respecting for # some determinant of the form T, x , where x is either T or F, and any valuation consistent with a rightextensional gcr must ‘treat falsehoods alike’ in the sense of respecting for # some determinant of the form F, x , when it comes to consequence relations, an asymmetry, predictable in the light of the Strong/Weak Claim discussion of 1.14 (p. 67), emerges. The former point continues to hold for valuations consistent with a left-extensional consequence relation (since (LE) remains intact), while for a right-extensional consequence relation , all we are guaranteed (having replaced (RE) with (RE )) is that when Γ C, there exists a -consistent valuation which respects some determinant of the form F, x , and verifies Γ but not C. The situation is a similar for # of arity > 1. In the case of gcr’s when # is binary, for example, the four conjuncts of (E2 ) above dictate, respectively, that every consistent valuation respects some determinant of the form T, T, x

(“treats first-component-true, second-component-true cases alike”), some determinant of the form T, F, x (“treats first-component true, second-component false cases alike”), some determinant of the form F, T, x , and some determinant of the form F, F, x , while only the first of the four conjuncts is available for consequence relations to force such respect from every consistent valuation, and the remaining cases must be rephrased conditionally along the lines of (RE ), with the attendant weakening of the semantic effect. We defined extensionality according to a gcr in terms of (Exti ) which does not make sense for consequence relations, so let us here adopt a definition which does make sense: we will say that a (singulary) connective # is extensional according to the consequence relation when # is both left- and right-extensional

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according to . A potentially confusing fact here is that the obvious ‘purification’ of the condition of the ↔-extensionality of # according to a consequence relation , as defined in 3.21, actually gives something weaker than this combination of left- and right-extensionality. We call this purified version of the notion defined earlier ‘weak extensionality’. That is, say that # (again, for simplicity of exposition, assumed singulary) is weakly extensional according to the consequence relation when for all Γ, A, B in the language of : (WExt) Γ, A B and Γ, B A imply Γ, #A #B. This could equally well be called “congruentiality with side formulas” for consequence relations, congruentiality – the subject of §3.3 – being the special case in which there are no side formulas (i.e., the case of Γ = ∅). Observation 3.23.1 If # is extensional according to a consequence relation , then # is weakly extensional according to . Proof. Suppose that # and satisfy (LE) and (RE ), and that Γ, A B and Γ, B A; we must show that in that case, Γ, #A #B. From the fact that Γ, A B and the instance of (LE) to the effect that A, B, #A #B, it follows by (M) and (T) that (1) Γ, A, #A #B. And we similarly use the fact that Γ, B A and this instance of (LE) to infer by (M) and (T): (2) Γ, B, #A #B. Now Γ, #A, #B #B, by (R) and (M), so along with what has just been derived, namely that Γ, #A, A #B ( = (1)) and Γ, #A, B #B ( = (2)), we have all of the antecedents of an instance of (RE ), taking #B as the C in our statement of (RE ) and the present Γ ∪ {A} as the Γ figuring therein, we infer that Γ, #A #B, as was to be shown. A full converse to 3.23.1 is not available: Observation 3.23.2 Weak extensionality of # according to a consequence relation implies left-extensionality according to , but it does not imply rightextensionality. Proof. For formulas A, B, put Γ = {A, B} and note that we have Γ, A B and Γ, B A by (R) and (M), so, assuming (WExt) we infer (LE): A, B, #A #B. Thus weak extensionality implies left-extensionality. To show that weak extensionality does not imply right-extensionality, we note that according to the consequence relation IL of 2.32, ¬ is weakly extensional; indeed Γ, ¬A IL ¬B follows from the hypothesis that Γ, B IL A alone, without the need to assume also that Γ, A IL B. Yet the condition (RE ) is not satisfied. Taking Γ = ∅, A = ¬p, B = p, C = p ∨ ¬p gives a counterexample, since we have ¬p IL p ∨ ¬p, and ¬p IL p ∨ ¬p (again), and p IL p ∨ ¬p; yet ¬¬p IL p ∨ ¬p.

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Exercise 3.23.3 Show that, 3.23.2 notwithstanding, if → is a connective in the language of a consequence relation , and is →-classical, then any connective # weakly extensional according to is right-extensional according to . (Hint: begin by assuming that Γ, A C and Γ, B C and Γ, #B C, and from the first two parts of this assumption infer that Γ ∪ {C → A}, A C and Γ ∪ {C → A}, C A, and similarly for B.) Thus – special circumstances, as in 3.23.3, aside – weak extensionality is genuinely a weaker concept than extensionality ( = left- & right-extensionality) for consequence relations. The picture would be simplified if weak extensionality, implying left-extensionality but not right-extensionality, turned out to be just equivalent to left-extensionality. Admittedly the proof given of 3.23.1, showing extensionality to imply weak extensionality made use of both (LE) and (RE ). This leaves open the possibility that appeals to the latter condition were avoidable. That simplification is not available, in view of the example in the following: Exercise 3.23.4 Check that ¬ is, according to the consequence relation determined by the three-element matrix of Łukasiewicz (Figure 2.11a, p. 198), left-extensional, but that, since (where is this consequence relation), while we do have p ∧ ¬p q ∧ ¬q, we do not have: ¬(p ∧ ¬p) ¬(q ∧ ¬q). Note that this gives a counterexample to weak extensionality (taking Γ = ∅). Thus, by itself, left-extensionality does not imply weak extensionality. (Since Γ = ∅ in the example of 3.23.4, what is actually illustrated is the failure of a weaker property than either of these, namely the property of congruentiality – the topic of §3.3; we return to the example in that setting as 3.31.5(i), p. 487.) We summarize our findings diagrammatically in Figure 3.23a. The poset of implicational relationships between the concepts that have been in play various properties of connectives according to consequence relations looks like this, where ϕ ψ when any connective which has property ϕ according to a consequence relation must have property ψ according to . Remark 3.23.5 All of the properties in play in our recent discussion (and appearing in Figure 3.23a), defined as possessed by a connective according to a consequence relation, can be lifted to apply to consequence relations themselves. If every (primitive) connective in the language of has the property ϕ according to , we will speak of itself as having the property ϕ. Thus for example, if the language of has two connectives # and # , and each is left-extensional according to , we will say that is left-extensional. The same transfer of terminology will be made in the case of properties defined for connectives according to gcr’s. Connectives weakly extensional according to a consequence relation are simply called extensional some authors: for example Czelakowski and Pigozzi [2004]. A consequence relation which is weakly extensional, in the present terminology, is called by these and others working in abstract algebraic logic Fregean. The following provides practice with this terminological transfer, and will be of use to us below. (If in doubt, consult the proof of 3.31.1, p. 485.)

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left extensionality

/ // // // // // / weak extensionality / / // // // // // //

right extensionality

extensionality

Figure 3.23a

Exercise 3.23.6 Show that if is a left-extensional consequence relation then for every formula C (of the language of ) containing a propositional variable for which formulas A and B are uniformly substituted to give respectively C(A) and C(B), we have: A, B, C(A) C(B). (This should be proved by induction on the complexity of the formula C. The C(·) notation for contexts was introduced after 3.16.1l.) To show that we have indeed ended up with an appropriate notion of extensionality for consequence relations, we should check that it is related to pseudo-truth-functionality in the same way as the property of being fully determined (for consequence relations) is related to truth-functionality; the latter was given in 3.13.9(iii) on p. 391. Having included in our discussion sufficiently many clues, we omit the proof: Theorem 3.23.7 A connective # is extensional according to a consequence relation iff there exists a class V of valuations w.r.t. which # is pseudotruth-functional and which is such that = Log(V ). The existential quantification over classes of valuations here and in 3.13.9(iii) deserves some comment. The following discussion—down to the end of 3.23.8— supplements that in 3.13.8 (p. 391). A concept whose primary application is to classes of valuations may be ‘transferred’ to apply derivatively to (generalized) consequence relations in any number of ways, three of which in particular we list here. Suppose we have some property ϕ of classes of valuations. The ‘logical transfer’ log(ϕ) is to be a corresponding property of (generalized) consequence relations. Or, more precisely, since we shall consider three different ways of obtaining the logical analogue of a property of valuation-classes, we define three ways, log i (i = 1, 2, 3) of transferring concepts across to (generalized) consequence relations, over which relations the “” here ranges:

458

CHAPTER 3. TRUTH-FUNCTIONAL, EXTENSIONAL, . . . has property log 1 (ϕ) ⇔ ∀V [ = Log(V ) ⇒ V has property ϕ] has property log 2 (ϕ) ⇔ Val () has property ϕ has property log 3 (ϕ) ⇔ ∃V [ = Log(V ) & V has property ϕ]

Now it is clear that for any and any property ϕ of classes of valuations (for the language over which is a consequence relation or gcr): (*)

has log 1 (ϕ) ⇒ has log 2 (ϕ) ⇒ has log 3 (ϕ)

Notice that taking ϕ as the property of being a class of valuations w.r.t. which (some connective) # is truth-functional gives log 3 (ϕ) as the property of being a consequence relation according to which # is fully determined (3.13.9(iii)), and similarly, substituting “pseudo-truth-functional” for “truth-functional” gives log 3 (ϕ) with “extensional” in place of “fully determined”, by 3.23.7. Exercise 3.23.8 (i) Show that for a gcr , and a property ϕ of classes of valuations, the following are equivalent: (2) has log 2 (ϕ) (3) has log 3 (ϕ). (1) has log 1 (ϕ) (ii ) Show that for each of the implications in (*) above, we can find a consequence relation and a property ϕ of classes of valuations for which the converse of that implication fails. We close with an application of some of the concepts reviewed in this subsection to the area of unital matrix semantics (from 2.13). Theorem 2.13.1 (p. 221) gave necessary and sufficient conditions for a logic in Fmla to be determined by a unital matrix, as Theorem 2.12.9 (p. 217) had done for a logic in Set-Fmla (or in Set-Set) to be determined by a collection of (arbitrary) matrices. Here we consider the Set-Fmla case, with reference to a collection of unital matrices. The key concept for the proof of 2.13.1 was that of the S-synonymy of formulas A and B, symbolized there by A ≡S B. S was to be the set of formulas constituting the Fmla logic whose determination by a unital matrix was at issue, and we saw that the S-synonymy in S of all S-provable formulas (S’s being monothetic, for short) was a necessary condition of such determination. Rather than speaking of logics in Set-Fmla, we use the ‘consequence relations’ terminology with which we have been working in this chapter. Instead of the requirement of S-synonymy of all elements of S, what we need here is Cn(Γ)synonymy of formulas, for varying choices of Γ, where Cn is the consequence operation associated with the consequence relation we are interested in. (It would be misleading to think of this as the a special case in which S is chosen to be Cn(Γ), since typically we are thinking of S, but not Cn(Γ), as closed under Uniform Substitution – though we will be requiring that Cn is substitutioninvariant.) If we impose the condition of left-extensionality on , then by 3.23.6, we have A, B, C(A) C(B), for all formulas A and B, the C( ) notation being understood as in that Exercise. If this condition is satisfied then for any Γ with Γ A and Γ B, we have Γ C(A) iff Γ C(B), so that where Cn(Γ) = {D | Γ D}, if A, B ∈ Cn(Γ) then A ≡Cn(Γ) B. Thus the sets Cn(Γ) play the role of S in Theorem 2.13.1 (p. 221). Theorem 3.23.9 A necessary and sufficient condition for it to be the case that a consequence relation is determined by some collection of unital matrices,

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is that (i) is substitution-invariant and (ii) is left-extensional and (iii) for any formula B such that B there exists a formula A such that A B. Proof. Necessity: It is left to the reader to verify that a consequence relation cannot be determined by a collection of unital matrices unless it satisfies (i), (ii), and (iii). The latter two conditions correspond, respectively, to the existence of at most one, and at least one, designated value in each of the matrices concerned. Sufficiency: We define the collection C of matrices to be the class of matrices MΘ where MΘ = (AΘ , DΘ ), AΘ being constructed for any non-empty set Θ of formulas like the algebra A in the proof of 2.13.1, but using the relation ≡Cn(Θ) in place of ≡S , and similarly DΘ = {[B] | Θ B}, where [B] is the synonymy class of the formula B under the relation ≡Cn(Θ) . (See the proof of 2.12.9, p. 217.) Note that DΘ contains at least one element, because we consider only Θ = ∅; that it contains at most one element follows from (ii) thus: Suppose that [B], [B ] ∈ DΘ . This means (1) Θ B and (2) Θ B . To show [B] = [B ], we need to show that for an arbitrary context C(B) such that (3) Θ C(B), we have (4): Θ C(B ). But (4) follows by several appeals to (T) from (1), (2), and (3), and this instance of 3.23.6: B, B , C(B) C(B ), guaranteed for left-extensional . It remains to be verified that is determined by C, i.e., that Γ B iff for each M ∈ C, Γ |=M B. This is left to the reader to argue, following the pattern of the corresponding part of the proof of 2.13.1, with a twist invoking (iii) for the Γ = ∅ case. (See the proof of 3.23.13 for a more informative description, which can be adapted to the present case.) For an application of this result, consider the consequence relation determined by the Kleene matrix K1,2 from 2.11. As the subscripting indicates, this matrix has two designated values. One may wonder if the same consequence relation is nonetheless determined by some collection of unital matrices. However, this consequence relation—call it 1,2 —is not left-extensional: for example p, q, ¬p 1,2 ¬q. (So, more specifically, ¬ is not left-extensional according to 1,2 .) Thus 3.23.9 returns a negative answer to our question: no determining class of unital matrices is to be found. (As a matter of fact, dual reasoning shows that the gcr, 1 , determined by K1 is determined by no class of matrices each having only one undesignated element, since this gcr is not right-extensional: ¬p 1 ¬q, p, q. But we are not discussing gcr’s here.) Although what is more pertinent to the issue of extensionality is the “at most one designated value” half of the definition of a unital matrix, it is worth paying some attention to the “at least one designated value” half, which enters Theorem 3.23.9 via condition (iii) on our consequence relations . Let us begin by noting that a contraposed form of the condition more evidently expresses the requirement that there are no ‘mere followers’ according to , where a mere follower (for ) is a formula which is a consequence by of (the unit set of) each formula but not of ∅. (This terminology is from Humberstone [1996a], p. 59.) Exercise 3.23.10 Show that if some formula is a mere follower for a consequence relation , then is atheorematic. (This term was defined in 2.11.6.)

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The contraposed form of (iii) says that no formula B is a mere follower: (‘No Mere Followers’) If A B for all A, then B. Rather than address, as 3.23.9 does, consequence relation determined by a collection of matrices, let us make life simpler by considering the consequence relation |=M determined by a single matrix M. Observation 3.23.11 For M = (A, D), |=M satisfies the No Mere Followers condition if and only if D = ∅. Proof. ‘If’: Suppose D = ∅ and |=M B, with a view to showing that for some A, A |=M B. (Here we use the contraposed form of the condition, as in (iii) of Theorem 3.23.9.) Since |=M B, there is some M-evaluation h with h(B) ∈ / D. Choose some propositional variable pi not occurring in B and vary h without changing its assignment to the variables in B to an evaluation h with h (pi ) ∈ D, / D, this gives pi |=M B. as we may since D = ∅, since h (B) = h(B) ∈ ‘Only if’: Suppose D = ∅ with a view to showing that |=M violates the No Mere Followers condition. Since A = D ∪ (A D) and A is non-empty, we must in this case have A D non-empty, so an evaluation assigning one of its elements to q, say, shows that |=M q. (Here A is the universe of A.) The No Mere Followers condition is then violated because for every formula A we have A |=M q, as we can never have h(A) ∈ D.

Exercise 3.23.12 (i) Alongside condition (iii) of Theorem 3.23.9, repeated here for convenience, compare the condition (iii)* on consequence relations: (iii) for any B such that B there exists a formula A such that A B. (iii)* for any B such that B there exist formulas A, C, such that A C. Show that for any of the form |=M , (iii) is satisfied if and only if (iii)* is satisfied. (ii ) Are (iii) and (iii)* similarly equivalent for arbitrary consequence relations ? (Proof or counterexample required. The comments which follow may be of assistance.) Let us notice that the second condition listed under part (i) of the above exercise amounts to saying that if is trivial it is inconsistent, as these notions were defined for consequence relations in 1.19.2 (p. 92). To connect with the special valuations in play in that discussion, we observe that for M of the form (A, ∅), the bivalent valuation vh induced by every M-evaluation is vT . We return to the more general setting of determination by classes of matrices to conclude the present discussion. Observation 3.23.13 A consequence relation is determined by a collection of matrices each of which has at least one designated value if and only if satisfies conditions (i) and (iii) of Theorem 3.23.9.

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Proof. One can adapt the proof of 2.12.9 (p. 217) sketched in the text following the statement of that result and use as the determining collection of matrices the set {MΘ | Θ is a -theory}. Here for a given -theory Θ, the matrix MΘ has for its algebra the algebra of formulas of the language of and for its set of designated elements the set Θ, but discard from this bundle those MΘ for which Θ is empty – since we want every matrix in the determining class to have at least one designated value. (We give only the completeness half of the proof here.) It needs to be checked that whenever Γ B, there is some matrix MΘ in the bundle with Γ ⊆ Θ and B ∈ / Θ. In the original proof of 2.12.9, we could take Θ as Cn (Γ) itself, but here we need to take special measures to guard against the possibility that Cn (Γ) is ∅, since in this case the required MΘ has been discarded. But condition (iii) from 2.32.9 (p. 315) comes to the rescue, since when Γ B for Γ = ∅ this condition tells us that there is a formula A with A B, so we can use Cn ({A}) as the Θ for which MΘ is an invalidating matrix with a non-empty set of designated values.

3.24

Hybrids and the Subconnective Relation

We return from consequence relations to gcr’s and in particular to that we called $ and showed (3.22.4, p. 450) to be Log(VI ∪ VF ). By general Galois connection facts (recalled for (2) of 1.12.4(i), p. 60) we have Log(VI ∪ VF ) = Log(VI ) ∩ Log(VF ), so $ = Log(VI ) ∩ Log(VF ), and we can regard $ as it behaves according to $ as possessing precisely the logical features common to connectives associated with the identity truth-function and the constant-false truth-function. We call such a connective a hybrid of those whose common features it possesses. Here, a relevant feature is specifically one captured by a gcr-statement, such as that for all A, #A A, which is satisfied by as Log(VI ) as well as by as Log(VF ). The reference to gcr’s makes that definition specific to Set-Set, but it is intended to apply mutatis mutandis to Set-Fmla and Fmla, and some of our examples will be drawn from those frameworks. Of course, here the term connective is used in its ‘logical role’ sense, and, as suggested in 1.19, we could perhaps more perspicuously say that $, $ is a hybrid of $, 1 and $, 2 , where 1 = Log(VI ) and 2 = Log(VF ). Continuing to speak of connectives in that sense and to use this ordered pairs representation of them, the above reference to common logical properties amounts to forming the meet (greatest lower bound) of two connectives, with the underlying partial ordering being the following ‘subconnective’ relation: #, is a subconnective of # when for all Γ, Δ: Γ Δ implies Γ Δ ; here Γ and Δ are the results of replacing # by # in the formulas in Γ and Δ. Of course it is presupposed here that # and # are connectives of the same arity. It also needs to be assumed, for this definition to make sense, that all connectives (with the possible exception of #) – now in the ‘syntactic operation’ sense of connective – in the language of are connectives of the language of . (More generally, we should wish to avoid this last assumption, and make the subconnective relation itself relative to an

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association of connectives of the language of with those of the language of . However, we will not need such subtleties for present purposes.) As a simple example, intuitionistic negation is a subconnective of classical negation, these being understood respectively as ¬, IL and ¬, CL , where the subscripted “” refers to the consequence relations concerned. When, as here, the subconnective relation holds in one direction but not the other, we can emphasize this by calling (e.g.) intuitionistic negation a proper subconnective of classical negation. Thus, for a further example, as it behaves according to the Fmla logic KT is a proper subconnective of as it behaves according to S4 – a simple reflection of the fact that the former is a proper sublogic of the latter. These are intuitively satisfactory results of the definition in that they reflect the fact that the of S4 (or ¬ in classical logic) is governed by a proper superset of the logical principles governing the of KT (or ¬ in intuitionistic logic, respectively). But we similarly get the untoward consequence that intuitionistic (or KT) ∧ is a proper subconnective of classical (or S4) ∧, since (in respect of the “proper”) for example p ∧ ¬¬q CL q while p ∧ ¬¬q IL q. Intuitively one would think of this as a by-product of the fact that ¬ satisfies fewer principles in the IL case, with ∧ answering to the same basic principles. To bring this out is it best to consider just the fragments concerned: the proper subconnective relation holds between intuitionistic and classical negation in the pure negation fragments of the consequence relations concerned, while this is not so in the pure conjunction cases (where the fragmentary consequence relations coincide). Naturally this strategy is less widely applicable to the case of logics in Fmla, where, applied to IL and CL in the negation and the conjunction cases, the pure fragments are equal (because empty) in both cases. Returning to Set-Fmla and Set-Set, one might instead consider sequent-to-sequent rules in a proof system for purposes of making such comparisons, though we do not pursue that avenue here. Certainly just looking at the provable sequents and asking about what rules are admissible is not a promising line of enquiry (as we see in 3.24.16–17, in a related context). The subconnective relation need not relate only primitive connectives of the languages of two logics. For an example in which we take the same logic twice over, we could ask whether ¬¬ is a subconnective of in some modal extension of IL. (This question is one suggested by the title of Došen [1984]. For treatments of intuitionistic modal logic, see the notes and references to §2.2, which began on p. 296.) A particularly interesting discussion in this vein is provided by Zolin [2000], where the subject is (mono)modal logics in Fmla with CL as the underlying non-modal logic (as in 2.21 above, but not restricting attention to normal modal logics). Derived 1-ary connectives of the language involved here (with as the sole non-modal primitive) are sometimes called modal functions (of one variable) – e.g., in Hughes and Cresswell [1968] – but Zolin calls them modalities (a term used by Hughes and Cresswell in a more restrictive sense). The following is from pp. 861f. of Zolin [2000]: Given a modal logic L, it is natural to measure its expressive power by a number of “distinct” modalities in L. However, there are (at least) two different approaches to formalise the quoted word. According to the first, or internal, approach, modalities are identified if they are equivalent in L, i.e., if the equivalence of formulas they are induced by is a theorem of L. (. . . )

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The second, or external approach prescribes not to distinguish between modalities having an identical “behaviour” over L. (. . . ) In general, to each modality ∇ we assign a ∇-translation tr∇ of formulas by putting tr∇ (A) to be the result of replacing all occurrences of the symbol in A by the operator ∇. Further, we define a logic L(∇) of a modality ∇ over a logic L as the set of all formulas whose ∇-translations are theorems of L. (. . . ) Thus the external approach prescribes, given a logic L, to identify, modalities having equal logics over L; we call these modalities analogous over L.

The main logics considered by Zolin are, if not normal, then at least congruential in the sense that provably equivalent formulas are synonymous. (See further §3.3 below.) So modalities which are equivalent in the sense given in the above passage are always analogous (over the logic concerned), though not in general conversely. Zolin supplies many examples, the simplest of which have the form: and are analogous over – or in, as it may seem more natural to put it – many familiar normal logics, including K, KT, and KTB, in none of which are these modalities equivalent (by contrast with the case of S4 or indeed any extension of K4!: nomenclature for modal logics here as in 2.21). An immediate consequence is that in the logics listed, m and n are analogous for any m, n ∈ N. Modalities which are analogous (in Zolin’s sense) in L are, in our terminology, 1-ary connectives each of which, as it behaves in L, is a subconnective of the other. A definition of the subconnective relation in the style of Zolin but without restricting attention to the mutual subconnective case is not hard to come up with: ∇1 is a subconnective of ∇2 in L just in case L(∇1 ) ⊆ L(∇2 ). We shall see more of the mutual subconnective case later, as we consider Łukasiewicz’s “twins” simile; further modal examples appear at 3.24.18. We return to the topic of hybrids, and for the moment, observe that for the purposes of a discussion of this topic, several simple points can be made in terms of the meet of two gcr’s (or consequence relations) in the language with just one connective #. In this case the subconnective relation between #,

and #, amounts to ⊆ , and talk of #, as a hybrid of #, and #, amounts to saying, as in the above $ example, that = ∩ . Similarly 3.22.6 shows that for the extensional gcr’s mentioned there as 1 and 2 , 1 = Log(VV ) ∩ Log(VF ) and 2 = Log(VV ) ∩ Log(VF ) ∩ Log(VI ). The logic of a completely undetermined singulary extensional connective—the least gcr on a language containing such a connective—is (Log(Vf | f ∈ {V, F, I, N}). The singulary connective in question, as treated by this gcr, is a hybrid of all the singulary fully determined connectives: a generic one-place connective whose behaviour is precisely the common behaviour of those with which the singulary truth-functions are associated. There has been some tendency in the literature to regard the connective in this case as not really being a connective at all, but a variable ranging over connectives (or their semantic values, such as truthfunctions) – in the words of Łukasiewicz, who exemplifies this tendency – as a ‘variable functor’ (Łukasiewicz [1951]). Reasons for not following suit may be found in the final section of Humberstone [1986]. We should note that although we raise the issue of hybrids within a section on extensionality because hybridizing fully determined connectives yields extensional results, there is no reason why hybrids in general should be extensional.

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Although any hybrid of fully determined connectives is extensional, the converse is not so, as we learn from the examples in 3.22.7–8. That is, making the simplifying assumption that , , have only one connective # in their languages, while it does follow from = ∩ that if # is fully determined according to and according to , then # is extensional according to , it does not similarly follow that if # is extensional according to , then # is fully determined according to and according to . However, it does follow that there exists a family {i }i∈I of gcr’s with = i∈I (i ) and # fully determined according to each i , since we may take as our family the set of all ⊇ according to which # is fully determined. Exercise 3.24.1 Justify this last assertion, showing that is the meet (intersection) of the described. One disadvantage of the “hybrid” terminology is that it evokes no natural dual term. Reverting to the connectives-as-logical-roles way of speaking, it makes sense not only to consider a connective satisfying precisely the conditions satisfied by each of two (or more) connectives, but also to consider a connective satisfying just such conditions as are satisfied by at least one of two (or more connectives). For reasons reviewed in 3.12, if the initially given connectives are fully determined (and in different ways), this will send any satisfying this union of logical principles toward the top of the lattice of gcr’s in its language; but interesting—let us say for the moment—‘sums’ of less than fully determined connectives can be formed. For instance, each fully determined k-ary connective can be viewed as a sum of 2k partially determined connectives, each satisfying precisely one of the determinant-induced conditions. The terminological dangers here are that if we spoke of sums, we should naturally wish to use product rather than hybrid as the dual term. But both sum and (especially) product have a quite different usage in matrix-based approaches to logic, as we recall from §2.1. The need for care is especially great since the idea of products in the latter setting coincides with our current conception of hybrids in the framework Fmla, though not in Set-Fmla or Set-Set. (The same applies to sum in its matrix sense and its ‘dual to hybrid’ sense.) This matter deserves our attention. The notion of a direct product of matrices was introduced in the discussion leading up to 2.12.1, which—as we shall see presently—justifies the assertion just made about Fmla. Examples of taking the direct product of a matrix with itself were depicted in Figure 2.12a (p. 213). When people talk of taking the product of one connective with another, they have in mind tables for the two connectives and computing coordinatewise a table for the product. Following the conventions of our discussion in 2.12, we use the labels 1, 2, 3, 4, for T, T , T, F , F, T , F, F . The product of the two-valued table for ∧ with that for ∨ is given in Figure 3.24a. (More carefully formulated: what the diagram depicts is product algebra {T, F}, ∧b ) ⊗ ({T, F}, ∨b ).) Now if we expand a language containing some boolean connectives, (say) ∧, ∨, →, and ¬, with a connective – call it, as on the above table, ∧∨ – for which this table gives the associated matrix operation (‘four-valued truth-function’) then we find that if B(∨) and B(∧) are formulas the first obtained from some formula B containing ∧∨ in the first case by replacing each occurrence of this connective by ∨ and in the second case by making such a replacement with

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1 1 1 3 3

2 1 2 3 4

3 3 3 3 3

4 3 4 3 4

Figure 3.24a: The Product of Conjunction and Disjunction

∧, then B(∧∨) is valid in the product matrix just in case each of B(∧), B(∨), is valid in the two-element boolean matrix. One can derive this observation from Theorem 2.12.1 by reformulating it so that the additional connective ∧∨ is taken into account from the beginning, as an additional connective alongside the boolean connectives listed above. Let M∧ be the expansion of the twoelement boolean matrix (in the appropriate sense—see above) got by taking the operation associated with ∧ again as the interpretation of ∧∨, and M∨ the similar matrix in which it is the ∨ operation that puts in a second appearance to interpret ∧∨. Finally, call the expansion of the product of the boolean matrix with itself, expanded by the operation described in the above table, M∧∨ . Then Theorem 2.12.1 yields the result that a formula is valid in M∧∨ iff it is valid in M∧ and in M∨ . An example of such a formula is: (p ∧∨ q) → (q ∧∨ p). Digression. The analogous result for direct sums of matrices (2.12.3) yields the dual property for ‘sums’ of connectives; algebraically this is the product as just defined, but with the more generous policy for designating elements of the product algebra described before 2.12.3. Calling the direct sum of the matrices M∧ and M∨ just described M∧+∨ we have that a formula is valid in M∧+∨ just in case it is valid either in M∧ or in M∨ . End of Digression. The situation is quite rosy, then, for construing products as hybrids in Fmla. The moral of 2.12.6, however, is that the two ideas come apart in Set-Fmla, and even more so in the case of Set-Set. Putting the matter in terms of (generalized) consequence relations, we saw there that while the consequence relations determined by M and by M are both included in that determined by M ⊗ M , the latter can be more extensive than the intersection of the former pair. For gcr’s, we do not even have the inclusion just mentioned. This was illustrated in 2.12.5, which is itself a case of a product connective. Reworking and renotating that example somewhat, to bring out this aspect of it, let the language of have two binary connectives ① and ②, and let V be the class of all valuations over which these connectives are associated respectively with the truth-functions mapping x, y to x and x, y to y, for all x, y ∈ {T, F} (the functions proj12 and proj22 , that is). is, in particular, to be the gcr Log(V ). We compare this with the gcr determined by the product of the two two-element matrices here implicitly described, in a language with one binary connective (though we could also, as with the above ∧, ∨, ∧∨ example, keep ①, ②, as well as adding this new connective) we shall call simply ⊗ (which can be thought of as abbreviating “① ⊗ ②”); call this gcr ⊗ . Then we have p ① q p, q and p ② q p, q; yet p ⊗ q ⊗ p, q. So ①, and ②, do not have ⊗ , ⊗ as a hybrid, because the latter connective does not satisfy all the logical principles satisfied by the former pair. (For more on this example, see 5.35.) By 2.12.6(i) this kind of failure cannot occur for consequence relations, but products can fail

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as hybrids because they do not satisfy only the logical principles satisfied by the pair to be hybridized – a special case of what was mentioned under 2.12.6(i) with the words “though in general validity in the product does not imply validity in each of the factor matrices”. We can illustrate this for the case of products of boolean connectives with the aid of ∧∨: Examples 3.24.2(i) Noting that for any formula A, the formula A ∧ ∨¬A always takes the undesignated value 3 ( = F, T ), for the consequence relation ∧∨ determined by the matrix M∧∨ expanded by a table for the product of two-valued ¬ with itself we have: p ∧∨ ¬p ∧∨ q even though we do not have both p ∧ ¬p CL q and p ∨ ¬p CL q (because, specifically, we do not have the latter). 3.24.7, below, shows that such counterexamples require the presence of extraneous connectives (here ¬). (ii ) A simpler example, not involving an extraneous connective (such as ¬ in (i)) is the following, from Rautenberg [1989a], p. 532: take the product ∨ of inclusive and exclusive disjunction. (We discuss the latter in 6.12, p. 780.) A∨A can never take the designated value T, T

because of the second factor (i.e., for x = T, F, we have x x = F). So adapting the above notation, we have p∨ p ∨ q while this does not hold for the hybrid since it does not hold for ∨. Remarks 3.24.3(i) The example under 3.24.2(i) shows, if we throw in not only negation but also implication that ∧∨ is not →-classical (or even →-intuitionistic) since we have p ∧∨ ¬p ∧∨ q while ∧∨ (p ∧∨ ¬p) → q. (ii ) Rautenberg remarks in the reference cited under 3.24.2(ii) that there are precisely three substitution-invariant consequence relations (on the same language) extending ∨ (excluding the trivial case of for which p q), namely the expected strengthenings to the case in which the binary connective involved behaves like ∨, and that in which it behaves like , and – the new case – the consequence relation determined by the product matrix, this last being the smallest substitutioninvariant consequence relation extending the hybrid (the intersection of the two expected strengthenings) which satisfies the inset condition under 3.24.2(ii). The next illustration of products not serving as hybrids concerns a singulary connective, and we may as well take this to be ¬. The function associated with ¬ over BV is of course N, alias ¬b (3.14). We can form the product of N with I; let be the consequence relation determined by the product matrix in this case (assuming, for simplicity, that there are no other connectives in the language of ). Then we have: Example 3.24.4 Understanding as just introduced, we have p, ¬p q, since when p has the designated value 1 (alias T, T ), ¬p has the undesignated value 3 ( = F, T ). But although q is true on all ¬-boolean

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valuations verifying p and ¬p (there being none), q is not true on all valuations over which I is associated with ¬ and each of p, ¬p is true. So does not give precisely the common properties of (connectives with which are associated) the truth-functions N and I: the product goes beyond the hybrid in respect of its logical behaviour. Remark 3.24.5 Working in Fmla, so that products and hybrids coincide, and in a language containing ∧, ∨, → and ¬, Parsons [1966] considered the formulas which were true on every valuation in BV {∧,∨,→} on which N was associated with ¬ as well as true on every valuation in BV {∧,∨,→} on which V was associated with ¬. The idea was to have a logic in Fmla standing to CL as Minimal Logic (8.32) stands to IL. (This theme had also been pursued in Kanger [1955]. We take it up in more detail in 8.34.) We return to the ∧∨ table of Figure 3.24a, picking up two issues it raises. The first of these concerns the status of ∧∨ as a hybrid of classical conjunction and disjunction, and the second, the possibility of taking the product of ∨b with ∧b (in that order) rather than, as in 3.24a, that of ∧b with ∨b . The hybrid status of ∧∨ in Set-Fmla was settled negatively by 3.24.2. However that example conspicuously featured ¬ alongside ∧ and ∨, leaving open the possibility that if we consider just ∧ and ∨, their common properties according to CL are given by the consequence relation the product matrix determines. And indeed, this is the case, as we report in 3.24.7. This reflects a general fact about product matrices in which patterns of designation in the factors do not ‘interfere’ with each other; to make this precise, we need the following definition. Let us say that a formula A is satisfiable in a matrix M =(A, D) just in case for some M-evaluation h, we have h(A) ∈ D. With “for every” in place of “for some” here, we have the definition of A’s being valid in M, and it is only for the sake of registering that fact that we give the above definition. What we are really interested in is the satisfiability of sets of formulas, defined as follows. A set of formulas Γ is satisfiable in M (as above) just in case for some M-evaluation h, we have h(Γ) ⊆ D. This makes the satisfiability (in M) of the formula A equivalent to the satisfiability (in M) of the set {A}. But because of logical relations between the formulas involved, the satisfiability (in M) of Γ is not equivalent to – and in particular, is not implied by – the satisfiability (in M) of each A ∈ Γ. Observation 3.24.6 (Rautenberg.) Let M1 = (A1 , D1 ) and M2 = (A2 , D2 ) be matrices for a language L in which exactly the same sets of formulas of L are satisfiable. Then, where 1 , 2 , are the consequence relations determined respectively by M1 , M2 , and M1 ⊗ M2 : = 1 ∩ 2 . Proof. ⊇ 1 ∩ 2 : given by 2.12.6(i), without the need of the special hypothesis of the present Observation. ⊆ 1 ∩ 2 : We show that ⊆ 1 , using the hypothesis that the same sets of formulas are satisfiable in M1 as in M2 , and in particular that any set of formulas satisfiable in M1 is satisfiable in M2 . (The converse similarly gives us the conclusion that ⊆ 2 .) Suppose Γ 1 A. Then for some M1 -evaluation

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/ D1 . Since Γ is thus satisfiable in M1 , h1 , we have h1 (Γ) ⊆ D1 and h(A) ∈ by our hypothesis, there exists an M2 -evaluation h2 with h2 (Γ) ⊆ D2 . Now consider the M1 ⊗ M2 -evaluation h1 ⊗ h2 . (See discussion following 2.12.8 for this notation.) For this evaluation we have h1 ⊗ h2 (Γ) ⊆ D1 × D2 , and (regardless of whether h2 (A) ∈ D2 ), h1 ⊗ h2 (A) ∈ / D1 × D2 , so Γ A. Note that if M1 = M2 , the condition that in M1 and M2 the same sets of formulas are satisfiable is met automatically, thus subsuming the earlier 2.12.7 (from p. 216) under the present account. Corollary 3.24.7 Where M is the product matrix of Figure 3.24a, the consequence relation determined by M is the intersection of those determined by M∧ and M∨ . Proof. The conditions of Obs. 3.24.6 are met here, the same sets of formulas – namely all sets of formulas – being M∧ - and M∨ -satisfiable. We turn now to the matter of taking the product of ∨b with ∧b rather than in the reverse order; that order was depicted in Figure 3.24a (p. 465). The new order is as in: ∨∧ *1 2 3 4

1 1 2 1 2

2 2 2 2 2

3 1 2 3 4

4 2 2 4 4

Figure 3.24b: The Product of Disjunction and Conjunction

Corollary 3.24.7 holds equally well for the consequence relation determined by this matrix (taking the four-valued product tables for ∧ and for ∨) in the language with connectives ∧, ∨, and ∨∧. The latter thus emerges, exactly as did ∧∨, as hybridizing the former pair. This raises a certain difficulty, first noted in Łukasiewicz [1953], for thinking about hybrids. To bring this out, let us review more explicitly what it is for a connective # to be a hybrid of two connectives #1 and #2 according to a consequence relation in a language containing all three (being presumed to be all of the same arity). For any set of formulas Γ ∪ {B} of this language, we take this relation to hold when: (Hyb.)

Γ B iff Γ(#1 ) B(#1 ) and Γ(#2 ) B(#2 ).

Here, as in our earlier explication of this relationship for Fmla, A(#i ) is the result of replacing every occurrence of # in A by #i (i = 1, 2), and Γ(#) = {C(#) | C ∈ Γ}. (We could write the above condition more succinctly thus: for all σ, σ ∈ iff both σ(#1 ) ∈ and σ(#2 ) ∈.) Note that if we start with a consequence relation 0 in the language with connectives #1 and #2 but without #, this uniquely fixes how # is to behave according to the extension of 0 satisfying (Hyb.). There naturally arises, then, a tendency to want to speak of # as the hybrid of #1 and #2 (according to ). But the dangers of this way of speaking are evident if we consider the simultaneous presence of another

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connective # , which also qualifies a hybrid of #1 and #2 . This is precisely what Figures 3.24a and 3.24b suggest. In the language containing ∧, ∨, ∧∨ and ∨∧ we can consider the consequence relation determined by the matrix whose tables for ∧ and ∨ are the usual product tables, and whose tables for ∧∨ and ∨∧ are as in 3.24a and 3.24b. According to the consequence relation so determined, ∧∨ is a hybrid of ∧ and ∨, and so is ∨∧. In other words, as in (Hyb.) above, a -statement involving ∧∨ is correct just in case the result of putting ∧ for ∧∨ and also the result of putting ∨ for ∧∨ are both correct, and the same goes for the results of making such replacements in a -statement involving ∨∧. Thus the logic (in Set-Fmla) of ∨, ∧, and ∧∨ looks just like the logic of ∨, ∧, and (instead:) ∨∧. But if both product connectives are present, they do not form equivalent compounds (from the same components). For, where is the consequence relation concerned, p ∧∨ q p ∨∧ q, since we do not have p p ∧ ∨ q (as p q). (Alternatively, by inspection of the tables: when p and q have the values 1 and 3, respectively, p ∧ ∨ q and p ∨∧ q take these same values respectively.) Adapting what Łukasiewicz said on noticing this phenomenon in a related context – described below in the Digression on the Ł-modal system – we can liken ∧∨ and ∨∧ to a pair of twins, easily distinguished when together, but hard to tell apart when seen separately. Somewhat less picturesquely, we borrow a description from our forthcoming discussion in 4.34 of inter- vs. intra-systemic notions of uniqueness: the effect of (Hyb.) above is to select a unique logical role for the hybridizing connective # there displayed, but this role need not be one which is uniquely filled. Small wonder, then, there arises a question about the propriety of speaking of ‘the’ hybrid of two connectives. Example 3.24.8 The non-equivalence of A ∧∨ B with A ∨∧ B according to (as lately conceived) notwithstanding, there are some interesting logical relations between the two products to notice; here we, cite by way of example the ‘absorption’ principles: (p ∧∨ q) ∨∧ p p and (p ∨∧ q) ∧∨ p p. A more complicated replacement procedure than (Hyb.) above makes sense of these: consider the results of replacing ∧∨ and ∨∧ by ∧ and ∨ respectively and then the results of replacing these product connectives by ∨ and ∧ respectively. We have not said anything of the pure logic of a conjunction-disjunction hybrid, where ‘pure’ means that the connectives ∧ and ∨ so hybridized are not present in the language. To get away from the matrix connotations of “∧∨” and “∨∧”, we will use “◦” for the hybridizing connective. We considered in 3.13.7 (p. 390) the consequence relation and the gcr determined by the class of all valuations respecting the determinant F, F, F shared by ∧b and ∨b . These truth-functions share another determinant also, namely T, T, T . If we amend the syntactic description of the (generalized) consequence relations given there to take this into account, we should be led to a provisional description of the gcr for ◦ as the least gcr satisfying A, B A ◦ B and A ◦ B A, B. But this is insufficient for present purposes, since it does not build in the extensionality of

470


◦ according to . Thus we should add also the requirement that satisfies the condition (E2 ) from 3.22.1(ii), p. 449. Then by the techniques of 3.22, we can show Observation 3.24.9 Where meets the conditions just described, if Γ Δ there is a valuation v assigning T to all formulas in Γ, F to all formulas in Δ, and associating with ◦ either the truth-function ∧b or the truth-function ∨b . Accordingly, we have here a description of the gcr according to which ◦ behaves as a hybrid of (classical) disjunction and conjunction. A similar description, using 3.13.7(i) and the condition of extensionality for consequence relations (alongside A, B A ◦ B) gives a description of the consequence relation according to which ◦ acts as such a hybrid. An elegant syntactic description of this consequence relation in question has been given in Rautenberg [1989a], starting from a somewhat different point, namely, the observation that both ∧ and ∨ are (according to {∧, ∨}-classical ) commutative, idempotent, and associative. Thus ◦ should also exhibit these ‘semilattice’ properties. (We shall discuss the application of this algebraic terminology to connectives in 3.34, p. 497.) Rautenberg asks what else we need to require of ◦ in order for the common properties (in the sense of (Hyb.) on p. 468 above) of ∧ and ∨ to be captured, and finds that the needed supplementation is given by the condition: A, B ◦ C A ◦ B, for all formulas A, B. (See also Rautenberg [1989b].) Digression. (The Ł-Modal System.) The situation in which Łukasiewicz came to use the above ‘twins’ simile was à propos of modal logic. The story is quite interesting in its own right, and since it belongs in a discussion of extensionality, it is worth telling within the present section. If we start off by thinking of as representing logical or metaphysical necessity, one obvious principle is that A should be a consequence of A. This gives a determinant F, F whose induced condition, A A, any consequence relation with a chance of recording the logic of necessity should satisfy. But no such determinant T, x deserves comparable respect, since A could be thought of as standing in for a necessary truth—which would suggest setting x = T, but A might also be thought of as a contingent truth, suggesting x = F. Thus there is no way of completing the truth-table in a proposed truth-table account of . At least, this is clear if only two truth-values are allowed. Having apparently exhausted the means available for semantically describing modal concepts in two-valued terms, the idea naturally arises of considering a three-valued logic, with values for necessity, contingency, and impossibility: 1, 2, and 3, respectively (say). This is so close to the intended (‘future contingents’) application of Łukasiewicz’s three-valued logic, discussed in 2.11, that it will come as no surprise that Łukasiewicz himself suggested, in the 1920’s, such an approach to modality in general. The suggested table for in Łukasiewicz [1930], p. 169, took the value 1 to 1 and each of the values 2 and 3 to 3. Unfortunately, we know from the discussion in the notes to §2.1 (‘Philosophy’: p. 269) that before the merits of that suggestion are considered, things have already gone wrong for the non-modal connectives ∧ and ∨ (setting aside questions about Łukasiewicz’s →), since whether or not (e.g.) a conjunction is contingent cannot be determined on the basis of the contingency, necessity, or impossibility of the conjuncts, in particular because a conjunction with two contingent

3.2. EXTENSIONALITY

471

conjuncts may be contingent (if, say, we take each conjunct to be the same statement) and may be impossible (as when we take one conjunct as the negation of the other). A solution that comes to mind would be to have instead of three values, four, so that the contingencies are subdivided into two classes, since after all the problem with p ∧¬p receiving a value other than impossible is resolved by noting that whichever conjunct is contingently true, the other is contingently false: we can think of the two ‘intermediate’ values as representing these two cases of contingency. The algebra this suggestion leads to is the direct product of the two-element boolean algebra with itself. We can still keep Łukasiewicz’s 1930 idea of expanding the algebra by an operation for which takes the top element (again designated) to itself and all other elements to the bottom element. Łukasiewicz [1953] embodied a change of mind on this score. The three-valued scheme described above is replaced by an expansion of the four-element boolean algebra with a different table for from that just described. What he chose was a table which interpreted by the product of I with F, on the grounds that these are the two singulary truth-functions for which such an interpretation will save the above-noted -principle A A. Actually, Łukasiewicz worked in Fmla rather than Set-Fmla, so the point should better put as the claim that every →-boolean valuation on which I or F (but not V or N) is associated with verifies the formula A → A, for any A. The logic in Fmla determined by the matrix we have just described he called the Ł-modal system. The effects of this choice are unfortunate, as most subsequent commentators have agreed, since it is no less a desideratum of a formalization of alethic modal logic that it should respect the principle that necessity implies truth, as that it should not render extensional. (Likewise for , in terms of which this objection is forcefully put at p. 124 of von Wright [1957].) Yet, since is interpreted by a product of truth-functions, this is just what we end up with: all instances of the extensionality-like schema (Ł)

(A ↔ B) → (A ↔ B)

belong to the Ł-modal system. (In fact, the same goes for the schema in which the two occurrences of “↔” in (Ł) are replaced by “→”: a sort of ‘poor man’s K axiom’, and from time to time confused with the real thing, as in the claim in Hocutt [1972] that the epistemic logic of Hintikka [1962] is committed to it, with representing “a knows that. . . ”.) The very consideration which makes a simple two-valued truth-table account inappropriate tells against this schema: consider A contingent and B necessary. The notes may be consulted for further information on the Ł-modal system. Our purpose in introducing it here, apart from the above point about the inadequacy of logics according to which is extensional when the intended interpretation is as necessity, was to note the context in which Łukasiewicz made the analogy about twins looking identical when seen apart even though easily distinguishable when together: he considered the effect of treating not by the product of I with F but of F with I. (In fact, the discussion centres on , which Łukasiewicz writes as Δ, rather than on .) Łukasiewicz further observed – something he regarded as at least mildly paradoxical – that we could add two separate (actually ) operators subject to these two interpretations and they would then be like a pair of twins. To quote Łukasiewicz verbatim (Borkowski [1970], p. 370), he says of these two operators that they

472

CHAPTER 3. TRUTH-FUNCTIONAL, EXTENSIONAL, . . . are undistinguishable when they occur separately, but their difference appears at once when they occur in the same formula. They are like twins who cannot be distinguished when met separately, but are instantly recognised as two when seen together.

The end-of-section notes (p. 483) supply references to further discussions of this aspect of the situation. The Ł-modal system is not a normal modal logic (in Fmla, in the sense of 2.21), since it is not closed under Necessitation—indeed it contains no formulas of the form A. We can ask, out of interest, which normal modal logics are extensional (and are therefore, if the above reasoning is accepted, ill-suited to the formalization of alethic modality). The answer is that there are three such, namely, the two Post-complete normal modal logics, KVer and KT!, according to each of which is fully determined, and also their intersection, according to which is not fully determined but is still extensional. Concerning this last logic, note that, since KVer = ML({Firr }) and KT! = ML({Fref }), where the frames here are the single-point irreflexive and single-point reflexive frames, respectively, KVer ∩ KT! = ML({Firr }) ∩ ML({Fref }) = ML({Firr } ∪ {Fref }) = ML({Firr , Fref }) = KTc and further, that KTc = K + (Ł). (By the latter notation is meant the normal extension of K by axiom-schema (Ł); often “⊕” would be used here instead of “+”.) End of Digression. Let us return to the topic of hybridizing (classical) conjunction and disjunction in Set-Fmla with no other connectives present. We summarised a syntactic characterization of this logic, due to Rautenberg, before the above Digression, and we have seen a semantic characterization of it as the logic determined by the product matrix of Figure 3.24a on p. 465 (or that of Figure 3.24b, p. 468), in view of 3.24.7. That does not make it a four-valued logic, however, since for all we know there may be a determining matrix with fewer elements for the same consequence relation. And indeed, Rautenberg [1989a] notes that a threeelement matrix suffices for this purpose, namely, as he describes it (p. 536), “the 3-element totally ordered semilattice with the designated element in the middle”: in other words, take the matrix consisting whose algebra is provided by the ∧-table of Kleene or (three-valued) Łukasiewicz, discussed in 2.11, with 2 rather than 1 (or 1 and 2) as designated. We prefer to relabel the values so that 1 is the sole designated element, so let us interchange the labels “1” and “2”. This presents, as one of its advantages, an opportunity the making contact with the important contribution of Sobociński [1952], in which the tables of Figure 3.24c appear – though with a different choice of designated elements (on which, see the Digression after 3.24.11 below). What matters for present purposes is just the ∧-table. For our statement of Rautenberg’s result, we need to introduce some (temporary) notation. Accordingly, let ◦ be the consequence relation on the language with one connective, namely (binary) ◦, which is determined by the matrix with just the ∧-table (rebaptized as the ◦-table if desired) from Figure 3.24c, and let ∧ and ∨ be the consequence relations on this same language in which ◦ is interpreted as conjunction and disjunction respectively; i.e., they are the consequence relations determined, respectively, by the class of valuations v satisfying

3.2. EXTENSIONALITY ∧ *1 2 3

1 1 1 3

2 1 2 3

3 3 3 3

473 ∨ 1 2 3

1 1 1 1

2 1 2 3

3 1 3 3

→ 1 2 3

1 1 1 1

2 3 2 1

3 3 3 1

¬ 1 2 3

3 2 1

Figure 3.24c: Unital Version of Sobociński’s Matrix

v(A ◦ B) = T iff v(A) = v(B) = T and by the class of valuations v satisfying v(A ◦ B) = F iff v(A) = v(B) = F. The following proof makes use of the notion of a submatrix and of a basic result for this notion (4.35.4, p. 598). Theorem 3.24.10 With ◦ , ∧ , ∨ as just defined we have ◦ = ∧ ∩ ∨ . Proof. First we show that ◦ ⊆ ∧ ∩ ∨ . To show that ◦ ⊆ ∧ , note that the submatrix ∧-reduct of Figure 3.24c with universe {1, 3} is isomorphic to the two-element boolean ∧-matrix, so 4.35.4 gives the result. For the case of ◦ ⊆ ∨ , we appeal to the same result, but this time noting that the submatrix with universe {1, 2} is isomorphic to the boolean ∨-matrix. Next, we show that ∧ ∩ ∨ ⊆ ◦ . With ∧∨ as in 3.24.2(i) except that the sole connective is the binary ◦ (interpreted using the matrix of Figure 3.24a, (p. 465) we have ∧ ∩ ∨ ⊆ ∧∨ by 2.12.6(i), so it suffices to show that ∧∨ ⊆ ◦ . This follows by another appeal to 4.35.4 since the submatrix of that depicted in 3.24a with elements {1, 2, 3} is isomorphic to that consisting of the ∧-table (which we use to interpret ◦ here) of Fig. 3.24c. Thus the Set-Fmla ‘common logic’ of conjunction and disjunction is threevalued. Theorem 3.24.10 may arouse curiosity on the following grounds. We introduced the ∧-table of Figure 3.24c by quoting Rautenberg’s words “the 3element totally ordered semilattice with the designated element in the middle” and then relabelling the elements of the three-element Kleene matrix for ∧. But if we had done a similar relabelling in the case of the Kleene matrix for ∨, what we would have ended up with was the ∨-table of Figure 3.24c, and – here comes what seems curious – the proof of Theorem 3.24.10 could not have been conducted with the matrix consisting of just this ∨-table. So the question is: why, when we seem to be discussing an issue that is symmetrical in ∧ and ∨, is there this asymmetry? The ∨-reduct of the matrix of Figure 3.24c could not be used in the way we used the ∧-reduct because, writing “◦” rather than “∨” for the connective concerned (for continuity with the above proof) and ◦ , for the consequence relation determined by the ∨-reduct of the matrix (of 3.24c) we have p ◦ p ◦ q (as well as q ◦ p ◦ q), and so have drifted outside of the common logic of conjunction and disjunction (since this is -statement not correct when ◦ is replaced by ∧ and ◦ by CL ). Indeed to avoid a collapse into the (classical) logic of disjunction instead of the common logic, we must invalidate such componentto-compound inferences in the same we as we must, if we are to avoid a collapse into the logic of conjunction, invalidate the compound-to-component inferences. Consider how this works in the case of the interpretation of ◦ by means of the ∧-matrix extracted from Figure 3.24c, as in 3.24.10. We invalidate the sequent

474


p p ◦ q by putting h(p) = 1, h(q) = 3, since h(p ◦ q) = 3 in that case, and we invalidate the sequent p ◦ q p by putting h(p) = 2, h(q) = 1, and then the unusual behaviour of the value 2 gives h(p ◦ q) the designated value 1. To obtain similar effects with the ∨-table of Figure 3.24c, we need to change our designation policy and include 2 alongside 1 as a designated value. (That gives Sobociński’s own matrix – see the Digression below.) Then we could invalidate p p ◦ q by setting h(p) = 2, h(q) = 3, and invalidate p ◦ q p by setting h(p) = 3, h(q) = 1. But no matrix with two designated elements can be isomorphic to a submatrix (such as the ∧∨ product matrix) with only one: so the proof of Theorem 3.24.10 would not go through with this change in designation. Nor would the analogous result be correct, for while a submatrix argument readily establishes that any sequent valid in the matrix we are now considering is classically valid when ◦ is interpreted as conjunction as well as when it is interpreted as disjunction, the converse fails, as we see with Example 3.24.11(i). Examples 3.24.11(i) The sequent p, q ◦ r p ◦ q is valid in the two-valued boolean matrix for ∨ as well as in that for ∧, but is not valid in the matrix consisting of the ∨-table of Figure 3.24c, with 1 and 2 as designated elements. (Put h(p) = 2, h(q) = 3, h(r) = 1 and we get h(q ◦ r) = 1, so both left-hand formulas receive designated values, while h(p ◦ q) = 3 – the undesignated value.) (ii ) Considerations of duality may suggest that there is a problem raised by (i) for our earlier conclusion in 3.24.10 that the ∧-table from Fig. 3.24c, with the value 1 as designated, determines the common consequence relation of classical conjunction and disjunction. However, dualizing the example fully takes us outside of the arena of consequence relations, the logical framework Set-Fmla, and into Set-Set. Reversing the example under (i) we get p ◦ q p, q ◦ r, a sequent which is classically valid whether ◦ is interpreted (throughout) conjunctively or disjunctively, but which is indeed invalid in the ∧-table matrix, as we see by putting h(p) = 2, h(q) = 1, h(r) = 3, giving h(p ◦ q) = 1, while h(q ◦ r) = 3, so both left-hand formulas receive undesignated values while the right-hand formula takes the designated value. The sequent mentioned under 3.24.11(i), is mentioned in Rautenberg [1989a], p. 533, as part of the basis for a proof system for the consequence relation hybridizing classical conjunction and disjunction, alongside sequents securing that ◦ is a semilattice connective (is, in other words, associative, commutative and idempotent). The required sequent-to-sequent rules are the usual structural rules and Uniform Substitution. In view of Example 3.24.11(ii), one may wonder about a similar systematization of the generalized consequence relation hybridizing these connectives, which we shall turn to after some remarks relating the foregoing material to other pertinent literature. Digression. Sobociński [1952], the original source for the tables in Figure 3.24c (p. 473), uses them in a matrix with the two designated values 1 and 2 rather than with just the value 1 designated. He is working in Fmla and would thus have no logic if only the value 1 had been designated (since assigning 2 to each variable would give every formula that value). Any one of the binary connectives may be taken as primitive alongside negation so as to define the remainder

3.2. EXTENSIONALITY

475

using De Morgan style equivalences and the equivalence of A → B (A ∨ B, A ∧ B) with ¬A ∨ B (¬A → B, ¬(A → ¬B), respectively). (These equivalences are indiscriminately valid in the resulting matrix, in the sense of 2.15, so do not depend on the choice of designated values, and make the equivalent formulas not just equivalent but synonymous according to the logic – in any logical framework – determined by the matrix such a choice yields.) Sobociński in fact took → and ¬ as primitive and wanted to sketch a logic staying as close to the classical logic of these connectives as possible while avoiding the more egregious paradoxes of material implication. (See 7.13.) The resulting logic turns out to be the {→, ¬}-fragment of the “semi-relevant” logic RM, though in that context the above conjunction and disjunction are the intensional (or multiplicative) versions of these connectives and would accordingly be denoted by “◦” (no connection with the ad hoc use of “◦” in our discussion for whatever binary connective is currently under discussion as a candidate hybrid) and “+” respectively. So we could equally well say we were dealing with the {→, ¬, ◦, +}fragment – the multiplicative fragment – of RM, and it would be better to do so since RM with the additive connectives present is not itself tabular. Anderson and Belnap [1975], p. 149, express surprise at this contrast between the multiplicative fragment and the full system: but see Avron [1987].) Further information and bibliographical references may be found in Anderson and Belnap [1975], pp. 98, 148; for an interesting application of the Sobociński matrix together with its unital cousin from Figure 3.24c, see Hösli [1992]. This features a multiplicative version of the additive-style Kleene matrix treatment of conjunction and disjunction (the additive analogue appearing in 2.12.4 above). End of Digression. Let us turn to the promised task of hybridizing conjunction and disjunction in Set-Set. A particularly simple proof system is that given in the following result; note that since Log(V1 ∪ V2 ) = Log(V1 )∩ Log(V2 ) (1.12.4(i), p. 60), and here for V1 and V2 we choose V∧ and V∨ , these being the classes of valuations on each of which the connective ◦ is associated with ∧b and ∨b respectively. Thus the result tells us we have a proof system for the hybrid of classical conjunction and disjunction. Theorem 3.24.12 The proof system with, alongside the structural rules (R), (M) and (T), the zero-premiss rules (I), (II), (III) and (IV), is determined by the class of valuations V∧ ∪ V∨ : (I) A, B A ◦ B; (III) A ◦ B, C A, C ◦ D

(II) A ◦ B A, B; (IV) A ◦ B B ◦ A.

Proof. Soundness: It suffices to check that any instance of the sequent-schemata (I)–(IV) holds on any valuation in V∧ as well as on any valuation in V∨ . Completeness: Let be the (finitary, by 1.21.4(ii), p. 109) gcr associated with this proof system. Then for any unprovable sequent Γ Δ we have Γ+ ⊇ Γ, Δ+ ⊇ Δ with Γ+ Δ+ and every formula belonging to exactly one of Γ+ , Δ+ , by 1.16.4 (p. 75). We must show that the valuation v defined as assigning T to exactly the elements of Γ+ , v ∈ V∧ ∪ V∨ . Suppose, for a contradiction, that v∈ / V∧ ∪ V∨ . Since v ∈ / V∧ , there exist A and B for which the condition that v(A ◦ B) = T iff v(A) = v(B) = T is violated, and since v ∈ / V∨ , there exist


476

C and D for which the condition that v(A ◦ B) = F iff v(A) = v(B) = F is violated. So we have either (a) v(A ◦ B) = T while v(A) = F or v(B) = F or (b) v(A) = v(B) = T while v(A ◦ B) = F; And also either (c) v(C ◦ D) = T while v(C) = v(D) = F or (d) v(C) = T or v(C) = T while v(C ◦ D) = F. Possibilities (b) and (c) are excluded by rules (I) and (II) respectively, given the way v was defined. (To take the case of (c) in more detail: the assignments supposed for v here mean that C ◦ D ∈ Γ+ while C, D ∈ Δ+ , contradicting the fact that Γ+ Δ+ , in view of rules (I) and (M).) So we have only to rule out the case in which (a) and (b) obtain. Because of the “or”s here, there is a further division into subcases: (1) v(A ◦ B) = T while v(A) = F; v(C) = T while v(C ◦ D) = F (2) v(A ◦ B) = T while v(A) = F; v(D) = T while v(C ◦ D) = F (3) v(A ◦ B) = T while v(B) = F; v(C) = T while v(C ◦ D) = F (4) v(A ◦ B) = T while v(B) = F; v(D) = T while v(C ◦ D) = F Subcase (1) is directly excluded by rule (III), and it is clear that the other cases are excluded by the combined effect of (III) and the commutativity rule (IV). Note that by the ‘method of common consequences’ (3.13) we could convert the basis given for the above proof system into one in Set-Fmla for hybridizing conjunction and disjunction in that framework – though the result would not be as elegant as that mentioned from Rautenberg [1989a] in the discussion following 3.24.9. It is interesting to compare the joint effect of rules (I) and (II) used to present the proof system figuring in 3.24.12, which certainly seem to have something to do with idempotence of ◦ – wanted since each of ∧, ∨, is idempotent – with what one would normally take to be the direct expression of this feature, namely the combination of (I) with (II) : (I)

AA◦A

(II)

A ◦ A A.

(These last rules make ◦ idempotent in the sense that for any consequence relation satisfying them we have A A ◦ A. On the propriety of using the term idempotent here, even though A and A ◦ A remain, for any A, distinct formulas, see 3.34 – p. 497.) Clearly, (I) and (II) follow from (I) and (II), since we may take instantiate the “B” in the latter to the schematically represented A. The converse leads us back into the issue of extensionality, about which we have had little to say since the opening remarks of the present subsection, and on whose application to the case of n-ary connectives for n > 1 rather little was said in (especially) 3.22, and we take up the issue here for n = 2 by relettering the conditions collectively called (E2 ) in 3.22.1(ii) from p. 449. We have also written “◦” in place of “#” to connect with the present discussion. TTx Rule: A, B, A ◦ B, C, D C ◦ D TFx Rule:

A, A ◦ B, C B, C ◦ D, D

3.2. EXTENSIONALITY

477

FTx Rule: B, A ◦ B, D A, C ◦ D, C FFx Rule: A ◦ B A, B, C ◦ D, C, D The labelling here is intended to recall the notation “ T, T, x ”, etc., denoting, where x ∈ {T, F} one or other of the two determinants which tell us how a binary truth-function is to treat the case in which its first and second arguments are T (and whose value is in that case given by x). The TTx rule just tells us that one of the two possible determinant-induced conditions for this case is satisfied, without telling us which. (This terminology is from 3.11.) In other words, it tells us that like cases are to be treated alike, when the two components are both true: the compound must be true in all such cases, or false in all such cases. Similarly, the TFx Rule tells us that all cases in which the first component is true and the second is false are to be treated alike. And so on for the remaining cases. Any hybrid of binary connectives satisfying any particular determined induced condition (i.e., in which x is specified as T or F) must accordingly satisfy all four of the above rules. Rules (I) and (II) from 3.24.12 go further and tell us which determinant is to be respected in the case of the TTx and FFx Rules, namely, x = T and x = F respectively. With these rules we have captured that common aspect of the truth-functions ∧b and ∨b embodied in the following partial table, dashes indicating unspecified values: A T T F F

◦ T – – F

B T F T F

Figure 3.24d

We could, of course, have given a 2-by-2 array (“matrix”) representation of this information, as with the 3-by-3 tables of Figure 3.24c and elsewhere. We return to the issue of the blanks on its second and third lines, presently, for the moment noting how the idempotence issue raised above is to be resolved. We saw that (I) and (II) followed from (I) and (II) and raised the question of a converse derivation. It will be enough if we concentrate on the case of (I) and (I) . As a special case of the TTx Rule, we have A, A, A ◦ A, A, B A ◦ B We have included redundant repetitions so as to make it clear how the TTx Rule has been specialized to the above schema (namely, with A playing the role not only of the “A” there but also of “B” and “C” – the current B playing the “D” role). Eliminating this redundancy (since we are in Set-Set after all, and not Mset-Mset or Seq-Seq), we have A, A ◦ A, B A ◦ B. And from this and (I) , we have by the Cut Rule (T), a derivation of (I), the cut formula being A ◦ A. For extensional connectives, then, idempotence in the usual (I) -with-(II) sense, is equivalent to the (I)-with-(II) variant of idempotence.

478


Exercise 3.24.13 (i) Describe a gcr in whose language ◦ is a binary connective, satisfying (I) and (II) but neither (I) nor (II). (Suggestion: By the preceding discussion, we need to avoid an extensional for this. Modal logic supplies many such examples. As a first step, we might consider, against the background of the normal modal logic S5 (in Set-Set), say, defining A ◦ B as (A → B) ∧ A. This gives A ◦ A A and A, B A ◦ B but does not deliver A ◦ B A, B, since A ◦ B A. Consider varying ∧ to another boolean connective, however.) (ii ) Derive the TTx, TFx, FTx and FFx Rules from the basis given of the proof system in 3.24.12. (iii) Consider the connective we shall write as ∧¬, to be treated as primitive despite its orthographic structure, which is intended to be suggestive of its intended effect as forming a compound from A and B, in that order, which is equivalent to A ∧ ¬B. We do not envisage either ∧ or ¬ to be present, however, in the language of the gcr determined by the class of all valuations v satisfying the condition that v(A ∧ ¬B) = T iff v(A) = T and v(B) = F for all formulas A, B. Show that the hybrid of ∧¬ as it behaves according to and (classical) ∧ is ‘axiomatized’ in Set-Set by the following five principles, alongside the standard structural rules, the following, in which as usual we denote the hybridizing connective by ◦: A ◦ B A; A ◦ B, C ◦ D, D B; A, B, D, C ◦ D A ◦ B;

A ◦ B, C B, D, C ◦ D;

and A, B, C A ◦ B, D, C ◦ D.

(Follow the example of the proof 3.24.12 and show that this proof system is sound and complete w.r.t. the class of valuations v such that either for all A, B, v(A ◦ B) = T iff v(A) = v(B) = T or for all A, B, v(A ◦ B) = T iff v(A) = T and v(B) = F.) We return to Figure 3.24d, depicting a partial truth-table for an idempotent connective. Some of what is shared by conjunction and disjunction but not by all such connectives remains unrepresented in the table. The fact that the truth-functions associated on boolean valuations with these connectives are both commutative remains unrepresented, and would be made explicit only by some indication – say by inserting the same variable over truth-values (x, for example) in place of both dashes – that the two blanks are to be filled in the same way. (This does not give a mere consequence of commutativity, but the whole of it, since in general the claim that some function of two variables, f (x, y) say, is commutative is equivalent to the claim that for all x and y with x = y, f (x, y) = f (y, x), since we have such an identity automatically for the case in which x = y. This is reflected in the fact that in a tabular representation of a commutative operation, any change on the descending diagonal preserves the commutativity of the operation represented.) This consideration may lead one to the conclusion that of rules (I)–(IV) in Theorem 3.24.12, (III) was redundant. Rules (I) and (II) give us idempotence, after all, and (IV) enforces commutativity, thereby restricting the options for completing the table of Figure 3.24d to two: one giving ∧b (put F into both blanks) and the other giving ∨b (fill both blanks with T). Do we, then, also need (III), or is it derivable from (I), (II) and (IV)?

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The line of thought just presented, suggesting that (III) is redundant, makes the tacit assumption that we are dealing with an extensional connective: without this there is no reason to expect, for a given valuation, ◦ to be associated with any truth-function on that valuation. (Recall the connection between extensionality and pseudo-truth-functionality given by 3.22.5(i), p. 451.) Indeed, given the extensionality we can derive (III) from the ‘strengthened idempotence’ combination of (I) and (II) alongside the commutativity schema (IV), as we now show. The rule (T) (with (M)) is applied twice here with the cut-formula indicated on the right of the line marking its application: (TFx Rule)

(II)

A, A ◦ B, C C ◦ D, B, D

A ◦ B A, B

A ◦ B, C C ◦ D, B, D Cut D

(I) Cut A

C, D C ◦ D

A ◦ B, C C ◦ D, B

The terminal sequent above is not quite (III), which should have “A” rather than “B” on the right; but relettering, we see we have a proof of B ◦ A, C C ◦ D, A. (III) itself then follows by (M) and (T) from (IV), with A ◦ B as the cut-formula. On the other hand, without the help of an extensionality assumption, one cannot derive (III) from (I), (II) and (IV) as we show with an example. The example is of some interest in its own right, since it features a connective satisfying (I) and (II), like the Set-Set hybrid of conjunction and disjunction, and like the Set-Set hybrid of the first and second projections mentioned briefly in the discussion before 3.24.2 (and pursued at greater length in 5.35 below). Like the former and unlike the latter, the following example is also commutative (satisfies (IV), that is). Example 3.24.14 By a pair selection function on a set U we mean a function f such that for all a, b ∈ U , f ({a, b}) ∈ {a, b}. We write “f (a, b)” for “f ({a, b})”, and include the possibility that a = b, in which case accordingly f (a, b) = a = b. Viewed, as the above abbreviative notation suggests, as a function of two (not necessarily distinct) arguments, a pair selection function is accordingly a commutative idempotent binary operation which is in addition a quasi-projection or conservative operation, meaning that its value for a given tuple (here: pair) of arguments is always one of those arguments. For the current application, consider f as a pair selection function on the set of formulas of the language generated from the usual stock of propositional variables with the aid of the binary connective ◦. Consider the gcr determined by the class of all valuations v satisfying the condition that for some pair selection function f we have: for all formulas A and B, v(A ◦ B) = v(f (A, B)). Note that ◦ and satisfy (I), (II) and (IV). Rule (III) is not satisfied however, since, for example, p ◦ q, r p, r ◦ s – consider f and v with f (p, q) = q, f (r, s) = s, v(q) = v(r) = T, v(p) = v(s) = F. Remarks 3.24.15(i) Although all that was needed for the preceding example was the soundness of the proof system with the structural rules and (I), (II) and (IV) w.r.t. the class of valuations there mentioned – viz. the class

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CHAPTER 3. TRUTH-FUNCTIONAL, EXTENSIONAL, . . . of all v for which there exists some pair selection function f satisfying the condition that for all formulas A and B, v(A ◦ B) = v(f (A, B)) – this proof system is also complete w.r.t this class of valuations. For we can extend any given unprovable Γ Δ maximally to Γ+ ⊇ Γ, Δ+ ⊇ Δ with Γ+ Δ+ and define v as the characteristic function of Γ+ and f by the stipulation that for all A, B, f (A, B) = A if v(A) = v(A ◦ B), and f (A, B) = B otherwise. By (IV) this does not depend on the order of A, B, and only, as required, on the set {A, B}; and using (I) and (II) one can verify without much trouble that the condition (on arbitrary A, B) that v(A ◦ B) = v(f (A, B)), is satisfied. (ii ) The gcr associated with the proof system based on (I), (II), (IV), is not only non-extensional, it is non-congruential, in a sense to be made precise in the following section, but which is illustrated for the present case by the fact that although p ◦ p p, we do not have (p ◦ p) ◦ q p ◦ q. This gives the whole thing a very syntactic look to it. We could avoid this feature by adding a rule taking us from A B and B A to A ◦ C B ◦ C (which secures congruentiality in view of (IV)), and fix up the semantics by having the pair selection functions f select one of every pair of ‘Lindenbaum propositions’ [A], [B]. Here (as in 2.13.1, p. 221) [C] is {D | C D}, and we understand the condition on valuations, now appearing as v(A ◦ B) = v(f ([A], [B])) in accordance with the convention that v([A]) = v(A). (Can this stipulation lead us into inconsistency in the face of the possibility that [A] = [B] while v(A) = v(B)? Not if v is consistent with , the gcr associated with the revised proof system. In that case there is no such possibility.) Further details are left to the reader. (iii) By an ordered pair selection function (alias “conservative binary operation”) on a set U we mean a function f such that for all a, b ∈ U , f (a, b) ∈ {a, b}. (We could equally well write “f ( a, b )” for “f (a, b)”.) By the methods of (i) above we can show that the gcr on a language with ◦ the gcr determined by the class of all valuations v for which there is an ordered pair selection function f on the set of formulas with v satisfying v(A ◦ B) = v(f (A, B)) for all A, B, is the least gcr satisfying (I) and (II). (iv ) The preceding semantic apparatus can be given a suggestive recasting, somewhat in the style of Kripke models for modal logic, letting a model be a triple (F, v, f ) in which F is a set of pair selection functions on the set of formulas, v an assignment of truth-values to the atomic formulas, and f ∈ F . Given such a model we define truth for the language with (binary) ◦ as its sole connective, annotated with the aid of “|=”, thus: (F, v, f ) |= pi iff v(pi ) = T; (F, v, f ) |= A ◦ B iff (F, v, f ) |= f (A, B). Calling a sequent C1 ,. . . ,Cm D1 , . . . , Dn valid when for every such model (F, v, f ) with (F, v, f ) |= Ci for all i (1 i m) we have (F, v, f ) |= Dj for some j (1 j n), we have a redescription of the semantic characterization under (iii), whose interest consists in its amenability to the language we obtain by adding 1-ary connectives and , extending the truth-definition by:

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(F, v, f ) |= A iff for all g ∈ F , (F, v, g) |= A; (F, v, f ) |= A iff for some g ∈ F , (F, v, g) |= A. We are then in a position to observe some conjunctive (disjunctive) behaviour on the part of the compounds (A ◦ B) ((A ◦ B), resp.), as we illustrate with the case. If C is a ◦-free formula then whether or not (F, v, f ) |= C is independent of f , from which it follows that for ◦-free A, B, the sequent A, B (A ◦ B) is valid. If we make the further assumption that F ‘provides enough functions’ in the sense that for all formulas A, B, and all f ∈ F with f (A, B) = A (f (A, B) = B) then there exists g ∈ F with g(A, B) = B (resp. g(A, B) = A), then the sequents (A ◦ B) A and (A ◦ B) B are valid when A, B are ◦-free. The “for all g” in the clause for , as applied to a formula of the form A ◦ B may appear some somewhat overblown, since the only respect in which the ‘current’ f and an alternative g ∈ F (available if F provides enough functions) differ relevantly is in that one chooses one of A, B from the pair A, B , an the other chooses the other. The set of sequents of this language is not closed under Uniform Substitution, we should note, since p p is valid while p ◦ q (p ◦ q) is not. Finally we note that a particularly small F satisfies the “provides enough functions” conditions, namely F = {proj12 , proj22 }. With this F in mind, where f is either of its elements we adopt the convention that f * is the other. The we can rewrite the above findings for the clause for thus (F, v, f ) |= A iff (F, v, f ) |= A and (F, v, f *) |= A. (For we would have “or” on the right-hand side here.) We return to related issues in 5.35. We conclude our discussion of hybrids with a couple of conceptual warnings. The first concerns the subconnective relation. As we have remarked, hybridizing several connectives amounts to isolating their greatest common subconnective. We repeat a warning from Humberstone [1986a] that the subconnective relation is not to be confused with the relation of being at least as strong as – or indeed with the converse of this relation – in the sense that conjunction (according to any reasonable logic, say conceived as a consequence relation ) is at least as strong as disjunction. Indeed, since the relation here – and we mean the fact that p ∧ q p ∨ q – does not obtain in the reverse direction, we may as well as that conjunction is stronger than disjunction. (Cf. 4.14.1, p. 526, and surrounding discussion.) But neither ∧ nor ∨, behaving according to CL , say, is a subconnective of the other. ∧ satisfies the condition, when taken as #, that (for this choice of ) A # B A for all A, B, while ∨ does not, and ∨ satisfies the condition that A A # B, which ∧ does not, so neither of these satisfies all the (unconditional) -conditions that the other does. (See further Williamson [1998a].) This parenthetical “unconditional” leads to our second warning. We return to the present point in 3.23.18. Suppose that we have hybridized the connectives #0 and #1 as they behave according to gcr’s 0 and 1 , respectively, each for simplicity in a language with no other connectives, in the shape of # as it behaves according to . In other words, identifying #0 , #1 , and #, we have = 0 ∩ 1 . It follows that any conditional condition – such as those amounting to the admissibility of various sequent-to-sequent rules – satisfied by 0 and 1 , must be satisfied by . But the converse is not the case:

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Example 3.24.16 Let ∧ and ∨ be the gcr’s on a language with one binary connective ·, determined by the classes V∧ and V∨ respectively, the former comprising all those valuations associating ∨b with · and the latter all those valuations associating ∧b with ·. Thus = ∧ ∩ ∨ presents · as the hybrid of conjunction and disjunction. First, forget about our earlier syntactic characterization of this particular hybrid. Then observe, in view of the exercise below, that for all Γ, Δ: Γ Δ implies Δ Γ, though a corresponding implication is not satisfied by either ∧ or ∨ (let alone both). Exercise 3.24.17 With notation as above, show that if a premiss sequent for the following ‘rule of symmetry’ (see Coro. 2.15.7 on p. 253, and discussion following): ΓΔ ΔΓ is V∧ -valid (holds on each v ∈ V∧ , that is), then the conclusion of that application of the rule is V∨ -valid, and similarly if a premiss is V∨ -valid, then the corresponding conclusion is V∧ -valid. (Hint: See p. 93, 1.19.3– 4.) Thus we may conclude that for any Γ, Δ ∈ ∧ ∩ ∨ , we have Δ, Γ ∈ ∧ ∩ ∨ . Of course we could also have argued the above point – that our conjunctiondisjunction hybridizing gcr is symmetric – syntactically, since we have already seen syntactic characterizations (3.24.9, 3.24.12) making for an easy inductive argument (on length of proofs). We return to the earlier warning about not confusing subconnective relation with one of logical strength. We illustrated this with the case of ∧ and ∨, the first of which was stronger than the second though neither was a subconnective of the other. Let us close with some examples from modal logic illustrating further possible combinations of comparability and incomparability in respect of inferential strength and the subconnective relation. Examples 3.24.18(i) Take the extension of the smallest normal bimodal logic, say as the inferential consequence relation in which 1 enjoys the properties of in monomodal K while 2 enjoys the properties of in monomodal KT. (A similar example was given near the start of this subsection.) In this logic, 1 is a subconnective of 2 , and the two connectives are incomparable with respect to strength. We can extend the logic to require that 1 is stronger than 2 . We can equally well extend it so as to make 2 stronger than 1 . Model-theoretically, this amounts in the first of these extensions to considering the inferential consequence relation determined by the class of frames W, R1 , R2 with R2 reflexive and R2 ⊆ R1 and in the second to that determined by the class of frames W, R1 , R2 in which R2 is reflexive and R1 ⊆ R2 . (ii) Begin instead with the modal logics KB and KT, combined as under (i) into a normal bimodal logic in 1 and 2 respectively. The i are now incomparable w.r.t. the subconnective relation, as well as w.r.t. inferential strength. We can again extend this logic as under (i) so as to keep the two i incomparable w.r.t. the subconnective relation, while making 1 inferentially stronger than 2 or vice versa.

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It is quite natural to reach for modal logic in order to show how some of the notions under consideration here come apart. As was noted after 3.13.17 (p. 397), in (non-modal) classical logic no connective is ever a proper subconnective of another – a situation which, interestingly enough, does not obtain in the case of intuitionistic logic (8.24.7, p. 1245). Although the points illustrated in 3.24.18 are obvious enough, there is a terminological provocation for the confusion we are warning against, namely that of the monomodal logics K and KT mentioned in 3.24.18(i)the latter is said to be stronger than the former, meaning simply that it is a proper extension of the former. This might help encourage the idea that if connectives answering to the demands of these logics on are both present in a combined logic, the one with the KT properties should be described as stronger. It is only a small further step to confuse this with what we have put by saying that the one connective is inferentially or logically stronger than the other. Rather than invite such confusions, we have preferred to put matters in terms of the subconnective relation rather than introduce a second notion of strength alongside inferential strength – for example distinguishing an ‘internal’ from an ‘external’ notion of strength, to echo the contrast in the passage quoted from Zolin [2000] early in this subsection. Incidentally, the same linguistic contrast as we find there is to be seen in the following passage from Byrd [1973], p. 183, discussing some epistemic and doxastic logics from Hintikka [1962]: Knowledge in the sense of the strong epistemic operator and true belief in the sense of the strong doxastic operator have the same logic when the two notions are considered in isolation. Similarly, knowledge in the sense of the weak epistemic operator and true belief in the sense of the weak doxastic operator have the same internal logic.

The same passage also illustrates, incidentally, the potentially dangerous use commented on above of the term ‘strong(er)’, to mean: obeying further principles (in the present instance, the modal axiom 4). The technical content of the point Byrd is making here is more succinctly available as Lemma 5.9 of Zolin [2000]. (Byrd’s task is made harder because Hintikka [1962] doesn’t identify by a simple axiomatic description which logics he is concerned with. Incidentally, Byrd’s point would appear to have been anticipated in an 1966 conference paper by Jack Barense, judging from a remark on p. 142 of Johnson Wu [1972].) An application of the same internal/external contrast to an issue in tense logic appears in Byrd [1980], note 2.

Notes and References for §3.2 Extensionality is treated in greater detail—though with some notational and terminological differences (e.g., over the definition of ‘pseudo-truth-functional’)—in Humberstone [1986]; there is a tendency for this notion, or more accurately, for its semantic analogue, pseudo-truth-functionality, to be confused with truthfunctionality (see Carnap [1943], §12, for example). Satisfaction of the inset condition on in 3.23.6 is called ‘Strong Replacement’ in Wójcicki [1988], p.236. Theorem 3.23.9, connecting (what we call) left-extensionality and determination by a class of unital matrices, is in Czelakowski [1981]; see also Blok and Pigozzi [1986], Rautenberg [1981], [1985]. What we call weak extensionality (WExt) in


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3.23 is called congruentiality in the papers by Rautenberg just cited, as well as in his [1989a,b]; in §3.3 we study a weaker property under this name. Products of (the tables for) distinct connectives—mentioned in 3.24—are used in Rasiowa [1955], for material implication and material equivalence, and in Setlur [1970b] for material implication and its converse. See also Łukasiewicz [1953] and pp. 101f. of Rescher [1969]. An application to issues in the theory of truth appeared in Herzberger [1973]. Łukasiewicz’s completeness proof for his axiomatization of the Ł-modal system (Łukasiewicz [1953]) relies on a feature of the system not mentioned in our Digression on that topic in 3.24, namely, the use of rejection; a clean proof may be found in Smiley [1961]. See also Chapter 1 of Prior [1957] for an early discussion. Smiley [1963a] and Porte [1979] note that if we expand the non-modal language of (e.g.) the present section by a zero-place connective (propositional or sentential constant) Ω, then the logic determined by all boolean valuations, placing no constraints on Ω at all, can be regarded as containing the Ł-modal system exactly if A is taken to be defined by Ω ∧ A. This is what would be expected if one notes that valuations verifying Ω will verify A, so defined, if and only if they verify A ( as I), and those falsifying Ω will falsify A ( as F). More on this theme will be found in Sotirov [2008]. An excellent discussion of the Ł-modal system, with numerous further references, is provided by Font and Hájek [2002]. Some comments on the ‘twins’ analogy of Łukasiewicz – critical of the latter’s suggestion that the situation they present is somewhat paradoxical – may be found in Krolikoski [1979b]; see further Humberstone [2006c]. Section 4 of Malinowski [2004] also touches on the original modal Łukasiewicz twins example (as does Malinowski [1997]). For more on hybrids and direct products, including ∧∨ and ∨∧, see Humberstone [1995c]; 5.35 treats another special case in some detail.

§3.3 CONGRUENTIALITY 3.31

Congruential Connectives

As in 3.21, we give a biconditional-based introduction to our topic for this section, eliminating the use of “↔” presently. Again we make the assumption that we have ⊇ Log(BV↔ ), though this time we shall not need to assume that is a gcr; even after we rid the following account of its dependence on “ ↔”, we shall end up with something that applies quite directly to consequence relations no less than to gcr’s. Suppose # is a k-ary connective in the language of the (generalized) consequence relation ; then we can ‘verticalize’ the ‘horizontal’ replacement condition (↔-Exti ) from 3.21, and call # ↔-congruential in the i th position (where i k) according to if for all B1 , . . . , Bk ,C: (↔-Congi )

Bi ↔ C ⇒ #(B1 , . . . , Bk ) ↔ #(B1 , . . . , Bi−1 , C, Bi+1 , . . . , Bk )

Since we suppose ⊇ Log(BV↔ ), this amounts to saying that formulas which are logically equivalent according to can always be substituted, one for another, into the ith place in a #-compound to yield logically equivalent compounds. Clearly ↔-extensionality in a given position according to implies ↔congruentiality in that position according to , by appeal to (T). Parallelling

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our discussion of the former concept in 3.21, we may proceed to define # to be ↔-congruential according to if # is ↔-congruential in each position according to , and to observe that this is equivalent, given the assumption that ⊇ Log(BV ), to saying that: (↔-Cong)

B1 ↔ C1 and . . . and Bk ↔ Ck imply #(B1 ,. . . ,Bk ) ↔ #(C1 ,. . . ,Ck ).

Notice that the conditions (↔-Congi ) and (↔-Cong) in fact make sense also for logics in Fmla (with A meaning, as usual, that A belongs to the logic in question), as well as for consequence and generalized consequence relations; however, this cannot be said for the result of purging these concepts of their dependence on ↔, to which we turn in the following paragraph. As with extensionality, we can also consider (↔-Cong) as a special case of a condition of the notion of # -congruentiality of a connective # according to ; this and some further generalizations were mentioned à propos of Fmla in 2.13. (The references there cited were Porte [1980] and Czelakowski [1981].) For the promised purification (‘purging’), we define a k-ary connective # is congruential in the i th position according to if the condition (Exti ) from 3.22 is satisfied for all B1 ,. . . ,Bk ,C, when Γ = Δ = ∅, and that # is congruential (according to ) if this is the case for i = 1, . . . , k. For singulary #, this can be expressed as: A B ⇒ #A #B, though, as in the case of extensionality, the second occurrence of “” is actually redundant. We make this simplification in the following statement of the condition of congruentiality in the ith position according to , included for ease of reference: (Congi )

Bi C implies #(B1 ,. . . ,Bk ) #(B1 ,. . . ,Bi−1 , C, Bi+1 , . . . , Bk )

When every connective in the language of a (generalized) consequence relation is congruential according to that relation, we call itself congruential, in keeping with the convention introduced in 3.23.5 (p. 456). However, as was remarked in 3.15, we can also say that ’s being congruential amounts to the following: any formulas which are logically equivalent according to are synonymous according to . That this formulation implies congruentiality as defined by the convention of 3.23.5 (p. 456), is immediate. The converse requires proof; the following is sometimes called the ‘substitutivity of provably equivalent formulas’: Theorem 3.31.1 If every connective of a language L is congruential according to a gcr (or consequence relation) on L, then where C(A) and C(B) are any two formulas differing in that the first has A occurring in one or more places where the second has B as a subformula, then A B implies C(A) C(B). Proof. We may regard C(A), C(B) as the results of uniformly substituting A, B, respectively for some propositional variable pi occurring in a formula C. Let s1 be the substitution which replaces every occurrence of this variable in a formula by A, and s2 , the substitution which similarly replaces pi by B. Then we prove by induction on the complexity of C that A B ⇒ s1 (C) s2 (C),

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which gives the result. Basis. C is of complexity 0. Then C is pi and s1 (C) is the formula A itself, s2 (C) the formula B itself, and there is nothing to prove. Inductive step. Supposing the result to hold for all formulas of lower complexity than some given C = #(D1 ,. . . , Dk ) for k-ary #, we must show that it holds for C. So assume A B. Then by the inductive hypothesis, the various Dj (1 j k) being of lower complexity than C, we have for each j: s1 (Dj ) s2 (Dj ) from which by the congruentiality of # according to it follows that #(s1 (D1 ), . . . , s1 (Dk )) #(s2 (D1 ), . . . , s2 (Dk )) But the formula represented on the left here is none other than s1 (#(D1 , . . . , Dk )), alias s1 (C), and that on the right is just s2 (#(D1 ,. . . , Dk )), i.e., s2 (C), so we have shown that s1 (C) s2 (C).

Corollary 3.31.2 (i) If is a congruential consequence relation then A B implies Γ C ⇔ Γ C , where Γ and Γ differ at most in having zero or more occurrences of A in formulas of Γ where B appears in formulas of Γ , and C and C differ in having zero or more occurrences of A in C replaced by occurrences of B in C . (ii) If is a congruential gcr relation then A B implies: Γ Δ ⇔ Γ Δ , the (·) notation understood again as in (i). Remarks 3.31.3(i) The characterization of the synonymy of A and B from 3.15 according to was as in 3.31.2. (ii ) The use of Uniform Substitution in the proof of 3.21.1 should not lead the reader to confuse the replacement of one formula in another by an equivalent (i.e., -related) formula of which this Theorem speaks, with the replacement involved in passing from a formula to one of its substitution instances. The latter replacement must be uniform—every occurrence of the replaced variable must be replaced, that is—while in the former case, we speak of the replacement of one or more occurrences, so the replacement may be, but is not required to be, similarly uniform. And the latter replacement must be the replacement of a propositional variable by an arbitrary formula, whereas in the former case there is no constraint on the complexity of the replaced formula. The consequent of Theorem 3.31.1 for a particular (established by showing that the antecedent holds for ) is sometimes called the Substitutivity of (provable) Equivalents for . Digression. The distinction in 3.31.3(ii) between Uniform Substitution and the Substitutivity of Equivalents is the analogue in sentential logic of a distinction between two replacement properties of equational logic. The former corresponds to the fact that any equational theory containing an equation involving the


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variable x (say) contains also the results of replacing x uniformly by any term t. The latter corresponds to the fact that any such theory containing t = u (or t ≈ u, in the notation mentioned in 0.24.7, p. 32) for terms t, u, and also some equation in which t occurs, also contains any result of replacing (not necessarily uniformly) t by u in that equation. End of Digression. In 3.22 (p. 449), we made use of the notation “Γ / A B / Δ” in the course of purifying our ↔-based account of extensionality. It may be helpful here to redescribe this relationship between Γ, Δ, A and B, by saying when it obtains that A and B (Γ, Δ)-equivalent (according to ). (Note that we have to work with this notion of (Γ, Δ)-equivalence and cannot get away with just saying Γ, A B, Δ, meaning in general by Θ Θ that Θ Θ and Θ Θ. Why not? What would this amount to?) Then by an argument just like the proof of 3.31.1 we can show that if all connectives in the language of a gcr are extensional according to , then so are all contexts provided by that language: Theorem 3.31.4 If every connective of a language L is extensional according to a gcr on L, then where C(A) and C(B) are any two formulas differing in that the first has A occurring in one or more places where the second has B as a subformula, if A and B are (Γ, Δ)-equivalent according to , for Γ, Δ ⊆ L, then so are C(A) and C(B). Of course, the notion of equivalence figuring in 3.31.1 is, in the present terminology, the special case of (∅, ∅)-equivalence. Further à propos of the distinction in 3.21.3(ii), p. 447, between substitution invariance and congruentiality, the question can be asked as to the status of the latter property as a desideratum for logics. Some claims for the former property (alias closure under Uniform Substitution, if we think of the logic as a collection of sequents) were sympathetically reviewed at the end of the Appendix to §1.2, though we noted that some of the motivation for such a condition leads only to the weaker condition of closure under uniform substitution of variables for variables, and also listed (in the notes to §1.2, under ‘Uniform Substitution’, p. 191) several examples of apparently well-motivated logics lacking the property. For congruentiality, a feature of many familiar consequence relations, including CL and IL , we can certainly say that this makes for convenient treatment, since it erases any potentially troublesome distinction between logical equivalence and synonymy. We leave the reader to judge whether this feature represents a necessary condition for being a ‘reasonable’ logic. Such a judgment needs to be informed by a familiarity with some examples of non-congruential logics, so we turn to some examples. Examples 3.31.5(i) The consequence relation determined by the three-element Łukasiewicz matrix (Figure 2.11a, p. 198) is non-congruential: see 3.23.4, p. 456, for a counterexample to left-extensionality which also constitutes a counterexample to congruentiality. (ii ) The consequence relations determined by the two three-valued Kleene matrices of 2.11, K1 and K1,2 are both non-congruential. The counterexample mentioned under (i) serves in the case of K1 , since no → is involved and K1 and the Łukasiewicz matrix differ only on →. For K1,2 , change ∧ to ∨ in that example.

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Note that it would have made no difference to these examples if we had spoken of the gcr’s rather than the consequence relations determined by the matrices mentioned in 3.31.5. The following examples involve modal logic; for the first, understand by the ‘model-consequence relation’ the C-model-consequence relation for C = the class of all frames (2.23): Example 3.31.6 The model-consequence relation is not congruential. For example, p and p are equivalent according to this consequence relation but q → p and q → p are not. The model-consequence relation is somewhat unfamiliar, though it will occupy us in 6.32, and the remainder of our modal examples rely on something more like the inferential consequence relation. By contrast with 3.31.6, (like the boolean connectives) is congruential according this relation; or more accurately, according to the C-inferential consequence relation for any class C of frames. We say “something more like” because we will now consider a variation on the concept of a frame from §2.2, so no straightforward transposition of the terminology of that section will be ideal. We shall also work the first example in Fmla. The new notion of a frame we need is that of a frame with distinguished element, by which is meant a structure (W, R, x) where (W, R) is a frame in the old sense and x ∈ W ; we call x the distinguished element of the structure. There are several jobs such an element can be called on to perform. It can, for example, enter into the truth-definition in some way. We will see this in Example 3.31.8. It can also enter—either as well or instead—into the specification of a notion of validity. The latter is our current interest. Let us say that a formula is valid on (W, R, x) just in case it is valid at x in the frame (W, R), these notions being understood as in §2.2. It is easy to see that the formulas valid on every frame are precisely the formulas valid on every frame with distinguished element. If we consider the class of frames with distinguished element (W, R, x) in which that element is reflexive (i.e., xRx), we get a different set of formulas validated than if we consider the class of all reflexive frames. In the latter case, the set of formulas is given by the Fmla system KT. In the former case, we do not get a normal modal logic at all, since, for example, p → p belongs to the set by (p → p) does not. (The latter formula can be falsified at a reflexive point if that point has accessible to it an point which is not reflexive.) The set of formulas valid on all frames with reflexive distinguished elements is clearly a modal logic however (containing all tautologies and closed under Uniform Substitution and tautological consequence) and, though not normal, is a nonnormal modal logic of a special kind called by Segerberg [1971] – from which the present example is taken – quasi-normal. (The terminology is adapted from a usage in Scroggs [1951], and before that from McKinsey.) This means that not only are all tautologies in the logic, but so are all theorems of the system K. In fact, as the reader may care to prove by a minor adaptation of the canonical model method, a syntactic description of this logic, which we shall call K(T), is as follows: K(T) is the smallest modal logic extending K and containing all instances of the schema T. Example 3.31.7 The Fmla logic K(T) is not ↔-congruential. For instance: (p → p) ↔ (q → q) ∈ K(T), but (p → p) ↔ (q → q) ∈ / K(T).


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What is happening here, in semantic terms, is that the distinguished point of the frames always ends up giving the same truth-value (and in this case, the value T) to A ( = p → p) and B ( = q → q), in any model on the frame, but this need not be so for A and B, since the takes us away to consider other points in the frame. (The reader may care to check that in fact any quasi-normal modal logic which is congruential must be normal.) Our second example is of a bimodal logic in which the distinguished element enters into the definition of truth for the second primitive modal operator (the first being ), which we shall write as “A”. This is mnemonic for “Actually”, and we think of the distinguished point as representing the ‘actual world’ of the frame. The clause in the truth definition is predictable: where M = (W, R, x, V ), M |=w AA iff M |=x A, for all formulas A. We still have a choice as to whether to let the distinguished point enter into the definition of validity. If we do, let us call the result “real world validity”; if we don’t, let us speak of “general validity”. That is, we say that a formula A is real-world valid on the frame (W, R, x) if for all models M = (W, R, x, V ) we have M |=x A, and that A is generally valid on the frame if for all models M = (W, R, x, V ) and for all points w ∈ W , we have M |=w A. Example 3.31.8 The set of formulas real-world valid on every frame (W, R, x) is not ↔-congruential, since Ap ↔ p belongs to this set but Ap ↔ p does not; on the other hand the set of formulas generally valid on every frame is congruential. (Ap ↔ p does not belong to this set, since it can be falsified at non-distinguished worlds.) Restricting attention to the class of frames (W, R, x) in which R is universal, the logics of actuality advocated in Hazen [1978] and in Crossley and Humberstone [1977] axiomatize, respectively, the class of real-world valid and the class of generally valid formulas. Remark 3.31.9 An alternative way of setting out the above semantics which is more convenient for some purposes is to take the models to be the familiar (W, R, V ) Kripke models but to evaluate a formula for truth in such a model w.r.t. a pair of worlds. Instead of writing, as above (W, R, x, V ) |=w A, we write (W, R, V ) |=xw A. We are thinking of the upper and lower index as representing what Lewis [1973], §2.8, q.v. for some interesting applications, the reference world (for the present application: the world taken as the actual world) and the world of evaluation (the world at which we are interested in the truth-values of formulas from the perspective supplied by the reference world). This is called two-dimensional modal logic, and we will have occasion to draw on its resources from time to time below. (See in particular the discussion between 7.17.4 and 7.17.5, – pp. 1015–1021 – and the Digression following 8.32.4, p. 1260); general references include Segerberg [1973], Marx and Venema [1997], Gabbay, Kurucz et al. [2003]. For a survey of related work, somewhat differently motivated, see Humberstone [2004a], and for more on the real-world vs. general validity contrast, Hanson [2006].)

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3.32


Some Related Properties

A pair of conditions stronger than congruentiality have made their way into the literature. Say that a k-ary connective # is monotone in the i th position (1 i k) according to a consequence relation or gcr when: Bi C implies #(B1 ,. . . ,Bk ) #(B1 , . . . , Bi−1 , C, Bi+1 , . . . , Bk ), and that # is antitone in the i th position according to when: C Bi implies #(B1 , . . . , Bk ) #(B1 , . . . , Bi−1 , C, Bi+1 , . . . , Bk ), in both cases the schematic letters ranging over arbitrary formulas of the language of the (generalized) consequence relation . Obviously if # is either monotone or antitone in the ith position according to then it is congruential in that position according to , and further if in each position # is either monotone or antitone (not necessarily the same for different positions) then # is congruential (tout court) according to . (Some say “monotonic” here, for “monotone”, but we wish to avoid confusion with monotonicity in the sense of (M), talk of ‘nonmonotonic logics’ etc.; for the same reason, we use “monotony” for the associated noun. Instead of monotone and antitone, others say “upward monotone” and “downward monotone”, respectively.) Examples 3.32.1According to the consequence (or generalized consequence) relation Log(BV {∧,∨,→,¬,↔} ): (i) ∧ and ∨ are each monotone in both (i.e., first and second) positions. (ii) → is antitone in the first position, monotone in the second. (iii) ¬ is antitone (in its one and only position). (iv) ↔ is neither monotone nor antitone in either of its two positions. (v) According to any -normal consequence relation (2.23, p. 291), is monotone, as is . Notice that any connective compositionally derived from connectives which are monotone according to is monotone according to , and similarly with ‘antitone’, where we understand by these terms: monotone (antitone) in each position. In mentioning under 3.32.1(iv) above, we have applied described as monotone a compositionally derived connective since we are taking A as the formula ¬¬A; here the monotony of follows from that of (which itself is a trivial consequence of -normality), together with the double use of the antitone ¬. On the other hand, it is not true that every connective compositionally derived from connectives each either monotone or antitone according to must itself be either monotone or antitone – though, as already noted, it will be congruential – according to We can remain with modal logic for our first counterexample. Examples 3.32.2(i) Consider the singulary derived connective defined by: A = ¬A ∧ ¬¬A, in the light of the consequence relation (for example) KT (2.23). (If we read “” as “it is necessarily the case that”, we may read “” as “it is contingent whether”). is monotone according to KT , while ¬ and ∧ have their boolean ‘tonicities’ as in 3.32.1, but c is neither monotone nor antitone according to KT . In this case, we could


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take as primitive and define necessity by A = A ∧ ¬A, but that is entirely due to choosing KT rather than something weaker as the underlying logic. For the general case, see Humberstone [1995d], Kuhn [1995], Zolin [1999], Humberstone [2002b]. The first of these papers provides references to work going back to the 1960s in which contingency or noncontingency (symbolized by , where A and understood as A ∨ ¬A) is taken as a modal primitive, rather than necessity or possibility. More recently, the idea of taking as primitive, not “it is contingent whether A” but rather “it is contingently the case that A” (understood as A ∧ ¬A). Marcos [2005] writes this as • A and explores the expressive power of the language in which it is the sole non-boolean primitive. (ii ) Suppose we take ↔ as compositionally derived rather than (as in 3.16) primitive, with definition A ↔ B = (A → B) ∧ (B → A). For a {→, ∧}-classical , each of the connectives used in this definition is, in each of its positions, either monotone or antitone according to , while this does not hold for (either position of) the connective defined (as mentioned in 3.32.1(iv)). Exercise 3.32.3 (i) Suppose that is an ∧-classical consequence relation according to which a singulary connective # is both monotone and antitone; show that all formulas of the form #A are equivalent according to . Show that the same conclusion follows from the each of the following suppositions: that is ∨-classical; that is →-classical; that is -classical; that is ⊥-classical. (ii) Let be a gcr with binary # in its language, and satisfying the condition that A A # B for all formulas A, B, of that language. Show that if # is right-extensional in its first position according to , then # is monotone in its first position according to . (iii) With and # satisfying the conditions of the first sentence of (ii), does it follow from the further hypothesis that # is instead leftextensional according to , that # is monotone in its first position according to ? (Give a proof for an affirmative answer, a counterexample for a negative one.) (iv) Now suppose that is a gcr with binary # in its language, and satisfying the condition that A # B A for all formulas A, B, of that language. Does if follow from the further hypothesis that # is leftextensional in its first position according to , then # is monotone in its first position according to ? (Explain why, or show why not.) The above concepts can be generalized to allow ‘side formulas’, so that, a k-ary connective # is monotone with side formulas in the ith position according to a gcr iff for all formulas B1 ,. . . ,Bk , C and all sets of formulas Γ, Δ, we have: Γ, Bi C, Δ implies Γ, #(B1 ,. . . ,Bk ) #(B1 ,. . . ,Bi−1 , C, Bi+1 , . . . , Bk ), Δ, and is monotone with side formulas according to when this condition is satisfied for each i (1 i k). To obtain notions of monotony with side formulas

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according to consequence relations as opposed to gcr’s, simply delete the references to Δ. Taking the singulary case by way of illustration, # is monotone with side formulas according to a gcr when for all Γ, Δ, A, B (in the language of ), Γ, A B, Δ implies Γ, #A #B, Δ. This strengthens the condition of extensionality according to , since the antecedent asks only that Γ, A B, Δ and not also that Γ, B A, Δ. Like that condition, monoton(icit)y with side formulas can also be expressed by means of an unconditional requirement, a variation on (E); recall, for part (i) of the following exercise, that we separated its conjuncts as (RE) and (LE). Exercise 3.32.4 (i) Show that a singulary connective # in the language of a gcr is monotone with side formulas according to iff for all formulas of that language we have: (trimmed LE) B, #A #B and (trimmed RE) #A #B, A (ii ) Show that if is the least gcr in its language satisfying these two conditions, that Val () = VV ∪ VF ∪ VI , where this notation is understood as in 3.22.6, in which (trimmed LE) also put in an appearance, though not under that name. (Cf. also Humberstone [1986], p.45, where the equivalence of the two ‘trimmed’ conditions is noted for extensional .) (iii) Show that if a singulary connective # in the language of a finitary consequence relation is monotone with side formulas according to , then it satisfies the following condition: Γ, Δ A implies Γ, #Δ #A for any set Γ, any non-empty set Δ and any formula A, where #Δ = {#D | D ∈ Δ}. (Note that the analogous result for merely monotone # does not hold.) Clearly, though we give it no further attention, the property of being antitone (in a given position) can be generalized to the ‘with side formulas’ case, for consequence relations and gcr’s alike: simply interchange the “B i ” and “C” in the definition above (deleting references to Δ for the consequence relations case). Though we do not go into the matter here, the idea of monotony with side formulas has a semantic analogue in the form of a certain notion (from Post [1941]) of monotone truth-function, briefly mentioned at 4.38.7 below (p. 621). There is of course no need to introduce any separate terminology for ‘congruentiality with side formulas’, since that—see §3.2—is what extensionality is (or, for the case of consequence relations, what was called weak extensionality in 3.23). Digression. A further point to note à propos of (iii) of the above Exercise is that the condition Δ = ∅ is indeed needed. The threefold distinction between monotony, regularity, and normality familiar from modal logic (Chellas and McKinney [1975], Chellas [1980], elaborating on some terminology from Segerberg [1971]). For singulary #, if A1 ,. . . ,An B always implies #A1 , . . . , #An #B for n 0 we have that # is normal according to (as in the discussion of in 2.23); if the implication holds whenever n 1, this is put by saying that # is regular according to ; and if the implication holds for n = 1, that # is monotone according to . (This differs from our definition in the present subsection by applying only when there are finitely many formulas


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on the left.) Such terminology can be generalized in an obvious way to give position-relative concepts for k-ary #, with or without side formulas (and with or without the finiteness restriction just remarked on). Thus for example, k-ary # is regular in the ith position if when n 1, A1 ,. . . ,An B always (i.e., for all A1 , . . . , An , B, C1 , . . . , Ck ) implies #(C1 , . . . , Ci−1 , A1 , Ci+1 , . . . , Ck ), . . . , #(C1 , . . . , Ci−1 , An , Ci+1 , . . . , Ck ) #(C1 , . . . , Ci−1 , B, Ci+1 , . . . , Ck ). Similarly, k-ary # is monotone with side formulas in the ith position if (for n 0) Γ, A1 , . . . , An B always implies Γ, #(C1 , . . . , Ci−1 , A1 , Ci+1 , . . . , Ck ), . . ., #(C1 , . . . , Ci−1 , An , Ci+1 , . . . , Ck ) #(C1 , . . . , Ci−1 , B, Ci+1 , . . . , Ck ). In the terminology here introduced one can make the point of 3.32.4(iii) by saying that if k-ary # is regular with side formulas in the ith position if and only if # is monotone with side formulas in that position, according to any finitary . End of Digression. The properties of being monotone and being antitone according to a (generalized) consequence relation are each stronger than that of being congruential according to . We will now work our way toward a discussion of the relation between congruentiality and the ‘monothetic’ property introduced for 2.13.1 (p. 221). The terminology we shall use is that most naturally suggested by the Fmla framework, in which, as we have already noted, congruentiality can be taken for as satisfaction of the condition (↔-Cong) from 3.31; here we suppose that our Fmla system S is closed under tautological consequence and Uniform Substitution, and writing “S A” for “A ∈ S”, note that for S to be congruential (or more explicitly, ↔-congruential) is for the following to be the case, for arbitrary formulas A, B, and contexts C(·): (1)

S A ↔ B implies S C(A) ↔ C(B)

(The context notation “C(·)”, as in 3.23.6 – p. 457 – is from the discussion following 3.16.1, p. 424.) Now the notion of synonymy in a set of formulas S, called S-synonymy in 2.13, requires something looking rather different, namely, that for all A, B, C, we have (2)

S A ↔ B implies S C(A) iff S C(B).

But this difference is more apparent than real: Exercise 3.32.5 Show that for sets of formulas S meeting the above conditions on Fmla systems, (1) holds for all A, B, C iff (2) holds for all A, B, C. Remarks 3.32.6(i) In fact the reader will have been able to show that for a given A and B, what follows the “implies” in (1) holds for all C iff what follows the “implies” in (2) holds for all C. (ii ) For this purpose, much less stringent conditions on S than were imposed above suffice: namely, that the language of S has some binary connective # with the property that for all A, B: (a) If A # B ∈ S then A ∈ S ⇔ B ∈ S, and (b) A # A ∈ S.

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We recall from 2.13 that a logic S in Fmla S is monothetic when for all A, B, C: (3) S A and S B imply S C(A) iff S C(B). Note that, unlike our ↔-based notion of congruentiality for Fmla, (3) makes no reference to the presence of, or prescribed logical behaviour on the part of, any specific connective such as ↔. However, one way of seeing that a logic in Fmla is monothetic is to find a connective # in its language, for which (2) above, with # for ↔, is satisfied, and for which S A and S B jointly imply S A# B. Since S A# B in turn implies (by the analogue of (2)) that A and B are S-synonymous, this conclusion follows from the assumption that each belongs to S. We can use ↔ as # for classical logic in Fmla as well as for intuitionistic logic in Fmla, its different logical behaviour in the latter logic notwithstanding. Clearly there is no way of choosing such a # for the logic R of relevant implication (2.33), since, as we saw when perusing its implicational fragment, BCI logic, in 2.13, this logic is not monothetic. (See the discussion following 2.13.25.) If we move outside the framework Fmla, into Set-Fmla and Set-Set, we still have the idea of a theorem (or ‘thesis’), as a formula A for which A belongs to the logic, and so we can make sense of a monothetic logic in one of these richer frameworks as again a logic in which any two theorems are synonymous: but the latter now requires interreplaceability in arbitrary sequents salva provabilitate. (In other words, if A and B and also Γ Δ are in the logic—with Δ of the form {D} for the Set-Fmla case—then so is Γ Δ , where Γ and Δ differ from Γ and Δ in having some occurrences of A, in subformulas of elements of these sets, replaced by B.) When a Set-Fmla or Set-Set logic is closed under the familiar structural rules, we transfer, as usual, the “monothetic” terminology to apply to the associated consequence relation or gcr. Indeed, in our discussion of extensionality, we had to notice the resemblance between monotheticity and the latter property (in the preamble to 3.23.9, p. 458); for left-extensional consequence relations we have, in particular (3.23.6, p. 457) the following for all contexts C(·): (4) A, B, C(A) C(B) which should be compared with a reformulation of (3), above, expressing monotheticity: (5) A, B, and C(A) imply C(B). Although we have written it across rather than (rule-style) down the page, (5) is clearly a ‘vertical’ form of the ‘horizontal’ (3)—cf. 1.25. (Alternatively put: (3) and (5) are respectively metalinguistic Horn sentences of what in 1.12 we called the first and second type). Our purpose in bringing up monotheticity here, however, is to draw attention to a simple connection between this property and congruentiality: Observation 3.32.7 Any congruential (generalized) consequence relation is monothetic. Proof. Let be a congruential (generalized) consequence relation, and suppose that A and B. Then by (M), A B and B A, so by congruentiality, A and B are synonymous according to , making monothetic.


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Our discussion of the non-monothetic BCI logic (2.13, 7.25) does not conflict with 3.32.7, despite the logic’s being congruential in approximately the sense of ↔-congruential (A and B being synonymous whenever A → B and B → A are both provable). This is because the provability of B does not imply the provability of A → B (or A, that of B → A, so that the analogue of the appeal to (M) in the above proof is not available.

3.33

The Three Properties Compared

In this chapter we have noted the following concepts and their deductive relations, as applied to an arbitrary connective according to a (generalized) consequence relation: fully determined ⇒ extensional ⇒ congruential Neither of the indicated implications can be reversed: each concept, taken as applying to the treatment of a given connective according to a given gcr is stronger (in general) than that appearing to its right. We have already seen examples of the irreversibility of the first “⇒”; congruentiality without extensionality is illustrated for example by the case of as it behaves according to KT (or S4 , or . . . ). To get a better feel for the chain of implications here, we have also been considering the beginnings of a parallel chain, whose links are semantic: concepts defined in terms of classes of valuations and functions. (Of course, might itself have been presented in such semantic terms, as Log(V ) for some class V of valuations; but it might not, and in any case the concepts of full determination, extensionality, and congruentiality apply to connectives paired with gcr’s solely in virtue of how those gcr’s relate various formulas. In this sense we can see them as ‘syntactic’ concepts even when has not itself been defined in terms of a proof system as in 1.21.) This parallel chain begins with the implication: truth-functional ⇒ pseudo-truth-functional. These are semantic analogues of the first two terms of the above chain. To complete it, we need a semantic analogue of congruentiality. We can bring congruentiality into the fold by a slightly more complex definition than those of truth-functionality and pseudo-truth-functionality. Where V is a class of valuations for a language containing an k-ary connective #, define, for any formula A, the truth-set in V of A, denoted HV (A), by: HV (A) = {v ∈ V | v(A) = T}, and put: H = {V ⊆ V | ∃A V = HV (A)}. (The “H” notation here is adapted from Beth [1959], esp. Chapter 19. No doubt we should strictly write something like “HV ” rather than just “H” as above; but on the few occasions on which this notation is used, the relevant class of valuations will be clear from the context.) Remark 3.33.1 Note that we do not here have quite the same notion of ‘truth set’ as in 2.21, where the truth set of a formula, relative to a model, was the set of points in the model verifying that formula. Although for each such point x, there is a valuation vx assigning T to precisely the formulas true at x in the model, we allow vx = vy even when x = y. Now say that a k-ary connective # is truth-set functional w.r.t. V if there is a function g:Hk −→ H such that for all B1 , . . . , Bk :

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CHAPTER 3. TRUTH-FUNCTIONAL, EXTENSIONAL, . . . HV (#(B1 , . . . , Bk )) = g(HV (B1 ), . . . , HV (Bk ))

We can now verify that pseudo-truth-functionality (and therefore also truthfunctionality) w.r.t. V implies truth-set functionality w.r.t. V . As we saw in 3.22, if # is pseudo-truth-functional w.r.t. V then every valuation v ∈ V associates with # a truth-function fv . In terms of these fv we can define a truth-set function g associated with # as in the above definition of truth-set functionality by setting: g(HV (B1 ), . . . , HV (Bk )) = {v ∈ V | fv (v(B1 ), . . . , v(Bk )) = T} Thus we have the chain of implications, for a given # w.r.t. a given V : truth-functional ⇒ pseudo-truth-functional ⇒ truth-set functional Our two chains, syntactic and semantic, are link-for-link equivalent, as we saw for the first two links in §3.1 and §3.2, and as we show for the third in part (iii) of the following summary of the three relativizations: Theorem 3.33.2 For any gcr and any connective # in the language of : (i) # is fully determined according to if and only if # is truth-functional w.r.t. Val(); (ii) # is extensional according to if and only if # is pseudo-truth-functional w.r.t. Val(); (iii) # is congruential according to if and only if # is truth-set functional w.r.t. Val(). Proof. (i): This is 3.11.5 (p. 379). (ii ): This is 3.22.5(i) (p. 451). (iii): Note that A B iff HV al( ) (A) ⊆ HV al( ) (B), so A B iff HV al( ) (A) = HV al( ) (B). Supposing for illustration that # is singulary, and further that # is truth-set functional w.r.t. Val () and that the function in question is g: H −→ H, we can argue that if A B then HV al( ) (A) = HV al( ) (B), so g(HV al( ) (A)) = g(HV al( ) (B)). Therefore HV al( ) (#A) = HV al( ) (#B), so finally, #A #B. Conversely, suppose that # is congruential according to , and define g:H −→ H by g(HV al( ) (C)) = HV al( ) (#C) for all formulas C. Now, g is well defined by this stipulation since if HV al( ) (C) = HV al( ) (C ) then we have HV al( ) (#C) = HV al( ) (#C ), by congruentiality, and clearly it associates a truth-set function with # over Val (). As we saw in 3.13.8 (p. 391), 3.13.9, 3.22.5(ii), we can rephrase all parts of 3.33.2 as: Theorem 3.33.3 For any connective # in a language L and any class V of valuations for L: (i) # is fully determined according to Log(V ) if and only if # is truth-functional w.r.t. V


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(ii ) # is extensional according to Log(V ) if and only if # is pseudo-truthfunctional w.r.t. V (iii ) # is congruential according to Log(V ) if and only if # is truth-set functional w.r.t. V. Log here is to be taken as the gcr determined by V ; for consequence relations, as we have seen, parts (i) and (ii) of 3.33.2 and 3.33.3 fail; since congruentiality is a condition whose upshot is the same for gcr’s and consequence relations – there being no involvement with empty or multiple right-hand sides – part (iii) remains intact here, and precisely the above proof of 3.33.2(iii) establishes: Theorem 3.33.4 For any consequence relation and any connective # in the language of , # is congruential according to if and only if # is truth-set functional w.r.t. Val(). Exercise 3.33.5 Show that 3.33.3(iii) is also correct when Log(V ) is interpreted as the consequence relation determined by V . The semantical treatment of connectives about which no more is assumed than congruentiality (according to this or that consequence relation or gcr) is quite interesting in its own right, though no more will be said about it here. A more highly structured treatment is provided by the ‘referential semantics’ of Wójcicki [1988], Chapter 5. (See also further Wójcicki [2003].) That chapter also includes (in §5.4) a discussion of the neighbourhood semantics for modal logic, which is very much to the point here – though preferably generalized so as to apply to connectives of greater arity than the usual 1-ary . This topic is also covered in Chellas [1980], where the name “minimal models” is used for neighbourhood models.

3.34

Operations vs. Relations

The connectives are operations which turn formulas into formulas (1.11). We often want to describe them using terminology which does indeed apply to operations, but not in quite the way suggested by this description, which alludes to the algebra of formulas (of a language). For instance, people say that conjunction (∧) is commutative. Now it is not literally true that the formulas A ∧ B and B ∧ A are the same, except in the special case in which A is the same formula as B. This follows from the conception of languages as absolutely free algebras of their similarity types. Clearly the intention is to evoke another algebra, namely the Lindenbaum algebra – or, as we usually prefer to say, the Tarski–Lindenbaum algebra – of formulas (2.13.1, p. 221) associated with the language on specification of a particular logic. In 2.13.1 we had in mind specifically logics in Fmla, but the notion of synonymy there used to collect formulas into equivalence classes has more general application. If is a (generalized) consequence relation, we can denote by ≡ the relation of synonymy according to . The equivalence class of a formula A w.r.t. this relation is again denoted by [A]. As in our discussion in §2.1, we can transfer the operations of the algebra of formulas to this algebra— call it A—by setting, for k-ary #: (*) #A ([A1 ],. . . ,[Ak ]) = [#(A1 ,. . . ,Ak )]

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which makes good sense because ≡ is guaranteed to be a congruence relation on the algebra of formulas. In fact, this is the etymology of the term “congruential”, since for a congruential the relation holding between formulas A and B when A B is the relation ≡ . Remark 3.34.1 At least one early advocate of ‘algebraic logic’ (Halmos [1956]) proposed that logics be identified with congruence relations on the algebra of formulas. This is too extreme: it would for example, identify any gcr with the gcr −1 (see 1.19.4, p. 93; cf. further Humberstone [2005a], including Proposition 3.14 there, for a more general version of this point and the end of note 9 for the point just made). Now we can explain what someone claiming ∧ to be commutative has in mind by saying that if is a congruential ∧-classical (generalized) consequence relation, and A the quotient algebra of ≡ -equivalence classes, then the operation ∧A (conjunction as an operation on the ‘propositions’ distinguished by ) is commutative: [A] ∧A [B] = [B] ∧A [A], for any formulas A, B. This is of course the case, since A ∧ B and B ∧ A are synonymous according to any such , and so [A ∧ B] = [B ∧ A]; indeed we used facts of this type to connect up classical sentential logic in Fmla with boolean algebras in 2.13, and the Fmla-Fmla logic Lat with lattices in 2.14. If the above restriction to congruential ∧-classical (generalized) consequence relations strikes the reader as excessive in an explication of what is meant by calling conjunction commutative – and this is really a matter of taste (or terminological preference) – then we can consider instead a different equivalence relation, namely that holding between A and B when A B, again denoting A’s equivalence class (under this new relation) by [A], and note that [A] ∧A [B] = [B] ∧A [A], provided that is ∧-classical. Here we rely on the fact that, even if is not congruential, this proviso secures that ∧ is congruential according to . However, the presence of other non-congruential connectives renders the “A” superscript somewhat problematic, in that we cannot make a similar definition in the style of (*) above for them, with this new understanding of “[ · ]”. Accordingly, when the possibility of non-congruential connectives is being considered, we might drop the strategy of seeing talk of commutativity for ∧ as literally attributing this property to some (related) operation, and as simply the claim that A ∧ B B ∧ A for all A, B, and some (understood) . (There is of course no weakening involved if we replace “” by “” in this instance.) For the remainder of this discussion, we will simply forget about non-congruentiality, so that either the explication of terms such as commutative given in this paragraph or that of the preceding paragraph can be taken as adequate. Similarly, for the remainder of the present subsection, we take (CL , etc.) to be a consequence relation rather than a gcr. Here we further illustrate this use of terminology for binary operations: Examples 3.34.2(i) According to CL , ∧ and ∨ are commutative, associative and idempotent, and ↔ is commutative and associative. (Note that the provability of A#A, as with # = ↔, has nothing to do with idempotence, which would require instead the provable equivalence of A # A


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and A, for arbitrary A. This mistake is occasionally found in print, such as at p. 144 of Brown [2001].) (ii ) All the above claims go through for IL in place of CL , except for the claim of associativity for ↔. (See 7.31.13(i), p. 1136.) The terminology of 3.34.2 is very common in the literature, and as we have seen, it is perfectly reasonable to apply terms for properties of binary operations to binary connectives, relative to logics, since we can understand this terminology derivatively via other binary operations (∧A etc.) to which it applies quite literally. This brings us to the topic raised in the heading of the present subsection: operations vs. relations. For, still sticking with the case of binary connectives, there is also evident in the literature a potentially confusing—if not actually confused—tendency to use terminology in describing such connectives which is appropriate instead to binary relations, rather than binary operations. Thus, instead of saying that conjunction and disjunction (according to suitable ) are commutative, some writers will say that conjunction and disjunction are symmetric. (See the notes to this section, which begin on p. 508, for further examples and references.) Similarly, material implication will be described as transitive. Sometimes, one and the same text will describe the latter as transitive and the former pair as commutative. In the case of Quine [1951], the very same connective (conjunction) is described as transitive (p. 58) and associative (p. 59). Similarly, at p. 385 of Restall [1993a], one and the same thing (a binary operation corresponding to a kind of conjunction) is described as “idempotent, symmetric, and associative”. How does it come about that descriptions one would have thought applicable to entities of disjoint classes – binary relations and binary operations – are being applied to the same thing: a binary connective? We shall try and say something to reduce the potential for confusion by finding for binary connectives some associated binary relations which have the corresponding property of relations (e.g., symmetry) when the connective concerned has the binary operational property (commutativity, in this case). Although, as we shall see, it is possible to justify the use of the (binary) relational terminology, it still seems to be a mark of unclarity to mix the operational and relational vocabulary together in the same breath, as in the above examples (and those given in the notes). Digression. The reverse of the error complained of here, namely that of applying terminology appropriate to binary operations instead to binary operations can also be found in the literature. For instance Church [1956], p. 281, lists alongside a Reflexive Law and a Transitive Law for equality, a “Commutative Law”. Given Church’s interests, on can imagine a defence being mounted on Fregean lines: a binary relation is really a ternary operation, in the case of equality, taking a pair of individuals to a truth-value. The usage hardly seems conducive to clarity, nonetheless. End of Digression. First, we should note that our heading “operations versus relations” is itself somewhat misleading, since any n-ary operation can be regarded as a relation, namely an (n+1)-ary functional relation. So what is potentially confusing, more precisely, is that we have n-ary operations (for n = 2 in the cases on which we have focussed) being spoken of as though they were, not (n + 1)-ary relations, but n-ary relations. To dispel the mystery, we examine first what might be called a ‘local’, and then a ‘global’ way of justifying such transference. In each

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case, there is a semantic and a proof-theoretic way of proceeding. We begin with the local case, thinking about it in semantic terms, where locality amounts to dependence on and (globality to independence of) particular valuations. If, over some class of valuations in which we are interested, some truthfunction is associated (3.11, p. 376) with a connective, then it is natural to transfer terminology applicable to the former so that it becomes derivatively applicable to the latter. This is certainly what some have had in mind in saying that conjunction is a commutative connective: namely, that the truthfunction associated with ∧ – which we refer to when convenient as ∧b – over the class of boolean valuations (or indeed any class of ∧-boolean valuations) is a commutative binary operation. Here the problem with which we are concerned does not arise, since the ‘binary relations’ terminology has not crept in. It remains to justify describing, e.g., the binary operation →b associated with → on →-boolean valuations, as transitive. This manifestly illustrates the problematic feature of describing (binary) operations in (binary) relational terms, though now, at the level of truth-functions rather than of connectives. The solution is to note that whenever we have a function whose values are truth-values (even if their arguments are not—though in the cases which concern us here, arguments and values are all truth-values), we can regard that function as the characteristic function of a set. For instance, if we have a function from animals to truth-values, we have the characteristic function of a set of animals, and can recover the set as comprising those animals mapped to T by the function, or, given the set, define the function as that mapping its members to T and its non-members to F. Similarly, if we have a function which maps (ordered) pairs of animals to truth-values, we have something interchangeable with a set of pairs of animals, i.e., a binary relation among animals. Finally then, if, as in the case of binary truth-functions, we have a function from pairs of truth-values to truth-values, we have on our hands the characteristic function of a binary relation between truth-values. Now this relation, in the case of the truth-function →b is indeed, quite literally, a transitive relation (as the reader should check). Hence the naturalness of describing the binary operation →b with the aid of a piece of terminology defined for binary relations. Similarly, the binary relation of which a commutative binary truth-function is the characteristic function is indeed a symmetric relation. Thus our original problem is resolved by transferring this terminology one step further, so that it applies to # rather than #b (to take the special case in which BV is the class of valuations in which so special an interest is taken that reference to it comes to be suppressed). Remarks 3.34.3(i) The preceding discussion shows that we may regard an nary truth-function as (in effect) an n-ary relation amongst truth values, not that we must; the fact that such a reconstrual is possible does not show that there is anything wrong with the original description of it as an n-ary truth-function. (Pace Welding [1976], esp. p. 157f.; see further Simons [1982].) (ii ) A related qualm is raised in Medlin [1964], p. 184f.: “The essential step in developing the theory of truth-functions is to regard sentences as names. There is a general objection to doing this: that if sentences are names it is impossible to assert anything.” Medlin goes on to weaken the second comment here, and we need not evaluate it. But the first


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should not be allowed to pass: all we need is that sentences (possibly relative to certain contextual parameters) have truth-values, not that they name their truth-values. The development in 3.11 of the theory of truth-functions for the formal languages of sentential logic made use of the notion v(A) for formulas A: no special assumption was required to the effect that v(A) assigned to A something standing as the bearer of a name to that name. (See Medlin’s paper for references to Frege’s writings, including Frege [1892], urging the correctness of the latter analogy.) Similarly, calling the truth-value (on v) of A the extension of A (relative to v), alluded to in 3.21 (p. 444), need not be construed as building in more than is wanted of the name–bearer relation. (iii) We warn the reader that aside from the usage – transferable to a binary connective if desired – of “idempotent” we take for granted, according to which such a function f is idempotent just in case f (x, x) = x (for x ∈ {T, F}), there is a different usage to be found in several of Menger’s works cited in the notes to §3.1 (p. 443) according to which f is idempotent just in case f (f, f ) = f , which is Menger’s shorthand for: f (f (x, y), f (x, y)) = f (x, y), for all x, y ∈ {T, F}. The latter condition is weaker than idempotence as we have been understanding it. Menger’s terminology is related to another usage of the term idempotent, applying to one-place functions g when g ◦ g = g (where ◦ symbolizes function composition). (iv ) The latter use of the term idempotent is more general that the example of functional composition might suggest, since in any semigroup (indeed any groupoid), even if it is not idempotent in the sense that the result of ‘multiplying’ any element by itself gives that element again (as in 0.13), there may be particular elements with this property, in which case they are called idempotent elements. Now if we consider the semigroup of binary relations on a set under composition (relative product), this results in applying the term “idempotent” to a binary relation R for which R ◦ R = R (or R2 = R, as this is often put). These are the relations which are both transitive (R2 ⊆ R) and dense (in the sense of 2.21.8(vi), p. 281, which amounts to: R ⊆ R2 ). In this case, unlike those reviewed earlier, the application to binary relations of terminology appropriate to binary operations of course reflects no trace of confusion. Similarly, in discussing relational structures with a single binary relation (such as the frames for monomodal logic) there is no confusion involved in transferring the term “reflexive” (or “irreflexive”) from relations to elements of the domain bearing the relation in question (resp., not bearing it) to themselves. Can the same be said of the terminology used in Anderson [1997] (e.g., p. 418), where alongside the usual terms associative and commutative for properties of binary operations, idempotence is referred to as reflexivity? The above semantic route to the justification—or, at least, explanation—of the use of terminology apt for n-ary relations to describe certain n-ary operations can be put in more ‘syntactic’ or proof-theoretic terms by speaking of the consequence relation, or the set of sequents, determined by the class of valuations one had in mind. Similarly, such a semantic perspective may be arrived at

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by starting with a proof-theoretically presented and considering Val (). For instance, if for some such we have (for all A, B, C, and a given binary #): A # B, B # C A # C then if a truth-function is associated with # over V , that truth-function may be described as ‘transitive’ when attention is restricted to V ∩ Val(). (We return to this in 3.34.7 below.) Indeed, we may drop the reference to truthfunctions altogether here, as well as the (explicit) reference to truth-values. Given a consequence relation with binary # in its language, for any -theory # to relate formulas A, B, (of that language) just Θ, define the binary relation RΘ in case A # B ∈ Θ. Then the -condition above can be paraphrased by saying # is transitive. (Recall that the here is that for any -theory Θ, the relation RΘ to be a consequence relation. If one starts instead with a gcr, one may take the induced consequence relation – i.e., that consequence relation agreeing with in the sense explained after 3.13.9 on p. 391 above.) This perhaps justifies talk of transitivity, even though it is not a single binary relation whose transitivity # is at issue – since we have to consider the relations RΘ for varying Θ. For a concrete illustration: Example 3.34.4 Take as K (for instance, as introduced after 2.23.3: this is the inferential or truth-local consequence relation associated with K), and # as strict implication; then we have the above pattern: A B, B C K A C. (Recall that “A B” means “(A → B)”, etc.) But of course the truthvalues of formulas A B) etc., at a point in a Kripke model are not fixed by the truth-values of A, B, at that point. (Syntactic formulation: strict implication is not fully determined according to K .) And, as noted, is transitive for each K -theory Θ. this amounts to the claim that RΘ In describing the above transferences of relational terminology to operations as ‘local’, the intention is to recall the local/global contrast of 1.25; the term horizontal, from 1.26, would be equally appropriate here. On the global – or ‘vertical’ – side of the picture, there is the binary relation R # , for a given binary connective # in the language of the consequence relation , defined to relate formulas A and B (in that order) just in case A # B. (If we are thinking of a logic in Fmla, take as any consequence relation agreeing with the set # # above, R # is RΘ for of formulas concerned.) In terms of the relations RΘ Θ = Cn (∅). The transitivity of this relation amounts not to the horizontal -condition above but to its vertical companion, here written in rule notation as a reminder of the reason for the horizontal/vertical terminology here, though we are still using “” rather than the “” that would be more appropriate: A#B

A#C

A#C So this gives a second way of applying relational terminology to connectives. By contrast with the first (‘local’) way, now in saying that # is, for example, transitive or reflexive (according to ) one means that the associated relation R # is – this time a single binary relation – quite literally transitive (or reflexive,


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symmetric,. . . ). In practice, there is a tendency to use this device more freely with some connectives than others. For example, when # is →, it is natural to speak of (logical) implication, meaning the relation R # ; but while people are happy to speak of A’s implying B (a specific being understood), though is no equally natural tendency to speak of A’s ‘disjoining’ B when A ∨ B. (Our way of putting this – e.g., in 8.11 below, also indirectly in 3.18 above – is to say that A and B are subcontraries according to . This is the account of subcontrariety, with a similar treatment of other ‘logical relations’, at p. 69f. of Lemmon [1965a].) In fact the rule called ‘Transitivity’ or ‘(RTRans)’ in 1.29.2 (p. 158) concerns the this vertical condition, the transitivity of the relation R →CL ; by contrast the discussion of Transitivity in connexion with indicative or subjunctive conditionals in 7.15 (in the Digression on p. 992) and 7.18 respectively pertains to the horizontal form considered earlier. (For this horizontal form, when the connective in question is interpreted conditionally, the horizontal form of inference is traditionally known as Hypothetical Syllogism. Of course, the horizontal/vertical contrast is easily overlooked as a result of focussing exclusively on CL , in view of the structural completeness of this consequence relation – 1.29.6(i), p. 162.) Semantically, when is congruential, a description in terms of the truth set functions of 3.33 may be given, as in the local case touched on above in terms of truth-functions, but we leave the interested reader to provide this. Note that for # to be, e.g., symmetric in the present sense is for A # B to imply (and be implied by – though this addition is redundant) B # A, for all A and B, which is weaker than the earlier ‘local’ condition, namely that we should have A # B B # A for all A, B. (Commutativity as a rule of proof vs. commutativity as a rule of inference, we might say.) Remark 3.34.5 With Example 3.34.4 we noted that local versions of symmetry, transitivity, etc., are not fully recovered by looking at the binary relations derived by the inverse of the characteristic function construction from appropriate truth-functions, since there may be no such truth-functions. However, we can still approach such cases relationally, adapting the above definition of the relations Rv# . Consider the binary # relations RM,x defined, for a model (as in §2.2 for the current application) M, a point x therein, and a binary connective # (not necessarily primitive) thus: # B iff M |=x A # B. A RM,x

For # as , 3.34.4 may be summarized in semantic terms by saying that # for any M = (W, R, V ) with x ∈ W , the relation RM,x is transitive. What a model and a point therein give rise to of course is a particular valuation – see the valuations vxM from 2.21 (p. 279) – so the more general way of describing what is going on in the local case is as defining a binary relation Rv# for any binary connective # (of some language) and valuation v (for that language); for all formulas A, B, we define Rv# (A, B) iff v (A # B) = T. (Here and for the remainder of this subsection, we use the notation “Rv# (A, B)” rather than the infix notation “A Rv# B” to make the super-

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CHAPTER 3. TRUTH-FUNCTIONAL, EXTENSIONAL, . . . and subscripts more readily visible.) We can then say that, if v is ∧boolean, Rv∧ is a symmetric relation, and so on. Evidently, the fully general definition of such relations should allow n-ary #, saying that when v is a valuation for a language with such a #, we have the relation Rv# defined by: Rv# (A1 , . . . , An ) iff v (#(A1 ,. . . ,An )) = T. # theme, for suitably chosen background This is a variant on the earlier RΘ , for which Θ is a -theory comprising the formulas true on v. (In the notation of p. 775 below, Θ = T r(v).)

Exercise 3.34.6 (i) As already remarked (with # as ∧), instead of defining # to be commutative according to , as we did earlier, when we have for all A and B: A # B B #A, we could equally well have used just the direction here. This does not mean that for a given , and for a particular choice of A and B in the language of , if A # B B #A then A # B B #A. Give a counterexample to the latter implication (specifying a particular , A, B). (ii ) Returning to the universally quantified formulation as in the first sentence of (i), is it true that if satisfies the schema A # (B # C) (A # B) # C then # is associative according to , in the sense that A # (B # C) (A # B) # C for all A, B, C? Give a proof or else a counterexample. (iii) How is the answer to (ii ) affected if we add the additional assumption that # is also commutative according to ? Exercise 3.34.7 (i) Let us return to the earlier (horizontal) example of the ‘transitivity’ of binary # according to a consequence relation : A # B, B # C A # C. Taking as CL and # as any connective definable in the language of CL, show that the above condition is satisfied (for all A, B, C) whenever the following ‘reflexivity’ condition is satisfied (for all A): A # A. (Note that because reflexivity is an unconditional condition, the vertical/horizontal distinction is nullified.) (ii) Would the same be true if we took in (i) as IL and # as any connective definable in terms of the intuitionistic primitives →, ∧, ∨, and ¬ (or ⊥)? (Justify your answer.) The relations R # can be given a description in terms of (sets of) valuations. If V is a set of valuations for the language of , we can define the binary relation RV# by stipulating that for all formulas A, B, of that language: RV# (A, B) for all v ∈ V , Rv# (A, B).


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Then R # = RV# whenever = Log(V ) (for example, when V = Val()). In this notation, Lemmon’s proposed treatment – alluded to above (p. 503) – of the traditional logical relations of contrariety, subcontrariety, contradiction and # equivalence, for example, identifies them with the relations RBV obtained by taking A # B as respectively ¬(A ∧ B), A ∨ B, A B and A ↔ B. Whether described in ‘global valuational’ terms (with the V subscript as just defined), or more syntactically in terms of vertical transitions (subscripted ), it is clear that these binary logical relations need to be distinguished from their valuationally local or ‘horizontal’ analogues. We have been concentrating in the contrast in respect of transitivity of the relations of the two kinds, but here we give an example involving symmetry. Example 3.34.8 Let us consider the derived binary connective # of the language of propositional modal logic (§2.2) defined by setting, for all formulas A, B: A # B = A ∧ B, and its behaviour according to the (‘inferential’) consequence relation KT , mentioned on p. 294. (The corresponding gcr, similarly notated, appeared in 3.21.4 on p. 448; this notation for the current consequence relation appeared most recently at the start of 3.33, and before that in 3.23.2(i) on p. 490.) Note the following contrast: (1) For all A, B: KT A # B ⇒ KT B # A; (2) It is not the case that for all A, B, we have A # B KT B # A. For (1) we use the fact that since B KT B, we have KT A # B ⇒ KT B, and the fact that the consequences by KT of the empty set (alias the theorems of the Fmla logic KT) are closed under necessitation, so that KT A # B ⇒ KT A, as well as the ∧-classicality of KT to draw the conclusion (1) from these facts. (For the first of these facts, the weaker property that KT B implies KT B, so in virtue of 6.42.2 – on Denecessitation – below (p. 873), we could actually have used K to make points (1) and (2), instead of KT .) Rephrasing (1) in relational terms: the relation R #KT is symmetric. Alternatively, where VKT is the set of valuations v of the form vxM for some Kripke model M with a reflexive accessibility relation and x among its elements: the relation RV#KT is symmetric. For (2), we use the local valuational reformulation: not all the relations Rv# for v ∈ VKT are symmetric. Probably no-one would described the connective # as commutative (according to KT ) on the basis of (1), since this would too strongly suggest, at least for congruential consequence relations such as KT (by contrast with the corresponding model-consequence relations, for example, which are not congruential), that A # B and B # A should be freely interchangeable in -statements (as (2) shows them not to be here). The valuationally local and global relations contrast in respect of a property of binary relations to be discussed in 5.33 below (p. 729), where R ⊆ S × T is defined to be and -representable just in case there exist S0 ⊆ S, T0 ⊆ T such that for all s ∈ S, t ∈ T : Rst if and only if s ∈ S0 and t ∈ T0 .

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(In fact the terminology there used is “∧-representable”, and we write sRt in the definition, rather than Rst, as here.) With this concept in mind, it is interesting to consider the following remark from Kennedy [1985], p. 91, in which Lemmon’s # proposed 16 logical relations – the relations RBV for all possible choices of binary # for which there is a truth-function #b – is under critical discussion. The R in this quotation refers to the case in which #b is the second projection function (proj22 ): But there is not much to be said for R’s being a relation with immediate deductive utility. For one thing, R’s status as a relation of any sort is marginal: the conditions necessary and sufficient for R’s holding between two items are statable entirely in terms of ‘internal’ properties of just one of those items.

One might extend this line of thought to come to view as not genuinely relational any and -representable binary relation, in which account may be taken not only of one but of both relata but again only in respect of what Kennedy here calls internal properties – properties describable without reference to the other relatum: in the above definition, membership in the sets S0 , T0 . (Actually this consideration would lead to the broader class of what in 5.33 are called monadically representable relations, but we do not go into that here.) Which of # mentioned above are and -representable and to that extent the 16 relations RBV only spuriously relational? The answer is given by the following six choices of #, rendering A # B CL-equivalent respectively to: (1) A ∧ B, (6) B,

(2) A ∧ ¬B, (7) ¬A,

(3) ¬A ∧ B, (8) ¬B,

(4) ¬A ∧ ¬B, (9) ⊥,

(5) A, (10) .

To see how this is so with a few examples, let L be the set of all formulas and Taut the set of tautologies (BV -valid formulas), with Taut the set of formulas whose negations are tautologies. Then we have (for all A, B): # (A, B) iff A ∈ Taut & B ∈ Taut, RBV

when # is understood in accordance with (2) on the above list. In this case, the S0 and T0 from the definition of and -representability are Taut and Taut respectively (while S = T = L). Similarly, if # is understood in accordance with (7) or (9) respectively, we have the following attesting to and -representability: # RBV (A, B) iff A ∈ Taut & B ∈ L, # (A, B) iff A ∈ ∅ & B ∈ ∅, RBV

though either of the &-conjuncts in this last case could be replaced by anything else, since the other already guarantees that the condition cannot be satisfied. The reader is left to verify that the other cases on the list of 10 given above can be dealt with similarly, and that the list is exhaustive. Remark 3.34.9 The remaining 6 binary truth-functions are those whose corre# sponding relations RBV Kennedy [1985] regards as embodying the traditional core of logical relations, which he is keen to single out as worthy of special consideration among the 16 which Lemmon suggests are all equally significant bona fide relations. However, Kennedy’s route to this


507

conclusion is somewhat different, and embodies the erroneous claim – p. 91 – for the surviving six relations that when R is one of them, “if A bears R to B then an ideal logician who knows the truth-value of A will be able to come to know, by deduction, the truth-value of B, or vice ∨ (A, B) and the truth-value, on some given versa”. For example if RBV valuation, of A is T, there is no way an ideal logician can deduce from this what B’s truth-value is – or vice versa, supposing the truth-value of # relations for the surviving B to be T. Nor is it at all clear that the RBV 6 options for # correspond to the traditional logical relation, since these are usually characterized so as to be mutually exclusive for a given A and B, by means of an additional negative condition. Thus contrariety, relative to , would correspond not to its merely being the case that # ¬(A, ∧ B) (which is what the appropriate RBV delivers for the case of =CL ), but to this alongside the additional requirement that A ∨ B, and similarly for some of the other cases. (Several alternative formulations are available for making this point: 8.11.1 on p. 1165 below puts it slightly differently. See also 6.31.10–11, p. 848.) Although not bearing on this dispute between Kennedy and Lemmon, the question of which boolean # give valuationally local relations Rv# are, and which are not, and -representable for all boolean v is also worth pondering, but we will not do so here – especially as the question comes up later as Exercise 5.31.3(iv) (p. 710). For a more interesting contrast, in any case, one should compare the analogous notion of or -representability – ∨-representability, to use the language of 5.33 – as it applies on the one hand to the global and on the other to the local relations. There is one residual issue it is worth touching on (even if we cannot resolve it fully) before we leave the matter of relations vs. operations. The binary operation account of, e.g., ∧ has been taken as fundamental here, since it is this account which fits with the ‘languages as algebras’ treatment of 1.11 (p. 47). Yet this means that our symbol “∧” is one which takes two names to make a name, whereas “and”, of which this symbol is supposed to be a formal representative, takes two sentences to make a sentence. In terms of categorial grammar (5.11.2, p. 636), this makes the former of category N/N, N and the latter of category S/S, S. Thus it may seem that all of our efforts to explain the application of n-ary relational terminology to n-ary operations still fail to make contact with the familiar connectives of interpreted languages, since these are not themselves symbols for operations. We should recall from 1.11 the rapprochement with these ‘familiar connectives’ afforded by the view of languages as – specifically – (absolutely) free algebras. This allows, as we noted there, a ‘concrete’ construal of languages, the conjunction of A with B (to stick with our example) being formed by interposing a symbol between A and B. It is this interposed symbol which is of category S/S, S; what “∧” stands for in this case is the operation which maps the pair A, B, to the resulting concatenation of symbols, and so it is itself of category N/N, N. However, since the interposed connectives-as-symbols and the connectives-as-syntactic-operations are in a one-to-one correspondence, it is only a matter of routine rephrasing to turn a comment about the one into a comment about the other. For a discussion of the excessive fuss that is sometimes made about this and related distinctions, see Belnap [1975]. Whether or not one thinks the fuss in the present case is excessive, the contrast would

508


remain between expressions of types S/S, S and N/N, N on the one hand, and those of type S/N, N on the other (to stick with the binary case for illustration), in respect of whether the category (or type) of the compound is the same as that of the components, or different.

Notes and References for §3.3 Terminology. The term “congruential” is due to D. Makinson; Wójcicki [1988] uses “self-extensional” (though actually writing “selfextensional” – a practice which has since spread somewhat) for this concept, and gives a matrix-oriented discussion of its application. (In Wójcicki [2003] the hyphen appears in several places internally, though it is missing from the title.) The term “classical” was used for this property in Segerberg [1971], though Segerberg [1982] uses “congruential” (reserving “classical” for a different property). A different use of “congruential” was mentioned in the notes to §3.2 (p. 483). A useful hierarchy of properties of modal logics, including congruentiality (under the name “classicality”) and normality, and some intermediate properties, may be found in Chellas and McKinney [1975]. The Operational vs. Relational Vocabulary Issue. In 3.34, it was claimed that several logic texts use terminology appropriate to binary relations while discussing some binary connectives, perhaps at the same time using terminology apt for binary operations while discussing others. Here we give examples of such confusion-inviting (if not confusion-displaying) practices, aside from Quine [1951] and Restall [1993a], mentioned in the text. Fitch [1952] describes strict equivalence (which he calls strict coimplication) as reflexive, symmetric and transitive (p. 79), having referred earlier (p. 33) to the ‘commutative law for conjunction’; the latter, according to Fitch, may also be called the ‘rule of symmetry of conjunction’. Similarly, Leblanc [1955] describes material implication and the material biconditional as transitive, but conjunction and disjunction as commutative (p. 17); in a later footnote (p. 188) appended to a discussion of properties of binary relations, he says that the word “commutative” is sometimes used in place of the word “symmetric(al)”, and in a discussion of the relation of identity includes a parenthetical “or commutativity” to the description “the law of symmetry”. McCawley [1993], p. 87, says that it follows from certain observations that the English expression if and only if is asymmetric, meaning that is it non-commutative – so over and above the confusing application of relational terminology here, we also have the wrong relational terminology being applied: recall that a relation’s not being symmetric is far from the same as its being asymmetric. Similarly, Geach [1976], p. 193, refers casually to “the simple case of two-place symmetrical connectives”. The older use of “symmetrical” for “symmetric” is one which carries its own dangers – cf. the end-of-section notes to §1.1, under ‘Consequence Relations’ (p. 100) – but the present point is simply that the word commutative is what is really wanted here. In the case of Quine [1951], terms such as reflexive, symmetric, etc. which are customarily applied to binary relations, and terms such as idempotent, associative, etc., customarily applied to binary operations, are stipulatively redefined so as to apply to binary connectives; our discussion in 3.34 may be thought of as reducing the element of stipulation involved in this new application, by showing the systematic nature of the transfer of terminology. There is a similar


509

stipulative element in the discussion of Strawson [1952]. At p. 206 of this work, Strawson describes, in terms of symmetry, both the fact that from the disjointness of one class from a second, the disjointness of the second from the first follows, and the fact that from the disjunction of one statement with another there follows the disjunction of the latter with the former. (Actually, Strawson says “clause” rather than “statement”, for reasons of his own.) He notes that in the latter case, the disjunction symbol (“∨”) is not a binary relation symbol but a binary connective and comments: Nevertheless, the formal analogy—viz., the legitimacy of pivoting two similar bits of a sentence about a third dissimilar bit—is certainly present. The nomenclature of ‘symmetrical relational statements’ is harmless so long as we put no more weight upon it than this.

In fact, Strawson’s discussion here relates as much to the theme of the final paragraph of 3.34 (‘connectives vs. operations’, we might say) as to the main theme of operations vs. relations.

Chapter 4

Existence and Uniqueness of Connectives §4.1 PHILOSOPHICAL PROOF THEORY 4.11

Introduction

In this chapter, we will discuss the existence (§4.2) and uniqueness (§4.3) of sentential connectives in the sense of connective in which connectives are not simply syntactic operations, but operations building formulas with certain inferential properties and relationships (as in 1.19). The present section fills in some of the background to that discussion, and the current subsection begins at the beginning, which is to say: with Gentzen. Recall that Gentzen [1934] introduced both the natural deduction approach and the sequent calculus approach to logic, relating these to each other and to the more traditional ‘Hilbert’ formalisms. The latter consist of axiomatic presentations of logic in Fmla. Gentzen’s sequent calculi were developed in Set-Set for classical logic and Set-Fmla0 for intuitionistic logic (1.27 and 2.32 respectively: as we have noted, strictly the “Set” here should be “Seq”, for historical accuracy). The natural deduction approach in both the classical and intuitionistic cases, he pursued, ‘naturally’ enough (though not inevitably: 1.27) in Set-Fmla. It is the latter approach (and framework) with which we are concerned here, with its characteristic organization of introduction and elimination rules for each connective (and quantifier – though these do not enter our discussion). Of this mode of organization, Gentzen made a very suggestive remark, the precise elaboration of which has greatly exercised Prawitz, Dummett, and other writers in the tradition of what we have chosen to call philosophical proof theory: the project of analyzing proofs with a view to throwing light on the meanings of the logical constants rather than for the derivation of technical results (consistency, decidability, etc.). Gentzen’s remarks on introduction and elimination rules began with the following sentence (we continue the quotation in the notes, p. 626): The introductions represent, as it were, the ‘definitions’ of the symbols concerned, and the eliminations are no more, in the final analysis, than the consequences of these definitions. (Gentzen [1934], p. 80.)

511

512 CHAPTER 4.

EXISTENCE AND UNIQUENESS OF CONNECTIVES

The “as it were”, as well as the scare-quotes around “definitions”, are an acknowledgment that in no standard sense of the term are we dealing with ordinary definitions here; and the ‘final analysis’ gestured toward still remains to be agreed upon. We will track some moves made by Prawitz in the desired direction, concentrating on what is known as Positive Logic, which is to say: the {∧, ∨, →}-fragment of intuitionistic logic; we will often use “PL” to abbreviate “Positive Logic”. A natural deduction proof system for PL – PNat, we can call it (as mentioned in 1.23) – may be obtained from Nat (or INat) by taking only the introduction and elimination rules for the three connectives listed. Note that the logical framework in which this example—and indeed most of our discussion in the present chapter—is couched, is Set-Fmla rather than Set-Set. (See Remark 1.21.1(i), p. 106.) First, a terminological issue must be addressed. The problem is over the word semantics. It has associations with concepts such as truth and reference, connecting language with the world. It also has associations with meaning. The double association presents no difficulty to those who see the basis of an account of meaning in a specification of the truthconditions of sentences, proceeding, inter alia, through the assignment of references to subsentential expressions. (For simplicity we ignore here such things as tone and conventional implicature: see 5.11.) But if one holds a theory according to which the central concepts to be used in elucidating meaning are not those, such as truth and reference, purporting to connect the linguistic with the non-linguistic, then a decision must be made as to whether semantics is to be deemed the study of meaning or rather that of the connections now discarded as not central to that study. A theory on which the meanings of the connectives are given by introduction and elimination rules, or (as suggested by the quotation from Gentzen) by the introduction rules alone, or indeed—another possible position—by the elimination rules alone, then it is not the connections between sentences and extra-linguistic reality that we need to have in focus for the study of meaning, so much as the connections—specifically, the inferential connections—between some sentences and others. For most of this Chapter, we will follow the convention that “semantic” means: pertaining to (the theory of) meaning—even when proof-theoretic (and so on an alternative convention, purely ‘syntactic’) considerations are in play. Accordingly, semantic vocabulary, such as “valid”, will not be deemed inappropriate when introduced (as in 4.13) on the basis of proof-theoretic concepts: we are discussing proof-theoretic semantics, rather than (e.g.) model-theoretic semantics, so we had better not mean ‘model theory’ by the term semantics. One should also bear in mind Heyting-style (‘BHK’) explanations of the meanings of the connectives (2.32) in terms of the conditions under which this or that construction counts as proving an assertion with the connective in question as its main connective. On the present convention, this counts as (informal) semantics. Actually, the above warning about how ‘truth’-oriented semantics should be in order to deserve the name is somewhat overblown as applied to such apparatus as the valuational semantics (for example) of Chapter 1 involves. In this case valuations were just characteristic functions of sets, with no suggestion as to the basis on which a formula gets into the sets of interest, and hence takes the value T on the associated valuation. Indeed, we have no qualms below (Lemma 4.13.2, p. 518) over availing ourselves of such apparatus in the course of an exposition of (one type of) proof-theoretic semantics.

4.1. PHILOSOPHICAL PROOF THEORY

4.12

513

Normalization of Proofs in Positive Logic

Prawitz [1965], Chapter 2, extracts from the sentiment expressed by Gentzen in the quotation from early 4.11 (given more fully in the notes below, p. 535) what he calls the Inversion Principle, introducing this by the observation that “an elimination rule is, in a sense, the inverse of the corresponding introduction rule: by an application of an elimination rule one essentially only restores what had already been established if the major premiss of the application was inferred by an application of an introduction rule”. (There follows a more precise statement of the Inversion Principle, which we omit here. We also omit any discussion of an earlier proposal in this vein, by P. Lorenzen.) Note that Prawitz is thinking of premisses and conclusions here as formulas rather than sequents, and his whole discussion takes place in a setting in which proofs are certain trees of formulas; the ‘major’ premiss of an elimination rule is that instantiating the schema containing the connective to be eliminated, in a schematic representation of the rule (as a formula-to-formula rule, with associated instructions about the manipulation of assumption dependencies.) Indeed Prawitz’s arguments very much exploit our quasi-geometrical grasp of operations performed on such trees. One such operation is the transformation of one proof-tree into another by reduction steps, which articulate the idea of an elimination rule’s restoring what one would have had before the application of an introduction rule. Since we are imagining the elimination rule to be applied to the result of an introduction rule, it must be the same connective that is introduced as is then promptly eliminated. (The formula with that as its main connective is called a maximum formula – think of the idea of a ‘local maximum’ in complexity – by Prawitz, as was mentioned in the Digression on p. 125 above.) A reduction step simplifies the proof by removing this pattern of #-introduction followed by #-elimination, and a proof to which no such reduction is applicable is said to be normal. (The discussion which follows gives only the gist of the procedures involved. See Ungar [1992], esp. Chapters 1 and 6, for details and several complications. This topic is closely related to the Curry–Howard isomorphism briefly introduced in the Digression starting on p. 165, but we are not going into the connection here. See further the references cited there.) For example, if (∨E) is applied to a formula A ∨ B which has itself been inferred by (∨I), the conclusion C of the ∨-Elimination, must already have been derivable from A (as well as from B), in which case we could more directly have use this derivation instead of going via A ∨ B. Figure 4.12a gives a diagrammatic representation, somewhat in the style of Prawitz, but incorporating the information that A has been derived with (the help of) assumptions Γ, that C has been derived from A with the aid of Δ, and C from B with the help of assumptions Θ (cf. our formulation of (∨E) in 1.23). The triple occurrences of “·”, “◦”, and “∗” stand in place of whatever the particular derivations concerned may be. If the original application of (∨I) had from B to A ∨ B, instead of from A to A ∨ B, as in Figure 4.12a, then the reduction step would of course subjoin the derivation schematically indicated by the “*”s of C from B (with Θ as ancillary assumptions) to that of B from Γ. The reduction transformation applied to a subtree representing the successive application of (∧I) and (∧E) is more straightforward, and a diagram is not needed. If A ∧ B has been obtained by (∧I), then we must already have had a

514 CHAPTER 4. Γ · · · A A∨B


A (with Δ) ∗

B ◦

∗ ∗ C

◦ ◦ C

Γ · ·

(with Θ) −−−−−−−→ reduces to

· A (with Δ) ∗ ∗ ∗

C

C Figure 4.12a: Reduction for ∨

derivation of A, as well as one of B, which derivation will serve in place of the original which proceeds to derive A (or B) from A ∧ B by (∧E). Note that if A depended originally on assumptions Γ and B on assumptions Δ, then whereas in the original (unreduced) proof, A ends up depending on Γ∪Δ whereas in the reduced proof, A depends on Γ. (A similar point about diminishing dependencies applies to the case of disjunction treated above.) Thus in order to claim that applying a reduction step to a derivation of A from Γ, results in something which again constitutes a derivation of A from Γ, it is necessary to understand “derivation of A from Γ” to mean “derivation of A from assumptions all of which are included in Γ” rather than “derivation of A from precisely the assumptions Γ”. Now this point has an obvious repercussion for the topic of assumption-rigging as this features in a system such as Nat, which consists in exploiting the increase in assumption-dependencies that results from the introduction-then-elimination pattern now to be reduced away. The need to exploit this possibility is due to the stringent formulation of conditional proof as (→I) in Nat, which requires that the would-be consequent depends on (inter alia) the would-be antecedent before the rule can apply. Such a rule could not then continue a derivation in which, in unreduced form, A → B has been inferred from B as depending on A, since the dependence in question has been rigged by a successive (∧I) from A and B, and (∧E) to B. See the Nat proof of p q → p in 2.33 for an example. Accordingly, Prawitz uses a different form of →-introduction rule from our (→I), which we will call (→I)P ra and which can be formulated in sequent-tosequent form thus (Prawitz [1965], p. 23): (→I)P ra

Γ, A B Γ {A} A → B

Using this rule, the proof of, for example, p q → p, proceeds straight from p p to p q → p by one application of the rule, since {p} {q} is just {p}. This means that the reduction-steps for ∧ (and also for ∨, in connection with which here a similar phenomenon occurs: Prawitz, p. 84) do not interfere with subsequent applications of the → rules. Of somewhat broader interest is the fact that the use of (→I)P ra in place of (→I) means that all those sequents containing → as their only connective which are provable in intuitionistic or


515

minimal logic can be proved without applying rules for other connectives. (This of course leaves the logic of relevant implication without the foundation it has in Lemmon-style systems; Prawitz addresses the topic separately, in Chapter 7 of Prawitz [1965].) As for → itself, the last connective we shall consider, the pattern to be disposed of by reduction consists of (→I)P ra followed by (→E), there being no change called for to the latter rule from Nat. The process is pictured in Figure 4.12b. (One should bear in mind that the antecedent A indicated in the “ . . . ” subproof may not actually occur, since it is (→I)P ra that we are supposing to have been used to obtain A → B in the unreduced proof.)

Δ

Δ ∗

· ·

∗ ∗

∗ ∗

B

∗ A

A ·

(with Γ)

A→B B

−−−−−−−→ reduces to

A (with Γ) · · · B

Figure 4.12b: Reduction for →

Although Prawitz [1965] considers minimal, intuitionistic, and classical logic, we do not wish to treat ⊥ or ¬ here, since the application of the introduction/elimination distinction is somewhat problematic in their case (see 4.14), and there are certain complications in the reduction steps. Some remarks on these connectives may be found in the Digression at the end of 4.14. The proof system as unfolded to this point, our natural deduction system PNat with (→I) replaced by (→I)P ra , is quite adequate to the current expository needs. (The rules are accordingly (R), (∧I), (∧E), (∨I), (∨E), (→I)P ra and (→E).) As an ad hoc designation for the system, we choose “Pra”. By successively applying the reduction steps described above, any proof from assumptions of a formula (in Pra) can be converted into a normal proof, to which no reduction step applies, and the possibility of such normalization enables Prawitz to obtain certain results for the logics (mentioned above) he considers which were previously derived, for sequent calculus formulations, as corollaries to Gentzen’s Cut Elimination Theorem (1.27). (The relationship between the natural deduction and sequent calculus results is discussed in several chapters of Ungar [1992], as well as references there cited.) Our present interest is not in the consequences of the fact that every proof can be normalized, however, but in the extent to which the process of reduction throws light on Gentzen’s remark quoted in 4.11. We are to think of the introduction rules (#I) as giving the meaning of the connective #, and the elimination rules (#E) as justified on the basis of the meaning so given: ‘the consequences of these definitions’, as Gentzen put it. The elimination rules are certainly not derivable from the introduction rules, so derivation cannot

516 CHAPTER 4.


be the kind of justification that is at issue. This leads to the somewhat more indirect style of justification underlying Prawitz’s inversion principle, with the #-elimination rules serving only to ‘restore’ what would have to have been established if the formula from which the occurrence of # in question is eliminated had itself been inferred by a #-introduction rule. (Cf. Exercise 4.12.1 below.) The reduction steps outlined above do give some content to this rather vague suggestion, though there are some questions left over as to precisely how in general, supposing them not to be antecedently given, to determine appropriate elimination rules given only that such-and-such a connective has so-and-so introduction rules. (See, further, 4.14.) But for semantic notions to be reconstructed on the basis of Gentzen’s suggestion, one would like to see a notion of validity defined which a sequent possesses in virtue of the meanings conferred on its constituent connectives by their introduction rules. In his writings since 1965 (see notes), Prawitz has experimented with several suggestions as to how such a notion might be characterized, and 4.13 gives the flavour of some of this work, though not the details. Exercise 4.12.1 Consider the language whose only connective is ∧, and a proof system on this language with the rules (∧I), (M), and (R) restricted to yielding only those sequents A A in which A is atomic. Show that the rule (∧E) is not derivable in this system but that it is admissible.

4.13

A Proof-Theoretic Notion of Validity

We continue to work with the language of PL, though shortly a more drastic diminution of vocabulary will be with us, as we set → aside. In keeping with Gentzen’s idea of the priority of introduction rules, let us say that a set Γ of formulas introductively entails a formula A if A can be derived from Γ without using any of the elimination rules. We will symbolize this by “I ”. Thus Γ I A when for some finite Γ0 ⊆ Γ, the sequent Γ A is provable in the subsystem of Pra which drops the rules (∧E), (∨E), and (→E). Note that we have A I A; more generally, in fact, the relation I is a finitary consequence relation. Taking for granted certain assumptions Γ, the idea is to replace the usual notion of truth (e.g., on a valuation) with that of being introductively entailed by Γ. This gives rise to a new consequence relation, namely that holding between Δ and B when, for any Γ, if all formulas in Δ are introductively entailed by Γ, then so is B. We want this relation to hold between Δ and B when, in particular, a derivation of B from Δ (alternatively put: a proof of Δ B) has made use of the elimination rules of Pra. Take the case of a sequent of the form A ∧ B A. As with the Inversion Principle, we want to reason: the only way to have obtained A ∧ B introductively from Γ would be to have already had A (introductively from Γ). But there is a slight hitch here, since it might have been—unless there are restrictions to prevent this—that A ∧ B itself belonged to Γ, in which case it would be introductively entailed by Γ, on our definition, since no elimination rules are involved. A first reaction might be to change the definition, and disallow (R), as well as the elimination rules. But this would mean that no propositional variable could be introductively entailed by any set Γ, giving the unwanted result that Δ would have everything as a consequence by the above definition, as soon as Δ contained some propositional variable. Therefore, we restrict Γ to being what, roughly following Prawitz, we call an


517

atomic base, which in the present context, may be taken to mean: a set of propositional variables. (Cf. the restriction on (R) in Exercise 4.12.1 above.) And, to emphasize this—though very much in contrast with Prawitz, who uses this letter to range over parts of proofs—we will reserve “Π” to stand for atomic bases. We are now prepared for our definition. A sequent Γ A is proof-theoretically valid if for any atomic base Π such that Π I C for each C ∈ Γ, we have Π I A. We will concentrate, for reasons which will become clear in due course, on the subsystem of Pra which does not use the →-rules, and from the language of which the connective → is absent. Thus in the notation “I ”, for present purposes, the superscript “I” is an allusion to the rules (∧I) and (∨I) only. We claim, as 4.13.3 below, that the provable sequents of this {∧, ∨} logic are precisely the proof-theoretically valid sequents. Let us consider the ‘soundness’ half of this claim first. It is clear that sequents instantiating the schema (R) are proof-theoretically valid: for any atomic base Π, if Π I A then Π I A. The rule (∧I) preserves proof-theoretic validity, since if (1) Γ A is proof-theoretically valid, and (2)

Δ B is proof-theoretically valid, and

(3)

we have an atomic base Π with Π I C for all C ∈ Γ ∪ Δ,

then, by (1), Π I A, and by (2) Π I B, in which case one application of (∧I) yields the conclusion, Π I A ∧ B. This, of course, is not very edifying, since we have in effect ‘justified’ the rule of (∧I) by appeal to that rule’s presence in the system. (However, compare the remarks on (→I) after Theorem 4.13.3 below.) But it is what is to be expected in an account whose distinctive aim is to justify specifically the elimination rules, given the introduction rules. So we turn to (∧E), in the form ΓA∧B ΓA (Clearly, ∧-elimination to the second conjunct will be similar.) Assuming the proof-theoretic validity of the premiss-sequent, we must show that any atomic base which introductively entails each of Γ introductively entails A. So let Π be an atomic base with Π I C for each C ∈ Γ. By the validity of the premisssequent, then, Π I A ∧ B. Now the only derivation using only introduction rules (and (R)) of A ∧ B from a set of atomic formulas must derive this formula by (∧I) from A and B, these formulas themselves being already introductively entailed by the set; so Π I A. Exercise 4.13.1 Show that the rules (∨I) and (∨E) preserve proof-theoretic validity. We are also interested in showing the completeness of the present proof system for the current semantics: that every proof-theoretically valid sequent is provable. Here we take advantage of having excluded →, whose presence would have caused a divergence from classical logic (consider such principles as Peirce’s Law, from 2.32), and the fact that the ∧ and ∨ rules given suffice for proofs of all tautologous sequents in those connectives. Thus, if Γ B is unprovable,

518 CHAPTER 4.


there is a {∧, ∨}-boolean valuation v which verifies all of Γ but not B. Our job is to turn such a v into an atomic base which introductively entails the formulas in Γ but does not introductively entail B. Lemma 4.13.2 If A is any formula all of whose propositional variables are included in some set Π of variables, and v is some boolean valuation, then, where Πv = {pi ∈ Π | v(pi ) = T}, we have: v(A) = T ⇔ Πv I A. Proof. This is a straightforward induction on the complexity of A.

Theorem 4.13.3 All and only the provable sequents of our natural deduction system for ∧ and ∨ are proof-theoretically valid. Proof. ‘All’: by the discussion, above, of the preservation of proof-theoretic validity by all the rules. ‘Only’: Suppose that Γ B is not provable, and accordingly, not tautologous. Let v be a boolean valuation with v(Γ) = T and v(B) = F. Putting Π for the set of all propositional variables in Γ∪ {B}, define Πv as in 4.13.2. By that lemma, Πv I C for all C ∈ Γ but Πv I B. So Γ B is not proof-theoretically valid. Let us turn now to the exclusion of → from our recent discussion. The sticking point turns out not to be the elimination rule, but the introduction rule: somewhat surprisingly, no doubt, for an approach on which the heavy reliance on introductive entailment might seem to render every introduction rule ‘self-justifying’. The difficulty arises because (→I)P ra is an assumptiondischarging introduction rule. (The same difficulty would arise with (→I).) We treat the easy case first, showing that the passage from Γ A → B and Δ A to Γ, Δ B preserves proof-theoretic validity (use of the introduction rule for → being understood as now allowed for introductive entailment). Suppose that (1) Γ A → B is proof-theoretically valid, and (2) Δ A is proof-theoretically valid, and (3) we have an atomic base Π with Π I C for all C ∈ Γ ∪ Δ, with a view to showing that Π I B, so demonstrating that Γ, Δ B is prooftheoretically valid. By (1) and (3), Π I A → B. Therefore Π, A I B. (Note: although Π ∪ {A} need not be an atomic base, since we can make no assumption about the form of A, the relation I was defined for arbitrary sets of entailing formulas.) By (2) and (3), Π I A. So, since Π, A I B, we have Π I B, as was to be shown. To see the problem with the introduction rule, let us consider the special case in which, for distinct formulas A and B, and some formula C, we are passing by (→I)P ra from A, B C to A B → C. (This particular transition would also be an instance of (→I), and of the rule (→ I)d considered in 2.33.) Suppose that (1) (2)

A, B C is proof-theoretically valid, and we have an atomic base Π with Π I A.


519

To show that A B → C is valid, we will need to show that Π I B → C. Necessary and sufficient for this purpose is showing that Π, B I C. If we could argue in the following way, success would be assured: (i)

Π ∪ B I A (by (2)),

and (ii )

Π ∪ {B} I B (since B ∈ Π ∪ {B});

therefore (iii)

Π ∪ {B} I C, from (i) and (ii) by (1).

Steps (i) and (ii) are perfectly correct; but there is a mistake in (iii), since to invoke (1) to obtain this conclusion, we should have to know that Π ∪ {B} was an atomic base, which we do not know: B could be of any complexity. In particular, since B need not be atomic, it may, perhaps with the help of A, have yielded C (as recorded in (1)) by means of some elimination inferences. The following example from Dummett [1991] illustrates this possibility: r (p ∧ q) → (p ∧ r). The premiss-sequent for conditional proof was in this case r, p ∧ q p ∧ r. The latter sequent is of course (by 4.13.1) proof-theoretically valid, though it is not ‘introductively valid’, i.e. r, p ∧ q I p ∧ r, which shows that r (p ∧ q) → (p ∧ r) is not even proof-theoretically valid. Exercise 4.13.4 Justify the claims just made, (i ): r, p ∧ q I p ∧ r; and (ii ): that this shows that r (p ∧ q) → (p ∧ r) is not proof-theoretically valid. Thus, even setting aside problems associated with negation, some emendation of the notion of proof-theoretic validity with which we have been working is needed even to extend Theorem 4.13.3 above to the whole of PL. Suggestions have been made on this score by Prawitz [1985] and Dummett [1991] (see p.257), involving the imposition of an inductive organization in terms of the complexity of formulas in sequents; see also Chapter 13 Tennant [1987b].

4.14

Further Considerations on Introduction and Elimination Rules

Whether one takes the introduction rules to give the meanings of the connectives they govern and to deliver suitable elimination rules as justified on the basis of the meanings so conferred, or vice versa, one will hold that at least together, introduction and elimination rules determine meaning. We have of course, thus far, only the vaguest general conception of what an introduction rule or an elimination rule has to be like in general for it to qualify as such, together with a few paradigm cases for the desired generalization to take off from: the introduction and elimination rules for ∧, ∨, and →. These rules—in the last case, whether we consider (→I) or (→I)P ra —have some striking properties which we can usefully describe in the following terminology of Dummett’s (somewhat adapted here – see the Digression below). A rule whose schematic statement requires mention of only one connective, we call pure; if in addition it mentions that connective only once, the rule is simple. The rules just alluded to are pure and simple. For such a rule, there is an obvious way of saying what it is for the rule to be an introduction rule, namely, that the sole occurrence of the connective in question should be in the schematically represented conclusion-formula of the statement of the rule. In the more accurate sequent-to-sequent description of

520 CHAPTER 4.


the functioning of the rule, this means: that the connective in question should appear as the main connective in the formula-schema appearing to the right of the “” in the sequent-conclusion of the statement of the rule. (For convenience we will often employ the formula-to-formula terminology, however, a formulaconclusion being as just explained, and a formula-premiss being a formula on the left of the “” in a sequent-conclusion of an explicitly sequent-to-sequent formulation of the rule.) The desirability of pure and simple introduction and elimination rules is emphasized in §4 of Corcoran [1969] – not that he uses this terminology. Digression. The terminology in Dummett [1991], from pp. 256f. of which the following remarks come, is actually slightly different from that described above: . . . let us call a rule ‘single-ended’ if it is either an introduction rule but not an elimination rule, or an elimination rule but not an introduction rule. (. . . ) A rule may be called ‘pure’ if only one logical constant figures in it, and ‘simple’ if any logical constant figuring in it occurs as principal operator of the sentence.

It is not quite clear what is meant here by “the sentence” (or as it would be put in the present setting, “the formula”): which sentence? There are several involved – any one of the premisses, to say nothing of the conclusion. So the proposed definition of simple needs repairing, and this could be done in various ways. One could, for example, allow a (pure) rule to count as simple if every occurrence of the connective involved (in a schematic formulation of the rule) is a main occurrence. Then, even for a rule that is what Dummett calls singleended, if it is an elimination rule with more than one premiss, it can qualify as pure even though not pure according to the stricter interpretation of the term given before this Digression. For example a rule governing a binary connective ∗ which took us from premisses A ∗ B and B ∗ A to conclusion A—or more explicitly, from Γ A ∗ B and Δ B ∗ A to Γ, Δ A—would count as a simple elimination rule on the repaired conception of purity currently being entertained, since (by contrast with, say ‘Double Negation Elimination’ – (¬¬E) from 1.23, that is) ∗ figures only as the principal connective in the schematic formulation just given of the rule. Of course this is not a simple rule on our preferred use of the phrase, since it figures in that formulation more than once. See also the discussion of ‘elementary rules’ in Humberstone and Makinson [forthcoming]. End of Digression. When we turn our attention to the ¬-involving rules of (say) Nat, from 1.23, we find several conspicuous failures of purity and simplicity. Since it is easy to see those conditions as desiderata—a failure of purity, for example, suggesting that the meanings of the connectives cannot be explained one by one in terms of their inferential behaviour—these cases may give us pause. The rule (RAA), as stated in 1.23, for example, is impure, since it involves both ¬ and ∧. However, we can easily fix this: Purified (RAA)

Γ, A B

Δ, A ¬B

Γ, Δ ¬A

though we notice that the resulting rule is still not simple, in view of the double occurrence of “¬”. (Lemmon’s own presentation of the system, in [1965a], uses


521

another impure rule, Modus Tollens, involving both ¬ and → (see 1.23.6(i)), as well as the non-simple rule (¬¬I)—or (DN) as he calls it (combined with (¬¬E)). The above purification of (RAA) is suitable in the presence of (M); more generally, one would want to allow the formula represented by “A” to appear in one or other, but not necessarily both, of the premiss-sequents.) The other primitive rule in our presentation of Nat was (¬¬E) which is again not a simple rule, leaving us wondering why the role of ¬ could not be characterized in more ‘atomic’ terms, with a general account of the inferential behaviour of formulas of the form ¬A explaining why, when A is itself of the form ¬B, B should follow from ¬A. (Compare the use, as primitive rules in natural deduction system, of such rules as ‘Commutativity’ which would sanction the inference of B ∨ A from A ∨ B, depending on the same assumptions. Rather than stipulating, out of the blue, the correctness of such a transition, a proof system using only the standard introduction and elimination rules for ∨ shows its correctness to follow from those fundamental principles.) Of course, in this last case, a natural reaction—given our current concern with philosophical proof theory—would be to regard the defect as a symptom that all is not well with classical logic. The special status given to the natural deduction approach should also be noted here, since the (# Left) and (# Right) rules of Gen, for example (1.27), are certainly pure and simple. Actually, it is not so much the choice of approach (natural deduction vs. sequent calculus) as the choice of framework (Set-Fmla vs. Set-Set) that is at work here: cf. Exercise 1.27.2, p. 141. Various additional restrictions no doubt come to mind; for example, that if the rule discharged assumptions then the discharged assumptions should be subformulas of the conclusion-formula. The →-introduction rule(s) have this property.) A ‘rule’ version of Peirce’s Law would lack it: Γ, A → B A ΓA The above rule is what Dummett [1991] calls oblique, meaning that the connective it governs (here →) occurs in the schematic formulation of the rule in the position of a discharged assumption. (Our main discussions of the above rule may be found in 7.21 and 8.34.) One might further demand that rules be ‘general’ in each of two respects: (1) Generality in respect of constituent-formulas. Here we mean that the connective introduced or eliminated by a rule exhibits, in a schematic formulation of the rule, its most general possibility of occurrence. If it is a binary connective, for example, it should appear in that formulation flanked by distinct schematic letters. (2) Generality in respect of side formulas. By this is meant that in a schematic formulation of the rule, set-variables should range over arbitrary sets of formulas, and should all be distinct; or else that, given such structural rules (from the set {(R), (M), (T)}) as are part of the presentation of the proof system to which the proposed rule is added, the rule should be interderivable with a rule meeting this condition. Condition (1) would be violated by a rule introducing a ternary connective # thus: from Γ A and Δ B to Γ, Δ #(A, B, A). The failure of generality

522 CHAPTER 4.


(in respect of constituent-formulas) consists in not saying what is required for the introduction of # in its most general possibility of occurrence, namely, in a conclusion-formula of the (general) form #(A, B, C). Condition (2) would be violated by a rule introducing a connective ⇒, say, by a rule licensing the transition from A B to A ⇒ B, since this is in effect to set the value of Γ as ∅ in the fully general formulation: from Γ, A B to Γ A ⇒ B. Here we are assuming that the logical framework is Set-Fmla; if it were Set-Set, then full generality would require that arbitrary side formulas Δ be allowed on the right (alongside B) also. For another example, see the discussion surrounding 4.32.2 below. And violations of the condition of generality in respect of side formulas are built in as part of the rationale of the rules employed in the ‘Basic Logic’ of Sambin, Battilotti and Faggian [2000]. These generality requirements have obvious analogues in the case of rules other than pure and simple introduction rules. Indeed in the case of Sambin et al., the rules are sequent calculus Left and Right insertion rules, and the authors take it as a desideratum (called ‘visibility’) that the compound into which the connective has been inserted should not be accompanied by side formulas. (Visibility in this sense is akin to the Display Property of Belnap’s ‘Display Logic’, references to which are given at 1.21.2: p. 107.) A pure and simple elimination rule, similarly, should license the passage from (arbitrary) premisses containing the connective eliminated as main connective, together with additional premisses (as in (→E)) and perhaps discharge of assumptions (as in (∨E)). More restrictive suggestions have appeared in the literature, taking (∨E) as their paradigm, especially à propos of the Gentzeninspired project of saying in general terms how a given collection of introduction rules determines as appropriate a certain set of elimination rules: see Prawitz [1978], for example. The basic idea is similar to that of Zucker and Tragesser [1978], briefly described below. The detailed working out of Prawitz’s proposal has a certain defect, concerning the elimination rule the procedure dictates for →, but we will not go into details here, referring the reader to Schroeder-Heister [1983], [1984a] and Prawitz [1985] for corrective discussion. Schroeder-Heister’s suggestion, in the interests of getting such a determination-procedure, involves extending the idea of natural deduction so that as well as formulas, formulato-formula transitions, can also be ‘assumed’ and then ‘discharged’ in proofs; these are called assumption-rules. The idea is to allow, for the sake of argument, the inferability of certain formulas from others. Suppose we represent such a transition (from the Ai to B) by A1 , . . . , An /B. Then the →-elimination rule dictated by Schroeder-Heister’s modification of Prawitz’s suggestion would be: A/B · · · C

A →B C

which turns out equivalent to the usual (→E) rule, given the way the apparatus is set up. Zucker and Tragesser [1978] also say what makes it the case that a given collection of introduction rules ‘determine as appropriate’ certain elimination


523

rules, though they specifically refrain from suggesting a general recipe for transforming sets of pure and simple introduction rules into a set of pure and simple elimination rules which are ‘appropriate’, calling a connective (with given introduction rules) stable when such elimination rules are available. (An example of instability will be given presently.) According to Zucker and Tragesser, the meaning conferred upon a connective by an introduction rule is that of a conjunction of conditionals, each of whose antecedents is a conjunction of the formulas discharged (as assumptions) by the application of the rule when the rule requires their consequent to have been derived from them. In the case in which no such assumption is discharged, we have, not a conditional, but only the consequent. So if (in sequent-to-sequent notation) the premiss-sequents for the conclusion-sequent Γ #(A, B, C, D) are Γ, A, B D and Γ C the corresponding conjunction is: ((A ∧ B) → D) ∧ C. For each such introduction rule for a connective, there is such a conjunction of conditionals (or ‘degenerate’ conditionals, consisting only of their would-be consequents, such as “C” in the example given). To obtain a representation of the meaning conferred on the connective by the set of all its introduction rules, we disjoin these conjunctions (using ⊥ for the empty disjunction, since this set might be ∅). Thus, suppose that the # of our recent example was also governed by a second introduction rule, with premiss-sequent Γ, D B. The corresponding conditional here would be D → B, so, supposing these two rules to be the only introduction rules for #, meaning of #(A, B, C, D) would be represented by the disjunction (((A ∧ B) → D) ∧ C) ∨ (D → B). (Note: we have not said what is to count as an introduction rule for present purposes. The reader is referred to Zucker and Tragesser for their—somewhat restrictive—conception as to what this category comprises.) If we had stuck with just the first introduction rule imagined for # above, there would have been a straightforward set of elimination rules of the preferred form (i.e., pure, simple, and meeting the generality conditions): Γ #(A, B, C, D) ΓD

ΓA

Γ B

and

Γ #(A, B, C, D) ΓC

which ‘unpack’ the meaning of #(A, B, C, D) as the conjunction of (A ∧ B) → D with C. We do not here give the precise description of this process, since a careful and general account of such matters would require some apparatus whose introduction would not be invoked elsewhere. (What we would need for a more precise account – and what perhaps the reader has already noted to be woefully absent – is a range of meta-schematic variables over the schematic formula- and set-variables “A”, “B”, . . . “Γ”, “Δ”,. . . ) However, the present example illustrates the procedure, the intention behind which is given by Zucker and Tragesser’s answer to the question: what is the point of the E(limination)-rules? (Similarly “I” abbreviates “introduction”.) This quotation is from p. 506 of their [1978]: It is (we propose) this: they stabilize or delimit the meaning of the logical constant concerned, by saying, in effect, of the given I-rules: “These are the only ways in which this constant can be introduced”. Without such E-rules, one would have the option (so to speak) of changing the meaning of a constant by adding new I-rules (thereby adding new disjuncts to its meaning). The E-rules prevent this (i.e., remove this option), and it is in this sense that they stabilize the meaning.

524 CHAPTER 4.


With our modified version of the above # example, giving #(A, B, C, D) the meaning of (((A ∧ B) → D) ∧ C) ∨ (D → B), there appears to be no such set of stabilizing elimination rules (of the preferred type), making the connective # ‘unstable’ as Zucker and Tragesser put it, though they give a simpler example of the phenomenon, presented below. What is important for their own purposes is that the IL connectives →, ∧, ∨, and ⊥ themselves are stable, and it is in terms of these that the meaning-representing disjunctions of conjunctions of conditionals can all be written: a kind of expressive completeness result for the doctrine that the introduction rules give the meanings of all connectives. (There is a similar result for a suitably constrained set of sequent calculus style rules in Kaminski [1988], Theorem 8.) We turn to Zucker and Tragesser’s example of instability, which concerns the case of the following rules, governing a ternary connective (we shall write as) z, for which again there appears to be no suitable set of elimination rules: Γ, A B

ΓC

Γ z(A, B, C)

Γ z(A, B, C)

(We saw this example in the setting of sequent calculus rather than natural deduction in 2.32.14, 2.32.15 above – p. 322.) According to the meaning Zucker and Tragesser’s account, summarised above, says these introduction rules confer upon z, z(A,B,C) means the same as (A → B) ∨ C. The first rule supplies the first disjunct and the second, the second. It is important to note that it is not the fact that the given rules are amongst the introduction rules for z that leads us to say this, but the fact that the given rules exhaust the set of introduction rules: additional rules would lead to additional disjuncts. (So a more careful formulation of the point that introduction rules determine meaning would be that the meaning of # is given by the totality of its introduction rules together with the fact that those rules comprise that totality: see the quotation above.) Further, since the intention is to make sense of Gentzen’s suggestion that— to revert to the looser way of putting the point—introduction rules determine meaning, within the specific context of intuitionistic logic (which, when motivated along these lines, Zucker and Tragesser call ‘inferential logic’), we have to understand the “→” and “∨” in the meaning-explicating (A → B) ∨ C as having the logical powers intuitionistically accorded to them. This excludes the possibility that an appropriate elimination rule might be: Γ z(A, B, C)

ΔA

Θ, B D

Θ , C D

Γ, Δ, Θ, Θ D That would have been a suitable elimination rule if the meaning conferred upon z(A, B, C) had been that of A → (B ∨ C). The latter is classically, but not intuitionistically, equivalent to (A → B) ∨ C. Using Schroeder-Heister’s device (though he does not employ this sequent-to-sequent representation of rules), the lacuna of the missing elimination rule for z can be filled by: Γ z(A, B, C)

Δ, A / B D Γ, Δ, Θ D

Θ, C D


525

The reader will have to decide whether allowing such ‘assumption rules’ as in A/B here really buys the desired purity that was absent had one simply written the formula A → B. (There is also an interesting question as to whether, in allowing rules a premiss-like function in arguments along with formulas, one is playing into the hands of the tortoise in Carroll [1895]. We will not pursue these matters further here, however; a guide to the literature on this topic can be found in §5.3 of Moktefi [2008].) Whatever the difficulties there may be about finding suitable elimination rules, there is no difficulty in finding – without resort to such devices – a complete and manageable set of sequent-to-sequent rules for z, as we shall see in 7.22.11–7.22.13. (For more on Schroeder-Heister’s system, see Avron [1990] and §2.2 of Wansing [2000], where similar ideas are traced to work published in German in the 1960s by F. von Kutschera; Hazen [1996] traces the ‘assumption rule’ idea in this system back to Fitch [1952], where provision is made for it, though it is not exploited. See again the discussion following 7.22.10 on p. 1077 below.) Setting aside the details of the various proposals alluded to above, what should not escape our notice is the general idea of some introduction rules ‘determining as appropriate’ this or that elimination rule, or set thereof. We have seen some examples of what appropriateness might consist in in this connection. For example, the condition suggested by 4.12 would be that the application of an elimination rule (proposed for a connective) to the conclusion of an introduction rule (for that connective) should never yield anything not obtainable prior to the introduction in question. Or again, appropriateness might (à la 4.13) be cashed out semantically in terms of the preservation of a kind of validity characterized proof-theoretically in terms of the introduction rules. Dummett speaks of a ‘harmony’ between the introduction and elimination rules for a given connective for this kind of mutual appropriateness, and many suggestions have been made—including those just listed—as to how to explicate this informal notion. (Dummett also considers the idea of taking the elimination rules as meaning-conferring, as determining an appropriate set of introduction rules; also, more accurately, Dummett [1991] divides up the idea of mutual appropriateness into harmony and something he calls stability – cf. stability à la Zucker and Tragessert [1978], as above – but we do not go into details here.) We will mention a suggestion from Tennant [1978], as much for the sake of introducing a certain way (in terms of deductive strength) of thinking about connectives, as for its bearing on the topic of harmony. Tennant’s initial proposal (in Tennant [1978]), was to work some familiar ‘strength of propositions’ considerations into the form of a criterion of harmony. (Familiar, for example, from the references cited after 4.14.1, p. 526.) By a proposition, here, is meant an equivalence class of formulas (w.r.t. the relation , where is the consequence relation under consideration, cf. 2.13: we assume congruentiality, so that this suffices for synonymy). According to these considerations the conjunction of two propositions is the strongest proposition following from them (taken together), and also the weakest proposition from which each of them follows. Talk of comparative strength here is to be understood in the following terms: one proposition is stronger than another if it entails (has amongst its consequences) that other; colloquially one would naturally include also “but is not entailed by it”, though we will not require this here. “Weaker” expresses the converse relation. Thus the strongest proposition with a given property is a proposition with that property from which all others with

526 CHAPTER 4.


the property follow. So, saying that conjunction forms the strongest proposition following from both of them, amounts to saying that A ∧ B follows from A and from B—which is the upshot of (∧I)—and that A ∧ B implies anything else C which follows from both of them. “Both of them” here means “both of them taken together” rather than “each of them (taken separately)”. (∧E) secures this part of the claim, because if A, B C (for some ∧-classical ), then since A ∧ B A and A ∧ B B, we have. by (T), that A ∧ B C. In similar terms (see 4.14.2) we can see why a conjunction also qualifies as expressing the weakest proposition from which its conjuncts both follow. We list here, in Tennant’s terminology ([1978], p. 75, but with the notation altered for consistency with this book), analogous characterizations of the other connectives of PL, including the case of ∧; for simplicity of exposition, such locutions as “the proposition A” are used rather than “the proposition expressed by A” (or “[A]” in the notation of 2.13): Examples 4.14.1(i) A ∧ B is the strongest proposition implied by {A, B}. A ∧ B is the weakest proposition which implies A and implies B. (ii ) A ∨ B is the strongest proposition implied by A and implied by B. A ∨ B is the weakest proposition which implies C when A implies C and B implies C. (iii) A → B is the strongest proposition implied by Γ {A} when Γ implies B. A → B is the weakest proposition which, with A, implies B. (Cf. also Grzegorczyk [1972]; the idea goes back to Tarski: see the end of 6.11.1 below for further references; see especially Popper [1948], note 11, on the ‘superlative’ aspect of the present case.) Tennant goes on to say that “for each operator the verification of the first claim depends on a consideration of the elimination rule, while the verification of the second claim depends on a consideration of the introduction rule”; but this is not quite right: as our working, above, of the case of ∧ illustrates, in fact a consideration of the introduction and the elimination rule—or some rules equivalent to these, given the structural rules—is required for each claim. We examine this point more closely by considering the case of conjunction. Remark 4.14.2 Each of the claims (i)–(iii) of 4.14.1 consists of a ‘basic’ part, saying that a connective has certain inferential properties, and a ‘superlative’ part, saying that it is the weakest or that it is the strongest connective with those properties. (Strictly, these properties are ascribed to compounds formed by that connective.) In the case of ∧ the basic claim in the first characterization under 4.14.1(i) is that A ∧ B follows from {A, B}, which we can express directly as a schema (zero-premiss sequent-to-sequent rule): A, B A ∧ B. The superlative claim for this case is one to the effect that A ∧ B is the strongest formula with this property. We can put this with the aid of the one-premiss Set-Fmla sequent calculus rule (1.27) (∧ Left): Γ, A, B C Γ, A ∧ B C


527

Note that the “Γ” has been included for generality in respect of side formulas: taken literally, the superlative claim does not have this generality (and our treatment of this case in the discussion preceding 4.14.1 did not include it). We turn to the second characterization of conjunction in 4.14.1(i). The fact that a conjunction implies each of its conjuncts – the ‘basic’ claim here – is again zero-premiss expressible, by A ∧ B A and A ∧ B B, and the ‘superlative’ claim by: ΓA

ΓB

Γ A∧B Note that this rule is equivalent (in virtue of (M)) to the usual introduction rule (∧I), formulated with “Δ” in the second premiss-sequent, and “Γ, Δ” in the conclusion-sequent. (The same point applies, of course to the rules cited in our discussion of Zucker and Tragesser above.) Again, a literal reading of the comparative strength characterization would have “C” in place of “Γ”, here, and so not be fully general in respect of side formulas (for Set-Fmla). This would result in a genuine weakening of the rule: see 4.32.3 (p. 587). By way of conclusion: the basic and superlative parts of the first characterization of conjunction in 4.14.1(i) are tantamount, respectively, to introduction and elimination principles for ∧, whereas for the second characterization, it is the other way around. Note: such a double analysis in terms of rules is not possible for ∨ or →. In the former case, the first characterization has a basic part amounting to: A A ∨ B and B A ∨ B (‘introduction’ principles), and a superlative part amounting to a version of (∨E): from Γ, A C and Γ, B C to Γ, A ∨ B C. (We insert “Γ” for generality; without it, we have a variant of the rule (∨E)res from 2.31, p. 299.) The second characterization under 4.14.1(ii), however, has no expression in the form of a rule for its superlative part—in the absence of a device such as Schroeder-Heister’s assumption-rules—though its basic part is given by the above elimination principle. (See 4.31.8, p. 584, for a comment on this situation.) In the case of →, it is the superlative part of the first characterization that does not admit of expression in the form of a rule. The above problem with Tennant’s discussion aside, we quote in full (but changing Tennant’s “Λ” to “#” again for conformity with the present notation) what he calls the Principle of Harmony ([1978], p. 74): Introduction and elimination rules for a logical operator # must be formulated so that a sentence with # dominant expresses the strongest proposition which can be inferred from the stated premisses when the conditions for #-introduction are satisfied; while it expresses the weakest proposition which can feature in the way required for #-elimination.

The talk of premisses here is to be understood in accordance with a formulato-formula description of how the rules work rather than of premiss-sequents (1.23). A revised version of the above principle appears in Tennant [1987b] (and there are some further considerations in Chapter 10 of Tennant [1997]; see also Tennant [2005]. We will not discuss the nature of the revision here, or any of various other proposals explicitly concerned to provide an account of harmony.

528 CHAPTER 4.


In general, it would appear that, far from providing a logic-neutral vantage point from which to assess alternative logics, the informal idea of harmony can be explicated in several different ways so as to favour correspondingly different logics. For example, in Dunn [1986] harmony considerations are held to favour relevant logics over classical and intuitionistic logic (and likewise Chapter 9 of Read [1988]); in Tennant [1987b], Chapter 23, a kind of (in that author’s words) intuitionistic relevant logic comes to be favoured on such grounds; Weir [1986] argues that while traditional explications of harmony favour intuitionistic logic, there is an alternative explication to hand according to which the classical introduction and elimination rules indeed harmonize – provided one gives up on the desideratum of purity for one’s rules; see also Milne [1994b]. Read [2000a] offers a treatment in the same direction making no such concessions, using a Set-Set natural deduction system (cf. 1.27 above). Milne [2002], on the other hand, replaces the multiple right-hand sides with disjunctions, thereby again sacrificing purity. See also the numerous index entries under ‘harmony’ in Dummett [1991]. (For a characteristically idiosyncratic discussion, see Slater [2008].) We shall give a slightly fuller account of how a notion loosely describable as one of harmony (not that he uses this term, or is attempting to explicate this notion) emerges from a particularly rich discussion of some of these issues: Peacocke [1987]. In this case the intention is to relate considerations of philosophical proof theory to classical logic; we describe only a small number of Peacocke’s suggestions here. (See further Peacocke [1993], and the Digression on p. 1170.) Rather than fixing once and for all on introduction or elimination rules as playing a preferred role in meaning determination, Peacocke allows the question of such a choice to depend on the particular connective (or quantifier) at issue. He suggests that in the case of conjunction, both the introduction and elimination rules have equal priority, that for the universal quantifier it is the elimination rule, and that for the existential quantifier and for disjunction, it is the introduction rule that plays this role. Though Peacocke [1987] does not discuss the case of implication (or conditionals, we should rather say), it seems plausible to have this role played by the elimination rule (‘Modus Ponens’, as it is often called, though we generally to reserve this label for the rule of that name in Fmla), with the introduction rule having a secondary status. Digression. While it seems hard to attribute understanding of if to anyone who assents to something of the form “If S1 then S2 ” as well as to S1 , but not to S2 , we have a fictional prototype in Carroll [1895], and Williamson [2006b] draws attention to evidence that, at least under experimental conditions, only 97% of apparently fluent speakers of English endorsed the Modus Ponens (or (→E)) transition; see also Williamson [2007], esp. Chapter 4, where a plausible line of thought leads to the conclusion (p. 97) that “there is no litmus test for understanding”. (See also Williamson [2003] for this phrase – p. 277 – and an overlapping discussion. Before its appearance in the latter setting, Williamson writes: “The social determination of meaning requires nothing like exact match in use between different individuals; it requires only enough connection in use between them to form a social practice. Full participation in that practice constitutes full understanding.” Evidently the bar has been set rather low for what constitutes full participation in a community of users of conditionals if acceptance of reasoning in accordance with Modus Ponens is not required of a full participant. Some suggestions as to how to reconcile this consideration


529

with the experimental evidence appear at the end of this Digression.) A similar point may be found in Davis [2005], and broadly similar views about logical principles in general have also been urged by C. Cherniak – for reference to and discussion of which, see Evnine [2001]. We ignore these facts here, as well as the explicit criticism of Modus Ponens to be found in McGee [1985], on which for further see Williamson [2007], pp. 92–94, Weatherson [2009], §7. From the discussion of Jackson’s theory of indicative conditionals in 7.11 below it will be clear how, at least on that account, to describe McGee-style counterexamples to Modus Ponens: preservation of assertibility is not preservation of truth (on such accounts). While, as noted, Peacocke [1987] does not discuss conditionals, Peacocke [2005] does. Rather than endorsing the above suggestion giving primacy to the elimination rule, or the Gentzen proposal according primacy (as in all cases) to the introduction rule, or treating them on a par (as in Peacocke [1987] and [2005] for the case of conjunction), in Peacocke [2005], p. 176, we have the following, in reply to Davis [2005]: Davis says, rightly in my view, that ordinary thinkers do not find such rules as modus tollens or contraposition primitively compelling. I would not write immediate acceptance of them, nor even modus ponens, into the possession condition for the (or for an) indicative conditional. In cases in which, for example, the conditional behaves in the way identified by Stalnaker (. . . ), I would say that a thinker’s understanding of it consists in tacit knowledge of the content stated in Stalnaker’s semantical clause.

The reference is to what appears in our bibliography as Stalnaker [1968]; see 7.15 and 7.17 below for exposition. Quite why Peacocke wants to treat the case of the conditional so differently is not immediately clear, unless it is simply the claimed implausibility of requiring Modus Ponens transitions as primitively compelling for possession of the concept involved. Naturally, one would need to consider the empirical support for that claim, attending in particular to the distinction between finding a transition compelling and acknowledging that one does, as well as to the distinction between experimental designs which tap normal cognitive resources and those which instead trigger reasoning in “puzzle mode” (Levesque [1988]). Williamson [2007], p.104f., discusses related dichotomies and their bearing on the view (to which, as already intimated, he is hostile) that any particular inference pattern must be found primitively compelling for the possession of a logical (or other) concept. End of Digression. Further, we have not correctly described the role of the rule(s) given priority in Peacocke’s account by using the phrase ‘meaning determination’, so that on a narrow interpretation of what counts as philosophical proof theory, this account is a close cousin rather than a special case. Rather, Peacocke’s idea is that finding transitions in accordance with a certain rule, or set of rules, primitively compelling (or ‘primitively obvious’) – in the sense of making the transition directly rather than via intermediate steps, and of not feeling any need (or finding any room) for a justification of that transition – is partially constitutive of possessing the concept of (e.g.) conjunction, disjunction, or implication. Thus, in the case of (∧I) and (∧E), in which both rules have the favoured status in question, it is suggested that unless applications of these rules are found primitively compelling by a certain subject, then the subject does not express by “∧” the concept of conjunction. (Note that something needs to be

530 CHAPTER 4.


said about rules like (∧I) – sequent-to-sequent rules which cannot be regarded as formula-to-formula rules – as to what finding their application primitively compelling consists in. See Hodes [2004] for this and several further refinements of Peacocke’s picture.) In the case of disjunction, the introduction inferences are those required to be primitively compelling, (∨E) having a secondary status—on which more below, for possession of the concept of disjunction. (“Secondary”, rather than “derivative”, we say, in order to avoid the misleading suggestion that we are here dealing with rules derivable from the given rules.) There are in fact two respects in which talk of the meaning of a connective being determined by certain rules would be unfaithful to Peacocke’s views. The first is that just touched on: it is not that a meaning is conferred upon a connective by the demand that it obey the rules in question, but, rather, that unless the inferences licensed by the rules are found primitively compelling by a subject, then what the subject means cannot be (e.g.) and, or, etc. It is a matter of the conditions for correctly attributing the concept of (e.g.) conjunction to a subject, rather than what it takes for a proposition to be conjunctive. The second respect in which talk of meaning determination is misleading is that Peacocke wishes to employ the distinction from Frege [1892] between sense and reference, a useful though controversial distinction within the area pretheoretically demarcated by talk of meaning. (We assume the reader has some familiarity with this distinction.) The above description of which concepts a subject possesses is intended to lie on the ‘sense’ side of the sense/reference divide. On the ‘reference’ side Peacocke takes the reference (or ‘semantic value’, as he sometimes says) of a connective—at least for the cases we have been considering—to be a truth-function. (The idea of letting the rules record the sense and a truth-table record the reference of a connective can also be found in Wagner [1981]. See further Hodes [2004].) It is this aspect of the account which makes for the connection with classical logic alluded to above, as will become clear when we note the status of Peirce’s Law as a ‘limiting principle’ below. Since the sense of an expression should determine its reference, Peacocke needs to provide a recipe as to how this happens, when the sense is given by a set of rules governing a connective # whose status is that the inferences they sanction are found primitively compelling by one attributing the sense in question to #, and the reference is some truth-function to be associated with #. We turn to a description of the recipe offered. The notions of strength and weakness which featured in 4.14.1, 2 (and surrounding discussion) as applied to propositions have a natural application to truth-functions. Suppose f and g are n-ary truth-functions. Then f is at least as strong as g when for all x1 , . . . , xn (xi ∈ {T, F}) f (x1 ,. . . ,xn ) = T ⇒ g(x1 , . . . , xn ) = T Sometimes, for brevity, we say “stronger than” in place of “at least as strong as”. “Weaker than” signifies the converse relation, with strongest, weakest, understood in the obvious way in terms of these relations. Example 4.14.3 Using the notation of 3.14, ∧b is stronger than ∨b (and not conversely); looking at the conventional truth-table representations of these functions, any row issuing in a T in the former case issues in a T in the latter. (Alternatively put: each determinant for the former with T in its terminal position is a determinant for the latter.)


531

The fixing of an appropriate truth-function as the reference of a connective # then proceeds in two stages. First, we find the truth-functions which make the rules having a distinguished position in specifying the concept expressed by # truth-preserving (i.e., the rules, inference in accordance with which has to be found primitively compelling by a subject possessing that concept). Secondly, from amongst these candidate truth-functions, we select the strongest, in the case that the rules are introduction rules, and the weakest, in the case that the rules are elimination rules. Several aspects of this recipe are somewhat problematic, but we first illustrate the procedure in the case of disjunction, on the assumption that it is the rule(s) of ∨-Introduction that for this case are deemed to have the distinguished status. We will also illustrate the procedure for the case of implication, on the assumption, this time, that the elimination rule is similarly privileged. Examples 4.14.4(i) We can represent the class of truth-functions which can be associated with ∨ in such a way that the inferences A A ∨ B and B A ∨ B are truth-preserving by a partial truth-table: A T T F F

∨ T T T –

B T F T F

The three determinants depicted here are those which a valuation must respect (to use the language of §3.1) for sequents of the two forms mentioned to hold on that valuation. If we had, for example, an F instead of a T in the third row, then the passage from B to A ∨ B would not be guaranteed to preserve truth. We have now completed Stage 1 of the procedure described above. To find the strongest truth-function from amongst those containing the three determinants we have arrived at, which is what we are directed to do by the recipe, since we are considering ∨-Introduction, we replace the blank in the fourth row by an F. This gives ∨b as the truth-function determined. If, instead, we were to put T in that position, getting the constant true binary truth-function, we should have something properly weaker than ∨b by the above definition of “is at least as strong as”. (Indeed, we should have the weakest of all two-place truth-functions.) (ii ) Let us now consider the case of (→E); to secure truth-preservation for inferences in accordance with this rule, we must associate with → some truth-function containing the determinant T, F, F , since we must arrange matters so that whenever a conditional and its antecedent are true, the consequent is also. Thus Stage 1 issues in the partial truthtable: A T T F F

→ – F – –

B T F T F

532 CHAPTER 4.


As we are dealing here with an elimination rule, we complete the table in such a way as to obtain the weakest truth-function containing the determinant indicated; in other words, instead of replacing, as with 4.14.4(i), the blanks by F, we replace them with T. Of course the result of this is →b . It was mentioned that certain aspects of the general description of Peacocke’s procedure here illustrated are problematic. In particular, there are two questions we shall (scarcely more than) raise here. The first of these potential problems arises as follows. The recipe speaks of what to do when the rules given priority (in terms of what is found ‘primitively compelling’) are introduction rules, and of what to do when they are elimination rules. What if some rules of each kind are deemed to have this status? Peacocke’s preferred treatment of the concept of conjunction, in which both the introduction and elimination rules have the status in question, does not raise any difficulty, since they straightforwardly determine the desired truth-table. In general, though, difficulties could arise at either Stage 1 or Stage 2 of the procedure. A difficulty of the former sort would be one in which not even the partial truth-table Stage 1 is intended to issue in could be constructed, because of clashes in entries dictated by the introduction and elimination rules. The classic example of such clashes (‘pathologies of overdetermination’, as we put it in 3.12) is Prior’s example of Tonk, to be discussed in 4.21. From Peacocke’s perspective, such clashes would not be consistent with the idea that the connective in question refers to a truth-function: for, recall, his intention is not to explicate semantic notions proof-theoretically, but to say what it is, on the basis of proof-theoretic facts (concerning primitively compelling transitions), that makes this rather than that truth-function the reference of a connective. (See the notes to §4.2 – p. 576 – for a comment on Peacocke’s discussion of Tonk.) Clashes in the entries of the partial truth-table at Stage 1 aside, there is a difficulty raised by Stage 2, for the current prospect of mixed introduction and elimination rules as having the special status. Recall that this is the stage at which a partial truth-table is to be completed. By way of illustration: Example 4.14.5 Suppose that the only inferences which it is deemed must be found primitively compelling for possession of the concept expressed by some binary connective # are those recorded in the introduction and elimination rules A#B A B (#E) (#I) A#B A In other words: as for conjunction, except that we have dropped the elimination rule to the second conjunct – and the above notation suppresses the assumption dependencies made explicit in the sequent-tosequent formulation of these rules. To secure truth-preservation, these rules force the partial truth-table: A T T F F

# T – F F

B T F T F


533

Line 1 is dictated by (#I), and lines 3 and 4 by (#E). Now, when it comes to replacing the blank, we cannot straightforwardly apply Stage 2 of the general recipe, since this would dictate an F for the strongest truth-function compatible with the three determinants we have so far, and a T for the weakest truth-function. Now we are told to seek strength when the partial table is based on introduction rules, and weakness when it is based on elimination rules: but we are told nothing for the ‘mixed’ case. The second problematic aspect of the discussion concerns its focus, at Stage 1 of the recipe, on rules from formulas to formulas. These can be treated as zero-premiss sequent-to-sequent rules or as conditions on consequence relations of the type considered in 3.13, so the idea of recording in a partial truth-table the requirement that the rule ‘holds on’ (or that the condition is ‘respected by’) a valuation is straightforward. What is less straightforward is what would become of Stage 1 if a rule whose applications were supposed to be found primitively compelling were not such a zero-premiss rule, or even a rule (e.g. (∧E) as formulated for Nat) which was interderivable with such a rule given (R), (M), and (T). The rules which come to mind here are the assumption-discharging natural deduction rules such as (∨E) and (→I). These two cases are themselves interestingly different. For the former, there is an equivalent zero-premiss Set-Set rule, namely A ∨ B A, B (‘equivalent’ meaning interdeducible, given the structural rules, as applied in Set-Set), in terms of which, if desired, Stage 1 could be executed. (This assumes we want to associate with ∨ some truth-function, of course. In the absence of the Strong Claim of 1.14 (p. 67), this is not mandatory even given a commitment to both (∨I) and (∨E) in Set-Fmla; we shall see many ‘non-truth-functional’ treatments of disjunction for logics formulated with these rules in 6.43–6.46.) For the latter example, as we shall show at 4.34.3, no such Set-Set zero-premiss trade-in is available. Of course, any problems here could be avoided if one had an a priori argument that no connective could be such that grasping its sense was partially constituted by finding primitively compelling those inferences which were applications of an assumption-discharging rule, or indeed of any rule other than one which is equivalent to a zero-premiss rule in Set-Fmla. (Certainly such cases do not arise in Peacocke’s discussion.) This concludes what we have to say on the problematic features of the recipe illustrated in unproblematic cases by 4.14.4. Peacocke defines a limiting principle for a set of introduction rules to be a principle not derivable from those rules “and which is validated by the maximally strong semantic assignment [sc. truth-function, for the cases of interest to us] to the logical constant in question which validates all the principles in [the given set of rules]” ([1987], p. 162), with a similar understanding in the case of a set of elimination rules, changing “strong” to “weak”. Thus in the case of {(∨I)}, we obtain – via 4.14.4(i) on p. 531 – (∨E) as a limiting principle, indeed as one from which all others follow. It is in this sense that Peacocke’s procedure can be regarded as articulating a notion of harmony—of describing the determination for any set of introduction (elimination) rules an appropriate set of elimination (introduction) rules. It is worth seeing how this differs from the purely proof-theoretic approach to this question reviewed above (e.g., in the quotation from Tennant on p. 527). In the case of disjunction, there is no difference, since we arrive at (∨E) as an appropriate elimination rule, from which other

534 CHAPTER 4.


limiting principles follow. So the reference to the partial and completed truthtables in 4.14.4(i) make no difference to the outcome: we could have conducted the discussion in terms of strongest and weakest propositions, as before, without bringing in the truth-functions. In the case of implication, treated under 4.14.4(ii), there is a difference, since the introduction rule (→I) does not yield all of what Peacocke calls the limiting principles to which (→E) gives rise via his procedure. For example, Peirce’s Law counts as a limiting principle which we know not to be derivable from (→I) and (→E) taken in conjunction. So the reference to truth-functions is not just providing a certain explanatory perspective on the relation between what rules are (to quote the title of the section of Peacocke’s paper in which these matters are expounded) ‘underived, unobvious, but justifiable’ on the basis of those rules in the ‘primitively compelling’ category: rather, this involvement of the truth-functions actually makes a difference to the verdicts delivered by that procedure. We obtain, not PL, but the {¬, ⊥}-free fragment of CL. (The restriction on connectives here is for expository convenience, rather than one imposed by Peacocke: see the final sentence of the following Digression.) Digression. We close with a few comments on the question of how the above treatments of the ‘positive’ connectives are affected by the presence of negation. To retrace the development of 4.12 with ¬ presumed available, we should note, following Prawitz [1965], that the proof exhibited as Example 1.23.2 (p. 119), of the sequent p ∨ ¬p in Nat contains (at line 8) a formula which is simultaneously the conclusion of an introduction rule, provided we agree to regard (RAA) as a rule introducing ¬, and the premiss of an elimination rule, provided that we regard (¬¬E) as a rule eliminating ¬. Further, this proof cannot be normalized so as to avoid that feature. However, as Prawitz points out, if ⊥ is added to the language (of positive logic) and ¬A treated as simply A → ⊥, rather than as the result of applying a primitive singulary connective to A, a formulation can be given of classical logic in which every provable sequent does have a normal proof. In fact, this move helps with the non-simple purified (RAA) above, which would have figured in a purified version of our presentation of the intuitionistic system INat in 2.32, since this rule is no longer one of the primitive rules. We still have to describe the rules for ⊥, and what come to mind are, first, an intuitionistic ⊥-rule: from Γ ⊥ to Γ A and, secondly and somewhat less obviously, a (provisional) classical ⊥-rule: from Γ, ¬A ⊥ to Γ A. The latter rule takes us from ¬¬A, ¬A ⊥ (i.e.: (A → ⊥) → ⊥, A → ⊥ ⊥) to the distinctively classical ¬¬A A (instantiating Γ as {¬¬A}). In fact, to avoid having to use both these rules in a proof system for classical logic, Prawitz makes use of the same device that converted (→I) into (→I)P ra , allowing the ‘discharge’ even of unused assumptions: classical ⊥-rule: from Γ ⊥ to Γ {¬A} A Now the intuitionistic rule represents a special case of the classical rule, subsuming only such application of the latter as have ¬A ∈ / Γ. We will discuss the appearance of ⊥ in definitions of negation in §8.3. (Peacocke [1987] treats negation by adding to the notion of the primitively compelling inference, the notion of ‘primitive incompatibility’, not discussed here.) End of Digression.


535

Notes and References for §4.1 In 4.11, we gave only the first sentence of the following quotation from Gentzen [1934], p. 80: The introductions represent, as it were, the ‘definitions’ of the symbols concerned, and the eliminations are no more, in the final analysis, than the consequences of these definitions. This fact may be expressed as follows: In eliminating a symbol, we may use the formula with whose terminal symbol we are dealing only “in the sense afforded it by the introduction of that symbol”.

Discussing the matter in terms of conditions on consequence relations rather than specifically introduction (or elimination) rules, Popper [1947a], [1947b], [1948] pursues the idea that the various connectives are defined by their distinctive logical roles. See Schroeder-Heister [1984b] for a very useful critical discussion, with comprehensive references to Popper’s work and the reactions it provoked. Prawitz’s various attempts since Prawitz [1965] to use normalizationrelated concepts in the articulation of a proof-theoretic conception of validity may be found in Prawitz [1971], [1972], [1973], [1974]; Prawitz [1975], [1978] and [1981] are in a similar vein. See also Prawitz [2006] and Schroeder-Heister [2006]. An introductory review of some aspects of the literature alluded to here, as well as of topics – e.g., the work of P. Martin-Löf – not touched on in §4.1, is provided by Sundholm [1986]; some issues arising from reliance on this work (actually, in a different paper of Sundholm’s) are raised in Dummett [1994]. For a sceptical response to proof-theoretic semantics à la Prawitz and Dummett, see Stirton [2008], and for an exploration of the applicability of similar ideas in the case of classical logic – traditionally taken to fare less favourably than intuitionistic logic when viewed from this perspective – see Sandqvist [2009]. Numerous technical issues arising out of normal forms in natural deduction are presented in Ungar [1992]. See further §§5–6 of Negri and von Plato [2001]. Also not touched on in our coverage is the relationship between normalization of natural deduction proofs and normalization of λ-calculus terms; see Troelstra and Schwichtenberg [1996], especially Chapters 1 and 6 for information on this, as well as the references given for the Digression beginning on p. 165. For a broader perspective on Peacocke [1987], some aspects of which were summarized in 4.14, see Peacocke [1990]. (For a reconsidered position, see Peacocke [2008], especially Chapter 4.) A less refined presentation of similar views may be found in Appendix A of Harman [1986a]; see also Harman [1986b]. (A broadly similar approach, with several distinctive features of its own, is the ‘inferentialism’ of Robert Brandom, expounded in Brandom [1994], [2000]. We quote from the first of these in the discussion following 8.11.4, as well as hearing more from Peackocke there: see the Digression on p. 1170. The phrase inferential semantics is often used to apply to all approaches along these lines, including within Zucker and Tragesser [1978] to describe the authors’ own position; however, the same phrase is also used, in an unrelated sense, in Tennant [2010].) The priority, in terms of what is to be found primitively compelling, of (∨I) and (∃I) in Peacocke’s treatment, over the corresponding elimination rules, contrasts intriguingly with the suggestion at p. 71f. of Tennant [1987b] that we think of disjunction as originally motivated by the need to link A with B in the antecedent of a conditional with consequent C in such a way as to form some-

536 CHAPTER 4.


thing equivalent to the conjunction of A → C with B → C. (There is a similar suggestion in Jennings [1986]; see the end of 6.14.) If such a quasi-historical account is taken to correspond to the cognitive dependencies involved, this is in effect to give priority instead to the elimination rule, which Tennant goes on to argue determines as appropriate the usual introduction rule(s). It would appear somewhat obscure how such differences of opinion are to be adjudicated. Mention should also be made – as in the Digression on p. 528 – of Williamson [2006b], esp. p. 33, in which the whole idea of requiring certain inferential transitions to be found primitively compelling as a necessary condition for the possession of the concept of conjunction, disjunction, etc. (or for understanding and, or, etc.), is subjected to thoroughgoing criticism. See also Williamson [2007], as well as Davis [2005] and the reply, Peacocke [2005], as well as Peacocke [2008], note 1 on p. 114. Rules described as general in respect of side formulas in 4.14 are sometimes called local rules – e.g., in Hacking [1977].

§4.2 EXISTENCE OF CONNECTIVES 4.21

The Conservative Extension Proposal

One may wonder whether a connective could be conceived of with this or that inferential behaviour. For example, one might wonder whether we could have a 1-ary connective § with the property that ¬A and §§A were equivalent (for all A). Any logic providing such an equivalence – say conceived of as a consequence relation ⊇ CL according to which §§A ¬A for all A in the language of – would have to be what is called in Humberstone [2000a] (profoundly) contraclassical, meaning that no compositional translation (as there defined) τ exists with the property that for all formulas A1 , . . . , An , B of the language of : A1 , . . . , An B only if τ (A1 ), . . . , τ (An ) CL τ (B). (Note that we do not require “if and only if” here – a faithful translational embedding of into CL . The compositionality of τ here means that there is in the language of CL some formula C(p), in which p is the only variable occurring, for which τ (§A) = C(τ (A)). See 6.42.3, p. 873, for another such example.) The reason for this is that there is no truth-function f with the property that f composed with itself gives the two-valued negation truth-function (¬b or N, as we have variously called it). Still, one may ask about the prospects for a as just described. One might even insist, perhaps to ensure that § is treated as a genuine connective in its own right and not just one half of an orthographic block, “§§” – a fancy new way of writing “¬” – that § should be congruential according to . If this demand were satisfied, one might be persuaded that there really was such a connective as §. As it happens, we will not return to this example in the present chapter or in our discussion of negation (Chapter 8), though further information on it may be gathered from sources given in the end-of-section notes (under ‘Demi-negation’, p. 576). The kind of question, just illustrated, as to whether or not there exists a connective satisfying certain specified conditions gained prominence as a result of Prior [1960], in which the conditions were given in terms of introduction and elimination rules for a proposed (or better: supposed) new binary connective Tonk. (The present discussion was foreshadowed in the remarks immediately

4.2. EXISTENCE OF CONNECTIVES

537

after 1.18.9, p. 86.) As an antidote to some views of the type considered in §4.1, according to which providing such rules can be seen as giving the meaning of the connective they govern, Prior considered a stipulation to the effect that the meaning of Tonk was to be that conferred upon it by the pair: (Tonk I)

A A Tonk B

(Tonk E)

A Tonk B B

Here, as in Example 4.14.5, the rules have been presented in the formula-toformula format; the intention is to indicate the sequent-to-sequent rules passing from Γ A to Γ A Tonk B, and from Γ A Tonk B to Γ B. Thus we have a combination of one of the ∨-introduction rules and one of the ∧-elimination rules stated for this new connective Tonk. Of course by applying (Tonk I) and then (Tonk E) we can get a proof of the sequent A B for arbitrary formulas A and B, and by drawing attention to this untoward conclusion, Prior is attempting to call into question the idea that one can simply stipulate that a connective is to have whatever meaning is dictated by its being governed by such and such rules. In the present instance, it seems, there is simply no such meaning to be had. (Note that views deriving from Gentzen’s, according to which the totality of introduction rules governing a connective encapsulate the meaning of that connective, and then determine as appropriate a particular collection of elimination rules, are not vulnerable to this attack, since (Tonk E) would not be amongst those delivered by any plausible account as appropriate elimination rules on the basis of {(Tonk I)}. See 4.14 for example. Prior cites as potential targets for his argument Popper [1947a] and Kneale [1956].) Note, in passing, that the above example shows how in Set-Set, no uniform reading of the commas employed to display sequents can be given which represents the comma as a sentential connective (for example, as a binary connective for which the idempotence, associativity, and commutativity equivalences hold). For by (R) and (M), we prove A A, B and also A, B B. If “A, B” could somehow be construed as a formula compounded from A and B by a connective “,” then (T) would yield: A B. Prior’s example equips Tonk with just enough of the properties such a would-be comma connective needs to have for this reasoning to go through. (Cf. 1.16.1.) See the notes to this section (‘Comma Connectives’: p. 577) for pointers to the literature in which commas (inter alia) are construed as something called structure connectives rather than sentence (or formula) connectives. In an influential response to Prior’s discussion, Belnap [1962], attention is drawn to the fact that when pondering the question of whether there can intelligibly be presumed to exist a connective answering to certain rules, one is considering this question from the perspective of some given logic, taken as at least the correct logic for the connectives in its language. Digression. We will call someone viewing a given logic as correct in this way an adherent (or partisan) of the logic in question, but the same caveat as at the start of 2.31 applies here: a more refined discussion would arguably make all such talk application-relative, to accommodate the position of the logical pluralist – see Beall and Restall [2006] – according to whom different applications call for different logics and there need be no such thing as a once-and-for-all correct logic. See also the review Humberstone [2009b]. End of Digression.

538 CHAPTER 4.


Setting aside the considerations of the above Digression, then, and identifying the given background logic with a Set-Fmla proof system or even—as Belnap prefers—its associated consequence relation , Prior’s discussion assumes that it is not the case for all formulas A, B, in the language of , that A B. (We are assuming that the proof systems under consideration are closed under (R), (M) and (T).) And this is reasonable enough if this is supposed to be (a formalized version of) a logic we are prepared to treat as correct. Thus, the untoward effect of adding the above rules for Tonk is that in the consequence relation, + , associated with the extension of the original proof system by (Tonk I) and (Tonk E), we shall have A + B even for Tonk-free formulas A and B. So Belnap’s conclusion is that what goes wrong in the case of the Tonk rules is that they yield a non-conservative extension of the original logic. Now, a logic here could either be taken to be either a proof system or the associated consequence relation (since we are in Set-Fmla), and as was indicated in 3.18, this makes a difference to whether imposing certain conditions governing a new connective yields a conservative extension or not. For the present example, we can think of this notion as applying at the level of consequence relations, and say that a consequence relation 2 whose language properly includes the language of a consequence relation 1 extends the latter relation when 2 ⊇ 1 , and that the extension is conservative when for any set Γ ∪ {C} of formulas of the language of 1 : Γ 2 C only if Γ 1 C. (Since 2 ⊇ 1 , we could equally well put this with “if and only if” in place of “only if”.) In other words, a consequence relation is extended conservatively when, even though new cases of one formula’s following from others are admitted, this never happens when all of the formulas involved belong to the old language. In practice several ways are encountered of specifying the extension 2 whose conservativeness over 1 is at issue, but the commonest is that which defines 2 to be the smallest consequence relation on its language extending 1 and satisfying certain rules. Though this is not always the situation (see for instance 4.22), when it is, the collection of rules concerned—some of which typically involve (inter alia) the new vocabulary of the language of 2 —is itself described as extending 1 (or perhaps as extending some set of rules which present a proof system whose associated consequence relation is 1 : cf. §1.2 Appendix, p. 180) either conservatively or non-conservatively as the case may be. Often there will also be an explicit understanding that we are to consider 2 as the smallest congruential and/or substitution-invariant consequence relation closed under the rules specified. Digression. The notion of conservative extension made precise above is clearly only one of several informal ideas of ‘non-disruptiveness’ of extensions that could be considered. For example, we could add a new rule to a proof system which involves only connectives already present, though not all of those. Then the extension may ‘disrupt’ relations amongst the connectives not involved (not ‘governed’ by the rule, in a sense to be made precise in 4.33), in the sense that new sequents involving only those connectives may be provable in the extended system which were not provable in the original. Although there is, informally speaking, something non-conservative about such an extension, it would not


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count as a non-conservative definition by the technical definition given above. We shall not have occasion to consider this notion further. Another example of arguably untoward disruptiveness is what we shall be calling failures of the condition of ‘conservation of synonymy’ in 8.23; again, such failures do not amount to non-conservative extension in the technical sense. End of Digression. It is perhaps clear from the above description that conservative extension in Set-Fmla may be considered without the restriction to consequence relations, so that logics not closed under the appropriate structural rules can be treated. (An example will come up in the discussion following 4.21.3.) In this case the references to consequence relations above should be replaced by references to the collection of sequents of the logic(s) concerned. Again, we may say, derivatively, that one proof system (or presentation thereof) is a conservative extension of another when the collections of provable sequents stand in the conservative extension relation. It is clear also how the concept of conservative extension should apply to logics in Set-Set (or to gcr’s) and in Fmla: simply put “Δ” for “C”, or erase the “Γ”, respectively, in the explanation given in the preceding paragraph. More explicitly: adding rules governing (perhaps inter alia some (typically) new connective extends a proof system non-conservatively when new sequents become provable with the aid of these rules, some of which do not contain the connective in question (as we saw in 3.18, when the →Introduction rule was taken still to be in force in the presence of the new rules). The double life of the notion of conservative extension – applying to consequence relations and also to proof systems – is much like the similarly double life led by the notion of structural completeness, and behind it lies the same explanation. (See the discussion following 1.29.6, p. 162.) A fuller treatment of the topic of conservative extension will be found in Humberstone [2011b], which provides a more nuanced discussion of issues glossed over here. Belnap’s proposal, then, is that a new connective may be introduced as subject(ed) to certain rules—and so, a connective in the ‘logical role’ rather than ‘syntactic operation’ sense (1.19) be acknowledged to exist—provided that the rules in question yield a conservative extension of the unextended logic one takes to be correct for the formulas of the unexpanded language. The vague word “correct” here has two sides to it, a soundness side (only those sequents that ought to be provable are provable) and a completeness side (all those sequents that ought to be provable are provable) which in valuational, model-theoretic, many-valued, or algebraic semantics amount respectively to capturing only, and all, the valid sequents for some given notion of validity. In §4.1 we mentioned some proof-theoretically defined analogues of these notions which would be more likely to appeal to someone pressing the current theme of giving the meaning of a new connective by saying what rules it was to obey. It is clear that the side that is doing the work here in making Belnap’s criterion of existence plausible is the completeness side: it is only if we are satisfied that all the sequents of the original language that ought to be provable are already provable, that we will have an objection to an extension merely on the grounds that it is nonconservative. (Belnap [1962] has been differently interpreted elsewhere; see the end-of-section notes under ‘Tonk’, p. 576.) Thus, for example, if we think of the subsystem of Nat based on the introduction and elimination rules for ∧, ∨, and →, as the original logic, we know that adding the rules for ¬ produces a non-conservative extension, since they

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make possible a proof of previously unprovable, yet ¬-free, sequents such as (p → q) → p p (Peirce’s Law: 2.32), p → r, (p → q) → r r (a sequent version of the intuitionistically unacceptable part of the condition of →-classicality from 1.18, p. 84, discussed in a variant form under the name (Subc)→ ) in 7.21.7, p. 1060, and elsewhere), as well as (p → q) ∨ (r → p), or indeed this last sequent in the more specific (‘LC’) form (p → q) ∨ (q → p). But this would not bother someone wielding tautologousness as the preferred notion of validity for sequents, since the original proof system was obviously incomplete w.r.t. this notion. The problem such examples raise is, however, acute for a theorist who thinks of the validity of a sequent as arising from the meanings of the connectives it involves, and of the meanings of those connectives being given by their (standard) introduction and elimination rules in a Set-Fmla natural deduction system. This theorist simply cannot endorse classical sentential logic, in view of such supernumerary sequents as cited above, and regardless of any initial inclinations as to the acceptability of the classical ¬-rules, must deny the intelligibility of the connective they purport to govern in the light of the ¬-free sequents they bring (non-conservatively) in their wake. We saw already, in 3.18, how the standard natural deduction proof system for IL was non-conservatively extended by rules governing new connectives to reflect what we called ‘intuitionistically dangerous’ pairs of determinants. §4.3 (esp. 4.32) will provide several further examples of non-existence claims justified by an adherent of some logic which would be non-conservatively extended by the admission of a connective with certain logical powers. Here we begin with an example inspired by Lucas [1984], which mounts a light-hearted attack on intuitionistic logic, and includes (p. 68f.) the remark à propos of the law of double negation, that “it is inherent in a natural language, which is used to listen as well as to talk with, that there should be a principle of cancelling, and of cancelling cancellations”. Now while of course we do not in general have, for = IL , ¬¬A A, so that the outer occurrence of “¬” cannot itself do the work of ‘cancelling’ the inner occurrence, this raises the question of whether or not some other connective might be envisaged as intuitionistically intelligible—even if not already definable in terms of the usual intuitionistic connectives—which did have this ¬-cancelling property: Examples 4.21.1(i) Suppose that we extend IL by a singulary connective # which has the property that for any formula A, we have #¬A A for the consequence relation of the extended logic; not only do we assume this for our ⊇ IL , but let us suppose further that is congruential. Then the extension in question is not conservative. For, by 2.32.1(iii) from p. 304 (the ‘Law of Triple Negation’), ¬p ¬¬¬p, since ⊇ IL ; thus applying # to both sides (by congruentiality) and peeling off the prefix “#¬”, we get: p ¬¬p. This is perhaps not an especially telling non-existence argument because of the hypothesis of congruentiality, which is essential: see 8.23 for the (non-congruential) logic of strong negation, a connective with precisely this ¬-cancelling power. (But see also the comment preceding 4.22.5 below, for the rationale for such congruentiality conditions.) Most of the examples that follow do not depend on assuming congruentiality. (ii) From an algebraic perspective the example discussed under (i) can be described as the question of the conservativity of extending IL with


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a connective serving as a left inverse for ¬. This naturally suggests the corresponding question about a right inverse for ¬, which we address here despite its no longer being motivated by the idea of cancelling a negated claim to recover the claim negated. (For more on left and right inverses for 1-ary connectives, see the discussion after 4.22.23 below.) Here we want singulary # satisfying, according to some ⊇ IL : ¬#A A for all formulas A in the language of . Again the Law of Triple negation answers our question negatively, subject to a further hypothesis isolated below. Doubly negating both sides of the above equivalence, we get: ¬¬¬#A ¬¬A. By Triple Negation, the left-hand side is -equivalent to the left-hand side of the earlier equivalence, so the right-hand sides would have to be equivalent for arbitrary A, including #-free formulas (e.g., p). So the extension is non-conservative over IL . But what entitles us to apply ¬¬ to both sides of the original equivalence, as we have just done? Is this not just another assumption of congruentiality for ? Well, a less radical hypothesis will serve here, unlike in (i), namely: that is substitution-invariant. Then we can use the fact that ¬p IL ¬¬¬p and the fact that IL ⊆ to conclude that this equivalence, and hence its substitution instance with arbitrary #A for p, holds for . A well-known fact about classical and intuitionistic logic is that implications with conjunctive consequents behave like conjunctions of implications: C → (A ∧ B) (C → A) ∧ (C → B). In the discussion after 1.29.16, we put this by saying that for the logics in question, ∧ is consequent-distributive. On the other hand, according to these consequence relations, implications with disjunctive antecedents behave like – not disjunctions, but again – conjunctions of implications (A ∨ B) → C (A → C) ∧ (B → C). This suggests the question of what happens if the compound in A, B, here is both antecedent and consequent: i.e., if we consider ↔-formulas rather than →-formulas. Here we pose that question: Exercise 4.21.2 (i) Investigate the possible existence, for extending IL or for extending CL , of a binary connective ∗ satisfying: (A ∗ B) ↔ C (A ↔ C) ∧ (B ↔ C). (Suggestion: make trouble for even the left-to-right direction alone of this -condition by considering what happens when C is A ∗ B.) (ii ) Make a similar investigation of the prospects of ∗ satisfying (A ∗ B) ↔ C (A ↔ C) ∨ (B ↔ C).

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In a similar vein, but returning to conditionals rather than biconditionals, note now the discrepancy between classical and intuitionistic logic in that for = CL (but not for = IL ) we have both of these equivalences: C → (A ∨ B) (C → A) ∨ (C → B) (A ∧ B) → C (A → C) ∨ (B → C). This naturally suggests: Exercise 4.21.3 (i) Investigate, from the point of view of an adherent of IL, the possible existence of a connective ∗ satisfying: C → (A ∗ B) (C → A) ∨ (C → B). (ii ) From the same perspective, consider the existence of a connective ∗ satisfying: (A ∗ B) → C (A → C) ∨ (B → C). (Hint: show that the weakest extension of IL having ∗ as in (i) or (ii) includes the intermediate logic LC (2.32).) We return to the theme of this exercise and the discussion leading up to it in 4.22. Several examples of non-conservative extension appeared in §2.3. The proof system RMNat (2.33) based on (R), (→I), and (→E) was non-conservatively extended by the addition of the rules (∧I) and (∧E), since with their aid, though not before, we could prove the ∧-free p q → p. Another interesting example of non-conservative extension by means of apparently uncontentious principles governing conjunction (which might have been expected to be amongst the ‘safest’ additions one could make) turned up in work by R. A. Bull and A. N. Prior in the study of the implicational fragment of IL ( = the implicational fragment of PL) and some of its extensions in the 1960’s. (See Prior [1964b], and Kirk [1981] for a semantic gloss on what is going on in this example; see also Humberstone [2005b], [2011b].) The logical framework is Fmla, in which Bull described two logics LIC and OIC. The former adds to a complete axiomatization (using Modus Ponens and Uniform Substitution)) of the →-fragment of IL the formula: ((p → q) → r) → (((q → p) → r) → r); the result is an axiomatization of the →-fragment of the intermediate logic LC. For OIC we add instead as a new axiom: (((p → q) → q) → r) → (((p → q) → r) → r). It turned out – and again, the details may be found in Prior [1964b] – that although (the set of theorems of) OIC is properly included in LIC, if we work in the {∧, →}-fragment instead, with the IL-provable conjunction-implication principles available, then the two logics coincide. In other words, while in OIC there is no proof of the LIC axiom ((p → q) → r) → (((q → p) → r) → r),


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such a proof becomes available if we help ourselves to IL-properties of ∧, so that OIC is non-conservatively extended by the addition of principles stating those properties ((p ∧ q) → p, etc.). (This example of non-conservative extension raises some puzzles, discussed in Humberstone [2005b]. For further information, see Humberstone [2011b].) We now return to Set-Fmla. As noted in 2.31, the system QNat for ∧ and ∨, using (∨E)res as the elimination rule for the latter connective, was non-conservatively extended by the addition of (→I) and (→E), since with the aid of these rules we could prove the QNat-unprovable and →-free p ∧ (q ∨ r) (p ∧ q) ∨ (p ∧ r). Examples have already been given (Peirce’s Law, etc.) of how INat is non-conservatively extended by the addition (giving Nat) of the new rule—this time, one governing already a connective already present—(¬¬E). We turn to another well-known example, drawn from intuitionistic predicate logic. (The rules governing the quantifiers, for example in Lemmon [1965a], are the same for IL as in CL, i.e. do not differ as between quantified INat and quantified Nat.) Example 4.21.4 The ¬-free sequent ∀x(B ∨ Fx ) B ∨ ∀x(Fx ) is provable in classical but not in intuitionistic predicate logic, for any closed formula B. (We return to this example in 4.21.9.) Remark 4.21.5 This undermines the ∀:∧ analogy (mentioned in 2.32) for IL. The sentential analogue of 4.21.4 (taking the ∀ as a conjunction with two conjuncts) is (B ∨ p) ∧ (B ∨ q) B ∨ (p ∧ q), which, like any other sequent provable in CL involving only ∧ and ∨, is provable in IL. (If we were considering predicate logic—as indeed we do briefly in a moment—we could add ∃ to this list also, though the list could not be further extended by others from amongst the standard primitive connectives and quantifiers of intuitionistic logic. See Theorem 1 and comments following it in Leblanc and Thomason [1966].) Thus, however this analogy is spelt out (no easy matter, in fact) so that it holds for the provable sequents of CL, it must fail – even for CL – when we consider instead the proofs of those sequents. Or, more accurately, that this is so as far as proof theory in the natural deduction approach and the Set-Fmla framework is concerned. For, from the intuitionistic unprovability of ∀x(B ∨ F x) B ∨ ∀x(F x), we know that any classical proof, in say (quantified) Nat, will need to appeal to (¬¬E), even though negation does not appear in the sequent under consideration. (Of course B may contain ¬; but it need not.) Yet no such detour through ¬ is required for the proof of the sentential analogue of this sequent (in CL or IL). So the analogy between ∧ and ∀ is less close, even for classical logic, when we turn our attention to how sequents are proved, than it appears when we consider only which sequents are provable. Incidentally, an explanation of why the inference represented by the sequent in Example 4.21.4 should be regarded as intuitionistically unacceptable may be found in Dummett [1977], p. 30f.; it requires an account of the assertibility conditions of universally quantified statements, not provided in 2.32. Digression. It can happen that a proof system for a predicate logic is nonconservatively extended by the addition of rules which extend the corresponding

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sentential proof system (i.e., that obtained by deleting the quantifier rules from a suitable presentation of the predicate-logical proof system) conservatively. In the case of IL this is illustrated in 4.21.9 below. Another example of how the nonconservativity of a sentential modification can show up other than at the level of sentential logic is through its interaction with the rules for identity (“=”); see Humberstone [1983b]. End of Digression. Having already mentioned one example (that of RMNat) in which the usual conjunction rules give a non-conservative extension, we include a somewhat different illustration of this same possibility. (The example can be ‘dualized’ so as to provide a corresponding illustration for disjunction: see the references cited below.) Example 4.21.6 Consider the language with ∨ (binary), O1 , O2 (both singulary), and the proof system in Set-Fmla on this language with the structural rules (R), (M), (T), and the rules (∨I) and (∨E), together with the ‘monotone’ rules from A B to Oi A Oi B (i = 1, 2), and the zero-premiss rule A O1 A ∨ O2 A. If we add ∧ to the language and (∧I), (∧E) to the proof system, we can prove p, q O1 p ∨ O2 q, in which ∧ does not appear, though this sequent was not provable in the original ∧-free system. (For further discussion and ramifications, see Humberstone [1985], [1990], [2005b] and [2011b].) The example with which the first part of following Exercise is concerned is the impure rule governing a singulary connective O, as well as ¬: Γ, ¬A OA ΓA If we ask whether Nat would be conservatively extended by the addition of such a rule for the new connective O, we can see that the answer is affirmative, since OA can already be interpreted either as ⊥ or as A, and so interpreted we have a derivable rule of Nat. For IL, however, the situation is different, as is indeed suggested by the ‘oblique’ (4.14) treatment this rule accords to ¬: Exercise 4.21.7 (i) Show that the above rule gives a non-conservative extension of INat (2.32). (ii ) Consider a similar rule but with “¬A” on the left of the premisssequent replaced by “A”; show that the extension of Nat by the addition of this rule would be non-conservative. Our final examples of conservative and non-conservative extension in the present subsection involve the concept of the ‘residual’ connective corresponding to a given binary connective, a concept taken from Dunn [1991]. (Actually, the present concept is not quite either Dunn’s ‘left residual’ or ‘right residual’, though we shall not go into these matters here. Further, the name residual is normally given to the result of applying the operation in question, rather than, as below, to this operation.) Given two binary connectives #1 and #2 in the language of a consequence relation , we say that #2 is a residual of #1


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(according to ) when for all A, B, C: A B #2 C if and only if A #1 B C. For example, → is a residual of ∧ according to CL as well as IL . It is not hard to see that, to within equivalence () of the compounds they form from given components, the residual of a connective is unique, and similarly there is at most one connective of which any given connective is the residual. (In the terminology of §4.3, the following pair of rules, the double line, as in 1.28, indicating the upward and downward direction: A B #2 C A #1 B C uniquely characterize each of #2 and #1 in terms of the other.) Thus we can say that according to CL (or IL ) that → is the residual of ∧. On the other hand, there is no connective of which ∧ itself is the residual according to these consequence relations, and the two-way rule just schematized would non-conservatively extend all but the most pathological proof system (taking #2 as ∧, and #1 as some new connective). In fact, the upward direction alone is sufficient for this effect, with (∧E) being all that is needed for ∧. For, starting with A #1 B A #1 B, we proceed to apply the above rule (upward), obtaining A B ∧ (A #1 B), whence A B. Since A and B are arbitrary formulas here, the effect is as bad as the addition of the Tonk rules. A similar proof will answer (i) of the following Exercise. For part (iii), we need the following concept. Say that #2 is a generalized residual of #1 (according to ) when for all formulas A, B, C, and sets of formulas Γ (in the language of ): Γ, A B #2 C if and only if Γ, A #1 B C. Exercise 4.21.8 (i) Show that if is an ∨-classical consequence relation according to which some connective is a residual of ∨, then A B for all formulas A and B (in the language of ). (ii) Show that according to CL , if A #1 B = A ∧ ¬B then #1 has ∨ as its residual, and that this is not so for IL . (iii) Show that if is →-intuitionistic, extends IL , and has some connective with ∨ as its generalized residual, then extends IL nonconservatively. (Hint: start with the fact that p, r → r p ∨ q, and so p, (r → r)#1 p q, where #1 is the connective with ∨ as its residual. Since is →-intuitionistic, it follows that (r → r) #1 p p → q. It remains only to move the p on the left across to the right and delete the “r → r”, recalling that IL p ∨ (p → q).) (iv) By contrast with →, which we have already noted is a residual of ∧ according to CL , show that no connective has ↔ as a residual in any conservative extension of CL . Digression. The notion of residuation is algebraic in origin. A structure (A, ·, →, ), in which (S, ·) is a commutative semigroup expanded by a binary operation →, together with a partial order (on S), is said to be residuated (with → as the residual of ·) when for all a, b, c ∈ A we have: a b → c if and only if a · b c.

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(There are again left and right versions of this concept which we are allowing ourselves to blur by building in commutativity for ·.) Similar structures arise also as algebras in their own right when the partial order in question is ‘equationally definable’. The most popular case comprises the residuated lattices which are the expansions of bounded lattices by binary operations ·, →, satisfying the above equivalence when is the usual induced lattice ordering. (See 0.13.1, p. 8, and preceding text; semilattices would do equally well for this purpose.) Authors differ as to whether the top element, 1, of the lattice should also be an identity for the semigroup operation. If not, then the latter can have its own identity element e, the semigroup reduct will normally be taken as a commutative monoid instead, and 1 and e are algebraic analogues of the T and t constants of such developments as were given in 2.33 above. As that suggests, residuated monoids and residuated lattices have figured prominently in the algebraic semantics of substructural logics. See Blok and van Alten [2002], Raftery and van Alten [2000] (where an additive notation is used for the monoids), Ono [2003], and – for a book-length exposition – Galatos, Jipsen, Kowalski, and Ono [2007]. End of Digression. 4.21.8(iii) above shows that we cannot conservatively extend INat with rules like those inset for #1 and #2 above but with a variable “Γ” for side formulas sitting on the left above and below the double line. (This would be a rule, unlike that given above, fully general in respect of side formulas for Set-Fmla.) This leaves open the question of whether such an extension is possible for the original rule, without the “Γ”. Part (ii) says that A ∧ ¬B does not have A ∨ B as its residual, leaving it open that some other derived connective of IL might do the job. In fact there is none, though one can be conservatively added. Appropriate truth-conditions for such a #1 , which we shall write as →d (usually called dual intuitionistic implication – see the Digression in 8.22: p. 1225) in Kripke models can be provided by deeming A →d B true at a point x in a model (W, R, V ) iff for some y ∈ W with yRx, A is true at y but B is not true at y. The dual intuitionistic negation of A, studied (under the notation ¬d A) in 8.22, then amounts to →d A. The conservativeness of the extension is easily established by the method of 8.22.5, which is to say: by providing a semantics w.r.t. to which the original system (INat) is complete and a semantics w.r.t. to which the extended system (INat + the earlier inset rules with #1 as →d and #2 as ∨) is sound, with the former and latter semantics agreeing as to the validity of sequents not containing the new connective. This result does not hold in the presence of the usual quantifier rules, however, illustrating the point made in the Digression after 4.21.5 above. See the papers by C. Rauszer cited in the notes to §8.2; the present point is emphasized in López-Escobar [1985], along with the proof in 4.21.9. The first appearance of dual intuitionistic implication known to the present author is Moisil [1942], where it is called exception (in French); see (2.1) and (2.2) in the review Turquette [1948]. It appears again in Johansson [1953], where it is called subtraction, as it is in Crolard [2001], [2004]. See also Brunner [2004] and Wansing [2008]. There is an extensive discussion in Wolter [1998], where the term “coimplication” is used – though this use of the prefix “co-” to indicate duality risks a confusion with the use of the term coimplication in some quarters – compare the use in Anderson and Belnap [1975] of the term “co-entailment” to mean mutual implication, or the use (remarked on in the notes to §3.3, p. 508) by Fitch of “strict coimplication” to mean strict


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equivalence. Wansing [2008] employs the same terminology. There are a few asides on the topic in 8.22. (Wansing [2010] calls dual intuitionistic negation co-negation, for which there is no corresponding risk of confusion.) Readers with no interest in matters concerning quantification should skip the following Remark. Remark 4.21.9 (López-Escobar [1985].) With →d having ∨ as its residual, we can prove the sequent(-schema) mentioned in 4.21.4 as not being provable in intuitionistic predicate logic; how each line follows from its predecessor will be clear enough, so we have not annotated the proof: ∀x(B ∨ Fx ) ∀x(B ∨ Fx ) ∀x(B ∨ Fx ) B ∨ Fa ∀x(B ∨ Fx ) →d B Fa ∀x(B ∨ Fx ) →d B ∀x(Fx ) ∀x(B ∨ Fx ) B ∨ ∀x(Fx ) (Here B is some formula not containing the parameter a – otherwise just choose a different parameter, so the step from the third to the fourth lines is justified.) Accordingly what was a conservative extension at level of sentential logic no longer has this status for predicate logic. For another example of this phenomenon, see 4.38.10 (p. 626). We conclude our introduction to the topic of conservative extension by mentioning a concept connected both with this and with the notion of purity of rules, from 4.14; for the idea of presentations (or bases) of proof systems, see the Appendix to §1.2, p. 180. A presentation of a proof system in Set-Set or Set-Fmla has the separation property if its rules are pure and any provable sequent has a proof using (apart from the structural rules) only such rules as govern the connectives appearing in the sequent. Derivatively, a logic in any of the other three senses distinguished (in terms of criteria of individuation) in the Appendix to §1.2 – p. 180 – has the separation property (or ‘is separable’) if it can be presented via a basis with the separation property. Separability is a mark of the amenability of a logic to a presentation which separately gives the inferential powers of its various connectives in such a way that all its provable sequents are just the results of these separate accounts, as they pertain to the connectives in any given sequent, interacting to yield a proof of that sequent. The connection with conservative extension is clear enough. The fact that adding the rules of Nat governing negation ((¬¬E) and (RAA), preferably taken in the purified form given in 4.14) produces new ¬-free sequents (Peirce etc.) not previously provable, cited above to show the non-conservativeness of this extension to the ¬-free subsystem, reveals a lack of the separation property for the total set of rules. We should also note that the separation property follows for any sequent calculus proof system for which the Cut Elimination theorem holds – see the end of 1.27 – via the subformula property. (This point was made already in the discussion after 2.33.25, p. 362.) Digression. In Fmla, the separation property is understood as a less stringent condition, in view of the pervasive role of (typically) implication as a connective to lubricate the deductive machinery. It is generally assumed that all languages under consideration possess →, and that Modus Ponens is a rule of any proof

548 CHAPTER 4.


system. An axiomatization is then deemed to have the separation property if any provable formula has a proof from axioms involving, apart from →, only such connectives as appear in the formula in question. End of Digression. Issues of separation will not be discussed (as such) further here; some references have been included in the notes, for those wishing to follow up the topic.

4.22

Conditions not Corresponding to Rules: An Example

Not even every universally quantified ‘syntactic’ condition on a consequence relation (or gcr, or set of formulas), can be replaced by a requirement of closure under certain rules, since such a condition may not lend itself to reformulation as what in 1.13 (p. 63) we called a metalinguistic Horn sentence ‘of the second kind’, or a set of such sentences. Here we illustrate this possibility as arising through a condition with an irremediably disjunctive consequent. (Note: this amounts to a metalinguistic disjunction of -statements and so has nothing to do with the distinction between consequence relations and gcr’s.) After that we continue in the same vein as 4.21, with conditions which, the title of the present subsection notwithstanding, do correspond to rules (4.22.4 onwards). If is an ∨-classical consequence relation then a formula follows from a disjunction if and only if it follows from each disjunct: A ∨ B C iff A C and B C. (This example was anticipated in the discussion after the proof of Obs. 0.14.7 on p. 15.) Consider the possibility of a binary connective $ which forms compounds from which a formula follows iff it follows from – not each but instead either – component. That is, impose the following condition on a consequence relation whose language has $ as a binary connective: ($)

A $ B C ⇔ A C or B C

(We use the same notation as in the discussion between 1.13.1 and 1.13.2 (p. 63), where a stronger version of the above condition was cited, having an additional set variable “Γ” for ancillary assumptions on the left-hand side.) We can put the contrast between $ (subject to ($)) and ∨ (for ∨-classical ) neatly in terms of the consequence operation associated with (1.22.1, p. 113): (1) Cn(A ∨ B) = Cn(A) ∩ Cn(B)

(2) Cn(A $ B) = Cn(A) ∪ Cn(B)

(Note that we write “Cn(A)” for “Cn({A})”.) The Cn formulation of ($) as (2) here may look a little suspicious, since unions of Cn-closed sets are not generally Cn-closed; but if the condition is indeed satisfied, then since Cn(Cn(C)) = Cn(C) for any formula C and any consequence operation Cn, applying Cn to both sides gives the result that (3) Cn(A) ∪ Cn(B) = Cn(Cn(A) ∪ Cn(B)) for any formulas A and B. (The new rhs could be written Cn(A) ∪˙ Cn(B), if preferred, in the style of 0.13.5, p. 10.) Thus from (2)’s being satisfied it follows that the condition (4) is also satisfied: (4) Cn(A $ B) = Cn(Cn(A) ∪ Cn(B)). Now by 0.13.4(i) from p. 9, the right-hand side can be simplified further: (5) Cn(A $ B) = Cn({A} ∪ {B}).


549

But this is clearly just another way of writing (6) Cn(A $ B) = Cn({A, B}) which says that the consequence relation associated with Cn treats $ ∧-classically. (See further, 5.11. Note that we have not claimed that (1) above is equivalent to the ∨-classicality of Cn, which it is not; see further 6.11.) Any earlier suspicions should at this point be re-awakened, since the conditions of ∧-classicality already characterize any connective satisfying them uniquely—as indicated at the start of 4.31 below—and clearly do not have (2) (or ($)) as a consequence. The untoward effects of this are recorded in 4.22.1 below. We are in a situation analogous to that to be treated in greater detail in our discussion of unique characterization, under the heading of rules which are stronger than needed for that purpose (4.32, 4.34) except that, as we now indicate, the condition currently at issue cannot be construed as the condition of being closed under a set of (sequent-to-sequent) rules. In terms of a proof system for which is the associated consequence relation, the ⇐-direction of the condition ($) amounts to the admissibility of the SetFmla rules: AC A$B C

and

BC A$B C

(In fact such rules are admissible for a Set-Fmla logic closed under the structural rules just in case A $ B A and A $ B B are provable, so there is no admissible/derivable contrast here.) But the ‘disjunctive consequent’ feature of the ⇒-direction of ($) means that there is in this case no corresponding rule (or set of rules). Put another way: there will be in general no such thing as the least consequence relation extending a given consequence relation 0 and satisfying ($). Thus, in particular, if 0 embodies a logic we deem correct for the connectives in its language, we cannot directly employ the conservative extension test to see if we should acknowledge the existence of a connective $ satisfying ($), since no extension has thereby been specified about which to ask if that extension is conservative (over 0 ). Slightly indirectly, though, we may use the conservativeness criterion to rule on the existence of $, since we can show that any extension of 0 which satisfies ($) is non-conservative, on the assumption – plausible enough if 0 is a consequence relation we are at all inclined to favour as correct – that there are formulas A and B for which neither A 0 B nor B 0 A. That ruling is negative: as we shall now show, ($) makes a consequence relation strongly connected, in the sense of 1.14.8 (p. 70). Observation 4.22.1 For any consequence relation satisfying ($) and any formulas A, B, in the language of , either A B or B A. Proof. Letting R be the binary relation holding between formulas A and B just in case A B (for as here); since R is a pre-order and ($) provides disjunctive combinations on the left, the result follows by 0.14.7. Returning to the situation in which we extend 0 , assumed not strongly connected, to any obeying ($), then, we need only select A and B in the language of 0 for which A 0 B and B 0 A, and conclude from 4.22.1 that the extension is non-conservative.

550 CHAPTER 4.


To relate the content of 4.22.1 to our earlier discussion ((2)–(6)), recall that (2), a reformulation of ($), implied (6), giving ($) all the expected inferential properties of conjunction. Thus in particular, we have the (∧E) principle that each of A, B is a consequence of A $ B, so since all of A $ B’s consequences are consequences of one or other of A, B, it follows that one of these must be a consequence of the other. Alternatively, if we concentrate on the (∧I) principle that A $ B follows from A together with B, then by the fact that Cn({A, B}) = Cn(A) ∪ Cn(B), since each is Cn(A $ B), A $ B must follow either from A or from B, so both A and B follow from either of them alone. (Compare 1.21.7.) Exercise 4.22.2 Consider a binary connective ∗ and the following condition on consequence relations: (∗)

Γ B ∗ C ⇔ Γ B or Γ C

Prove the result of substituting (∗) for ($) in 4.22.1. (Taking Γ in (∗) as {A} gives a condition dual to ($).) Exercise 4.22.3 The original Tonk example, and 4.22.1, 4.22.4, indicate the untoward results of certain stipulations – but there do at least exist consequence relations and connectives meeting those stipulations. Show by contrast that there can be no consequence relation and connective in the language of for which we have (for all sets Γ of formulas and individual formulas A of that language): Γ A ⇔ Γ A. Exercise 4.22.4 Show that the smallest normal bimodal logic (in Fmla: 2.22) with modal operators 1 and 2 is non-conservatively extended by the smallest normal trimodal logic containing all instances of the schema 3 A ↔ (1 A ∨ 2 A). (Hint: show that the extended logic proves the previously unprovable and 3 -free formula (1 p → 2 p) ∨ (2 q → 1 q).) With this last example, we have moved away from the title of the present subsection, in that we are no longer concerned with sequent-to-sequent (or claim to -claim) conditionals which may or may not be ‘strict Horn’ in form. We have moved from the vertical to the horizontal, in Scott’s metaphor. (See 1.26.) A horizontalized version of the earlier condition ($), in which we replace the metalinguistic conditionals by object-linguistic conditionals →, was in fact considered in the preceding subsection; it appeared in 4.21.3(i), but we repeat the formulation here: (A ∗ B) → C (A → C) ∨ (B → C) In general we may distinguish, in such →-distribution equivalences, the ‘inner’ connective (appearing in the scope of →) from the ‘outer’ connective. The inner connective may appear in the antecedent or in the consequent, giving respectively (1) and (2), in which #1 is the inner connective and #2 is the outer connective: (1) (A #1 B) → C (A → C) #2 (B → C) (2) C → (A #1 B) (C → A) #2 (C → B)


551

(We start the numbering afresh here, since there will be no further cause to refer back to the earlier (1)–(6). For a more systematic treatment of the patterns exemplified by (1), (2) and the like, see the theory of ‘distribution types’ in Dunn [1991], and §6.2 of Goré [1998a]. See also the remarks on intraposed/extraposed versions of connectives at the end of the present subsection.) Two kinds of existence questions arise in such a context. One might be interested, for a given #2 , in whether or not there exists a connective #1 behaving as required by (1) or (2). This was the theme of Exercise 4.21.3, which posed this question for (1) and (2) (in parts (ii), (i), resp.) with #2 given as ∨, understanding as IL and assuming the question to be settled by the conservative extension criterion. (In 4.23 we will query the adequacy of this criterion, but the conservative extension phenomenon is of sufficient interest in its own right to be worth pursuing in any case; of course one could pose a similar question with a different connective in place of →; for example, 4.21.2 had ↔ playing this role.) The second kind of question asks instead, for a given choice of #1 , whether or not there exists (sc. there can conservatively be added) a connective #2 behaving as required by (1) or (2). We will consider, inter alia, some instances of this second question, taking the inner connective #1 as ∧ or ∨. For the choices just mentioned of #1 we can consider the following forms of (1)–(2), in which, in conformity with the discussion of 4.21, we write ∗ for #2 , the outer connective of interest: (1∨) (1∧) (2∨) (2∧)

(A ∨ B) → C (A ∧ B) → C C → (A ∨ B) C → (A ∧ B)

(A (A (C (C

→ → → →

C) C) A) A)

∗ ∗ ∗ ∗

(B (B (C (C

→ → → →

C) C) B) B)

In the case of all four equivalences, CL allows us to ‘solve for ∗’ in the sense of providing an already available connective, some of which were already mentioned in the discussion before 4.21.2: for (1∨) we may take ∗ as ∧; for (1∧) we may take ∗ as ∨; for (2∨) we take ∗ as ∨ again; and for (2∧), ∗ is ∧. What about IL? For (1∨) and (2∧) the same solutions are available as in CL. Let us consider the remaining cases, (1∧) and (2∨), in this context. An easy argument shows for the case of (2∨) that the extension of IL with any connective ∗ satisfying this condition is non-conservative—provided that we require ∗ to be congruential according to the extension concerned—since it brings in its wake the additional strength of the intermediate logic LC (cf. 4.21.3), as the proof which follows reveals. When attending to the existence of such ‘outer’ connectives, the requirement of congruentiality is a natural one: little interest attaches, after all, to the suggestion that we simply avail ourselves, in the shape of (C → A) ∗ (C → B), of a new notation for what we would otherwise write as “C → (A ∨ B)”, with the parts of the notation not accorded their customary logical force. Observation 4.22.5 For any congruential consequence relation ⊇ IL , if satisfies (2∨) for all A, B, C: C → (A∨B) (C → A)∗(C → B) then ⊇ LC . Proof. We assume a formulation of IL with primitive (otherwise pick p → p for this role). Let be as in the statement of 4.22.5; then by (2∨),

552 CHAPTER 4.

EXISTENCE AND UNIQUENESS OF CONNECTIVES p → (q ∨ r) (p → q) ∗ (p → r);

by congruentiality, the rhs can be replaced by ( → (p → q)) ∗ ( → (p → r)). By (2∨) again, this is equivalent to → ((p → q) ∨ (p → r)), and hence to (p → q)∨(p → r). Thus for we have the ∗-free p → (q∨r) (p → q)∨(p → r), which does not hold for = IL , giving nonconservativity; the conclusion as given in 4.22.5 then follows by 2.32.10(i) (p. 318). The case of (1∧) presents greater difficulty. (It may be useful to consider whether a ∗ behaving as (1∧) dictates is conservatively addable to the language of IL before reading on. We work our way to answering this question over several paragraphs; the solution appears in 4.22.7.) As already remarked, for CL we may simply take ∗ as ∨; the result is to provide ammunition for the attack on material implication as a representation of natural language conditionals, . . . Suppose we consider the analogous conservative extension question for strict implication, here symbolized by , in a normal modal logic in Set-Fmla (2.23): (3)

(A ∧ B) C (A C) ∗ (B C).

In terms of , this reads: (4) ((A ∧ B) → C) (A → C) ∗ (B → C). Much as in the proof of 4.22.5, we need to rearrange the ∗-components on the rhs (assuming congruentiality) so as to re-invoke (4) a second time. But now, instead of aiming at a reformulation in which a common antecedent is exhibited, we need a common consequent. Such a recasting is available by exploiting the double negation equivalences, since the non-modal aspect of the logic is classical: (¬(A → C) → ⊥) ∗ (¬(B → C) → ⊥). The “ → ⊥” format is used so that we have ⊥ as a common consequent, rather than just writing, equivalently, ¬ . We now need to be specific about which modal consequence relation is, and for definiteness, though far from arbitrarily, let us take as S4 (in the notation of 2.23). Then the ∗-compound inset above is componentwise equivalent to (¬(A → C) → ⊥) ∗ (¬(B → C) → ⊥), or, in the “” notation: (¬(A C) ⊥) ∗ ¬(B C) ⊥. We can re-invoke (4) (or (3)), now with ¬(A C) and ¬(B C) for the A and B there, and ⊥ as C, to conclude that this is in turn equivalent to (¬(A C) ∧ ¬(B C)) → ⊥, which is to say: (A → C) ∨ (B → C). But we do not have, for example, ((p ∧ q) → r) S4 (p → q) ∨ (p → r), so by detouring through the properties of ∗ given by (3) or (4), we have nonconservatively extended S4. (The least extension of S4 for which the above


553

-principle does hold is the ‘Trivial’ modal logic KT!.) The choice of S4 was described as far from arbitrary. Inspection of the detail of the argument reveals that the nonconservativity result holds for any extension of K4! which is not an extension of KDc ; it therefore applies, for example in the case of S5 no less than S4. Since, as we have just seen, the strict implication analogue of (1∧) is prone to extend non-conservatively certain reasonable modal logics, one may well expect a similar result for the case of IL: the Kripke semantics for intuitionistic no less than for modal logic interprets the implication at issue by means of universal quantification over accessible antecedent-verifying points in the model, demanding that each such point verify the consequent. This makes the prospects for (1∧)’s conservatively extending IL seem bleak. It is instructive to address the case of (1∧) which we get by selecting C as ⊥, and rewriting the ___ → ⊥ as ¬___; this gives a De Morgan style principle as to whose conservativity over IL we may inquire: (5)

¬(A ∧ B) ¬A ∗ ¬B.

In CL we may take ∗ in (5) as ∨, though this is not so for IL. (In fact the least ⊇ IL for which it holds is KC , as was noted in 2.32.10(iii), p. 318.) However, for as IL , we do have: (6) ¬(A ∧ B) ¬¬(¬A ∨ ¬B), and this means that we already have ∗ available in the language of IL for which (5) holds, namely taking C ∗ D as ¬¬(C ∨ D). Does this help with the original problem of (1∧)? If we think of the occur→ ⊥, we note that the more general form (7) holds rences of ¬ in (6) as for as IL : (7)

(A ∧ B) → C (((A → C) ∨ (B → C)) → C) → C.

But this only means that we have a ternary connective # for which: (8)

(A ∧ B) → C #(A → C, B → C, C),

namely, taking #(p, q, r) as ((p ∨ q) → r) → r, and what we wanted was a binary connective ∗ as in (1∧). Remark 4.22.6 The point just reached may seem less informative than it is, since it might be thought that what we have on the left of (8) is after all itself a ternary compound. If we write # (p, q, r) for (p ∧ q) → r, then with # as above, we have simply equated a # -compound with a #compound, and since both # and # are ternary (derived) connectives, nothing has been gained. This is not quite accurate, however. For (8) does show something non-trivial, namely that for any formulas A, B and C, the proposition (in the sense of the discussion following 2.13.1, p. 221) expressed by (A ∧ B) → C is uniquely determined by the propositions expressed by A → C, B → C, and C (according to IL). The analogous claim in the context of modal logic would not be correct, for example. (See the Digression which follows.) Digression. We show that there is no way of conservatively extending S4 or S5 by the addition of a congruential connective #3 satisfying

554 CHAPTER 4.

EXISTENCE AND UNIQUENESS OF CONNECTIVES (A ∧ B) C #3 (A C, B C, C).

If there were, we should have as a special case the binary connective #2 arising by taking C = ⊥, for which ¬(A ∧ B) #2 (¬A, ¬B). Putting ¬A, ¬B, for A, B, and adopting infix notation for #2 , this would mean that we had, quite generally: (A ∨ B) A # B. In subsequent discussion, beginning after 4.22.11 on p. 556 below, we will call such a #2 an extraposed version of ∨ (across ), looking at the present example in more detail there (p. 558). In view of the S4-equivalence of A with A and B with B, by congruentiality, the equivalence just inset allows us to pass to (A ∨ B) (A ∨ B) which is not satisfied (in the direction) by S4 (or S5 ). End of Digression. Although (7) does not solve the problem of finding amongst the connectives definable in IL a binary ∗ for which (1∧) is satisfied, it does feature a construction which leads to such a solution (and so trivially to an affirmative answer to the conservative extension question), namely the pattern (D → E) → E (more on which appears in 4.22.10). It is simply a matter of finding this pattern in the right place: Observation 4.22.7 The condition on ∗ imposed by (1∧) above is satisfied, for as IL if we take D ∗ E as (D → E) → E (for any formulas D, E). Proof. It is a routine exercise (in proof construction in INat, or using the Kripke semantics for IL) to check that for as IL (A ∧ B) → C ((A → C) → (B → C)) → (B → C) for all A, B, C.

Thus the question of the existence, from the perspective of IL, of a congruential connective behaving as (1∧) requires ∗ to behave has now been settled affirmatively, in the strongest possible sense: not only is such a connective conservatively addable to IL – such a connective is already present in IL. We can use the (D → E) → E pattern further, in a slightly different way, to return a negative answer to the question of whether IL is conservatively extended by (1∧) for ∗ assumed not just congruential but monotone. (See 3.32.) For this result (4.22.9), we need a Lemma. Lemma 4.22.8 If in the language of ⊇ IL there is a connective ∗ satisfying (1∧) and congruential according to , then (p → q) ∗ (q → p). Proof. By (1∧), since for ⊇ IL we have (p ∧ q) → (p ∧ q) appealing to (1∧) with A = p, B = q, C = p ∧ q, we get (p → (p ∧ q)) ∗ (q → (p ∧ q)), the two implications in which are -equivalent respectively to p → q and q → p, the result follows by congruentiality.


555

Observation 4.22.9 If ⊇ IL and ∗ satisfies (1∧) and is monotone according to , then non-conservatively extends IL . Proof. If ∗ is monotone according to , it is certainly congruential according to , so we have: (p → q) ∗ (q → p). Since ⊇ IL , A (A → B) → B, so monotony gives (((p → q) → r) → r) ∗ (((q → p) → r) → r) By (1∧) and (T), then: ((((p → q) → r) ∧ ((q → p) → r)) → r Since this formula is not IL-provable and is ∗-free, the extension is not conservative. The formula cited at the end of this proof could be transformed into the equivalent form (((p → q) ∨ (q → p)) → r) → r, in which form it is easily seen to yield the original LC axiom (p → q) ∨ (q → p) upon substitution of this formula for each occurrence of r. Another variant: ((p → q) → r) → (((q → p) → r) → r) was used in Bull [1962] to axiomatize the implicational fragment of LC, as was mentioned in our discussion in 4.21 (the example of OIC and LIC after 4.21.3). Note that this last, purely implicational formula, would be written as (p → q) (q → p). in the notation of 1.11.3 (p. 50), where it was called the deductive disjunction of p → q and q → p. Remark 4.22.10 What we have been calling the (D → E) → E pattern deserves some simple notational embodiment of its own. Let us write ¨ E. The “∨” part of this notation serves as a resuch a formula as D ∨ minder that classically, though not intuitionistically, such a compound is equivalent to the disjunction of D with E. In an intuitionistic con¨ E as the pseudo-disjunction of D text we will sometimes refer to D ∨ with E. Disjunction was defined this way for classical logic in Russell [1937], p. 17. (This work was written in 1900 and first appeared in 1903. Ninety years later logic books were being published displaying their authors’ ignorance of the equivalence underlying this definition: see Humberstone [1995a].) The two dots on top are meant to be reminiscent the two occurrences of E in (D → E) → E. Note that, the ¨ is not commuleft-right symmetry of the notation notwithstanding, ∨ tative (in IL). Disjunction (∨), we noted – foreshadowing the present notation – in 3.15.5(iii) on p. 420, is definable in the intermediate logic ¨ B) ∧ (B∨ ¨ A). (The latter is abbreviated LC: A ∨ B can be defined as (A∨ to a notational variant of “A ∨ B” but with a single dot over the “∨” at p. 67 of Gabbay [1981]. The definability of disjunction along these lines ¨ , see the following was mentioned in Dummett [1959b].) For more on ∨

556 CHAPTER 4.


Exercise, as well as 6.24.11–12, 7.22, and 9.25; there were also some references to this connective in 2.13, especially at 2.13.13(ii) (p. 234). Note that unlike , mentioned at 1.11.3 (p. 50) and immediately before ¨ genuinely counts as a (compositionally) derived the present Remark, ∨ ¨ in (alias ‘defined’) connective. Because of the non-commutativity of ∨ IL it would be tricky to adapt the terminology of subcontrariey to this ¨ B was IL-provable, case and call A and B pseudo-subcontraries when A ∨ though perhaps the awkwardness could be reduced if the formulas concerned were described as mutual pseudo-subcontraries in this case; de Bruijn [1975] calls formulas so related orthogonal. In the general case an order-sensitive formulation would handle the difficulty: A is a pseudosubcontrary of B when the above disjunction is provable. ¨ is Exercise 4.22.11 (i) Show that the smallest intermediate logic in which ∨ ¨ ¨ p, commutative is CL itself. (Hint: Substitute p → q for q in p ∨ q q ∨ and note that the resulting sequent, when unpacked in terms of →, takes us from the intuitionistically acceptable Contraction Principle to the notoriously classicizing Peirce’s Law. An alternative though less informative argument puts ⊥ for q, making for a derivation of ¬¬p p; this is less informative because it takes us outside the implicational fragment of IL.) In the discussion leading up to 9.25.3 (p. 1320), we ¨ B and B ∨ ¨ A are not intuitionistically equivalent shall see that while A ∨ for arbitrary A and B, they are for any formulas A, B, which are what we there call “head-linked”. ¨ in place of ∨ is not admis(ii ) Show that a rule like (∨E) except with ∨ ¨ sible in IL. (Hint: Use part (i) of this exercise and the fact that the ∨ analogues of the two (∨I) rules are admissible.) Many of the examples of existence questions raised in this and the previous subsection fall under the following general pattern. Suppose that #E and #I are n-ary connectives (not necessarily primitive) of the language of a consequence relation and that O is a 1-ary connective. (“O”: mnemonic for “Operator”.) Then if for all formulas A1 , . . . , An in the language of we have O#I (A1 , . . . , An ) #E (OA1 , . . . , OAn ), we call #E an extraposed version of #I across O, and #I an intraposed version of #E across O (according to ). Thus for =CL , the De Morgan equivalences say (i) that ∨ is an extraposed version of ∧ across ¬ (equivalently: ∧ is an intraposed version of ∨ across ¬) and (ii) that ∧ us an extraposed version of ∨ around ¬ (equivalently: ∨ is an intraposed version of ∧ across ¬); see the discussion of (5) and (6) before 4.22.6 above for related questions, as well as the Digression before 4.22.7. Fixing on a consequence relation 0 and connectives #E and O we can always ask whether there exists a #I satisfying the condition inset above, either already in =0 or in some conservatively extending . We can ask similarly if there exists (either endogenously – i.e., already definable – or in some conservative extension) an extraposed version of a given #I across a given O. (We could also ask whether, for a given #I and #E there exists an O satisfying the inset equivalence.) Note that we speak just of an extraposed version of a given connective (across a given O): there is no suggestion of


557

uniqueness, which will be the chief business of the following section – not that any particular attention is given to extraposed/intraposed connectives there. The terminology just introduced could be extended so that O is allowed to be a 1-ary context rather than just a 1-ary connective; earlier examples like that discussed after 4.22.4 (originally in 4.21.3(i)) ask about the existence of a connective ∗ satisfying (A ∗ B) → C (A → C) ∨ (B → C), which in the present terminology amounts to asking whether, for an arbitrary → C in place of the 1-ary connective O, there formula C, taking the context is an intraposed version of ∨ across this context. (We spoke of inner and outer connectives instead of intraposed and extraposed connectives, and left the role of O tacit.) For the following discussion we reserve ∗ for use as the extraposed or intraposed version of a 1-ary connective, and # (with infix notation) in the binary case. And there will be occasional informal talk of extraposed negation or intraposed disjunction, etc., in place of the official “extraposed version of ¬”, “intraposed version of ∨”, etc. Digression. There is a structural resemblance between the #E /#I definition above and the standard definition of a homomorphism from algebra A to algebra B (given in 0.23) as a function h: A −→ B such that every n-ary operation symbol f of the relevant similarity type satisfies, for a1 , . . . , an ∈ A: h(f A (a1 , . . . , an )) = f B (h(a1 ), . . . , h(an )), with h playing the role of O and f A and f B playing the roles of #I and #E , except that, put in terms of algebras: there is only one algebra involved in the present case, namely the Lindenbaum algebra of the logic under consideration (the algebra underlying the Lindenbaum matrix – see the proof of 2.11.5, p. 204). Of course we can have a homomorphism from an algebra to itself, making ‘only one algebra involved’, but this – an endomorphism of the algebra – is not what we have in the #I /#E case, since #I and #E are (in the general case) different fundamental operations of that algebra. The possibility that #I = #E is not excluded by the definition, and when #I = #E = # one says that the O commutes with or distributes over # (in the logic in question). The aptness of the former term is evident from the case in which # is 1-ary, since then it means that the order in which O and # are applied to a given formula is immaterial: the results are equivalent (and so in a congruential logic, synonymous). One danger in the latter terminology is a certain ambiguity, since some authors describe O as distributing over # not just when O#(A1 , . . . , An ) #(OA1 , . . . , OAn ), for all A1 , . . . , An , but when just the “-direction of this equivalence is satisfied (again for all formulas Ai ). Returning to the former terminology, what we really want for elements a, b, of a groupoid to commute is that ab = ba (where the groupoid operation is indicated by juxtaposition). For this we should consider what will be called here the Lindenbaum monoid of 1-ary operations for a given logic. Its elements are the 1-ary (fundamental or derived) operations of the algebra of the Lindenbaum matrix of the logic, mentioned above, and its own fundamental binary operation is composition (“◦”, but omitted in favour of simple juxtaposition here), with

558 CHAPTER 4.


the identity map as neutral element. Now if O and # satisfy the inset condition above (and is congruential) then the corresponding Lindenbaum operations really do commute in the Lindenbaum monoid. End of Digression. Here we illustrate such existence questions in the area of normal modal logic, i.e., with the inferential (or “local”) consequence relations S corresponding to various normal modal logics S in Fmla. (That is, A, B S C, for instance, just in case (A ∧ B) → C ∈ S.) Note that the restriction to normal modal logics means inter alia that ∧ is an extraposed version (and an intraposed version) of itself across ; but normality here means -normality, and the possibility of other non-boolean primitives (to serve as extraposed or intraposed versions of the boolean connectives is not ruled out). Now, a modal case was already considered in the Digression between 4.22.6 and 4.22.7, in which we saw that S4 (and indeed, even S5 ) provided no # satisfying (A ∨ B) A # B, which is to say, no extraposed version of ∨ across . Since in what follows will always play the O-role, we will omit the “across ” part of subsequent formulations. Thus we say that 1-ary ∗ is an extraposed or an intraposed version of ¬ (according to ) according as ∗A ¬A

or

∗A ¬A,

respectively (for all A, in each case). If ∗ is an extraposed version of ¬ for , according to which is normal and further, ∗ is congruential (needed for lines (3) and (4) here, lines (2) and (4) using the key equivalence of “¬” with “¬”, and line (3) the normality of )), then we can reason as follows: (1) (2) (3) (4)

(A ∨ B)

¬(¬A ∧ ¬B) ∗(¬A ∧ ¬B) ∗(¬A ∧ ¬B) ∗(∗A ∧ ∗B).

Similarly, if ∗ is an intraposed version of ¬ for -normal , we have: (1) (2) (3) (4)

A ∨ B)

¬(¬A ∧ ¬B) ¬( ∗ A ∧ ∗ B) ¬(∗A ∧ ∗B) ∗ (∗A ∧ ∗B).

The justifications here are much as in the preceding derivation, mutatis mutandis, though in this case, we don’t need an explicit congruentiality proviso, since the relevant replacements are made in the scope of and the boolean connectives rather than of ∗, so they are covered by the (-)normality of . (Alternatively, one could change the definition of extraposed/intraposed so as to require synonymy rather than mere equivalence of the O#I A and #E OA forms.) Observation 4.22.12 (i) If ∗ is a congruential extraposed version of ¬ for (normal modal) , then provides an extraposed version of ∨. (ii) If ∗ is an intraposed version of ¬ for (normal modal) , then provides an intraposed version of ∨.


559

Proof. The first derivation above shows that defining A # B as ∗(∗A ∧ ∗B), makes # an extraposed version of ∨, establishing (i); for (ii), the second derivation gives # with A # B = ∗(∗A ∧ ∗B) as an intraposed version of ∨. The conditions on ∗ in parts (i) and (ii) here evidently guarantee, by similar reasoning, the existence of extraposed and intraposed versions, respectively, of # for each boolean connective #, and thus (again, respectively) that every fully modalized formula is equivalent to one in which no occurrence of has a boolean connective in its scope, and to one of the form A. To illustrate with the latter property, writing ∗ and # for intraposed versions of ¬ and ∨, consider the formula q → q, which we begin by writing using the official modal primitive, , subsequent lines being transformations justified by the appropriate ‘intraposition’ steps: ¬q ∨ ¬¬q ∗q ∨ ∗∗q (∗q # ∗∗q). Since we considered extraposing disjunction in the earlier Digression, an intraposed version of ∨ is due some attention of its own; for the reader’s convenience, the relevant equivalence to be satisfied (by the inferential consequence relation associated with a normal modal logic) is written explicitly here: (A # B) A ∨ B. Thus disjunctions of formulas come to be treated as themselves formulas in the presence of such a #. It doesn’t take much imagination for find a setting in which this leads to trouble: Example 4.22.13 The logic the normal modal logic K4c (see 2.21.8(vi), p. 281) proves A → A for all formulas A, but not (for all A, B) (A ∨ B) → (A ∨ B); the latter’s unprovability (e.g. with A = p, B = q) can easily be checked semantically, using the fact that this logic is determined by the class of all dense frames. Putting the former fact in terms of the inferential (i.e., locally truth-preserving) consequence relation K4c , we have A K4c A. Now, if we had a intraposed version # of disjunction behaving as in the equivalence above, we should have as a special case of this last principle, that (A # B) K4c (A # B), and making replacements in this, as justified by the above equivalence, would give: (A ∨ B) K4c A ∨ B, which, as already observed (in connection with the →-form in Fmla) is not the case. Thus adding an intraposed version of disjunction would extend K4c non-conservatively.

560 CHAPTER 4.


Digression. If we think of the epistemic interpretation of , with A read as saying that a (some fixed cognitive agent) knows that A, then intraposed disjunction is at first sight like the “or” that appears in whether -constructions, since “a knows whether p or q” amounts to much the same as “a knows that p or a knows that q”. (Thus whether -disjunction is intraposed ∨ w.r.t. the epistemic operator in question.) There are many complications, however, one of them being the presumption conveyed by the whether -sentence that exactly one of p, q, is true; a second is that one would want an explanation of the absence of this construction with verbs like believe in place of know ; thirdly, there are occurrences of whether in what seems to be the same sense of the word but in which no such treatment would be plausible – e.g., to take an example from Karttunen [1977], when it occurs in construction with depend on (though Karttunen is making a slightly different point with this example). Detailed discussion of the semantics of whether and the role of or in its scope can be found in the Lewis and also Groenendijk and Stokhof references given in the first Digression in 6.11 below, p. 770. End of Digression. From 4.22.13 we see that there is no intraposed version of disjunction already definable in K4c , since even the addition of a new candidate is non-conservative. Thus no such intraposed disjunction is definable in K either; but this does not imply that adding such a connective (more accurately, principles governing such a connective to justify its being called an intraposed version of disjunction) would be non-conservative over K (or “over K ”, if you prefer). Indeed this turns out to be the case, as we shall see below (4.22.23). First there is an interesting effect of the canonical accessibility relation for normal modal logics S having an intraposed version of disjunction in the sense that for any formulas A and B there is a formula C satisfying: C A ∨ B. Naturally we have no hesitation in denoting, for a given A, B, an arbitrary such C as A # B, but the point of this is that we make no demand that a new connective is available and remain neutral on the question of whether the frames (W, R) of our modal logic need expanding by additional machinery in case there is. We assume only that the truth at and membership in a maximal S-consistent set, considered as a point in the canonical model with (WS , RS , VS ), coincide (Lemma 2.21.4 above). Rather than use, e.g., Set-Fmla sequents, we use the S-provability of the corresponding biconditional in what follows to record the -equivalence recorded as inset above (allowing the discussion to remain in Fmla). Observation 4.22.14 If a consistent normal modal logic S satisfies the condition above – that for all A, B, there exists a C behaving as the intraposed disjunction of A with B (i.e., C is equivalent to A ∨ B), according to S , then RS satisfies the condition on a binary relation R. For all x, y, z ∈ WS : If R(x) ⊆ R(y) ∪ R(z) then R(x) ⊆ R(y) or R(x) ⊆ R(z). Proof. We denote by A # B any C behaving as described (for a given A, B). Suppose that RS (x) ⊆ RS (y) ∪ RS (z) for x, y, z ∈ WS . By 2.21.9 (p. 282), this implies that for every formula D ∈ y ∩ z, we have D ∈ x. For a contradiction, suppose that RS (x) ⊆ RS (y) and RS (x) ⊆ RS (z). Thus there exist u, v ∈ WS


561

with (1) xRS u, (2) not yRS u, (3) xRS v, (4) not zRS v. (2) and (4) give formulas A and B with A ∈ y, A ∈ / u, B ∈ z, B ∈ / v. Since A ∈ y, we have A ∨ B ∈ y and thus (A # B) ∈ y, and since B ∈ z, we have, similarly, (A # B) ∈ z. Thus A # B is a D for which D ∈ y ∩ z, so by the above result recalled from 2.21.9, for this choice of D we have D ∈ x. But this means that A ∨ B ∈ x, which is impossible since the first disjunct’s belong to x clashes with (1), as A ∈ / u, and the second disjunct’s belonging to x clashes with (3), as B ∈ / v.

Exercise 4.22.15 Show that for any frame (W, R), if for all formulas A, B, there is a formula C for which the equivalence C ↔ (A ∨ B) is valid on the (W, R), then (W, R) satisfies the condition on R mentioned in 4.22.14. (Suggestion: show how, if the condition is not satisfied, we can choose V (p) and V (q) so as to rule out any choice C for which this equivalence is valid when A and B are taken as p and q respectively.) Because the intraposed version of disjunction is harder to think about than the intraposed version of negation, and the presence of the latter guarantees the presence of the former (4.22.12(ii)), we switch to considering this. A first point worthy of note is a correspondingly simpler version of the first-order condition on accessibility to that figuring in 4.22.14. Observation 4.22.16 If a consistent normal modal logic S satisfies the condition that for all formulas A there is a formula B behaving as the intraposed negation of A (in the sense that B is equivalent to ¬A), then the canonical accessibility relation RS satisfies the following condition on binary relations R. For all x, y ∈ WS : R(x) ⊆ R(y) implies R(x) = R(y). Proof. Clearly it suffices to show, on the hypothesis that RS (x) ⊆ RS (y), that RS (y) ⊆ RS (x) for the selected S, in which the formula B (or any such formula, for a given A) described will be denoted by ∗A. So suppose, for a contradiction, that RS (x) ⊆ RS (y) while for some z ∈ WS , yRS z but not xRz. Since not xRS z, there is some formula A with A ∈ x, A ∈ / z, so since yRS z, A ∈ / y, meaning that ∗A ∈ y. Since RS (x) ⊆ RS (y), every -formula in y is in x (a simplified version of the point used from 2.21.9 used in the proof of 4.22.14), so ∗A ∈ x, which implies that ¬A ∈ x contradicting the fact that A ∈ x (together with the consistency of x).

Exercise 4.22.17 (i) Show that S5 already has an intraposed version of negation and describe it/one. (That is say, for any formula A, which formula ∗A, in the language with boolean connectives and , satisfies the condition that ¬A S5 ∗A.

562 CHAPTER 4.


(ii) Show that S4 would be extended non-conservatively by the addition of a new connective ∗ serving as the intraposed version of negation. (iii) (Analogous to 4.22.15.) Is it the case that for any frame (W, R), if for all formulas A, there is a formula B for which the equivalence B ↔ ¬A is valid on the (W, R), then (W, R) satisfies the condition on R mentioned in 4.22.16. (Justify your answer with a proof or a counterexample, as appropriate.) Observation 4.22.6 should not encourage the idea that adding a new connective ∗ to the language of K, the smallest (-normal) extension of K in which ∗ served as an intraposed negation would be determined by the class of frames there mentioned. This would be a non-conservative extension of K, so whatever was said about the semantics of ∗, there would be a problem with the ‘soundness’ half of that idea. Let us put this in terms of formulas rather than -conditions. The schema to be added would be: ¬A ↔ ∗A, and even just the ←-direction of such an equivalence would render provable the schema D, as is the case with any axiom of the form B → ¬C, since by the monotone property of (in any normal modal logic) we have ⊥ → B provable and also ¬C → ¬⊥, rendering ⊥ → ¬⊥, and hence ¬⊥ itself – a minor variant on the usual D axiom-schema – also provable. So the envisaged logic would have theorems not valid on all frames satisfying the condition that R(x) ⊆ R(y) implies R(x) = R(y), a condition satisfied automatically by nonD-validating frames in which R(x) = ∅ for all points x. A more refined treatment calls accordingly for a condition which does imply seriality, such as the following, written here as a first order formula (to emphasize which, we write Rwx, →, etc., rather than our usual informal wRx, ⇒, etc.): ∀w∃x(Rwx ∧ ∀y(Ryx → ∀z(Ryz → Rwz))). By way of explanation: this condition emerges from tense-logical considerations as a condition on frames (W, R), quantifiers ranging over W – though we write (−1 ) and (−1 ), rather than Prior’s G (H) and F (P), as in the discussion between 2.22.8 and 2.22.9 above (p. 286f.) – which would guarantee the validity (on frames satisfying the condition) of the equivalence −1 ¬A ↔ ¬A, in which an intraposed version of ¬ appears as the modality −1 ¬. Note that the ←-direction here is automatic, as a substitution instance of the generalpurpose (i.e., Kt -provable) tense-logical bridging axiom q → −1 q (or, in the traditional notation: q → GPq), once the “¬A” is rewritten as “¬A”. (This illustrates the mistake of thinking that tense logic is of interest only for temporal applications: every binary relation – and thus in particular the accessibility relation of any frame – has a converse, after all.) Theorem 4.22.18 If a normal (mono)modal logic is determined by a class of frames satisfying the “∀w∃x” condition inset above, then it is extended conservatively to a bimodal logic with the addition of a new 1-ary connective ∗ serving as a intraposed version of negation, i.e., by the additional axiom schema: ∗A ↔ ¬A.


563

Proof. Interpret ∗ in models M = (W, R, V ) by means of the following clause in the inductive definition of truth (at a point x ∈ W ): M |=x ∗A iff for some y, z ∈ W with yRx, yRz: M |=z A, making no change from the usual Kripke semantics in respect of the interpretation of other connectives ( included). Then every formula of the form ∗A ↔ ¬A is easily seen to be valid on the frames of such models provided they satisfy the earlier “∀w∃x” condition. So, letting S and S + be, respectively, a logic (in Fmla, for ease of exposition) determined by a class of frames meeting that condition and its extension by the ∗-schema just cited, we have the latter as a conservative extension of the former, since if a ∗-free formula B is not provable in S, B is false at some point in some model on such a frame, and remains so when the model is considered as a model for S + (i.e., with the above clause for ∗ in force), since connectives other than ∗ are not affected, and B is therefore not provable in S + .

Remark 4.22.19 Since the equivalences ∗ A ↔ ¬A are true throughout the models here considered, we can take the S + of the proof as the smallest -normal extension of S, and the phrase “extended conservatively to a bimodal logic” in the statement of 4.22.18 to mean “. . . bimodal logic in and ∗ in which continues to be normal”. (One might add also “and in which ∗ is congruential”, on the similar grounds.) Exercise 4.22.20 (i) Show that the “∀w∃x” condition under consideration here implies the earlier condition mentioned in 4.22.16. (As already noted, it is not implied by that condition, as non-serial frames can satisfy the latter condition.) (ii) Discuss the prospects for strengthening Theorem 4.22.18 by strengthening the “if” to an “if and only if”. To bring 4.22.18 to bear on KD, which we have seen is the weakest normal modal logic with any of being conservatively extended by making a new 1-ary connective ∗ serve as an intraposed negation, we borrow ahead from Chapter 9. Theorem 9.24.1 there tells us that as well as being determined by the class of all serial frames (something shown by a canonical model completeness proof), KD is determined by the class of all serial frames satisfying the following ‘unique predecessor’ condition on frames (W, R): For all x, y, z ∈ W , if xRz and yRz, then x = y. Now, any frame satisfying this condition automatically satisfies the “∀w∃x” condition mentioned in 4.22.18 – since for any w we can take any x ∈ R(w) as the promised x, w being x’s sole predecessor – giving us the following consequence of 4.22.8: Corollary 4.22.21 The smallest extension of KD in which (a new 1-ary connective) ∗ is an intraposed version of ¬ is conservative. Let us return to intraposing disjunction, and follow a similar path at a brisker pace, this time inspired by the tense-logical:

564 CHAPTER 4.

EXISTENCE AND UNIQUENESS OF CONNECTIVES −1 (A ∨ B) ↔ (A ∨ B),

where, as before, the ‘active’ half is just the →-direction, and again we could equally well put this using instead of ↔. We also need the following condition on frames (W, R): ∀w∀x, y ∈ R(w)∃z ∈ R(w)∀u(uRz ⇒ (uRx & uRy)). Theorem 4.22.22 If a normal (mono)modal logic is determined by a class of frames satisfying the condition above condition inset above, then it is extended conservatively to a bimodal logic with the addition of a new binary connective # serving as a intraposed version of disjunction, i.e., by the additional axiom schema: (A # B) ↔ (A ∨ B). Proof. As in the proof of 4.22.18, it suffices to show how to interpret the new # in models whose underlying frames (W, R) satisfy the above condition. To this end, we stipulate, where M is a model on such a frame, that for all v ∈ W , for all formulas A, B: M |=v A # B iff for some u ∈ W with uRv: R(v) |= A or R(v) |= B, where “R(v) |= A” (etc.) means: for all v ∈ R(v), M |=v A. One then checks that with this truth-definition in force every instance of (A # B) ↔ (A ∨ B) is valid on any (W, R) meeting the above condition (which is only needed for the → direction, in fact). A mild variation on Theorem 9.24.1, appealed to above à propos of KD, is available (see Sahlqvist [1975], which supplies the proof for 9.24.1 also) to the effect that K is determined by the class of all frames satisfying the unique predecessor condition above: for any w ∈ W , |(R−1 (w)| 1. (I.e., we just drop the “serial” part of the corresponding completeness result for KD.) But one verifies without difficulty that this condition implies that given just before 4.22.22, giving the following (in which the first occurrence of “extension” can be taken either to include “-normal” or otherwise, as per Remark 4.22.19): Corollary 4.22.23 The smallest extension of K in which (a new 2-ary connective) # is an intraposed version of ∨ is conservative. The not uncommon notation “−1 ” (and “−1 ”) used here for what in Prior’s notation for tense logic would be “H” and “P” is intended to recall the use of the notation “R−1 ” for the converse of the binary relation R, rather than to have any connection with the inverse a−1 of a group element a (as in 0.21). The latter makes sense for any monoid (though it is not what semigroup theorists usually have in mind when they say “inverse”), and thus derivatively for 1-ary connectives. Specifically, in the case of , let us say that 1-ary operators L and R are respectively a left inverse and a right inverse of according to depending on whether: LA A (all A)

or

RA A (all A).

(These correspond to what are literally left and right inverses of in the Lindenbaum monoid of the logic concerned, as explained in the Digression after


565

4.22.11 above; the question of the existence of left and right inverses for ¬ in IL was addressed above in 4.21.1, p. 540.) Two interesting existence-related questions which we could have pursued here instead of concentrating on extraposed and intraposed versions of connectives are (i) when a normal modal logic already has definable in its vocabulary a left or a right inverse (for ), and (ii) when, in the absence of such already available inverses, the logic can be extended conservatively by their addition. As these questions have already received an extensive airing in print (see Humberstone and Williamson [1997]), it did not seem useful to revisit the issues here. (For further references, see the discussion immediately after 5.23.3, in the course of which the “(·)−1 ” notation is briefly used for inverses of connectives.) Let us instead pause only to notice the connections between such inverses and the intraposed/extraposed theme; it turns out that two uses of the superscripted “−1” are not as unrelated as might initially appear. On the former point, we have: Observation 4.22.24 (i) The existence of a right inverse for guarantees the existence of intraposed negation across in any modal logic. (ii) The existence of a left inverse for guarantees the existence of extraposed negation across in any modal logic. Proof. For (i), we want ∗ with ∗A equivalent to ¬A, so take ∗ as R¬, where R is a right inverse of in the logic concerned. For (ii), we want ∗ with ∗A equivalent to ¬A, so take ∗ as ¬L, where L is a left inverse of . As to the latter point, note that the tense-logical equivalences used to motivate intraposed negation and disjunction were respectively: −1 ¬A ↔ ¬A

−1 (A ∨ B) ↔ (A ∨ B),

though in the former case we wrote “¬” rather than “¬”; or, in Prior’s original notation (for the tense operators), to which we revert here: GP¬GA ↔ ¬GA

GP(GA ∨ GB) ↔ (GA ∨ GB),

the first of which could equally well be put as: GPFA ↔ FA. The case of KD, as discussed in the lead-up to Coro. 4.22.21, suggests a simple common generalization of both forms: GPA ↔ A, which (bi)modally defines the class of frames (W, R) satisfying: For all w ∈ W , there exists x ∈ R(w) such that for all u ∈ W , uRx ⇒ u = w. The work here is all done by the →-direction of the tense-logical schema, of course, since the ←-direction is valid on all frames, but writing the schema in biconditional form reveals G as a left inverse of P, and P as a right inverse of G, in (the Lindenbaum monoid of) the normal extension of Kt by the schema in question. The condition on frames just given, which says that each point has a successor of which the former point is the sole predecessor, is weaker than the condition mentioned à propos of 4.22.21 – that every point has a successor and no point has more than one predecessor (though the conditions are equivalent

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as applied to finite frames). By 0.21.1(ii) from p. 19, the existence of a right inverse for G makes the corresponding operation in the Lindenbaum monoid of this logic surjective, which we express in §9.1 by saying that the connective G itself is ‘universally representative’ in this logic, while the existence of a left inverse for P makes the corresponding operation injective, leading to a kind of cancellation property for the connective: closure under the rule passing from PA ↔ PB to A ↔ B. Note that this is equivalent to closure under the rule passing from HA ↔ HB to A ↔ B. See further parts (ii)–(iv) of Exercise 6.42.8 below, and, for further information, Humberstone and Williamson [1997]. Exercise 4.22.25 (i) Picking up the theme of 4.22.24(i): does the existence of a right inverse for guarantee the existence of intraposed disjunction across in any modal logic? (Note that this says “any modal logic”, not “any normal modal logic”, so 4.22.12 from p. 558 cannot be exploited.) (ii) Show, by exhibiting syntactic derivations, that the normal extension of Kt by GPq → q (which is the same as the normal extension of Kt by all formulas GPA ↔ A) coincides with the normal extension of Kt by F(Hq ∨ H¬q).

4.23

Other Grounds for Claiming Non-existence

There are many things one might mean, with a particular background logic in mind, by the question of whether or not there exists a connective with such and such properties, apart from whether postulating such a connective yields a conservative extension of the given logic. Most obviously, this question could be taken as the question: does there exist, amongst the primitive connectives of the language of the given logic, a connective behaving as specified, according to that logic? Or again, we could weaken primitive to primitive or derived (“definable”: 3.15). Clearly Belnap’s proposal, described in 4.11, was not made with such considerations is mind. His question is: under what conditions can adherents of the original logic concede the existence and intelligibility of a previously unfamiliar connective (in the inferential-role sense of “connective”)? But with this – as we might put it, partisan – question in mind, the answer that, provided that the connective could be conservatively added, the concession should be made, does not seem like the whole story. In other words, presuming correctness for the background logic, the non-conservativeness of a proposed extension by rules governing a certain connective is at best a sufficient – and not also a necessary – condition for holding that there exists no connective behaving as those rules dictate. We give an example to substantiate the claim just made, from the area of relevant logic, treated in the framework Fmla (2.33). T. Sugihara had a somewhat different conception of relevance from those operative in our discussion there, regarding a logic as ‘paradox-free’ if it contained neither a weakest nor a strongest formula: neither a formula provably implied by every formula nor a formula provably implying every formula. (See Anderson and Belnap [1975], p. 335, for details and discussion of Sugihara’s proposals.) Now, whatever philosophical grounds one might have for the idea that it is a mistake to countenance such formulas, they are hardly to be imagined shaken by the observation that a logic lacking them—for example, the system R of 2.33—can be conservatively extended by the addition of sentential constants T and F as there, governed


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by principles making them respectively, weakest and strongest formulas in the extended system. (Recall that we speak of T and F rather than of and ⊥ in such a context, to underline the T vs. t, F vs. f distinctions.) If the original reasons were good reasons, then they are reasons for doubting the intelligibility of these particular zero-place connectives (playing the given logical roles), and the fact that inferential relationships between formulas of the unextended language are not disrupted by their postulation is neither here nor there. We can change this example slightly so that it applies (ironically) to Belnap’s own criterion of relevance, imagined as altered so as not to rule out the above constants: a necessary condition for the implication with A as antecedent and B as consequent to count as relevant (or for a logic proving such implications to count as a relevant logic) is that either A and B have a propositional variable in common, or else A or B (or both) contains at least one variablefree subformula. For consider the (putative) singulary connectives T1 and F1 , conservatively addable to R with axioms A → T1 B and F1 A → B. The extension violates the (amended) criterion of relevance since it contains, for example, p → T1 q. Why should anyone enamoured of that criterion concede the existence, in the current ‘partisan’ way of taking that phrase, of T1 and F1 ? The extension may leave things as they were as far as the original connectives are concerned, but it runs counter to the very reasons for having them that way in the first place. (Here we are playing up the ‘subject matter’ conception of relevance rather than the ‘use’ conception.) Yet another example, in the same area, of how all not need be regarded as well, the conservativeness of an extension notwithstanding, is the reaction of Belnap and Dunn [1981] to the connective which has come to be called Boolean negation in discussions of relevant logic. (See 8.13.) Although the Fmla system R is conservatively extended by adjunction of axioms governing this connective, which we will write as (to distinguish it from the De Morgan negation denoted by “¬” in 2.33): (A ∧ ∼A) → B

A → (B ∨ ∼B)

which, notice, immediately violate the Belnap relevance criterion, Belnap and Dunn say, at p. 344 of their [1981]: “The gist of our reply for the relevantist is that he does not have to, indeed should not, recognize the legitimacy of Boolean negation”. (Note: we have reversed the usage of the symbols “¬” and “∼” from the relevant logic literature, Belnap and Dunn [1981] included. The present usage may be found in Priest [1990], which defends the Belnap–Dunn illegitimacy verdict. See also Priest [2006b], Chapter 5.) The moral of these examples is that it is not promising to look for a purely formal criterion, such as the conservative extension criterion, as providing a condition both necessary and sufficient for acknowledging the existence of a connective with given logical powers. Reasons for disquiet with any connective having the powers in question can come from anywhere; that its postulation disturbs a logic deemed correct for the connectives already in play is one— but only one—such reason. The situation is thus no different from reasons for denying the existence of a connective satisfying any condition, not necessarily a condition formulated as a syntactic constraint on consequence relations. One asks what things would have to be like if there were such a connective, and any reason for believing that things are not like that is a reason for saying that there is no such connective. A well-known example falling under this rubric is David Lewis’s argument against the existence of a binary connective ¢ with

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the property that for any probability function P and any formulas A and B, P (A ¢ B) = P (B/A). Several writers had suggested, thinking of the probability functions here as registering degrees of belief, that the (indicative) conditional in English was a connective ¢ for which the above equation held; but Lewis [1976] showed that the existence of such a ¢ implied that at most three statements were probabilistically independent. This is an unacceptable result, so we conclude that there is no such ¢. (A somewhat similar example, ‘probabilistic conjunction’, will come up in 5.14.) We can drive the above moral home by exhibiting a logic conservatively extended by principles governing each of two proposed new connectives though not conservatively extended by the simultaneous addition of both sets of principles. Thus since for each of two things to exist is the same as for both of them to exist, the conservative extension test cannot be both necessary and sufficient for the existence of a connective (with given logical powers). We turn to the example. McKay [1985] gives an extension of Positive Logic (in Fmla) with an additional singulary connective, governed by various axioms and rules, and shows that any non-degenerate matrix validating all theorems of the extension is infinite. For present purposes, we need only a small part of this extension, and take no interest in this ‘infinite model property’ (which, incidentally, Meyer and Slaney [1989] thought they were the first to illustrate). In particular, we take the following theorem-schema of McKay’s logic, which involves the additional connective we shall write as “O”: (O)

(A → OA) → A

We imagine the language of Positive Logic (with connectives ∧, ∨, and →: recall from 4.11 that this is the fragment of intuitionistic propositional logic in those connectives) expanded by the new connective O. The extension we are concerned with is the least collection of formulas in this language containing all theorems of PL, all instances of the schema (O), and closed under Modus Ponens and Uniform Substitution. McKay proves that his system, which properly includes the present system, conservatively extends PL: so the present system is certainly also conservative over positive logic. Call it the O-system. Intuitionistic logic with ⊥ as a primitive connective, also conservatively extends PL, the ⊥-classicality schema (⊥)

⊥→A

playing the role of (O) in obtaining this extension. But, as McKay notes (for his system rather than the smaller O-system), the addition of both O, subject to (O), and ⊥, subject to (⊥), is not conservative over PL. In fact the resulting system is inconsistent; letting B be an arbitrary formula, we have: (1) (2) (3) (4) (5)

⊥ → O⊥ (⊥ → O⊥) → ⊥ ⊥ ⊥→B B

by (⊥) by (O) 1, 2 Modus Ponens by (⊥) 3, 4 Modus Ponens

Thus while the proposed new connectives, ⊥ and O, taken one at a time, each yield conservative extensions, taken together they have the most nonconservative effect there could be: not only does something in the original


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{∧, ∨, →}-language previously unprovable now become provable, everything does! Thus, as noted above, one deeming PL correct for the connectives it deals in and also accepting the necessity and sufficiency of the conservative extension criterion of existence is in an invidious position: this theorist must say, using the sufficiency part (contested above) of the criterion that ⊥ exists (satisfying (⊥)), since this gives a conservative extension, and that O (satisfying (O)) exists, which joint admission amounts to saying that both exist, in direct conflict with the (more plausible) necessity part of the criterion since the joint extension is non-conservative. Observation 4.23.1 (A Faulty Observation!) Suppose S is a Fmla proof system in a language containing ∧ and →, and that the theorems and derivable rules of S include those of the {∧, →}-fragment of R. Let c be a zero-place connective absent from the language of S, added to that language to give the language of S + , with additional axioms c → A for all A in the language of S + . Then S + is a conservative extension of S. Proof. (Putative proof, that is.) Given a proof A1 . . . , An of a c-free formula An in S + , construct the sequence of formulas A∗1 ,. . . ,A∗n by taking A∗i to be the result of substituting B1 ∧ . . . ∧ Bk for each occurrence of c in Ai , where B1 ,. . . ,Bk are all the formulas for which some appeal to the c schema of S + is made in the proof of An , i.e., some instance c→ Bj of that schema is invoked. We claim that A∗1 , . . . , A∗n is a proof in S of A∗n , since each use of the c-schema goes over into a formula (B1 ∧ . . . ∧ Bk ) → Bj for some j (1 j k) which is provable in S (being provable in R). But, since An does not contain c, A∗n just is the formula An . So S + is a conservative extension of S.

Exercise 4.23.2 The above would be a convenient and quick proof that adding F with the usual (⊥) or ‘c-like’ axiom schema conservatively extends the system R itself, if the argument were valid. But the example from McKay shows that the argument is invalid (or we could take c as ⊥ and S as the O-system). Where precisely does the argument go wrong? A propos of the point made (against Belnap) with the aid of McKay’s example we remark that it would not have been possible to make the same point by considering instead the fact that PL is conservatively extended (to IL) by adding ⊥ with the schema ⊥ A, as well as being conservatively extended by adding ⊥ with the schema ⊥, though clearly the combined extension would not be conservative. This would be unfair because the simultaneous presence of connectives playing these two inferential roles—as long as we do not also assume (gratuitously) that it is the same connective in both cases—produces a conservative extension. The two roles, ⊥-classicality and -classicality, are played by ⊥ and respectively, so having notated the occupant of the former role by “⊥”, we should not have used the same notation for the occupant of the latter role. The case of the O-system is different from this because the rule (O) makes no direct reference to ⊥, and nor does (⊥) make any such reference to O, so there is no assumption that one and the same role is being doubly occupied. Of course, there is a kind of indirect reference in both directions in the sense that for the combined system, the schematic letters (such as “A”) entering into

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the formulation of (O) and (⊥) are taken to subsume formulas constructed with the aid of ⊥ and O respectively: this is a point which a general account of what a rule actually is must attend to, and we postpone giving it the required attention until 4.33. Exercise 4.23.3 A special case of (O) is the following: (p → Op) → p; show that whatever formula B may be, neither PL nor IL has the formula (p → B) → p as a theorem. [Hint. It is easy to show this first for CL—say, in the form, “no sequent of the form p → B p is tautologous”, and then use the fact that PL ⊆ IL ⊆ CL.] We conclude by putting the example of the O-system into a broader setting. Given a consequence relation − , with a (possibly derived) binary connective # in its language, a typical range of ‘existence questions’ that come up concern the existence of a 1-ary connective O for which, either taking as − itself or as some conservative extension of , one of the following four conditions is satisfied for all formulas A (in the language of ): (1a) OA # A OA

(1b) A # OA OA

(2a) OA # A A

(2b) A # OA A.

Obviously further variations on this theme can be thought up, especially if we consider # of artity 2, but this will do for the present discussion. Note that if # (binary, as above) is commutative according to , then the (a) and (b) forms here are equivalent, and that if # is idempotent then we may take OA as -equivalent to A. The (1) forms are what in Humberstone [2006b] are called fixed point equivalences, which arise naturally from a consideration of Curry’s Paradox, presented in the notes to §7.2 below – p. 1123 onwards – which also explain the ‘fixed point’ idea: see p. 1126. The (2) forms make the connective # universally representative (according to ) in the sense explained in §9.1. Here we consider taking # as →, in which case (for familiar ) neither of the conditions just mentioned – commutativity and idempotence – is satisfied. Retaining the same numbering, that is, we consider: (1a) OA → A OA

(1b) A → OA OA

(2a) OA → A A

(2b) A → OA A.

For continuity with the earlier discussion, let us consider starting from PL, or even, discarding ∧ and ∨, just the implicational fragment of IL, as the initial consequence relation − . (1b) and (2a) pose no difficulties, since keeping as − itself, they are satisfied when OA is defined as A → A. But (1a) and (2b) are another matter. (We expect trouble from the fact that OA occurs ‘negatively’ on the left in (1a) but ‘positively’ on the right, and likewise for A itself in (2b).) Now, (1a) is treated in the Curry’s Paradox discussion in the notes to §7.2, in connection with the rules (which appear there with “O” written as “#”): (i)

Γ OA

(ii )

Γ OA → A

Γ A → OA Γ OA which are giving in an →-purified form also in that discussion, here reformulated as conditions on a consequence relation not only presumed to extend the current choice of − but also to be →-intuitionistic, as defined on p. 329. (This


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is not needed for the argument but for the purified version to be equivalent to that with →.) Example 4.23.4 Suppose a consequence relation satisfies the following condition, for all formulas A and some 1-ary connective O in its language: Γ OA ⇔ Γ, OA A. We can then reason, for (1) OA OA (2) OA, OA A (3) i.e., OA A (4) OA (5) A

arbitrary A: By (R) From 1 by the ⇒ direction above since {OA, OA} is the same set as {OA}. From 3 by the ⇐ direction above From 3, 4 by (T).

Thus ends up inconsistent (since Γ A for all Γ from (5) by (M).) This would be blocked in the transition from (2) to (3) by treating what is on the left of the “” – which would now no longer stand for a consequence relation, for this reason – as a multiset rather than a set, and disallowing the structural rule of Contraction – or more properly speaking, the analogous structural condition (cf. (M) vs. (M), etc.) – that would then be needed to legitimate this transition. (See 7.25.) Let us turn from (1a) to (2b). The -direction of (2b) was addressed already in 4.23.3, where, adjusting the result to the case of − as the pure implicational fragment of IL (or PL) there is no O definable satisfying the condition in question in − , but from what we have seen of McKay [1985] (distilled into comments about ‘the O-system’ above), we know that we can extend this − conservatively to a satisfying the condition. What about the full two-way condition (2b), though? Again, to purify the condition so that it addresses only “O”, we assume that is →-intuitionistic, to proceed with: Example 4.23.5 Much as in the case of Example 4.23.4, we reformulate (2b) for (as just described) thus: Γ A ⇔ Γ, A OA. As in 4.23.4 from the ⇒-direction we obtain (for all A): A OA. From this, using the ⇐-direction with Γ = ∅, we conclude that A, whence Γ A as in 4.23.4, for all Γ, A. In other words, no consistent can satisfy (2b). Thus in particular, for the announced choice of − , the proposed extension is non-conservative.

4.24

Non-partisan Existence Questions

We have seen that while there can reasonably be claimed to exist no connective obeying rules which non-conservatively extend a logic deemed correct, there are also other kinds of grounds for returning a negative verdict on such existence questions. The reference to deeming a certain logic correct here indicates that we are thinking of the existence question as being entertained by a partisan of the logic concerned. (All that is relevant here to such endorsement of a logic

572 CHAPTER 4.


is that its provable sequents encode precisely the arguments one endorses; in §4.3 we shall emphasize a further aspect of deeming a logic correct, in respect of which sequent-to-sequent rules are derivable.) In the last parts of 4.22 and at the start of 4.23, some ‘impartial’ existence questions were also mentioned: do inverses for such-and-such exist in a given logic, intraposed and extraposed version of various connectives, etc. One might for example, wonder – without taking side on the correctness of intuitionistic logic – whether there is any single derived connective in that logic which could be taken as primitive and allow for the definition of the standard primitives ∧, ∨, ¬, and →. (See 3.15.8, p. 423, for the answer to this question.) The interest of such questions as that just raised turns in part on the extent to which the connectives ∧, ∨, ¬, and →, are ‘expressively complete’ for intuitionistic logic. In the case of classical logic, the truth-table explanations of the intended meanings of the connectives lead to taking functional completeness (3.14) as an explication of the vague idea of expressive completeness. For intuitionistic logic, the absence of an analogous association with truth-functions leaves room for argument as to what measure of expressive power to adopt. (See the last two paragraphs of 3.15.) One could use the Kripke semantics for IL to motivate the idea of a ‘possible meaning’ for an intuitionistic connective; this is done in McCullough [1971], where the range of meanings is delimited in such a way that the four connectives cited do turn out expressively complete. As was mentioned in 4.14, a similar conclusion is reached by Zucker and Tragesser [1978] for ∧, ∨, → and ⊥, by concentration on the idea that meanings for connectives are given by natural deduction introduction rules. Here we give a couple of further cases of such non-partisan existence issues. Example 4.24.1 Given a binary connective # a consequent-relative version of # consequence relation on a language with → and # in its language, is the ternary connective whose application to A, B and C we write as A# C B such that for all formulas A, B, C: (A # C B) → C (A # B) → C. The intent of the definition is really that the left and right hand sides here should be synonymous according to whatever logic is under discussion, rather than that we have a consequence relation (congruential or otherwise) in mind. So the “D E” notation should be understood in connection with a substructural logic in Fmla as the joint provability of the implicational formulas D → E and D → E, and the logic chosen in such a way that this secures the synonymy of D and E. For example we note in 5.16 (discussion leading up to 5.16.10 on p. 672) that in this sense the pure implicational fragment of linear logic, alias BCI logic, provides consequent-relative fusions, since the condition (A ◦ C B) → C (A ◦ B) → C, is satisfied for all A, B, C, with “” understood as abbreviating the provability of → and ← implications, when (A ◦ C B) is taken as (A → (B → C)) → C. And in 1.19.11, we found that the implicational fragment of IL provided consequent-relative disjunctions, satisfying: (A ∨ C B) → C (A ∨ B) → C,


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since A ∨ C B could be taken as (A → C) → ((B → C) → C). We could approach a similar idea without explicit appeal to the presence of →, and for a logic as a set of sequents regard the equiprovability of Γ, A# C B C and Γ, A # B C (for all Γ, A, B, C, as qualifying A# C B as a succedent-relative version of #. Returning to the implicational form, one could also consider the more general case of n-ary # and the n+1-ary consequent-relative versions for n other than 2. One might also explore the issue of antecedent-relative instead of consequent-relative versions of connectives, and so on. None of this is done here. For the above example, it is important to recall (from 2.33) that when, as in the BCI -fusion case, A B is just another way of saying that A → B, this last “” indicating provability in a substructural logic without K such as BCI or BCIW logic or their extensions to accommodate additional primitive connectives beyond →, from the hypothesis that B, one cannot conclude that A B. These logics are not monothetic and make room for interesting and nontrivial logical relations among the provable formulas. Against this backdrop, we might ask whether the implicational fragment of linear or relevant logic (or more precisely ILL or R) has the property that any two theorems have something that might reasonably be regarded as their disjunction. This would be a formula that behaved like their least upper bound, logically, with the ordering according to which a formula A bears the relation to B just in case A → B is provable. A simpler version of this question just asks whether any two BCI theorems have an upper bound: something or other they both provably imply. Essentially this question was raised by Bob Meyer in correspondence (in 2006) and because it asks about the existence of formulas standing in prescribed relations to some given formulas, we include it as Example 4.24.2. (In fact Meyer had a slightly different formulation in mind, asking about the existence of upper bounds in certain ordered BCI -matrices.) Example 4.24.2 Even though not fully deserving to be notated using a disjunction symbol (since we are not asking for a least upper bound), for this example, we write A ∨* B for a compound of A and B satisfying the condition that if A and B are both provable – for definiteness, BCI provable – then so are A → (A ∨ * B) and B → (A ∨ * B). (Think of the “*” as a reminder of the marking of designated values in a matrix, thereby suggesting the restriction that it is only for theorems A and B that we expect these implications to be provable.) This is actually somewhat stronger than the demand alluded to above in connection with Meyer, since it asks us to provide an ‘upper bound’ for (or common consequence of) A and B formed by application of a binary connective to A, B. We have already seen one route to an affirmative answer here, in 2.33.8 (p. 343) where we saw that any two formulas in R have a whole range of formulas they successively imply, the proof making use only of BCI -available resources. These formulas successively implied by A and B (i.e., formulas such that A provably implies that B implies the given formula) are what appear in Example 4.24.1 as the formulas A ◦ C B for any choice of C. By permuting antecedents (the “C” of “BCI ”) we thus have not only: BCI A → (B → (A ◦ C B))

but also

BCI B → (A → (A ◦ C B)).

574 CHAPTER 4.


Thus by the hypothesis that each of A and B is provable, Modus Ponens gives BCI B → (A ◦ C B) and also BCI A → (A ◦ C B): so for any choice of C, we may take the purely implicational compound, A ◦ C B, of A and B as A ∨ * B. Remarks 4.24.3(i) The above solution to Meyer’s problem was not Meyer’s own preferred solution, which instead consisted in the observation that every BCI -theorem is an implication, so given BCI -provable A and B we can think of them as being respectively A0 → A1 and B0 → B1 and then use the BCI -theorem: (A0 → A1 ) → ((B0 → B1 ) → ((A1 → B0 ) → (A0 → B1 ))). Thus if each of A and B is provable, then as before Modus Ponens gives us the conclusion that each provably implies (A1 → B0 ) → (A0 → B1 ), which accordingly constitutes an upper bound for them. (ii) The reason Meyer gave for favouring the upper bounds described in (i) is that the argument for their existence goes through against the backdrop of a weaker implicational logic than BCI, namely BB logic. For this purpose the phrase “as before” in (i) – meaning “as in 4.24.2” – is not quite accurate since we do not have C available to permute antecedents, and what is called for is the observation that each of the original and the permuted form is BB -provable. It was with this transposition to the BB case that the wording “every BCI -theorem is an implication” was used in (i), since this remains so for the BB case, while the stronger claim that every formula is BCI -provably equivalent to an implication would fail in this weaker setting. Note that the Meyer style upper bounds for A and B, taken in that order, as in (i), and for B and A, in that order, are quite different; the logical relations between them remain obscure (to the author). Notice also that we have not presented a connectival route from A and B to their Meyer style upper bound, since there is no way to recover, by means of →, the antecedent and consequent of an implicational formula from that formula, so the “A ∨* B” notation, as introduced in 4.24.2, would be out of place. (iii) However, if we return to BCI itself, we can use the ‘canonical’ equivalence of an arbitrary formula A with (A → A) → A, we can treat the A0 and A1 of Meyer’s proposal as respectively A → A and A itself and define our upper-bounding connective ∨* in the following way, in which for brevity, the abbreviation |A| (from Blok and Raftery [2004]) is used for A → A. This turns the inset schema under (i) into: (|A| → A) → ((|B| → B → (A → |B|) → (|A| → B))), allowing us to provide a Meyer-inspired A ∨ * B in the form of (A → |B|) → (|A| → B)). (iv) As noted in 4.24.2, really to live up to its name, our upper bound forming ∨* should yield least upper bounds for any pair of theorems, which may be taken to mean that for provable A, B, we should not only have A → (A ∨* B) and B → (A ∨* B) provable, but it should also be the case that for any C with A → C and B → C both provable, we have (A ∨* B) → C provable. On the existence of any implicationally definable


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‘theorem disjunction’ connective in the logics here under discussion, the author has no information. There may be some residual curiosity as to why we are restricting attention to provable formulas in BCI (or in BB ) having an upper bound or common (implicational) consequence. What about arbitrary formulas? Exercise 4.24.4 Show that there is no formula A for which p → A and q → A are both BCI -provable. (Hint: think about the equivalential fragment, in Fmla, of CL.) The hint for this exercise shows us also that there is no formula A for which both A → p and A → q are BCI -provable; thus not every pair of formulas have a lower bound in BCI. This leaves open the possibility that, as with upper bounds, every pair of BCI -theorems might have a lower bound (so that we might contemplate the availability of a ‘theorem conjunction’ connective “∧*”). But such lines of inquiry are left to the interested reader to pursue. Finally, let us mention the approach of Gabbay [1977], who lists several desiderata for a proposed new intuitionistic connective and presents examples of additional connectives satisfying them. Naturally, that the principles giving the logical role of the new connective should conservatively extend IL is one of them. But, in the further interest of keeping the extension in the spirit of IL, he also requires that it possess (inter alia) the Disjunction Property (6.41). Gabbay also demands that the principles added ‘uniquely characterize’ the new connective, in a sense which will occupy us in §4.3, and which also goes back to Belnap [1962], where thanks are given to H. Hiż on the matter. The idea is that a particular logical role should be specified for the new connective proposed, rather than a logical role which could be played by more than one connective obeying the principles in terms of which the proposal is cast. The ideas of existence and uniqueness are usually regarded as dual: there is at least one . . . vs. there is at most one . . . , whereas in this case Gabbay’s existence question is itself a question as to existence of a unique connective satisfying certain conditions. Accordingly, we defer further discussion of this issue until 4.37 (and 4.38). At its most impartial, the question of whether there exists a connective with such-&-such logical behaviour is simply the question of whether there exists a consequence relation (or gcr, or Fmla logic, etc., depending on the logical framework in terms of which the discussion is conducted) according to which some connective exhibits that behaviour. The consequence relation need neither stand in some preassigned relation (conservative extension, for example) to a favoured consequence relation, nor be thought of—as in the intuitionistic case just considered—as favoured by, if not the theorist asking the question, then at least some other group. In this sense, there is certainly such a connective as Tonk, for example, since the description of the consequence relation associated with the (Tonk I)–(Tonk E) proof system is perfectly consistent, even if it is unlikely to be of interest for anything other than illustrating this very point. It is to be emphasized that it is the consistency of our description of the proof system that is at issue here, not the consistency of the putatively described proof system itself. If the only non-structural rules involved are the Tonk rules, then this too is a consistent system, since although all sequents Γ B are provable if Γ = ∅, not every sequent is provable: with Γ = ∅, none is.

576 CHAPTER 4.


The associated consequence relation is a consequence relation analogous to the gcr we called CV, the constant-valued gcr, depicted in Figure 1.19a, p. 91, as the join in the lattice of gcr’s (on an arbitrary but fixed language) of the gcr’s we there called Yes and No, and having for its consistent valuations the constant valuations vT and vF . (For consequence relations, the distinction between No and CV disappears.) But the present point is that even in the case of an inconsistent consequence relation, there is no reason (from the present non-partisan perspective) to deny the existence of a connective inducing that inconsistency: for example, a zero-place connective c satisfying (for all A) the condition(s) c A and c In the terminology of 3.24, c is a hybrid of (classically behaving) and ⊥.

Notes and References for §4.2 Demi-Negation. This heading is used in Humberstone [1995c] to refer to the connective § envisaged in the opening paragraph of 4.21. It is there discussed in the setting of gcr’s rather than consequence relations so that its behaviour can be subjected to pure rules which have the effect of making two applications of § amount to one application of (classically behaving) ¬; see further Humberstone [2000a]. Such a connective turns out to have been discussed in the quantum physics literature under the name of the “square root of negation”. References may be found in 4.3 of Humberstone [2010b]. Tonk. Discussions of the Prior/Belnap interchange and some of the issues raised by Tonk may be found in Wagner [1981] – criticizing both parties – and Hart [1982]. See also Tennant [1978], §4.12, for (in effect) a diagnosis in terms of failure of harmony (4.14) between (Tonk I) and (Tonk E), and Peacocke [1987], though with the latter’s claim (p. 167) that the “semantical objection to tonk is that there is no binary function on truth values which validates both its introduction and its elimination rules” misleadingly suggests that all the example illustrates is the non-existence of a truth-functional connective governed by those rules (i.e., a connective which is truth-functional w.r.t. some determining class of valuations). This correction can also be found in Read [1988], p. 168. The earlier papers Stevenson [1961], Prior [1964a], are still worth reading. More recent discussions may be found in Cook [2005] and Bonnay and Simmenauer [2005], Tennant [2005], Wansing [2006a], and §3 of Humberstone [2010b]. It is also possible to consider Tonk in the sequent calculus (as opposed to natural deduction) approach; see 4.37.9. The interpretation of Belnap [1962] on the existence question for connectives adopted in the present section has been that conservative extension of a favoured logic is necessary and sufficient for those favouring it to acknowledge the existence of a connective governed by rules yielding such an extension. By contrast, Garson [1990], p. 155, says that “Belnap [1962] defended the claim that natural deduction rules define the meaning of a connective, provided at least that those rules form a conservative extension of the structural rules of deduction.” (See also §6 of Garson [2001].) Non-conservatively extending the purely structural (e.g., (R), (M), (T)) basis of a proof system is certainly a spectacular kind of non-conservativeness, one exhibited by the Tonk example; cf. the


577

remarks on ‘pathologicality’ following 3.12.4, p. 384. Our discussion in 4.21 of Belnap’s criterion for the existence of a connective with given inferential properties – namely that conferring such properties on a new connective conservatively extended an initially favoured logic – is inspired by some but not all of the literal content of Belnap [1962]. In particular, it represents one way of interpreting the phrase “antecedently given context of deducibility” appearing on, p. 131; on the following page of the article undue prominence – leading to Garson’s gloss above – seems to be given specifically to the structural rules antecedently endorsed. It is certainly true that in the case of Tonk we end up with the presumably not antecedently endorsed zero-premiss structural rule A B, but this is just a feature of this special case, on the interpretation of Belnap assumed in §4.2. (A propos of newly derivable structural rules for not just the zero-premiss case, we introduced the concept of structural nonconservativity in 2.33.35, p. 369.) Comma Connectives. The scant remarks in 4.21 on this subject may usefully be supplemented with a perusal of some of the literature on Display Logic – see 1.21.2 (p. 107) – in which they would be counted as structure connectives (as opposed to sentential connectives), because they build things called structures from structures, rather than formulas from formulas. To those references we may add Restall [2000], beginning with §2.2 thereof. As with other developments described in 1.21.8, p. 111, this exercise may be theoretically interesting but it is conducted at such a distance from of abstraction from ordinary reasoning as to make the occasionally eyebrow-raising multiple succedents of Set-Set seem thoroughly natural. Conservative Extension. As was mentioned in 4.24, Belnap credits the idea of considering questions of existence and (§4.3) uniqueness of connectives to H. Hiż; Belnap [1962] sometimes (unfortunately) speaks of consistency, meaning conservativeness. (The two coincide only if the logic to be extended is Postcomplete.) For one angle on the relation between conservative extension and unique characterization, see Došen and Schroeder-Heister [1985]. The need for care in discussions of conservative extension, as to what precisely is at issue, is emphasized in Sundholm [1981] (criticizing I. Hacking) and Chapter 10 of Tennant [1987b] (criticizing R. Grandy and J. P. Burgess). The points made in 4.23 about McKay’s logic appear in §6 of Humberstone [2001]. Read [1988], §9.3, and [1994], pp. 225–230, discuss aspects of Belnap’s conservative extension condition. The latter discussion involves a slip: “Is classical logic a conservative extension of its negation-free fragment?”, asks Read ([1994], p. 228). The term fragment here should be subsystem – showing that in any case the question only makes sense relative to a particular proof system (for classical logic). The negation-free fragment of CL (in whatever logical framework) by definition simply consists of all classically valid negation-free sequents (of the framework in question); thus Read’s question (as formulated) does make sense, but, contrary to his intentions, it has a trivially affirmative answer. Separation Property. The following references will prove useful for those wishing to investigate this topic, briefly touched on at the end of 4.21: Leblanc [1963], [1966], Leblanc and Meyer [1972], Hosoi [1966b], [1974], McKay [1968b], Rousseau [1970b], Maksimova [1971], Homič [1980]. For historical information

578 CHAPTER 4.


(especially concerning Wajsberg’s seminal role in the exploration of this issue), see Bezhanishvili [1987], and, for a particularly clear philosophical discussion, §2 of Bendall [1978] (and, less fully, §2.11 of Wójcicki [1988]). New Intuitionistic Connectives. Apart from the papers cited in 4.24, Bessonov [1977], Goad [1978], de Jongh [1980], Section III of Kreisel [1980], Kreisel [1981], Troelstra [1981], and Yashin [1985] provide general discussion and specific examples. We will consider various such examples below; for instance, two additional styles of negation (apart from intuitionistic ¬) will be considered in 8.22, 8.23. The latter subsection, on intuitionistic logic with strong negation, describes a general semantic setting in which to ask which semantically specified connectives make reasonable sense; §§4 and 6 of Wansing [2006b] offers an answer to this question, extending the work of McCullough [1971], already mentioned in 4.24. More discussion of the general topic of new intuitionistic connectives will be found in 4.37, 4.38, where further references are given. Upper Bounds in BCI. The points attributed to Bob Meyer before, within, and after Example 4.24.2 were expressed by him in email communications in 2006 (three years before his death) simultaneously with Sam Butchart, Tomasz Kowalski, John Slaney and myself.

§4.3 UNIQUENESS OF CONNECTIVES 4.31

Unique Characterization: The Basic Idea

From Belnap [1962] there emerged the idea of saying under what conditions one should be prepared to concede the existence of a connective playing a certain inferential role; in §4.2 we explained (and criticized) Belnap’s own proposal as to what those conditions were. As well as considering the existence of connectives, Belnap also touched, in passing, on the question of their uniqueness. (See the notes, which begin at p. 626, for further references.) From the perspective of what we have called (§4.1) philosophical proof theory, according to which the meaning of a connective is given—under suitably favourable conditions—by specifying its inferential role, an affirmative answer to the existence question leaves open the appropriateness of this talk of the meaning: unless that role is such that it can be occupied by only one connective, no unique meaning has been conferred. Actually this is too strong a formulation: presently we shall come to weaken it so as to demand that the role can be occupied to within synonymy by only one connective; and, in the absence of the assumption of existence, the “only one” should read “at most one”. (We call connectives synonymous when they form synonymous compounds from the same components.) Now we have already encountered the kind of reasoning which would establish such uniqueness conclusions, both in our discussion of algebraic preliminaries (§0.2) and, closer to the present topic, in our discussion of consequence relations (§1.1). In the former case, we were at pains to emphasize (0.21) that the conditions on identity elements in semigroups – indeed in arbitrary groupoids – were sufficiently strong that any elements satisfying those conditions in a given semigroup were identical. In the latter case, by way of foreshadowing the

4.3. UNIQUENESS OF CONNECTIVES

579

material of the present section as well as cleaning up some terminological infelicities, we noted (1.19) the possibility of two connectives each of which behaved ∧-classically according to a given consequence relation; we describe this possibility, and what it shows, in the following paragraph. Here, our attention is on those conditions on consequence relations (or gcr’s) which can be expressed in the form of sequent-to-sequent rules in proof systems. The derivability (indeed, admissibility) of such a rule induces a closure condition of a certain sort – on which there will be more to say below (p. 594) – on the associated consequence relation. More precisely, the phenomenon of unique characterization of connectives which will occupy us in this section arises when a connective is governed by rules which so constrain the logical behaviour of compounds formed by that connective that any other connective supposed governed by exactly similar rules would form compounds synonymous with those formed by use of the originally given connective. We are to understand this claim of synonymy in terms of a combined proof system presented by the original rules and the duplicated rules for the new connective. The structural rules (R), (M) and (T) are presumed available. We concentrate on the natural deduction approach to begin with, and when using Lemmon-style annotations (eschewing explicit occurrences of “ , that is) an appeal to (R) is as usual indicated by writing the word “Assumption” on the right. For example (in Set-Fmla) if we set alongside the rules (∧I) and (∧E) precisely analogous rules for a new ‘copy-cat’ connective ∧ , (∧ I) and (∧ E), the former (by way of example) being given by: ΓA

(∧ I)

Δ B

Γ, Δ A ∧ B

Then we can prove A ∧ B A ∧ B thus, for any formulas A and B: 1 1 1 1

(1) (2) (3) (4)

A∧B A B A ∧ B

Assumption 1 ∧E 1 ∧E 2, 3 ∧ I

Similarly, we can prove the converse sequent by making use of (∧ E) and then (∧I). Thus, where is the consequence relation associated with this combined ∧ + ∧ proof system, we have A∧B A∧ B, for any formulas A and B. (A more specific version of unique characterization can be found in Zucker and Tragesser [1978], p. 509, where it is specifically required that for a new connective # with the same – or rather reduplicated with # in place of #, both connectives being n-ary for some n – introduction rules as #, we should be able to prove #(A1 , . . . , An ) # (A1 , . . . , An ) for all formulas A1 , . . . , An . We do not here consider the merits of this proposal – cast as part of Zucker and Tragesser’s conditions of suitability for a set of elimination rules for a given set of introduction rules – or of any alternative which is similarly tied to the natural deduction approach. We prefer a notion of unique characterization which is ‘approach-neutral’, including in the following exercise the case of sequent-calculus rules, and below, in 4.31.3, the axiomatic approach.)

580 CHAPTER 4.


Exercise 4.31.1 (i) Show that a similar reduplication of the rules (∨I) and (∨E) for a new connective ∨ allows the proof of A ∨ B A ∨ B, and the converse sequent, for any formulas A, B. (ii ) Do the same for the rules (→I) and (→E). (iii) Do the same for the rules (EFQ) and Purified (RAA) (4.14). (iv ) Go through the set of Set-Set sequent calculus rules governing ∧, ∨, →, and ¬ in the system Gen (1.27) and show that the pair of rules (# Left, # Right) governing each connective gives an equivalent connective when reduplicated and combined with the original rules. (Note: In practice – certainly for the cases mentioned in part (iv) here – and no doubt in theory too, given a suitable general characterization of Gentzen-style sequent calculus rules, the unique characterization property of the rules (# Left, # Right) is the same as the regularity property for these rules, as introduced for 1.27.11 above. See also the Digression following 4.37.5; the case of ¬ is treated in the second paragraph of 4.32.) Now the equivalence of ∧ and ∧ compounds (with the same components), noted in the discussion leading up to the above exercise, does not guarantee their synonymy, since may not be congruential. But since our question is whether the rules (∧I) and (∧E) uniquely characterize ∧, one simple policy would be to consider the combined {∧, ∧ } proof system to have a language with no other connectives besides ∧ and ∧ ; in this case congruentiality does follow. Such a simplification is not possible in the case of impure rules which govern not only the connective whose unique characterization is at issue, but also some other connective; even in the case in which this other connective (or connectives—for there may be more than one) is guaranteed by the rules in question to be congruential, we shall sometimes describe the given connective as uniquely characterized by the rules in terms of that other (or those others), as reminder of its (their) presence. By way of illustration, consider, for a language containing a singulary connective O and a binary connective $, the following rules, stated here as formulato-formula rules, but to be understood as in Set-Fmla with the conclusion depending on the union of the assumptions on which the premisses depend. Example 4.31.2 Rules governing two connectives: A

OB A$B

A$B

A$B

A

OB

The upshot of these rules is to make A $ B amount to what, in the presence of (classical) ∧ would be written as A ∧ OB. As with ∧, if we reduplicate these rules governing $ by parallel rules governing a new binary connective $ , we can prove A $ B A $ B and conversely (for any A, B). Note that we are supposing that the reformulation of the rules leaves O intact. They do not, for example, invoke a duplicate singulary connective O . In such cases, it is natural to say that the rules uniquely characterize $ in terms of O. We will make precise the idea, deployed informally here, of a rule’s governing a particular connective in 4.33. On the explication proposed, the rules in 4.31.2


581

govern O no less than they govern $, so it would be wrong to say that these rules uniquely characterize the connectives they govern, tout court, and potentially misleading even to say that they uniquely characterize $, for which reason we sometimes add some such rider as “in terms of O”. Now nothing in this last example presupposes that O is congruential. In fact, we might suppose that the rules 4.31.2 are the only rules governing O, and then the combined $-with-$ proof system certainly won’t prove, for example, the sequent O(p $ q) O(p $ q). So we have to say, strictly, that these rules don’t even uniquely characterize $ in terms of O, since they give equivalent, but not synonymous, $- and $ -compounds. One could say that the rules uniquely characterize the connective to within equivalence, as opposed to (as is required by the official definition) to within synonymy. However, we will mostly ignore this nicety, and will in any case concentrate on pure rules for which such complications don’t arise. To reduce complications, we will also always assume that any rules for the framework Set-Fmla or Set-Set always include, whether or not these are explicitly listed under those to be reduplicated, the frameworkappropriate versions of (R), (M), and (T). Before leaving the impure cases behind, we give a more interesting example than 4.31.2, this time in Fmla, of the relativized notion of unique characterization. The system Kt was introduced at the end of 2.22. Observation 4.31.3 The rules used to present the basic tense logical system Kt uniquely characterize H in terms of G and the other connectives of the language of that system. Proof. Note that (1) B → HFB and (2) FHA → A are derivable in Kt ; therefore (1) and (2) = FH A → A are derivable in the combined {H, H } system; so, putting H A for B in (1), we get, in this system, (3) H A → HFH A. Since the rule allowing H to be prefixed to antecedent and consequent of a provable implication is derivable in Kt , we can apply it in the combined system to (2) , giving (4): HFH A → HA. Now, (3) and (4) have as a tautological consequence: H A → HA; the converse is proved similarly. Exercise 4.31.4 (i) Why does the fact that the formula cited in the last sentence of the above proof is a tautological consequence of (3) and (4) show that that formula is provable in the combined system? (ii ) How does the fact that H A provably implies and is provably implied by HA in the combined system, for any formula A, show that HA and H A are synonymous in that system? One reason for including the phrase “in terms of G” in the formulation of 4.31.3, whose absence we described above as potentially misleading, is to avoid confusion with another possible claim, to the effect that H and G are uniquely characterized as a pair by the cited rules, in the sense that if those rules are reduplicated by simultaneously replacing H and G by surrogates H and G , the combine system yields H and H synonymous and G and G synonymous. (This would be false, as with O and # from Example 4.31.2.) We shall, however, having nothing to say in what follows on this topic of simultaneous reduplication. Turning our back now on impure rules and the relative notion of unique characterization, let us consider some negative examples:

582 CHAPTER 4.


Exercise 4.31.5 (i) Show that the rules (∧E) do not uniquely characterize ∧. (ii ) Show that the rules Γ A

Γ A#B

ΔB

Γ, Δ A # B

Δ, A B

Θ, B C

Γ, Δ, Θ C

do not uniquely characterize the connective # they govern. (Hint: think about the rules governing ∧ and ∨ in Nat.) To amplify the hint given for part (ii) here: note that we could associate either of the truth-functions ∧b and ∨b with # and the rules supplied would be validitypreserving – in fact they would preserve the property of holding on any given valuation making either association of a truth-function with #; therefore we could associate one with # and the other with # in the combined system, revealing their non-synonymy. This method is not always available: Example 4.31.6 Take the following pair of zero-premiss rules governing a singulary connective #: A ###A

###A A

Only one of the four singulary truth-functions I, V, F, N can be assigned to # in the manner of the previous remark, namely I. Yet those rules do not uniquely characterize # (and we do not have A #A), since we have not built into the stock of rules anything to guarantee extensionality. To see that we do not have uniqueness, consider functional Kripke frames (W, f ), as in 2.22.9 (p. 288), with the added condition that f (f (f (x))) = x for all x ∈ W . Clearly every such frame validates all sequents instantiating the above schemata. Adding reduplicated forms of the schemata for # and considering frames (W, f, f ) in which each of f , f is similarly ‘of period 3’, gives a class of frames for the combined logic, any one of which having f = f will invalidate #p # p and its converse. (Note, incidentally, that the combined logic is congruential.) Remark 4.31.7 The absence of any rules securing extensionality in 4.31.6 is crucial to the example. In their presence, by 3.22.2 (p. 450), we have not only A ###A and ###A A, but also #A ###A and ###A #A, and therefore A #A and the converse sequent, which together suffice for unique characterization. We close this introductory subsection by making a connection between conditions on algebraic operations which uniquely specify those operations—briefly alluded to in the second paragraph—and unique characterization of connectives. (The fact that connectives are operations in the algebra of formulas is not to the point here: see 1.19.) We have already seen (4.31.1(iii)) that the rules governing negation in INat—or more accurately a version of that system with a ‘purified’ version of (RAA)—uniquely characterize that connective. For comparison here we should recall from 0.21.5(ii) on p. 22 that a bounded distributive lattice can be equipped with an operation of pseudocomplementation in at most one way. To qualify as such an operation, we further recall, singulary ¬ must satisfy two conditions, namely that for all lattice elements a and b: (i)

a ∧ ¬a = 0

and

(ii )

a ∧ b = 0 ⇒ b ¬a.


583

If we suppose that these conditions are satisfied not only by some given operation ¬ but also by an operation ¬ the question of whose distinctness from ¬ we leave open, then we find that question answered in the negative since ¬ a will, in virtue of what we might call (i) , verify the antecedent of (ii) when taken as b, implying the consequent: ¬ a ¬a. Reasoning instead with (i) and (ii) , we get: ¬a ¬ a. Hence ¬a = ¬ a. ‘Uniqueness by antisymmetry’, we could label the phenomenon illustrated here. (Note that the condition of distributivity turns out to have been irrelevant to the present point, and was included only because pseudo-boolean algebras are required satisfy it.) In terms of -based algebraic semantics, as we called it in 2.14, the correspondence between connectives and operations in suitable (other) algebras, and between “” and “”, shows us that we are just working through the uniqueness argument for intuitionistic negation in an algebraic setting. For the conditions (i) and (ii) are just versions suited to that setting of the rules (EFQ) and (RAA), respectively. Similarly, the cases of conjunction and disjunction correspond to meet and join in an arbitrary lattice, thought of as arising (0.13) from a poset, with ordering , in which every pair of elements have a greatest lower bound and a least upper bound, we have ‘uniqueness by antisymmetry’ for these operations. (Strictly, the sentential analogue of this is for the case of congruential logics; all we have from 4.31.1(i) and the discussion preceding it is a “” conclusion, not sufficing in the absence of congruentiality for the synonymy of the formulas involved.) Conditions (i) and (ii) on pseudocomplementation above are algebraic analogues of a ‘basic’ and a ‘superlative’ condition respectively, in the terminology of 4.14.2 (p. 526). The first says that the pseudocomplement of a has a certain property—namely ‘meet’ing with a at 0—while the second says that it is the greatest element to possess this property. Note that keeping the basic condition the same, but dualizing the superlative condition (to a ∧ b = 0 ⇒ ¬a b) would also allow for a ‘uniqueness by antisymmetry’ argument; though in this case, the operation defined would be less interesting: it would be the constant function mapping every element to 0. However, if the basic condition is also dualized we again obtain something of algebraic interest, sometimes called dual pseudocomplementation. The corresponding singulary connective will occupy us in 8.22. (And in 8.11, we shall reformulate the above basic + superlative analysis thus: intuitionistic negation forms the strongest contrary of what is negated, whereas this connective, ‘dual intuitionistic negation’, forms instead the weakest subcontrary.) Similarly, in the case of the greatest lower bound characterization (in terms of antecedently given ) of the meet of two elements, we have a ‘basic’ part (“lower bound”) and a superlative part (“greatest such”). This gives a slightly different slant on the uniqueness-by-antisymmetry phenomenon noted for this case above, since it enables us to relate it to the case of conjunction as treated in 4.14.1(i) from p. 526, and in more detail in 4.14.2. Note that it is specifically the second characterization of conjunction in 4.14.1(i) that corresponds to the ‘greatest lower bound’ characterization of the meet operation. The conjunction implies each of its conjuncts, and is the weakest formula with this property. (Recall that the logical “weaker” corresponds to lattice-theoretic “greater”: see Figure 2.13a for a visual rendering of this correspondence.) The first characterization has no such analogue because we can’t replace the comma in the direct translation of A, B C, namely “a, b c” by “∧” in advance of having this operation

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available, and as it stands, this translation does not make sense because is a binary relation between lattice elements whereas the consequence relation relates, not formulas, but sets of formulas, to formulas. Remark 4.31.8 The above consequence statement A, B C corresponds to a sequent A, BC in Set-Fmla. In Fmla-Fmla (as for the system Lat of 2.14) again we have only the second of the above basic/superlative characterizations of conjunction. For disjunction, in Set-Fmla, we again have only one of the such characterization expressible, as mentioned in the ‘note’ following 4.14.2; here, because of the framework chosen, we can no more write “A B, C” than in the lattice-theoretic setting, we could write “a b, c”. (Of course, people do sometimes write something like this, but meaning simply that a b and a c.) Still, all we need for uniqueness by antisymmetry in the case of connectives is one basic + superlative description at our disposal, and it is clear that we have such descriptions for the connectives mentioned in 4.31.1. In the terminology of Zucker and Tragesser [1978], as reviewed in 4.14, we could describe the effect of the ‘superlative’ component in these cases as stabilizing the content of the ‘basic’ component (though without the bias in favour of elimination rules effecting this stabilization); formulations emphasizing more strongly the element of unique characterization in such bipartite descriptions may be found in Došen and Schroeder-Heister [1985].

4.32

Stronger-than-Needed Collections of Rules

The present subsection draws out the significance of considerations of unique characterization for the philosophy of logic. In particular, we shall see that there are difficulties for a position, which has tempted several writers, maintaining that proponents of different logics on the same language must attach different meanings to the connectives. Let us illustrate the issue by reference to the difference between the treatment accorded to negation by classical and intuitionistic logic. Part (iii) of 4.31.1 shows that the rules governing ¬ in a natural deduction formulation of intuitionistic logic in Set-Fmla are sufficient to characterize that connective uniquely. In classical logic similarly presented, there is an additional rule ((¬¬E), as in Nat in 1.23) which renders provable additional sequents, in the light of which we say that the classical collection of rules is stronger than the intuitionistic collection, and in particular, therefore, that those rules are stronger than is needed for unique characterization (of, in this case, ¬). We will sometimes refer to this feature simply by saying the rules are ‘stronger-thanneeded’. The same phenomenon is evident in the sequent calculus treatments of these logics. Recall that, for the intuitionistic case, we apply similarly formulated rules for the classical case but in the framework Set-Fmla0 rather than Set-Set. (See 2.32.) Then the point is that we can argue: (1) (2) (3)

AA ¬A, A ¬A ¬ A

(R) 1 (¬Left) 2 (¬ Right)


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and similarly, mutatis mutandis, for the converse sequent, without venturing out of Set-Fmla0 . However, let us return to natural deduction in Set-Fmla, since in this case, the formulation of the rules has a more evidently plausible claim as embodying the inferential practices of those reasoning in (rather than theorising about) the two logics. The tempting position mentioned in our opening paragraph would have the proponent of classical logic urge upon the intuitionist the suggestion that classically ¬ means one thing, something (presumably) in virtue of which meaning such classical principles as the law of excluded middle and—a better example here, since extraneous connectives are not drawn in—the rule (¬¬E), are correct, while intuitionistically, ¬ means something different, in virtue of which meaning, such principles are not correct. In the celebrated phraseology of W. V. Quine, a popularizer of this kind of view, the intuitionist (or proponent of any other ‘rival’ to classical logic) sets out intent on ‘denying the doctrine’ but in the end is only ‘changing the subject’ (Quine [1970], p. 81). Since the parties to such a dispute mean different things by the connectives they employ, the appearance of a conflict is, on this view, the illusory product of not recognising that they are talking at cross-purposes. The notes (p. 626 onward) to this section quote a passage from Popper [1947a] which gives a particularly clear expression of this attitude; Popper [1948], on the other hand, brings to bear the considerations of unique characterization which show what is wrong with it. If people are under the mistaken impression that they are disagreeing, and this mistake is due to their attaching different senses to one and the same expression, then the appropriate move is to disambiguate the expression, and continue on from there. Thus, in our particular case, the ‘different meanings’ theorist is in effect suggesting that there might be distinguished two separate singulary connectives, say ¬c and ¬i , for classical and intuitionistic negation, respectively, each governed by its own set of rules: the classical ¬-rules for ¬c and the intuitionistic ¬-rules for ¬i . “Live and let live”, urges this theorist, adding ‘while I, as a proponent of classical logic find that ¬c renders the properties of my use of “not” satisfactorily, your favoured rendering will be in terms of ¬i .’ But it will be clear from what has gone before that this plea for tolerance should be rejected by the intuitionist. After all, the intuitionistic rules governing ¬ are already sufficiently strong so as uniquely to characterize that connective; writing “¬i ” doesn’t change this. Any other connective ¬ in the same logic governed by these rules whether or not it is governed by additional rules will collapse into (end up synonymous with) that connective: in particular, this will happen if we choose ¬ as ¬c . So, as even someone impartial in respect of the difference of opinion between the classicist and the intuitionist here will acknowledge, if the disambiguation ploy is attempted and the two separate negations are allowed to rub shoulders in the same logic – and if we do indeed have two separate concepts of negation here, then why shouldn’t they? – the ploy defeats its own purposes, since in this combined logic, there is now no difference between them after all. Setting aside this impartial spectator, consider now how much more seriously disastrous the proposal to recognize the separate existence of ¬c and ¬i is for the intuitionist. Since in the combined logic, the two connectives are synonymous (i.e., form from the same components, synonymous compounds), all the distinctively classical features of ¬c will spill over onto ¬i : in particular, we will be able to prove ¬i ¬i p p. Adding the connective ¬c to the language and the classical rules governing it to the proof system, then, gives rise to a non-

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conservative extension of the original intuitionistic system (INat, as it might be). Accordingly, in the spirit of §4.2, there simply is no such connective as classical negation: its existence (with the assigned logical properties) is intuitionistically unintelligible. The offer of ¬c alongside ¬i is one most strenuously to be declined. Exercise 4.32.1 (‘Classical Negation as a Trojan Horse’) Actually exhibit a proof of the sequent ¬i ¬i p p in the proof system presented by purified (RAA) and (¬¬E) for ¬c and purified (RAA) and (EFQ) for ¬i . The moral of the above example is rather general: if your favoured logic has rules which uniquely characterize a connective, then any connective putatively governed by some stronger collection of rules which includes all your rules should be denied intelligibility, if that excess strength consists in allowing the proof of sequents whose analogues for your own connective are not already in your favoured logic. One might consider a more aggressive position: why should stronger-than-needed rules be countenanced at all? If the meaning of a connective is already fixed by certain rules, then what justification can there be for logical principles which do anything more than simply exploit the meaning so fixed? A treatment of the issues here would take us outside our brief, since it would involve the question of what makes a piece of vocabulary (such as a connective) a piece of logical vocabulary, and, as remarked in the Preface and in the Appendix to §1.2 (p. 180 onward), we are not pursuing this “What is a logical constant?” question. (The idea that a logical constant should be specifiable by uniquely characterizing rules is frequently given a sympathetic airing—for example, in Belnap [1962], Gabbay [1977], Zucker and Tragesser [1978], and Došen [1989b] and Dummett [1991]. There is often – as in Došen’s paper, as well as in Tennant [1987b] – the attractive suggestion that the connectives be susceptible of an ordering in which the ‘earliest’ are uniquely characterized by pure rules, and any impure rules for a connective involve, in addition to that uniquely characterized, only connectives earlier in the ordering.) But a purely ad hominem consideration of this, by way of a response from the intuitionist, as we have crudely been calling one who deems IL to be the correct logic (for the connectives ∧, ∨, →, and ¬), will illuminate a key feature of the above ¬c /¬i discussion. The ad hominem attack on the proponent of IL who declines the invitation of ¬c on the above spill-over grounds (as illustrated in 4.32.1), we may imagine to come from the quantum logician, and since the point concerns disjunction, let us take the latter in particular to endorse only the restricted elimination rule (∨E)res given in 2.31 (p. 299), for the reason that the unrestricted rule (∨E) leads, together with the ∧-rules of systems such as Nat to a proof of the unwanted Distribution Principle(s). 4.31.1(i) asked for a proof that the usual natural deduction ∨-rules uniquely characterize ∨; since the restriction on (∨E)res need not be violated in showing this (and will not be unless unnecessary detours have been taken), the rules (∨I) and (∨E)res uniquely characterize the connective they govern, and the original pair, (∨I) and (∨E) – at least when taken in combination with the ∧-rules – fall into the stronger-than-needed category. (If we defined comparative strength of collections of rules in terms of derivability of sequent-to-sequent rules rather than of provability of sequents, we would not need to add this last qualification.) Thus our quantum logician’s


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charge against the intuitionist is that if collections of rules are to be repudiated when they any stronger than is needed for the unique characterization of the connectives they govern, IL is itself in bad shape as far as ∨ is concerned. Let us pause before considering a reply on the part of the intuitionist, to acknowledge that the quantum logician has exactly the same reasons in respect of ∨ as the intuitionist has in respect of ¬, to decline the offer of an ambiguity resolution. The quantum logician cannot sensibly be saying, or be talked into conceding, that while classical (or intuitionistic: recall that IL is ∨-classical in the sense of 1.13) disjunction is a perfectly coherent concept, it just happens not to be the disjunction concept which QL formalizes by “ ∨” and takes a special interest in. The same spill-over problems as before beset any such position, and an attitude of intolerance is called for: Exercise 4.32.2 (‘Classical Disjunction as a Trojan Horse’) In the proof system with rules (∧I), (∧E), and (∨I) and (∨E) for a connective ∨c , and (∨I) and (∨E)res for a different connective ∨q (the subscripts suggesting classical and quantum), prove the sequent: p ∧ (q ∨q r) (p ∧ q) ∨q (p ∧ r). Returning to the imagined debate on disjunction between the proponent of QL and that of IL, one can readily imagine an objection of the following form on the latter’s part, on being asked to surrender the additional strength of (∨E) over (∨E)res : it is true that (∨E) is stronger-than-needed, but the restricted rule violates the condition of generality in respect of side formulas mentioned in 4.14, and (∨E) is what we get by turning (∨E)res into a rule abiding by that condition. Suppose—though no argument has actually been provided to give this generality condition the status of a desideratum—its having such a status is not contested by the quantum logician. Then, since the effect of this condition depends very much on the logical framework chosen, by moving from Set-Fmla to Fmla-Fmla, its negative verdict on (∨E)res could be sidestepped. Of course that rule would then itself be even further restricted, since the set-variable “ Γ” in its formulation (see 2.31) would have to be replaced by a formula variable. Analogous changes would need to be made in the other rules. An additional point of interest to note here is that if the quantum logician were to resist this change of framework—say, not being persuaded that all rules should be general in respect of side formulas—then the ad hominem move against the intuitionist would backfire, since QL itself has, in respect of ∧, rules which are ‘stronger than needed’ for the unique characterization of that connective. A glance at the proof, in the opening paragraph of 4.31, of A ∧ B A ∧ B shows that this proof would go through with a ‘restricted’ version of (∧I) which we may as well call (∧I)res : from C A and C B to C A ∧ B. Taken as a rule in Set-Fmla, this rule violates the side-formula generality condition, though of course it does not do so in Fmla-Fmla. In Set-Fmla, the collection consisting of (∧E) and (∧I)res is weaker than the more familiar pair of rules for ∧, since, for example, we cannot prove on their basis p, q p ∧ q. This follows from the following Exercise 4.32.3 Show that the consequence relation associated with the proof system based on rules (∧E), (∧I)res , and (R), (M), (T), is left-prime. (Left-primeness was introduced for 1.21.5, p. 110; the point of the above exercise is made in Koslow [1992], p. 129f. See also the notes to this section, under the heading ‘Restricted ∧-Classicality’: p. 629.)

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The moral of the above discussion is that pressing the objection against stronger-than-needed collections of rules when the extra strength comes from greater generality in respect of side formulas implicitly involves taking a stand on the issue of which logical framework is to be preferred. Of course, we already know from Gentzen’s having given an articulation of the difference between IL and CL (using the sequent calculus approach) in terms of the distinction between Set-Set and Set-Fmla0 , that disputes as to which logic is correct are highly sensitive to framework choice. This issue would appear to deserve more discussion than it usually receives. Some further aspects of the stronger-than-needed situation presented by the case of ¬c here will be taken up in 4.34, after we have had a closer look at the notion of rule in play in the discussion.

4.33

Intermission: Rules Revisited

In 1.24 we characterized n-premiss rules for a given language and in a given logical framework as (n + 1)-ary relations holding between whatever count according to that framework as sequents, composed out of formulas of the language in question. This was not intended as a definition in the sense of a statement of necessary and sufficient conditions, since there was no desire to deny the plausibility of imposing additional conditions on the relations involved. Rather, the intention was to suggest what kind of thing a rule is. However, that characterization was provisional, being especially well suited to the discussion of rules in its immediate vicinity. It will perhaps be clear from the way rules have been spoken of in the present section that a refinement is needed. For we have insisted that a connective governed by certain rules in a language continue to be governed by those same rules in an expanded language. This led to requiring the rules (∧I), (∧E), to remain in force in the language containing not only ∧ but an additional connective ∧ , governed by analogous rules. It is essential to the treatment of uniqueness for connectives that such talk should make sense. This means that we should not tie a rule down to a particular language in the manner of the provisional characterization. Since we want to re-explicate the notion of a rule, let us call a rule in the sense of 1.24 an application-set, since the old explication in effect identifies a rule with the set of its applications. In the case of an n-premiss rule, these application-sets are all sets of (n+1)-tuples of sequents of a particular language, so to jettison the dependence on a single language, we should allow that one and the same rule have different application-sets for different languages. This leads to the following idea: a rule is a function from languages to application-sets. Again, the intent is to say what kind of thing a rule is best thought of as being, rather than to be very selective amongst things of that kind one would be happy to consider as bona fide rules. It is clear that this conception is the one we have in effect been working with in treating schematic formulations such as that of (∧ I) at the start of 4.31. The rule depicted there takes us, within any language L, from sequents Γ A and Δ B, where Γ ∪ Δ ∪ {A, B} ⊆ L, to the sequent Γ, Δ A ∧ B. Calling the rule ρ, we have: ρ(L) = { Γ A; Δ B; Γ, Δ A ∧ B | Γ ∪ Δ ∪ {A ∧ B} ⊆ L}. (Recall from 1.24 the use of semicolons in place of commas between sequents.)


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Here we have been slightly more careful, in requiring that A ∧ B ∈ L. This implies that A and B belong to L but its not implied by it, of course; nevertheless, we want the application-set of a rule, as interpreted for a particular language, not to take us outside the language as we pass from premisses to conclusion. We take the line that if L does not have the connective ∧ , then ρ(L) = ∅; an alternative would be to treat ρ as undefined for such L and explicate rules as partial functions from languages to application-sets. We can, with the above understanding of rules in place, still continue to speak of the derivability of a rule in a proof system or its validity-preservation according to some notion of validity, even though the proof system will be couched in, and the notion of validity tailored to, some particular language L. In this case, where ρ is the rule in question, we understand these locutions as pertaining specifically to the application-set (or ‘rule’ in the older sense) ρ(L). From the schematic formulation of a rule, it is usually clear which rule has been formulated: its application-sets for a given language comprise precisely the instantiations of the schema in that language. We note that the idea makes perfectly good sense even for rules which are not schematic or substitutioninvariant, the most famous case of which is the rule of Uniform Substitution. Taken as a rule in the present sense, this rule maps any language to the set of pairs σ, σ of sequents of that language such that σ results from σ by uniformly substituting formulas of the language for propositional variables in σ. Apart from its making good sense of the way we are accustomed informally to speaking of rules, and in particular legitimating enough of that usage for the needs of treating unique characterization, this proposal allows as a by-product a convenient explication of what it is for a rule to govern a connective. We say: a set Φ of connectives is the set of connectives governed by a rule ρ when for all languages L, ρ(L) = ∅ iff all the connectives in Φ are amongst L’s connectives. When # ∈ Φ for such a Φ, then ρ governs (perhaps inter alia) the connective #. (On the alternative ‘partial function’ account mentioned above, we would put “ρ(L) is defined” for “ρ(L) = ∅” here.) (One could consider a more general notion of a rule, allowing the same rule to have application-sets as values relative not only to languages, but also to logical frameworks. However, that further modification will not be made here.) As was said above, it is usually clear from the formulation with schematic letters of a rule, what the intended rule in the present language-independent sense is. However there can be room for doubt; we defer raising examples illustrating this possibility until 4.36, where it will be illustrated in the area of modal logic. In the meantime, problematic cases will not need to concern us.

4.34

Stronger-than-Needed Collections of Rules (Continued)

Two residual worries over the discussion of ¬i and ¬c in 4.32 deserve to be addressed. They concern (i) a possible misunderstanding of the notion of unique characterization and (ii) a suggestion that the intuitionist can after all make sense of ¬c alongside of ¬i . To show the incorrectness of the latter suggestion, we shall draw on the richer conception of rules introduced in 4.33. But first, let us attend to the possible misunderstanding. Intuitionistic negation behaves one way, and classical negation behaves another way. For example, we have A ¬¬A for any A when = CL but we

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have (in general) only the left-to-right direction of this -claim for = IL . How then, can it be said that the intuitionistic rules governing negation characterize that connective uniquely, when classically, negation obeys all these rules (and more besides) and yet exhibits quite different behaviour? After all, we are dealing with connectives in the ‘logical role’ sense here, and surely disparities such as that over double negation just alluded to show that quite distinct roles are being played by ¬ in IL and CL? Of course these observations are quite correct. Thus it is a question of clarifying the sense in which talk of unique characterization is not undermined by conceding them. For the needed clarification, we should distinguish an inter systemic from an intrasystemic notion of uniqueness. Comparing two logics, we may say that the role played by # in one of them is the same as the role played by # in the other. This is an intersystemic (or ‘interlogical’) comparison. If additional connectives are present in both, then such a comparison is only intelligible modulo a correlation of the connectives in the two sets. For simplicity, take the case in which no other connectives are present, so that we do not have to deal with this correlation. Then sameness of logical role amounts to each of #, and #, ’s being a subconnective of the other in the sense of 3.24 (p. 461), where and are the consequence relations associated with the logics concerned. By contrast, the notion of unique characterization of connectives we have been working with is intrasystemic in nature: we ask whether nonsynonymous connectives could both obey certain rules within the same logic. Once a logical role has been picked out uniquely, by any description satisfied only by that role – and here one could take roles to be equivalence classes of ordered pairs consisting of a logic and a connective under the equivalence relation holding between #, and # , when # plays the same role in as # plays in (as explicated above: the mutual subconnective relation) – the further question arises as to whether the role so isolated is itself capable of being played by more than one connective (up to synonymy) within a single logic. It is the latter question which “Is # uniquely characterized?” asks. (Yet another uniqueness-related notion is mentioned in the notes to this section, under the name Troelstra–McKay supervenience. See the division of the notes headed ‘A Supervenience Relation’, on p. 629.) One cautionary note needs sounding here, on this question “Is # uniquely characterized?”, brought out by the follow-up: uniquely characterized by what? Recall (§1.2, Appendix, p. 180) that there are numerous different notions of what a logic is, appropriate to different purposes. We have been treating unique characterization by collections of rules, so if one wishes to speak of a logic uniquely characterizing a connective in its language, one had better be thinking of logics as collections of rules. Just looking at the collection of provable sequents is not enough. If “IL” is thought of as a name of the set of intuitionistically provable Set-Fmla sequents – or what comes to the same thing, of the consequence relation IL – then there is no intelligible question as to whether IL uniquely characterizes ¬, unless these rules are taken as zero-premiss rules each consisting of a single sequent. Zero-premiss rules of the more customary sort, sequent-schemata, offer an alternative and more interesting ‘collection of rules’ construal of IL. And this collection of rules does not uniquely characterize ¬, as it happens. (See Humberstone [2006a], pp. 67–71.) The same holds for the case of ∨ (see 4.35), though not for ∧. The point, to which we shall return after broaching the second of our two residual worries, is that the effect of (RAA) and


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(∨E) in the uniqueness arguments for ¬ and ∨ (supplied in response to 4.31.1) cannot be achieved using just the zero-premiss rules, even in conjunction with (M) and (T) whose ubiquitous presence (along with the zero-premiss (R)) we are always assuming for our combined #-with-# logics. The second worry can be put like this. Suppose one wanted, as an adherent of classical logic, to study intuitionistic and classical negation in a logic on a single language, notated (as in 4.32) respectively as ¬i and ¬c . The hope would be that the two connectives would remain logically distinct in such a combined setting. One might be thinking along the following lines. Does the Kripke semantics (2.32) for intuitionistic logic not afford an obvious way of interpreting a classical negation, which as before, we write as ¬c , as well as that treated in 2.32, which we shall now write as ¬i ? While for the truth at a point in a Kripke model of a formula ¬i A we require that A fail to be true at all ‘accessible’ points, for the truth of ¬c A we require instead only that A fail to be true at the point in question. Then, using the definition of validity as truth at every point in every model, the class of valid sequents would appear to provide precisely the desired ‘mixed’ logic, in terms of which one can, for example, make sense of the oft-heard claim that intuitionistic negation is ‘stronger’ than its classical counterpart by noting that while ¬i p ¬c p is valid (on the above semantics) the converse sequent is not. And so on. Now we have our puzzle: for does not the argument for the ¬i /¬c collapse, based on the stronger-than-needed rules governing ¬c , preclude precisely this possibility of a simultaneous logic of non-synonymous ¬i and ¬c ? The solution to this puzzle is again to recall that what does or does not uniquely characterize a connective is a collection of rules. We already know that (RAA) and (EFQ) uniquely characterize the connective they govern. Thus these two rules cannot be validity-preserving for both ¬i and ¬c , since in the combined logic inspired by the above Kripke semantics, these two connectives are not synonymous. The casualty is in fact (RAA) for ¬i . For simplicity, let us consider the ‘non-purified’ form of the rule, as in 1.23. In showing, for 2.32.5 (p. 309), that this rule, which takes us from Γ, A, B ∧¬B to Γ ¬A, preserves truth in a Kripke model (and a fortiori validity on a frame, and a fortiori again, validity over the class of all frames – this being the property 2.32.8, from p. 311, assures us is co-extensive with intuitionistic provability), we need to know that all formulas in Γ are persistent. That was no problem for IL, in view of 2.32.3 (p. 308), which assured us that every formula in the language was persistent. But of course in the new version of the Kripke semantics, designed to accommodate ¬c alongside ¬i , this is no longer so. Formulas of the form ¬c A are not guaranteed to be persistent. Therefore (RAA) for ¬i no longer preserves validity. This dissolves the puzzle. The worry just addressed arose from purely technical considerations. The fact that the rules for intuitionistic negation uniquely characterized that connective, and that the rules for classical negation were stronger, meant that in a combined logic of the #-with-# style with all these rules, the two connectives would come to be synonymous. One could only regard the symbols “¬i ” and “¬c ” as notational variants of each other. Yet the notion of validity afforded by the modified Kripke semantics promised precisely a composite classical-intuitionistic logic in which the two negations cohabited peacefully without collapsing into one. This seemed like a contradiction. The resolution came by noting that the composite logic was after all not that got by pooling the two collections of

592 CHAPTER 4.


rules: one of the rules – (RAA) for ¬i – had at the very least to be restricted in some way for the composite logic. A very simple syntactic restriction suffices, in fact, namely adding a proviso to the effect that no formula in Γ (recall the premiss sequent: Γ, A B ∧ ¬B) should contain occurrences of ¬c which are not in the scope of occurrences of ¬i . This will ensure that all formulas in Γ are persistent, thus allowing a soundness proof to go through, and it turns out not to be too restrictive to enable completeness also to be proved, for the composite {¬i , ¬c , ∧} logic w.r.t. the mixed Kripke semantics. (Classical and intuitionistic implications A → B may be taken as defined, respectively by ¬c (A ∧¬c B) and ¬i (A ∧ ¬c B).) The details may be found in Section II of Humberstone [1979]. (See also Barnes [1981] for a correction.) In fact the description of the mixed Kripke semantics above left open an important detail, namely: are propositional variables to be treated as persistent, or not? In the work just cited, they are so treated; if one does not require this, further changes would be needed to the proof theory just described. The puzzle we have been considering concerns the compatibility, as it was put above, of certain purely technical claims. We drew some philosophical morals from the uniqueness results for negation and disjunction in 4.32: first, the conclusion that ambiguity claims popular in some quarters could not be sustained, and secondly, that the intuitionist and the quantum logician would be well advised to adopt an attitude of intolerance toward—deny the very intelligibility of—certain look-alike connectives with stronger logical powers. Let us look again at this advice with the above Kripke semantics in mind. It was the intuitionist who was being advised, in the ¬i / ¬c case, to adopt the attitude of intolerance – not the classicist, while it was the latter who arranged the marriage of ¬i and ¬c in the mixed Kripke semantics whose offspring included the explication of the plausible-sounding claim of greater strength for ¬i over ¬c that we noted above. We can imagine the classicist urging the intuitionist that—for example, in the interests of seeing clearly what, if anything, was at issue between them—it is highly desirable that some composite logic be developed in which their contrasting conceptions of negation could be displayed and compared, and that since the mixed Kripke semantics offers this opportunity, the possibility of endorsing an (RAA) rule weaker than that traditionally endorsed, should be seriously considered. That is, the intuitionist is being told that while there is nothing wrong with (RAA) when all formulas involved are composed out of the familiar intuitionistic connectives, it will need to be restricted when distinctively non-intuitionistic connectives, such as ¬c , are also present. These suggestions should be resisted! Recall that the philosophical perspective which gives content to questions of existence and lends interest to questions of uniqueness is the idea that the rules governing a connective, or some special subset thereof (e.g., introduction rules), give it the meaning it has. From this perspective, the suggestion that to avoid non-conservative extension when other putative connectives are added, one abandon or restrict some of these rules is absurd. The nonconservativity of the envisaged extension serves only to underline the ‘putative’, and unless independent grounds are produced for showing the original logic—IL in this case—to be incorrect, no grounds for change have been offered. Consider an illustration of this point which is neutral between CL and IL. Suppose aliens land on earth speaking our languages and familiar with our cultures and tell us that for more complete communication it will be


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necessary that we increase our vocabulary by the addition of a 1-ary sentence connective V, say, the exact upshot of which we will eventually get the hang of but concerning which we should note immediately that certain restrictions to our familiar inferential practices will need to be imposed. As these Venusian logicians explain, (∧E) will have to be curtailed. Although for purely terrestrial sentences A and B, each of A and B follows from their conjunction A ∧ B, it will not in general be the case that VA follows from VA ∧ B, or that VB follows from A ∧ VB. (Or they tell us that in English-as-extended, a sentence of the form “A and VB” must not be taken to have VB as a consequence, etc.) How should we respond? By coming to suspect that they don’t know us that well after all, perhaps. Whatever they think “∧” (“and”) means, they’ve got it wrong if they don’t know that A as well as B, for any A, B, whatever, have to be consequences of a compound of which they are components for that compound to qualify as their conjunction. Perhaps they have misunderstood “and” as the equivalent of some special Venusian mode of sentence composition, which of course we will be glad to hear about, along with their singulary “V”: but mess with (∧I) and (∧E) and all you have is a failure to communicate. Exercise 4.34.1 “The above argument against revising the logic of conjunction in the light of the aliens’ advice is no different from rejecting the existence of negative integers out of reluctance to give up the expectation on the part of those unfamiliar with them, that m + n must always be greater than or equal to m.” Discuss. The above example of the rule (∧E) applies equally to assumption-discharging rules such as (RAA) and (→I). The reaction of the intuitionist to the suggestion that these rules be restricted should be the same as the terrestrials’ reactions to the Venusians. The revised conception of rules offered in 4.33 is intended to capture this universal and open-ended aspect: the rule (→I) is a function assigning to any language L containing → the set of pairs Γ, A B; Γ A → B

for Γ ∪ {A, B} ⊆ L. Remark 4.34.2 The special feature of ¬c which caused a problem for (RAA) and (→I) was that it allowed the formation of compounds which were, in terms of the Kripke semantics, non-persistent. We have suggested that such connectives be regarded as intuitionistically unintelligible. Although the title of Clarke and Gabbay [1988] speaks of an ‘intuitionistic basis’ for nonmonotonic reasoning, the novelty (see also Gabbay [1982]) consists in the addition of a singulary connective M for which the definition of truth lays down that MA should be true at a point in a Kripke model (as in 2.32, with accessibility relation R) if and only if at some R-related point, A is true. Since such formulas then fail to be persistent, the above rules of intuitionistic logic are compromised. (Here we are thinking of IL as a logic at level 1 of the hierarchy from the Appendix to §1.2, pp. 180 and 186.) The role of persistence for IL can be thought of as reflecting what counts as a (mathematically significant) proposition for intuitionists; similar conditions – cf. 6.43–6.47 – play a like role for other alternatives to CL. The idea of tracing differences between logics to differences in the conception of admissible propositions has been emphasized especially by van Fraassen, e.g., in van Fraassen [1973], [1988]; see also Hazen [1990a].

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The fact that rules which cannot be rewritten as zero-premiss sequent-tosequent rules may yet be described as uniquely characterizing a connective they govern shows that it would not be appropriate to think of a consequence relation itself (or even a gcr) as uniquely characterizing this or that connective in a language. The closest one can come to the present notion in these terms is to talk of a set of conditions on consequence relations—or on gcr’s—as having this effect (as indeed we did in 1.19). Those that we are considering especially here, to amplify the comments in the second paragraph of 4.31, are those conditions which have the form of metalinguistic Horn sentences of the second type distinguished in 1.13. The discussion continues in the terminology of rules. The point of the rules mentioned in 4.34.2 is that they, no less than the zero-premiss rules, record the inferential practices of a community. (Here we are taking rules like (∧I) and (∧E) as in effect zero-premiss rules in view of their equivalence, given (R), (M), and (T), to sequent-schemata A, B A ∧ B; A ∧ B A; A ∧ B B.) The disposition to argue from a conjunction to one of its conjuncts is not more worthy as a component of one’s use of (or commitment to) a particular logic than the disposition to pass from C (as depending on assumptions A and B) to B → C (as depending on A); see the introduction in 1.23 of this way of describing sequent-to-sequent transitions. In this particular case, of course, it is an introduction rule that is at issue, so the point being stressed here is consilient with Gentzen’s suggestion that we think of the introduction rules as giving the meanings of the connectives they introduce. In the following subsection, we will discuss the case of an elimination rule, namely (∨E), whose effect for purposes of unique characterization, cannot be captured with zero-premiss rules in SetFmla. We can do this in Set-Set, to be sure (by means of A ∨ B A, B) but that does not deprive the point of interest, because in the first place, this shows the general sensitivity of these questions to the choice of framework, and in the second place because Set-Set, for all its theoretical utility, is hardly the first framework that comes to mind for describing the inferential practices of a community, and in the third place because even that shift would not work in the case of other connectives; see the discussion of normal modal logics in Blamey and Humberstone [1991]. A further indication that reaching for Set-Set will not allow us to consider as rules governing connectives just zero-premiss rules, which does not emerge in the case of disjunction, on which CL and IL are in agreement, comes from the case of implication. While →-classicality for gcr’s can be described in terms of zero-premiss rules (cf. 1.18.4: p. 85), the property of being →-intuitionistic, which we now understand as at the start of 2.33 (p. 329) except that “” refers to a gcr rather than a consequence relation, cannot be. The following is due to Dana Scott (unpublished lecture notes): Observation 4.34.3 There is no set of unconditional -requirements on a gcr such that is →-intuitionistic if and only if satisfies those requirements. Proof. Let be a gcr which is →-intuitionistic but not →-classical. Then for some Γ, A, B, Δ, we have Γ, A B, Δ but Γ A → B, Δ. For any set of unconditional conditions on gcr’s satisfied precisely by those which are →-intuitionistic, if meets those conditions then so does any gcr ⊇ . But then take as given by: Θ Ξ ⇔ Θ Ξ, Δ (Δ as above). Note that Γ, A B but Γ A → B, contradicting the →-intuitionistic nature of and therefore


595

refuting the hypothesis that there is a set of unconditional requirements on gcr’s satisfied by all and only those which are →-intuitionistic. We can give a similar argument for the case of consequence relations, using disjunction to code commas on the right; for more on the theme of extensions of IL according to which → is not →-intuitionistic, see Humberstone [2006a]. Observation 4.34.4 There is no set of unconditional -requirements on a consequence relation such that is →-intuitionistic if and only if satisfies those requirements. Proof. As for 4.34.3, except that, where is the restriction of the consequence relation IL to formulas in the connectives → and ∨, we now define by: Θ C ⇔ Θ C ∨ p. It is easily verified that is a consequence relation and that ⊇ . So since is →-intuitionistic, it must satisfy any of the envisaged unconditional requirements on consequence relations, as must (as ⊇ ). But while p q (since p q ∨ p), we do not have p → q (since (p → q) ∨ p): so is after all not →-intuitionistic. Unconditional requirements on consequence relations (on gcr’s) correspond to zero-premiss sequent-to-sequent rules in Set-Fmla (in Set-Set). However, 4.34.3, 4 do not obviously imply that there are no such rules in the frameworks concerned which uniquely characterize intuitionistic implication. The latter result (for Set-Fmla) may be found in §4 of Smiley [1962a], in a somewhat different terminology (see notes). Smiley uses a matrix argument, as we shall see in the following subsection for the case of disjunction. (Unique characterization by zero-premiss rules means: by such rules taken together with the non-zeropremiss structural rules (M) and (T), since we always presume these available.) There is an implication in the converse direction, however. If we could find a set of intuitionistically acceptable zero-premiss rules which uniquely characterized →, then we could infer that the corresponding unconditional requirements on a consequence relation were necessary and sufficient for its being →-intuitionistic. (Why?) Using similar reasoning we could infer from the result for disjunction in 4.35 that no set of unconditional requirements on a consequence relation is necessary and sufficient for a consequence relation satisfying them to be ∨classical; this could then be transferred to →-classicality by using the classical synonymy of (A → B) → B with A ∨ B. Of course both of these results fail for gcr’s (1.16.5 on p. 77, 1.18.4 on p. 85). We have emphasized throughout this section that what does or does not uniquely characterize a given connective is a collection of rules, and our discussion of late, as well as in the following subsection, shows that one cannot restrict one’s attention to zero-premiss rules. One moral to draw from this is that it would be a sign of confusion to ask, without further ado, about the relationship between a connective’s being fully determined (in the sense of 3.11, 3.13) and its being uniquely characterized. For a connective is fully determined relative to (‘according to’) a (generalized) consequence relation, whereas a connective is

596 CHAPTER 4.


uniquely characterized relative to (‘by’) a set of rules, which (to take the SetFmla case) can be thought of as a set of conditions on consequence relations. The difference between these relata means that more care (‘further ado’) is required in formulating the questions in this area. We can certainly transform a consequence relation into a set of conditions on consequence relations. For example, given a consequence relation we can consider the condition (or onemembered set of conditions) on arbitrary consequence relations that they should extend the given . More in line with our reconceptualization of rules in 4.33 would be to transform into the collection of rules comprising precisely those rules ρ such that whenever A1 , . . . , An B, the rule ρ(L), for any language L extending the language of , consists of those sequents s(A1 ), . . . , s(An ) s(B) for s a substitution in L. This yields only zero-premiss rules, of course, and the rules so obtained by starting from as the smallest ∨-classical consequence relation do not uniquely characterize ∨, as we shall see in 4.35. The potentially confused question, raised above, about the relation between full determination and unique characterization, however, suggests a somewhat different approach. (We remain in Set-Fmla.) For, being fully determined – which, the reader will recall is a syntactic mark of (roughly) being “amenable to (though not necessarily requiring) truth-functional treatment” (p. 391: 3.13.9(ii),(iii)) – is a matter of satisfying certain determinant-induced conditions on consequence relations. These induced conditions can then be thought of as rules which may or may not together uniquely characterize the connective concerned. In fact, taken together, when the connective is fully determined according to the given consequence relation, they will have this effect, as we note in Remark 4.34.5 Consider the claims (1) and (2): (1) “If # is fully determined according to a consequence relation then the set of determinants concerned (considered as rules) uniquely characterize #.” and, conversely (2) “If # is uniquely determined by a collection of rules then # is fully determined according to the consequence relation associated with the proof system with that collection (as well as the structural rules) as a basis.” Claim (1) here is correct, as the reader is invited to verify, whereas (2) is not correct: consider for example ¬ and the rules governing it in INat. We have unique characterization here, a fact that has occupied us at some length in this section, but ¬ is not fully determined according to IL . (Of course, we could equally well have chosen → to illustrate this; see further 4.34.6 below.) Apart from full determination, Chapter 3 considered the weaker properties of extensionality and congruentiality. The latter pair (especially extensionality) will receive some attention below, in 4.37. We close with a further illustration pertinent to claim (2) above. Example 4.34.6 Consider a sentential constant z governed by the Set-Fmla rule: A z. This rule uniquely characterizes z. (Why?)


597

If we consider the proof system with the usual structural rules and this single rule, then where is the associated consequence relation we have A z for all A, though z (and indeed, for no formula A do we have A). It follows that z is not fully determined according to , since we have neither z (the condition induced by the determinant T ) nor z A (the condition induced by F ). (Such a z is called a mere follower in the discussion between 3.23.9 and 3.23.10: see p. 459.)

4.35

Unique Characterization by Zero-Premiss Rules: A Negative Example

We have seen that (∨I) and (∨E) uniquely characterize disjunction. Given the presence of (R), (M) and (T), the zero-premiss rules AA∨B

and

BA∨B

are interderivable (taken together) with (∨I). But (∨E) cannot similarly be reduced (Set-Fmla) to equivalent zero-premiss rules. (The contrast is of course with Set-Set, where we have: A ∨ B A, B.) And the zero-premiss rules derivable in the proof system based on (∨I) and (∨E) do not, as we shall see, uniquely characterize disjunction. To show this, we use a matrix argument, following Smiley [1962a], p. 430f. which (as was mentioned at the end of 4.34) features a similar argument for intuitionistic implication in Fmla. In 2.11, the direct product construction was reviewed for matrices, but matrix analogues of the algebraic notions of subalgebra and homomorphism were not considered. Before proceeding with the argument, we must make good that omission. We give slightly more detail than absolutely necessary for the present applications, so as to fill out the picture and make connections with the material in §2.1. Several candidates come to mind for extending the homomorphism concept from algebras to matrices, differing over what they require in respect of the designated values. All have appeared in the literature in various places as appropriate notions of matrix homomorphism. Given a function f : A1 −→ A2 , we describe f as a matrix homomorphism from M1 = (A1 , D1 ) to M2 = (A2 , D2 ) iff f is a homomorphism from the algebra A1 to the algebra A2 which—and here come three alternative proposals as to how to complete the definition—is further such that (i) if a ∈ D1 then f (a) ∈ D2 , for all a ∈ A1 or (ii ) if a ∈ / D1 then f (a) ∈ / D2 , for all a ∈ A1 or (iii) a ∈ D1 iff f (a) ∈ D2 , for all a ∈ A1 . We say that f is •

a designation-preserving matrix homomorphism, if (i) is satisfied

•

an undesignation-preserving matrix homomorphism, if (ii) is satisfied

•

a strong matrix homomorphisms, if (iii) is satisfied.

The first of these gives us a simple preservation result, whose proof is left to the reader; when we speak of a homomorphism from one matrix onto another, we mean that as a homomorphism from the algebra of the first to that of the second, the mapping in question is surjective:

598 CHAPTER 4.


Observation 4.35.1 If there is a designation-preserving homomorphism from M1 onto M2 , then any formula valid in M1 is valid in M2 . Corollary 4.35.2 If M1 = (A, D1 ) and M2 = (A, D2 ) and D1 ⊆ D2 , then any formula valid in M1 is valid in M2 . Proof. The identity mapping from A to itself is a designation-preserving homo morphism from M1 to M2 if D1 ⊆ D2 , so 4.35.1 applies. Of course the above Corollary follows immediately from the definition of validity in a matrix, since if a formula cannot receive values outside of D1 and D1 ⊆ D2 , then it cannot receive values outside of D2 ; nevertheless, this is as good a point to record the fact as any. Remark 4.35.3 Corollary 4.35.2 tells us that validity of a sequent in Fmla is preserved on passage from a matrix to a matrix differing only in having more values designated. The analogous claim for sequents of Set-Fmla (or Set-Set) would not be correct, as is shown by the example of the two Kleene matrices (2.11.2(i), p. 201). Undesignation-preserving homomorphisms can also be used for a ‘backward’ preservation result (cf. van Fraassen [1971], p. 90) to the effect that if such a homomorphism exists from M1 to M2 then any formula valid in M2 is valid in M1 . Note that we do not require ‘onto’ here. This can be used to show that the ‘Ł3 -tautologies’ (2.11) are all of them classical two-valued tautologies, by taking f (T) = 1, f (F) = 3, whereas cardinality considerations rule out the possibility of a surjective homomorphism from the two-element to the three-element matrix. (Note also that there is no way of representing the two-valued boolean matrix as a homomorphic image of the three-element Łukasiewicz matrix because of the behaviour of the operation associated with negation in the latter: in particular the fact that ¬2 = 2.) Alternatively, we may obtain this conclusion about the boolean and (three-valued) Łukasiewiczian tautologies by an adaptation of the subalgebra concept to the matrix setting. We say that (A, D) is a submatrix of (A+ , D+ ) when A is a subalgebra of A+ and D = A ∩ D+ . Again the proof of the following is left to the reader: Observation 4.35.4 If M is a submatrix of M+ then any formula valid in M+ is valid in M. Indeed, not just for Fmla but for Set-Fmla and Set-Set, we can assert that any sequent valid in M+ is valid in M. For example, on the set {1, 3} of elements of the Łukasiewicz 3-valued matrix (Figure 2.11a, p. 198) the operations corresponding to ∧, ∨, →, and ¬ behave in the subalgebra with that set as universe exactly like the corresponding operations in the two-valued boolean matrix, we can see that the latter matrix is isomorphic to the submatrix with 1 and 3 as elements, and thereby derive the conclusion from 4.35.4 that all Ł3 -tautologies are (classical) tautologies. The term isomorphic here is to be understood as follows: matrices are isomorphic if there is an isomorphism between their algebras which is a strong matrix homomorphism in the above sense; clearly isomorphic matrices validate the same formulas—indeed, the same sequents in any of our logical frameworks. In fact,


599

we can say (again without proof) something stronger, completing our review of the materials needed for proceeding: Observation 4.35.5 If there is a strong matrix homomorphism from M1 onto M2 , then the same sequents (in Fmla, Set-Fmla, or Set-Set) are valid in M1 as in M2 . Now let us consider the language whose only connective is ∨, and the consequence relation ∨ on this language associated with the (∨I), (∨E) proof system described above. This usage is consistent with that introduced following 1.13.7 (p. 67), since ∨ is indeed the smallest ∨-classical consequence relation (on its language). Another description available to us is that ∨ is determined by the class of all ∨-boolean valuations, which can be rephrased in matrix terminology as: is determined by the two-element boolean matrix whose sole operation is given by the usual truth-table for disjunction (see 2.11.1, p. 200). We now give two four-valued characterizations, one in terms of the product of the boolean matrix just described with itself, and depicted on the left of Figure 2.12a (p. 213). Call this matrix M1 . The second matrix, with elements {1, 2, 3, 4}, has again only the value 1 designated, and for ∨ we have: 1 if either x = 1 or y = 1 x∨y = 4 otherwise. We call this matrix M2 . Theorem 4.35.6 ∨ is determined by M1 and also by M2 . Proof. ∨ is determined by M1 : this is a special case of 2.12.8. ∨ is determined by M2 : the function f with f (1) = T and f (2) = f (3) = f (4) = F is a strong matrix homomorphism from M2 onto the two-element boolean matrix, so the result follows by 4.35.5. We want to show that the zero-premiss rules derivable in the proof system based on (∨I) and (∨E) do not uniquely characterize ∨. The reduplicated system relevant to settling this question of course has non-zero-premiss structural rules (M) and—more to the point—(T); but the rules laid down specifically for ∨ and its duplicate ∨ are those corresponding to the sequents provable on the basis of the (∨I)–(∨E) system, in the following way: take any such provable sequent and replace the distinct propositional variables occurring in it by distinct schematic letters to obtain a sequent-schema and add also the schema resulting from this one by substituting ∨ (throughout) for ∨. Actually, we prefer to write “∨1 ” and “∨2 ” rather than “∨” and “∨ ”. Thus since in the (∨I), (∨E) system, whose associated consequence relation is ∨ , we have r ∨ q (p ∨ q) ∨ r, we have in our stock of new schemata: C ∨1 B (A ∨1 B) ∨1 C

C ∨2 B (A ∨2 B) ∨2 C

We take these (which of course include (R) as the basis of a new proof system we shall call DD (‘Double Disjunction’), alongside rules (M) and (T). The language of DD includes both ∨1 and ∨2 , so naturally we take any instance

600 CHAPTER 4.


of the schemata just catalogued as an application of that schema, considered as a one premiss rule. The unique characterization of ∨ by zero-premiss rules derivable from (∨I) and (∨E) then comes down to the question of whether all sequents of the forms A ∨1 B A ∨2 B and A ∨2 B A ∨1 B are provable in DD. We shall show that they are not. (Došen and Schroeder-Heister [1988], call the use of (∨E) for either of ∨1 , ∨2 , as applied to a sequent derived by (∨I) for the other of those connectives ‘mixing’, and remark on p. 557 of a uniqueness argument which would employ (∨E) that “this mixing seems to be unavoidable”; so the discussion to follow shows that the mixing is indeed unavoidable.) To that end we define a new matrix M in terms of those, M1 and M2 , described above. M is to have the M1 table for ∨1 and the M2 table for ∨2 . (Thus the elements are again 1, 2, 3, and 4, with 1 designated.) Theorem 4.35.7 Every sequent provable in DD is valid in M. Proof. Every instance of one of the zero-premiss rules is valid in M, because every such sequent is a substitution instance of a sequent provable from (∨I), (∨E), with “∨” replaced by “∨1 ” or else by “∨2 ”, the latter being valid in M (4.35.6) and Uniform Substitution preserving validity in any matrix (2.11.4: p. 203). Further, the rules (M) and (T) preserve validity in any matrix. We are now in a position of verify that ∨ is not uniquely characterized by its zero-premiss rules: we show that p ∨1 q p ∨2 q is not provable in DD. This shows, incidentally, that the consequence relation DD associated with DD is not ∨-classical in its treatment of ∨1 (1.19), since clearly p DD p ∨2 q and q DD p ∨2 q, even though p ∨1 q DD p ∨2 q. Observation 4.35.8 The sequent p ∨1 q p ∨2 q is not provable in DD. Proof. In view of 4.35.7, it suffices to show that this sequent is not valid in M. For this, consider the M-evaluation h with h(p) = 2, h(q) = 3. This gives h(p ∨1 q) = 1, designated, but h(p ∨2 q) = 4, undesignated.

Exercise 4.35.9 Show that the role of the four-valued matrix M in the above proof could be played instead by the three-valued matrix: ∨1 *1 2 3

1 1 1 1

2 1 2 1

3 1 1 3

∨2 1 2 3

1 1 1 1

2 1 2 2

3 1 2 3

Note that the table on the right is the Kleene–Łukasiewicz table for disjunction. (Hint: the table on the left is a submatrix of M1 , so 4.35.4 applies; for that on the right, note that the crucial feature of M2 exploited in our discussion was that a disjunction received a designated value iff one of the disjuncts did.)


601

Further aspects of the logic of disjunction in Set-Fmla related to the above developments, though not specifically bearing on matters of uniqueness, will emerge in 6.46. Remark 4.35.10 If instead of considering the consequence relation ∨ , we consider the consequence relation on the language whose connectives are ∨ and also ⊥ which is determined by the class of all ∨-boolean valuations, it is natural to expect that the least consequence relation extending in turn this consequence relation by addition of all sequents ⊥ A (for all formulas A) is determined by the class of all {∨, ⊥}-boolean valuations. Using the table on the left of Exercise 4.35.9 above, Rautenberg [1981] shows that this is not so: the sequent p ∨ ⊥ p remains unprovable, for instance, as was mentioned shortly after 1.13.2. (Since only one binary connective is involved, we write “∨” rather than “∨1 ”. We must say how to deal with ⊥: we stipulate that h(⊥) = 3 for every matrix-evaluation h. This makes all sequents ⊥ A valid in the resulting matrix, while p ∨ ⊥ p is invalidated by the assignment h(p) = 2.)

4.36

Unique Characterization in Modal Logic

The main reason for broaching the topic of unique characterization in modal logic here is to illustrate some general issues arising out of the relation between the schematic description of a rule and the rule (in the sense of 4.33) so described. As in §2.2, only normal modal logics will be under consideration. We can begin with the Set-Fmla treatment of the logic K from 2.23, and the question of whether the rule (K) of that subsection: (K)

A1 , . . . , An B A1 , . . . , An B

uniquely characterizes the connective . If we add another rule, (K ), governing another singulary connective , directly analogous to (K), can we then prove— with any help we might need from (R), (M), and (T)—all sequents of the form A A and their converses? Exercise 4.36.1 Provide a negative answer to the above question by considering bimodal frames (W, R, R ), as in 2.22, in which R and R interpret respectively and , and neither R nor R is a subrelation of the other. We can similarly conclude that the collection of rules by which S4 was presented in 2.23 does not uniquely characterize , since we can have a bimodal frame with two distinct accessibility relations, each of them being reflexive and transitive. To escape such reasoning, we would need a modal logic S with Fr (S) described by conditions on its accessibility relation so strong that if (W, R) meets those conditions and so does (W, R ), then R = R . For example, if R is constrained to be the identity relation, we will be in this situation, since there is only one identity relation on a given set. Rules to add to (K) – which in fact becomes redundant in their presence – would be the zero-premiss rules: T:

A A

and

Tc :

A A.

602 CHAPTER 4.


Exercise 4.36.2 Show that the above rules uniquely characterize . Since there is likewise only one empty relation on a given set, we could consider the following variation on the schema Ver ( = ⊥) from 2.21: Ver

A

from which, again, (K) is derivable. (Note that the class of frames for the 2.21 system KVer comprises those whose accessibility relation is empty; “⊥” has been replaced by “A” in the interests of purity.) Exercise 4.36.3 Show that the above rule Ver uniquely characterizes . (Hint: remember that (R), (T), and in particular, (M), are always available.) The above examples of unique characterization in modal logic are not very promising perhaps, since in each case a truth-functional interpretation for is possible; in the terminology of 3.14, we interpret as I for the example of 4.36.2 and as V for that of 4.36.3. Exercise 4.36.4 Why do F and N not similarly put in an appearance here? The modal logics in which we have seen uniquely characterized are very strong – they are, as has already been mentioned, the two Post-complete normal modal logics. In looking for uniqueness amongst weaker systems, we may be inspired by the fact that not only are the identity relation and the empty relation unique on a given set, but so are the relations of non-identity and universality. We follow up this (as it happens, abortive) suggestion for the latter case, recalling (2.22.7, p. 286) that, just as KVer is determined by the class of empty-relational frames, S5 is determined by the class of universal frames. For this case, we abandon the ideal of purity, since we are considering a rule (5: A A, alongside those for KT) involving and hence also ¬ (since we took A to be the formula ¬¬A). We may as well suppose that all the boolean connectives (from 2.22) are present and ask after unique characterization in terms of them. The hitch in the line of reasoning gestured toward here is that the completeness result for S5 w.r.t. the class of universal frames (2.22.7 above) used the fact that point-generated subframes of equivalence-relational frames are universal. But this fact has no analogue in the bimodal setting: the ancestral of the union of two equivalence relations is not in general an equivalence relation. And after all, we can quash the present line of thought definitively exactly as in 4.36.1: one set can support two distinct equivalence relations. (Thus here we have one repercussion of the fact – 2.22.6, p. 286 – that ‘universality’ is not modally definable.) We obtain our first illustration of the not entirely straightforward way in which the formulation of a rule picks out the rule formulated, in the current, language-independent sense of rule, by considering a syntactic argument attempting to counter the above, semantically derived, verdict of non-uniqueness in the present case. The argument would invoke the rule mentioned in 2.23.6 (p. 295), q.v. for a definition of fully modalized : (I)

A1 , . . . , An B A1 , . . . , An B

Provided each Ai is fully modalized.


603

Our objector would argue for uniqueness of on the basis of this rule and the rule (E) of 2.23.4—or equivalently, the above rule T—by applying it to the T instance A A, to obtain A A, on the grounds that the left-hand formula is indeed ‘fully modalized’. The derivation of the converse sequent would proceed, as usual, mutatis mutandis. But is the formula A ‘fully modalized’ in the sense relevant to an application of (the above form of) (I)? Our official definition of full modalization in 2.23.6 required the occurrence of each atomic formula within the scope of some occurrence of : no mention here of . So the argument just sketched fails. A version of the above argument might succeed if full modalization were defined in terms of occurrences of atomic subformulas within the scope of modal operators, and the notion of a modal operator were spelt out in some language(and preferably also logic-)independent way. Rather than speculate as to how such a development might go, we can more profitably pursue an alternative route, sharing with that set aside here the idea of more fully exploiting the language-independence of rules than we have to this point: this fuller exploitation will be evident below as we isolate the concept of a ‘language-transcendent’ rule. The starting point for this alternative route is the logic of identity. It is clear that a notion exactly like that of unique characterization for connectives applies to the identity predicate ‘=’, as governed by the rules which in the terminology of Lemmon [1965a] appear as: (=I) t = t

(=E)

Γ t=u

Δ A(t)

Γ, Δ A(u)

In these formulations, t and u are arbitrary terms, A(t) a formula with one or more occurrences of t, and A(u) any result of replacing one or more of those occurrences by u. Reduplicating the rules, so as to govern a surrogate = , we can argue: 1 1

(1) t = u (2) t = t (3) t = u

(R) (= I) 1, 2 (=E)

To obtain line (3) here, we are substituting u for the second occurrence of t in line (2), so t = t is A(t). So the combined set of rules suffice for a proof of t = u t = u and, as usual, the converse sequent. (Unique characterization of identity along the above lines can be found in Quine [1970] and Williamson [1988b], amongst other places.) Now let us consider a binary connective ≡, which we may think of as expressing propositional identity (reading “A ≡ B” as “the proposition that A is the same proposition as the proposition that B”), governed by rules analogous to those above for =: (≡I)

A≡A

(≡E)

Γ A≡B

Δ C(A)

Γ, Δ C(B)

where C(B) is any formula differing from C(A) in having one or more occurrences of A replaced by B. (We return to these rules and the topic of proposi-

604 CHAPTER 4.


tional identity in 7.32.) A transcription of the above uniqueness argument for = establishes Observation 4.36.5 The rules (≡ I) and (≡ E) uniquely characterize ≡. To make the desired connection with modal logic, and in particular with the system S4, we need to note that strict equivalence in any normal modal logic – assuming for simplicity the availability of some functionally complete set of primitive boolean connectives – satisfies a rule like (≡I), with (A ↔ B) replacing A ≡ B, and that S4 is the weakest normal modal logic satisfying the analogous ‘strict equivalence’ version of (≡E); this is observed in Cresswell [1965] and Wiredu [1979], for example. To stick with the modal primitives familiar from §2.2, we rewrite the above rules as: ( ↔I) (A ↔ A)

( ↔E)

Γ (A ↔ B)

Δ C(A)

Γ, Δ C(B)

These rules, alongside (K), and any complete set of rules for the boolean connectives, suffice for the provability of precisely the sequents of S4 (as understood in 2.23.4, p. 295). Observation 4.36.6 The collection of rules just mentioned uniquely characterizes . Proof. For the consequence relation associated with the mixed {, } proof system using the rules in question together with their reduplicated analogues for , we have (A ↔ ), (A ↔ A) (A ↔ ) by ( ↔E); but the first formula on the left is a -consequence of A, the second is an instance of ( ↔I), while the formula on the right has A as a consequence. Thus: A A; a corresponding argument gives A A. The present collection of rules for S4 sidesteps the negative verdicts obtained along the lines of 4.36.1 for the more usual rules for S4 by virtue of the rule ( ↔E), which, like (≡E), exploits the current language-independent conception of rules in a particularly dramatic way. Call an n-premiss rule ρ locally based at a language L if L is the least language for which ρ(L) = ∅, and for all

∈ ρ(L ) there exists some substitution s (on L ) L ⊇ L, for all σ1 , . . . , σn+1 such that σi = s(σi ), for i = 1, . . . , n + 1, with σ1 , . . . , σn+1 ∈ ρ(L). In other words, a rule is locally based at L if it has applications relative to L, and every application of the rule in any larger language arises by substitution from some such application. By contrast with the ‘usual’ rules, the rules ( ↔E) and (≡E) are language-transcendent, i.e., not locally based at L for any L. This constitutes the second of our two illustrations of the ‘unstraightforwardness’ of the relation between the schematic formulation of a rule and the rule thereby formulated. To substantiate the claim of transcendence in the case of (≡E), it suffices to consider the application of this rule needed for the proof of 4.36.5:

4.3. UNIQUENESS OF CONNECTIVES A≡B A≡B

605 A ≡ A

A ≡ B A ≡ B for arbitrary formulas A and B. Clearly this application does not arise by substitution from an application of the rule in the language without ≡ ; the same is so for the application of ( ↔I) alluded to in the proof of 4.36.6. In fact, though we shall not prove this, exclusive use of non-transcendent rules in SetFmla or Set-Set does not allow for to be uniquely characterized except for the cases mentioned earlier: KT! (4.36.2) and KVer (4.36.3). In more elaborate frameworks, however, it is possible to give formulations even of K in which is uniquely characterized. As we shall not go into the details here, the interested reader is referred to Došen [1985a], and Blamey and Humberstone [1991], as well as the background supplied by Belnap [1982]. See also the Digression after 8.13.10 below for an illustration of this issue of framework sensitivity as it arises for the unique characterization of negation in some substructural logics.

4.37

Uniqueness and ‘New Intuitionistic Connectives’

The discussion of proposed new connectives for IL in 4.24 was put to one side because of the involvement of unique characterization in some aspects of this topic. Accordingly, we are now in a position to touch on those aspects. The scare quotes in the title of the present subsection are intended to suggest that no precise explication is agreed on as to what would or would not constitute a connective increasing the expressive power of the conventional language of IL (as in 2.32, say) in a manner broadly consilient with intuitionistic thought. One requirement on the addition of a new connective, with certain inferential powers, which is agreed to is that the addition should conservatively extend IL (4.21), and another – to justify the description as new – is that the connective in question should not be definable (3.15) in terms of the conventional intuitionistic primitives. Bowen [1971] was impressed with the possibility of introducing new connectives meeting the above two conditions via sequent calculus rules (in the framework Set-Fmla0 , as for IGen in 2.32) for which a Cut Elimination theorem could be proved (1.27). He described three such connectives, all having a somewhat ‘negative’ flavour; they are converse non-implication, an intuitionistic analogue of the Sheffer stroke (3.13.5, p. 388), and an intuitionistic analogue of the dual of the Sheffer stroke (“neither–nor”: 3.14.4). The negative flavour surfaces in the fact that one or more of the components ‘changes sides’ as we pass from premiss(es) to conclusion in the rules – contralateral as opposed to ipsilateral cases, in the terminology of Humberstone [2007b] – and in the abundance of empty right-hand sides, as we illustrate here with the rules for converse non-implication, to be symbolized as “n”: (n Left)

Γ, B A Γ, A n B

(n Right)

Γ, A

ΓB

Γ AnB

Exercise 4.37.1 Show that, using the rules (including (T)) of IGen, the above pair of rules are interderivable with the two zero-premiss rules: (nE)

A n B ¬(B → A)

and

(nI)

B ∧ ¬A A n B.

606 CHAPTER 4.


Now we can show that we have a ‘new’ connective on our hands here; the particular translations invoked in the proof arise from the schemas in 4.37.1. Observation 4.37.2 After extending IGen with the above rules, we can find no n-free formula C such that both p n q C and the converse sequent are provable. Proof. Let τ1 and τ2 be translations from the language of the present proof system to the language of IGen which translate propositional variables by themselves, and act as the identity translation for the connectives ∧, ∨, ¬, and → (in the sense that τ1 (A ∧ B) = τ1 (A) ∧ τ1 (B), etc., and similarly for τ2 ), and for formulas A n B behave thus: τ1 (A n B) = ¬(τ1 (B) → τ1 (A));

τ2 (A n B) = τ2 (B) ∧¬τ2 (A).

We take these translations to act on sequents of the language with n formula by formula, giving sequents of the language without n. It is clear that if a sequent σ is provable in the present proof system, then each of τ1 (σ) and τ2 (σ) is provable in IGen, since the provability of the translations of the premiss-sequents of (n Left) and (n Right) implies the provability (in IGen) of the conclusion-sequent. Thus if, in the extended system with n, we can prove both p n q C and C p n q, for n-free C, we have τ1 (p n q) and τ1 (C) provably equivalent in IGen, as well as τ2 (p n q) and τ2 (C). But since C does not contain n, τ1 (C) = τ2 (C) = C, so τ1 (p n q) and τ2 (p n q) are likewise provably equivalent in IGen; however, these two formulas are respectively ¬(q → p) and q ∧ ¬p, which are not in fact so equivalent (in particular ¬(q → p) IL q ∧ ¬p, since ¬(q → p) IL q). Therefore there is no such formula C provably equivalent in the extended system to p n q. In general, then, formulas A n B are not synonymous with n-free formulas in Bowen’s logic, and the new connective n is not definable in terms of the usual intuitionistic primitive connectives. But the materials deployed in the proof of n’s novelty reveal very clearly that n is not uniquely characterized by the proposed rules. We can think of the proof as showing that no derived connective of the language of IL (the language of IGen) has all and only the inferential properties common to the derived connectives used to specify τ1 and τ2 , it is obvious that if all we want is derived connectives having all such properties, then either of those between which – as we see most clearly from 4.37.1 – the connective n is ‘sandwiched’, will do perfectly well. If we interpret A n B à la τ1 , as ¬(B → A), then n obeys (n Left) as well as (n Right), as is also the case for the τ2 -style interpretation of A n B as B ∧ ¬A. Thus if the rules are reduplicated for a surrogate n , in the combined system we do not have n and n synonymous, since in general, as noted in the above proof, we do not have ¬(B → A) IL B ∧ ¬A. (Compare 4.31.5(ii), and the hint offered there.) Thus: Observation 4.37.3 The rules (n Left) and (n Right) do not uniquely characterize n. Similarly, the connective # from 4.31.5(ii) could be shown not to be definable in terms of the usual boolean connectives, in the logic extending CL by the rules


607

there given. Those rules amount to showing that A # B is ‘sandwiched between’ A ∧ B and A ∨ B, so to adapt the proof of 4.37.2, we should consider translations τ1 , τ2 , with τ1 (A # B) = τ1 (A) ∧ τ1 (B) and τ2 (A # B) = τ2 (A) ∨ τ2 (B). Its undefinability notwithstanding, one would be unlikely to think of such a # as a ‘new classical connective’ whose addition was mandated by a desire for expressive completeness. It is, no doubt, with such considerations in mind that Gabbay [1977] (and [1981], Chapter 7, §4) demands that not only should the addition of a new intuitionistic connective be effected by rules which give something not definable in IL though conservatively extending it, but also that the proposed rules (or axioms) uniquely characterize the connective in question. A similar non-uniqueness holds for one of Bowen’s two other connectives, |, the (intuitionistic) Sheffer stroke and its ‘dual’ ↓, as we shall see presently. These connectives are sometimes called nand and nor, respectively. Especially the first of these is potentially problematic in the intuitionistic case, and indeed we shall see below (4.38.5 – p. 618 below – and discussion following) that there is really no such thing as the intuitionistic Sheffer stroke – even setting aside the issue to be raised here about Bowen’s attempt to address | proof-theoretically. Bowen’s rules for | and ↓ are: (| Left)

(↓ Left)

ΓA

Γ B

(| Right)

Γ, A | B ΓA Γ, A ↓ B

ΓB Γ, A ↓ B

(↓ Right)

Γ, A

Γ, B

Γ A|B

Γ A|B

Γ, A

Γ, B

Γ A↓B

Like | (cf. 3.13.5 on p. 388, 3.14.4 on p. 405), ↓ has, in discussions of CL, a boolean interpretation, namely as that function ↓b associated with ↓ over the class of those valuations v satisfying, for all formulas A and B, v(A ↓ B) = T iff v(A) = F or v(B) = F. (↓b provides the unique solution to Exercise 3.14.4(i).) In accordance with our regular practice, we call such valuations ↓-boolean. Note that the above rules for | preserve the property of holding on v, for any | -boolean valuation v, and that the ↓-rules do likewise for ↓-boolean v. So any sequent provable using these (and the structural) rules is tautologous. (Of course, we do not expect the converse, since we are exploring possible new intuitionistic connectives.) Neither are we here interested in anything like the functional completeness properties enjoyed by these connectives in the context of CL, on which see, for example §9 of Quine [1951], where | and ↓ are called alternative denial and joint denial respectively. In fact the different connotations of the labels “nand” and “alternative denial” serve quite well to mark the distinction in IL, which is about to prove crucial, between ¬(A ∧ B) on the one hand and ¬A ∨ ¬B on the other. (This distinction will occupy us here, through much of 4.38, and also in 8.24.) Corresponding to 4.37.1, we have the following: Exercise 4.37.4 (i) Show that, using the rules (including (T)) of IGen, the rules governing | above are interderivable (| E): A | B ¬(A∧B) and (| I): ¬A ∨ ¬B A | B. (The “I” and “E” are for Introduction and Elimination, since we have horizontalized some impure natural deduction rules here, in the case of (| E), for instance, the rule taking us from premiss ΓA | B to conclusion Γ ¬(A ∧ B).)

608 CHAPTER 4.


(ii ) Do likewise for the above ↓ rules and the schemata: (↓E) A ↓ B ¬(A ∨ B) (↓I) ¬A ∧¬B A ↓ B. Parts (i) and (ii) here give us that the new compounds are sandwiched between compounds in the standard vocabulary of IL, but there is a difference between the two cases. The left-hand side of ( | I) is (in general) properly stronger than the right-hand side of ( | E) in IL; so the situation in this case is analogous to that of n above. But the sandwiching formulas under (ii) are intuitionistically equivalent. It immediately follows that ↓ is uniquely characterized by any complete set of rules for IL supplemented by the above ↓-rules, since each of ↓ and a connective ↓ for which those rules are reduplicated will form from formulas A and B a compound provably equivalent to ¬(A ∨ B) (alias ¬A ∧ ¬B). But of course, although we now have unique characterization, we also have definability, and so not a ‘new’ connective in the sense currently under investigation. (This is acknowledged at Theorem 7 of Bowen [1971].) Exercise 4.37.5 (i) In fact the rules for ¬, ∧, and ∨ do not need to be invoked for a proof of uniqueness in the case of ↓. Show that (↓ Left) and (↓ Right) by themselves uniquely characterize the connective they govern. (ii ) Show that the compounds between which A | B is sandwiched in 4.37.4(i) are equivalent in the intermediate logic KC. (iii) Conclude that | is not uniquely characterized by the rules (| Left) and (| Right), by reasoning analogous to that given for 4.37.3, and show that in the system extending IGen by those rules, this connective is not definable, by adapting the proof of 4.37.2. Digression. With the examples of n and |, then, we see that the Cut Elimination property for a collection of sequent calculus rules (shown by Bowen) is no guarantee of uniqueness for the connectives governed by those rules. (Indeed, as we shall soon see, this issue is completely irrelevant to uniqueness and bears instead on the existence question.) Although we are not, in this book, going into technical proof theory in the Gentzen tradition, we offer the following comment on this situation. As remarked in the note at the end of Exercise 4.31.1, it is this reducibility in the complexity of (R)-formulas – what we are calling the regularity of the operational rules concerned – rather than the analogue for the case of (T)-formulas – cut-inductivity as it is put in the discussion before 2.33.27 (or more generally under 2.33.27(ii), p. 365) – that makes for unique characterization by sequent calculus rules. For of course with the ∧ rules reduplicated for a new connective ∧ , we simply replace the appeal to (∧ Right) in the above proof by one to (∧ Right), and have a proof of B ∧ C B ∧ C, and use (∧ Left) followed by (∧ Right) for a proof of the converse sequent—as the reader will have done in addressing 4.31.1(iv). In general sequent calculus rules (# Left) and (# Right) will characterize # uniquely if and only if the assumption that (R) holds for the components of a #-formula implies that (R) holds for the formula. End of Digression. Although, as the above Digression indicates, regularity in the current sense – a term introduced with this meaning in Kaminski [1988]. Kaminski there considers unique characterization as an excessive demand on new intuitionistic connectives, and has a different complaint against n and |, as introduced by Bowen.


609

This objection is that these are neither ↔-extensional (to use our terminology from 3.11, which corresponds to weak extensionality according to a consequence relation, as defined in 3.23) nor ↔-congruential (3.31). (Of course not being ↔-congruential implies not being ↔-extensional. Kaminski also claims that ↓ shares these defects, but this is not correct.) We should note that officially we introduced this vocabulary for gcr’s and – more to the point here – consequence relations, and some explanation is required since we are now working with SetFmla0 rather than Set-Fmla. Thus the notion of an associated consequence relation is not quite in place. However, we can easily fix this by supposing every empty right-hand side to be filled by ⊥, assuming also the rule to pass from Γ ⊥ to Γ C, for arbitrary C; this replaces appeals to (M). Then, by the consequence relation associated with a proof system in Set-Fmla0 , we mean that associated with the Set-Fmla proof system so obtained. Then, where is the consequence relation thus associated with Bowen’s system, the ↔-extensionality point is that we do not have, for # = n or # = |, A1 ↔ A2 , B1 ↔ B2 (A1 # B1 ) ↔ (A2 # B2 ) for arbitrary formulas A1 , A2 , B1 , B2 . Note that we are taking ↔, as well as (below) →, to have its (IL-)customary inferential powers, so that the above formulation is in fact equivalent to that given by Kaminski (in a different notation, and under the name ‘extensionality’ – which corresponds to weak extensionality in the sense of 3.23 above): (1)

A1 ↔ A2 , B1 ↔ B2 , A1 # B1 A2 # B2 .

To see what is going on here, let us fix on (↔-)extensionality in the first position, thus identifying B1 with B2 , and for definiteness, focus on the case of # = |. What we don’t have, then, is: (2)

A1 ↔ A2 , A1 | B (A2 | B)

What we expect from the ‘negative’ nature of | is, more specifically, the following version of the condition of being ‘antitone with side-formulas’ (in the first position), from 3.32: (3)

A2 → A1 , A1 | B (A2 | B)

To see why (3) fails, it is helpful to recall the ‘sandwiching’ formulas for | from 4.37.4(i). From A1 | B, there follows, according to , ¬(A1 ∧ B), and from this together with A2 → A1 , ¬(A2 ∧ B). Now we reach an impasse, since what we should need to obtain A2 | B on the right is ¬A2 ∨ ¬B, rather than the (intuitionistically) weaker ¬(A2 ∧ B). We can reformulate this explanation of the failure of (3) in model-theoretic terms, which, further, do not require us to suppose that the language of has connectives ¬, ∨, and ∧. Indeed, while we are doing this purifying, we may as well go further, and suppose that | is the only connective in our language. This means we return to Set-Fmla0 , since we exclude ⊥; also, since (3) above includes →, we consider a purified version, formulated here as a rule: (4)

Γ, A2 A1 Γ, A1 | B A2 | B

Before looking at (4) from a semantic perspective, we quote Kaminski’s comment ([1988], p. 309) that the sequent corresponding to (1) “must be valid in any

610 CHAPTER 4.


reasonable semantics” even though it is not provable in Bowen’s extension of IGen. In fact, serious attention has been given to connectives which are not even congruential—let alone extensional—as useful additions to the vocabulary of IL; a conspicuous example is (what is called) strong negation: see 8.23. Transposing the ‘sandwiching’ phenomena for | into the setting of Kripke’s semantics for IL leads to the following novelty – which may indeed disqualify as not ‘reasonable’ this semantic treatment (though matters will get somewhat worse presently, when we note a Persistence-related problem). A model M = (W, R, V ) is as in 2.32, though we have eliminated all the connectives in play there. For our new connective | , we do not provide a clause in the inductive definition of truth (at a point), but rather merely impose a two-fold constraint, namely that for all formulas A and B, and all points x ∈ W : (N) M |=x A| B only if for all y ∈ W such that xRy, M |=y A or M |=y B. (S) If either M |=y A for all y ∈ W such that xRy, or M |=y B for all y ∈ W such that xRy, then M |=x A | B. (N) and (S) are so labelled because they give, respectively, necessary and sufficient conditions for the truth of A | B at a point in a model, with no single condition stipulated as both necessary and sufficient. Faced with any such proposed conditions, one needs to check that they can be consistently imposed, since the sufficient condition’s being satisfied implies that the necessary condition must be. In more detail, abbreviating what follows “only if” in (N) to “ϕN (x)”, and what follows “if” in (S) to “ϕS (x)”, (N) and (S) together tell us that ϕS (x) ⇒ |=x A | B

and

|=x A | B ⇒ ϕN (x).

(Here the “M” is omitted for readability.) Thus the two conditions can only be imposed if their consequence ϕS (x) ⇒ ϕN (x). is satisfied. In this case, we are all right, since evidently for any x, the above condition is satisfied by x. The use, as here, of separate necessary and sufficient conditions for truth (at a point in a model) is a departure from the usual pattern of model-theoretic semantics, but it is not altogether unfamiliar: we have in effect seen a precisely analogous phenomenon in the valuational semantics of partially determined connectives (in §3.1). For example, the conditions induced on a consequence relation by the two determinants T, T, T and F, F, F , for a binary connective #, give the complete Set-Fmla logic (presented under 4.31.5(ii) above) for the class of valuations v such that v(A) = v(B) = T ⇒ v(A # B) = T (‘sufficient condition’) and v(A#B) = T ⇒ v(A) = T or v(B) = T (‘necessary condition’). As with the case we are now considering, the joint imposition of these conditions on valuations does not make # extensional (or even congruential) according to the consequence relation determined by the class of valuations satisfying the conditions. For some discussion of this phenomenon in the setting of modal logic, see §6 of Humberstone [1988a]. The point to note is that even though truth has been constrained rather than defined here, the notion of validity on a frame remains perfectly definite, namely truth at every point in every model on the frame, satisfying the above conditions,


611

as well as those from 2.32. The latter include not only the conditions on the relation R, but also the condition (Persistence); we return to this condition below. Theorem 4.37.6 Every sequent valid on all frames is provable using the structural rules and (| Left), (| Right). Proof. We use the canonical model technique as for 2.32.8 (p. 311), deleting the condition of primeness on our deductively closed consistent sets of formulas, and now understanding consistency in the sense of including no subset Γ for which the sequent Γ is provable in the present proof system. It remains to show that (N) and (S) are satisfied, if truth at x is identified with membership in x. For (N), suppose A | B ∈ x and xRy (i.e., x ⊆ y). Then A ∈ / y or B ∈ / y, since otherwise {A, B, A | B} ⊆ y, which would contradict the consistency of y, since the sequent A, B, A | B is easily proved using (| Left). For (S), suppose A | B ∈ / x. We must find y1 ⊇ x, y2 ⊇ x, in the model, with A ∈ y1 , B ∈ y2 . But we can take y1 as the deductive closure of x ∪ {A} and y2 as the deductive closure of x ∪ {B}, since if the first set is inconsistent we get Γ, A for some Γ ⊆ x, so Γ A | B by (| Right), which contradicts the deductive closure of x since Γ ⊆ x and A | B ∈ / x. A similar argument works for the case of x ∪ {B}. Theorem 4.37.6 is a completeness result for Bowen’s | -system; what about soundness? There is no difficulty in showing that (| Left) preserves the property of holding at a point in a model (appealing to (N) and the reflexivity of the relation R), and hence, the properties of holding in a model and of being valid on a frame. In the case of (| Right) we do have a problem. Let us take the case in which we use this rule to pass from Γ, A to Γ A | B. We can start by supposing that the conclusion-sequent fails at a point x, so that x verifies all of Γ but not A | B. Then, by (S), we conclude that x bears R to points y1 and y2 verifying respectively A and B. We want the point y1 here to show that Γ, A fails at y1 , since we know A is true at y1 : but for this we need to know that all the formulas in Γ are true at y1 . Thus we need all formulas to be persistent for this part of the soundness proof to work. We have laid this down for propositional variables in the condition (Persistence), and in 2.32.3 (p. 308) we found this to be sufficient for extending the property to all formulas under consideration there. However, because of the bipartite (i.e., (N) with (S)) treatment of | , there is a problem about providing the inductive part of the argument, that if A and B are both persistent, so is A | B. For suppose that A | B is true at x1 and x1 Rx 2 but that A| B is not true at x2 . We cannot get a contradiction out of these suppositions, because the first tells us (by (N)) that no point R-related to x1 verifies both A and B, and the second (by (S)) that some points y1 and y2 , each R-related to x2 , verify, respectively A and B. Neither y1 nor y2 need verify both A and B, so we have nothing contradicting the fact that no point R-related to x1 (as y1 and y2 are, by the transitivity of R) verifies both. Accordingly, we have to resort to brute force, and stipulate that | -formulas are persistent, requiring a model M = (W, R, V ) to satisfy not only (N) and (S) above, but also

612 CHAPTER 4. (P)


M |=x A | B and xRy imply M |=x A | B, for all formulas A,B, and all x, y ∈ W .

By “valid on the revised semantics” will be meant: valid on every frame, in the sense of holding in every model satisfying (N), (S) and also (P), on that frame. An issue as to the consistency of this additional stipulation arises, as before with ϕS (x) ⇒ ϕN (x): now, with (P), our stipulation implies that the condition (ϕS (x) & xRy) ⇒ ϕN (y). As before, however, there is no possibility of a frame element x’s not satisfying this condition (which implies the original condition, ϕS (x) ⇒ ϕN (x), in view of the reflexivity of R). The following sequent is valid on the revised semantics which but invalid on the original semantics (i.e., without condition (P)): q | r, p | (q | r) p | s Note that this sequent is a substitution instance of a sequent, namely q, p | q p | s which was already valid on the original semantics. Uniform Substitution, then, does not preserve validity on the original semantics. The respect in which propositional variables are treated as ‘special’ (cf. 9.21) is of course that they are subject, unlike | -formulas on the original semantics, to a persistence condition. Having imposed this condition as (P), we have now fixed up the problem with soundness for Bowen’s proof system (in which both the sequents inset above are provable). For completeness, we need only check that the canonical model meets condition (P): but this is automatic, since we take (the canonical) R as ⊆. Thus: Theorem 4.37.7 A sequent σ is provable using, alongside the structural rules, the rules (| Left) and (| Right) if and only if σ is valid on the revised semantics. Now that we have got our semantic account to match the deliverances of the proof system, we can return to the question of (4) above, restated here for convenience: (4)

Γ, A2 A1 Γ, A1 | B A2 | B

Now suppose that the premiss-sequent of (4) holds in a model (W, R, V ), understood as on the revised semantics, but that the conclusion-sequent fails at some point x ∈ W . Then all of Γ are true at x, along with A1 | B, so that no y ∈ R(x) verifies both A1 and B, but A2 | B is false at x, so that for some y1 , y2 ∈ R(x), A2 is true at y1 and B is true at y2 . Since all formulas are persistent (this being the revised semantics) all of Γ are true at y1 , since they were all true at x. So, since the premiss-sequent is supposed to hold in our model, A1 is true at y1 . If we knew that y1 = y2 , we should be able to obtain a contradiction, since we would then have a point in R(x) verifying both A1 and B. But of course we have no grounds for thinking y1 = y2 , and accordingly no contradiction is


613

forthcoming. This completes our examination of (4) from the model-theoretic perspective. Similar reasoning shows that, in spite of the symmetrical treatment of A and B in the present semantics for A | B, commutativity fails for | in Bowen’s system: Exercise 4.37.8 (i) Show that p | q q | p is invalid on the revised semantics (and hence unprovable in Bowen’s system, by 4.37.7). (ii ) Add a new singulary connective O to the language, for which it is stipulated in the semantics that OA is true at a point iff A is true at that point. Then, to illustrate non-congruentiality, show that p | q Op | q is invalid on the revised semantics. Both (i) and (ii) here are intimately related to the non-regular nature of the proof system, in the sense introduced in the Digression following 4.37.5 (Kaminski’s sense). In the case of (ii), the mere fact that we have a different formula on the right—with Op in place of p—blocks the argument for the validity of (R): if a given formula is true at a point then that formula is (of course) true at that point. Now that we have changed the formula in passing from left to right, we are reduced to showing that if the condition necessary for the truth of the left-hand formula (given by (N)) is satisfied then the condition (given by (S)) sufficient for the truth of the right-hand formula is: but this is precisely what is not in general guaranteed. Moving to a more proof-theoretic view of the situation, we note that if A | B OA | B could be proved in Bowen’s system (supplemented by the obvious rules for “O”, prefixing this connective to any formula on the left or on the right of a sequent), then the proof could be mimicked for the case in which instead of “OA”, we had simply “A”. That would mean that the an appeal to (R) for the formula A | B could be bypassed. More information on the relations between (weak) extensionality, regularity, and persistence may be found in Kaminski [1988]. We drop the subject here (though see further the end-of-section notes, which begin on p. 626), as our reason for describing Bowen’s rules was to illustrate failure of unique characterization. However, our discussion has left several other loose ends, and we shall tidy some of them up in the following subsection. We close with a few final comments on the subject of regularity and Cut Elimination, picking up the theme of the Digression following 4.37.5. The duality between existence and uniqueness for connectives is mirrored by the duality between these two properties of (systems of) rules governing those connectives. This is described in the following terms by Girard, who calls (R) the ‘identity axiom’, and after stating it with “C” as a formula-variable on the left and right says of the Cut Rule (alias (T): again “C” represents the cut-formula) that it . . . is another way of expressing the identity. The identity axiom says that C (on the left) is stronger than C (on the right); this rule states the converse truth, i.e., C (on the right) is stronger than C (on the left). Girard et al. [1989], p. 31.

Note that “is stronger than” here means “is at least as strong as” (cf. 4.14). These notions of relative strength can be cashed out, as in Chapter 3 of Girard [1987b] in terms of three-valued logic (following an idea of Schütte), or, alternatively, by using the ‘valuation-pairs’ device of Humberstone [1988a]. However, we are

614 CHAPTER 4.


not here going into the ‘Semantics of the sequent calculus’ (to quote the title of Girard’s chapter). In view of the relationship, noted in our Digression, between regularity and unique characterization, it will now come as no surprise to see a similar relationship between existence (or more accurately: conservative extension of the system of structural rules) and Cut Elimination, or more precisely, cut-inductivity. There is a failure of harmony—to adapt that term from its use in 4.14 in respect of natural deduction rules—between Bowen’s sequent calculus rules (| Left) and (| Right). The former rule, as we saw at 4.37.4(i), confers on A | B the property of being at least as strong as ¬(A ∧ B), while the latter renders this compound no stronger than ¬A ∨ ¬B. As we have seen, there is some slack between these two extremes (in IL) capable of being taken up in more than one way – whence the non-uniqueness. The dual possibility would be realized by a pair of rules (# Left) and (# Right) which, instead of leaving a gap between the upper and lower limits of the deductive strength of #-compounds, make them overlap. This is what happens when we convert Prior’s Tonk example (4.21) from the natural deduction approach to the sequent calculus approach, as observed in Hacking [1979], though see Sundholm [1981] for a more careful formulation of the issues involved, and §3 of Humberstone [2010b] for variations on this theme: Example 4.37.9 The conversion in question can be undertaken still in SetFmla0 , so that Δ here has at most one element: (Tonk Left)

Γ, B Δ

(Tonk Right)

Γ, A Tonk B Δ

ΓA Γ A Tonk B

Then the application of (T) cannot be eliminated from the following proof (set out in tree form), top nodes being cases of (R), even on the hypothesis that any cut on A or B can be (making for a failure of cutinductivity):

Tonk Right

AA

BB

A A Tonk B

A Tonk B B

AB

4.38

Tonk Left Cut [A Tonk B]

Postscript on ‘New Intuitionistic Connectives’

Certain issues raised by the examination in 4.37 of Bowen [1971]’s participation in the project of expanding the traditional language of IL deserve further comment. The first of these concerns the ‘sandwiching’ phenomenon. We will mention a device of Gabbay [1977] for obtaining new connectives by this method which yields, by contrast with Bowen’s case, uniquely characterized connectives. Secondly, we say a few words about the general question of new intuitionistic connectives which are specifically intended to be intuitionistic analogues of (primitive or derived) connectives of CL. In terms of the traditional intuitionistic connectives, we saw in 4.37.4(i) that Bowen’s | forms compounds A | B intermediate between ¬A ∨ ¬B and ¬(A ∧ B), implied by the former and implying the latter. This was reflected in our semantic treatment of his proof system in the distinction between necessary and sufficient


615

conditions for the truth of | -formulas. (A similar situation arose for n – 4.37.1 on p. 605 – though we did not look at this example semantically.) We need to note that this sandwiching of a new primitive connective between existing derived connectives does not inevitably lead to non-uniqueness (à la 4.37.3, 4.37.5(iii)), as long as more is said by the logical principles governing the new connective than that it is thus sandwiched. If not too much else is said, the result can be a conservative extension of IL in which the new connective is nonetheless not definable in terms of the old primitives. Example 4.38.1 Gabbay [1977], working in Fmla, gives the following axiomschemata governing a new singulary connective # to supplement any basis in Fmla (with Modus Ponens as a rule of proof) for IL: (A1) #A → (B ∨ (B → A)) (A2)

A → #A

(A3) #A → ¬¬A (A4)

(#A → A) → (¬¬A → A).

Gabbay proves that the resulting logic conservatively extends IL, that # is not definable in terms of the remaining connectives in it, and that the above axioms uniquely characterize #. These three properties are three from a list of six he suggests should be possessed when IL is extended by principles governing anything deserving to be called a new intuitionistic connective. The list is reduced to five in Gabbay [1981]. The other two of this reduced list of conditions are that the extended logic should have the Disjunction Property (see 6.41, p. 861), and that it should be ‘not non-classical’, a property we shall call being ‘an intuitionistic analogue of a classical connective’ below. An additional condition that one might with some plausibility impose is that the extended logic should be congruential. In the context of Fmla, we may take this to amount to a requirement of ↔-congruentiality (where ↔ is the intuitionistic biconditional). By appeal to 4.38.2(ii) below, the present example of Gabbay’s can be seen to satisfy this condition. As was mentioned in 4.37, so-called ‘strong negation’ does not satisfy it, a theme on which we shall elaborate in 8.23. Digression. The property dropped from the list of desiderata in the passage from Gabbay [1977] to Gabbay [1981] relates to definability in ‘second order intuitionistic sentential logic’, which is to say: IL with propositional quantifiers. Propositional quantification lies outside of the scope of the present work, save for asides such as this one, and we shall say nothing further of this condition. A criticism of any such requirement may be found at p. 311 of Kaminski [1988]. Kaminski also questions the unique characterization requirement, on the grounds that it would exclude modal-style operators added to IL, though rather oddly he endorses the extensionality condition we saw in 4.37, which is likewise inappropriate for such operators (pace Kaminski [1988], p. 322, where is described as extensional in – an intuitionistic version of – S4: the condition should be weakened to one of congruentiality—which Kaminski calls ‘weak extensionality’). See further our discussion of Gabbay’s ‘not non-classical’ condition below. In 4.37, we noted that ‘strong negation’ (8.23, p. 1228) fails even

616 CHAPTER 4.


to be congruential (according to the consequence relation SN there described), though it has often been urged as a ‘new intuitionistic connective’. As for the Disjunction Property, the rather less widely discussed ‘dual intuitionistic negation’ (8.22, p. 1222) gives a logic lacking this property. Further criticisms of Gabbay’s “mixed bag of conditions” may be found in Kreisel [1980], p. 90, where the description just quoted appears. There are also some remarks on the subject in §4 of Humberstone [2001], and a thorough treatment of aspects of Gabbay’s conditions is provided by Caicedo and Cignoli [2001]. End of Digression. Exercise 4.38.2 (i) Show that (A1)–(A4) in 4.38.1, alongside the rest of IL, uniquely characterize #. (Hint: put “# A” for “B” in (A1); if further assistance is required, consult Gabbay [1977], p. 129, where what we usually write as # and # in a uniqueness proof appear as #1 and #2 . Note that “#1 ” in line 6 from the base of the page cited is a misprint for “#2 ”. The same misprint occurs in the proof on p. 132 of Gabbay [1981].) (ii ) As well as (A1)–(A4) above, Gabbay lists an additional schema: (A → B) → (#A → #B). If this were independent of the others, we should have the arguably invidious situation of ‘stronger than needed’ logical principles encountered in 4.32. Show, however, that this schema is derivable from (A1)–(A4). (Hint: put “#B” for “B” in (A1), and treat the axioms as zero-premiss rules in an extension of INat, assuming A → B and #A. We want to derive #B from this. From the above instance of (A1) we get #B ∨ (#B → A), so it remains only to derive #B from the second disjunct. This disjunct itself gives #B → B in view of the assumption A → B. The reader is left to exploit the remaining assumption, in conjunction with (A3), and complete the proof by appeal to (A2) and (A4).) Now the point about sandwiching is illustrated by (A2) and (A3) which sandwich #A between A and ¬¬A. We still have unique characterization, because we have additional principles, namely (A1) and (A4). Thus the latter pair of schemata cannot each hold both for the result of replacing #A by A and for the result of replacing #A by ¬¬A, on pain of vulnerability to a nonuniqueness argument in the style of that given for 4.37.3. (In fact, (A4) is not IL-provable under the former replacement, and (A1) is not provable under either replacement.) Gabbay’s discussion in the works cited is considerably more general than the case of singulary # would suggest. What he actually shows is that for any formula C = C(p1 , . . . , pn ) of the language of IL, containing just the propositional variables indicated, which is not intuitionistically implied by its double negation, we can add a new n-ary connective # subject to the following generalizations of (A1)–(A4), in which, as explained for n-ary contexts in general (after 3.16.1 on p. 424 above) C(A1 , . . . , An ) is the result of uniformly substituting A1 , . . . , An for p1 , . . . , pn , respectively, in C: (1) #(A1 , . . . , An ) → (B ∨ (B → C(A1 , . . . , An ))) (2) C(A1 , . . . , An ) → #(A1 , . . . , An ) (3) #(A1 , . . . , An ) → ¬¬C(A1 , . . . , An )

4.3. UNIQUENESS OF CONNECTIVES (4)

617

(#(A1 , . . . , An ) → C(A1 , . . . , An )) → (¬¬C(A1 , . . . , An ) → C(A1 , . . . , An )).

The resulting extension of IL will possess all the properties claimed for the extension by (A1)–(A4) in 4.38.1; the latter constitute the special case (for which Gabbay gives an interesting Kripke semantics not reviewed here) in which n = 1 and C(p1 ) is just p1 (alias p) itself. Gabbay also makes the following observation, in which the reference to Bowen’s system as described in 4.37 is to the sequent calculus with the operational rules governing | given on p. 607. Observation 4.38.3 Bowen’s proof system for IL with | (described in 4.37) is obtained (in Fmla) by taking n = 2 and C(p, q) as ¬p ∨ ¬q, and using only (2) and (3). Proof. Given 4.37.4(i), we need only check that ¬¬(¬p ∨ ¬q) IL ¬(p ∧ q); but the latter follows from 2.32.2(ii) (p. 306).

Exercise 4.38.4 (i) Find a formula C(p, q) for which Bowen’s proof system for n (from 4.37) has a Fmla version axiomatized by (2) and (3). (ii ) The cases we have just considered are cases in which two CLequivalent formulas, one of which IL-implies but is not IL-implies the other, are such that the IL-weaker of the two is IL-equivalent to the double negation of the stronger. Find two CL-equivalent formulas between which IL holds in only one direction, but for which the weaker is not IL-equivalent to the double negation of the stronger. From (ii) here, it follows that there are other options for sandwiching a formula between classically equivalent but intuitionistically non-equivalent formulas than those covered by the (A2)-(A3) style of sandwiching, though we shall not consider such possibilities further. Instead, we turn to the question of what makes a connective—governed by certain logical principles in an extension of IL—an intuitionistic analogue of a connective of CL (again understanding “connective” in the inferential role sense). After a discussion of this question, we shall close the subsection with a look at some new zero-place connectives for IL which are associated with intermediate logics. It is at this point that we need to introduce Gabbay’s idea of a new intuitionistic connective’s being ‘not non-classical’, alluded to at the end of 4.38.1. This is the idea that the proposed extension of IL with the new connective should, when this extension is itself further extended by the addition of the schema ¬¬A → A (or any other principle which would turn IL into CL) ‘collapse’ the new connective and some (primitive or derived) connective of CL, the latter formulated in some language with a functionally complete set of connectives (3.14, p. 403). We can apply this idea not just to new connectives, but to any connective, including those of IL. Accordingly, let us say that when # is an n-ary connective of the language of some S ⊇ IL, that # is an intuitionistic analogue of the n-ary connective # of CL if the smallest logic extending S and containing ¬¬A → A for all formulas A (of the language of S) contains #(p1 , . . . , pn ) ↔ # (p1 , . . . , pn ). By connective is meant, throughout the definition, primitive or derived (‘compositionally derivable’) connective. The above discussion has been tailored to the case of logics in Fmla, and presumed that the connectives → and ↔ are present; the adaptations needed to

618 CHAPTER 4.


remove these restrictions are routine, and need not be entered into. The interest of new intuitionistic connectives which are analogues of classical connectives in the above sense is in focussing our attention on additions whose novelty is specific to the fact that it is intuitionistic logic whose language they enrich. Of course, one can add new connectives—such as modal operators (cf. the notes to §2.2, under ‘Intuitionistic and Relevant Modal Logic’: p. 296)—with some of the virtues listed under 4.38.1, but these do not present the appearance of being ‘on all fours with’ the traditional intuitionistic primitives ∧, ∨, ¬, → (and perhaps ⊥, , ↔). One way of focussing on what is different about such additions is to note that they—unlike the candidates considered by Bowen, Kaminski, Gabbay, and the writers listed in the notes to §4.2 (under ‘New Intuitionistic Connectives’)—are not intuitionistic analogues of any classical connectives. Examples 4.38.5(i) Each of #1 –#6 below is an intuitionistic analogue in IL of classical conjunction: A A A A A A

#1 #2 #3 #4 #5 #6

B B B B B B

= = = = = =

A∧B ((A → B) → A) ∧ ((B → A) → B) ¬¬A ∧ B A ∧ ¬¬B ¬(A → ¬B) ¬(¬A ∨ ¬B).

Such a list, which does not purport to be exhaustive, is misleading to the extent that it contains redundancies: #1 and (the Peirce-inspired) #2 are equivalent, for example, in the sense that for any formulas A, B, the compounds A #1 B are IL-equivalent, and the same goes for #5 and #6 and any other candidate, e.g. ¬¬A ∧ ¬¬B, which is a negated formula or is (as in this case) IL-equivalent to such a formula – by Glivenko’s Theorem. (See 2.32.2(ii), p. 306.) (ii ) The connective | in Bowen’s extension of IL is an intuitionistic analogue of the classical Sheffer stroke, as is the connective | in Gabbay’s uniqueness-conferring ‘completion’ of this system by his schemata (1) and (4) (see 4.38.3). (iii) The connectives #1 –#5 below are intuitionistic analogues of the classical Sheffer stroke: A A A A A

#1 #2 #3 #4 #5

B B B B B

= = = = =

¬(A ∧ B) ¬A ∨ ¬B ¬A ∨ (A ∧ ¬B) (B ∧ ¬A) ∨ ¬B ¬(A ∧ B) ∧ (A ∨ ¬A) ∧ (B ∨¬ B).

Here we have avoided duplicating representatives from the same ILequivalence class (the kind of duplication – or multiplication – commented on under (i) above). In particular various candidates which are negated formulas and therefore just give #1 all over again, such as ¬(¬¬A ∧ B) are not listed, or formulas IL-equivalent to negated formulas, such as A → ¬B or ¬¬B → ¬A. As in (i), we do not enter into considerations of exhaustiveness here. See further the Digression below.


619

If we wish to study an IL version of the classical Sheffer stroke, we have, in view of part (iii) above, a further choice to make. In the remaining cases, one may check by doing proofs in INat, or by considering Kripke models.) If we wish our intuitionistic Sheffer stroke to be an additional primitive connective, the equations listed under (iii) should be rephrased as stipulations to the effect that in the presence of the other IL connectives the two sides of the equation are provably equivalent (since that makes them synonymous in IL). For example, in Tennant [1979], it is #1 from (iii), thus reinterpreted, that is chosen. (Tennant’s approach is natural-deductive; he also indicates suitable rules for the classical Sheffer stroke. Other natural deduction treatments of the latter may be found in the references cited in the notes (p. 630) to this section. It is a good exercise to try and formulate suitable sequent calculus rules for this analogue of the Sheffer stroke. If in doubt, consult Riser [1967] or Bowen [1972], or p. 302 of Hacking [1979] (see also 8.24.12) – and reformulate these authors’ rules in Set-Fmla by deleting the set variable on the right.) What, if anything, should guide one’s choice here? A significant feature of the classical | is its functional completeness, and accordingly its Kuznetsov completeness (3.15) for CL. (Note that ‘strong’ functional completeness in the sense of Example 3.14.1 and the surrounding discussion, which we do not have for the present case, is not required for this – only the weaker kind we called functional completeness sans phrase.) But we know from 3.15.8(ii) (p. 423) that no binary connective definable in IL enjoys the latter property for IL, so this cannot favour one candidate over another. In any case, we should like to know in general if there is any ground for preferring one rather than another intuitionistic analogue of a classical connective as the ‘correct’ intuitionistic incarnation of that connective. Some further attention will be given to this (admittedly vague) question in 6.12 (p. 780), when we come to consider the illustrative case of intuitionistic exclusive disjunction; the discussion begins two paragraphs before 6.12.4. Of course there is an even more general issue here for all the ‘alternatives to CL’ described in §2.3, which we can summarise in the slogan: the weaker the logic, the more discriminations it makes. This theme is emphasized, for example, in Nelson [1959] as well as in other sources mentioned in Humberstone [2005a], where it is observed that the rough and ready slogan just cited is not universally correct. (The topic of relevant-logical analogues of classical connectives is raised in 8.43.) However, so that we may appeal to it in our discussion in 6.12, we include here some remarks on the Kripke-semantical treatment of intuitionistic connectives. This will also enable us to note one point of view – which we shall associate with the name of G. F. Rousseau – from which the Sheffer stroke (or any other boolean connective) has a uniquely natural or ‘preferred’ intuitionistic analogue. The inductive definition of truth at a point in a Kripke model for IL was given for formulas with connectives ∧, ∨, →, and ¬ in 2.32 by means of four clauses [∧], [∨], [→], and [¬] in the definition of truth (at a point in a model). The former pair attended to the truth-values of subformulas at the point in question, while the latter pair ‘looked ahead’ to points R-related to the given point. It is, however, possible to give an equivalent definition in which each clause is of the latter type, and if we do so, we can bring out more clearly the relation between the truth-conditions of compound formulas and the truth-functions associated on boolean valuations with these connectives. Such a reformulation may be found in Rousseau [1970b], in which appears the following recipe for obtaining

620 CHAPTER 4.


a condition for the truth of formulas in Kripke models from a truth-function. (The label topoboolean for such conditions is not used by Rousseau; we do not go into its etymology here: see the remarks à propos of McKinsey and Tarski [1948] below.) This recipe is most conveniently described in its general form with the aid of the (bivalent, but not necessarily boolean) vx associated with a point x in a model M = (W, R, V ); recall that this function assigns T to a formula A just when M |=x A, and that a more explicit notation (“vxM ”) may sometimes be desirable – and was indeed employed in 2.21 – to register the dependence on the model concerned. Given an n-ary truth-function f , we define the topoboolean condition induced by f to be the following condition on points x in models (as above): for all y such that xRy, f (vy (A1 ), . . . , vy (An )) = T, for all formulas A1 , . . . , An . Naturally, we are interested in the case in which the Ai are the components of a compound formula, and so we understand by the topoboolean condition for #, the topoboolean condition induced by #b , where # is any boolean connective (so that #b is the truth-function associated with # over the class of boolean valuations). We shall use [#]tb as a label for the topoboolean condition for #. Examples 4.38.6(i) The condition [→]tb is as follows: [→]tb

M |=x A → B iff for all y ∈ R(x),(vy (A) →b vy (B)) = T.

In other words, the condition requires that the corresponding material implication be true at all y ∈ R(x). But this is just another way of saying that all such y verifying A should verify B, i.e., the standard clause [→]. [¬]tb works out similarly as a very mild variant on [¬]. (ii ) For disjunction an explicitly topoboolean treatment would be as follows: [∨]tb

M |=x A ∨ B iff for all y such that xRy,(vy (A) ∨b vy (B)) = T.

The latter equation amounts to: y verifies A or y verifies B; so we may think of [∨]tb as a clause in the definition of truth saying that for a disjunction to be true at a point x, it is necessary and sufficient that for every y ∈ R(x), at least one of the disjuncts should be true at y. This time, we have obtained something rather different in appearance from [∨], because we now have a ‘forward-looking’ condition (or ‘upwardlooking’, if one is thinking of R as ). But if A ∨ B is true at a point in a Kripke model, with truth defined using [∨], then one or other of A, B, is true at that point, and hence, all formulas being persistent, that formula—and so, at least one of A, B—is true at all ‘later’ points, so that the topoboolean condition is satisfied. Working in the converse direction, it may appear that A ∨ B can come out as true à la [∨]tb in virtue of one or other disjunct being true at every later point, without it being the same disjunct in every case: and hence that the condition [∨] is not satisfied. But this is to forget that the relation R is required to be reflexive: thus if every y ∈ R(x) verifies A or else B, x itself verifies A or else B. Thus the rhs of [∨] is after all satisfied. Further, as we shall note below, the topoboolean clauses also guarantee that all formulas are persistent, so that at least either A is true at all y ∈ R(x), or else B is. [∧]tb presents a similar situation, and we leave the reader to verify its equivalence with [∧].


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We have appealed here to the Persistence property for compounds formed by the connectives topobooleanly interpreted. In each case, this is an immediate consequence of the transitivity of the relation R in a frame, since for such a compound to be true at a point is for all R-related points to meet a certain condition. Thus if any x ∈ R(x) failed to verify such a formula, it would be in virtue of some y ∈ R(x ) not meeting the condition: but y ∈ R(x) also. Note that now no appeal is made to the inductive hypothesis that the components (of whichever compound is under consideration) are themselves persistent, whereas – as noted in the proof of 2.32.3 (p. 308) – such appeal was needed for Kripke’s formulation of the truth-definition in the case of ∧ and ∨. Remark 4.38.7 It may be of interest to consider when a simplification of the kind employed by Kripke (in the definition of truth) is possible. That is: under what conditions can the topoboolean ‘forward-looking condition’ on points x to the effect that for all y ∈ R(x), we have f (vy (A1 ), . . . , vy (An )) = T, be replaced by its ‘local’ analogue, a condition to the effect that f (vx (A1 ), . . . , vx (An )) = T? We will say that ξ ζ, for truth-values ξ, ζ, unless ξ = T and ζ = F. (In §3.1 and elsewhere, we use x, y, to range over {T, F}, but such a usage would be confusing here, since these variables now range over points in a model.) Define an n-ary truth-function f to be monotone when f (ξ1 , . . . , ξn ) f (ζ1 , . . . , ζn ) whenever, for each i (1 i n), ξi ζi . We leave the reader to verify that a necessary and sufficient condition for a Kripke-style simplification in the truth-definition to be possible (with an attendant complication in the proof of the ‘Persistence Lemma’) in the case of a boolean connective # is that the function #b is monotone. (This also makes # a connective which is ‘monotone with side formulas’ according to Log(BV ), in the sense of 3.32.) Although in 3.14 it was said that the notion of expressive completeness cannot be cashed out for IL as for CL as functional completeness since there was no ‘intended’ truth-table interpretation for the former logic, we can see that a question in the vicinity of the usual functional completeness question does after all arise, namely: is it the case that for every truth-function f there is a derived connective of IL (with ¬, ∧, ∨, → primitive) # such that for all models M = (W, R, V ), all formulas A1 , . . . , An and all x ∈ W , we have M |=x #(A1 , . . . , An ) iff for all y ∈ R(x), f (vy (A1 ), . . . , vy (An )) = T. Rousseau [1968] answers this question affirmatively: but here we shall merely illustrate the situation with the case of | b . The induced topoboolean condition on points x for A | B to be true at x is that for all y ∈ R(x), vy (A) | b vy (B) = T, i.e., that no point R-related to x should verify both A and B. In this case we have for the derived connective #, A # B = ¬(A ∧ B) (or equivalently, A → ¬B). Thus if we look for the preferred intuitionistic analogue of a classical connective by applying Rousseau’s procedure (i.e., use the topoboolean condition) to the truth-function associated therewith on boolean valuations, we have in #1 of 4.38.5(iii) the intuitionistic Sheffer stroke. Very definitely out of the running is #2 , for example, which forms from A and B the compound ¬A ∨ ¬B, between which and A #1 B (alias ¬(A ∧ B)) we saw Bowen’s | to be sandwiched in 4.37. Indeed the truth-condition assigned by the Kripke semantics to #2 is not the topoboolean condition induced by any truth-function; a simpler example of

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this phenomenon is given by the case of #A = ¬¬A: for no truth-function f is vx (A) = T iff for all y ∈ R(x), f (vy (A))= T. (This makes the Rousseauinspired notion of functional completeness somewhat anomalous as a notion of expressive completeness, since without even considering the possibility of adding new connectives, we already have derived connectives which do not express the topoboolean conditions induced by truth-functions.) A full justification of the perspective from which we have seen one particular candidate emerge as the preferred intuitionistic analogue of the Sheffer stroke will not be attempted here. We content ourselves with remarking that the conditions [#]tb are embodiments in the Kripke semantics of conditions going back to the ‘topological interpretation’ of IL (McKinsey and Tarski [1948]); a description of this interpretation, and its relationship to algebraic semantics in terms of Heyting algebras and to Kripke’s model-theoretic semantics may be found in the references cited at the end of §2.3. Rousseau [1968] presents (what we have called) the topoboolean conditions in these terms, which (like Rousseau [1970b]) provides considerable evidence of the fruitfulness of this approach. Without going into such details, we can at least observe that one clear advantage of the approach is that it does straightforwardly make intuitionistic →, ∧, ∨, ↔, ¬, and ⊥ the preferred intuitionistic analogues of classical →, ∧, etc. (via the topoboolean conditions induced by →b , ∧b , etc.); this is a model-theoretic route to the same conclusion that we obtain in philosophical proof theory by taking the meanings of such connectives as given by uniquely characterizing collections of rules. Digression. The topoboolean approach aside, there is another respect in which, for given A, B, the formula ¬(A ∧ B), rather than some other compound classically but not intuitionistically equivalent to it, can be seen to have a special status: namely, that this formula is IL-weakest amongst such classically equivalent formulas. For if C CL ¬(A ∧ B) then, since C CL ¬(A∧B), we have by 2.32.2(ii) (p. 306) that C IL ¬(A∧B). Here we are exploiting the fact that this particular IL analogue of A | B is a negated formula. One may ask similarly, if there is an IL-strongest formula classically equivalent to ¬(A ∧ B). In this case, we can answer negatively, since, more generally whenever we have a formula C such that C CL D for some formula D (any ‘consistent’ formula C), we can find a classically equivalent but intuitionistically stronger formula: just take C ∧ (pi ∨ ¬pi ), where pi is any propositional variable not occurring in C. In particular, then, for ¬(p ∧ q), we have the IL-stronger formula ¬(p ∧ q) ∧ (r ∨ ¬r). However, this leaves open the possibility that there may be an IL-strongest formula amongst those containing just the variables p and q which are classically equivalent to ¬(p ∧ q). And so there is: namely p#5 q, understood as in 4.38.5(iii). (A proof of this claim can be given using Observation 8.21.1, p. 1215.) Thus, the weakest and strongest compositionally derivable connectives of IL which are intuitionistic analogues of classical | are respectively #1 and #5 from 4.38.5(iii). End of Digression. We have not had occasion to consider specifically any zero-place candidates (propositional constants) for the title ‘new intuitionistic connective’. To rectify this situation we conclude our discussion with a look at some intuitionistic analogues of the classical (nullary) Verum connective . Of course, since this


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connective is itself a familiar supplement to the stock of primitives of IL, we can say that is one such intuitionistic analogue of classical . But, more interestingly, so is the zero-place $, governed by the principles ($1) and ($2) below, for which, in the interests of continuity with the above discussion of Gabbay, we remain in Fmla. By way of clarification, we should remind the reader of our definition of the ‘intuitionistic analogue’ relation holding between a connective # of the language of some S ⊇ IL, and # of CL, namely that ‘classicizing’ S by addition of ¬¬A → A for all formulas A (of the language of S) leads to the provability of #(p1 , . . . , pn ) ↔ # (p1 , . . . , pn ). In looking for intuitionistic analogues of classical | , we took as S, IL itself. The definition allows S to be a stronger logic in the same language (an intermediate logic, or the inconsistent logic); but, more to the point here, we also allow S to be cast in an extension of the language of IL. We take advantage of this possibility in describing our new connective $ in the following system as an intuitionistic analogue of classical . (In IL itself, there is—to within provable equivalence— only one compositionally derivable connective which is an intuitionistic analogue of classical , as follows from the discussion of variable-free formulas below. However, if, by analogy with the use of “T1 ” in 4.23, we let 1 be that singulary classical connective with which the constant-true truth-function V (for Verum: see discussion following 3.14.5, p. 406) is associated over BV, then there are infinitely many pairwise non-equivalent intuitionistic analogues of 1 amongst the derivable connectives of IL. Representatives of each (IL-)equivalence-class are explicitly given in Nishimura [1960]. They are the elements of the Rieger– Nishimura lattice, mentioned above in Remark 2.13.4.) However, to return to the case of our new zero-place connective $, we lay down the following pair of axioms (the first of which is really an axiom-schema): ($1)

$ → (A ∨¬A)

($2) ¬¬$.

Here, we adapt an example of Smetanič, mentioned in Bessonov [1977], changing a singulary connective to a zero-ary connective. What Bessonov demands of a new intuitionistic connective # (assumed n-ary) is that supplementing IL with the new principles involving that connective leads to a conservative extension of IL, but if any formula B in the connectives ¬, ∧, ∨, →, ↔, , ⊥ is chosen then the further extension of the proposed system by #(p1 , . . . , pn ) ↔ B is a non-conservative extension of IL. (Yashin [1999], who also considers this 0-ary, case calls such maximally IL-conservative extensions of IL Novikov complete. See also Yashin [1998].) Since in the present instance, n = 0, we are demanding that the further addition, for any formula B of IL, of ($3)B

$↔B

alongside ($1) and ($2), is not conservative over IL. The assumption here is that the extensions of IL under consideration are all closed under Modus Ponens and Uniform Substitution (exactly as when we are considering intermediate logics in Fmla). Note that this way of explicating the ‘novelty’ idea for a new intuitionistic connective is more demanding that the explication in terms of undefinability which featured in 4.37 (e.g., 4.37.2). It is not true that for (a Fmla version of) Bowen’s logic for | , we get a non-conservative extension when we ‘identify’ | with some connective derivable from the standard primitives: each of the identifications given by p | q ↔ ¬(p ∧ q) and by p | q ↔ (¬p ∨ ¬q) leads to a conservative extension of IL. The stronger demand made by Bessonov

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is, however, satisfied for $ as above. To see this, the reader should first check that any formula containing no propositional variables (built up, that is, by the singulary and binary connectives from the atomic formulas and ⊥) is provable in IL iff it is provable in CL. (Hint: show how to trade in a point in a Kripke model falsifying such a formula, for a boolean valuation doing the same.) Now for the proposed extension of IL by ($1), ($2) and ($3)B for some $-free formula B, we distinguish two cases, according as to whether CL B or not. In the first case, IL s(B), where s maps every propositional variable to (or to ⊥), since CL s(B) and s(B) is a variable-free formula. This same substitution gives $ ↔ s(B) from ($3)B , so (by the ←-direction), $ belongs to the resulting logic, and hence by ($1), A ∨ ¬A also, for any formula A. Thus the extension is non-conservative. In the second case, in which CL B, we can find (by 8.21.6) a substitution s such that CL ¬s(B), so this time (in fact, whether or not s replaces all variables of B, by 2.32.2(ii), p. 306) IL ¬s(B), and by ($3)B we get ¬$ in the extended logic, which is non-conservative over IL since the resulting logic is then inconsistent (in virtue of ($2)). Note that the above argument can be reworked without the device of variable-free formulas, by judicious use of (e.g.) p ∧ ¬p in place of ⊥, and its negation in place of . In the case of $, not only do we have a new intuitionistic connective in the sense of Bessonov; this connective is uniquely characterized: Exercise 4.38.8 (i) Show that $ is uniquely characterized by IL + ($1), ($2). (ii ) Show that the system IL + ($1), ($2), is sound w.r.t. the class of finite Kripke frames (W, R) in which $ is stipulated to be true at a point x iff R(x) = {x}. (Here it is important to take a Kripke frame to have a reflexive, transitive, and antisymmetric relation R.) (iii) Show that the above system is complete w.r.t. the class of all Kripke frames. (Hint: use the method of canonical models, showing that for a point x in the canonical model for the present system $ ∈ x iff x bears R to no y = x.) (iv ) Show that for any $-free formula A, CL A iff the formula $ → A is provable in the present system. Note that the conjunction of (ii) and (iii) in the above exercise can be strengthened to the claim that the system under consideration is determined by the class of all finite frames (with $ interpreted as in (ii)). However, a proof of this would require us to go into the fact that IL has the finite model property, as this notion was introduced for the sake of Obs. 2.13.5 above (p. 228); this is briefly touched on below, in 6.43. In place of the schema ($1) above, we could equivalently (given the present understanding of what it takes to be a logic in Fmla) have chosen a particular formula, e.g., p ∨ ¬p. For any formula $-free formula A, let us now understand by ($1)A the formula $ → A. This is of interest especially for the case of A such that CL A while IL A. Such formulas are potential axioms for intermediate logics S, of which we have just seen the extreme case S = CL. Following Bessonov [1977] – except that we are now working with a zero-place rather than a one-place connective – we can also consider the ‘properly’ intermediate logics, which in Fmla are all of the general form IL + Γ, in which Γ comprises the set of new axioms for S. In 2.32, we explicitly described only two such intermediate systems, KC and LC, each of which could be presented as IL + Γ for Γ a


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singleton. In general, if Γ is finite, we can conjoin its elements to give a single formula. But there are intermediate logics which are not thus finitely axiomatizable (the best known being perhaps Medvedev’s ‘logic of finite problems’: see notes: p. 630), so for full generality we must take the form IL + Γ. Each such intermediate logic gives rise to a (distinct) new intuitionistic connective in the present sense (i.e., Bessonov’s sense). For consider the logic IL($Γ) which extends IL by {($1)A | A ∈ Γ} and ($2). By the reasoning given before 4.38.8, whenever IL + Γ is an intermediate logic, $ is a new connective in IL($Γ). And 4.38.7(iv) also transfers to this more general setting: Theorem 4.38.9 For any intermediate logic IL + Γ and any $-free formula B: B is provable in IL + Γ if and only if $ → B is provable in IL($Γ). We do not give a proof of this here. (A related result appears, with proof, as Theorem 1 of Bessonov [1977].) It follows from 4.38.9 that $, IL($Γ) is a subconnective of $, IL($Δ) iff IL + Γ is a subsystem of IL + Δ. Thus this way of encoding intermediate logics as new intuitionistic connectives provides us with as many candidate new connectives as there are intermediate logics: which is to say, uncountably many. Such theoretical interest aside, however, we should not wish to claim that these all constitute genuine ways of filling out any expressive incompleteness from which IL might be held to suffer. Conspicuously, there is no claim of unique characterization for $ in the systems $, IL($Γ) in general, in the way that we saw such a claim to hold for the case of Γ = {p ∨ ¬p} in 4.38.8(i). Note, however, that certain of the other desiderata imposed on new intuitionistic connectives— such as, in particular, congruentiality and (more strongly) extensionality—are vacuously satisfied simply because we have here a zero-place connective. We close by looking at another kind of candidate new intuitionistic connective, introduced by a special case. Say that a formula B anticipates a formula A according to a consequence relation , presumed to have → in its language, when B → A A. We are interested here in as IL and consider a connective, here written as a, with the properties that for any formula A, aA is the strongest formula anticipating A. (This example is studied in Humberstone [2001].) One zero-premiss sequent-to-sequent rule together with one one-premiss rule governing a (as well as →) give voice to this idea: (*) aA → A A

(**)

B→AA aA B

Although these rules uniquely characterize a in terms of → (in the sense of §4.3) – a fact which especially easy to see when (*) is replaced – equivalently, given the IL rules governing → – by the upward direction of (**), a is not definable in the language of IL with the customary primitives. As Hendriks [2001] observes, this is a special case of a general scheme for candidate new 1-ary connectives # to be introduced for given two-variable formulas C(p, q) (of the language of IL), namely as subject to the condition that for all formulas A, B, we have #A B if and only if C(A, B) where, as usual, C(A, B) is the result of substituting A and B uniformly for p and q in C(p, q). (C thus represents a context of two variables, as explained in

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3.16.1, on p. 424.) The anticipation operator above is the special case in which, writing # as a, C(p, q) is the formula (q → p) → p. Hendriks investigates the question of which cases C(p, q) allow # to be introduced conservatively when subject to the inset condition above, and for which cases this would be non-conservative, as well as related questions. As it happens, while a as just described – or equivalently, as given by the rules (*) and (**) – is conservative over propositional IL, in the case of intuitionistic predicate logic, the situation is different, as was pointed out to the author by Brian Weatherson, whose proof of that fact is adapted here. (We met this phenomenon already – rules conservative over propositional but not over quantified IL – in 4.21.9, on p. 547.) Observation 4.38.10 The rules given for a above yield a non-conservative extension of intuitionistic predicate logic. Proof. We use a version of the rules for (intuitionistic) predicate logic in Lemmon [1965a], with a sequent-to-sequent presentation. The first line of the formal derivation below is justified by the fact that in propositional IL we have (p ∨ ¬p) → ¬q ¬q by Glivenko’s Theorem, and any proof of this sequent (e.g. in INat) can by appropriate replacements turned into a proof of the first line here. (1) (2) (3) (4) (5) (6)

(F c ∨ ¬F c) → ¬¬∀x(F x ∨ ¬F x) ¬¬∀x(F x ∨ ¬F x) a¬¬∀x(F x ∨ ¬F x) (F c ∨ ¬F c) a¬¬∀x(F x ∨ ¬F x) ∀x(F x ∨ ¬F x) a¬¬∀x(F x ∨ ¬F x) ¬¬∀x(F x ∨ ¬F x) a¬¬∀x(F x ∨ ¬F x) → ¬¬∀x(F x ∨ ¬F x) ¬¬∀x(F x ∨ ¬F x)

As explained (1), by (**) (2), ∀I (3), ¬¬I (4) →I (5), by (*)

The reference to (*) in annotating line (6) is to a vertical form of (*), evidently derivable from it, taking us from aA → A (depending on assumptions Γ) to conclusion A (depending again on Γ). The non-conservative effect of the a rules is evident from the unprovability, in intuitionistic predicate logic, of the sequent (or the formula) at line (6) – a well-known counterexample to Glivenko’s Theorem for that logic (already mentioned as such in our discussion immediately after 2.32.1, p. 304).

Notes and References for §4.3 Oversights and Insights. A propos of our discussion of stronger-than-needed collections of rules, we quote from Popper [1947a], p. 220, a mistaken view of the matter (Popper has just suggested separate notations for classical and intuitionistic—and other—negations, along the lines of our ¬c and ¬i ): All these concepts can co-exist, in one and the same model language, as long as they are distinguished. Thus there is no quarrel between alternative systems. The rules of inference pertaining to the various concepts of negation are not identical, to be sure. But this is very satisfactorily explained by the simple fact that the various concepts have been given a meaning by rules which are not identical.


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The correct view of the matter is expressed in Popper [1948], p. 113, where it is noted that classical and intuitionistic negation collapse in a combined system, in view of the fact that the latter is governed by (as we put it) a collection of uniquely characterizing rules all of which also govern the former. This view can also be found in, for example, Harris [1982]. Unaware of these sources, the present author made this point as the main philosophical moral in the paper abstracted as Humberstone [1984a], on which 4.31–4.36 are based. The main technical point of the paper was an attempt at correcting a tendency in the literature to think of unique characterization as the analogue for connectives of what is called implicit definability for predicates and function symbols in a first order language. (Such an assimilation may be found in Zucker and Tragesser [1978], Hart [1982].) That point—which relies on the possibility of unique characterization by what in 4.36 are called language-transcendent rules, has since been made with admirable clarity in Schroeder-Heister and Došen [1988] and in note 5 of Williamson [1988b], to which discussions the reader is referred. However, as Belnap has suggested in correspondence, it may be harmless to speak of uniquely characterized connectives as implicitly defined as long as some such phrase as “second order” is included, meaning here second order propositional logic. To illustrate why this terminology might come to mind, consider the claim that added, say to the language of Positive Logic with the schema ⊥ → A, ⊥ is thereby uniquely characterized (in terms of →), since for a reduplicated new ⊥ we should have not only ⊥ → ⊥ , in virtue of the schema just given, instantiating A to ⊥ , but also the converse, in virtue of the reduplicated version of this schema. Now, when we say “in virtue of the schema just given” it is clear that we interpret the schematic letter “A” now as ranging over all formulas of the expanded language with not only → and ⊥ but also ⊥ as connectives. In our discussion (4.33) this point is made by introducing the abstract, languagetranscendent conception of rules. Alternatively, we may think of the original schema ⊥ → A as something along the lines of ∀p(⊥ → p), with an explicitly quantified propositional variable, which in the setting of an expanded language naturally allows a ∀-Elimination inference to the result of replacing “p” by any formula of that language. While this explains what lies behind the terminology of second order implicit definability, there are some awkward features. For example, → was brought into the above story in what might seem an unnecessary fashion, since we can consider the uniqueness question with Set-Fmla sequents in mind. Now we have the zero-premiss rule, or sequent schema ⊥ A and the same point about this rule’s having different applications in different languages applies, whereas to mimic the propositional quantifier account, we should now have to invoke “ ‘∀p(⊥ p)”: but this does not make sense, since quantifiers bind variables in formulas, not in sequents. (As it happens, the desired effect in the present case could be achieved by writing instead “ ⊥ ∀p(p)”, but that is just a special feature of this case.) The main thing for unique characterization is not the level of the quantification but that the quantification should be unrestrictedly universal. To take the first order version of the example of the ⊥-characterizing rule ⊥ A (or, if preferred, the rule to pass from Γ⊥ to ΓA, is the fact that a poset P = (P, ) boasting a least element 0, one satisfying 0 x for all x ∈ P , has only one such element. This is what we called uniqueness by antisymmetry in the discussion following 4.31.7. This does not mean we cannot adjoin a new least element 0 , say, with 0 < x for all x ∈ P , and have 0 = 0 . The point is that having passed

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to this new poset P with universe P = P ∪ {0 } and ordering as indicated, the element 0 is no longer a least element, since we do not have 0 x for all x ∈ P (as this fails for x = 0 ). We could similarly extend a logic in which ⊥ A is provable for every formula A of the language of that logic, by expanding the language to accommodate a new constant ⊥ and stipulating that ⊥ A for each formula A of the expanded language. In the extended logic we do not have ⊥ ⊥ , however, since we are not treating ⊥ A as a language-independent (0-premiss) rule in the sense of 4.33, because the schematic letter “A” is not interpreted as ranging universally over whatever set of formulas is under consideration. This is called the abstract conception of rules in Humberstone [2011b], where more discussion of this topic may be found. In the text, it was mentioned that the earlier Popperian view described above as incorrect has also been promulgated by Quine; a discussion of some of Quine’s views bearing on this issue may be found in Morton [1973], Haack [1974], especially Chapter 1, Berger [1980], and Paoli [2003]. The discussion in 4.21 of Belnap’s criterion for the existence of a connective with given inferential properties – namely that conferring such properties on a new connective conservatively extended an initially favoured logic – is inspired by some but not all of the literal content of Belnap [1962]. In particular, it represents one way of interpreting the phrase “antecedently given context of deducibility” appearing on, p. 131; on the following page of the article undue prominence seems to be given specifically to the structural rules antecedently endorsed. It is certainly true that in the case of Tonk we end up with the presumably not antecedently endorsed zero-premiss structural rule A B, but this is just a feature of this special case, on the interpretation of Belnap assumed in §4.2. (Cf. the concept of structural nonconservativity, introduced in 2.33.35 – p. 369.)

Reading. Historically, the concept of unique characterization for connectives emerged in Belnap [1962], with a credit, as has already been mentioned, to Hiż, and in Smiley [1962a]. In the latter paper, the term is “functionally dependent”, and to be more precise, we should say that functional dependence for Smiley is a matter of unique characterization of a connective by zero-premiss Set-Fmla rules (the discussion is actually cast in terms of substitution-invariant consequence relations) in terms of the remaining connectives in the language of the proof system. For a similar development, see Tokarz [1978]. In the framework Fmla an analogue of Smiley’s functional dependence is unique characterization in an extension by axioms – non new non-zero premise rules allowed – of a given logic with additional connectives. (One should not confuse functional dependence in Smiley’s sense and the notion of a functional dependency from the area of relational databases – though the latter itself is of considerable logical interest, as explained in Humberstone [1993b].) Caicedo and Cignoli [2001] call connectives uniquely characterized by axioms like this implicit connectives when they are not definable in terms of the connectives of the original logic, and study in particular the case in which the original logic is IL: the weak or ↔-extensionality condition mentioned a propos of Kaminski in 4.37 turns out to be crucial here. (Caicedo and Cignoli use the term “compatible” for such connectives, defined for the illustrative 1-ary case by requiring that #A ↔ #B should be a consequence of A ↔ B. See further Caicedo [2004], [2007], and Ertola, Galli and Sagastume [2007].) The topic of unique characterization has


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more recently found its way into introductory expositions and texts, such as Martin [1989], p. 33f., where the problem of cohabitation for ¬c and ¬i is also noted – as it is in Hand [1993], for example. Restricted ∧-Classicality. Koslow [1992] is mentioned à propos of Exercise 4.32.3 from p. 587, on (∧I)res and left-primeness. We could call a consequence relation restrictedly ∧-classical if for all A, B, we have A∧BA

and

A ∧ B B,

and for all A, B, and C, whenever C A and C B, we have C A∧B. (So this second condition corresponds to the rule (∧I)res .) Roughly speaking, Koslow’s ‘simple’ treatment of conjunction amounts to this condition of restricted ∧classicality on a consequence relation, while his ‘parametric’ treatment amounts to ∧-classicality. (This is rough because Koslow works in the framework Set1 Fmla rather than Set-Fmla. We encountered the same need to give this restricted version of (∧I) in 2.14 for the system Lat, couched in Fmla-Fmla.) A similar distinction, differently expressed, is drawn in Cleave [1991]. The discussion of conjunction by Koslow and Cleave leaves much to be desired; see Humberstone [1995a], pp. 478–9, as well as Humberstone [2005a] for a semantic treatment of restricted (or ‘non-parametric’ conjunction) in Proposition 3.10 thereof. A Supervenience Relation. A notion which might naturally be described in terms of uniqueness, though it is distinct from that we have treated under the heading of unique characterization is the following what we shall here call ‘Troelstra–McKay supervenience’. If we have a class C of logics in the same language and Φ ∪ {#} is a set of connectives of that language, with # ∈ / Φ, then # is Troelstra–McKay supervenient on Φ over C when any two logics in C which agree on their Φ-fragments agree on their Φ ∪{#} fragments. (Agreement on a Φ-fragment means proving precisely the same sequents when attention is restricted to sequents – of whatever logical framework is in use – all of whose connectives are in Φ.) McKay [1968b], p. 314, observes, taking intermediate logics to have primitive connectives Φ = {∧, ∨, ¬, →} and working in the framework Fmla, only for # = ∧ do we have # Troelstra–McKay supervenient on Φ {#} over the class of intermediate logics. (Actually, McKay uses the phrase ‘strongly undefinable’, for the negation of what we are calling Troelstra–McKay supervenience. It is certainly stronger than the claim that the connective in question is not definable in IL in terms of the remaining connectives.) In this case, to repeat, logics are taken in Fmla, and part of the definition of intermediate requires closure under Uniform Substitution and Modus Ponens. See also Wroński [1980], Minari [1986a]. Final Notes. After 4.33, refining the concept of a rule, was written, the author found a similar sentiment expressed in similar language – though with rather a different application in mind, the emphasis being on rules for quantifiers and languages differing in their stock of names (or individual constants), in Hodes [2004], at p. 145 of which the following appears:

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To my knowledge, the literature in logic on rules only considers rules governing particular languages. But it is important to conceive of a rule, and with it of a logical concept, as a language transcendent object.

Observation 4.34.3 and its proof are taken from (unpublished) lecture notes of Dana Scott distributed at Oxford in the early 1970s. The proof of 4.37.2 is based on that of Theorem 1 in Kaminski [1988]. References to the treatment of the Sheffer stroke in the sequent calculus approach were given in 4.38; for natural deduction treatments, see Siemens [1961], Price [1961], Webb [1970], Tennant [1978] (pp.63, 101–106) as well as Tennant [1979], and Segerberg [1983]. For historical information (Peirce, Sheffer, Nicod), see Church [1956], note 207, and note 1 on p. 49 of Quine [1951].) The dual connective (nor,“↓”) appears in infinitary as well as multigrade form in Wittgenstein [1922] (written in the former case as “N ” and in the latter as “|” (sic.)); for discussion see Geach [1981], Fogelin [1982]. Appeal to the possibility of a multigrade interpretation of “neither . . . nor ” is made in Borowski [1976], in reply to Halbasch [1975]. (More on multigrade connectives can be found, in the discussion following 6.12.3, p. 783). Brief mention was made of Medvedev’s logic of finite problems in 4.38, a non-finitely axiomatizable extension of the logic KP discussed in 6.42. For information and references to the literature on this elusive intermediate logic, see Medvedev [1966], Prucnal [1976] and [1979], Maksimova et al. [1979], Szatkowski [1981], Shekhtman and Skvortsov [1986], Miglioli et al. [1989], and (for analogous modal logics) Shekhtman [1990]. A simple semantic description of the logic in terms of Kripke frames may be found at p. 53f. of Chagrov and Zakharyaschev [1997].

Chapter 5

“And” All I had when I began Of Human Bondage was the conjunction “and.” I knew a story with “and” in it could be delightful. Gradually the rest took shape. Woody Allen [1981], p. 87.

§5.1 CONJUNCTION IN NATURAL LANGUAGE AND IN FORMAL LOGIC 5.11

Syntax, Semantics, Pragmatics

Before passing to the three aspects of the study of natural language alluded to in the title of this subsection, a word by way of reminder should be said on the subject of logic. Here syntactic and semantic considerations also arise, in the shape of proof-theoretic or formal conditions on (generalized) consequence relations and of conditions on valuations. For the case of conjunction, our primary concern in the present chapter, these have taken the forms, respectively, of isolating ∧-classicality as a property of consequence relations (and gcr’s), and ∧-booleanness as a property of valuations. In the former case, it is convenient to use a formulation in terms of consequence relations, or, more accurately, in terms of the associated consequence operations. To say that Cn is ∧-classical, or is the consequence operation associated with an ∧-classical consequence relation, is to say that for any formulas A, B of the language of Cn, we have Cn({A ∧ B}) = Cn({A, B}). In the latter case, a convenient formulation is available in terms of the truth-set notation “HV (·)” from 3.33. The notion of an ∧-boolean valuation essentially treats conjunctive formulas by means of intersection, in the sense that for any V ⊆ BV ∧ , we have HV (A ∧ B) = HV (A) ∩ HV (B), for all formulas A and B of the language concerned. In the case of conjunction, the syntactic notion of ∧-classicality and the semantic notion of ∧-booleanness are, as we have seen, especially intimately related; even remaining at the level of consequence relations (without, that is, passing to generalized consequence relations), we have not only the Weak Claim (from 1.14: p. 67) that ∧-classicality suffices for determination by some class of ∧-boolean valuations, but the Strong Claim that only such valuations are 631

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consistent with an ∧-classical consequence relation. Thus it is of no importance whether we consider ∧-classicality, as a (‘syntactic’) condition on consequence relations, or ∧-booleanness, as a (‘semantic’) condition on valuations. (A more detailed exploration of the above formulations will be given in the course of a comparison with the case for disjunction, in 6.11; see in particular the discussion starting at 6.11.1.) Some of our discussion asks whether the only respectable candidates for being regarded as notions of conjunction (or formalizations of and ) answer to these conditions. A negative answer in the case of valuations must take the form of rejecting either or both of the inclusions (for V of interest): HV (A ∧ B) ⊆ HV (A) ∩ HV (B)

HV (A) ∩ HV (B) ⊆ HV (A ∧ B).

In syntactical terms this amounts to rejecting, respectively, either or both of the rules (∧E), (∧I), in a natural deduction proof system (whose associated consequence relation is to be determined by the class V in question). Here preference is given to the syntactic or proof-theoretic way of conceiving of the suggestions concerned, for which reason 5.14 and 5.15 orient the discussion around proposed rejections, respectively, of (∧I) and (∧E). (In the terminology of 1.19, this amounts to urging as a rendering of “and” some connective which is not treated ∧I -classically or, respectively, not treated ∧E -classically, by a favoured consequence relation.) For 5.12 and 5.15, both rules are problematic. (This section does not exhaust every version or variety of conjunction that has ever been considered; a conspicuous omission is mentioned in the notes (p. 673), under the heading ‘Strong Conjunction’.) Before proceeding to any of these matters, however, it is best to begin by signposting some traps into which those who venture into the ‘logic of ordinary language’ have been known to fall. Such a venture requires a rough grasp of the three topics listed in our title; ‘syntax’ there means, incidentally, the syntax of natural language, and not the study of syntactic matters in the sense employed above. It has proved helpful to regard facts about the usage of expressions in the speech of a community as explained in part by the meanings of those expressions and in part by general considerations concerning what speakers are up to in communicating. The former – semantic – factors, and the latter – pragmatic – factors, both need to be taken into account. For example, someone who said “The blood of some human beings contains haemoglobin” would normally be taken as implying that the blood of some human beings also lacks haemoglobin, or at least that for all the speaker knew, this was the case. For, anyone believing that all human blood contains haemoglobin does something misleading in telling others that some human blood does. What the speaker implies in this case is no part of what is implied by the content of what is said: it is not a semantic matter. Rather, it is from the fact that the speaker has said this – together with the presumption that one does not without good reason say something less informative on a topic under discussion than one might have said at little or no extra cost in sentence length, that the speaker is inferred not to have been in a position to make the stronger (“all”) statement. H. P. Grice, who pioneered the study of such pragmatically based inferences, summarises the maxim the speaker is assumed to be following as: “Be Informative” (Grice [1967]). When an utterance is taken to convey something by virtue the operation of such general conversational maxims – all of them special cases of a supermaxim: “Be cooperative” – the speaker, the utterance, or even the sentence uttered, is described as conversationally implicating the something in question. The noun

5.1. CONJUNCTION IN LANGUAGE AND LOGIC

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form is implicature, rather than implication. This helps reinforce the contrast between something’s being (to return to our example) amongst the implications of the statement that some human blood contains haemoglobin, and its being merely implicated by the making of that statement. A conversational implicature can be ‘cancelled’ by the speaker’s saying anything to block off the inference a hearer might otherwise make. For example, one might say “Some – indeed all – human blood. . . ”, or “Some, if not all, human blood. . . ”. Another maxim of Grice’s, “Be Orderly”, is intended to govern such matters as the narrating of events, for which it dictates that the order of narration should match the order of the events narrated. If you say “Leo was fired and Miranda had a nervous breakdown”, the implicature arising in virtue of that maxim can be cancelled by adding “but not in that order”. (Thus the “and” in the original sentence cannot mean “and then”; for more on “and then”, see the following subsection.) Grice used such considerations to show that it was not appropriate to postulate a special non-commutative temporal conjunction in English, and to complain that its meaning was not adequately captured by the logician’s “∧”. The locus classicus for complaints of this nature is Strawson [1952]. An earlier venture into the area is represented by Reichenbach’s distinction between what he calls adjunctive and connective interpretations of the standard truth-tables (Reichenbach [1947], pp. 28–34), which appears to have elicited more interest on the part of psychologists then of philosophers (e.g., Bellin and Lust [1975]) and about which nothing will be said here. The general moral is that not every feature of use is an aspect of meaning. The same reply is appropriate to those who would discern special “and so”, “and by the same token” or “and while we’re on the subject” senses of and. (For variety, we often use italics in place of quotation marks around “and”, “or”, “if”,. . . – but only when it is clear from the context that the word in question is being mentioned rather than used. This tends to be rather less clear in the case of “and” than with the others.) Exercise 5.11.1 “The attempt to represent English and by ∧ as the latter connective behaves according to an ∧-classical consequence relation is clearly hopeless, since no-one would say “Dolphins are mammals and dolphins are mammals”, yet according to the proposed treatment this is simply equivalent to (and so should be no odder than) saying “Dolphins are mammals”, since we have A A ∧ A.” Discuss. As well as conversational implicature, Grice spoke of conventional implicature. This is a feature of meaning, though in a certain – as Grice puts it, favoured – sense of the phrase “what is said”, a report of what is said need not incorporate these features. The usual spelling out of the favoured sense is in terms of truth-conditions. What is conventionally implicated does not bear on the conditions for the truth of what is said. The classic example is the distinction between “and” and “but”. The conventional implicature in the case of the latter connective is that of some kind of contrast between what precedes and what follows the but. The details of this particular implicature are notoriously hard to spell out (see notes); a somewhat similar implicature attends the use of although, which is again usually treated as forming compounds for whose truth the truth of the two components is necessary and sufficient. The truth of those components (the subsentences S1 and S2 in “S1 although S2 ”) is not sufficient

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for the appropriateness of asserting the compound, however. And perhaps it is reasonable to hold this also for the case of and itself (Rundle [1983a]). It is interesting to see both issues of ordering and the and/but contrast, taken by Grice as illustrating the phenomena of conversational and conventional implicature respectively, mentioned in the same breath in the following passage from D’Arcy Cresswell: On arriving in Christchurch, my native city, I had several newspaper-men to see me, one of whom, although he carefully reported all my answers to his questions, as to the success of my book, and my doings in England and whom I had met there, and whether I still thought there were no poets nowadays, yet he so carelessly changed their order as to give an air of conceit to some of my news, as if I put myself first; as even a ‘but’ or an ‘and’ out of place can do. Cresswell [1939], p. 167f.

Obviously the exact characterization of conversational and conventional implicature is a matter requiring delicate handling, and this is no place for the lengthy excursion into the philosophy of language which any such detailed treatment would call for. Grice’s own substantive theory of conversational and conventional implicature has been subjected to intensive critical scrutiny and been founding wanting (see Davis [1998] and Bach [1999] respectively), though the problematic details will not infect – it is to be hoped – our own minimal use of this apparatus, which involves nothing more than acknowledging the phenomena his examples illustrate, whatever theoretical account one might end up endorsing of these phenomena. (See further Saul [2001], [2002], and Potts [2005].) The intention above has been simply to evoke a familiar picture of the terrain: semantics is the study of meaning, with a core pertaining to aspects of meaning bearing on truth-conditions, and additional ingredients conventionally signalling certain expectations, contrasts, and other aspects of the speaker’s attitude to the propositional content of what is said. Such signals include conventional implicatures, as well as various other phenomena (see next paragraph). Given a sentence with certain semantic properties, there are pragmatic factors influencing the suitability of uttering that sentence to make an assertion. There will be the assumption on the hearer’s part that one is abiding by the norms of co-operative conversation, which gives rise to an assertion’s having various conversational implicatures. We shall see several further examples of the latter phenomenon in subsequent chapters. It should be noted that something of essentially the same character as conversational implicature arises in non-linguistic communication, even on a very broad understanding of the latter term. For example, if one is watching a movie in which children are seen enjoying themselves in a playground, but one is given a view of them from a position even slightly occluded by out-of-focus foliage, then one takes what is being shown as indicating (‘implicating’) that the children are being watched by a concealed viewer, even without any candidate for that role being or having been explicitly depicted in the movie up to that point. Reconstructing the process that engenders this interpretation, one reasons that if we were merely being shown that the children are playing on the swings, and so on, then we would have been an unoccluded view of them. So there must be some special reason for viewing them from a position behind some bushes, and the most natural explanation for this is that we are sharing the point of view of some concealed witness to their play.


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Of course a similar phenomenon can occur with static depictions – paintings, drawings, still photographs – as is pointed out in Abell [2005], q.v. also for some remarks on the attenuated sense of “communication” involved here. Much of the interpretation of psychological experiments by cognitive psychologists with no background in pragmatics has been vitiated by under-acknowledgment of implicature effects, as Schwartz [1996] points out (with illustrations). ‘Various other phenomena’ apart from conventional implicatures: here are some examples. There is the matter of contrastive stress: “Harry doesn’t much care for politics” vs. “Harry doesn’t much care for politics”. Relatedly, we have the choice of focussing on certain material as the main content of what is said, and placing other material in the background as incidental. To illustrate this for the case of conjunction, consider the difference between “Rob joined the company in 1960 and is due to retire next year” and “Rob, who joined the company in 1960, is due to retire next year”. Semantics and pragmatics were two of the subdivisions made by Charles Morris of the study of ‘signs’: signs in relation to the non-linguistic reality they signify, and signs in relation to sign-users. (The former characterization will be toned down in 5.13. References to the origin – and some subsequent metamorphoses – of Morris’s trichotomy may be found in Sayward [1974].) The boundary between semantics and pragmatics is a recurrent locus of contention. For some of the issues here, see Kamp [1978] and the various papers in Bianchi [2004], including the editor’s own contribution. The third subdivision Morris labelled ‘syntactics’: the study of signs in relation to other signs. The usual term is (as above) syntax, of course. This term, especially in the adjectival form, syntactic, has two kinds of application in logic, only one of which is the formal analogue of the syntax of natural languages, namely as covering the formation of compounds from their components (conditions of well-formedness for terms and – more significantly for our purposes – formulas). The other use in logic is to cover the construction of proofs in proof systems, so that in this usage, syntactic means more or less the same as proof-theoretic. This tendency leads to potential trouble, as remarked in 4.11 (see p. 512), when it comes to the suggestion that there might be such a thing as proof-theoretic semantics, with its proposal that being governed by certain inference rules its constitutive of what various connectives mean, and so is more naturally described as a semantic rather than a ‘purely syntactic’ matter. Another terminological issue raised by the comparison between the syntactic vocabulary customary among linguists and grammarians on the one hand, and logicians on the other. The issue arises in connection with the word and ; it leads us to consider a further question: the subordinating/coordinating distinction. Taking the first point first: we need to note is that the term “conjunction” has been used in syntactic work by linguists – especially traditional grammarians – to name a part of speech comprising not only what logicians think of as conjunction but also their disjunction, implication, and so on. The latter usage coincides with the logicians’ use of “connective”, at least for connectives of arity 2. Naturally, we follow the latter usage here, since it would be very confusing in a discussion of logical matters to describe or, for example, as a conjunction. For presumably similar reasons, this terminology is now used by some linguists; for example, Hopper and Traugott [2003], p. 4 describe while as a temporal connective. (Of course there is also a non-temporal meaning – cf. the same authors at p. 91 – illustrating the theme of their book.)


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Moving to the second point: amongst what grammarians and linguists have in the past called conjunctions, and in particular those we would call binary connectives, a division is usually made into subordinating and coordinating. The idea is that a coordinating connective links two sentences (or ‘clauses’) to form a sentence, whereas a subordinating connective attaches itself to one, forming a unit in its own right (a ‘subordinate clause’), which unit then attaches to another sentence to yield the resulting sentence. From the various conjunctive connectives mentioned above, for example, and and but would be taken to be coordinating, with although classified as subordinating. The precise details of how this distinction is handled will depend on the kind of syntactic theory espoused, but there are many obvious marks of the distinction accessible to any speaker of English. For example, one can equally well say “S1 although S2 ” and “Although S2 , S1 ”. (Similarly with when, because, before, if, . . . ) Coordinating connectives, on the other hand, display no such mobility – one cannot, for instance, say “And S2 , S1 ” instead of saying “S1 and S2 ”. In spite of the evident significance of this contrast for the syntax of English (and other natural languages), it is not clear that the subordinating/coordinating distinction is of any semantic significance. If one takes one’s core semantic theory for a language to provide semantic values (whether truth-values, propositions, or something else) for all its sentences on the basis of the semantic values of their parts, the appropriate semantic value for a connective will be a function from the semantic values for sentences to the semantic values for sentences. (This is the constraint of compositionality on a semantic theory, referred to in 2.11.11, p. 208; see Lewis [1972], or the other items mentioned in the notes under this heading to §2.1, p. 271, for implementation and discussion.) In the case of a binary connective #, the same information will be provided if a two-place function f is supplied, which delivers the semantic value of #(S1 , S2 ) by: value(#(S1 , S2 )) = f (value(S1 ), value(S2 )) or a one-place function f from sentence values to functions-from-sentencevalues-to-sentence-values, with value(#(S1 , S2 )) = g(value(S2 )) where g(x) = f ((value(S1 ))(x). This is a special case of the observation made by Schönfinkel and Curry that as long as one is prepared for extensive encounters with functions whose values are functions, only one-place functions need ever be used. Remark 5.11.2 In what is called categorial grammar, the syntactic properties of expressions are described by assigning category labels to them. We have certain basic category labels, for example S (for sentences) and N (for names), and for any labels α, β1 , . . . , βn , allow the derived category label α/β1 , . . . , βn , for an expression which takes n expressions of categories β1 , . . . , βn to give an expression of category α. (For example, a monadic predicate is of category S/N, since it takes a name to make a sentence, and a binary sentence connective is of category S/S,S.) Then the upshot of the preceding discussion is that for purposes of semantics, there is no relevant distinction between the category labels α/β1 , β2 and (α/β1 )/β2 . (See further Humberstone [1975].) As far as our treatment of connectives in formal languages is concerned, however, it is worth


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noting that we cannot trade in a binary connective like ∧ for an infinite set of singulary connectives #A , one for each formula A, and write #A B for A ∧ B, since if #A were really a 1-ary connective (primitive or derived) the only proper subformulas of #A B would be the subformulas of B. Rather, in terms of our discussion – see (1.11.2 on p. 49 and) 3.16.1 . (Of course for paron p. 424 – #A denotes the 1-ary context A ∧ ticular choices of A, namely those constructed entirely from primitive connectives, with at least one of them being, of necessity, nullary we will have a genuine derived connective here, as with the context → ⊥ as a derived negation connective, discussed in §8.3, inter alia.) Remark 5.11.3 There is a one-way version of the bilateral notion of equivalence of categorial labels such as the α/β1 , β2 and (α/β1 )/β2 of the previous remark: abbreviating these two respectively to γ1 and γ2 , we can state the equivalence of the labels by saying that γ1 ⇒ γ2 and γ2 ⇒ γ1 , where γi ⇒ γj means that any expression of category γi also functions as an expression of category γj . There are many plausible general principles governing this relation ⇒, one of which is associated with Geach [1972b]: α/β1 , . . . , βn , ⇒ (α/γ)/(β1 /γ, . . . , βn /γ). Putting n = 2, and β1 = β2 = S, γ = N, this principle correctly predicts that any binary sentence connective—such as and —can also be used to join two predicates to make a predicate. (Such principles, including that cited here, may be found originally in Lambek [1958], though Lambek’s discussion is more oriented toward thinking of sentences as composed by concatenating expressions, whereas Geach’s, which we follow here, prefers a function-and-argument analysis. In fact Geach [1972b] does not endorse the above ⇒-principle itself, but something similar; see Humberstone [2005e] for details, and van Benthem [1988], [1991], as well as Zielonka [2009] and references there cited for further discussion of logical interest.) The semantic side of this particular principle is most simply illustrated for the case in which n = 1: α/β, ⇒ (α/γ)/(β/γ). Suppose that f is a function from semantic values for expressions of category β, from – we shall say for brevity – β-values, to α-values. Then f induces (‘canonically’) a function g from β/γ-values to α/γ-values, which serves accordingly as a semantic value for expressions of category (α/γ)/(β/γ), in the following manner. Let h be a β/γ-value (a function from γ-values to β-values, that is). Take g to be the composition f ◦ h. Thus for any γ-value c, g(c) = f (h(c)). Exercise 5.11.4 The above ⇒-principles are reminiscent of principles governing → in weak implicational logics. For instance, the principle α/β ⇒ (α/γ)/(β/γ) looks like: B → A (C → B) → (C → A), when “α/β” is replaced by “B → A” (etc.). We could use categorial grammar – devised as a formal theory of syntax – to supply what is in effect a semantics for such logics, if the formulas of the formal language are thought of as interpreted by categories of expressions of some (other) language. Urquhart’s treatment of relevant implication (2.33) seems especially suggestive in this regard: instead of reading “x |= B → A” as saying “B → A holds on the basis of the information x”, we read it as saying “x is an expression of category α/β’, where α and β are the


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categories assigned to A and B. Recall that there is a binary operation · in Urquhart’s models with x |= A → B just in case for any y with y |= A, we have x · y |= B; i.e., x is of category β/α just in case for any expression y of category α the expression we get by combining x with y is of category β. Of course we may not want the ‘combining’ operation to have all the semilattice properties of · in Urquhart’s semantics for R, the implicational axioms of which are the subject of this exercise. Which of the following implicational principles are plausible given the categorial-grammatical interpretation just sketched? (For more in this vein, see van Benthem [1991], Restall [1994b].) (i) (ii )

AA A → B (C → A) → (C → B)

(iii)

A → (A → B) A → B

(iv )

A → (B → C) B → (A → C)

(Here we have turned the main “→” into a “” in transcribing Identity, Prefixing, Contraction, and Permutation, in accordance with the explanation given above.) (v) How should sequents with several formulas on the left be interpreted in the present setting, such as those of the form A → B, A B? Finally, on the matter of what is of interest for linguistic syntax and what is of interest for (at least sentential) logic, we note that “and” can occur linking constituents which are not themselves sentences, as in “Harriet and Susan are widows” and “Harriet is honest and kind”, where there is an obvious relationship to sentential conjunction, as well as in “Harriet and Susan are sisters” where (on one reading) the relationship is less obvious. It would not be appropriate to list amongst possible objections to the rule (∧E) – some of them reviewed in 5.15 – the observation that so far is “Leo and Laura were alone in the house” from entailing “Leo was alone in the house” that it is actually inconsistent with this supposed consequence: the would-be premiss in this case is of course not equivalent to the sentential conjunction “Leo was alone in the house and Laura was alone in the house”. We touch on some of these uses of nonsentential conjunction again below (in 5.15, in fact). There are also numerous constructions of a more restricted and sometimes even idiomatic kind, as in “. . . between x and y”, “I’ll try and outrun him”,. . . Our concern is only with sentence conjunction. Even more specifically, since interrogative and imperative sentences can also be conjoined (though not with each other), to compare the “∧” of the preceding chapters with and, we should restrict attention to conjunctions of declarative sentences. Likewise for the connectives we discuss in succeeding chapters. Lastly, it should be noted – so that this too may be set aside – that, especially in spoken language, there is a use of and, or, but, etc., to begin a sentence in order to link it appropriately with the preceding (or following) discourse. The labels “discourse connective” and “pragmatic connective” have been used to apply to expressions playing this role—though primarily under such names theorists have had in mind such expressions as nevertheless and therefore; studies include van Dijk [1979], Warner [1985], and Chapter 6 of Schiffrin [1987].


5.12

639

Temporal and Dynamic Conjunction

Considerations of conversational implicature were urged in 5.11 (following Grice) to tell against postulating a special and then sense of and. This does not prevent us from considering and then itself as a kind of temporal conjunction, and in fact logical treatments of two types have been proposed for such a connective. Let us write “A ∧ B” with the intended reading “A and then B” or “A and next B”. The first type of treatment, from von Wright [1965], who works with the latter reading, turns out to be less interesting, from a logical point of view, than the second. Von Wright had in mind a picture of time as a discrete linear ordering, with A ∧ B (actually he wrote “AT B”) true at a point if A was true at that point and B at its successor. Rather than explicitly studying this semantic characterization, what he did was to lay down some axioms for a proposed logic in Fmla. Along with Modus Ponens, he used a rule allowing the replacement of provable equivalents within the scope of ∧ . The axioms, to be added to any basis sufficient for the classical tautologies (with Modus Ponens as a rule of proof), were all instances of the following schemata: (A1)

((A ∨ B) ∧ (C ∨ D)) ↔ [((A ∧ C) ∨ (A ∧ D)) ∨ ((B ∧ C) ∨ (B ∧ D))]

(A2)

((A ∧ B) ∧ (A ∧ C)) → (A ∧ (B ∧ C))

(A3)

A ↔ (A ∧ (B ∨¬B))

(A4) ¬(A ∧ (B ∧¬B)) (In fact, von Wright used individual axioms and a rule of Uniform Substitution, rather than schemata as here; also, a simplification from von Wright [1968] has been incorporated.) We may take the primitive connectives of the language of von Wright’s system to be precisely those figuring in the above schemata. Exercise 5.12.1 (i) Show that for any formula A, where t is any tautology (e.g., q ∨ ¬q), the formula (A ∧ B) ↔ (A ∧ (t ∧ B)) is provable. (ii ) Show that the converse of schema (A2) is derivable. (iii) Show that the following is a derived rule of the system: (A1 ∧ . . . ∧ An ) → B

((C ∧ A1 ) ∧ . . . ∧ (C ∧ An )) → (C ∧ B) Remark 5.12.2 If the logic of ∧ is recast in Set-Fmla it is easy to see that this connective is what in 5.31 we call ∧-like. In Fmla, this amounts to satisfying the condition that the conjunction of A ∧ B with C ∧ D provably implies A ∧ D, for all formulas A, B, C, D. See §5.3 for further details. Von Wright’s system struck him originally as something of a departure from traditional tense logic in the style of A. N. Prior. What he says in von Wright [1965], p. 203, is this: In Professor Prior’s tense-logic the ingredients new relative to traditional systems of logic are two tense-operators, roughly of the type of the quantifiers and the modal operators. In the system which is studied here the new

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CHAPTER 5. “AND” ingredient is a logical constant more reminiscent of the truth-connectives. It is a kind of asymmetrical conjunction.

Presumably by ‘truth-connectives’ von Wright intended to allude to the binary connectives, since Prior’s G and H behave syntactically like ¬ as much as like . (Von Wright’s choice of the term asymmetrical in preference to non-commutative raises some issues of its own; 3.34 (p. 497 onward) discusses the seepage of terminology appropriate to binary relations across to binary connectives and operations. Even setting aside such considerations, the right word here would be “non-symmetric”.) It was soon shown, however, in Clifford [1966], Åqvist [1966], that von Wright’s system is just a variant of the Prior-style systems, using a singulary operator , with the semantics and proof theory described in Example 2.22.9, p. 288. (The same observation was made in Wajszczyk [1994] for a restricted variant of von Wright’s logic, in which the and next connective is not allowed to appear within its own scope.) Reading A as “next (it is the case that) A”, we can define A ∧ B as A ∧ B, and derive the above axioms (and rules). Conversely, we can introduce such a into von Wright’s language by defining A as t ∧ A, “t” being understood as in Exercise 5.12.1. Indeed the point of that exercise was to render the above relationship between von Wright’s system and the system of mentioned at the end of 2.22.9 seem unsurprising. (For example, part (iii) tells us, substituting t for C, that our next operator has a normal modal logic.) In the passage quoted below, the symbol “S” is used in place of “ ”; it comes from Segerberg [1967] (p. 45), where the Clifford–Åqvist observation also appears. The idea underlying the present paper is simply that von Wright’s notion ‘and next’ may be profitably broken up into its constituents ‘and’ and ‘next’: the former is taken care of by ordinary conjunction and the latter by the new constant S.

Thus von Wright did not after all introduce a distinctively binary temporal connective: the temporality could be isolated in a singulary operator working in combination with non-temporal conjunction (∧). It is for this reason that his work was described above as providing only an uninteresting treatment of temporal conjunction. Behind that work, however, there lay a more promising intuition, evident in what von Wright says about associativity. In von Wright [1965] there appears a note of gratitude to Åqvist for correcting a mistake in an earlier draft of the paper, which included an associativity axiom for “∧ ”: (A ∧ (B ∧ C)) ↔ ((A ∧ B) ∧ C). Using the semantics of 2.22.9, we can see that for the lhs to be true at a point x in a model M = (W, f, V ), we require M |=x A and M |=f (x) B ∧ C, i.e., M |=f (x) B and M |=f (f (x)) C, while for the rhs, we require M |=x A ∧ B, hence: M |=x A and M |=f (x) B, and also M |=f (x) C. Thus the rhs is in fact equivalent to A ∧ (B ∧ C) rather than to A ∧ (B ∧ C), as remarked in Clifford [1966]. But setting aside these formal details, the original idea of associative temporal conjunction is intuitively appealing. Think of saying “S1 , and then S2 ,


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and then S3 ” (or “and next”). Such a narration is like putting three beads on a string, and it is hard to see what difference there could be between on the one hand, first placing bead a on the left of b and then putting both of them to the left of c, and, on the other hand, placing b to the left of c and then putting a on their left. The upshot is the same, pictorially: a — b — c. Think of the string as a stretch of history we are trying to describe, which in this case decomposes into an a-part followed by a b-part followed by a c-part. The above considerations lead to our ‘more interesting’ approach to temporal conjunction. Without going into excessive detail on the formal semantics, suppose that what corresponds to the frames of §2.2 are linearly ordered sets of points, subsets of which containing all points between any two points they contain are called intervals. Departing from the ‘instant-based’ approach of traditional tense logic, we evaluate formulas w.r.t. these intervals. If one interval x immediately precedes, without overlapping, another interval y, their union is also an interval, which we call x + y. Given a model M on a frame of this sort, we enter the following clause in the inductive definition of truth over an interval: [Interval ∧ ]

M |=x A ∧ B iff ∃y, z x = y + z & M |=y A & M |=z B.

A proposal along these lines may be found in Dowty [1977] and Humberstone [1979]. In fact in the latter paper, intervals are not construed as sets of instants, but taken as entities in their own right (following Hamblin [1971]). It is clear that the (partial) operation + is associative in the sense that where either of x + (y + z) and (x + y) + z is defined, so is the other, and they are equal. Further this secures associativity for ∧ in the sense that if either of A ∧ (B ∧ C) and (A ∧ B) ∧ C is true over an interval, so is the other. Unlike von Wright’s treatment, this semantic treatment (we do not go into proof-theoretic matters) makes ∧ ‘unbiased’, whereas for von Wright, A ∧ B is biased specifically in favour of the first conjunct, which it implies. However, a discussion of this feature would require us to enter into greater details on how truth over an interval is related to truth over a subinterval thereof, than would be appropriate here. In 6.45 we shall see that a clause of the same form as [Interval ∧ ] above can find its way quite naturally into the semantics of disjunction. (See also the Digression on ‘Dyirbal disjunction’ on p. 795.) Although our main business has been temporal conjunction, various other non-commutative versions of conjunction have also been proposed in the literature. These include an idea based on ‘lazy evaluation’ in computer programming (the three-valued proposal of McCarthy [1963], with what has been called sequential conjunction, serving as the prototype), as well as logics – see Mulvey [1986], Bignall and Spinks [1996] – with algebraic semantics given by ‘skew Boolean algebras’, to mention only a couple (to say nothing of non-commutative linear logic, in which it is multiplicative rather than additive conjunction that is non-commutative). In a rather different vein, motivation has come from that not only do some events spoken about occur before others (leading to temporal conjunction), there is the fact that whatever is said is said in some definite order. Asymmetries then arise in the processing of discourse – for example in respect of such things as anaphoric relationships – and some theorists have proposed a kind of semantic account according to which ∧ is emerges as non-commutative as a result. See Groenendijk, Stokhof and Veltman [1996] for a good illustration; it is with this work in mind, as well as the work of Gärdenfors mentioned below, that we


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include the term dynamic in the heading of the present subsection. (Groenendijk and Stokhof describe not only dynamic conjunction but also similarly anaphoramotivated dynamic disjunction; for some discussion, see pp. 128–130 of Simons [2000].) We give only the barest outlines here. The authors allow a non-scopal binding of variables by quantifiers, so that not only (1) but also (2) counts as well-formed: (1) ∃x(Fx ∧ Gx )

(2) ∃x(Fx) ∧ Gx.

The final “x” in (2) simulates certain occurrences of natural language pronouns anaphoric on quantifier phrases. Though (1) and (2) are equivalent, the result of interchanging the two conjuncts on (2) would not be equivalent to (2) since the semantics is designed (speaking very roughly) to have existential quantifiers introduce new objects into the discourse which later occurrences of the relevant variables then refer to. Until the objects in question have been introduced, such back-reference is not available, giving rise to non-commutativity. Remark 5.12.3 The above is not the only source of non-commutativity for Groenendijk et al. [1996]; thus on p. 195 they say “Whereas p ∧ ¬p is consistent, ¬p ∧ p is inconsistent”, illustrating this with substitutions for p which do not involve quantifiers and individual variables. (The term inconsistent has a somewhat specialized sense for the authors, it should perhaps be mentioned.) We now present a version of Gärdenfors’s semantics for sentential logic based, as he puts it, on the dynamics of belief. The sources for this discussion are Gärdenfors [1984], [1985], [1988], as well as Pearce and Rautenberg [1991], which should be consulted for further details – especially as here we shall consider only the treatment of conjunction. Where K is a non-empty set and P is a set of functions from K to K which contains the identity function – call it ι (iota, that is) – and is closed under composition (symbolized, as usual, by ◦), the pair (K, P) will be called a Gärdenfors frame if the following further pair of conditions are satisfied for α, β ∈ P, K ∈ K: (G1) α(α(K)) = α(K) (G2) α(β(K)) = β(α(K)). In other words, not only must P be closed under ◦ but ◦ (restricted to P) must be idempotent and commutative. (Of course, ◦ is always associative). To make the connection with the process of belief revision (the ‘dynamics of belief’), think of K as a set of possible belief states for some rational agent, and of P as a set of possible transformations from one belief state to another. (This use of “K” has nothing to do with its use in connection with the Kleene matrices or the smallest normal modal logic, as in §§2.1 and 2.2 respectively.) In the present context, we are thinking of only one particularly simple kind of transformation: that which consists in expanding one’s belief state on learning—or coming to accept—a certain proposition, and we can regard α ∈ P as being a proposition. Thus we conceive of propositions primarily as maps from belief states to belief states rather than in the more familiar ‘static’ approach in which propositions


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are (e.g.) sets of possible worlds. After adding the proposition represented by α to the agent’s belief state K, the agent comes to be in the state α(K) in which that proposition is accepted. It may be, of course, that the proposition in question was already accepted in the prior belief state α, in which case α(K) is just K itself since there is no change to be made. Indeed, Gärdenfors defines a proposition α to be accepted in a belief state K if and only if α(K) = K. With this picture in mind, we can see the rationale behind conditions (G1) and (G2). Addressing (G1) first: whether or not α is accepted in K, it is certainly accepted in α(K), so there is no further change to be effected by applying α again. As to (G2), it is presumed that there is no difference to the end result occasioned by learning (or more accurately – since no question of correctness enters here – coming to accept) two propositions in one order rather than the other. There is a natural notion of propositional entailment available in this setting, relative to any given Gärdenfors frame (K, P). We shall say that, relative to such a frame, α1 , . . . , αn ∈ P (collectively) entail β ∈ P just in case: For all K ∈ K, β(α1 (α2 (. . . (αn (K)). . . )) = α1 (α2 (. . . (αn (K)) . . .). Or, to put the definition in terms of composition, just in case: β ◦ α1 ◦ α2 ◦ . . . ◦ αn = α1 ◦ α2 . . . ◦ αn . (If n = 0, take the rhs here as being ι. This is the only reason we included ι in our stipulation as to what a Gärdenfors frame was to be, though the reader can readily imagine it would play a greater role if we were dealing, below, with a language with the connective .) When the αi entail β in the above sense, we record this by using the notation α1 , . . . , αn G β where G = (K, P) is the Gärdenfors frame in question. Because of (G1) and (G2), the order and number of occurrences of the αi on the left here are immaterial, and we can think of the relation G as holding between the set {α1 , . . . , αn } of propositions and the individual proposition β. Exercise 5.12.4 (i) Show that the relation G satisfies the conditions (R), (M), and (T) from 1.12, restricting the Γ (in (M)) to finite sets. (ii ) Show that the definition given above of α1 , . . . , αn G β is equivalent to the following: For all K ∈ K, α1 (K) = K& . . . & αn (K) = K ⇒ β(K) = K. By defining a notion of model on the frames we have been considering we can interpret the sentential language whose sole connective is the binary ∧ . To turn a Gärdenfors frame (K, P) into a (Gärdenfors) model, we expand it by supplying an interpretation function ! · ! which maps formulas of the language to elements of P, subject to the condition that for formulas A, B, we have !A∧B! = !A!◦!B!. Further such conditions would be appropriate were we here investigating more than just the logic of conjunction. To give an example already anticipated above, for a semantic treatment of -classicality, we would impose the condition that !α! = ι. In the works cited above, Gärdenfors gives fairly simple conditions which lead naturally to a semantics matching IL-provability,


644

with an additional twist yielding a match with CL-provability. Since the ∧fragments of these logics are the same, the twist in question need not concern us here. On the current presentation of the semantics the match in question is achieved by defining a sequent A1 , . . . , An B to hold in a model (G, ! · !) just in case the entailment relation described above holds between the propositions expressed, i.e., just in case !A1 !, . . . , !An ! ϕ !B!. A sequent is valid on the frame G provided that it holds in every model (G, ! · !) on G. (Note that for convenience we work in Set-Fmla.) In view of 5.12.4(ii), these notions can be recast into a more familiar-looking form (‘validity as truth-preservation’) by noting that a sequent A1 , . . . , An B holds in (G, ! · !) if and only if for all K ∈ K: K |= A1 and . . . and K |= An imply K |= B where we write “K |= A” to abbreviate !A!(K) = K(meaning, in terms of the motivating idea, that the proposition expressed by A is accepted in the belief state K, relative to the model in question). It is not hard to see that every instance of the sequent-schema forms of the rules (∧I) and (∧E) holds in every model. To take the former schema A, B A ∧ B, this is a matter of checking that, where for an arbitrary model (G, ! · !) we denote !A! and !B! by α and β respectively: α, β α ◦ β. But by our definition, this amounts to checking that (α ◦ β) ◦ α ◦ β = α ◦ β, which, however further internal bracketing is supplied, is a trivial consequence of the semilattice properties of ◦ in Gärdenfors frames. A similar argument applies in the case of the (∧E) schema. This, and 5.12.4(i), give the ‘only if’ direction of: Theorem 5.12.5 A sequent is provable in the ∧-subsystem of the natural deduction system Nat supplemented by the rule (M) if and only if it is valid on every Gärdenfors frame. Proof. In view of the preceding remarks, the ‘if’ (completeness) direction is what remains to be established. For this purpose, we may use a ‘canonical’ Gärdenfors model, based on the frame (K, P) in which K consists of the deductively closed sets of formulas (-theories, where is the consequence relation associated with the proof system described in the Theorem) and P contains the identity map from K to K as well as, for each formula A, the map [A] defined by: [A](K) = K ∪˙ {A} ( = Cn (K ∪ {A}) ) for K ∈ K. Note that P is closed under composition, because [A] ◦ [B] = [A ∧ B]; and also that conditions (G1) and (G2) are satisfied. The model we need on this frame is then obtained by setting !A! = [A] for each formula A. An unprovable sequent A1 , . . . , An B fails to hold in this model because we can find K ∈ K with [B] ◦ [A1 ] ◦ . . . ◦ [An ](K) = [A1 ] ◦ . . . ◦ [An ](K), e.g. by taking K = ∅, or again by taking K = Cn ({A1 , . . . , An }).


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The main interest of the above is perhaps the very idea of treating conjunction, or any binary connective, semantically in terms of function composition. Suppose we define a generalized Gärdenfors frame as a pair (K, P) exactly as at the start of our discussion, but without imposing the conditions (G1), (G2). In the absence of those conditions, it is best to use the characterization of G given in 5.12.4(ii) to define this relation. We still have what ended up as a version of (∧I) in the above treatment, for all such generalized Gärdenfors frames: α, β G α ◦ β, when G is understood as just suggested. (For if α(K) = K and β(K) = K then the latter identity allows us to replace the left occurrence of “K” in the former identity by β(K), giving α(β(K)) = K, i.e., α ◦ β(K) = K.) Exercise 5.12.6 Give a natural deduction system in Set-Fmla for the set of sequents valid on all generalized Gärdenfors frames, defining validity in terms of models exactly as before (with !A ∧ B! = !A! ◦ !B!), except that we insist on the ‘preservation’ account of G from 5.12.4(ii). Give a soundness and completeness proof like that of 5.12.5 for your set of rules. In §6.6 of Gärdenfors [1988] there are described some applications in the field of belief revision for operations other than those recognised as propositions in our discussion of (ungeneralized) Gärdenfors frames. The main example of interest concerns operations α ¯ which are in some sense reciprocal to the α ∈ P above; while α(K) represents the expansion of K by means of α, α ¯ (K) represents a corresponding contraction: a belief-state resulting from deleting (if it is present) one’s commitment to α. The relation between expansion and contraction is the subject of much discussion in the theory of belief revision – as is evident from Gärdenfors [1988] and references therein. (Or see the more recent monographs Hansson [1999], Rott [2001].) A topic with a similar flavour, logical subtraction, in which an operation is sought for deleting the content of a proposition, will be the subject of §5.2. (Indeed it has been suggested to the present author by several people – including Peter Lavers and David Makinson, – that this is really the same topic; however, we do not pursue this line of thought in §5.2. See further the notes and references to that section, p. 707).

5.13

Why Have Conjunction?

If we set aside questions as to whether there are conventional implicatures in sentences formed by and, taking these to be irrelevant to the evaluation of arguments in which such sentences figure, we may plausibly suppose the features relevant in that connection to be those given (syntactically) by the ∧-classicality conditions on consequence relations, or (semantically) by a restriction of attention to the ∧-boolean valuations. In 5.14 and 5.15 qualms on this score will be aired, but for the moment let us just make this supposition. (Note that this is a point on which quantum logic, intuitionistic logic, and—suitably construed— relevant logic, all concur.) A question then comes up as to why we should want a connective with these properties. After all, instead of asserting S1 and S2 , we might just as well assert S1 and also assert S2 . So put, the question almost answers itself. (For the answer which follows, see Geach [1963]. Here we ignore the sometimes striking rhetorical and pragmatic differences between asserting two sentences and asserting their conjunction: see the start of 5.14 below.) For there


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is more to be done with conjunctions than assert them. We might, for example, wish to deny S1 and S2 ; that is: to assert its negation. Or again, we might wish to consider a conditional with a conjunctive antecedent. As this choice of “consider” suggests, the formulation in terms of asserting various things is not exactly to the point: we may for example merely wish to suppose (for the sake of argument, say) that not(S1 and S2 ), and this is not tantamount to separately supposing each of two things. The general point is that a given conjunction may occur embedded within the scope of other connectives or operators. To think about this issue, let us say, where is a consequence relation and Γ and Δ are sets of formulas in the language of , that Γ and Δ are collectively equivalent (according to ) provided that: Γ D for each D ∈ Δ and Δ C for each C ∈ Γ. In the case (of particular interest here) in which Γ = {C1 , . . . , Cn } and Δ = {D}, we adopt the shorthand of saying that C1 , . . . , Cn are collectively equivalent to D. Note that this is just the obvious precisification (for the case of formulas) of the informal notion – deployed for instance in 3.11.3(ii) – of several things (conditions, rules, formulas, . . . ) being collectively equivalent to something when they are, taken together, equivalent to it. Note that the definition given also makes sense – but the wrong sense – if is a generalized consequence relation, which is why it would be unwise to write “C1 , . . . , Cn D” to mean that the relation in question holds between C1 , . . . , Cn and D: for the “” direction here would suggest that D C1 , . . . , Cn , which is of course much weaker than intended; likewise in the general case of Γ and Δ being collectively equivalent and the notation “Γ Δ”. (There is nothing wrong with such a usage, however, when it is clear from the context that is a consequence relation rather than a gcr, so the potential ambiguity just mentioned does not arise. This use may be found in many writers, for example Došen and Schroeder-Heiser [1988]; it is almost unavoidable in AAL, without excessive circumlocution, as a perusal of the references given in 2.16 will make evident. See further Remark 1.19.8.) The train of thought of the preceding paragraph amounts to saying that the ‘point’ of having conjunction in a language is that there are formulas constructed by means of ∧ to which there is no collectively equivalent set of ∧-free formulas, according to some ∧-classical consequence relation. It is not enough to observe that for the formula (say) p ∧ q such a set (namely {p, q}) can be found. Naturally, for this enterprise to have any interest, we should make sure that the language of does not contain alternative resources for expressing conjunction (e.g., both ∨ and ¬, w.r.t. each of which is classical; see 3.14). Exercise 5.13.1 Supposing that is →-intuitionistic, ∨- and ∧-classical, find, where possible, a set of ∧-free formulas collectively equivalent to each of (i) p ∨ (q ∧ r)

(ii ) (p ∧ q) → r

(iii) p → (q ∧ r).

(Hint: in each case, it is possible.) Remark 5.13.2 The above exercise provides the materials for showing that if is the consequence relation of positive logic, then every formula has a set of ∧-free formulas collectively equivalent to it according to . (For the consequence relation QL of 2.31, a problem would arise over (i); in the case of relevant logic – in particular for R, discussed in 2.33 – (ii) would


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raise a problem: what might the ∧-free formulas be whose conjunction is equivalent to this formula?) There is actually no problem including negation also, so that the result just described holds for IL; in the case of a negated conjunction ¬(A ∧ B) we can always use the ∧-free equivalent A → ¬B. (However, the result does not extend beyond propositional IL: for intuitionistic predicate logic it is no longer true that for every formula there is a set of ∧-free formulas collectively equivalent to it. For example, consider the formula ∃x(F x ∧ Gx).) By contrast: Exercise 5.13.3 Show that, for = {∧,¬} , the least {∧, ¬}-classical consequence relation, no set of ∧-free formulas is collectively equivalent (according to ) to any of (i) ¬(p ∧ q)

(ii ) ¬(¬p ∧ q)

(iii) ¬(¬p ∧¬q).

Thus the above answer to the question of why we need an operation of conjunction – that we cannot always find a collectively equivalent set of conjunction-free statements – works especially well if we consider embedded occurrences under negation, rather than in the scope of conditionals or disjunctions. Of course, natural (and many formal) languages provided numerous other contexts with which to make the same point: It is possible that (S1 and S2 ); Harriet wonders whether (S1 and S2 ), etc. – to say nothing of quantificational contexts (cf. the parenthetical comment at the end of 5.13.2). If such contexts are absent, and conjunction is not definable in terms of the existing connectives, as in the {¬, ∨, →}-fragment of IL, it is an interesting question whether adding conjunction, to obtain full (propositional) IL provides an increase in expressive power. Interesting, but vague. We obtain a negative answer if we require (for expressive adequacy of the fragment), that for every formula of the full language, an equivalent formula (according to IL ) of the reduced language can be found, but a positive answer if we require merely that for every formula of the full language, a set of collectively equivalent formulas of the reduced language can be found. A case for making the more stringent demand, and thus returning a negative answer, is made out in p. 106f. of Caleiro and Gonçalves [2005], by means of simple examples (on p. 107). The authors are considering the question of how best to cash out the idea of two consequence relations (on possibly different languages) representing what is intuitively speaking the same logic, and are rejecting a proposal which would return an affirmative answer to this question whenever the consequence relations induce what they call isomorphic theory spaces. The proposal would render irrelevant the question at the level of individual formulas, of whether any two formulas have a conjunction. (Theories enter the discussion since to say that Γ and Δ are collectively equivalent according to a consequence relation is just to say that the -theories Cn(Γ) and Cn(Δ) coincide, where Cn is the consequence operation associated with .) Mill [1843] makes what, in the light of these considerations, seems a rather strange claim, namely that calling the conjunction of two propositions a complex (specifically: conjunctive) proposition is like calling a street a house, or like calling a team of horses a horse. For the first analogy, Mill presumably has in mind as a street not the roadway but a terrace or ‘block’ consisting of houses.

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These claims come from §3 of Book I, Chapter IV, and they will seem less strange if we recall from the start of Mill’s §2 of that Chapter, that he defined a proposition to be a ‘portion of discourse in which something is affirmed or denied of something’. In other words: what we might, thinking of predicate logic, call an atomic or negated atomic sentence. And it is quite true that the conjunction of two of those is not another one of those. But he seems not to want the point to hang on this triviality, since he allows disjunctive and conditional ‘propositions’. The latter are analysed into the required subject-predicate form by taking the it antecedent and consequent themselves as subjects, with the predicate “from may be inferred that . . . ”. (Actually the relational form is suppressed, since at the time Mill was writing, people were not so comfortable as we have since become about thinking of monadic, dyadic, etc., predications.) Disjunctions are then analysed as conditionals by taking the negation of the first disjunct as antecedent, and the second disjunct as consequent. (The logic assumed here is CL.) So there is after all a quite specific prejudice against conjunction. Here it is held that we have, not a single proposition, but only a collection of propositions. (These are (i)–(iii) in the example given in the notes to this section, headed “But”: p. 674; we have been ignoring (iii), as representing at best a conventional implicature associated with and, in the above discussion of the conjuncts as collectively equivalent to the conjunction). Here is Mill’s summing up of the point from his §3: We have seen that when the two or more propositions comprised in what is called a complex proposition are stated absolutely and not under any condition or proviso, it is not a proposition at all, but a plurality of propositions; since what it expresses is not a single assertion, but several assertions, which, if true when joined, are also true when separated.

The “stated absolutely” part of this quotation seems to allude to unembedded conjunctions, for which Mill’s ‘plurality of propositions’ is then a set collectively equivalent to the would-be conjunction. As we have seen, however, this by no means disposes of the embedded cases. Conspicuously, in considering only negated atomic statements (as we would put it), Mill has buried from view the most problematic case of embedding emerging from our review (5.13.1–3) above. The concept of collective equivalence can be employed to say what it is for a logic, conceived of here as a consequence relation, to ‘have (classical) conjunction’. This is not a matter of the language’s having a particular syntactic operation (∧ or whatever), but rather of the presence of a connective in the ‘logical role’ sense of 1.19. We can say that has conjunction when according to , for every pair of formulas there is some formula to which that pair is collectively equivalent (according to ). To within equivalence () there is only one such formula for a given pair (1.19, 4.31), so we may harmlessly speak – non-congruentiality aside – of that formula as the conjunction of those in the pair. Of course it may not happen that there is such a formula for every pair and we can still ask about the existence of a conjunction for a given pair: this is the syntactically local (as opposed to global) use of inferentially loaded connective terminology distinguished in the discussion between 1.19.4 and 1.19.5 – see p. 94. For formulas A and B, and ∧-classical the formula A ∧ B of course qualifies as their conjunction. (A somewhat weaker notion is available for gcr’s, as was noted in 1.19: a pair of formulas may have a conjunction-set according to a gcr


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, as defined in the discussion immediately preceding 1.19.5, without having a conjunction according to .) A typical example of occasionally available local conjunction is provided by the implicational fragment of IL. A dull range of cases are provided by A, B, for which B is an IL-consequence of A, thereby giving A itself the status: conjunction of A with B. More interestingly, for any A and B, the formulas A → B and (B → A) → A have (B → A) → B as a conjunction, even if neither ‘conjunct’ follows from the other. (See the Digression on p. 1068.) If we change from IL to CL (retaining the restriction to the → fragment) this example simplifies to: A → B and (B → A) → A (or A ∨ B, as we might now abbreviate it) have B as their conjunction – this last part of the simplification, courtesy of Peirce’s Law. Since we need not construe the language of in the concrete manner (mentioned in 1.11), we do not need this language to combine A with B by means of some particular word or group of words directly corresponding to English “and” (say) in order for it to provide (according to some favoured ) conjunction. When it is said, as it frequently is, that this or that natural language “lacks conjunction”, it is usually some claim as the latter that is being denied: see, for example, Gazdar and Pullum [1976], Hutchins [1980], p. 47, and—for a logically interesting discussion of some of relevant data—Gil [1991]. Numerous further references to natural languages apparently permitting no sentence embedding (coordinate or subordinate) may be found in Section 5 of Pullum and Scholz [2009]. The following remark from Bryson [1987] exhibits something like Mill’s surprising confidence in being able to tell the difference between a house and a street, or rather their semantic analogues in the analogy he is making. Bryson’s book is a style guide and at this point (p. 182) the author is advising against packing too much into a sentence: When an idea is complicated, break it up and present it in digestible chunks. One idea to a sentence is still the best advice that anyone has ever given on writing.

Unfortunately, the advice in question is only intelligible to the extent that one can make a determinate count of the number of ‘ideas’ expressed in a given sentence. (A similar difficulty attaches to evaluating the claim, widely repeated on the internet at the time of writing, to the effect that average person has about 60,000 thoughts per day.) To conclude, we reformulate the ‘collective equivalence’ characterization from 1.19 of what it is for formulas A and B to have a formula C as their conjunction according to a consequence relation into valuational terms. We define A and B to have a conjunction w.r.t. a class V of valuations for the language under discussion just in case: ∃C∀v ∈ V v(C) = v(A) ∧b v(B), where, as usual, ∧b is the standard conjunction truth-function. Evidently if is determined by a class of valuations according to which a given A and B have C as their conjunction in the sense that C serves as a witness for the existential quantification above, then C is a conjunction of A and B according to (in the sense that A and B are collectively equivalent to C according to ). We have


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called such formulations syntactically local because they looks at the existence of formulas C satisfying this condition for this or that pair of formulas rather than globally demanding that for every pair there should be such a C. But the inset formulation above draws attention to a semantic dimension of global/local variation; the condition given is of the ‘global’ form ∃C∀v ∈ V , demanding a single choice of C work for every choice of v from V . A variation allowing local valuation-by-valuation dependence of C on A and B: ∀v ∈ V ∃C v(C) = v(A) ∧b v(B), deserves at least brief consideration. Observation 5.13.4 For any language L with A, B ∈ L and any set V of valuations for L, the local condition just stated is satisfied. Proof. If V is empty, the condition is vacuously satisfied, so suppose v ∈ V , with a view to finding C such that v(C) = v(A) ∧b v(B). If v(A) = F, then choose C as A itself, and if v(A) = T, choose C as B instead, verifying in either case that we have v(C) = v(A) ∧b v(B) as required. The core of the this proof shows that for every valuation v any pair of formulas A, B, have a valuationally local or v-dependent conjunction To get some feel for the general situation, we conclude with an exercise which steps outside of the confines of the “and” theme for this chapter: Exercise 5.13.5 (i) Show that valuationally local disjunctions are equally free for the asking: that for any valuation v for the language under consideration, and any formulas A, B of this language, there is a formula C serving as their ‘valuationally local’ disjunction in the sense that v(C) = v(A) ∨b v(B). (ii) Show that for any valuation v for the language under consideration other than vT or vF , and any formula A of this language, there is a formula C serving as a ‘valuationally local’ negation of A in the sense that v(C) = ¬b (v(A)). (iii) Is the qualification “other than vT or vF ” needed in (ii)? (iv) Now consider valuationally local implications, is the qualification mentioned under (iii) needed for showing that for a valuation v and any formulas A, B, there is a formula C such that v(C) = v(A) →b v(B)?

5.14

Probability and ∧-Introduction

Our discussion has not questioned the propriety of ∧-classicality as a condition articulating the logical role of and. This condition amounts to a combination of what in a natural deduction approach would be two conditions: the correctness of the rule (∧I) and the correctness of the rule (∧E). But do these rules mirror the logical behaviour of ‘and’ ? Here we will make some remarks about the introduction rule, and in 5.15, the elimination rule. Rundle [1983a] gives the following example in which a sequence of assertions could not naturally be replaced by their conjunction. Someone says “You can’t expect him to be here yet. The traffic is so heavy.” As Rundle comments,


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‘the second assertion gives a reason for the first, and not a comment having the like status which “and” would imply’. Now the ‘implying’ going on here in the envisaged replacement is really implicating, and the implicature is arguably indeed a conventional rather than a conversational implicature. (We need not take sides on this.) That is enough to make the conjunction an inappropriate thing to say. But, to turn this example of oddity into a counterexample to (∧I) for ∧, we need it to take us from a pair of true premisses to a false conclusion. As it stands, there is a problem about the case, in the presence of the word “so”; in the conjunctive sentence, “You can’t expect him to be here yet and the traffic is so heavy”, this word takes on a significance different from what it had in the unconjoined original. Without attempting to analyze what is happening here, let us simply change the “so” to “very” to eliminate the complication. Rundle’s observation remains correct for this case: you don’t convey the traffic’s being heavy as a reason for not expecting whoever it is to be here by now, if you say it with and. (The change we are making is analogous to replacing “she” with “Mary” in order to dispel any worries about the commutativity of conjunction coming from the difference between “I respect Mary and she respects me” and “She respects me and I respect Mary”.) So modified, it would seem that even though the speaker would not have served her purposes by saying “You can’t expect him to be here and the traffic is very heavy”, she would not deny that this followed from what she had said. (Similar contrasts between the juxtaposition of two assertions and the corresponding conjunctive assertion may be found in Carston [1993], and Carston [2002], esp. Chapter 3. The prototype for such observations in the discussion in p. 80f. of Strawson [1952], in which some play is made with the notational similarity of the symbol “ ” – used there in place of “∧” for conjunction – to the period, or full stop, as a punctuation mark. The embeddability of conjunctions within the scope of other connectives marks a serious respect of disanalogy not noted by Strawson, independently of the various pragmatic niceties recalled above; see 5.13.) A more interesting worry about (∧I) comes from the way we glossed the ‘semantics’ division of the syntax/semantics/pragmatics trichotomy in 5.11, and in particular the central core of semantics, supposing that we go along with the received conception of such matters as conventional implicature, focus, contrast, etc., as relatively peripheral. Characterizing that core in terms of the relations language bears to extra-linguistic reality may well strike us as inappropriately buying in to some kind of realist metaphysics. It is one thing to say that (core) semantic matters pertain to truth, leaving it open exactly what account is to be given to this notion of truth, and another thing to say that only an account of truth as correspondence with the mind-independent facts is to be countenanced. For example, in 2.32, we mentioned, in the informal semantics for intuitionistic logic, the idea that having been proved serves as the appropriate explication of what truth (for mathematical statements) amounts to. Outside of a context like that provided by mathematics, in which the canonical way of being in a position to assert something is by possessing a conclusive justification (a proof), the usual move for those wanting to inject such epistemic considerations into our account of language, is to reach for probability theory, under its subjective interpretation. A probability short of 1 is typically the most we expect, by way of degree of belief, on the part of someone prepared to make an assertion with that belief as its propositional content. This leads to the following problem. If one’s degree of belief in A is suffi-


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ciently high to warrant assertion, and likewise for one’s degree of belief in B, one may be in a position to assent to the premisses of an application of (∧I) without being in a position to assent to the conclusion. For example, where Pr embodies the subject’s distribution of degrees of belief, and the value .9 is stipulated as required for assertion, we can have Pr (p) .9, Pr (q) .9, but Pr (p ∧ q) < .9; the assumption here is just that Pr is a function obeying the usual axioms of probability theory. On that assumption, we can construct such ‘counterexamples’ to (∧I) as long as our assertion-threshold (.9 in this instance) is less than 1. We should be a little more specific about what to take as the ‘usual axioms’ here, especially as we need probabilities to be assigned to linguistic expressions rather than to sets of possible outcomes. For definiteness, let us say fix on the language of (most of) §1.1, to be called L here, with connectives ∧, ∨, ¬, , ⊥. We use the relation of tautological consequence in formulating our conditions (‘axioms’) on what is to count as a probability function. Specifically, we say that Pr is a probability function on L if Pr is a function from formulas of L to real numbers in the interval [0,1] such that (P1)

Pr () = 1;

(P2)

Pr (¬A) = 1 − Pr(A), for all A ∈ L;

(P3)

Pr (A ∨ B) = Pr (A) + Pr (B) – Pr (A ∧ B), for A, B ∈ L;

(P4)

Pr (A) = Pr (B) whenever A CL B.

Exercise 5.14.1 For L as above, define the relation P P ⊆ ℘(L) × L by: Γ P P A iff for every probability function Pr and every real number r such that .5 < r 1, , Pr (C) r for each C ∈ Γ implies Pr (A) r. (‘PP’ stands for ‘Probability Preservation’). (i) Verify that P P is a consequence relation. (ii ) Is P P -classical? (Give reasons, here and below.) (iii) Is P P ⊥-classical? (iv ) Is P P ¬-classical? (v) Is P P ∨-classical? (vi ) Is P P ∧-classical? (vii ) Which of the above answers would be different if the reference to .5 in the definition of P P had instead been to 0? (viii) What if .5 were replaced by .999? We have already had occasion to observe, of course, that the answer to question (vi) of this exercise is No. More specifically, while we always have A ∧ B P P A and also A ∧ B P P B, in general A, B P P A ∧ B. Notice that, understanding the above definition of P P as not tied to the specific language L given, for it to be the case that for some binary connective #, # is conjunctively classical according to P P (in the terminology of 1.19), is precisely for it to be the case that for any probability function Pr, and any formulas A, B: Pr (A # B) = min(Pr (A), Pr (B)).


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For, if we could have Pr(A # B) less than the minimum of Pr (A) and Pr (B), this would mean that Pr(A # B) was less than both of them, and we should not have A, B P P A # B; and if we could have Pr(A # B) greater than the minimum of Pr (A) and Pr (B), then we would have a counterexample either to the claim that A # B P P A or the claim that A # B P P B, depending on which out of Pr (A) and Pr (B) was the lower. Not only is it not the case that we can satisfy (for all Pr, A, B) the equation inset above by taking ∧ as #, it is also clear (and well known) that there is actually no function f such that the for all Pr, Pr(A ∧ B) = f (Pr(A), Pr(B)): the probability of conjunctions is not determined by the probabilities of their conjuncts. Returning to the specific case of min, let us say that a connective satisfying the condition imposed above on # is a connective of probabilistic conjunction. We know that ∧, as it behaves according to an ∧-classical consequence relation, is not a connective of probabilistic conjunction, but what is to stop us from extending L by a new connective $ say, expressly stipulated to be a probabilistic conjunction connective? Plenty, as we show in 5.14.3. Some results need to be noted first: Exercise 5.14.2 For any probability function Pr and any formula C for which Pr (C) = 0, define a function Pr C (·):

P rC (A) =

P r(A ∧ C) P r(C)

(i) Verify that, so defined, Pr C (·) is a probability function (i.e., that it takes values in [0,1] and satisfies (P1)–(P4) above.) Note: Pr C (·) is said to arise from Pr by ‘conditionalizing on’ C. Common notations for are Pr C (A) are Pr (A|C) and Pr (A/C), the latter being that employed below, usually read “the probability of A given C”. When Pr (C) is 0, the effect of the above definition would require division by 0, so the probability of A given C, in this case, is undefined. (One could stipulate that he conditional probability is 1 in this case, but then conditional probabilities no longer behave like absolute probabilities: for instance Pr (A/C) and Pr (¬A/C) will have not sum to 1 with this convention in force. An alternative strategy with a considerable pedigree is to take conditional probability as the primitive notion of the theory. See Hájek [2003] and Makinson [2010] for discussion and references.) (ii ) Show that if Pr (¬C) = 0 then Pr (C/¬C) = 0. (iii) Show that if Pr (C/D) = 1 then Pr (C/E) Pr (D/E). (Hint: Assuming the antecedent, it suffices to show that Pr (C ∧ E) is greater than or equal to Pr (D ∧ E), since the inequality in (iii) has both of these divided by the same quantity (namely Pr (E)). From the antecedent, infer that Pr (¬C ∧ D) = 0. This implies that Pr(¬C ∧ D ∧ E) = 0. (Why?) But, omitting some parentheses here, we have Pr (D ∧ E) = Pr (C ∧ D ∧ E) + Pr (¬C ∧ D ∧ E). (Why?) Since the summand on the right is 0, Pr(D ∧ E) = Pr(C∧D∧E), which cannot exceed Pr(C ∧ E). (Why not?))


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(iv ) Conclude from (iii) that if Pr (C/D) = Pr (D/E) = 1, then Pr (C/E) = 1. (v) Show that Pr (C/D) = 1 iff Pr (¬D/¬C) = 1, assuming that both these conditional probabilities are defined. (Hint: Rewrite the latter equation as Pr (¬D ∧ ¬C) = Pr (¬C), and use (P4) to infer that Pr (¬(D ∨ C)) = Pr (¬C), and (P2) to infer that Pr(D ∨ C) = Pr (C); then appeal to (P3),. . . ) We return to the theme of probabilistic conjunction. Suppose that we have such a connective, denoted by $. That is, suppose that the language L above is enriched by this new (binary) operation, and that not only (P1)–(P4) above, but also (P5)

Pr (A $ B) = min(Pr (A), Pr (B))

are satisfied by all probability functions Pr and formulas A, B of the enriched language. Observation 5.14.3 Let A and B be formulas of the language just described and Pr0 be a probability function (all such functions being assumed to satisfy (P 5)) with 0 = P r0 (A) = 1, 0 = P r0 (B) = 1. Then we have: either Pr0 (A/B) = 1 or else Pr0 (B/A) = 1. Proof. Since (P5) and the assumption on A and B implies that Pr 0 (¬(A $ B) = 0, we can conditionalize Pr 0 on that formula; then (P5) gives (1) Pr 0 (A $ B/ ¬(A $ B)) = min(Pr 0 (A/¬(A $ B)), Pr 0 (B/¬(A $ B))) By 5.14.2(ii), the lhs of (1) equals 0; thus (2) Either Pr 0 (A/¬(A $ B)) = 0 or Pr 0 (B/¬(A $ B)) = 0. Next, note that by 5.14.2(ii) again, we have Pr 0 (A/¬A) = 0; therefore min(Pr 0 (A/¬A), Pr 0 (B/¬A)) = 0; so by (P5) we have: (3) Pr 0 (A $ B/¬A) = 0. By similar reasoning, we obtain (4) Pr 0 (A $ B/¬B) = 0. Transforming these equations so that they have 1 on the rhs, we get: (2 ) Either Pr 0 (¬A/¬(A $ B)) = 1 or Pr 0 (¬B/¬(A $ B)) = 1. (3 ) Pr 0 (¬(A $ B)/¬A) = 0; (4 ) Pr 0 (¬(A $ B)/¬B) = 1. By 5.14.2(iv), the first disjunct of (2 ), together with (4 ), gives: Pr 0 (¬A/¬B) = 1; similarly, the second disjunct of (2 ), with (3 ), gives Pr 0 (¬B/¬A) = 1. Since one or other of these equations must hold, given (2 ), we have, by 5.14.2(v), Pr 0 (B/A) = 1 or Pr 0 (A/B) = 1.

Remarks 5.14.4(i) The above observation can reasonably taken as showing that there is no such connective as probabilistic conjunction (4.23). For there is nothing anomalous about a probability function satisfying the conditions on Pr 0 for statements A and B in the hypothesis of 5.14.3, without its being the case that the conditional probability of either statement on the other is 1.


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(ii ) The above proof is a probability-theoretic analogue of the proof of 4.22.1, which was the reason for the choice of the same symbol (‘$’) in both cases. (iii) A notion of inductive probability (or support) was developed in Cohen [1977] according to which the probability of a conjunction (relative to certain evidence) is the minimum of the probabilities of its conjuncts (relative to that evidence): see Cohen’s p. 168. indexprobability!L. J. Cohen’s non-Kolmogorovian The idea is that in paradigmatic evidence-weighing situations, such as legal proceedings, this is how we are best understood as reasoning. The above difficulties for probabilistic conjunction do not make trouble for that hypothesis, since Cohen also challenges other parts of the received theory of probability ((P1)–(P4)) as not correct for such applications. (Cf. also Rescher [1976], and §3 of Hamblin [1959].) A conspicuous feature of the introduction of the consequence relation P P above was that we made use in its definition of the notion of a probability function, itself defined in part—in condition (P4) to be precise—by reference to an antecedently given consequence relation: that of tautological consequence. Of course, the intention here was to present a version of classical probability theory, where probability was assigned to formulas. The appropriate equivalence relation with which to constrain that assignment is accordingly that given by classical sentential logic. Yet we found that the defined consequence relation, P P , by no means coincided with the relation of tautological consequence. The point to notice here is that the only use of the tautological consequence relation, CL , made was in its Fmla-Fmla subrelation, since we demanded only that when each of two formulas followed from the other according to CL , those formulas should receive the same probability. And in this restricted field, the two relations in fact agree. That is, A CL B if and only if A P P B. The divergence comes from the form of the probability-preservation definition of the latter relation. We could change this, so as to arrive after a suitable quantification over all probability functions, back with the relation CL in terms of which they were defined. There are several ways of doing this. For example (following Adams [1975]) we could define the improbability of a formula A, Pr(A) on a probability function Pr, to be 1 − Pr (A), which is of course Pr (¬A), and say that: A1 , . . . , An 1 C iff for every probability function Pr, n Pr(C) i=1 (Pr(Ai ). (Adams’s word for improbability is uncertainty, and he says that when the above condition is met, the argument from the Ai to C is probabilistically sound. Adams [1996] provides a comparative discussion of kindred notions; Adams [1998] gives a textbook treatment of these topics as well as others on relations between logic and probability theory.) Alternatively, we could use a restricted version of probability preservation, looking only at the case where the probability is 1 (‘certainty preservation’): A1 , . . . , An 2 C iff for every probability function Pr such that Pr(Ai ) = 1 for i = 1,. . . ,n, we have Pr (C) = 1. Various definitions using conditional probability are also possible, for example:

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A1 , . . . , An 3 C iff for every probability function Pr and formula B, if Pr (Ai /B) = 1 for i = 1,. . . ,n then Pr (C/B) = 1. There is a certain difficulty about this definition because of the fact that the definition of conditional probabilities by quotients as in 5.14.2 makes them undefined when the probability of the conditioning formula is 0, but various alternative formulations are possible in which conditional probabilities are taken as primitive (with Pr (A) then defined as Pr (A/)) and never undefined: see Leblanc [1983], from which 2 and 3 are taken, for details and references. With such an account in place 1 , 2 , and 3 are all none other than CL . The interest of this is that it is possible to characterize probability functions (assigning values to formulas), conditional or unconditional, which do not make the prior appeal to CL , so that we can regard the theory of probability as furnishing a probabilistic semantics for CL . Various merits of this idea are canvassed in Field [1977]; Leblanc [1983] provides a thorough survey of the area. (For some interesting adverse comments, see Roeper and Leblanc [1999], p. 162.) Note that we do not include amongst various probabilistically defined consequence relations any which by definition treat C’s following from A1 , . . . , An as amounting to C’s following from A1 ∧ . . . ∧ An . For example, we do not regard as a serious contender alongside the definitions of 1 , 2 , and 3 above, the suggestion that this relation holds when for any probability function Pr, Pr (C) Pr (A1 ∧ . . .∧An ), or when for any (suitably understood) Pr, for all B, Pr (C/B) Pr (A1 ∧ . . . ∧ An /B). (See Chapter 5 of Makinson [2005] on these and related matters.) This would be analogous to a characterization, in non-probabilistic semantics, of the validity of the inference from A1 , . . . , An to C in terms of C’s being true whenever the conjunction of the Ai is true (as opposed to: when each Ai is true). It gives every appearance of sweeping the semantics of ∧ under the rug. A final point to note about probabilistic semantics is that there is no particular bias in favour of classical logic involved; if a non-classical probability theory is employed, a non-classical logic will be correspondingly vindicated as appropriate. This has been done for intuitionistic logic, for example, in van Fraassen [1981] and in Morgan and Leblanc [1983]; cf. further Roeper and Leblanc [1999], esp. Chapter 10. There remains something of a puzzle over the point with which we began in this subsection. Each of two statements can have a high probability without their conjunction having a high probability, however numerically precise one cares to make this talk of high probability. Indeed, the same is true for any consequence of the two statements taken together which would not follow from either alone: for example, a conditional and its antecedent. We locate the problem as one for ∧-Introduction (rather than, say, →-Elimination) because of the status of A ∧ B as the deductively strongest consequence of {A, B}: if the problem can be solved here, it is solved for all the other cases. (For example, the probability of C cannot be lower than that of A ∧ (A → C), even though it may be lower than each of Pr (A), Pr (A → C).) The problem is felt most sharply if we consider not so much the willingness to assert each of two conjuncts but not their conjunction, as whether we can believe them but not it. On the ‘normal modal logic’ approach to thinking about belief (doxastic logic: 2.21), this is impossible. Belief is taken to be an all-or-nothing affair, and any rational agent believing each of two things is assumed to believe their conjunction (and indeed


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anything else that follows from them: but restricting the point to conjunctions does not seem as extravagant an idealization). Yet when belief is thought of in terms of subjective probability, it becomes a matter of degree. The above problem about (∧I) arises because there is no way of precisifying the quantitative notion of belief-to-a-degree such that under that precisification, whenever two conjuncts have the favoured status, so does their conjunction. No way, that is, other than fixing on the degree 1: but that would be implausible since very few of the things we take ourselves to believe are such that there is nothing else of which we feel even more confident. We close by mentioning one appealing response to this problem, taken from Jackson [1987]. The key idea is to recognise that the notion of belief is vague. A high degree of subjective probability is required, but exactly how high is simply not determinate, in the same way that to count as tall a person has to have a considerable height—measured in, say millimetres—but exactly what that measure is, is not determinate. Although for any precisification of “believes” (in the form: believes to degree r), it is possible to ‘believe’ A and to ‘believe’ B, without ‘believing’ A ∧ B, we are never happy saying that we believe that A and believe that B but do not believe that A ∧ B, because any such precisification is unfaithful to the vagueness of the concept of belief. Similarly, although for any precisification of “tall” (in the form: is n millimetres in height), a person can count as tall when a neighbour whose height is lower by 1 millimetre does not count as tall, we do not ever describe people whose height differs by so small a margin as differing in respect of being tall. For this analogy to work, we need to know that when Pr (A) and Pr (B) are high, Pr (A ∧ B) isn’t much lower: is sufficiently high still for it too seem as unreasonable as in the tallness case to describe A and B as being believed while withholding this description from A ∧ B. We offer a few words of assurance on this score. If Pr (A) = r and Pr (B) = s, then the lowest possible value Pr can assign to A ∧ B is given by the function μ(r, s) where: (r + s) − 1 if r + s 1 μ(r, s) = 0 otherwise. The point here is the same for probability as for other measures: if you are told that 3/5 of the new recruits are women, and that 2/3 of the new recruits speak Russian, then you can infer that at least μ(3/5, 2/3) = 9/15 + 10/15 – 15/15 = 4/15 of the new recruits are women who can speak Russian, and no smaller estimate of the minimum is possible. Returning to the probability case, we have in general that μ(Pr (A), Pr (B)) Pr (A ∧ B) min(Pr (A), Pr (B)). One salient intermediate value for Pr (A ∧ B) is the product of the values Pr (A), Pr (B), and, as will be familiar, when it takes on that value A and B are said to be ‘probabilistically independent’. Our interest here is in the lower end of the scale however, and the behaviour of the function μ. Notice that if r and s are fairly high, say both being .9, the value of μ(r, s) is .8: again fairly high, but distinguishably lower. On the other hand, if these values are set very high, say at .99, then the value of μ(r, s) is (.99 + .99 – 1 =) .98: a value which is for most practical purposes, and certainly from an introspective point of view,

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indistinguishably lower. As we increase the probability deemed ‘high enough’ for a pair of statements to count as believed (without qualification) by a subject whose degrees of belief that probability records, so the amount by which the probability of their conjunction can fall below that value becomes more and more negligible. This explains why one is in practice never prepared to describe oneself or anyone else as believing a pair of conjuncts without believing their conjunction. Of course, repeated applications of (∧I) will turn negligible differences into significant differences, and we will find ourselves clearly on the other negative side of the (imprecise) border between belief and non-belief. The situation is parallel to that with people’s heights, where a one-millimetre difference cannot separate a tall person from a person who is not tall, but a succession of such differences will surely take us from the determinately tall to the determinately not tall. This concludes our elaboration of Jackson’s treatment of the datum that (∧I) is not probability-preserving. For the benefit of readers familiar with the Sorites paradox and the lottery paradox, we remark that (as Jackson notes) this treatment amounts to representing the latter as a special case of the former.

5.15

‘Intensional Conjunction’ and ∧-Elimination

As the scare quotes are intended to indicate, various authors have had various different ideas, expressed with varying degrees of clarity, when they have spoken of intensional conjunction. One of those ideas, namely, fusion in relevant logic, is far from unclear – whether or not one thinks it deserves to be called intensional conjunction. We will be turning to that in 5.16, after reviewing some work stemming from an older tradition here, as well as looking at some examples of more recent vintage (5.15.1–2). The common thread in writers who use the phrase intensional conjunction is that the meaning – or perhaps, one important meaning – of and is not captured by the usual truth-table account of “∧”. Of course, anyone who thinks there is a special conventional implicature associated with and will agree to this. What we have in mind here is not the view that the truth-table account is part of the whole story about what and means, but rather the view that that account has already taken a wrong step. We will concentrate on the objections surfacing in writings of this genre to the principle that the conjuncts of a conjunction follow from that conjunction. The older tradition mentioned begins with Nelson [1930]. In 5.13 we found Mill refusing to countenance conjunctive propositions, insisting that the conjuncts cannot be joined together. Nelson has something of the opposite problem, gluing them together so tightly that (∧E) is under threat: Naturally, in view of the fact that a conjunction must function as a unity, it cannot be asserted that the conjunction of p and q entails p, for q may be totally irrelevant to and independent of p, in which case p and q do not entail p, but it is only p that entails p. Nelson [1930], p. 447.

As the mention of relevance suggests, Nelson is interested in a notion of entailment not afflicted by the paradoxes of strict implication, which is to say the provability in any normal modal logic of such formulas as (p ∧ ¬p) q and p (q ∨ ¬q), which have made many reluctant to think of strict implication as a formalization of the informal idea of entailment. Nelson is worried specifically


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by the idea that if we were to agree that p ∧ q p we shall by committed by the principle that if A ∧ B C then A ∧ ¬C ¬B to saying that p ∧ ¬p ¬q. What relevant logic after the manner of Anderson and Belnap (2.33) does is to reject the principle here invoked, rather than denying that conjunctions entail their conjuncts. (The latter denial is characteristic of ‘connexive logic’: for references, see the notes to §7.1, under ‘Connexive Implication: p. 1056.) What of Nelson’s suggestion that the conjunction must ‘function as a unity’, though? Again, it may be helpful to recall (from 5.13) Mill’s analogies: an aggregate of houses or of horses not being, respectively, a house or a horse. Having agreed to countenance conjunctive propositions, the mereological analogy should be to deal in quantities of matter, rather than individuals falling under some kind with a principle of individuation. For example, we could consider quantities of mud. The aggregate of two quantities of mud (unlike that of two horses) is another quantity of mud. But from the fact that the first quantity weighs a certain amount, it does not follow that the combined aggregate weighs that amount: indeed, unless the second weighs nothing at all, it follows that the combined aggregate does not weigh that amount. Similarly, it might be held, from the fact that one proposition entails a certain proposition (itself, in Nelson’s example), it does not follow that the conjunctive ‘aggregate’ of that proposition with another will entail that proposition. However, this line would require some independent argument that the appropriate aggregation analogy has “ entails q” parallelling “ weighs exactly amount X ”, rather than “ weighs at least amount X ”. (We return briefly to Nelson in the following subsection, after 5.16.8 and after 5.16.9.) These mereological analogies do not seem, however, to bring out especially well the concerns Nelson was voicing in the passage quoted above, the closing words of which suggest that the problem arises because the second conjunct in the conjunction p ∧ q isn’t doing any work in entailing p. It’s as though one were to say that Bill and John are carrying the bookcase when Bill is carrying the bookcase and John is just looking on. Highlighting this parallel, Foulkes [1972] suggests that we accustom ourselves to saying such things as “p with q entails r”, and that we acknowledge a quasi-prepositional sense of and, paraphrased by such a “with” construction, alongside the usual connectival sense. The “p”, “q”, here stand in place not of sentences, of course, but of names—names of sentences or names of propositions. We may note on this score a vacillation in the passage quoted earlier from Nelson: the reason given for why we can’t in general say that “the conjunction of p and q entails p” – note the singular form of the verb entails – is that when p and q meet a certain condition (vaguely expressed in terms of irrelevance and independence), “p and q do not entail p”: this time the plural form of the verb appears (“do” rather than “does”). Our way of marking this distinction—taking it to apply to linguistic expressions (formulas or sentences)—has been in terms of the distinction between the claim that p, q p and the claim that p ∧ q p, the interchangeability of commas on the left and occurrences of “∧” being precisely the mark of ∧-classicality (for ). (We return to this matter of singular vs. plural, at the end of 5.16.) Since Nelson takes these as equivalent also, his objection already arises, prior to a consideration of the behaviour of conjunction, to the general principle – courtesy of (R) and (M) that for all A, B: A, B A. And what this suggests is that he is best construed as meaning something different by ‘entails’, perhaps the relation ∗ defined in terms of a consequence relation by:

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CHAPTER 5. “AND” Γ ∗ A iff Γ A and for no proper subset Γ of Γ: Γ A.

While there is no harm in calling our attention to such relations, they are not especially convenient for logical work, ∗ typically not being substitution-invariant (not that ∗ is even a consequence relation) even when is; for example, taking as CL , we have p, p → q ∗ q, while p, p → p ∗ p. See Woods [1969] for further remarks in this vein on Nelson’s work. On something of a lighter note: Example 5.15.1 Gahringer [1970] opens with the following remarks, suggesting that (∧E) is fallacious: The commonest of fallacies – one so common that it has no standard name – is committed on the assumption that from the truth of the conjunction of p and q one can always infer the truth of p or the truth of q independently. This is the fallacy indicated in condemning an argument as one-sided, or as resting on a part truth or oversimplification.

One should perhaps be careful to postulate fallacies so common as to have escaped baptism. What we have here is a confusion of semantic and pragmatic considerations. (See also Williamson [2007], note 17 on p.96.) A similar situation arises when a partisan in some conflict is asked by an interviewer, whether, for instance, the Palestinian suicide bombings (or martyrdom operations) in Israel are cases of morally unacceptable violence. The interviewee is typically reluctant to answer affirmatively, preferring to say something along the lines of “Both the Israeli army’s conduct of the occupation and the suicide bombers’ response are equally cases of morally unacceptable violence.” Simply assenting to one conjunct, as an affirmative answer would amount to doing, is thought to risk the audience’s overlooking the other and drawing asymmetrical conclusions as to the merits of the disputing parties – giving a one-sided picture, as Gahringer puts it. There is no ground here for supposing the interviewee to fail to appreciate that from the truth of the conjunction, the truth of each conjunct follows. Indeed, this is precisely what the interviewee is stressing. Our final example of what presents itself as a failure of (or threat to) the and elimination rule is taken from Hampton [1988a], where an account of conjunctive concept formation is given, which its author (see especially Hampton [1997]) as it happens calls an intensional approach – though here we will not be explaining exactly what such a description amounts to. (It certainly does not invoke what philosophers typically have in mind under the heading of intensionality.) Example 5.15.2 More conveniently than quoting from Hampton [1988a], where the example of the screwdriver is buried in a table (in Appendix D), let us look at the description of this example given in Prinz [2002], p. 57: Consider the conjunctive concept tool that is also a weapon. Intuitively, everything that falls under this concept should be both a tool and a weapon. Subjects in Hampton’s experiment judged differently. For example, they said that a screwdriver is a tool that is also a weapon, but denied that a screwdriver is a weapon!


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Hampton calls the phenomenon here described overextension of conjunctive concepts, though he conspicuously fails to consider the possibility that the relative clause construction involved (“that is also a weapon”) in fact yields a conjunctive concept, or to raise the possibility of an explanation that subjects, trying to make sense of the unusual wording, construed as meaning that a screwdriver is a tool that can also be used as a weapon, or “pressed into service” as one – to use the wording from Rips [1995], p. 82, where this point is made – which is not enough for it to count as a weapon properly speaking. There are, however, other cases in Hampton [1988a] for which a similar diagnosis may not be available (Chess as a game that is also a sport but not a sport sans phrase, for example), though again the earlier point applies: any evidence pointing to a failure of and -elimination would seem to be evidence against the suggestion that the relative clause construction is not to be interpreted conjunctively. Pertinent here would be asking the experimental subjects: Is a screwdriver both a tool and a weapon? (Is chess both a game and a sport?)

5.16

Conjunction and Fusion in Relevant Logic

An axiomatization of the implicational fragment of the relevant logic R (in Fmla) was given early in 2.33 (beginning on p. 326); the later parts of that discussion involved plenty of sequent calculus in multiset-based frameworks, but what follows has been organized not to presuppose familiarity with that. (For the more flexible multiset-based natural deduction approach, see 7.25.) Continuing to work in Fmla, as in the former discussion, let us say that a binary connective # satisfies the principle of importation according to a logic in a language with → if the downward direction of the following two-way formulato-formula rule is admissible. Portation: A → (B → C) (A # B) → C and that # satisfies the principle of exportation if the upward direction of this rule is admissible. Clearly in intuitionistic (and a fortiori in classical) logic ∧ satisfies both portation principles, which can be thought of together as a Fmla version of the residuation conditions given in the discussion following 4.21.7, p. 544 (# having → as residual). But exportation for ∧ breaks down in relevant logic, since then we could pass from the theorem of (full) R, (p ∧ q) → p to the characteristically absent implicational formula p → (q → p), alias K. We can, however, conservatively extend the implicational fragment by adding a new binary connective usually written “◦” and called fusion, and adjoining to the basis given in 2.33 for that fragment the double-barrelled rule Portation stated with ◦ as #. (This use of “◦” is as in 2.33, and has nothing to do with its usage for composition of functions, most recently seen in 5.12 above.) Exercise 5.16.1 Prove on the basis alluded to above: (i) (p → (q → r)) → ((p ◦ q) → r)


662 (ii ) ((p ◦ q) → r) → (p → (q → r)) (iii) p → (q → (p ◦ q)).

Entries (i) and (ii) here are ‘formula’ (or ‘theorem’) forms of the downward (‘importing’) and upward (‘exporting’) directions of the rule Portation. Of course they are provable for arbitrary formulas in place of the propositional variables; with p ◦ q in place of the third variable in (ii) we derive (iii). In fact, we could simply have replaced the upward direction of the rule by taking the schematic formulation of (iii) as an axiom-schema: A → (B → (A ◦ B)) which says, in the terminology introduced in 2.33, that the fusion of two formulas is successively implied by those formulas (an honour we noted could not be accorded to their conjunction). Digression. That last parenthetical reference to conjunction was to additive conjunction, to put it in the terminology of linear logic, rather than multiplicative conjunction, which is what fusion is – though having weaker properties in linear logic than in relevant logic, thanks to contraction, which gives us A → (A ◦ A). In our discussion of the additive/multiplicative terminology in 2.33 (p. 345 and following), mention was made of Avron [1997], in which it is suggested that we consider a consequence relation extending any of those traditionally associated with relevant logic by having A – and thus also B, since commutativity is not in question – as consequence of A ◦ B (for which Avron uses the “A ⊗ B” notation; also, the underlying Fmla logic, an extension of RM, not only A → (A ◦ A) but also the converse schema). After all, as long as consequence relation has the Modus Ponens property that an implication together with its antecedent have its consequent as a consequence, if for we have, as above: A → (B → (A ◦ B)) then we also have A, B A ◦ B; so if we also had A ◦ B A and A ◦ B B then our multiplicative conjunction would stand a fighting chance of being regarded as the ordinary ‘extensional’ or ‘truth-functional’ conjunction connective, a status traditionally reserved for its additive cousin ∧. Note that it is not being suggested that should satisfy the condition on the left here (for all A), or else it would satisfy the ‘exported’ version on the right: (A ◦ B) → A

A → (B → A).

Accordingly cannot satisfy the Deduction Theorem condition. The interested reader is directed to Avron [1997] and further references there given for a full discussion, which includes a semantic characterization of the proposed consequence relation, as well as some interesting remarks on the additive/multiplicative distinction and some adverse remarks on additive ∧ (which he writes as &) in these substructural logics. We return to the simpler setting of relevant logic in Fmla. End of Digression. The (upper) premiss of Portation says that C is successively implied by A and B; its conclusion, going downward, then says C is provably implied by A ◦ B. So not only is the fusion of two formulas amongst those successively implied by them, it is the deductively strongest formula with that property, provably implying all others. As a corollary, we have:


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Observation 5.16.2 The connective ◦ is uniquely characterized in terms of → by the above basis for the {◦, →}-fragment of R. A historical note: fusion originated in the weaker system E (see 9.22.1, p. 1303), in which the → is thought of as expressing (relevant) entailment, i.e. (speaking only approximately here) necessary relevant implication, by the definition A ◦ B = ¬(A → ¬B). The intended reading was thus of a kind of consistency of A with B. The Portation principles are not satisfied by ◦, so defined, in E. On the other hand, the above definition holds as an equivalence in (full) R with fusion, since each of the following provably implies and is provably implied by its successor, in R: (0) (A ◦ B) → C (1) A → (B → C) (2) A → (¬C → ¬B) (3) ¬C → (A → ¬B) (4) ¬(A → ¬B) → C. The point at which the (1)–(4) part of the above chain breaks down in E is the transition between (2) and (3), since that system does not have the schema Perm (or C, in the B, C, K etc., nomenclature) in unrestricted form – only in the special case in which the formula to be permuted out to the front is an →-formula. We return to the {◦, →}-fragment of R. Exercise 5.16.3 Show, by sketching axiomatic proofs (rather than by using the sequent calculus presentation from 2.33), that in R we have all instances of the following schemata provable: (i) (A ◦ B) → (B ◦ A) (ii ) (A ◦ (B ◦ C)) → ((A ◦ B) ◦ C) (iii) A → (A ◦ A). Exercise 5.16.4 For which of the above schemata is the converse schema also derivable in R? (Ponder this before proceeding.) The answer to 5.16.4 is that only in the case of (iii) is the converse schema not forthcoming in R. By Portation, that converse would be equivalent to A → (A → A) which is the distinctive Mingle schema used (in 2.33) to extend R to RM. Thus in RM, fusion is a semilattice connective – commutative, associative and idempotent (see 3.34, beginning on p. 497) – while in R itself, this last property fails. This was one of the reasons Urquhart’s semilattice semantics for R, reviewed in 2.33, came as a surprise when it appeared. More accurately, we should say: Urquhart’s semilattice semantics for the {∧, →} fragment of R. For the notion of validity defined did not coincide with provability in R when formulas containing ∨ or ¬ were also taken into account. (See 6.45.) The clause governing ◦ in the definition of truth suggested in Urquhart [1972] has the same form as [Interval ∧ ] from 5.12, explicating fusion in terms of the semilattice operation (for ‘combining’ pieces of information) in the same way as that clause explicates temporal conjunction in terms of concatenation of intervals: For any semilattice model (S, ·, 1, V ), any x ∈ S:


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x |= A ◦ B iff for some y, z such that x = y·z, y |= A and z |= B. Exercise 5.16.5 Check that the Portation rules preserve validity on this semantics. The close connection forged by the above clause between ◦ and · leads us to expect the idempotence of the latter to have some untoward effects, and indeed the R-unprovable formula ((A → B) ∧ (C ◦ A)) → B does emerge as valid on the present semantics, for if x |= A → B and x |= C ◦ A then x has the form yz (i.e., y · z) for y verifying C and z verifying A, and so xz |= B. But xz = (yz)z = y(zz) = yz = x, so x |= B, showing the above formula to be valid. (This example is adapted from Humberstone [1988], q.v. for some further discussion.) It would be interesting to know if there are semilattice-valid but R-unprovable formulas in only the connectives → and ◦. What remains surprising about Urquhart’s treatment is that the idempotence of · manages to avoid validating the converse of the schema in 5.16.3(iii) or its exported form, the Mingle schema. (Recall that a condition, in 2.33 called Q0 , on Valuations is employed to deal with RM – in that subsection, in the implicational natural deduction system RMNat.) We avoided touching on natural deduction formulations of R and RM in 2.33 because of a complication about how the commas on the left of “ ” were to be interpreted. On the one hand, we may think of them conjunctively, and on the other, as signalling a prelude to conditional proofs (think of them additively or multiplicatively, that is, as it was also put in 2.33). That is, we can think of A1 , . . . , An B as the analogue either of the formula (1) or of the formula (2) (1) (2)

(A1 ∧ . . . ∧ An ) → B A1 → (A2 → . . . → (An → B). . . )

Now that we have ◦ in the language, governed by the Portation rules, we can state this contrast more crisply, since we can rewrite (2) as (2 )

(A1 ◦ . . . ◦ An ) → B

Thus, the distinction is between interpreting the commas conjunctively and interpreting them ‘fusively’. Actually there is a slight problem with the latter remark since officially our commas on the left in Set-Fmla build sets of formulas, not sensitive to the distinction between one and several ‘occurrences’ of an element, whereas A ◦ A and A are not, as we have noted, equivalent in R, though they are in RM. For this reason, we will now contrast ∧ and ◦ as they appear in natural deduction rules for RM. That is, for expository simplicity, we remain in Set-Fmla rather than getting involved with multisets and the framework Mset-Fmla; in fact, the applications (5.16.6–7) of this proof system do not trade on the multiple occurrences issue, and we can get away with avoiding multisets because of the admissibility of the structural rules called Contraction and Expansion in 2.33 and 7.25; see the latter subsection (the discussion leading up to 7.25.23 on p. 1116) for natural deduction in the more general framework Mset-Fmla (and the former for sequent calculus in Mset-Fmla0 and MsetMset).


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Digression. The following is not a suitable way to proceed if other connectives – and in particular, ∨ – are to be taken into account, for which purposes it is better to allow two kinds of grouping or “bunching” (as indicated in Dunn [1986], Read [1988], by commas and semicolons) for formulas on the left, conjunctive and fusive. (For example, this allows a proof of the Distribution principles.) Here we stick with the fusive or multiplicative interpretation, as in 2.33, and our commas are what would be written as semicolons in the more discriminating notation just mentioned. End of Digression. The following rules for ∧ and ◦ are adapted from (Mset-Fmla) rules appearing in Avron [1988], Troelstra [1992] (p. 57); the appropriate rules for → are those given for RMNat in 2.33: in other words, just the familiar rules (→I) and (→E) of Nat. This is for the sake of simplicity: we can prove the Mingle schema using these rules, as noted in the discussion after 2.33.1, as well as the Contraction schema. The only primitive structural rule is (R). We have added a subscript “rel ” (for “relevant”) to the name of the ∧ introduction rule to avoid confusion with the (∧I) of Nat:

(∧I)rel

(◦I)

ΓA

ΓB

Γ A∧B

ΓA

Δ B

Γ, Δ A ◦ B

(∧E)

(◦E)

Γ A∧B

Γ A∧B

ΓA

ΓB

Γ A◦B

Δ, A, B C

Γ, Δ C

Note that the rule (◦I) is like Nat’s (∧I) and is precisely the ‘intuitionistic’ (i.e., no multiple right-hand sides) form of the linear logic sequent calculus rule (◦ Right)ms governing multiplicative conjunction, except that the Greek capitals here range over sets of formulas rather than multisets, so the present rule incorporates contraction (enabling us to prove A◦AA for any A). Similarly (∧I)rel looks like the rule (∧ Right)ms for inserting additive conjunction on the right in the sequent calculus for linear logic in 2.33, but again in the latter case here “Γ” stands for a set rather than a multiset of formulas. As already mentioned, we are trying to stick with Set-Fmla here in order to minimize inessential novelty for those not accustomed to working in Mset-Fmla. (See 7.25 for natural deduction without this simplification.) The additive aspect of (∧I)rel requires each conjunct to depend on the same set of assumptions before it applies. This difference would come to nothing if (M) were available, which it is not. (Because of this restriction on the ∧-introduction rule, the old assumption-rigging route – discussed in 1.23 and 1.24 – to deriving (M) is blocked.) We illustrate this proof system in action with a proof of 5.16.1(i). The annotations are done in roughly the style of Lemmon [1965a] (cf. 1.23.1, the word “Assumption” signaling an appeal to (R)).

Example 5.16.6 A proof of (p → (q → r)) → ((p ◦ q) → r)


666 1 2 3 1, 2 1, 2, 3 6 1, 6 1

(1) (2) (3) (4) (5) (6) (7) (8) (9)

p → (q → r) p q q→r r p◦q r (p ◦ q) → r (p → (q → r)) → ((p ◦ q) → r)

Assumption Assumption Assumption 1, 2 →E 3, 4 →E Assumption 6, 5 ◦E 7 →I 8 →I

In 5.16.2, we observed that Portation uniquely characterized fusion in terms of →. Now that we have some machinery for doing logic without →, we can drop this relativization. Observation 5.16.7 The above natural deduction rules uniquely characterize ◦. Proof. Where ◦ is a binary connective governed by rules as above for ◦, we have: 1 (1) A Assumption 2 (2) B Assumption 1, 2 (3) A ◦ B 1, 2 ◦I 4 (4) A ◦ B Assumption 4 (5) A ◦ B 3, 4 ◦ E This proves A ◦ B A ◦ B, the converse sequent being proved similarly. (The provability of sequents C D and D C suffices for the synonymy of C and D in this system. Note that this is all really just a re-working of the observation made in the discussion after 4.21.7, p. 544, to the effect that a binary connective and a residual thereof uniquely characterize each other.) We recall from the discussion preceding 4.31.1 that the rules for ∧ also uniquely characterize that connective. So we have two separate connectives each uniquely characterized by the rules governing it, but not (as in the case, for example, of intuitionistic and classical negation reviewed in 4.32) collapsing into each other. We can prove A ∧ B A ◦ B, though not the converse sequent. Note: this means that conjunctions are deductively stronger than fusions (of the same formulas) – it does not mean that the rules governing conjunction are stronger (in the sense of 4.32) than those governing fusion, which would entail the mutual collapse just mentioned. It is of course not the case that when σ is a sequent involving ∧ and σ results from σ by replacing occurrences of ∧ by ◦, the provability of σ implies that of σ ; for example, take σ as: p ∧ q p. (∧ is not, in the terminology of 3.24, a subconnective of ◦ in this logic.) Digression. The provability of A ∧ B A ◦ B in the present system is a parochial feature on which not too much stress should be laid. In Linear Logic the two types of conjunction—additive (∧) and multiplicative (◦) as they are usually called in that context be called—are independent. See the discussion in 2.33. End of Digression. In view of these points about uniqueness, the credentials of ◦ as a potential new logical constant are excellent, though the intelligibility of the ◦/∧ distinction depends no doubt on more sympathy for the enterprise of relevant logic


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than many theorists can muster. A ‘new’ connective? Not according to Read [1988], in which it is maintained that English and is ambiguous, having an extensional meaning, captured by ∧, and an intensional reading, captured by ◦ (in a suitable relevant logic, no doubt weaker than R, let alone RM). But this does not seem very plausible. In the first place, as long as the sentences S1 and S2 are unambiguous, one simply does not hear “S1 and S2 ” as ambiguous, even after being introduced to the logical properties of fusion (though of course one could train oneself to create a new sense for and by reading “◦” that way); in the second place (Burgess [1990]) even those who claim—not being persuaded by the Gricean considerations aired in 5.11—that there is an ambiguity, would be hard pressed to accept that there is a sense of and on which from “S1 and S2 ” the conjuncts S1 , S2 , do not follow. For the would-be fusion sense of and, of course, not only would both conjuncts not follow, neither would follow. (However, this is also the situation for the ‘temporal conjunction’ connective, ∧ , of 5.12, interpreted by [Interval ∧ ], so the parallel with Urquhart’s clause for ◦ may show this consideration to be not so powerful after all.) For the case of BCK logic, or more specifically of fuzzy logic, where multiplicative conjunctions do imply their conjuncts but are not implied by the corresponding additive conjunctions, the suggestion has been made (Novák and Lehmke [2006], note 5) that the additive form is more suited to phrasal conjunction and the multiplicative to sentential conjunction – but without much in the way of evidence (or, it might be added, of plausibility). For the specifically sentential case, no convincing example of a lexical ambiguity in the word and has been provided. The term “lexical” is inserted used to here to stress that structural (e.g. scope) ambiguities are not to the point. Vague talk of different uses of the word and (temporal conjunction, etc., as at the start of 5.12) usually modulates on inspection to a pragmatic rather a semantic treatment with genuinely different senses being postulated. Even if “S1 and S2 ” is unambiguous, one could still maintain that such expressions as “S3 follows from S1 and S2 ” exhibit an ambiguity which is somewhat like a straightforward scope ambiguity: a structural ambiguity rather than a lexical ambiguity. After the “follows from S1 and from S2 ” sense has been weeded out, we still have the distinction mentioned in 5.14 between singular and plural forms of the verb which comes out if we replace “follows from” by “is implied by”, and then take the active rather than passive form: “S1 and S2 implies S3 ” vs. “S1 and S2 imply S3 ”. The first of these needs an extra pair of quotation marks, one before “S1 ” and another after “S2 ” in order to give it a singular subject. The occurrence of and here is mentioned rather than used. Let us consider the second. An ambiguity still remains, according to (one tradition in) relevant logic, depending on whether the grouping of S1 with S2 as doing the implying is taken fusively (‘intensionally’) or conjunctively (‘extensionally’) – ‘multiplicatively’ vs. ‘additively’ in the terminology of linear logic (cf. 2.33: but note that the equivalence of these glosses is contested – see the Digression on p. 662). But acknowledging this still leaves us with no reason to regard the plain “S1 and S2 ” as ambiguous. Rather than enlisting and to perform the new function, or claiming that it already does so, the better course for those convinced of the intelligibility and utility of fusion – ably defended in different ways in Read [1981] and Mares [forthcoming] – would be to urge that the current sentence-connecting resources of natural languages be enriched. There is no plausibility to the claim that the word “and” (or its translation in another language) in such sentences as “Roses are red and violets are blue” (or their

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translations) is itself ambiguous. Exercise 5.16.8 In the following quotation from Read [1981], p. 305, Read’s use of “×” has been replaced by “◦” for conformity with the present notation: ‘That is, the joint truth of p and q suffices for that of p ◦ q, whence “◦” expresses some sort of conjunction.’ Discuss. In the paper just cited, Read harks back to Nelson [1930], citing with approval Nelson’s suggestion that for a conjunction (Read would say: for an intensional conjunction) to entail something, the joint force of the conjuncts is required. This was clearly the upshot of the passage quoted from Nelson in 5.15. There is a sense in which the provability of A ◦ B C in a relevant logic (Set-Fmla framework, natural deduction approach) does record the existence of a derivation of C from A and B in which each of A and B is used. But this is compatible with the provability of A C, whereas Nelson’s discussion suggests that C’s following from A taken by itself shows that C does not follow from A and B. Example 5.16.9 For a case in which A ◦ B C and also A C are provable in a relevant logic (and the example holds for any of the usual systems, including, for instance, R), in which in addition A and B are not equivalent, we may take A = C = p → q, B = p → p. This example is related to the observation from 5.15 that Nelson’s logic is not closed under Uniform Substitution, which shows that it is not any of the relevant logics, with ◦ for “and”. (As we recall from the Appendix to §1.2 – p. 180 – in some people’s opinion, this feature makes it no kind of logic at all, in fact.) For the above choice of A ◦ B C is a substitution instance of the sequent (p → q) ◦ (r → p) r → q in which omitting either of the fused formulas does make the sequent unprovable. On the other hand, it may be most charitable to disregard those unattractive aspects of Nelson’s discussion which lead to the failure of substitution-invariance, and with Read, to extract the suggestive passages and regard them as inchoate anticipations of the emergence of fusion in relevant logic. (Similarly, logics of ‘connexive’ implication—see the notes to 7.19—can be thought of as having Nelson [1930] as a precursor.) The point remains, however, that the aspects disregarded could be brought to bear on conjunction (∧, that is) and on fusion. Relevant logic affords us two notions of what it is for C to follow from A and B (to take just the two-premiss case). As has been mentioned, in more detailed treatments than we have been able to provide (Dunn [1986], Read [1988], for example) these are distinguished by writing a comma (‘extensional’ premisscombination) or a semicolon (‘intensional’ premiss-combination) between the “A” and the “B”. In terms of Urquhart’s semilattice semantics, the distinction emerges as one we might call static vs. dynamic consequence. (The latter terminology is not intended to recall the dynamic semantics for conjunction mentioned in 5.12 except in respect of the vague connotation of an involvement with application; see Slaney [1990] on this.) In the former case, we are asking whether, given the truth of A at x and B at x, C must be true at x. We stay put at the same semilattice element, in other words. In the latter (dynamic) case, we are asking whether, given the truth of A at x and B at y, C must be true at the product xy. And in either case, we can ask whether the hypothesis


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concerning B’s truth (say) was redundant – Nelson’s question. A ∧ B follows statically from A and B, and whatever follows statically from A and B follows statically from A ∧ B; A ◦ B follows dynamically from A and B and whatever follows dynamically from A and B follows dynamically from A ◦ B. The idea of two kinds of ‘bunching’ of formulas went into something of a remission with the dramatic emergence of linear logic from Girard [1987] onward, which treats these matters completely differently, as we saw in 2.33 with the deployment of the multiplicative/additive contrast and a sensitivity to the exact handling of multisets in premiss-sequents as one passes to a conclusion by one of the rules, depending as to which side of that contrast is pertinent. There have since been some signs of a revival, however, e.g., in O’Hearn and Pym [1999], Pym [2002] (mentioned above at 2.33.11(i)); these authors combine a distinction in types of bunching with the usual additive/multiplicative discipline of linear logic. Now that Girard’s name has come up, we should consider what might seem to be an ∧/◦ ambiguity claim similar to that made by Read. On one, perhaps uncharitable, interpretation, Girard is making the same claim – though explicated against the background of a weaker substructural logic than R, namely linear logic – viz. that there is an additive/multiplicative (as we now say, rather than extensional/intensional) ambiguity in “and” as a sentence connective, the two senses being represented by what we have been writing as ∧ and ◦. The more charitable interpretation would be that the examples Girard gives are simply suggestive of the different behaviour of these two connectives in linear logic (in either its classical or its intuitionistic form – CLL or ILL from 2.33). Let us look at one of those examples, and do so with Girard’s own notation: ∧ and ◦ appear as & and ⊗, and → as . The following passage is from p. 2 of Girard [1995]; the material also appeared in Girard [1993b], beginning on p. 65 (as well as in more or less the same wording – though with “superposition” for the “superimposition” appearing here – in Girard [1989], beginning on p. 73): In linear logic, two conjunctions ⊗ (times) and & (with) coexist. They correspond to two radically different uses of the word “and”. Both conjunctions express the availability of two actions; but in the case of ⊗, both will be done, whereas in the case of &, only one of them will be performed (but we shall decide which one). To understand the distinction consider A, B, C: A:

to spend $1.

B:

to get a pack of Camels.

C:

to get a pack of Marlboro.

An action of type A will be a way of taking $1 out of one’s pocket (there may be several actions of this type since we own several notes). Similarly, there are several packs of Camels at the dealer’s, hence there are several actions of type B. An action of type A B is a way of replacing any specific dollar by a specific pack of Camels. Now, given an action of type A B and an action of type A C, there will be no way of forming an action of type A B ⊗ C, since for $1 you will never get what costs $2 (there will be an action of type A ⊗ A B ⊗ C, namely getting two packs for $2). However there will be an action of type A B&C, namely the superimposition of both actions. In order to perform this action, we have first to choose which among the two possible actions we want to perform and then do the one selected. This is an exact analogue of the computer instruction if . . . then . . . else


670

. . . ; in this familiar case, the parts then . . . and else . . . are available, but only one of them will be done. Although & has obvious disjunctive features, it would be technically wrong to view it as a disjunction: the formulas A&B A and A&B B are both provable (. . . )

Now while this passage opens by tells us that the multiplicative and additive conjunction connectives of linear logic “correspond to two radically different uses of the word ‘and’,” what follows doesn’t support this at all, which is why the attribution of an ambiguity thesis for sentence-connecting “and” to Girard may be regarded as uncharitable. The example simply tells us how to think of the logical behaviour of the two connectives, as in the case of an analogy with chemical reactions two pages later in Girard [1995] (or one page later in Girard [1993b]); they illustrate the different form of the rules as well as the inappropriateness, in these applications, of the structural rules of weakening and contraction. In neither case do the schematic letters stand in place of statements; for instance, in the passage above, we are told to think of them as representing action-types. Sometimes expositions of the additive/multiplicative contrast in the literature on linear logic do give glosses in which statements occur in these positions, presumably out of the understandable feeling that this is appropriate for the formal analogues of sentence connectives – by contrast with Girard’s “times” and “with”, which can’t link sentences anyway – as in the Okada [1998], p. 255f., in which the cigarettes are replaced by other purchases, and we are to consider the following transitions (which would be represented as horizontal transitions in the sequent calculus of 2.33 – though see the comments below): C A D B

C A C B

C, D A ⊗ B

C A&B

Okada continues: When we apply “If one has $1 then one gets a chocolate package” and “If one has $1 then one gets a candy package” to the two premises of the left inference rule for ⊗, then we can conclude “one has {$1, $1} one gets (a chocolate ⊗ a candy), which means “If one has two $1’s (namely, $2) then one gets both a chocolate package and a candy package a the same times”, and if we apply the same two premises to the right rule for &, we can conclude “one has $1 one gets both (a chocolate & a candy)”, which means “If one has $1 then one gets either one of a chocolate package and a candy package as you like”. The original idea of having full sentences flank the conjunction connectives is in place initially (“one has $1”, “one gets a chocolate package”) but soon dissolves into the bizarre “a chocolate ⊗ a candy”, “a chocolate & a candy” constructions – to say nothing of “one has {$1, $1}”, where the braces are presumably intended for multiset formation. Sticking with an explicitly sentential format, the only only way of interpreting the implications “If one has $1 then one gets a chocolate package” and “If one has $1 then one gets a candy package” which has them both coming out true in the situation as envisaged is by taking “one gets” to mean “one can get”. On this reading, though, it does not follow that one can get both


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the candy and the chocolate (for essentially the reason that no plausible modal logic endorses the inference from p and q to (p ∧ q)). We shall encounter much similar modal interference in the discussion of disjunction (§6.1). (Another mistake encountered in informal use of the vending machine analogy is that of supposing that there is a special ⊗-like sense of and according to which “I have $1 and I have $1” literally means “I have $2”.) Let us note, incidentally, that Okada’s inference figure on the left doesn’t really make sense, since (alias → in 2.33) is a binary connective so we can’t have two formulas separated by a comma somehow collectively making up an antecedent. So perhaps we should reconstrue the transitions as having (or as Okada [1999] would write) in place of . Then there is the fact that the schematic “C” and “D” are actually instantiated to the same statement (“one has $1”) in the application. The point could therefore be made using the same schematic letter twice over, and we are face to face with the awkwardness of not having the structural rule of contraction – awkward, that is, for purported illumination of natural language argumentation. (R. K. Meyer has remarked in print somewhere – though unfortunately I have not been able to find the reference – that it is hard to accept the idea of an invalid argument expressed in, say, English, which become valid when one of the premisses is repeated. A more charitable view is taken in §8 of Meyer and Slaney [2002]: “But as our friend Robert Bull put it to us, ‘If I want to use a premiss repeatedly, I can write it down repeatedly’.”) In informal (and probably some published) expositions of linear logic, one sometimes sees the two occurrences of such a schematic letter instantiated, in the case of the first to “One has a dollar” and the second to “One has a second dollar”. This is of course highly dubious, since saying twice over that one has a dollar (i.e., instantiating the schematic letter consistently) is very far from saying that one has two dollars. Linear logic is a beautiful edifice, but the case remains to be made that it throws much light on the behaviour of natural language connectives in general, or conjunction in particular. Recalling that BCI logic constitutes the implicational fragment of the linear logic just under discussion, we conclude with some remarks on adding fusion or multiplicative conjunction to it (i.e., adding ◦ to the language and the Portation rules above). Exercises 5.16.1(i) and (ii) do not depend on contraction (W or Contrac) so they are available in this setting, meaning that in the logic thus extended, writing A B to indicate the provability of A → B and thereby cut down on occurrences of “→” and parentheses, rather than to invoke a particular consequence relation, we have the following: ∀A ∀B ∃D ∀C: A → (B → C) D → C, since for any given A and B we can choose D as A ◦ B. Something weaker can already be said without the addition of fusion (again with BCI as the logic concerned): ∀A ∀B ∀C ∃D: A → (B → C) D → C. Here we have a more ‘syntactically local’ claim, since the promised D is allowed to depend not only on A and B but also on C. Echoing the terminology of 1.19.11, we may read this as the claim that BCI logic provides consequentrelative fusions (‘in the antecedent’), and echoing the notation of that discussion, we can write the promised (pure implicational) D as A ◦C B to register the


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dependence on the consequent C. (This discussion was foreshadowed in slightly different terms in Example 4.24.1.) The key to defining ◦ C , suggested (as in the case of 1.19.11) by the second order-definitions in 3.15.5(v)–(vi) from p. 420 – see also 2.33.8 (p. 343) – is what we call the Veiled Law of Triple Consequents, namely the fact that for all A, B, C, we have: [(A → (B → C)) → C] → C A → (B → C), where, again the is understood in terms of BCI logic. Now, this may look initially like just the Law of Triple Consequents from 2.13.20(ii) from p. 239, where it was noted to be available in BCI logic, here put in the current notation: [(A → C) → C] → C A → C, but on closer inspection you will see that the Veiled Law of Triple Consequents is not of this form, since its leftmost occurrence of “C” is not itself the consequent of a conditional with A as antecedent, but is instead buried inside another implication which has A as antecedent. Now, if we had (the familiar binary) fusion at our disposal, we could ‘unveil’ the new principle as essentially a case of the Law of Triple Consequents, since we could rewrite the left and righthand sides of the veiled version as [((A ◦ B) → C)) → C] → C and (A ◦ B) → C, respectively. This consideration is of no help, however, since we are working without ◦. Nevertheless, proofs in the style of 5.16.6 for the two directions here (in particular the -direction, since the converse is just the BCI familiar Assertion principle) will present no difficulty. (Alternatively, one may use the axiomatic approach, or else the sequent calculus rules for → in linear logic, as described in 2.33.) Thus in any extension of BCI logic may define our consequent-relative fission connective (ternary) ◦ satisfying A → (B → C) (A ◦ C B) → C, by taking A ◦ C B = (A → (B → C)) → C. An easy way to see how the Veiled Law of Triple Consequents fails to be subsumed under the Law of Triple Consequents itself is to take C as f and rewrite → f as ¬ . This delivers the Law of Triple Negation (from p. 304: 2.32.1(iii)) in the latter case, i.e., the equivalence of ¬A with ¬¬¬A, whereas in the former case we get a ‘veiled’ version: the equivalence of A → ¬B with ¬¬(A → ¬B), which is evidently not itself a special case of the Law of Triple Negation. (Since we are in a substructural setting here, we have enforced the f -vs.-F distinction from 2.33 rather than simply writing (as in 2.32) the undiscriminating “⊥”.) Exercise 5.16.10 Here are three formulas written in Polish notation (as explained in the notes to §2.1, p. 269), all of them purely implicational and containing three adjacent occurrences of q: (i) CCCpqqq (ii) CCCrCpqqq (iii) CCrCCpqqq. Which of these formulas can be simplified to implicational formulas of lower complexity by appeal to the Law of Triple Consequents or to its ‘veiled’ form, or to the replacement of one subformula by another equivalent to it by either of these equivalences? Write down, in Polish notation, a simplified BCI -equivalent formula when it exists.


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Notes and References for §5.1 Note that the division headed ‘Restricted ∧-Classicality’ in the notes and references for §4.3 (p. 629) is also relevant to the current chapter. Strong Conjunction. We have not discussed this topic because of its intimate connections with themes beyond the scope of this book (such as the lambda calculus) and because the literature on the topic has been in a state of such flux – each contribution claiming to correct an error in a previously published paper – as not to have settled into any readily summarizable form. Introduced in Pottinger [1980], the idea, very roughly, was to regard a proof as a derivation of the strong conjunction of A with B, from given assumptions, not just when it establishes each of A, B, from those assumptions, but when it derives each of these conjuncts the “in the same way” from the given assumptions: one issue is how to make this phrase precise. Subsequent discussions include López-Escobar [1983], [1981a] Mints [1989], Barbanera and Martini [1994]. It was soon realized that this notion of strong conjunction is intimately connected with the idea of ‘intersection types’ in lambda calculus and combinatory logic. See Alessi and Barbanera [1991], Hindley [2002], Venneri [1994], Bunder [2002a], and references therein. Restall [1994b] is also pertinent (as well as references there given). Another proposal for a special kind of conjunction, not treated in our discussion, and, like strong conjunction, distinguishing itself in respect of the nature of the grounds for the conjuncts (though in a very different way), may be found in Boldrin [1995]. Multiplicative conjunction (fusion) is sometimes referred to as strong conjunction, e.g., in Cintula [2006], Novák and Lehmke [2006], but this is a potentially confusing usage, since while multiplicative conjunction is stronger than additive conjunction in the presence of (left) weakening and the absence of (left) contraction (the BCK or ‘affine’ setting), it is stronger in the presence of contraction and absence of weakening (relevant logic), and the two are incomparable in the absence of both (linear logic). Implicature. Apart from Grice’s own writings, excellent introductions to the topic of conversational implicature are provided by Unit 26 of Hurford and Heasley [1983] and by McCawley [1993], §9.2, with §9.3 devoted to conventional implicature. For a sophisticated treatment of conversational implicature, and a discussion of conventional implicature, see Jackson [1987], especially Chapter 5; a more sceptical stance is taken in Davis [1998], though it should be noted that some fine-tuning of Grice’s own criteria for conversational implicature is already indicated in Jackson’s discussion. (A critique of Grice’s account of conventional implicature has been provided by Bach [1999].) Grice’s use of the idea of conversational implicature to defend the usual truth-table accounts of the meanings of the connectives against the charge that they lead to anomalies, is challenged in Cohen [1971]; materials for a reply are assembled in Walker [1976]; again, Davis [1998] gives a critical discussion of the main examples. See also the discussion (focussed on and ) in Schmerling [1975], and (for various connectives) Gazdar [1979], Chapter 3 and 4, and Edgington [2006]. Pertinent earlier work also includes Lakoff [1971], Fillenbaum [1977]. Fillenbaum is especially interested in such constructions as “Come any closer and I’ll scream”, which do not appear semantically to be conjunctions in that someone saying such a thing would not regard themselves as (unconditionally) committed to “I’ll scream”. See also

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Clark [1993], Franke [2005], and the other references given in the Digression in 6.11 (p. 770) on somewhat similar ‘pseudo-imperatives’ (often regarded as disguised conditionals) with or. Linguistic Studies of Connectives. The coordinate vs. subordinate and sentential vs. subsentential (sometimes called phrasal ) distinctions are discussed in Hutchinson [1982], Chapter 7 of Wierzbicka [1980], and in several of the contributions to Haiman and Thompson [1988]. See also Gazdar [1980], [1981], Gunter [1984], van Oirsouw [1987] and Haspelmath [2004] (especially the editor’s overview chapter); semantically interesting subsentential and-constructions are discussed in Hoeksema [1988] and Krifka [1990]. An overview of coordination in the grammar of English – including semantic and pragmatic considerations – is provided by Chapter 15 of Huddleston and Pullum [2002], of which especially relevant for sentential logic is §2; for subordination, see their index entries under that entry, as well as ‘conjunction, subordinating’. A discussion of some aspects of non-sentential coordination can be found in Wälchli [2005]. See also Barnes [2007], pp. 170–172. The interplay between and and too (with some discussion of but) is described in Green [1973], Kaplan [1984]. Discussions of and which concentrate on pragmatic matters may be found in Lakoff [1971], Schiffrin [1986], and Carston [1993]. A more semantic orientation, hoping to explain which truth-functions are monolexically realized in natural languages, may be found in Gazdar and Pullum [1974] (or see Gazdar [1979]). “But”. Mill [1843], Book I, Chapter IV, §3, gives the ‘contrast’ formulation of what but involves, saying that “Caesar is dead but Brutus is alive” is tantamount to the fourfold assertion: (i) Caesar is dead; (ii) Brutus is alive; (iii) It is desirable that (i) and (ii) should be thought of together; (iv) between (i) and (ii) there exists a contrast. (If we delete (iv), we get Mill’s account of “Caesar is dead and Brutus is alive”). He elaborates on the reference to a contrast in (iv): “viz. either between the two facts themselves, or between the feelings with which it is desired that they should be regarded”. The latter disjunct undermines the oft-heard suggestion that in describing someone as poor but honest one represents oneself as expecting dishonesty from the poor. Dummett [1973], p. 86, makes a similar observation about this example, as well as giving another example of his own: If a club committee is discussing what speakers to invite, and someone says, “Robinson always draws large audiences”, a reply might be, “He always draws large audiences, but he is in America for a year”; the objector is not suggesting that a popular speaker is unlikely to go to America, but that, while Robinson’s popularity as a speaker is a reason for inviting him, his being in America is a strong reason against doing so.

It may be profitable to consider this example with the aid of a distinction between levels drawn in Sweetser [1990], explained in 6.13 below, with but in Dummett’s example signalling a contrast not at the ‘content domain’ level. Sweetser would not put it like this since she is not happy to acknowledge a content domain usage of but, saying at p. 104 of Sweetser [1990]: But what does it mean to say that A and B “clash” or “contrast” in the world? How can discordance or contrast exist outside of the speaker’s


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mental concept of harmony or contrast? In a sense, if two states coexist in the real world (and conjunction with but does present both conjuncts as true), then they cannot be said to clash at a real-world level.

However, this reasoning does not appear consistent with Sweetser’s acknowledgment of a content domain interpretation (as well as an epistemic and speech act interpretation) of although in example (6a) on p. 79 of Sweetser [1990]. Jackson [1987], p. 93, also provides a useful discussion of a variation on Dummett’s example. Somewhat similar to that example are some cases involving “Shaq is huge but he is agile” discussed at the top of p. 344 of Bach [1994]. Bach rejects the theoretical overlay involved in talking of conventional implicature for but – or for anything else – reviving instead the theory of Frege [1892] that a single unambiguous sentence expresses two propositions, in this case one such being that which would be expressed alone by the and -conjunction of the but-linked sentences and the other being that expressed by the following (which is however, not to be taken simply as a further conjunct – Bach [1994], p. 347): “There is a certain contrast between being huge and being agile”. This is no place to discuss the pros and cons of Bach’s proposal, though the point should at least be noted that the semantics and pragmatics of the quantificational “a certain” construction are themselves far from straightforward. (Potts [2005] also presents a ‘double proposition’ account of conventional implicatures.) Under the influence of Grice, one would think of Mill’s (iv) – let alone (iii) – as something which is not actually asserted, but conventionally implicated: its failure makes what is said inappropriate, rather than false. See Dummett [1973], pp. 86ff., and [1991], p. 121f. Dummett uses the term ‘tone’ for what distinguishes the meanings of and and but, as an alternative to the talk of colouring/illumination in Frege [1892]. It is this paper that suggested the postMillian view that what was required for appropriateness (such as (iv) above) need not be required for truth. For qualms about the overinclusiveness of the category of tone, see Rundle [1983a], and §49 of Rundle [1979]. The word but in English may function as a connective in two ways (“as a connective” to exclude the except sense of but), illustrated by the difference between “He isn’t an economist but a businessman” and “He isn’t an economist, but (at least) he is a businessman”, a pair taken from Dascal and Katriel [1977]; see also the discussion in Abraham [1979]. (These papers give examples of languages in which this distinction is lexically marked; however, the former but cannot be followed by a complete sentence—see §6.4.3 in Horn [1989] for this and related points. See any Spanish grammar for the distinction in that language between sino and pero, both translated in English as but, and Anscombre and Ducrot [1977] for the corresponding ambiguity question for French.) Conventional implicatures of expectation and contrast in general are discussed in Tannen [1977] and at monograph length in Rudolph [1996]. (See also our discussion of even in 7.12 and the references there and in the notes to §7.1, under ‘Focus Constructions’: p. 1055, to published literature.) More on “but”, in particular, may be found in Greenbaum [1969], Lakoff [1971] (which contribution is discussed in §4.2 of Sweetser [1990]), Osgood and Richards [1973], §3.4 of van Dijk [1979], Gunter [1984], and Blakemore [1989] and [2002] (see the many index entries in the latter work). Discussions of an archaic usage of but (as in “It never rains but it pours”) are provided by Anscombe [1981], Traugott [1997] and Jennings [1994a] – §6 of this last also including some interesting remarks about but in

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its contemporary usage (and connections with that of or ). We close with an interesting example – the non-commuting “but”s in the paraphrases offered – from Amis [1998], in an entry on “Few” (p. 68): One might have thought that only a foreigner would confuse few with a few, paraphrasable as ‘some but not many’ and ‘not many but some’; but if so one would have been mistaken.

An especially promising attempt to unify several of what have seemed to some to be distinct uses of but, along with many more references to earlier discussions than have been listed here, is to be found in Umbach [2005]. An interim conclusion reached by Umbach (p. 227) is worth quoting here: ‘there is no “concessive but” just as there is no “causal and ” ’. Concessive clauses, to use some traditional terminology, are subordinate clauses introduced by (al)though. (Concessive conditionals, by contrast, involve clauses introduced by even if ; see 7.12 for some discussion of these.) The link, apparently thus contested, between although and but is made especially clearly in West and Endicott [1976], where the entries under although and but equate “S1 but S2 ” with “S2 although S1 ”. Predictably, the proposed equivalence does not fare especially well with examples like that of Dummett about (in which, however, “S1 , but S2 ” would be closer to the mark than “S1 but S2 ”, and the but can be replaced by though without exchanging S1 and S2 . While “but” (and other contrast indicators) may not seem promising as a subject of formal logical study (proof systems, semantics, etc.), ventures into this territory may be found in Francez [1995] and in Meyer and van der Hoek [1996]. Categorial Grammar. Lewis [1972], Geach [1972b], Levin [1982] and Chapter 7 of van Benthem [1986], and Steedman [1996] provide interesting reading on this topic, which originated in Poland with Leśniewski and Ajdukiewicz in the 1930’s. More comprehensive references may be found in van Benthem [1991], Wood [1993]; Chapter 1 of the latter is a good source for information on different notational practices. See also the notes to §8.2 (p. 1251). Temporal Conjunction. 5.12 is based on the opening section of Humberstone [1974]. Readers interested in interval semantics for tense logic are referred to Roeper [1980], van Benthem [1982]. Von Wright’s temporal conjunction connective in his [1965], [1966], [1968], he reads as and next when it indicates immediate temporal succession of one state or event upon another (the only case considered in 5.12) and as and then when there is no suggestion of immediacy (i.e., roughly, for and later ). The claim that and is ambiguous between a purely truth-functional meaning and a temporal meaning (amongst others) may be found in Staal [1968]. Johnson-Laird [1969] notices that the cancellability of the “and then” implicature is diminished with various reduced forms of conjunctive sentences; it is perhaps easier to add “though not necessarily in that order” to “He put on his hat and he left the building” than to “He put on his hat and left the building”. Observations in this vein also appear in Posner [1980]. Why Have Conjunction? Subsection 5.13 is an elaboration of the discussion in Geach [1963]. The point about the ineliminability of conjunction from

5.2. LOGICAL SUBTRACTION

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embedded contexts (especially embedding under negation) may also be found at p. 68 of Grice [1967]. Probability. For discussion of – and references to the literature on – the Lottery Paradox and other aspects of the problem raised by the lowering of probability by deductive reasoning, see Hilpinen [1968]. For a comparative survey of alternative approaches to the mathematical treatment of rational degrees of belief, of which the subjective interpretation of probability theory is one prominent example, see Paris [1994]. In 5.14 we had occasion to refer to Roeper and Leblanc [1999], a book on whose title the authors appear to have had a change of mind. The title given with the publication details inside Roeper and Leblanc [1999] is what is listed in our bibliography as Probability Theory and Probability Logic, whereas on the spine—at least in the printed copy consulted by the present author—the title is appears as Probability Theory and Probability Semantics. The Veiled Law of Triple Consequents. This name was given in 5.16 to the BCI -equivalence shown here (in the notation of that discussion): [(A → (B → C)) → C] → C A → (B → C). I have since been shown an unpublished paper by Andrzej Wroński on BCK identities (equations holding in all BCK algebras) – Wroński [2006] – including the equational version of this equivalence, as well as that of the following generalization of it (which again, is BCI - rather than just BCK -provable): [(A1 → (A2 → (. . . (An → C) . . .))) → C] → C A1 → (A2 → (. . . (An → C) . . .)) Law of Triple Consequents is the n = 1 case, while the veiled version from 5.16 is the n = 2 case. (I am grateful to Matthew Spinks for telling me of the existence of Wroński’s paper. The actual purpose of the paper is indicated by its opening sentence: “The question how to axiomatize the identities of BCK -algebras arose when it became clear that BCK -algebras form a proper quasivariety.”)

§5.2 LOGICAL SUBTRACTION 5.21

The Idea of Logical Subtraction

We begin with a loose but suggestive analogy between arithmetic and sentential logic. An arithmetical equation a = b + c giving the value of a in terms of b and c, can be transformed into an equation b = a − c, giving the value of b in terms of a and c. (So we are taking arithmetic here as the arithmetic of the integers, including the negative integers.) Now suppose that according to some ∧-classical consequence relation , we have an equivalence of the form A B ∧ C. If we assume is also ↔-classical, we may prefer to think of this as recorded in the form A ↔ (B ∧ C). If the formulas involved are formulas of predicate logic, then when certain well-known conditions are met (Suppes [1957], Chapter 8), such an equivalence may be regarded as a definition of a function symbol or predicate symbol in A in terms of the vocabulary of B and C. (We review these conditions, as they apply in the case of defining predicate symbols, in 5.32.)

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Could we not transform such an equivalence into an equivalence—which may or may not meet the conditions on definition—giving B in terms of A and C? Let us write the result of this imagined transformation as B ↔ (A − C), or, in the -formulation, as: B A−C. For the moment, regard the choice of the symbol “−” as just a reminder of the arithmetical analogy. Following the terminology of Hudson [1975], we call what it purports to symbolize logical subtraction. Just as arithmetical subtraction ‘undoes’ the work done by addition, so logical subtraction undoes the work done by conjunction, explaining the appearance of this discussion in the present chapter. (In fact, from the point of view of some algebraic analogies, “logical division” might have been the more appropriate label, since conjunction is often treated as analogous to multiplication. But the “subtraction” terminology is established and in any case will be seen to have its own merits when we come to think of subtraction, or difference, in set-theoretic rather than arithmetical terms.) It appears that quite frequent use is made informally of a device corresponding to the above transformation. For example, suppose we are faced with a foreign speaker who has learnt a little English and who happens to have encountered already the words vixen and female, as well as being familiar with the general logico-grammatical vocabulary of English, but who does not yet know the meaning of the word fox. Then it would seem that we could convey to our learner (whose native language we do not speak) the meaning of fox by saying something along the following lines: something’s being a fox is like its being a vixen except that it does not need to be female. Or we might say: the requirements for being a fox are just the requirements for being a vixen minus the requirement of being female. In the notation of the preceding paragraph, this is an attempt to define “x is a fox” as “x is a vixen − x is female”. An example from the philosophy of science: a lawlike statement is to be understood as something meeting all the conditions for being a law-expressing statement with the possible exception of the requirement that the statement in question be true (Goodman [1965], p. 22). From (universal) algebra: a semigroup is a semilattice, but for the requirements of idempotence and commutativity. We do seem to have some feel for the intuitive idea of subtracting content in this way, and what we shall do in this section is to ask, in an experimental spirit, after the availability of a formal account of the procedure within the somewhat rarefied setting of sentential logic. (Positive suggestions in what follows should accordingly be understood as offered only tentatively.) First, we give another example from predicate logic. Example 5.21.1 Consider the formula: ∀x(Fx )−Fa. What would be a suitable reading of this? Well, what is required for its truth? For ∀x(Fx ) to be true, everything has to be (in the extension of the predicate) F . From this, we subtract the requirement that the individual a be F . So we are left with something amounting to: everything with the possible exception of a is F . Remarks 5.21.2(i) Note that we do not say “everything except a is F ”, which may suggest that a is not F . When we subtract B from A, we are simply removing the requirements for the truth of B, not also adding in the requirements for the truth of its negation. (For a discussion of the meaning of except phrases, see the references cited in connection with that under 7.12.7.)


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(ii ) If we were thinking of adding “−” to the language of first-order logic without identity, then we have, in ∀x(Fx ) − Fa, something otherwise inexpressible, though of course, with the aid of ‘=’, we can express it by: ∀x(x = a → Fx ). The suggestion is, then, that talk involving “dropping the requirement that ”, “with the possible exception of ”, etc., represents the workings of a new and widely overlooked connective, with as much claim as conjunction to be regarded as a logical constant. Since in this work we are not concerned with the general question of when something deserves to be regarded as a piece of logical vocabulary, we confine a defence of this aspect of the suggestion to asking why, if the adding of requirements is deemed to be work befitting a logical device (conjunction), is this not also the case for the dropping of requirements? Of course, all this talk of requirements for truth is so far merely suggestive metaphor; as we will see in 5.24, it is not as easy as one might imagine to cash out the metaphor. Before that, we will have to make some choices about what kind of account we want (5.22). But, before even that, some limitations of the kind of intuitive understanding we have been relying on need to be pointed out. Although with a little coaching, it is possible to see what is intended by some subtraction formulations, such as the examples so far given, in many cases, one would simply be baffled. In the first place, it would not be natural to say S1 − S2 unless S2 followed from S1 . We will return to this in 5.22 (and 5.26). But, even when S2 does follow from S1 , we will often be at a loss as to how to interpret S1 − S2 . The understanding that what is required for its truth is whatever is required for the truth of S1 minus whatever is required for the truth of S2 seems insufficient to bring anything to mind in these cases. Adapting an example from Jaeger [1973], due to R. H. Thomason, take S1 as “a walked slowly into the room” and S2 as “a walked into the room”. The ‘red’ example in the following passage quoted from Jackson [1977] pp. 4–5, also appears in Jaeger’s paper: It sometimes seems to be thought that we can sidestep the question of whether ‘sees’ has ‘success grammar’ or ‘existential import’, by arguing as follows: Let us grant that ‘see’ as used in current English licenses inferring ‘D exists’ from ‘S sees D’. But, for various reasons, this usage is philosophically inconvenient; hence we should conduct our discussion in terms of ‘see*’, where ‘see*’ means just what ‘sees’ means, except that ‘S sees* D’ does not entail ‘D exists’. There is, however, a fundamental problem with such a procedure. Consider someone writing on the secondary qualities who observes that ‘X is red’ entails that X is coloured, and decides to introduce the term ‘red*’ to mean precisely what ‘red’ means except that ‘X is red*’ does not entail that X is coloured. The question such a procedure obviously raises is whether the deletion of the entailment to ‘X is coloured’ leaves anything significant behind. And it is hard to see how to settle this question other than by considering whether ‘X is red’ may be analyzed as a conjunction with ‘X is coloured’ as one conjunct and some sentence, P, not entailing ‘X is coloured’ as the other. If it can, ‘X is red*’ means P; if not, ‘red*’ has no consistent meaning at all.


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This passage raises numerous issues, some of them touched on later. The point of quoting it here is in the first instance to call attention to the fact that we find ourselves at a loss to understand “a is red − a is coloured”. This same feature is presented when the entailments are sanctioned by the familiar logical properties of the connectives. Thus if, after reading the opening paragraphs of this subsection, you were to be asked what to make of “James and Nancy are lawyers − James is a lawyer”, presumably you would without too much hesitation reply that this amounts to “Nancy is a lawyer”. But what about “James was drowned − James or Nancy was drowned”? Presumably, this time, no clear answer would be forthcoming. (In a similar vein, compare the readily intelligible Example 5.21.1 with its dual form: Fa − ∃x(Fx ).) Roughly, we can say that the following. Although there is an entailment from disjuncts to disjunctions just as there is from conjunctions to conjuncts, we do think of the truth of p and q as required for the truth of p ∧ q, whereas we do not think of the truth of p ∨ q as required for the truth of p, so much as just something that would follow from the truth of p. We will not attempt to make anything of this informal suggestion here (though cf. the notion of introductive entailment in 4.13, the connective “⇒” of 5.27.5(ii), as well as Remark 7.12.4(i) on p. 961 concerning the logical application of the necessary/sufficient distinction).

5.22

Four Choices to Make

The above informal remarks leave very much underdetermined the shape of a formal theory of logical subtraction they might be taken to motivate. In the first place, it is not clear, though our discussion has tended to assume, that this formal account should provide the logic (or some range of logics) of a new binary connective. Secondly, if that issue is resolved in favour of such a connectival treatment of logical subtraction, we have to decide whether or not closure under Uniform Substitution is desirable for the resulting logic(s). Thirdly, an analogous choice would need to be made on the question of congruentiality, which brings with it the need to attend to what the underlying logic is, on top of which we envisage the logic of subtraction being superimposed. And finally, there is the question of what to do about the cases in which subtraction seems to make little or no intuitive sense, such as those reviewed at the end of 5.21. First Question: Is a Connectival Treatment Desirable? Well, what are the options? We will consider one possible alternative to taking “−” as a connective, and that is: to think of this symbol as belonging exclusively to the metalanguage, and denoting a function from pairs of formulas to formulas. In other words “A − B” is our name for a certain formula in whatever familiar connectives (∧, ∨, etc.) the object language provides. The role of an account of logical subtraction is then to say, given A and B, which formula this is. Here is a parallel, introduced originally as Example 1.11.1 (p. 49). Suppose the object language we are discussing contains ¬; then we might define the contradictory of a formula A, abbreviated to ctd (A) to be, where A is ¬B for some formula B, the formula B, and to be the formula ¬A should A not be of that form. If we think of languages in what in 1.11 was called the ‘concrete’ way, with formulas as strings of symbols, it is very clear why ctd is not a singulary connective. If you apply a singulary connective to a formula, you get a longer formula: but, if A is, say, ¬(q ∧ r), then ctd (A) is a shorter formula – namely, q ∧ r. It


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may seem that on the ‘abstract’ conception of languages, according to which a connective is just an algebraic operation on the universe of the language (as algebra), it will be hard to sustain this distinction: we certainly won’t be able to appeal to considerations of comparative length of formulas. But recall that 1.11 takes the algebras in question to be absolutely free algebras, ensuring the unique readability of formulas (and justifying the transfer of apparently concrete terminology such as ‘subformula’). In the example just given, this property would fail if we took ctd as a connective: then the formula q ∧ r would be both the result of applying ∧ to q and r, and also the result of applying ctd to ¬(q ∧ r). (See further 1.11.2 on p. 49, and then 1.11.3.) Other (‘naturally occurring’) examples like ctd will come to mind. For instance, let the object language have the connective → and consider the (partial) functions: ant(A → B) = A, cnsq(A → B) = B, cnv(A → B) = B → A. (The abbreviations are intended to suggest antecedent, consequent, and converse.) These are only partial functions since not every formula is of the form A → B; we could make them total by having them behave like the identity function on other formulas. What is the main connective of cnv(p → q)? We said this was to be the formula q → p, so the main connective is →, and cnv is not any kind of connective at all (but rather a non-connectival operation on implicational formulas). If, alternatively, we want to have cnv as a connective, then it will be the main connective of cnv(p → q) but this formula will not then be the formula q → p, on pain of non-unique readability. The best one could hope for would be to have the two formulas equivalent, though one lesson of 1.15 is that unless → is behaving very unfamiliarly, this hope is misguided. (We give an example of such unfamiliar behaviour in the BCIA logic of 7.25.) Like ant, cnsq, and cnv, the most natural proposal for a metalinguistic account of “−” would leave “A − B” undefined for many cases. We could stipulate that “A − B” denotes the formula A in those cases. Suppose that is the relation of tautological consequence, on a language containing all the boolean connectives. Then we might say: if A B ∧ C and ¬A then A − C is the formula B, provided that B and C meet a certain condition. The condition alluded to here is one of independence; we will review the exact form the condition should take, for a related proposal, in the following subsection. Since B and ¬¬B (for example) might both stand in this relation to C, the above definition does not single out a unique formula. But we can say that, relative to some ordering of all formulas, A − C is to be the first formula in the ordering which meets the conditions. That is a sketch of how the ‘non-connectival’ treatment of “−” would go. When we speak, on this account, of the formula (p ∧ q) − q, for example, we are in fact speaking of the formula p, and not of any formula containing q as a subformula. We shall not adopt such a treatment, however, not least because some of the motivating examples (e.g., 5.21.2(ii)) suggest that the interest of logical subtraction lies in the fact that subtractive formulas may not be equivalent to (let alone be identical with) formulas of the subtraction-free language. Indeed this is so with the opening “vixen” example in 5.21; the whole point was to get across the meaning of “fox(x)” by saying “vixen(x) − female(x)” to someone previously otherwise unacquainted with that term: a point which could hardly

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be achieved if, as the informal analogue of the present proposal would have it, in quoting someone as saying “vixen(x) − female(x)” what we are quoting the speaker as having said is simply “fox(x)”. Second Question: Closure Under Uniform Substitution? Having decided to treat “−” as a bona fide sentence connective, we have to decide whether or not the appropriate logic for this connective (and some or all the boolean connectives) should be closed under Uniform Substitution. Since we shall be working in Set-Fmla, this amounts to the question of whether the class of Set-Fmla sequents we wish to accept is so closed, or, equivalently, whether the associated consequence relation is to be substitution-invariant. For continuity with our treatment of the behaviour of other connectives, we will decide this question in the affirmative. But there is a considerable pressure from the current case to go the other way. We have mentioned more than once the possibility of holding (p ∧ q) − q to be equivalent to p. It is out of the question to do this for arbitrary formulas A and B in place of p and q, as will become clear in the following subsection, in which we will experiment with a non-substitution-invariant consequence relation. However, for the reason just given, as well as for the general reason favouring Uniform Substitution – namely, that we wish to treat propositional variables as representative of the class of formulas in general (Stalnaker [1977]), substitution-invariance will be a feature of our more favoured account(s), beginning with that of 5.25, though not of the proposal discussed in 5.23. (See further the Appendix to §1.2, which began on p. 180, and the references given in the notes to that section, headed ‘Uniform Substitution’: p. 191.) Third Question: Congruentiality Examples of non-congruential logics were given in 3.31, though it seems clear that unless there are very good reason to the contrary, a congruential treatment would be preferable to one that is not congruential. We would like our new connective itself to be congruential according to any proposed logic because we don’t want formulas we have agreed prior to its addition to have the same logical content, to turn out to behave differently in its scope. This is not a requirement of conservative extension (which we take without question as a desideratum: see §4.2), though its motivation is a similar distaste for the disruption of prior logical relations. And we would like antecedently familiar connectives, such as conjunction, not to suddenly become non-congruential in the presence of the new connectives (for a similar reason). This of course raises the question of what the ‘prior’ logic is. We generally take it that the strongest case for the acceptance of the status of logical subtraction would be made if its logical behaviour could be handled in a congruential extension of classical logic, though there are temptations to go below this. (One such – in the direction of relevant logic – will be mentioned in the next subsection.) In any case, much of our discussion will concentrate on the case in which the only other connective, aside from subtraction, is conjunction, so that we are in effect able to maintain a certain neutrality on this question, in view of the near unanimity amongst alternative logics on the appropriate behaviour of ∧. Fourth Question: What About Intuitively Unsubtractable Cases? The cases alluded to by this heading are those mentioned in 5.21. There are those occasioned by the presence of disjunction and certain other connectives,


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which we shall simply assume absent in 5.25 and 5.26, and there are those arising from the difficulty of making sense of A − B when B does not follow from A. These cases are probably best settled by abiding by the verdicts of a theory dealing satisfactorily with the clear cases; the accounts of 5.25 and 5.26 return different verdicts on this score.

5.23

Cancellation, Independence, and a ‘Stipulated Equivalence’ Treatment

The analogy with which we started in 5.21 involved thinking of logical subtraction as playing a role in relation to conjunction similar to that played by arithmetical subtraction in relation to addition. But of course conjunction (according to an ∧-classical consequence relation) is not very much like addition (on the integers) to begin with. For example, the latter is not idempotent. That is not an important obstacle to the use of the sign “−”, since it emerged that the real parallel justifying that use was with set-theoretic difference rather than arithmetical subtraction: we are talking about subtracting the content, or set of truth-requirements, of one statement from those of another. Now this places the (idempotent) operation of set union in the position occupied by addition in the arithmetical analogy, and makes conjunction behave like union. But set union and (∧-classical) conjunction of propositions lack a crucial property of their arithmetical prototype. The semigroup consisting of the integers under the operation + is a cancellation semigroup (0.21), while the cancellation laws are not satisfied by set-theoretic ∪ or logical ∧ . And it is only if we have such a cancellation property that we can make sense in the most straightforward way of “−” along the lines the implication with which 5.21 opened: (*) a = b + c ⇒ b = a − c. For such an implication simply entails the cancellation property: b + c = b + c ⇒ b = b , thus: (i)

b = (b + c) − c (Taking a as b + c in (*));

(ii )

b = (b + c) − c (Taking a as b + c, and assuming b + c = b + c);

(iii)

b = b (from (i) and (ii)).

Thus in particular, if we tried to treat logical subtraction by extending classical logic with a principle to the effect that (**)

A B ∧ C ⇒ B A − C

the extension would be non-conservative, since we can have B ∧ C CL B ∧ C without having B CL B . An uninteresting way in which this situation arises is if C is, for example r ∧¬r and B and B are any two formulas; this might prompt a premature move to a weaker logic, and in particular, to a relevant logic. But such a response is really beside the point since we could equally well illustrate the possibility by choosing B = p, B = p ∧ q, C = q, in which case the equivalence of B ∧ C with B ∧ C is contested by no-one, but the cancellation inference would yield p p ∧ q. (Of course we can derive q p ∧ q similarly, and hence conclude that p q.)


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In this last case, it is clear from an intuitive point of view what is going on: we cannot say that ((p ∧ q) ∧ q) − q is equivalent to p ∧ q, since peeling off the outer q from (p ∧ q) ∧ q does not subtract the content of q from the whole formula: it leaves an occurrence of q still sitting as a conjunct in the inner conjunction. What we have learnt (and this point is in Jaeger [1973]) is summed up in: Examples 5.23.1(i) We have an Unacceptable Cancellation Condition, notated here for visibility as though we were dealing with a rule (despite using “” rather than “” – so this is not really a legitimate notation): B ∧ C B ∧ D B B and therefore we cannot generally allow the following Incorrect Subtraction Principle: A B ∧ C A − C B (ii ) We add incidentally that CL does satisfy the following cancellationlike condition (again using ‘rule’ notation): B ∧ C B ∧ C

B ∨ C B ∨ C

B B by 0.13.7(ii) (consider the Lindenbaum algebra) or alternatively, by checking that the corresponding ‘horizontal’ form (p ∧ r) ↔ (q ∧ r), (p ∨ r) ↔ (q ∨ r) p ↔ q is a tautologous sequent. However, this is not of much help to us in finding a ‘correct’ subtraction principle. We should note also that a horizontalizing of the Unacceptable Cancellation Principle in (i) here gives the non-tautologous sequent (p ∧ r) ↔ (q ∧ r) p ↔ q. Before considering how to amend that principle, we digress for a moment on why this sequent is not tautologous. A simple answer to this question is given by the observation that its lhs is equivalent in CL (in IL, even) to r → (p ↔ q), so the inference recorded would amount to just deleting—with, of course, no justification—the antecedent r. With a view to illustrating cancellation phenomena more generally (and not just for the case of conjunction), we prefer to put the point as follows. In terms of boolean valuations, the point comes to this: from v(p ∧ r) = v(q ∧ r), for such a valuation v, it does not follow that v(p) = v(q). Thus, returning to subtraction, there is no question of there being a binary truth-function g such that f (∧b (x, y), y) = x; here, as in 3.14, we have written ∧b for the truth-function associated with ∧ over the class of ∧-boolean valuations (the x, y, of course range over the set {T, F}). In general, where f is a binary truth function, let us recall some definitions from 3.14.16 (p. 415), according to which a binary truth-function g is a first-argument reciprocal function for f, or a second-argument reciprocal function for f, respectively, if for all x, y: g(f (x, y), y) = x or for all x, y: g(f (x, y), x) = y.


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Exercise 5.23.2 List the 16 binary truth-functions. Say which of them have a first-argument reciprocal function, which have a second-argument reciprocal function, and what those reciprocal functions are for the cases for which they exist. Remarks 5.23.3(i) Note that g is a first-argument reciprocal function for f iff for all y the singulary functions gy and fy defined by gy (x) = g(x, y)

fy (x) = f (x, y)

have the identity truth-function for their composition. Thus (0.21.1(ii): p. 19) gy is a left inverse of fy , gy is a surjection, and fy an injection. (We could give an analogous description in the case of second-argument reciprocals.) (ii ) The above definition of reciprocal functions need not be restricted to truth-functions; in particular, if we apply it to the case of binary operations in the Lindenbaum algebra (2.13) of a logic, then we can say that the rule corresponding to the Unacceptable Cancellation Condition in 5.23.1(i) is admissible in the logic if and only if the operation corresponding to ∧ has a first-argument reciprocal function (which need not be a fundamental operation of the Lindenbaum algebra). As (i) and (ii) here suggest, we might also consider singulary connectives # and the cancellation property #A #B ⇒ A B, possession of which is necessary and sufficient for a conservative extension by adding a ‘left inverse’ connective #−1 such that #−1 #A A (for all A). We discussed the failure of this condition for ¬, with as IL in 4.21.1(i) on p. 540; for as CL , we may take ¬−1 as ¬, of course. (Note that the “ −1 ” notation is a bit misleading since we are only considering the availability of a left inverse; the right inverse question was addressed in 4.21.1(ii) – see p. 540.) Williamson [1988a] provides some discussion of cancellation for in normal modal logics; see also his [1990], as well as Humberstone and Williamson [1997], Williamson [1998b]. Left and right inverses for were discussed in the notation “L”, “R”, above in (and before) 4.22.24 (p. 565). We return now to the question of obtaining an ‘acceptable’ form of the Unacceptable Cancellation Principle from 5.23.1(i). We should think in terms of some relation of independence holding between B and C and also between B and C. If we can find such a relation, which when added as a proviso to the Unacceptable Condition allows us to infer that B and B are equivalent, then we will be able to apply it to the B and C figuring in the Incorrect Subtraction Principle to make sure that when A B ∧ C and we ‘subtract’ C from A we will get (to within equivalence) the same answer (A − C B) as we would get had we chosen a different formula (B ) instead of B for which the same hypothesis (A B ∧ C) also held. The semi-formal literature on logical subtraction contains allusions to the required notion of independence. In the quotation from Jackson [1977] in 5.21, there was the reference to “considering whether ‘X is red’ may be analyzed as a conjunction with ‘X is coloured’ as one conjunct and some sentence, P, not entailing ‘X is coloured’ as the other.” Of course here the difficult notion of analyzability is in play, which involves more than mere logical equivalence, but, ignoring this, the suggestion is that before we can make sense of subtracting “X is coloured” from “X is red”, we would need to have


686 X is red X is coloured and P

and only a weak notion of independence is employed, namely that P not entail “X is coloured”. More strongly, we could insist that before concluding from the fact that A B ∧ C that A − C B, we must have B and C independent in the stronger sense that neither entails the other. Jaeger [1973] suggests an independence requirement still stronger, demanding that in addition, the negation of neither should entail the other (that B and C should not be ‘subcontraries’ to use the traditional vocabulary: of course it is enough to pick either B or C and say that its negation should not entail the other, obtaining the same effect for ¬-classical ; in fact as he formulates it, Jaeger’s demand is that A − C and A should not be subcontraries). One might go even further, and add that neither B nor C entails the negation of the other. Imposing this strongest possible notion of independence from the spectrum of inferential notions of independence amounts to saying that all combinatorially possible truth-value combinations are logically possible: true-true, true-false, false-true, false-false. But even with this strongest restriction in force—call it complete logical independence—there is trouble for cancellation, and hence for subtraction, as we may illustrate by taking as CL again: Example 5.23.4 Whereas (for = CL ) we have (p ↔ q)∧q p ∧ q, we do not have p q. Yet p ↔ q and q are completely logically independent, and so are p and q. Thus subtracting q from the left hand conjunction in accordance with the Subtraction Principle of 5.23.1(i) with the added complete independence proviso yields something not equivalent to what we get if we subtract q from the right hand conjunction. (Note: we could equally well have chosen p ∧ (p ↔ q) q ∧ (p ↔ q) to make the point, subtracting in this case p ↔ q from both sides.) We turn, however, to an emendation of the Cancellation Condition which will serve the purposes of this ‘stipulated equivalence’ treatment of logical subtraction, as we may call it since we are seeking a condition on B and C which will allow us safely ( = conservatively) to stipulate that B ∧ C (or some equivalent A thereof) minus C is to be equivalent to B. We can find a clue from the relationship between the conjuncts p ↔ q and p of Example 5.23.4. What relationship, you may ask? For we chose p ↔ q, rather than, say, p → q (which also, when conjoined with p, gives something tautologically equivalent to p ∧ q), precisely to ensure the complete logical independence of the conjuncts. But although these formulas are thus independent, so that (as far as boolean valuations are concerned) the truth-values they receive are independent, the ways in which they get these truth-values are not independent. This is reflected syntactically by the fact that (or: is a reflection of the syntactic fact that) they have a propositional variable in common. If we insist that our conjuncts B and C are instead variable-disjoint (share no propositional variables), then their complete logical independence is ensured provided that they are neither tautologous nor inconsistent (have tautologous negations). It turns out that only the second part of this proviso needs to be imposed, and that, only on the conjunct to be cancelled, for an amended version of the Unacceptable Cancellation Condition. Acceptable Cancellation Condition:


687

B ∧ C B ∧ C B B Provided (i) C is consistent, and (ii ) B and C share no propositional variables, and nor do B and C. At least if we forget about Proviso (i) – see below – we can think of the above condition as a rule, which is the reason for the horizontal line (as in 5.23.1) rather than using ⇒ between the -statements. The premiss sequents, set out in rule format have the form B ∧ C B ∧ C and B ∧ C B ∧ C; we may take the conclusion sequent to be simply B B , in view of the symmetry between B and B in the premisses. In calling the condition ‘acceptable’, we mean simply that this rule is admissible in classical sentential logic (in Set-Fmla): Observation 5.23.5 The above Acceptable Cancellation Condition is satisfied taking as CL . Proof. Rewrite the ‘premiss’ B ∧ C B ∧ C as C B ↔ B ; note that the biconditional here shares no variables with C, given Proviso (ii), so, since C is consistent (Proviso (i)), the result follows by 2.11.8 (p. 205). The proof just given draws on the Shoesmith–Smiley cancellation condition, which involves ‘cancelling’ premisses (lhs formulas in a -statement), and can be applied to obtain a result concerning the cancellation of conjuncts: Corrected Subtraction Principle: A B ∧ C A − C B Provided (i) C is consistent, and (ii ) B and C share no propositional variables. Again, we should like to think of this as a rule in Set-Fmla for extending classical logic. The problem hinted at above à propos of proviso (i) arises here too. Since we think of rules as giving closure conditions on the class of provable sequents, we do not want a formulation alluding to consistency, which amounts (for A) to the negative claim that ¬A is not provable—or equivalently, given the present logic, to the claim that for some B, the sequent A B is not provable. The simplest way around this difficulty for present purposes is to work in a language in which such inconsistency cannot arise. The positive proposals we shall make on the logic of subtraction in the following subsections actually restrict attention to a language containing only ∧ and −. However, we could, still avoiding the presence of (CL -)inconsistent formulas, allow also (for example) ∨, →, and ↔. So let 0 be the relation of tautological consequence restricted to formulas in these three connectives, as well as ∧ and −. (Thus we have, for instance, p − q 0 p − q.) Then the sense in which we have ‘corrected’ our earlier subtraction principle is given by 5.23.6, whose proof (using 5.23.5) we again omit. Observation 5.23.6 The smallest consequence relation extending 0 and satisfying the above Corrected Subtraction Principle is a conservative extension of .


688

We shall not pursue further this idea of stipulating the equivalence of A − C with a formula B when the appropriate variable-disjointness relation holds between B and C in the conjunction B ∧ C antecedently given as equivalent to A, since this very condition precludes substitution-invariance for the resulting consequence relation. As was mentioned under that heading in our discussion (5.22) of choices to make, we will not in general be able to have (A∧B)−B A, whereas on the present account, we do have, taking A = p ∧ q, B = p, C = q, from the Corrected Subtraction Principle: (*)

(p ∧ q) − q p.

Now, as indicated by the references cited in the notes to §1.2 (‘Uniform Substitution’, p. 191), failures of substitution-invariance in proposed logics are by no means unheard of. At the end of the Appendix to that section, we observed that failures of closure under (uniform) variable-for-variable substitution are less defensible; but just such failures arise here. For example, substitute, q for p in (*) above. Such anomalies aside, rather than stipulating what is to be equivalent to what, it would seem preferable to have some systematic account of how subtraction works, on the basis of which such equivalences were explained. (Also, as stressed in 5.22, under ‘First Question’, we want to be able to use subtraction to increase our expressive power, rather than to give long-winded equivalents of things expressible without its aid.) It is to the task of providing such an account that we turn in 5.24. Exercise 5.23.7 A propos of the complete logical independence (in CL) of p and p ↔ q above, the following remark was made: “But although these formulas are thus independent, so that (as far as boolean valuations are concerned) the truth-values they receive are independent, the ways in which they get these truth-values are not independent.” For example, if we have a boolean valuation v with v(p) = F, then one a priori possible way of arranging matters so as to have v(p ↔ q) = T, namely by setting v(p) = v(q) = T, has been ruled out. Characterize a notion of partial boolean valuation which gives boolean valuations in the special ‘totally defined’ case, and for which you can show that if u and v are such partial valuations and A and B are variable-disjoint formulas with u(A) = x, v(B) = y (x, y ∈ {T, F}), then there exists a partial valuation w extending each of u, v, for which we have w(A) = x and w(B) = y. (This cashes out the somewhat metaphorical “ways they get these truth-values.” In saying w extends u we mean that for any formula C with u(C) defined, w(C) is defined and is equal to u(C).) Remark 5.23.8 Although one can make precise the talk of ways of assigning truth-values, along the lines indicated in 5.23.7, whether two formulas are variable-disjoint in the first place is a somewhat superficial feature of the relation between them, in the sense that if we think of interpreted (sentential) languages, one can give intertranslatable languages in which variable-disjointness is not preserved under translation. The relevant observations are due to Max Black and David Miller: see the notes to this Section, which begin on p. 707. (For further discussion of the bearing of this on logical subtraction, see Humberstone [2000d].)


5.24

689

Content Subtraction and Converse Implication

Concentrating on the relation between conjunction and logical subtraction with which we began in 5.21, we can see the need for some notion of the content of a statement, thought of as a set representing the set of requirements for its truth. We may distill from that informal discussion the principles (∧)

Content(A ∧ B) = Content(A) ∪ Content(B)

(−)

Content(A − B) = Content(A) Content(B)

The “” on the rhs of (−) is of course for set-theoretic difference (‘relative complement’): an operation itself often—especially formerly—denoted by “−” itself. Now all we need to know is what the sets concerned should contain. But before even contemplating that question, we can see that some principles concerning the relations between conjunction and subtraction ought to hold good: (Subt. 1) A A − B (Subt. 2)

A−A

(Subt. 3)

A − B, B A

(Subt. 4) A (A ∧ B) − B The first of these is acceptable, because if whatever requirements there are for the truth of A (‘Content(A)’) obtain, then so do whatever requirements are left over when we delete such as belong to Content(B). In the case of (Subt. 2), we are left with the empty set of requirements, which cannot therefore contain any which fail to obtain. For a neater parallel with (Subt. 4), we could write (Subt. 3) as: (A − B) ∧ B A. Here we have taken the content of B away from A and then put it back again, which could give us something stronger, but certainly can’t give us something weaker, than A back again. We can put all these considerations into explicitly set-theoretical terms, as we do here for (Subt. 4). The general point is that for any sets X and Y : X ⊇ (X ∪ Y ) Y . Thus, less cannot be required for the truth of A than for the truth of (A ∧ B) − B. Replacing “⊇” by “⊆” would be wrong. Some of X—Content(A), in the present instance—may also be in Y (Content(B)), in which case it will be deleted from X ∪ Y . We saw this above in the demonstration of the unacceptability of the two-sided version A (A ∧ B) − B of (Subt. 4). Now (Subt. 1)–(Subt. 4) conspicuously possess the following feature. If we replace every occurrence of “A − B” by “B → A”, we have conditions which must satisfy to be an →-classical, indeed even merely an →-intuitionistic, consequence relation. Further, there is an appealing general argument—whose conclusion we shall contest presently—for the claim that far from being some interesting new connective, logical subtraction is simply the converse of (some kind of) implication. As Hudson [1975] puts it, A − B, what we get by taking away the content of B from that of A, ought to be the weakest statement which, together with B, entails A. That, together with B, A − B entails A, appeared on our list above as (Subt. 3). What about the claim that A − B is the weakest such statement? As in 4.14.1 (p. 526), this means that for any C, if B, C A then C A − B. This is a form of (→I) or the rule of Conditional Proof, with


690

C in place of Γ. And (Subt. 4), again reading the “−” as an “→” in reverse, is a horizontalized form of (→E) or Modus Ponens. Hudson announces this as showing logical subtraction to be the converse of material implication. More modestly, we do have – give or take the worry about “Γ” vs. “C” – rules here making subtraction the converse of intuitionistic implication. (On that worry: if is ∧-classical and -classical, we can trade in Γ for the conjunction of its elements, taking this as when Γ = ∅.) Further, by unique characterization considerations as in §4.3, if any connective # is behaving →-classically according to , then A # B and A → B will be equivalent according to , so this correction to Hudson is irrelevant given only the intelligibility of material implication. Incidentally, for the uniqueness result, we do not need the presence of ∧ and −: Exercise 5.24.1 Show that (→E) and Hudson’s specialized form of (→I): From C, A B to C A → B are already strong enough to characterize uniquely the connective they govern. (Assume that (R), (M), and (T) are available.) Remark 5.24.2 Jaeger’s non-subcontrariety requirement (mentioned in 5.23) is not met by the converse of material implication, a point noted by Hudson, who argues that the requirement was undermotivated. Note that for intuitionistic implication, the requirement is satisfied. (Johansson [1953] refers to dual intuitionistic implication, last seen in 4.21.9 (p. 547) above, as subtraction; note that despite the terminology of 4.21.9 and elsewhere, this connective is really the converse of the dual of intuitionistic →.) However, even according to IL , the formula B → A does not behave in at all the ways our informal discussion in 5.21 leads us to expect A − B should behave. For example, even though in the interests of simplicity we have been keeping negation out of the picture, while we have ¬B IL B → A, nothing in that discussion leads us to expect that A − B should follow from the negation of B. From the fact that not every requirement for the truth of B obtains, how is it supposed to follow that every requirement for the truth of A which isn’t also a requirement for the truth of B obtains? (Alternatively, moving from intuitionistic to classical logic and from negation to disjunction, we have CL B ∨ (B → A): but we don’t want to say that either B or else A − B must be true.) Again, in 5.21 it was suggested that “a is a vixen − a is female” should be taken as equivalent to “a is a fox”. But if the former is understood as “a is female → a is a vixen” then it will follow, for example, from “All females are vixens”, from which “a is a fox” certainly does not follow. (Like the previous point this does not depend on specifically classical as opposed to intuitionistic implication.) Provisionally, then, we should try for a logic of subtraction which does not identify this connective with a converse implication, at least not one behaving intuitionistically (and so, a fortiori, not one behaving classically). One great merit of the converse implication proposal is that it gives us a systematic account of the semantics of subtraction, whether by →-boolean valuations as on the material implication version of the proposal, or by Kripke models (for example) as on the intuitionistic variant. The resulting logics are both congruential and closed under Uniform Substitution, making for simple and tractable objects of study. What we should like, then, is a similarly systematic


691

account which does not yield the untoward consequence just noted for this proposal. This returns us to the question of how to understand Content(A). We will be very unambitious, and continue to work in Set-Fmla over a language whose only primitive connectives are conjunction and subtraction. This will allow us to ignore the delicate question of how to extend the list of conditions (∧), (−) above, with analogous conditions for disjunction, negation, and the other connectives. (The delicacy of the situation with regard to disjunction was mentioned in 5.21; we do not naturally think of the truth of A ∨ B as something required for the truth of A.) Several formalizations of the notion of content have been offered in the philosophy of science, for explicating the idea of strength of hypotheses or informativeness of theories. For works by Popper and by Carnap in this vein, see the notes to this section. Here we mention a Popper-inspired measure of content, in line with the thought that the informativeness of a statement varies directly with the number of situations it excludes as incompatible with its truth. We could take over the Kripke models from §2.2, but forgetting about the accessibility relations, and say that relative to any one such model M = (W, V ), the content of A is the set {x ∈ W | M |=x A}. This gives (∧): Content(A ∧ B) = Content(A) ∪ Content(B), as desired, in view of the shift from truth-sets to falsity-sets. (If we take truth-sets, then conjunction aligns with intersection, of course, rather than union.) So let us see what we get for Content(A − B) by (−): Content(A − B) = Content(A) Content(B) = {x ∈ W | M |=x A&M |=x B} Thus, to have A − B true at w ∈ W , it is necessary and sufficient that w does not belong to the set mentioned on the right, which is to say, that either M |=w A or else M |=w B, which is to say, finally—and disastrously—that A − B is true at a world iff B → A is, where → is for material implication. For the reasons already given against identifying logical subtraction with converse material implication, we must look elsewhere for a representation of content. (A more helpful ‘possible worlds’ perspective will come up in 5.27.) Another measure of the content of a statement might be given by what follows from it. So suppose that we have a consequence operation Cn 0 defined on the subtraction-free fragment of our language, and we try to extend this to the whole language exploiting the fact that if (the consequence relation associated with) Cn 0 is ∧-classical, then we very nearly have the analogue of (∧): Cn 0 (A ∧ B) = Cn 0 (A) ∪ Cn 0 (B). As remarked at the end of 4.22, where we saw the oddity of this equation (for a hypothetical connective $ in place of ∧), the rhs is not guaranteed to be closed under Cn 0 , whereas the lhs is; a corrected form would be: Cn 0 (A ∧ B) = Cn 0 (Cn 0 (A) ∪ Cn 0 (B)) (Note that for brevity we write “Cn 0 (A ∧ B)” rather than “Cn 0 ({A ∧ B})”.) In the same way, we should not expect for the proposed extension—call it Cn 1 —of Cn 0 to the class of formulas in which “−” may appear between subtraction-free formulas: Cn 1 (A − B) = Cn 0 (A) Cn 0 (B) but rather the ‘closed’ version:


692

Cn 1 (A − B) = Cn 0 (Cn 0 (A) Cn 0 (B)) which bears some resemblance to (−). There is still a problem, however, since Cn 1 so defined won’t be a consequence operation; for example, we have not guaranteed that A−B ∈ Cn 1 (A−B). Finally then, and now at some considerable remove from (−), we stipulate that Cn 1 is the least consequence operation extending Cn 0 and satisfying, for all subtraction-free A and B Cn 1 (A − B) ⊇ Cn 0 (Cn 0 (A) Cn 0 (B)). This process is repeated to obtain a consequence operation Cn 2 on the set of formulas in which “−” occurs within the scope of (at most one occurrence of) “−”, and so on. So perhaps we have here a construction which simultaneously explicates of the idea of content and precipitates (as the limit of the Cn i ) a logic of subtraction. The logic precipitated, however, leaves much to be desired. In the first place, the consequence relation is not substitution-invariant. For example, we have p as a consequence (by Cn 1 ) of p – q, but we do not generally have (as indeed, we should not wish to have) A as a consequence of A − B. On the other hand, perhaps the non-invariant theme, in which the propositional variables are treated specially, distinct variables behaving like disjoint units of content, does not deserve the dismissive rejection we have given it. Of course, you may say, p – q should be a consequence of p: these two should be equivalent, since there is nothing in the content of p being subtracted here. Earlier, we saw that this way of thinking supports the view that (p ∧ q) – q and p should be equivalent; this was used to show the non-invariance since the same could not be said for (A ∧ B) – B and A in general. So, secondly, by way of objection to the above proposal, we note that it cannot do justice to this thought. One thing that follows (Cn 0 ) from p ∧ q but not from q, and therefore which ought, on the present treatment, to follow (Cn 1 ) from (p ∧ q) − q, is the formula p ∧ q itself, which in turn has q as a consequence, exactly what was not wanted from (p ∧ q) − q in a treatment in which the distinctness of propositional variables is given this special status. This last example suggests a variation; we leave it to the reader to ponder. By Cn −1 we mean the operation taking a formula A to {B | A ∈ Cn(B)}, for a consequence operation Cn. Exercise 5.24.3 Discuss the variation on the above proposal which results by replacing the extension condition Cn i+1 (A − B) ⊇ Cn i (Cn i (A) Cn i (B)) with Cn i+1 (A − B) ⊇ Cn i (Cn i (A) Cn −1 i (B)).

5.25

Requirement Semantics: First Pass

Rather than trying to reduce the notion of content to other familiar notions, we shall now simply give a formal version of the intuitive discussion of 5.21, in which requirements for the truth of formulas are taken as primitive. As in the preceding subsection, we restrict attention to the language with connectives ∧ and −. The basic ingredient in the current, more abstract, development is what


693

we shall call a requirement structure,by which we mean a pair (R, R+ ) of sets with R+ ⊆ R. If R = (R, R+ ), then we call the elements of R ‘requirements’ and describe those belonging to R+ as the requirements which obtain according to R. Note that we have not demanded that R+ (or even R) be non-empty. An R-assignment is a function R from formulas to subsets of R, satisfying the following conditions, to which we have given the same names as their informal prototypes from 5.24: (∧) (−)

R(A ∧ B) = R(A) ∪ R(B) R(A − B) = R(A) R(B)

Finally, for a requirement structure R = (R, R+ ) and an R-assignment R, we R defined a valuation vR (more explicitly we should say vR ) by vR (A) = T ⇔ R(A) ⊆ R+ Exercise 5.25.1 Check that for any R-assignment R, the valuation vR induced as above by R is an ∧-boolean valuation; conclude that the consequence relation determined by the class of all such valuations is ∧-classical. As usual, we say that a sequent (of Set-Fmla) A1 , . . . , An B holds on a valuation vR whenever if vR (Ai ) = T for all i(1 i n), then vR (B) = T. In the present semantics, what this amounts to is: if all the requirements in R(A1 ) ∪ . . . ∪ R(An ) obtain, then so do all the requirements in R(B). A sequent is valid when, for all requirement structures R it holds on every Rassignment. In this terminology, 5.25.1 amounts to saying that all sequents of the following forms are valid: A, B A ∧ B

A∧BA

A∧BB

or, equivalently, that the rules (∧I) and (∧E) preserve the property of holding on an arbitrary assignment-induced valuation vR . For emphasis we shall occasionally refer to this notion of validity as ‘validity on the requirement semantics’. Exercise 5.25.2 Check that all sequents of the following forms are valid; we have retained the labelling as for the corresponding -statements in 5.24: (Subt. 1)

AA−B

(Subt. 2)

(Subt. 3)

A − B, B A

(Subt. 4)

A−A A (A ∧ B) − B

Since we have managed to validate our sample of intuitively correct sequents involving subtraction, and the absence of any special treatment of propositional variables in the requirement structure semantics guarantees that the class of valid sequents is closed under Uniform Substitution, the possibility arises that the present notion of validity is just what we need for the semantics of logical subtraction. In fact, however, all we have succeeded in providing is a new semantics for the {∧, →}-fragment of classical logic in Set-Fmla, with the formula A → B appearing here as B − A. In the preceding subsection, reasons were given for dissatisfaction with any such converse-implication treatment of logical subtraction. However, as the semantics is something of a novelty, we will go to the trouble of showing that the set of sequents it treats as valid does


694

indeed coincide with those in the said CL fragment. Further, by an emendation in the form of the above clause (−), to be presented in 5.26, we can still validate, for example, all the sequent-schemata figuring in 5.25.2, without the collapse into converse material implication. (This description is an oversimplification even for the present—unmodified—version of the semantics, since we certainly don’t arrive at converse material implication in Set-Set: see below.) Indeed from a more careful examination than will be given in the present subsection, it will turn out that even the unmodified semantics can be incorporated into an account of a language featuring all the boolean connectives and visibly not issuing in the feared collapse. This will be explained in 5.27. The rules (→E) and (→I) alongside (∧I) and (∧E) and (R) provide a natural deduction proof system for the {∧, →}-fragment of IL (see 2.32). We have already seen that the ∧-rules preserve validity (in the current sense), as of course does (R). We relabel the → rules as (→I)− and (→E)− to indicate that we are treating them as rules governing “−”, and so with antecedent and consequent reversed: (→I)−

Γ, A B ΓB−A

(→E)−

ΓB−A

ΔA

Γ, Δ B

That (→E)− preserves validity is immediate from the considerations needed to show (Subt. 3) valid in Exercise 5.25.2 above. Subt. 1, 2 and 4 are easily derived with the aid of (→I)− . We deal with this rule in a pair of Lemmas toward a result (5.25.5) claiming that every sequent provable in the present {∧, −}-system is valid. Lemma 5.25.3 If R is an R-assignment for R = (R, R+ ) and R0 ⊆ R, then the function R defined by: R (A) = R(A) R0 , is also an R-assignment. Proof. It is routine to check that the conditions (∧) and (−) are satisfied by R if they are satisfied by R.

Lemma 5.25.4 The rule (→ I)− preserves validity on the requirement semantics. Proof. Suppose that Γ B − A does not hold on R, an R-assignment for R = (R, R+ ). Then for all C ∈ Γ, R(C) ⊆ R+ but not R(B − A) ⊆ R+ . Then for some r ∈ R(B − A), in other words (by (−)), for some r ∈ R such that (1) r ∈ R(B) and (2) r ∈ / R(A), we have: (3) r ∈ / R+ . Now adjust R to R by putting R (D) = R(D) R(A), for all formulas D; by 5.25.3, R is an R-assignment. Since R (D) is, if anything, smaller than R(D), and R(C) ⊆ R+ for each C ∈ Γ, we have R (C) ⊆ R+ . Further, R (A) = ∅, so R (A) ⊆ R+ . But by (1) and (2) r ∈ R (B), so by (3), we do not have R (B) ⊆ R+ . Thus the sequent Γ, A B does not hold on R .

Theorem 5.25.5 Every sequent provable in the proof system with rules (R), (∧I), (∧E), (→ I)− , (→ E)− is valid on the requirement semantics.

5.2. LOGICAL SUBTRACTION Proof. By Lemma 5.25.4 and the discussion preceding 5.25.3.

695

We now have a soundness result for our proof system. Since this is a proof system for conjunction and the converse of intuitionistic implication, from our earlier comment that the semantics renders “−” the converse of material (i.e., classical) implication, it is clear that the converse of 5.25.5 does not obtain. In 1.18 the missing part of the →-classicality condition was given in a form which suggests a rule allowing the passage from Γ, A C and Γ, A → B C to Γ C. Here, we prefer to consider boosting up the above basis by the addition instead of a zero-premiss rule, namely the schematic form of Peirce’s Law: (Peirce)

(A → B) → A A.

Exercise 5.25.6 Show that the above schema is derivable from the intuitionistic basis together with the two-premiss rule just mentioned, and that, conversely, given (Peirce), that rule is derivable. (For solution, see 7.21.7, p. 1060, in which the rule in question is called (Subc)→ .) For present purposes, we rewrite (Peirce), as with (→I), (→E): (Peirce)− :

A − (B − A) A.

All sequents instantiating the schema (Peirce)− are valid on the requirement semantics, since for any sets X and Y , X (Y X) = X; the claim follows by taking, for arbitrary requirement structure R and R-assignment R, X and Y as R(A) and R(B) respectively. Thus we have Theorem 5.25.7 A sequent is provable using (R), (∧I), (∧E), (→I)− , (→E)− , and (Peirce)− if and only if it is valid on the requirement semantics. Proof. ‘Only if’ (Soundness): From 5.25.5 and the remark preceding the statement of the Theorem. ‘If’ (Completeness): In view of the fact that we have a complete proof system for converse material implication, it suffices to show that any ∧-boolean valuation v verifying all of Γ but not C, for an unprovable sequent Γ C, and in addition satisfying (←)

v(A − B) = T iff v (B) = F or v (A) = T, for all A, B,

gives rise to a requirement structure and assignment R for whose induced valuation vR we have vR (Γ) = T, vR (C) = F. So, given v as above, let R = (R, R+ ) have R = {F}, R+ = ∅, and put {F} if v(D) = F R(D) = ∅ if v(D) = T for all formulas D. Check that R is indeed an R-assignment, and that for every formula D, vR (D) = v(D). Since v(Γ) = T and v(C) = F it then follows that Γ C is invalid on the requirement semantics. The second part of this proof shows how to construct a requirement structure invalidating any sequent not holding on all ∧-boolean valuations satisfying the


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condition (←) mentioned therein. (Of course there was nothing special about taking the truth-value F as a requirement: anything else would have done as well.) Similarly, we could replace the first part of the proof with a purely semantic argument showing how to obtain an ∧-boolean (←)-satisfying valuation which agrees with the valuation vR induced by a requirement structure assignment R. We have already noted (5.25.1) that vR is itself ∧-boolean. But vR will typically not satisfy the condition (←) in the above proof. To satisfy that condition, three things need to hold (we assume R is an R-assignment for R = (R, R+ )): (1) vR (A − B) = T ⇒ vR (B) = F or v R (A) = T (2) vR (A) = T ⇒ vR (A − B) = T (3) vR (B) = F ⇒ vR (A − B) = T Reminder: (1), for example, means: if R(A − B) ⊆ R+ then either not R(B) ⊆ R+ or else R(A) ⊆ R+ . It is routine to show that (1) and (2) here are always satisfied. Condition (3), however, need not be satisfied, since from the fact that not every requirement for the truth of B obtains, it does not follow that every requirement for the truth of A which isn’t also a requirement for the truth of B does obtain: there may be additional requirements for the truth of A, not amongst those for B, which likewise fail to obtain. Thus, as in the proof of 5.25.4, we must ‘adjust’ our given R-assignment R to another one, R , but this time we need to change R as well. The recipe is as follows. Given R, an R-assignment for R = (R, R+ ), we define R = (R , R+ ) by: If R = R+ then (R , R+ ) = (R, R+ ); if R = R+ then select r ∈ R R+ and set R = R+ and R = R+ ∪ {r}. We can then define an R -assignment R by: If R(C) ⊆ R+ then R (C) = R(C); R(C) = R otherwise. The reader is left to verify that vR (C) = vR (C) for every formula C, and that as well as conditions (1) and (2), now condition (3) is satisfied by vR . While on the subject of variations on the proof of 5.25.7, we should note the availability of an argument replacing the second (‘completeness’) part of the proof given, which argument does not go via the fact that the given basis is complete for (converse) material implication. The key observation is that since for any ∧-classical consequence relation we have, for all Δ, A, B: ΔA∧B⇔ ΔA&ΔB we also have: Δ A ∧ B ⇔ Δ A or Δ B, so that the function R defined by: R(C) = {Δ | Δ C}


697

obeys our condition (∧) on assignments. Thus in particular, if is the consequence relation associated with the present proof system, then (∧) is satisfied. The same cannot be said, however, for the condition (−). To get around this, we can consider sets Δ which are maximal consistent w.r.t. the present proof system, in a sense approximating that of the discussion following 2.21.2: such a Δ does not have every formula as a consequence (by ) though all proper supersets of Δ do. It can then be shown that taking the collection of all such maximal consistent sets as the R of a requirement structure R (whatever we say—see below—about R+ ) that Lemma 5.25.8 The function R so defined is an R-assignment. The proof of 5.25.8 requires us to show that (∧) and (−) are both satisfied. The case of (∧) has already been covered in the above remarks. For (−), we have to show that Δ ∈ R(A−B) ⇔ Δ ∈ R(A) & Δ ∈ / R(B), and so Δ A−B ⇔ Δ A & Δ B, or in other words: Δ A − B ⇔ Δ A or Δ B the tricky part of which is showing that if Δ B then Δ B − A. This is left to the interested reader. (Compare 7.21.7, p. 1060.) Given an unprovable sequent Γ A of the present proof system, we can provide a ‘canonical’ (relative to this sequent) requirement structure R by taking R as above, and defining R+ to consist of all those maximal consistent Δ such that not Δ ⊇ Γ. It follows from this that R(C) ⊆ R+ , for every C ∈ Γ: for suppose C ∈ Γ, and Δ ∈ R(C); then Δ C (by the above definition of R) and so Δ is not a superset of Γ; thus Δ ∈ R+ . However, R(A) is not included in R+ , since, letting Γ+ be some maximal consistent superset of Γ such that Γ+ A, we have Γ+ ∈ R(A) but Γ+ ∈ / R+ . To conclude our discussion, let us return to the failure of the condition (3) above for arbitrary vR . This means the formulas B and A − B can both receive the value F under such a valuation, whereas of course on an →-boolean valuation, B and B → A cannot both be false (are subcontraries, as we sometimes put it). Thus, for all their coincidence in the framework Set-Fmla, logical subtraction under the present semantic treatment and converse material implication have different logics in Set-Set. The sequent p, q → p is tautologous but the sequent p, p − q is not valid on the requirement semantics. To the worry that in so much as raising the topic of Set-Set sequents, we have introduced (in the shape of multiple right-hand sides) the element of disjunction we were hoping to avoid – for the reasons mentioned under ‘Intuitively Unsubtractable Cases’ in 5.22 – the reply is that the shift of framework does not give the effect of disjunction embedded in the scope of “ −”, and it is this that we were keen to avoid. However, even in Set-Fmla, having seen that A − B behaves like the material conditional B → A, we can use the possible definition of disjunction by A ∨1 B = (A → B) → B, to consider a defined connective A ∨1 B = B −(B − A) which is itself ∨-classical according to the consequence relation associated with our proof system. (Here we use the kind of terminology introduced in 1.19; we write “∨1 ” in place of “∨” since – as explained in 1.11 – the latter is reserved for


698

a particular syntactic operation which is assumed here not to be present. For the formula constructed out of A and B as in the earlier definition of A ∨1 B ¨ B discussed at as (A → B) → B we generally use the special notation A ∨ 4.22.10 – p. 555 – and elsewhere.) Again, it is significant to raise the matter of the difference between Set-Fmla and Set-Set, or, in the present setting, consequence relations and gcr’s. For the gcr given by the requirement semantics does not have ∨1 behaving ∨-classically; calling this gcr , we do not have, in general, A ∨1 B A, B, for example. (Using the above definition, we find that for any assignment R, R(A ∨1 B) = R(A) ∩ R(B): all these requirements could obtain without either all the requirements in R(A) or all those in R(B) obtaining.) Similar phenomena will occupy us at length in §6.4. Already in Set-Fmla, in which framework we continue to investigate logical subtraction, we may fear that the current shape of the requirement semantics is not doing justice to the motivating ideas. Even a collapse into converse intuitionistic implication, let alone material implication, is incompatible with the idea that— to cite an earlier example—from “All females are vixens” there follows (according to CL or IL ) “a is female → a is a vixen”, whereas we were hoping to have “a is a vixen − a is female” amount to “a is a fox”, which does not follow from that premiss. Nor—it may appear—is there any hope of keeping logical subtraction and converse implication apart, even though governed by analogous principles, since the principles in question (cf. 4.32) are strong enough for unique characterization. This suggests that we attend to the possibility of modifying the present semantics. However, we shall return to it in 5.27, and reconsider the appropriateness of the current pessimistic conclusions.

5.26

Requirement Semantics: Second Pass

Our experiment with logical subtraction cannot, in view of what has just been said, be regarded as an unqualified success up to this point. In this subsection, we will briefly indicate what appears to be a more promising line of development. In fact, it can be described as a variation on the requirement semantics of 5.25. To motivate this variation, let us recall the ‘intuitive unsubtractability’ of B from A when B does not follow from A. In the treatment of 5.25, the condition (−) said that R(A − B) was to comprise the requirements in R(A) which were not in R(B). Thus even though not every requirement for the truth of A was a requirement for the truth of B, which amounts, by 5.26.1 below (taking n = 1), to B’s not following from A, we still allow R(A − B) to be different from R(A). To disallow this would be to say that unless you can subtract all of B’s content (given by R(B)) from A’s, you are not to subtract any. To say this succinctly, let us introduce the following modified set-theoretic difference operation defined by: X Y X Y = X

if Y ⊆ X otherwise

Our modified requirement semantics uses this operation in place of ordinary settheoretic difference, and is otherwise as in 5.25. In other words, we work with requirement structures R = (R, R+ ) exactly as before, saying that a sequent is


699

valid when it holds on every valuation vR defined in terms of such a structure and an R-assignment R by: vR (A) = T ⇔ R(A) ⊆ R+ , except that now to qualify as an R-assignment, R must map formulas to subsets of R subject to the constraints: R(A ∧ B) = R(A) ∪ R(B)

(∧) (−)

R(A − B) = R(A) R(B)

For emphasis, we will refer to this notion of validity as validity on the modified requirement semantics. The observation alluded to above, which depends for its correctness of the definition of validity and has nothing to do with the shift from (−) to (−) , is as follows: Observation 5.26.1 A sequent A1 , . . . , An B is valid (on the original or on the modified requirement semantics) iff for every requirement structure R and every R-assignment R: R(B) ⊆ R(A1 ) ∪ . . . ∪ R(An ). Proof. The ‘if’ direction here is trivial; for the ‘only if’ direction, suppose we have R = (R, R+ ) with R an R-assignment for which the displayed inclusion is false. Put R = (R , R+ ) where R+ = R(A1 ) ∪ . . . ∪ R(An ). R is then an R -assignment on which the sequent A1 , . . . , An B does not hold. Turning now to matters pertaining specifically to the difference between (−) and (−) , we begin by noting that (→E)− is unscathed by the transition to (−) . To work through the its zero-premiss analogue, (Subt. 3): B − A, A B, suppose for some R-assignment R, R(B−A) ⊆ R+ and R(A) ⊆ R+ . R(B−A), alias R(B) R(A), is either R(B) R(A), in which case the considerations of 5.25 apply, or else it is R(B), in which case the result is immediate.

Exercise 5.26.2 Verify that (Subt. 1), (Subt. 2) and (Subt. 4), from 5.25.2, are valid on the modified requirement semantics, but that (Peirce)− is not. Like (Peirce), the rule (→I)− does not fare well under the current modification of the semantics. For example, applying this rule to the valid sequent p ∧ q p gives p − (p ∧ q). But if R(p) = {r1 } and R(q) = {r2 } where r1 = r2 , / R+ , vR (p − (p ∧ q) = F. then R(p − (p ∧ q)) = R(p) = {r1 }, so as long as r1 ∈ With an additional premiss-sequent, we can obtain a validity-preserving rule: modified (→I)−

Γ, A B

BA

Γ B−A

Exercise 5.26.3 Show that (Subt. 1), (Subt. 2) and (Subt. 4) are derivable from modified (→I)− .

700


Theorem 5.26.4 The proof system with rules (R), (∧I), (∧E), modified (→ I)− , and (→ E)− is sound w.r.t. the modified requirement semantics. Proof. It remains only to check that modified (→I)− does preserve validity, as claimed. So suppose that (i) Γ, A B is valid, and (ii) B A is valid. Letting R be an arbitrary R-assignment for an arbitrary requirement structure R, and writing R(Γ) for the union of {R(C) | C ∈ Γ}, we have, from (i), by 5.26.1: R(B) ⊆ R(Γ) ∪ R(A), from which it follows that R(B) R(A) ⊆ R(Γ). But by (ii) and 5.26.1 again, R(B) R(A) = R(B) R(A), so R(B−A) ⊆ R(Γ). Therefore the conclusion-sequent Γ B − A is valid. Remark 5.26.5 Without the detour through (ii) and (−) , this gives an alternative proof of 5.25.4. (Does this mean we did not need to shift R-assignments, from R to R , in that proof? Well, there we kept something else fixed, namely R, which has been tacitly shifted, via the appeal to 5.26.1, in the present argument.) It would be interesting to know if the proof system described in 5.26.4 is also complete, or, should the answer to that question be negative, what further rules might be added to obtain a complete system. However, our interest for the present lies more in some questions left over from the original, unmodified requirement semantics.

5.27

Requirement Semantics: Final Considerations

Our discussion of the original (unmodified) requirement semantics in 5.25 ended on a negative note, with talk of a collapse of logical subtraction into the converse of material implication. The modification of 5.26 dealt with one kind of qualm about what we have been calling intuitively unsubtractable cases, namely, that in which we try to subtract B from A even when B did not follow from A. However, 5.21 ended with a mention of another kind of intuitive unsubtractability, namely that arising when we take A as (e.g.) p and B as p ∨ q. Here B does follow from A but, since we do not think of the truth of p ∨ q as required for the truth of p, there is a difficulty in making sense of the subtraction. To avoid such difficulties, we have tried to steer clear of ∨ (as well as → and ¬) in setting up the requirement semantics. We could tackle them in either of two ways: by extending the requirement semantics so as to assign disjunctive (negated, etc.) formulas sets of requirements, or by thinking of the requirement semantics as embedded in some broader account which does not make any such assignments to these other formulas. The former option might be pursuable along the lines of van Fraassen [1969]: see the notes to this section. Below, we will pursue instead the latter option. We could call it the bipartite treatment, since formulas will fall into two classes: those for which sets of requirements are supplied, and those for which they are not supplied. Before getting to that, there are several preliminaries to attend to. First, let us consider more carefully the threat of collapse noted at the end of 5.25, where it was remarked that, apparently, if material implication was added to the language, the fact that it was uniquely characterized by appropriate additional rules would mean there was no hope of keeping apart converse implication


701

and logical subtraction. For we already justified the presence of the uniquely characterizing rules (→I)− and (→E)− for the latter connective on the basis of the requirement semantics. The case of (→E)− was dealt with, for example, in Lemma 5.25.4. But this negative conclusion is premature, since that Lemma was proved for the language whose only connectives were ∧ and −, and there is no reason to suppose that it would go through in the presence of additional connectives. (Note the role of Lemma 5.25.3 in the proof, exploiting features specific to the way requirements were assigned to conjunctive and subtractive formulas.) It is true that the logic of conjunction and subtraction, considered in isolation from other connectives, is precisely the same as that of conjunction and converse material implication, considered in similar isolation, but this in no way implies that when converse material implication and subtraction are considered together, alongside conjunction, the resulting logic will deliver the ‘collapse’: A − B B → A. (Two connectives, in separate logics, can be subconnectives of each other in the sense of 3.24, which began on p. 461, without being equivalent in a combined logic; a nice example of this is given in Byrd [1973], as was mentioned at the end of 3.24.) We will expect, of course, one half of this equivalence, namely A − B B → A, since this is (→I)-derivable from (Subt. 3), but any grounds for expecting the converse, B → A A − B, to follow from B → A, B A by appeal to (→I)− have been lost in the absence of a justification of the latter rule in the context of a language supporting → as well as −. (Recall that we use the “Subt.” labelling indiscriminately for sequent-schemata and the corresponding -conditions.) As we noted in the preceding paragraph, there is no reason to expect such a justification to be forthcoming. Apart from focussing suspicion on the rule (→I)− in the presence of the additional connectives, we can illustrate the need for extreme caution in extending the results of 5.25 to accommodate those connectives from another direction. We will need to note an additional valid schema for this purpose. Exercise 5.27.1 Show that all sequents in the language of 5.25 of the forms (Subt. 5) (A ∧ B) − B A − B (Subt. 6) A − B (A ∧ B) − B are valid on the requirement semantics of that subsection. The principle we need here is, specifically, (Subt. 5); by way of reminder, here is (Subt. 4): A (A ∧ B) − B. Observation 5.27.2 Let be a congruential consequence relation extending CL and satisfying (Subt. 4), (Subt. 5). Then B → A A − B for all formulas A, B, of the language of . Proof. Make the suppositions listed concerning . Then we have: (i) B → A ((B → A) ∧ B) − B as a special case of (Subt. 4). (ii) B → A (A ∧ B) − B since is a congruential extension of CL and (B → A) ∧ B and A ∧ B are CL-equivalent. (iii) (A ∧ B) − B A − B. (Subt. 5). Therefore: (iv) B → A A − B. From (ii), (iii), by (T).


702

With these indications as to what to avoid, we turn to the promised bipartite treatment. We shall describe a language L2 , for whose Set-Fmla sequents a notion of validity will be defined. The valid sequents constitute the logic in which the bipartite treatment issues. This ‘outer’ language is built around an ‘inner’ language L1 , whose formulas are assigned sets of requirements. The semantics for these inner formulas is as in 5.25, so that the only connectives we allow for them are conjunction and subtraction. Such formulas are then composed by the boolean connectives to yield the formulas of L2 , which are semantically evaluated ‘by truth-tables’ (i.e., by considering boolean valuations). So, for example, A − B will only be a formula of L2 when it is a formula of L1 : this means A and B themselves cannot be built up with the aid of negation or disjunction, for example, though A − B can itself be negated or disjoined with another L2 formula. In terms of the definition of language in 1.11, L2 is not strictly speaking a language at all, because of these restrictions on composition by the connectives. But we shall continue to refer to is as such. A precise specification of the class of formulas of this language follows shortly. From the above informal description, it will be clear that conjunction appears in two lights, as a device for compounding L1 formulas, and as a device for compounding L2 formulas. Greater clarity will be achieved if we have two connectives for performing these two roles, and since we require a boolean valuation for a language with ∧ as a connective to be ∧-boolean, we shall use ∧ for the latter role. Let us use “&” for the former role. This means that appeals to the definitions results of 5.25 should be reinterpreted with “&” replacing “∧”. For example, the condition (∧), on requirement assignments R becomes: (&) R(A & B) = R(A) ∪ R(B). The condition (−) remains as it was: (−)

R(A − B) = R(A) R(B).

(The ‘modified’ version, with (−) , will be touched on below, in 5.27.5.) We proceed with the inductive definition of L2 , whose basis is given by the formulas of L1 : (1) The propositional variables p1 ,. . . ,pn ,. . . are formulas of L1 . (2) If A and B are formulas of L1 , then so are A & B and A − B. (3) Nothing else (i.e., other than as justified by (1) and (2)) is a formula of L1 . (4) Any formula of L1 is a formula of L2 . (5) If A and B are formulas of L2 , then so are ¬A, A ∨ B, A ∧ B, A → B and A ↔ B. (6) Nothing else is a formula of L2 . Expressed more concisely, then, the L1 is the smallest class of formulas containing the propositional variables and closed under the L1 -connectives, while L2 is the smallest class of formulas containing the L1 -formulas and closed under the L2 connectives. We formalize the semantic ideas described above by isolating the notion of what we shall call a suitable valuation, meaning by this a boolean valuation for L2 which is of the form vR for some R-assignment R, for some requirement structure R = (R, R+ ). Recall that this amounts to saying that vR (A) = T


703

for any L1 formula A iff R(A) ⊆ R+ , where R satisfies the conditions (&) and (−) above. A sequent is valid iff it holds on every suitable valuation. Note that this definition of validity coincides with that of 5.25 when all formulas in the sequent in question are L1 -formulas. Thus, for example, (Subt. 1)–(Subt. 6) and (Peirce)− are all valid on the present semantics. Exercise 5.27.3 (i) Verify that p & q p ∧ q and the converse sequent are both valid. (ii ) Where (as in 5.25) A ∨1 B is the formula B − (B − A), for any L1 -formulas A, B, the sequent p ∨ q p ∨1 q is valid but its converse is not. The bipartite treatment does involve some compromise with two desiderata mentioned in 5.22: substitution-invariance and congruentiality. For example, while we have (as an instance of (Subt. 1) p p − q as a valid sequent, p ∨ p (p ∨ p) − q, which arises from it by Uniform Substitution (of p ∨ p for p) as well as by replacement of p by the equivalent formula p ∨ p, is not a valid sequent. (‘Equivalent’ here, meaning that p p ∨ p and the converse sequent are both valid.) Note, however, these replacements do not turn a valid sequent into an invalid sequent. Rather, they result in something which is not a sequent at all. The bipartite L1 /L2 set-up has the effect that replacing an L1 -subformula of an L2 -formula by an L2 -formula which is not an L1 -formula can destroy wellformedness. It is not hard to check that if σ is a sequent resulting from σ either Uniform Substitution or by a congruentiality-justified replacement, then validity is preserved in passing from σ to σ . The point just made about Uniform Substitution applies again when we are considering the instances of schematically formulated principles: the instantiation of schematic letters is constrained by the condition that the overall result should be a formula (or a sequent) of L2 . For example, line (i) of the proof of 5.27.2 above, renotated with “&” for “∧”, so as to have some hope of being a consequence-statement amongst L2 -formulas, looks like this: B → A ((B → A) & B) − B. But this is no longer the form of an instance of (Subt. 4), since it has →, an ‘outer’ connective, in the scope of &, an ‘inner’ connective. If we don’t make the change from “∧” to “&”, on the other hand, then we have another forbidden pattern, the appearance of the outer connective ∧ in the scope of the inner connective −. In fact, even without attending to the exact type of conjunction used, we can see there is something wrong with the rhs above, since → appears within the scope of −. The last line of the proof of 5.27.2 was: B → A A − B, and this will make sense for any L1 -formulas A and B. We should assure ourselves that no such unwanted result is forthcoming on the present semantics: Observation 5.27.4 The sequent q → p p − q is not valid. Proof. For r1 = r2 , put R = (R, R+ ) with R = {r1 , r2 } and R+ = ∅, and define the R-assignment R by: R(p) = {r1 }, R(q) = {r2 }. Then, since r2 ∈ / R+ , vR (q) = F, so (since vR is to be →-boolean) vR (q → p) = T. But vR (p − q)


704 / R+ . = F, as r1 ∈ R(p) R(q) and r1 ∈

In view of the validity of the sequent q → p, q p, we see, with the above example, how in the presence of →, the ‘Conditional Proof’ rule for −, called (→I)− in 5.25, can fail to preserve validity. We shall not look into the question of how to formulate suitable rules for a proof system in which precisely the valid sequents of L2 are provable. Rather, after giving an exercise for practice with a modified version (à la 5.26) of the semantics, we close by noting a ‘possible worlds’ model theory which is equivalent in respect of the induced notion of validity to the semantics in terms of requirement structures. Exercise 5.27.5 To treat the modified subtraction semantics of 5.26 in the present setting we change (−) above to (−) in the semantics for L1 formulas, keeping everything else the same. (i) On this modified semantics, are 5.27.1’s (Subt. 5) and (Subt. 6) valid? (ii ) Adjust the formation rules (1) − (6) given for L2 above to accommodate a new binary connective “⇒”, by adding: if A and B are formulas of L1 , then A ⇒ B is a formula of L2 . (Note that, unlike all the other connectives we have considered, this one cannot occur in its own scope.) Correspondingly, make the semantic stipulation that for vR to be a suitable valuation, we must have, for any L1 -formulas A, B, vR (A ⇒ B) = T iff R(B) ⊆ R(A). Is the sequent ¬(A ⇒ B) A ↔ (A − B) valid on the modified semantics? Is it valid on the unmodified semantics? Since we have said nothing as to what requirements, the elements of the sets R and R+ of a requirement structure, actually are, the requirement semantics of the present subsection and its two predecessors, may seem excessively abstract. We turn now to the provision of a (relatively) concrete example of a requirement structure, namely that obtained from a collection of possible worlds whose subsets (alias propositions) are the requirements. In fact, we may reformulate the semantics in these terms, and that will be our procedure. So, for the present discussion, understand by a possible worlds model a pair (W, V) in which W is a non-empty set, and V is a function assigning collections of subsets of W to propositional variables. (Note: collections of subsets, not subsets; hence the more elaborate lettering.) We extend V to all formulas of L1 by: V(A & B) = V(A) ∪ V(B)

V(A − B) = V(A) V(B)

where A and B are any L1 formulas; and, given x ∈ W , we define: (W, V) |=x A iff x ∈ RV(A), for any L1 -formula A. This last is our way of saying that a formula of the type to which sets of requirements are assigned is true at a world x when every requirement for its truth obtains at x. Remark 5.27.6 If we wish to use the modified version of the semantics, we replace the second condition on V by: V(A − B) = V(A) V(B). If the connective ⇒ of 5.27.5(ii) is to be present, whether or not the modification just made is in force, we should add:


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(W, V) |=x A ⇒ B iff V(B) ⊆ V(A) for any L1 -formulas A, B. We extend the (|=) truth-definition up through the formulas of L2 in the usual way: (W, V) |=x ¬A iff (W, V) |=x A; (W, V) |=x A ∨ B iff (W, V) |=x A or (W, V) |=x B; and so on, for the other boolean connectives. A sequent Γ B is valid when for all (W, V) and all x ∈ W , if (W, V) |=x A for each A ∈ Γ, then (W, V) |=x B. Given any possible worlds model (W, V) and any x ∈ W , we can obtain an equivalent requirement structure (R, R+ ) paired with an assignment R by putting R = ℘(W ), R+ = {X ⊆ W | x ∈ X} and R(A) = V(A) for all L1 formulas A. One sense in which the resulting requirement structure is equivalent is that, as may shown by induction on the ‘L2 -complexity’ of formulas (i.e., taking L1 formulas as of complexity 0, and in general counting the number of occurrences of boolean connectives in an L2 -formula to calculate the complexity of that formula): Lemma 5.27.7 With (R, R+ ), and R defined as above on the basis of (W, V), x, we have: (W, V) |=x A iff vR (A) = T, for all formulas A. It is easy to see that, while every possible worlds model together with a distinguished world essentially is a requirement structure, equivalent in the sense given by this Observation, the converse is not the case. For example, if |R| = 3, then R cannot be taken to be ℘(W ) for any W , since power sets are always either infinite or of cardinality 2n for some n. We shall see below (5.27.8) that a result analogous to 5.27.7, but starting with a requirement structure and requirement assignment, can all the same be achieved. We shall approach this result by giving first an interestingly fallacious construction purporting to establish it. Suppose we said that given (R, R+ ) and R we can obtain (W, V) and x ∈ W such that (W, V) |=x A iff vR (A) = T for all formulas A, by the following procedure. We just pick an object x, and let W = {x}, defining a function f from R into ℘(W ) by: f (r) = {x} if r ∈ R+ , f (r) = ∅ otherwise, for all r ∈ R, and setting V(A) = {f (r) | r ∈ R(A)}, for all formulas A. We then argue that (W , V) |=x A iff vR (A) = T, since the lhs here means that x belongs, for each r ∈ R(A), to the set f (r). Since this is in turn the case iff for all r ∈ R(A), f (r) = {x}, and thus—by the way f was defined—iff each such r ∈ R+ , it is indeed true that (W , V) |=x A iff vR (A) = T. Nevertheless, this would be fallacious if it purported to show how to obtain an equivalent possible worlds model for a given requirement structure (with assignment), since it has not been shown that V behaves as required for such a model. For this, we should need to show that for all A, B ∈ L1 , V(A & B) = V(A) ∪ V(B) and V(A − B) = V(A) V(B). That is, we should need to check that (i) {f (r) | r ∈ R(A) ∪R(B)} = {f (r) | r ∈ R(A)} ∪ {f (r) | r ∈ R(B)}, and that

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(ii ) {f (r) | r ∈ R(A) R(B)} = {f (r) | ∈ R(A)} {f (r) | r ∈ R(B)}. Now, while there is no difficulty in the case of (i), (ii) is not in general correct. (It is for this reason that we describe the purported proof as interestingly fallacious.) For suppose that X, some subset of W , belongs to the lhs of (ii); that is, that X = f (r) for some r ∈ R(A) with r ∈ / R(B). How would we argue that X belongs to the rhs? Certainly, X ∈ {f (r) | r ∈ R(A)}, but we need also that X ∈ / {f (r) | r ∈ R(B)}, and this means that X is not f (r) for any r ∈ R(B). We have this assurance only if we know that f is injective, which in the case of the present choice of f is far from being the case. (All non-obtaining requirements are mapped to ∅, and all obtaining requirements to {x}.) Note that, had the above argument not been fallacious, it would have delivered for each requirement structure paired with an assignment, an equivalent one-element possible worlds model (i.e., with |W | = 1), which in turn, by the proof given for 5.27.7, would allow us to convert this into an equivalent requirement structure with |R| = 2, |R+ | = 1. But an examination of the proof of 5.27.4 will reveal that the sequent q → p p − q can only be invalidated by a requirement structure in which there are at least two requirements which do not obtain. We turn to a corrected version of the fallacious argument. In view of the above discussion, all we need to do is make sure that the function f to be described is injective. We envisage the requirements in our given R to be indexed by some set I in such a way that for i, j ∈ I, i = j implies ri = rj . We pick for these distinct ri , as many new elements yi , distinct from each other as well as from some element x, and put W = {x} ∪ {yi }i∈I . (We could let the yi just be the ri .) Now define f , on the basis of (R, R+ ), as a function from R to ℘(W ) thus: f (ri ) = {x, yi } if ri ∈ R+ ; f (ri ) = {yi } otherwise. Note that this time, f is injective, so that we do have, when we set V(A) = {f (r) | r ∈ R(A)}, the condition V(A − B) = V(A) V(B). The rest of the original argument completes the proof of: Lemma 5.27.8 With (W, V) defined on the basis of (R, R+ ) and R as above, we have: (W, V) |=x A iff vR (A) = T, for all formulas A. This gives us all we need to know to claim the two semantic approaches are equivalent: Theorem 5.27.9 A sequent is valid on the possible worlds semantics iff it is valid on the requirement semantics. Proof. ‘If’: by Lemma 5.27.7. ‘Only if’: by Lemma 5.27.8.

This concludes our presentation of the ‘bipartite treatment’ of the language of “−” and the boolean connectives, and with it our experimentation with logical subtraction. Some will regard the experiment as less than wholly successful in view of that treatment. Attempts at a more homogeneous account are of course to be welcomed; we have seen some of the obstacles that would need to be overcome in that project. Many further loose ends have been left. What about proof theory for the system(s) of the present subsection? What about


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non-classical (e.g., intuitionistic or relevant-logical) treatment of the boolean connectives? What of subtraction in modal logic? And of course there are pressing philosophical questions too: is the (or ‘some’) idea of logical subtraction really coherent? If so, which is the preferable approach, the unmodified or the modified requirement semantics? Our intention has been to open up this area for investigation rather than to provide a definitive resolution of the issues arising.

Notes and References for §5.2 This section is a descendant of a paper, ‘Reciprocal Connectives’, presented to a seminar at the University of York (England) in 1972. Later versions, under various titles, were presented to the annual conference of the Australasian Association for Philosophy at the University of Wollongong in 1977, an A.N.U. (Research School of Social Sciences) Philosophy Program seminar in 1978, and a University of London philosophy colloquium (University College, London) in 1981. The last mentioned version was a paper entitled ‘Logical Subtraction: Problems and Prospects’, which was also circulated amongst several friends and colleagues. Helpful remarks concerning this material were made on these and other occasions by John (A.) Burgess, Martin Davies, David Lewis, David Makinson, Peter Lavers and John Slaney. Further development of some of the themes in this section may be found in Humberstone [2000d]. As was mentioned at the end of 5.12, some of those just thanked have suggested – something not taken up in our discussion – that what is called ‘contraction’ in the beliefrevision (or theory-change) literature might serve as something of a model for logical subtraction; see 3.12 of Fuhrmann [1997] for an airing of this thought in print, and Fuhrmann [1999] for further discussion. The published literature on logical subtraction (Jaeger [1973], [1976], Hudson [1975], and – especially the Appendix to – Hornsby [1980]) takes off from a question asked by the later Wittgenstein (“What is left over if I subtract the fact that my arm goes up from the fact that I raise my arm?”). The variabledisjointness notion of independence arrived at in 5.23 is arguably a notion of independence entertained by the early Wittgenstein; see Wolniewicz [1970], and also Miller [1974]. The use of ↔ in Example 5.23.4 was suggested by the latter paper, which in turn acknowledges earlier work by Max Black. (Cf., in this connection, the discussion of Figure 2.13a – p. 225 – and note 28 of Humberstone [2000a].) See further the discussion following 7.31.15 below (p. 1137). The unacceptability of the Unacceptable Cancellation Condition in 5.23.1(i) is a special case of the ‘Cancelling-Out Fallacy’ of Geach [1968]. It is interesting to note that in his elaboration of some remarks made by Hobbes, Gaukroger [1989] (p. 94) makes spontaneous use of the “−” notation for logical subtraction. A modification of this notation, and the term “logical subtraction”, is used by Peirce for a would-be operation reciprocal to disjunction (or, more accurately, union) rather than conjunction (Peirce [1867], p. 5). What we call logical subtraction, Peirce calls logical division. (Cf. the end of the opening paragraph of 5.21.) The example of 5.24.3 is close to a line of investigation pursued by Peter Lavers in unpublished work. At the start of 5.27, mention was made of van Fraassen [1969]. This paper develops two (mutually dual) theories of facts, one modelled on work of Bertrand Russell, according to which for a statement to be true some fact signified by the statement must obtain, and the other modelled on work of C. I. Lewis,

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according to which for a statement to be true every fact (or, as Lewis puts it, state of affairs) it signifies must obtain. The latter conception provides the kind of thing we looked for in 5.24 as a notion of the content of a statement. Van Fraassen shows how to associate sets of facts with formulas of a language whose connectives are ∧, ∨ and ¬ in such a way as to secure these results. This might therefore serve as an inspiration for an extension of the requirement semantics to cover all boolean formulas, as a more streamlined alternative to the bipartite treatment offered in 5.27. His proposals, however, lead to the assignment of different sets of facts for classically equivalent formulas, and indeed provide a novel semantic account of the consequence relation of relevant logic for the formulas concerned. We have not attempted to follow this path, not seeing how to adapt the account to the case of classical logic, and not being convinced that the attempt to accommodate logical subtraction is especially closely tied with any motivation for repudiating CL as the logic of the boolean connectives. Some subsequent thoughts on logical subtraction can be found in Humberstone [2000d], which also recapitulates some findings from the present section. Stephen Yablo has also shown me some potentially promising developments for this topic in work as yet unpublished.

§5.3 ∧-LIKE CONNECTIVES AND THE LIKE 5.31

#-Like and #-Representable Connectives

We will be concerned initially in the present section, and intermittently thereafter, with a property we call and -likeness, defining a binary connective # to be and -like, or as we shall actually say, ∧-like, according to a given (generalized) consequence relation if it satisfies a certain condition which, roughly speaking, allows for a compound A # B to be interpreted as though it were a compound O1 A ∧O2 B for some (fixed) singulary connectives O1 , O2 , with ∧ itself assumed to behave ‘∧-classically’. This has some distinctive repercussions for the inferential behaviour of compounds A # B and their interrelationships, even though these do not include what might first come to mind as the most familiar features of conjunctions: for instance they do not include the inferability of the components A, B, from the compound A # B, or the commutativity of #. The inferential behaviour of these compounds (according to a consequence relation ) is exhausted by a simple syntactic condition (on # and ) which we shall be meeting under the name “(∧-like)” in due course (p. 712). Where $ is any binary boolean connective, we adopt a similar understanding of #’s being $-like: simply replace the references to ∧ in the preceding informal explanation by references to $. It is this extension of the usage of “∧-like” that the words “and the like” in the title of the present section are intended to evoke. (As already mentioned, our initial focus will be on the case of $ = ∧, making the present chapter as good as any in which to include this discussion; this enables us to begin work in the simpler and more natural setting of consequence relations rather than gcr’s. The latter will, however, provide the most natural setting for a suitably ‘pure’ formulation analogous to (∧-like) for other connectives.) One could – though we shall not – consider a further generalization by allowing $ to be any binary connective or indeed to be an n-ary connective (n arbitrary). Restricting attention to the case of boolean $ – any binary con-

5.3. ∧-LIKE CONNECTIVES AND THE LIKE

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nective compositionally derivable from ∧ and ¬, we might equally well say – it turns out all sixteen cases (identifying CL-synonymous such connectives) can be reduced to three, by taking $ to be one of ∧, ∨, ↔: Observation 5.31.1 Suppose is ¬-classical as well as $-classical for all binary boolean $, and that O1 , O2 , are singulary connectives and # a binary connective in the language of . Then if A # B O1 A $ O2 B, for all A, B, we can find singulary P1 , P2 , and $ ∈ {∧, ∨, ↔}, such that for all A, B in the language of : A # B P1 A $ P2 B. Proof. An examination of (all sixteen) cases suffices. By way of example, if $ is → we can take $ as ∨, putting P1 = ¬O1 , P2 = O2 ; if $ is we may take $ as ↔ and choose P1 , P2 as before. (The boolean connective , for exclusive disjunction, is officially introduced in 6.12, p. 780, though already encountered more than once – e.g., at 3.14.5(ii), p. 406.) The three choices of $ described in 5.31.1 could have been put differently. We could for example have replaced ∨ by →, or ↔ by , or ∧ by the connective “∧¬” (say) which forms from A and B the compound A ∧ ¬B; and so on. But we stick with the representatives chosen. Let us say that the binary connective # is booleanly representable according to (meeting the conditions of 5.31.1) if there exist O1 , O2 , and boolean binary $ such that for all A, B: (1)

A # B O1 A $ O2 B.

We say that # is ∧-representable according to when for all A, B, (1) holds taking $ as ∧, ∨-representable according to when for all A, B, (1) holds taking $ as ∨, and ↔-representable according to when for all A, B, (1) holds taking $ as ↔. Then the content of 5.31.1 can be put by saying that any booleanly representable binary connective is ∧-representable, ∨-representable, or ↔-representable (according to any given ). The three kinds of representability just described are not mutually exclusive, as is most simply illustrated for the case in which O1 and O2 are themselves – something not presumed in the general definitions to this point – boolean connectives. We shall speak of fully boolean representability in such a case. An intermediate case arises when this is true of one of O1 , O2 , but not the other. We encountered such a case – a case of ∧-representability – in 5.12 (p. 639), with von Wright’s temporal conjunction (‘and next’), in which O1 turned out to be the identity (or ‘null’) connective while O2 , when made explicit as described in our discussion, was a non-boolean tense-logical connective. (The formulation alluding to the ‘identity connective’ should be taken with a pinch of salt: as explained after 3.13.10, p. 392, there can be no such connective. There is a perfectly good context, C(p) for which, given any formula A, C(A) is the formula A, namely take C(p) = A; and of course we can have a connective O1 with A equivalent to (or, better, synonymous with) O1 A for any A, relative to some background ).


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In the case of fully boolean representability, it is simpler to discuss the truth-functions associated with the connectives over boolean valuations. The terminology transfers straightforwardly; we use # and $ temporarily to stand for binary truth-functions and write ∧, ∨, and ↔ in place of the more explicit notation introduced in 3.14 (namely ∧b , ∨b , ↔b ); rather than make a similar usage of ¬, we employ the “N” of the “V, F, I, N”-notation also introduced there (after 3.14.5, p. 406). The truth-function # is $-representable if there exist f , g ∈ {V, F, I, N} such that for all x, y ∈ {T, F}: (2)

x # y = f (x) $ g(y).

We may refer to the rhs of (2) as a $-representation of # in terms of f and g (in that order). The functions f and g thus play the role played in our connectiveoriented discussion by O1 and O2 . The analogue of 5.31.1 in the present setting runs: Observation 5.31.2 Every binary truth-function is at least one of the following: ∧-representable, ∨-representable, or ↔-representable. Proof. Any binary # is #-representable, since we may take f = g = I. Thus it remains only to check, as for 5.31.1 (indeed it is the very same checking that is involved) that we can always modify f and g appropriately so that (2) holds for all x, y, with $ being one of ∧, ∨, ↔. To illustrate the kind of overlap alluded to above, note that (writing “Vx” for V(x), etc.): Vx ∧ Fy = Fx ∧ Fy = Ix ∧ Fy = Nx ∧ Fy = Fx ∨ Fy = Vx ↔ Fy. The four ∧-representations as well as the ∨-representation in terms of F and F, and the ↔-representation in terms of V and F, all give the same truth-function: the constant binary truth-function with the value F. This illustration has other features which are typical. Any truth-function which is both ∧-representable and ∨-representable, for example, is also ↔-representable; an even more general statement holds – see (i) of Exercise 5.31.3 below. Also, the fact that we have a constant truth-function in this particular instance or multiple representability is again no coincidence: see (ii ) of the exercise. Here, by multiply representable is meant, specifically, $- and $ -representable for two distinct elements $, $ of the set {∧, ∨, ↔}. Exercise 5.31.3 (i) Show that where $1 , $2 and $3 are ∧, ∨ and ↔, taken in any order, if a binary truth-function is $1 - and $2 -representable, it is also $3 -representable. (ii ) Show that a binary truth-function is multiply representable if and only if it is not essentially binary. (‘Multiply representable’ as explained immediately before this exercise, ‘essentially binary’ as in 3.14.10, p. 411.) Hint: for (i) and (ii ), an argument based on general principles would provide the more informative answer, but in its absence an exhaustive examination of the cases will suffice. Such an examination remains useful, however, in revealing that of the 16 binary truth-functions,


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10 are essentially binary, and of these 10, 4 are ∧-representable, 4 ∨representable, and 2 ↔-representable. (“Essentially n-ary”, as explained in 3.14.10(i), p. 411, and here applied to the case of n = 2.) (iii) What is wrong with the following claim: ↔ is ∧-representable because for all x, y ∈ {T, F}, x ↔ y = (x → y) ∧ (y → x)? (iv ) Picking up on the Hint above: list all binary truth-functions – not just those that are essentially – which are ∧-representable. Recall the generalized Post duality patterns α0 , α1 , . . . , αn for a truthfunction of n arguments, introduced in 3.14 (p. 410); each αi was either I or N. Closest to our present concerns are such patterns for the case of n = 2. Such a pattern α0 , α1 , α2 represents a mapping which turns a given binary truth-function f into the binary truth-function function g defined by: g(x, y) = α0 (f (α1 (x), α2 (y)). Similar patterns arise in the present context, more restricted in one respect and less so in another. Since allowing α0 to be N rather than I would, for example, turn ∧-representable f into ∨-representable g, we need to be more restrictive in requiring that α0 be I. The respect in which we should be less restrictive is that the remaining αi can be chosen from amongst {V, F, I, N} and not just from {I, N}. (However, the only effect of disallowing V and F would be to set aside the functions which are not essentially binary.) The ∧-representable truth-functions are then precisely those obtainable as g by the equation inset above when α0 , α1 , α2 is one of the patterns of the type just characterized and f is taken as ∧. Likewise, mutatis mutandis, for ∨-representable and ↔representable truth-functions. We leave the arena of fully boolean $-representability to return to the general case – boolean representability, that is – in which O1 and O2 are arbitrary 1-ary connectives (though $ is still restricted to being one of ∧, ∨, ↔). We seek a semantic analogue of $-representability according to a (generalized) consequence relation . By concentrating on the case in which $ is ∧, we will be able to conduct the bulk of the discussion taking to be a consequence relation rather than a gcr, because the -condition we call “(∧-like)” below has exactly one formula schematically represented to the right of the “”. After introducing this semantic version of, in particular ∧-representability, we provide a precise description of the notion informally introduced in our opening paragraph, that of being ∧-like (according to ). Fix on some language which provides a binary connective #, and let U be the class of all valuations for that language. We say that # is ∧-representable over V ⊆ U just in case there are functions f , g: V −→ U such that for all v ∈ V and all formulas A, B, of the language: (3) v(A # B) = T iff fv (A) = T and gv (B) = T. Here, in the interests of readability we denote the value of f (v) for argument A by fv (A) (and likewise for g and B), rather than the more cumbrous “(f (v))(A)”. Note that although we use the term “∧-representable” rather than “and -representable” (for the sake of brevity), we do not suppose that the language concerned has the connective ∧ or that the valuations in V are, even if ∧ is present, ∧-boolean. The crucial conjunctiveness on the right-hand side is given


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by the metalinguistic and appearing there. Definitions of ∨-representability and ↔-representability over V are to be understood as similarly worded except that metalinguistic analogues of boolean ∨ and ↔ replace this occurrence of and on the rhs. We could restate these definitions using the corresponding truthfunctions – and here it is useful to make explicit notational distinction between the boolean connective $ and the truth-function $b – in the following way: # is $-representable over V ⊆ U just in case there are functions f , g:V −→ U such that for all v ∈ V and all formulas A, B of the language (3 )

v(A # B) = $b (fv (A), gv (B))

We continue to concentrate, for the moment, on the $ = ∧ case. It is not hard to see that the (“fully”) boolean representability concepts introduced earlier, where O1 and O2 are themselves boolean and all valuations in V are boolean, emerge as special cases of the more general concepts just defined. For example, if O1 A is A itself and O2 B is ¬B, then taking $ in (1) as ∧, # satisfies (3) for all A and B, since we may take f (v) as v for all v ∈ V and g(v) as the unique valuation v¯ satisfying v¯(C) = v(C) for all formulas C. (¯ v appeared as –R v in 1.19.4, p. 93, and will appear again in 8.11.5, p. 1172.) These are, however, very special cases, since in general we do not require that fv (A), for instance, should be h(v(A)) for some truth-function h, as it is in these cases (and likewise for g and B). The general case may appear somewhat artificially general; the definition we have chosen is designed for the sake of its connection with the concept of ∧-likeness, foreshadowed above and explained precisely below. It is the latter concept whose naturalness will have to make up for any apparent artificiality in the characterization in terms of (3) above. If # is ∧-representable according to an ∧-classical consequence relation , then is determined by a class of valuations over which # is ∧-representable, since we can take any class V of ∧-boolean valuations determining and define f and g as functions from V to the class of all valuations for the language of by putting fv (C) = v(O1 C), gv (C) = v(O2 C), for all formulas C. The converse does not hold, since not only (classically behaving) ∧, but also O1 , O2 , satisfying (1), may be absent, and yet still be determined by a class of valuations over which # is ∧-representable. The crucial property of # and is rather that # should be ∧-like according to in the sense that the following condition is satisfied by all formulas A, B, C, D: (∧-like)

A # B, C # D A # D.

(Again, we remind the reader that a more accurate – though less concise – way of putting this would perhaps be to say “and -like” rather than “∧-like”.) By an obvious re-lettering and reordering, the above condition is of course equivalent to what we should have if we changed the “A # D” on the right to “C # B”. It is clear that if # is ∧-representable according to an ∧-classical , then # is ∧-like according to in the sense of the present definition, since this then amounts to the fact that for all A, B, C, D, we have (4)

O1 A ∧ O2 B, O1 C ∧ O2 D O1 A ∧ O2 D

immediately from familiar (∧E) and (∧I) properties. But, taking up an earlier point, need not itself have the resources to make the holding of the condition (∧-like) explicit in this manner. For an extreme example, suppose we take a


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binary connective # and stipulate that is to be the least consequence relation satisfying (∧-like) on the language whose only connective is #. # then counts as ∧-like according to , even though there is no explicit conjunctive representation such as (1). Although the syntactic notion of ∧-representability according to would not apply, however, our characterization in terms of valuational semantics would apply, as we now show quite generally. Theorem 5.31.4 A binary connective # is ∧-like according to a consequence relation if and only if is determined by some class of valuations over which # is ∧-representable. Proof. ‘If’: Suppose = Log(V ) and # is ∧-representable over V , but, for a contradiction that for some A, B, C, D: A # B, C # D A # D. Thus there exists v ∈ V with v(A # B) = v(C # D) = T while v(A # D) = F (as ⊇ Log(V )). Since # is ∧-representable, say with f and g as in (3) above, this means that fv (A) = gv (B) = fv (C) = gv (D) = T; thus fv (A) = T and gv (D) = T, contradicting the hypothesis that v(A#D) = F. ‘Only if’: Suppose that # is ∧-like according to . We must show that is determined by some class of valuations V over which # is ∧-representable. We know that we can take V as Val () and have determined by V (1.12.3: p. 58), so it remains only to show how to define f and g on V (though not necessarily taking values in V ) in such a way that (3) is satisfied for all A, B. We simply let f (v) and g(v), for v ∈ V , be the unique valuations such that for all formulas E: fv (E) = T iff for some formula E , v(E # E ) = T gv (E) = T iff for some formula E , v(E # E) = T. We must check that (3) is satisfied. The ‘only if’ direction of (3) is automatic from the way f and g have just been defined. For the ‘if’ direction, suppose that fv (A) = T and gv (B) = T, with a view to showing that v(A # B) = T. Since fv (A) = T, by the way f was defined, there is some formula A with v(A # A ) = T; likewise, since gv (B) = T, there is some B with v(B # B) = T. Since, relettering (∧-like), we have A # A , B # B A # B, and v ∈ Val (), it follows, as desired, that v(A # B) = T. Let us observe in passing that if is ∧-classical and # is ∧-like according to , then we can conservatively extend by the addition of O1 and O2 satisfying (1) above (with ∧ for $), the above proof revealing a ready-made semantic account of these operators: v(O1 A) = T iff fv (A) = T, v(O2 A) = T iff gv (A) = T. (Informal, unnumbered exercise: how exactly can we be sure the envisaged extension is conservative?) Note that the explicit presence of ∧ is not essential to the present point, since we could have written, in place of (1) with ∧ for $, three conditions to the effect that A#B O1 A, A#B O2 B, and O1 A, O2 B A # B. (Cf. the comment before 5.31.4 above.) Conditions such as (∧-like) above have a striking feature: every occurrence of a schematic letter in the formulation of the condition occurs at the same depth of embedding (under #) as every other occurrence of that letter does. Let us call such a principle flat. There is a contrast, for example, with such principles as (∧I), (∧E), stated here with “#” in place of “∧”, and which we may take in the sequent-schematic form:


714 (∧I)

A, B A # B

(∧E)

A ∧ B A and A ∧ B B

Here the letter “A” occurs at a depth of embedding (under #) of 0 (i.e., unembedded) on the left of (∧I) but at a depth of embedding of 1 on the right of the “” in (∧I) – and there is a similar depth discrepancy in the other direction in the case of (∧E). Indeed it is the very much the raison d’être of the ‘intelim’ – introduction and elimination, that is – rules of natural deduction (though a similar point holds for the Left and Right sequent calculus rules also, as in 1.27) that there should be such discrepancies since with such rules we are passing – to put the point rather imprecisely – from schematically represented components to compounds (introduction) or back (elimination). In addition to its ‘flatness’, the condition (∧-like) exhibits another conspicuous feature which we shall call ‘separatedness’. A condition on how consequence relations or gcr’s treat an n-ary connective # is separated if any schematic letter (formula variable) representing the ith component in some #-compound, occurs only in the ith position in any #-compound schematically dealt with by the rule. (Understood in such a way as to exclude its occurrence not just in other positions in #-compounds but also outside of such compounds, separatedness entails flatness.) Thus in our formulation of the condition (∧-like), “A”, which occurs in the first position of (binary) # in “A # B”, occurs elsewhere (in “A # D”) but only in the first position. Similarly, “D” occurs only in second position. (“B” and “C” only put in one appearance each, so the separation requirement is automatically satisfied in their case.) If we move to gcr’s so that we can have multiple right-hand sides, we can give another separated condition for binary connectives which behave at this ‘macro-level’ like disjunction: (∨-like)

A # B A # D, C # B

By an argument similar to that given in the proof 5.31.4, except that the way to define f (v) and g(v) for the present case is fv (E) = T iff for all formulas E , v(E # E ) = T gv (E) = T iff for all formulas E , v(E # E) = T we have the following: Theorem 5.31.5 Let be a gcr. Then for any binary connective # in the language of : # is ∨-like according to if and only if is determined by some class of valuations over which # is ∨-representable. Note that in this case we can also say “if and only if # is ∨-representable over Val ()”. (Compare 3.23.8 and surrounding discussion.) The separated nature of the conditions (∧-like) and (∨-like), given their linkage with boolean representability, is due to the fact that f and g are allowed to be distinct (in (3 ): likewise O1 and O2 in (4)). For a similar connection with ↔-representability, we have a double-barrelled separated condition on gcr’s, requiring both part (1) and part (2) below: (↔-like) (1): A # B, C # D, A # D C # B

(2): C # B A # B, C # D, A # D.


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We will refer to Parts 1 and 2 of this condition as (↔-like)1 and (↔-like)2 , respectively. The property of ↔-likeness has of course been specified in such a way as to provide, for gcr’s, a syntactic correlate of ↔-representability (over a determining class of valuations). This time the argument is somewhat different from that for the case of ∧-likeness, so we give it in full. Theorem 5.31.6 A binary connective # is ↔-like according to a gcr if and only if is determined by some class of valuations over which # is ↔representable. Proof. ‘If’: it is clear enough that if = Log(V ) and # is ↔-representable over V , the condition (↔-like) is satisfied. (We look at (↔-like)2 more closely in this regard in the discussion following 5.33.8 below.) ‘Only if’: Again we show that if satisfies (↔-like), then # is ↔-representable over Val (). So we must show how, for v ∈ Val (), to define the valuations f (v) and g(v) in such a way that for all formulas A, B, v(A # B) = T iff fv (A) = gv (B). (This is a minor reformulation of what (3) dictates for the present case: v(A # B) = ↔b (fv (A), gv (B)).) We define f and g after a subdivision of cases. Case (a): for no C, D, do we have v(C # D) = T. In this case put f (v) = vT , g(v) = vF . The requirement that v(A # B) = T iff fv (A) = gv (B) is satisfied because always v(A # B) = T and vT = vF . Case (b): for some C, D, we have v(C # D) = T. Then fix on a particular pair C, D, for which this is the case, and define f , g in terms of them thus (for all E): fv (E) = T iff v (E # D) = T gv (E) = T iff v (C # E) = T. We must show that for all A, B, if v(A # B) = T then fv (A) = gv (B), and conversely. So first, suppose v(A # B) = T, and fv (A) = T. Thus v(A # D) = T, by the way f was defined. But then, since we already know that v(C # D), we have v verifying all the formulas schematically represented on the left of the “” in (↔-like)1 , and v must verify the formula on the right (as v ∈ Val ()), i.e., v(C # B) = T, which means that gv (B) = T. We must also show that if v(A#B) = T and gv (B) = T, then fv (A) = T. Unpacking via the definitions of f and g again leads to an appeal to (↔-like)1 , in a suitably relettered form. This concludes the argument that if v(A # B) = T then fv (A) = gv (B), and we turn to the converse. So suppose fv (A) = gv (B) = T with a view to showing that v(A # B) = T; again unpacking the supposition leads through another appeal to (↔-like)1 , again appropriately relettered, to the desired conclusion. Finally, suppose fv (A) = gv (B) = F, with a view to showing that v(A # B) = T. Since fv (A) = gv (B) = F, we have v(A # D) = F and v(C #B) = F; recall also that v(C # D) = T. Now consider the following relettering of (↔-like)2 : C # D A # D, C # B, A # B. Since v ∈ Val (), the truth on v of the formula on the left and the falsity of the first two listed on the right force v(A # B) = T. The fact that (↔-like)1 is appealed to three times in the above proof and (↔-like)2 only once reflects no asymmetry in the subject matter, just an artifact of the choice of proof-strategy. To see this, the reader is invited to define f (v) and g(v) instead in terms of formulas C and D for which v(C # D) = F.


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Remark 5.31.7 The conditions (↔-like)1 and (↔-like)2 would seem to merit individual attention, whereas here we have just considered them in combination. If we consider IL instead of CL , we note that the first condition is satisfied, taking # as ↔ itself, but the second is not. (There is an issue about what gcr might be intended by such a reference to IL , on which see the discussion leading up to 6.43.8 (p. 899) below, but which we should like to avoid here: so simply consider the commas on the right as replaced by “∨”.) A similar point holds for strict equivalence in any of the normal modal logics reviewed in 2.21. Though we shall not give any further consideration to (↔-like)2 , we shall treat (↔-like)1 in 5.33, in the discussion from 5.33.5 to 5.33.8. An interesting way of writing the $-likeness conditions above ($ = ∧, ∨, ↔) for connectives # becomes available if we introduce an operation taking a pair of #-compounds to a #-compound. We shall borrow the splice notation “ ” from 0.26 for this purpose because of a family resemblance between the operations denoted in the two cases. (This is approximately the relation between transitivity and a ‘lazy’ version of transitivity introduced under the label [∧] in 5.33 below.) If E is the formula A # B and E is the formula C # D, then E E is to be, we stipulate, the formula A # D. Note that is not a new binary connective; the main (and perhaps sole) connective of (A # B) (C # D) is #, since what we have just given is simply a new way of referring to the formula A # D. (See the discussion in 1.11.1–4 (pp. 49–52) The present point is not simply that E E is not defined for formulas E, E , in general but only for #-compounds E, E . That would be compatible with ’s being a ‘partial’ connective – along the lines of partial operations in what would otherwise be algebras in the sense of 0.21. The point is rather that even in the cases for which it is defined, E E remains steadfastly a formula whose main connective is just #.) Similarly, where E is any #-compound, E E is not another formula equivalent to E but is the formula E itself, as is, for any other #-compound E , the formula (E E ) E – which we could just as well write without the parentheses since is associative. Then we can write (∧-like) as:

E1 , E2 E1 E2 ,

(∨-like) as:

E1 E2 E1 , E2 ,

(↔-like)1 as:

E1 , E2 , E1 E2 E2 E1 ,

(↔-like)2 as:

E2 E1 E1 , E2 , E1 E2 .

By an evident symmetry “E2 E1 ” and “E1 E2 ” could be interchanged in this last formulation. (Not that the operation is commutative, however: whereas (A # B) (C # D) is A # D, (C # D) (A # B) is C # B.) Our admonitions on the non-connectival status of notwithstanding, there is an intriguing resemblance between this latest formulation of (∧-like) and the ∧-Introduction style condition of ∧-classicality, laid down here for : E1 E2 follows from E1 together with E2 (by the lights of according to which # is ∧-like). This latest formulation of (∨-like) bears a similar relation to the ∨Elimination style condition of ∨-classicality (for a gcr). A similar remark does not apply in the case of the two ↔-likeness conditions, however. A propos of these last two, we may make the following observation. Instead of the formulation of (↔-like)1 given above, we could equally well have written

5.3. ∧-LIKE CONNECTIVES AND THE LIKE (↔-like)1 as:

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E1 E2 , E2 E1 , E1 E2 .

Though the equivalence of the two formulations is clear enough if one thinks explicitly in terms of the underlying #-formulas, it is worth demonstrating (at least one direction of) it just at the level of manipulating the representations. Starting with the earlier formulation of (↔-like)1 , and putting E1 E2 for E1 and E2 E1 for E2 (a replacement which is perfectly legitimate since the original formulation is supposed to hold for all E1 , E2 ), we get: (5)

E1 E2 , E2 E1 , (E1 E2 ) (E2 E1 ) (E2 E1 ) (E1 E2 ).

Rebracketing in view of the associativity of , and omitting some parentheses in the interests of readability: (6)

E1 E2 , E2 E1 , E1 (E2 E2 ) E1 E2 (E1 E1 ) E2 .

So by the idempotence of : (7)

E1 E2 , E2 E1 , E1 E2 E1 E2 E1 E2 .

Finally, as we noted above, E E E = E, so (7) can be further simplified to the second -formulation of (↔-like)2 above: (8)

E1 E2 , E2 E1 , E1 E2 .

Exercise 5.31.8 Show how, given the properties of already noted, the first -formulation of (↔-like)1 : E1 , E2 , E1 E2 E2 E1 can be derived from (8). We return to this operation and its algebraic properties in 5.33 (following 5.33.9). Exercise 5.31.3(i) asked for a proof that if a binary truth-function was both $1 - and $2 -representable, where $1 , $2 are any two out of ∧, ∨ and ↔, then it was also $3 -representable, where $3 was the remaining element of {∧, ∨, ↔}, and in 5.31.3(ii), for a proof that this situation arises precisely when that truthfunction is not essentially binary. One may wonder what becomes of these facts for $-likeness in place of $-representability. It is true that we have semantic characterizations of the relevant $-likeness in terms of $-representability over a determining class of valuations (Thms. 5.31.4–6), but the latter notion was significantly more general than the special case of ‘fully boolean representability’, as we called it. All that $-representability (in (3 )) required was that there should be functions f and g from valuations (in the determining class) to (arbitrary) valuations such that for all A, B, and all v in the determining class: v(A#B) = $b (fv (A), gv (B)). The special ‘fully boolean’ case (which is what the notion of $-representability of a truth-function, rather than of a connective, automatically builds in) is that f and g should satisfy the very strong condition that there are 1-ary truth-functions f0 and g0 such that for all formulas C: (9) fv (C) = f0 ((v(C))

gv (C) = g0 (v(C)).

ˆ defined on valuations in [Def. #] ˆ in 7.31 satisfy this condition The functions # (and for n-ary # where n 2, an analogous condition), but in our discussion


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here we have not required it – and indeed, couldn’t have done so without there arising a mismatch between $-representability over a determining class of valuations and $-likeness (since the theorems in question give necessary and sufficient conditions). The stronger condition here envisaged amounts–to take the case of f for illustration–to requiring that there should be f (and g. . . ) such that for all v in the determining class, and all formulas A and B: (10) v(A) = v(B) ⇒ fv (A) = fv (B) In fact, we did not even impose the much weaker condition, formulated here in terms of the “H” notation from 3.33, where V is the class of valuations in question: (11) HV (A) = HV (B) ⇒ fv (A) = fv (B) which would correspond to imposing a congruentiality requirement on the # claimed to be $-like iff determined by a V over which # is $-representable. Similarly, (10) corresponds to a requirement of being ‘fully determined’ (in the sense of 3.11) according to the in question. There is of course an intermediate case, in the shape of extensionality (§3.2), to the subject of which we shall return in 5.35. What effect does this more liberal perspective (not insisting on fully boolean representability) have on the results of Exercise 5.31.3? It turns out that part (i) is not affected, while part (ii) is. That is, if any two of the three claims in (12) are correct, so is the third; but in that case, it does not follow that # is not essentially binary: (12) # is ∧-like according to (gcr) ; # is ∨-like according to ; # is ↔-like according to . What does it mean to say, as here, that if all three claims are correct for # and , # can still be essentially binary according to ? We may take a connective to be essentially n-ary according to a consequence relation or gcr, when the associated Lindenbaum-algebraic operation (defined on -equivalence classes of formulas) has this property. (Cf. 3.34, p. 497 onward.) Remark 5.31.9 In the case of n = 2, this amounts to the following definition. Binary # depends on its first argument according to when it is not the case that for all A, B, C, we have A # C B # C and analogously for dependence on the second argument. Note that we could equivalently say (to take the first argument case) that it is not the case that for all A, B, C, we have A # C B # C. Finally, # is essentially binary (according to ) just in case # depends (according to ) on its first argument and also on its second argument. It will not be shown here that there are connectives # and gcr’s for which all claims in (12) are correct and yet for which # is still essentially binary according to (thereby showing that the analogue in the present setting of 5.31.3(ii) fails), since an example of precisely this kind arises naturally in 5.35 below, as we shall point out when we get there (5.35.10).


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We content ourselves here with showing that there is no similar breakdown for the analogue of 5.31.3(i). Let us remind ourselves of the three conditions involved: (∧-like)

A # B, C # D A # D

(∨-like)

A # B A # D, C # B

(↔-like): (↔-like)1

A # B, C # D, A # D C # B and

(↔-like)2

C # B A # B, C # D, A # D.

We recall that in place of “A # D” on the right in (∧-like), we could equally well have written “C # B”. Now (↔-like)1 follows from this by (M), ‘thinning in’ A # D on the left. Similarly, if we reletter (∨-like) by interchanging “A” and “C”, we obtain (↔-like)2 by thinning-in A # D on the right. Thus if # is both ∧-like and ∨-like according to , then # is ↔-like according to . Similarly, if # is ∨-like and ↔-like1 according to , we get, by an appeal to (T) as applied to those conditions without relettering, the relettered version of (∧-like) alluded to above. (Here A # D is the cut formula, and strictly speaking we should appeal explicitly to (M) and (T).) Exercise 5.31.10 Show that if # is ∧-like and also ↔-like2 according to , then # is ∨-like according to . The solution to this exercise, along with the foregoing discussion, establishes: Observation 5.31.11 If any two of the conditions listed under (12) above are satisfied, so is the third. This is the promised analogue of 5.31.3(i). Under the same hypotheses as figure in 5.31.11, there is an interesting ‘recombination of components’ property available (5.31.12(iii)): Exercise 5.31.12 (i) Show that if is a gcr according which # is ∧-like, then A1 #B1 , . . . , Am #Bm Δ iff A1 # Bp(1) , . . . , Am #Bp(m) Δ, where p is any permutation of {1,. . . ,m}. (ii ) Show that if is a gcr according to which # is ∨-like, then Γ C1 # D1 , . . . , Cn # Dn iff Γ C1 # Dq(1) , . . . , Cn # Dq(n) , where q is any permutation of {1,. . . ,n}. (iii) Conclude from (i) and (ii ) that if # is both ∧-like and ∨-like according to then: A1 # B1 ,. . . ,Am # Bm C1 # D1 , . . . , Cn # Dn iff A1 # Bp(1) , . . . , Am # Bp(m) C1 # Dq(1) , . . . , Cn # Dq(n) for any permutations p, q, as in (i), (ii ).


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We close with a special case of the notion of a flat condition, introduced above. Call a formula a simple conjunction if it is of the form pi ∧ pj , where we do not exclude the possibility that i = j. Let be the least consequence relation on the language with ∧ as sole connective, satisfying (with ∧ for #) the condition that ∧ is ∧-like according to and also that ∧ is commutative according to . Observation 5.31.13 With as just defined, and all formulas in Γ ∪ {A} being simple conjunctions, Γ A if and only if Γ CL A. A proof of the less obvious ‘if’ part of this result can be obtained by reflection on a necessary (and indeed sufficient) condition for Γ CL A where all formulas involved are simple conjunctions, in which case – taking Γ as finite without loss of generality – we may write the CL -claim as: q1 ∧ q1 , . . . , qn ∧ qn CL qn+1 ∧ qn+1 . Then by reflection on the behaviour of ∧-boolean valuations we can see that such a claim is correct (if and) only if qn+1 is one of the variables qi or one of the variables qi with i n, and likewise for qn+1 . Exercise 5.31.14 (i) Using the above condition, complete the proof of 5.31.13. (ii) Do the combined conditions of commutativity and ∧-likeness, figuring implicitly in 5.31.13, uniquely characterize the connective they govern (when expressed in the ‘rule’ format of §4.3)? (Explain your answer.) (iii) Replacing the reference to ∧ in the above definition of a simple conjunction, by one to →, we obtain the notion of a simple implication. Find conditions playing the role of (∧-like) and commutativity to fit a result for simple implications analogous to 5.31.13. (iv) What conditions would have been suitable had (iii) concerned the generalized consequence relation of classical logic, as opposed to the consequence relation (again, relating sets of simple implications)?

5.32

Non-creative Definitions

The present subsection departs from our usual concentration on purely sentential logic, though some of the points could have been cast in that setting instead. The reason for its appearance in the present section will emerge below (5.32.8) as we bring to bear the classification of 5.31 on the main connective of a (would-be) definition. Accordingly, we begin by reviewing the main lines of the theory of definition, as expounded for instance in Suppes [1957], Chapter 8, and with special attention to the introduction of new predicate symbols expanding the language of a first-order theory. (For definiteness, take the underlying logic to be classical predicate logic; for many points, quantified IL–for example–would do just as well.) Here we are envisaging a point in the development of a theory at which a new item of non-logical vocabulary is to be added by a definition which defines it in terms of existing vocabulary. The new item may be a predicate symbol or a function (‘operation’) symbol, but here we are concerned with the case in


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which it is a predicate symbol. It is traditional to extol two desiderata, embodying conditions first isolated by Leśniewski (who did not himself regard them as desiderata, however): non-creativity and eliminability. The criterion of noncreativity says that the extension we obtain after adding a definition should be a conservative extension – as this term applies to theories rather than (as in §4.2 and elsewhere) logics – of the original theory: that is, that nothing new should be provable which does not involve the newly added vocabulary. A definition which does result in the provability of some such formulas being provable is said to be creative. A rough and ready gloss on the criterion of eliminability runs as follows. The defined expression–that introduced by the definition–should be ‘eliminable’ in the sense that it is possible to prove any formula containing the newly defined expression equivalent to some formula not containing it (thereby ‘eliminating’ the defined expression). This is rough and ready because it would be preferable to have a formulation which does not allude – as this one does with the reference to equivalence – to any particular connective; but a theory-relative notion of synonymy (much as in the proof of 3.23.9) would serve us here. In any case, our main interest will be on non-creativity, since we have, with the discussion of synonymy, extensionality, and congruentiality, seen enough of the ‘eliminability’ side of things. One important point to note is that the two criteria pull in opposite directions: to be non-creative, a definition must not allow too much to be provable, while to satisfy the eliminability criterion, a definition must not allow too little to be provable. A second important point is that both criteria are satisfied by the familiar explicit definitions, which in the case of predicate symbols take the following form. Following Suppes’ discussion, we call the principles governing the construction of such definitions, rules for proper definitions. In the case of an n-ary predicate symbol P , a proper definition should take the form ∀v1 , . . . , ∀vn (Pv 1 . . . vn ↔

)

where some formula fills the blank and: (i) the variables v1 , . . . , vn are all distinct, (ii ) no variables other than v1 , . . . , vn occur free in the formula filling the blanks, (iii) the formula filling the blanks does not contain P . If restriction (i) or restriction (iii) is disregarded, there is a risk that the criterion of eliminability will not be satisfied. (In the case of (i), we will be able to use the rhs to provide an equivalent for Paa, for example–taking n as 2–if v1 is v2 – but not Pab, so this occurrence of “P ” is not eliminable. Violations of (iii) are circular definitions and when we try to use the rhs to replace the defined term P , we end up with something still containing P .) If restriction (ii ) or – again – restriction (iii) is disregarded, there is a risk that the criterion of non-creativity will not be satisfied. In the case of (iii), for example, we might add the would-be definition ∀x(Px ↔ (Rxx ∧¬Px )) to a theory whose non-logical vocabulary comprises just the two-place predicate letter R (for instance, the theory of an arbitrary transitive relation) in which case the P -free formulas ∀x(¬Rxx ) would emerge as a potentially – and for


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the instance just mentioned, actually – creative result. As for violations of restriction (ii ), it may not be clear what we have in mind here, since some texts (e.g., Lemmon [1965a]) disallow the presence of free variables in formulas; Suppes [1957] allows them, understanding them as tacitly universally quantified (by occurrences of ∀ at the start of the whole formula). Thus a typical violation of (ii ) would be the attempt to define the one-place predicate symbol P with the would-be definition: ∀x(Px ↔ Rxy) where R is a two-place predicate symbol in the original language of the theory. Here v1 is x but there is a disallowed free variable y on the right. The whole thing is interpreted as meaning: ∀y∀x(Px ↔ Rxy), which is potentially creative as a mode of introduction of P , since it has the P -free consequence ∀y∀z(Rxy → Rxz ), which may not have been provable in the original theory. (If it had been, then the upshot of the above ‘improper’ definition would be secured by the ‘proper’ ∀x(Px ↔ ∃y(Rxy)) or to the–relative to such a theory, equivalent–result of putting “∀y” for “∃y” here.) An interesting point – an ‘existence’ issue for quantifiers (cf.§4.2) – emerges incidentally from this example: Remark 5.32.1 Consider the possible existence of a quantifier Q such that for all formulas B(y) (possibly containing y free) and A (not containing y free): ∀y(A ↔ B(y)) A ↔ Qy(B(y)) If we take as classical (predicate) logical consequence, such a Q could be taken as “∀” if the “↔” were simply “→”, and as “∃” if it were, instead, “←”; but what about “↔” itself? Clearly, if we were able to define, in terms of the logical vocabulary of quantified CL, such a Q, then we could circumvent the restriction (ii ) and have a ‘proper definition’ satisfying that restriction but still–in view of the example given above– creative. Since, as already remarked, the conditions (i)–(iii) guarantee non-creativity, no such Q is definable. Exercise 5.32.2 Show that adding a new quantifier Q to classical predicate logic satisfying the condition at the start of 5.31.1 would result in a non-conservative extension. Hint: Put Qy(B(y)) for A. It helps to visualize this in a simple concrete case, so take B(y) as Py. The result of the substitution just suggested is that we have ∀y(Qy(Py) ↔ Py) and thus: Qy(Py) ↔ Pa


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as well as Qy(Py) ↔ Pb. It is not hard to see how to derive the Q-free but otherwise unobtainable ∃y(Py) → ∀y(Py) from here. Remarks 5.32.3(i) The appeal at the start of ‘hint’ in the preceding exercise to the provability of ∀y(Qy(Py) ↔ Py) presupposes that the latter is well-formed, whereas on accounts such as that of Lemmon [1965a] it is not, since a quantifier binding (some occurrences of) a given variable is not there allowed to lie in the scope of a quantifier binding (other occurrences of) that same variable; with such a restriction in force, the argument hinted at needs to allow relettering of bound variables, so that this had better be understood as part of the meaning of “quantifier”. (ii ) The mirror image of the condition in 5.32.1, with Q and ∀ interchanged, is also of considerable interest. Here one would ask whether there exists in classical predicate logic, or can be conservatively added thereto, a quantifier Q for which we have generally Qy(A ↔ B(y)) A ↔ ∀y(B(y)). To show the non-conservativeness of the extension one needs to assume that Q is ‘congruential’ in a sense illustrated by the following demonstration. (It certainly follows that no Q satisfying the condition just inset is already definable in terms of the usual first order logical vocabulary.) Begin as before, using the direction of the condition to conclude that Qy(∀y(Py) ↔ Py) (or instead Qx(∀y(Py) ↔ Px ), in case Lemmon’s restriction as in (i) above is in force; but we shall proceed with the original formulation). Now we use the congruentiality condition: since a biconditional is CL-equivalent to the result of negating both sides, we make a replacement and conclude that Qy(¬∀y(Py) ↔ ¬Py). But from this, using the direction of the condition on Q, we get: ¬∀y(Py) ↔ ∀y(¬Py), establishing non-conservativeness. (iii) The example we have just discussed under (ii ) shows that the Prenex Normal Form theorem–to the effect that every formula has an equivalent in which no quantifier occurs within the scope of a sentence connective–fails for the equivalential fragment of classical predicate logic (only ∀, ∃ and ↔ as logical vocabulary); for in general A ↔ ∀y(Py), with A some closed formula, lacks a prenex equivalent (in the fragment in question). Before the above interlude, it was remarked that the introduction of a new predicate letter P by means of the formula ∀y∀x(Px ↔ Rxy) – which can be thought of as a new axiom – was potentially creative, meaning that for some theories (formulated in a language without P ) it could lead to previously underivable P -free consequences. It is more convenient to work with the property complementary to potential creativity (in that sense), which we shall call universal non-creativity. That is, we define a formula, understood as introducing the new symbol P , to be universally non-creative just in case any theory (whose language does not contain P ) to which the formula is added as a new axiom proves no additional P -free theorems. Now we can put the point about the non-creativity criterion being satisfied by a proper definition like this:


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Observation 5.32.4 If D is a proper definition (satisfies the conditions (i)– (iii) above) of P, then D is universally non-creative. Proof. If D is a proper definition, it has the form ∀x(Px ↔ A(x)), where A(x) contains at most x free, and does not contain P . Suppose Γ is a first-order theory in a language without P . We must show that if Γ, D B and B does not contain P , then Γ B. For any formula C define C* to be the formula resulting from C by replacing every subformula of the form Pt, t any term (constant or variable) by A(t), the formula we get by putting t for such free occurrences of x as A(x) may contain. Note that for any set of formulas Δ and any formula E, if Δ E then Δ* E*, where Δ* = {C* | C ∈ Δ}, since is here taken to be just classical predicate-logical consequence. Returning to the main argument, then, suppose that Γ, D B, with Γ, D and B as above. By the principle just noted, we have Γ*, D* B*. But P does not occur in (any formula in) Γ or in B, so Γ* = Γ and B* = B; thus Γ, D* B. What is more, D* is just the formula ∀x(A(x) ↔ A(x)), so D*. Therefore (by (T)) Γ B, i.e. B is a theorem of Γ, as was to be shown. As was remarked earlier, condition (i) on proper definitions pertains to exclusively to the eliminability criterion; in any case, since we are considering for illustrative purposes only the case of a one-place predicate letter, there is no possibility of a transgression on this score. We should note the role of conditions (ii ) and (iii) in the above proof, since they were not explicitly mentioned. Consideration of the example given before 5.32.1 of a violation of (ii ) reveals that the implication Δ E ⇒ Δ* E* appealed to in the proof is not guaranteed to hold if * maps a formula C the result of putting A(t) for Pt throughout, when A(t) contains additional free variables (additional to t, that is, should t be a variable). The point at which satisfaction of (iii) is tacitly exploited is in the claim that D*, and again we can see this from an earlier example, the case of the would-be definition D of P by means of ∀x(Px ↔ (Rxx ∧ ¬Px )). In this case, D* is ∀x((Rxx ∧¬Px ) ↔ (Rxx ∧¬(Rxx ∧ ¬Px ))), and so we do not have D*. Remarks 5.32.5(i) It may be wondered why we are stressing the outright provability of the formula D*, when all we need for non-creativity in the case of a particular theory Γ is that Γ D* (since this together with the fact, as at the end of the proof of 5.32.4, that Γ, D* B, would suffice for the desired conclusion: Γ B). The reason is that we are considering universal non-creativity so we need to allow for arbitrary Γ – including in particular Γ = Cn (∅). (ii ) Notice that the crucial features of the proof of 5.32.4, namely that for the mapping * therein, we have Δ E ⇒ Δ* E* (all Δ, E), and that the definition D should be such that D*, are by no means restricted to the choice of classical first-order logical consequence as ; for example, intuitionistic predicate logic would do equally well.


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So much for requirements (i)–(iii) on proper definitions. We turn to an unnumbered requirement built into the form of a would-be definition, built into the schematic representation of a candidate definition by the use of the “ ↔”. Why this specific connective? Again, we shall focus on satisfaction of the noncreativity criterion rather than the criterion of eliminability. And we continue to illustrate with reference to the case of introducing a new one-place predicate symbol P into a theory with classical predicate logic as the background logic. Let us say that a formula ∀x(Px # A(x)) is a generalized definition of P provided that # is a binary boolean connective and A(x) a formula not containing P or any free variables other than x. This is a very slight generalization of the usual notion, in fact, and many definition-like modes of introducing new vocabulary which can be shown to be satisfy the criterion of non-creativity or that of eliminability, and perhaps both, fail to qualify as generalized definitions in this narrow sense. To see this, consider the obvious fact that if D E and D is universally non-creative, then E (which is, if anything, logically weaker ) is certainly non-creative. Examples 5.32.6(i) Let D1 be ∀x(Px ↔ Fx ) ∨ ∀x(Px ↔ Gx )). This is not a generalized definition but it follows from each of its two disjuncts, and each of these is a proper (‘explicit’) definition; thus D1 is non-creative in virtue of the point just made in the text, together with 5.32.4. (ii ) Let D2 be ∀x((Px ↔ Fx ) ∨ (Px ↔ Gx )). Then again, though not a generalized definition, D2 is universally non-creative for the same reasons as in (i). Cases like that of 5.32.6(i) have been of considerable interest in what is called the theory of definability rather than (our current concern) the theory of definition. (In the former case, one is interested in looking at a theory with P already in its vocabulary, and asking if the theory could be cut back to one into which P can be introduced by a definition; we are instead concerned only with what conditions such a definition should meet, without regard to whether the result of giving one coincides with some antecedently specified theory in a language already containing P . See Rantala [1991] for a discussion of both sorts of issue. The interest of such cases of 5.32.6(i) from the definability perspective is that they show P to be ‘piecewise’ or ‘model-wise’ definable: in each model of a theory in which D1 is provable, P is co-extensive with some P -free open formula.) As for 5.32.6(ii ), the reader may well wonder whether, although not itself a generalized definition, D2 might not be logically equivalent to a generalized definition. Rather than interrupt the proceedings here, we postpone consideration of this question until the end of this subsection. Now although Examples 5.32.6 illustrate the possibility of weakening a proper definition (a generalized definition in which # is ↔) and obtaining a non-creative formula which is not a generalized definition, it is noteworthy that the noncreativity of certain generalized definitions does follow straightforwardly from the point about ‘weakening’: Example 5.32.7 Suppose for binary # we have p ↔ q CL p # q; then the generalized definition ∀x(Px # A(x)) is universally non-creative, because it is a consequence (in classical predicate logic) of the explicit definition ∀x(Px ↔ A(x)); thus for instance: ∀x(Px → (Fx ∧ ∃y(¬Rxy))) is non-creative since, by 5.32.4,


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∀x(Px ↔ (Fx ∧ ∃y(¬Rxy))) is non-creative. Likewise for ∀x(Px ← (Fx ∧ ∃y(¬Rxy))) (i.e., ∀x((Fx ∧ ∃y(¬Rxy)) → Px )). Note that the implications here can be rewritten as disjunctions, to bring matters into line with our classification in 5.31. Since we already (in effect) know about ↔, we can sum up the situation with: Theorem 5.32.8 Suppose #b is essentially binary. Then the following are equivalent: (1) #b is ↔-representable or ∨-representable. (2) Every generalized definition of the form ∀x(Px # A(x)) is universally non-creative. Proof. Let #b be essentially binary. (1) ⇒ (2): if #b is ↔-representable then #b is ↔b or else . If D = ∀x(Px # A(x)) b has ↔ for #, then D is universally non-creative by 5.32.4; if # is , we can rewrite D so that # is instead ↔ again (put ∀x(Px ↔ ¬A(x)) and so again D is non-creative. If #b is ∨-representable, then we have one of four cases: (i) D is ∀x(Px ∨ A(x)) (ii ) D is ∀x(¬Px ∨ A(x)) (iii) D is ∀x(Px ∨¬A(x)) (iv ) D is ∀x(¬Px ∨¬A(x)). D in (i) and (iv ) follows from putting # = and so since the latter, as we saw already in the proof, is universally non-creative, so is the former. Similarly (cf. 5.32.7) D as in (ii ) and (iii) follows from what we get by putting # = ↔, and thus is universally non-creative. (2) ⇒ (1): Suppose that #b is not ↔-representable or ∨-representable. We must show how to find A(x) for which D = ∀x(Px # A(x)) is not universally non-creative. Since #b is neither ↔- nor ∨-representable, #b is ∧-representable, and further, must be one of the essentially binary cases in which (i) D is ∀x(Px ∧ A(x)) or (ii ) D is ∀x(¬Px ∧ A(x)) or (iii) D is ∀x(Px ∧¬A(x)) or (iv ) D is ∀x(¬Px ∧¬A(x)). In each case, put A(x) = Fx (F some monadic predicate letter distinct from P ), and note that each generalized definition listed is creative when adjoined to the theory Cn(∅). (We could just as well use Cn({∃x(Fx )}).) Thus with D as in (i), D ∀x(Fx ); with D as in (iv ), we have D ∀x(¬Fx ); etc. Thus, amongst the essentially binary boolean connectives, the risk of creativity is presented only in the ∧-representable case. Exercise 5.32.9 What happens in respect of universal non-creativity for generalized definitions ∀x(Px # A(x)) in which # is not essentially binary? (Recall that there are six cases to deal with.)


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Exercise 5.32.10 Give a ‘universal’ version of the criterion of eliminability and consider how, with reference to the classification {∧-representable, ∨-representable, ↔-representable} the # of a generalized definition ∀x(Px # A(x)) must be to secure this property. We are now in a position to take up a question raised earlier, as to whether the example D2 = ∀x((Px ↔ Fx ) ∨ (Px ↔ Gx )) of 5.32.6(ii ), was equivalent to a generalized definition. In fact, for all that we have said, D2 might be logically equivalent to a traditional explicit definition; that is, to a generalized definition ∀x(Px # A(x)) in which # is ↔ itself. The issue raised here is essentially one of sentential logic. Instead of asking when a universally quantified formula containing Px as a subformula, possibly several times, is equivalent to one of the form ∀x(Px # A(x)), with P not occurring in A(x), we may as well ask when a formula of sentential logic, A(p, q1 , . . . , qn ), containing only the propositional variables exhibited, is equivalent (in the sense of CL ) to a formula of the form p # B(q1 , . . . , qn ), where the second component, B, contains again only the sentence letters exhibited. (The subscripted qi are meant to be just so many distinct propositional variables; likewise with similar notations below. No connection with our official once-and-for-all enumeration of all the variables as p1 , . . . , pn , . . . Similarly, whereas usually we take p to be p1 of the latter enumeration, for the present discussion it is simply some variable distinct from the qi .) When some such equivalent can be found, let us say that p is extractible from the original formula A; in the desired equivalent, p # B(q1 , . . . , qn ), we describe p as occurring extracted. Thus, trivially, if p occurs extracted in a formula, then it is extractible from that formula, but in general the converse is not the case: Example 5.32.11 The variable p does not occur extracted in the formula (p ∨ q1 ) ∧ (p ∨ q2 ) but p is extractible from this formula since the formula is equivalent to p ∨ (q1 ∧ q2 ), in which p does occur extracted. (Recall that the notion of equivalence operative here is CL .) The significance of extractibility in the special case in which the main connective in the formula in which p occurs extracted is ↔, is that it is here that the * substitution argument relied in proof of 5.32.4 applies. Substituting A(x) for Px in D (to produce D*) here becomes a matter of substituting something-orother for p in B(p, q1 , . . . , qn ): we want to find the something or other–which we can do only if p is extractible, in which case the other component of the formula in which p occurs extracted, namely B(q1 , . . . , qn ), is the something-or-other which we can substitute for p. (Of course whether the whole of the argument will go through in general also depends on the particular # connecting p with this other component. One can glean from 5.32.8 what the prospects are for various choices of #.) The remainder of our discussion in this subsection will be devoted to extractibility as a topic of some interest in its own right. Now, erasing quantifiers and variables from D2 (of 5.32.6(ii)) to arrive at a propositional formula we get A(p, q, r) = (p ↔ q) ∨ (p ↔ r). (For convenience, we write q, r, for q1 , q2 .) Is p extractible from this formula? Recall an affirmative


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answer means that for some formula B we have A(p, q, r) CL p # B(q, r) where the “q, r” in parentheses indicates that only these variables appear in B. If so, then we can rewrite D2 as ∀x(Px # B(Fx, Gx )). Note that B(q, r) is itself a boolean compound of q and r, so we can think of the present question as whether, as well as #, there is a binary boolean connective # , say, for which A(p, q, r) p # (q # r). To this question, we can return a negative answer. For consider the truthtable of the formula A(p, q, r) (i.e., (p ↔ q) ∨ (p ↔ r)); it looks like this: A T T T F F T T T

(p, T T T T F F F F

q, T T F F T T F F

r) T F T F T F T F

(1) (2) (3) (4) (5) (6) (7) (8)

Of course, to check this table, one would need to write A(p, q, r) out as (p ↔ q) ∨ (p ↔ r), and do the calculation. From lines (1), (2), (4), (5) and (6), we can show that a representation in the form p # (q # r) is impossible. For suppose there is such a representation. First, consider lines (5) and (6); they tell us that F #b (T #b T) = F and F #b (T #b F) = T. It follows that T #b T = T#b F, since otherwise F #b (T #b T) and F #b (T #b F) would have to be the same. Thus one of T #b T, T #b F, is T and the other is F. It doesn’t matter which is which. Consider now lines (1) and (2), which tell us that T #b (T #b T) = T = T #b (T #b F). So, since one of T #b T, T #b F, is T and the other is F, we know that T #b x = T for x = T and for x = F. Thus in particular, taking F #b F as x, we must have T#b (F#b F) = T, which contradicts line (4) of the table. This contradiction establishes that p is not extractible from the formula (p ↔ q) ∨ (p ↔ r). Since, as we have just seen, there are no # and B(q, r) for which (p ↔ q) ∨ (p ↔ r) p # B(q, r), there is, in particular, no way of choosing B so that this equivalence holds when # is ↔, and this shows that our original formula D2 , ∀x((Px ↔ Fx ) ∨ (Px ↔ Gx )), is not logically equivalent to an explicit definition of P in terms of F and G. (If it were, then, the equivalence would mean the existence of a sentential analogue of the kind we have just ruled out.) The idea of extractibility is clearly a special case of the following more general notion. Suppose A(q1 , . . . , qm ,r1 , . . . , rn ) is a formula of a sentential language, built up from the propositional variables indicated. Then the question arises as to whether that language provides a binary connective # and a pair of formulas


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B(q1 , . . . , qm ), C(r1 , . . . , rn ), again in the variables indicated, with the property that for some consequence relation (or gcr) on the language in question, we have A B # C. In other words, can A (or the proposition expressed by A) be represented as a compound one of whose components draws on the one set of variables, the other on the other? The special case of extractibility of (say) q1 from A is the case in which m = 1. Exercise 5.32.12 Can the following formulas be represented in the form B # C for boolean #, with B constructed from p and r, and C from q and s? (Give an equivalent of the desired form if it exists; otherwise, a proof of non-existence.) (i) (p ↔ q) ∧ (r ↔ s) (ii ) (p ∧ q) ↔ (r ∧ s) (iii) (p ↔ q) ↔ (r ↔ s) (iv ) (p ∨ q) → (r ∧ s).

5.33

#-Representable Binary Relations

Here we look briefly at some analogues for binary relations of the properties of binary connectives distinguished in 5.31; for fuller discussion, consult the references in the end-of-section notes (p. 765). Where S and T are (not necessarily disjoint, or even distinct) sets, we call R ⊆ S × T respectively an ∧-representable, ∨-representable, or ↔-representable relation just in case the respective condition below is satisfied for some S0 ⊆ S, T0 ⊆ T : [∧-representable]

For all s ∈ S, t ∈ T : sRt iff s ∈ S0 & t ∈ T0 ,

[∨-representable]

For all s ∈ S, t ∈ T : sRt iff s ∈ S0 or t ∈ T0 ,

[↔-representable]

For all s ∈ S, t ∈ T : sRt iff s ∈ S0 ⇔ t ∈ T0 .

The first condition here – which we had occasion to consider already on p. 505 – could equally well be put more elegantly: R = S0 × T0 , for which reason (think of the ordered pairs here collected as the coordinates of points in the plane) ∧-representable relations are often referred to as rectangular relations. But this rather obscures the parallel with ∨-representability and ↔representability, which we are especially concerned to emphasize. And further, the whole notion of ordered pairs, as individuated by the principle that they are identical when their first coordinates are identical and their second coordinates are identical, ‘buries’ an employment of conjunction (rather than, say, disjunction or material equivalence) which deserves more explicit attention. We shall give it that attention in 5.34 (p. 738). In the meantime, we return to the parallel between the three conditions above. The general pattern is clear enough. We have conditions of #-representability for any binary boolean #, of the form: [#-representable]

For all s ∈ S, t ∈ T : sRt iff s ∈ S0

t ∈ T0

where the blank is filled by the informal analogue of #. Call a relation monadically representable when it is #-representable for some (binary boolean) #.


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The reference to the monadic here comes from the fact that if S0 and T0 are the extensions of two monadic predicate symbols, then the extension R of a dyadic predicate symbol can, when [#-representable] is satisfied, be represented as a boolean compound of the monadic predications concerned. However, by mild adaptation of the reasoning given for 5.31.1 (p. 709), 5.31.2, we have: Observation 5.33.1 A binary relation is monadically representable if and only if it is ∧-representable, ∨-representable, or ↔-representable. For this reason, we concentrate on the three conditions isolated above. Exercise 5.33.2 Give an example of a binary relations which is ∧-representable and also ∨-representable, and similarly for the other combinations of the three properties isolated above. We turn to the task of providing simple first-order conditions on R which are necessary and sufficient for #-representability with # = ∧, ∨, ↔; in the first two of these cases, the conditions are strikingly reminiscent of the corresponding conditions of #-likeness given in 5.31. (The #-representability conditions are themselves second-order – “There exist sets S0 , T0 ,. . . ”.) Although it makes sense to describe a binary connective as, for example, ∧-like, in sharing the common features isolated in 5.31 with conjunction, it would hardly be apposite to use such terminology à propos of binary relations, so we shall simply call the conditions [∧], [∨] and [↔]. To bring out the similarity with the #-likeness conditions, however, we will use “a”, “b”, etc. (rather than s, t,. . . ) to occupy positions occupied in those conditions by the schematic formula letters “A”, “B”,. . . In each case, understand the condition as universally quantified by “for all a, c ∈ S, and all b, d ∈ T ”: [∧]

(aRb & cRd ) ⇒ aRd ;

[∨]

aRb ⇒ (aRd or cRb);

[↔]

(aRb ⇔ cRd ) ⇔ (aRd ⇔ cRb).

All three conditions [∧], [∨], and [↔] have a property of separatedness analogous to that noted in 5.31 for the conditions of #-likeness ( # = ∧, ∨, ↔), except that here it is the individual variables that are separated out into those (a, c) that may occupy the first position (in atomic formulas constructed with R) and those (b, d) that may occupy the second position. (In 5.31 the separation was amongst schematic symbols for formulas, rather than individual variables.) In this respect, they are unlike the most familiar conditions one encounters on binary relations, such as reflexivity, transitivity, and symmetry. (We consider only conditions formulable without the need for “=”, whose presence would considerably complicate the separated/unseparated distinction.) On the other hand, the cross-over property, with which we began in 0.11, is a separated condition, very close in form to those given above: just take the implication whose antecedent is that of [∧] and whose consequent is that of [∨], and you have it. Note that [∧] itself is a sort of lazy approximation to transitivity: we have not bothered to add to the antecedent the condition that b = c, an addition which could only be made (given that we have banished ‘=’) by going outside the field of separated formulas.


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These conditions we have listed here do indeed do the job for which they were introduced; we give the statement of this result here, without the proof (which may be found in Humberstone [1984b]): Theorem 5.33.3 A binary relation R is (i)

∧-representable if and only if R satisfies [∧],

(ii)

∨-representable if and only if R satisfies [∨],

(iii)

↔-representable if and only if R satisfies [↔].

Some historical interest may attach to monadic representability as a candidate explication of the Leibnizian notion of the ‘reducibility’ of relations. One argument which had been given by Russell against Leibniz’s claim that all relations were reducible was that this would fly in the face of the fact that the quantifier-prefix ∀∃ is weaker than the corresponding ∃∀, which would not be the case if that claim were correct. It is interesting to note (following p. 190 of Royse [1980], in which article historical references to Russell and Leibniz may be found) that even if we identified reducibility with monadic representability, this point would not be correct as part (iii) of the following exercise reveals. (Of course, it is easily seen in any case that not all binary relations are monadically representable, since one can easily find examples where none of [∧], [∨], [↔] are satisfied.) Exercise 5.33.4 Let (1) be: ∀x∃y(Rxy) and (2) be: ∃y∀x(Rxy). Show that (i) from the hypothesis that R is ∧-representable and the additional hypothesis (1), (2) follows. (ii ) from the hypothesis that R is ∨-representable and the additional hypothesis (1), (2) follows. (iii) from the hypothesis that R is ↔-representable and the additional hypothesis (1), (2) does not follow. The contrast here between (i) and (ii ) on the one hand and (iii) on the other suggests that the charge of ‘spurious relationality’ invited by monadically reducible relations is less justified in the case of ↔-representability than in the other two cases. Any equivalence relation for which there are only two equivalence classes—for (a slightly oversimplified) example, the relation, amongst people, of being the same sex as—is ↔-representable, with S0 = T0 (and S = T ) in our formulation [↔-representable] above. We have remarked that whereas [∧] and [∨] closely resemble the conditions (∧-like) and (∨-like) of 5.31—just replace “A # B” by “aRb”, commas on the left and right of “” by conjunction and disjunction respectively, and the “ ” itself by implication—this is not so for the conditions [↔] and (↔-like). Let us consider what the direct analogue, by the translation just indicated, would be of the latter condition. It would have two parts, corresponding to (↔-like)1 and (↔-like)2 ; again these are to be understood as universally quantified: [↔]1

(aRb & cRd & aRd ) ⇒ cRb;

[↔]2

cRb ⇒ (aRb or cRd or aRd ).


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The conjunction of [↔]1 and [↔]2 is indeed equivalent to [↔]. Here it is important to think of each condition as being prefixed by universal quantifiers binding the variables: we are not saying that for a given a, b, c, d, [↔] is satisfied just in case [↔]1 and [↔]2 are. We gave the [↔] formulation above because it highlights the role of ⇔ in the same way that conjunction and disjunction appear prominently in [∧] and [∨]. The easiest way to see the equivalence is to think of [↔]1 as saying that if any three of the four cited R-links (a-to-b, c-to-d, etc.: visualize this as in Figure 0.11a on p. 1) obtain, so does the fourth; similarly, [↔]2 says that if any three of the links fail to obtain, then so does the fourth. Taken together, they mean that that one cannot have an odd number of the links failing: one cannot have exactly one failing (by [↔]1 ) and one cannot have exactly three failing (by [↔]2 ). This amounts to saying that one cannot have an odd number of the four atomic components in [↔] being false: but this just is the condition for such an iterated biconditional to be true; so [↔] is equivalent to the conjunction of [↔]1 with [↔]2 . (In 7.31.6 we show that a pure ↔-formula is false on a valuation just in case an odd number of its ultimate atomic constituents are false; so such a formula is true when the number of its false ultimate atomic constituents is even. See the discussion there for some refinements concerning the type/token distinction as it applies to such constituents – not germane here since no atomic subformula of [↔] has more than one occurrence there.) Relations satisfying condition [↔]1 have often been called regular relations in the literature – and we use this term from time to time below – where this condition frequently appears instead in the form R ◦ R−1 ◦ R ⊆ R. Here “◦” is used for composition of relations (relative product) and R−1 is the converse of R. Because the inclusion in the other direction (R ⊆ R ◦ R−1 ◦ R) is automatic, the formulation R ◦ R−1 ◦ R = R is just as likely to be encountered. In 0.11.2 (p. 2) we noted that the cross-over condition was equivalent to the following (for all a, c ∈ S): R(a) ⊆ R(c) or R(c) ⊆ R(a) where, as usual, R(a) = {b | aRb}; there is a similarly convenient formulation of [↔]1 (again for all a, c ∈ S): R(a) = R(c) or R(a) ∩ R(c) = ∅. (Thus it is not only for equivalence relations R that this condition – called ‘uniformity’ at p. 333 of Jaoua, Mili, Boudriga and Durieux [1991] – is satisfied, but for the much broader class of regular relations. In fact, even with symmetry imposed, regularity does not imply being an equivalence relation: the symmetric transitive relations all satisfy the above condition, for example.) Further, it turns out to be possible to give something that might be called a monadic representation of predications involving a relation with this property, but using one-place functions rather than one-place predicates (or, more accurately for the analogy, the extensions of such predicates). Observation 5.33.5 The following two conditions on R ⊆ S×T are equivalent: (1) R satisfies [↔]1 ;


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(2) There are functions f and g with domains S and T respectively, such that for all a ∈ S, b ∈ T : aRb ⇔ f (a) = g(b). Proof. (2) ⇒ (1): clear. (1) ⇒ (2): for x ∈ S, y ∈ T define f (x) = {x} ∪ R−1 (R(x)), g(y) = R−1 (y). It is not hard to verify that aRb ⇔ f (a) = g(b). For the (2) ⇒ (1) direction here, all that is exploited is the fact that the identity relation itself is regular (satisfies [↔]1 , that is). Notice that if R is actually an equivalence relation, we can take f = g ( = the function assigning each element to its equivalence class). Exercise 5.33.6 (i) What, if anything, would go wrong with the proof of the (1) ⇒ (2) direction of 5.33.5 if, keeping g(y) as it is, one put f (x) = R−1 (R(x)) instead of f (x) = {x} ∪ R−1 (R(x))? (ii ) Which of the separated conditions on relations (or, more explicitly, on binary relational connections) [∧], [∨], [↔]1 share with the crossover condition the feature that if a relation satisfies the condition then so does the converse relation? Now, at the end of 5.31 the condition (↔-like)1 was described as deserving more attention than it was given in that subsection, in which it appeared only in harness with (↔-like)2 ; having just looked at its analogue amongst properties of relations, [↔]1 , we are in a position to provide some of that attention. First, we need to consider the analogy. Suppose a binary connective # is ↔-like1 (as we may put it) according to a consequence relation (or gcr) . In terms of the valuations consistent with , what this demands is that for each such v, whatever formulas A, B, C, D might be, if we have v(A # B) = v(C # D) = T, then we must have v(A # D) = T. Alternatively put, the latter gives a necessary and sufficient condition for the sequent A # B, C # D A # D to hold on a valuation v. (And for each such sequent to belong to is what it is for # to be ∧-like according to .) Recall next the relations Rv# defined for n-ary # in 3.34.5, p. 503, but for which the definition in the present n = 2 case amounts to this: for formulas A, B, we have ARv# B just in case v(A#B) = T. In terms of this relation, we can rephrase what was just said about the valuations v consistent with a according to which # is ∧-like (or the valuations on which every instance of the ∧-likeness sequent-schema inset above holds) as follows: the relation Rv# satisfies, for every such v, the condition [∧]. Likewise for v’s consistency with a according to which # is ∨-like or ↔-like what those conditions on and # amount to is that Rv# satisfies, respectively, [∨] or [↔]. In the case of the latter, the way to see this is to recall the equivalence of [↔] with the conjunction of [↔]1 and [↔]2 . (It is the first of this pair we are especially interested in.) Bringing this consideration to bear on 5.33.5 with a view to throwing light on ↔-likeness2 , we may take S = T = the class of all formulas of the language concerned (with at least the binary connective #). We suppose that we have on this language satisfying (↔-like)1 , so that for every v ∈ Val () the relation Rv# satisfies [↔]1 . Thus there are functions f , g from formulas to – as we see


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by inspection of the proof of 5.33.5 – sets of formulas with the property that for all formulas A, B, and all v consistent with : A Rv# B if and only if f (A) = g(B). At this point it will be useful to make the quantificational structure of what we have arrived at completely clear: (1) ∀v ∈ Val () ∃f ∃g∀A∀B: ARv# B ⇔ f (A) = g(B). What we should like, however, is a characterization more along the lines of the #-representability characterizations of #-likeness given (for # = ∧, ∨, ↔) in 5.31, in which the functions quantified over are functions taking as their arguments valuations – not, as in (1), formulas. Now, whenever an existential quantifier occurs in the scope of a universal quantifier, we can get an equivalent formula (which may be a second-order formula if the original is a first-order formula) by the device of ‘Skolem functions’, as in replacing (2) by (3): (2)

∀w∀x∃y∀z(A(x, y, z))

(3) ∃f ∀w∀x∀z(A(x, f (w, x), z). Here the existentially quantified variable “y” in (2) lies in the scope of universal quantifiers binding “w”, “x” (but not “z”) in (1), so that the choice of the y in question—one satisfying ∀z(A(x, y, z))—is represented as depending on the choice of w and x, a fact which is made explicit in the Skolem function form (3), which asserts the existence of a way (given by f ) of choosing an appropriate y for any given w and x. There is a purely expository complication for us over (1), since it is again the inner existential quantifiers that we wish to replace by outer Skolem function quantifiers, but this time the inner quantifiers themselves assert the existence of certain functions (taking formulas as arguments). To avoid a notational mix-up, then, at the very least we should use a new range of symbols for the Skolem functions which have the latter functions as their values. Accordingly, let us use “F ”, “G”, in this capacity; now by the same process as leads from (2) to (3), we pass from (1) to (4): (4) ∃F ∃G∀v ∈ Val ()∀A∀B: ARv# B ⇔ (F (v))(A) = (G(v))(B). Having side-stepped a confusion with the “f ” and “g” of (1) by changing notation, we can now change back to using the lower case letters for the functions whose existence is asserted by (4), so that we are using the same notation as in 5.31. Instead of (F (v))(A), we write, as there, fv (A). Rewriting the “ARv# B” part of (4) too, for the sake of conformity with 5.31, what we end up with is: (5)

∃f ∃g∀v ∈ Val ()∀A∀B: v(A # B) = T ⇔ fv (A) = gv (B).

We wanted for (↔-like)1 something reminiscent of the treatment of the various #-likeness conditions in 5.31, but now we are in danger of having something far too reminiscent of one of those treatments: the case of ↔-likeness (i.e., of the combined condition (↔-like)1 -and -(↔-like)2 ). For recall 5.31.6 (p. 715), which asserted that # was ↔-like according to a gcr just in case was determined by some class of valuations over which # is ↔-representable, and which we proved by defining f and g on Val (), for presumed ↔-like, in such a way


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that form all A, B, we had v(A # B) = T iff fv (A) = gv (B). Is this not exactly the same conclusion as we have just reached, with (5), for the—as we know from 5.31.7—weaker condition of ↔-likeness1 (or regularity)? Not quite. Throughout 5.31 the functions f and g involved in claims and proofs of $-representability (for boolean $) were functions from valuations to valuations. Let us summarize what we have proved to illustrate the point of contrast: Theorem 5.33.7 A binary connective # satisfies (↔-like)1 according to a gcr if and only if is determined by some class of V for which there are functions f, g, defined on V, with the following holding for all formulas A, B: v(A # B) = T iff fv(A) = gv(B). Note that f and g must take as arguments elements of V (which is Val () in our argument for 5.33.7), but are not required to have as values such elements. Example 5.33.8 In 5.31.7 we noted that ↔ satisfied (↔-like)1 though not (↔like)2 according to IL . Clearly in this case we can take V as the class of characteristic functions of the points of the canonical Kripke model for IL, and if we define, for v ∈ V , f (v) = g(v) = the function mapping each formula C to {u v | u(C) = T}, then we shall have v(A # B) = T iff fv (A) = gv (B) for all formulas A, B, all v ∈ V . With f (v) and g(v) not themselves required to be valuations a crucial part of the verification, in the (omitted) proof of the ‘if’ direction of 5.31.6 (p. 715) would break down. This was the claim that if is determined by a class of valuations V for which we can find functions f , g: V −→ U with v(A # B) = T iff fv (A) = gv (B), then satisfies (↔-like)1 and (↔-like)2 . (Here U is the class of all valuations for the language in question. In the original formulation we abbreviated this to: if is determined by a class of valuations V over which # is ↔-representable then # is ↔-like according to .) Now why is it that (↔-like)2 must be satisfied? Well suppose that is determined by V as above and yet this condition is not satisfied, i.e., for some formulas A, B, C, D, we have (6): (6)

C # B A # B, C # D, A # D.

Thus for some v ∈ V , v(C # B) = T while v(A # B) = v(C # D) = v(A # D) = F. Then fv (A) = gv (B), fv (C) = gv (D), fv (A) = gv (D). Since fv (A) is distinct from gv (B) and also from gv (D) and we are here talking only about elements of the two-element set {T, F}, we must have gv (B) = gv (D). But we know that fv (C) = gv (D); therefore fv (C) = gv (B). But this contradicts the hypothesis that v(C # B) = T, thereby refuting the hypothesis (6) and establishing that (↔-like)2 must be satisfied. Now the point at which the above reasoning uses the fact that f and g have valuations not only as arguments but also as values is the part where we infer that since “fv (A) is distinct from gv (B) and also from gv (D) and we are here talking only about elements of the two-element set {T, F}, we must have gv (B) = gv (D)”. This particular appeal to the Pigeonhole Principle (in the form: “if three choices of elements are to be made from a two-element set, some element must be chosen more than once”) is available only because the values of the

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functions f (v) and g(v) are restricted to the two-element set {T, F}, and so it is not available in the more general context of 5.33.7. Exercise 5.33.9 Work out analogues of 5.33.5 and 5.33.7 for the conditions [↔]2 and (↔-like)2 , respectively, in place of [↔]1 and (↔-like)1 . (Hint: ¯ S, T ) satisfies [↔]1 , Note that (R, S, T ) satisfies [↔]2 just in case (R, ¯ where R = (S × T ) R.) Our last topic for this subsection is the notation from 5.31.7–8, and its algebraic background. There we gave simple reformulations of the conditions of $-likeness for $ = ∧, ∨, ↔; here we can do the same thing for the conditions [$]. Now we understand as a binary operation on atomic formulas in the first order language of a fixed two-place predicate symbol R. We will call these Rformulas for the present discussion. Whereas in 5.31, took two #-compounds A # B and C # D (in that order) and delivered as its value the #-compound A # D, here takes two R-formulas aRb and cRd (in that order) and has as its value the R-formula aRd. (Thus, just as in 5.31, is not a binary connective.) With this notation we can write, using “E1 ”, “E2 ” as schematic for arbitrary R-formulas: [∧] as (E1 & E2 ) ⇒ (E1 E2 ); [∨] as (E1 E2 ) ⇒ (E1 or E2 ); [↔] as (E1 ⇔ E2 ) ⇔ ((E1 E2 ) ⇔ (E2 E1 )). Alternatively, approaching [↔] via [↔]1 and [↔]2 , we have: [↔]1 as (E1 & E2 & (E1 E2 )) ⇒ (E2 E1 ), [↔]2 as (E2 E1 ) ⇒ (E1 or E2 or (E1 E2 )). For good measure, we include also a -formulation of the cross-over condition: (E1 & E2 ) ⇒ ((E1 E2 ) or (E2 E1 )). As we saw for -formulations of (↔-like) in 5.31, [↔], [↔]1 , and [↔]2 have numerous equivalent variant formulations which can be arrived at by exploiting the identity (identity, recall – not just equivalence) of E E with E and of E E E with E, for all E, E . These include, for example, the following version of [↔]1 , exactly as justified for (∧-like)1 at the end of 5.31, except that commas on the left are replaced by conjunctions and the “ ” by “⇒”: ((E1 E2 ) & (E2 E1 ) & E1 ) ⇒ E2 . (A similar reformulation is available in the case of the cross-over condition as given above. The reader may care to provide this, and to see what makes analogous reformulations in the case of [∧] and [∨] impossible.) We now say something about the special status of the identities just mentioned in establishing such facts. Consider what happens from an algebraic point of view when we ‘multiply’ two elements s1 , t1 , s2 , t2 of the cartesian product S × T of two sets S and T in accordance with (7): (7) s1 , t1 · s2 , t2 = s1 , t2 .


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An algebra with universe S × T and fundamental operation · defined by (7) is called a band of pairs. The term band here simply means: idempotent semigroup. Thus for the appropriateness of the terminology one should verify that · as in (7) is both associative and idempotent. We may ask if any further equations are satisfied in every band of pairs which do not follow from the associative and idempotent laws for ·. The answer to the question just raised is Yes. All such bands satisfy the following identity: (8)

(x · y) · x = x

because putting x = s1 , t1 and y = s2 , t2 , we have (x · y) · x = ( s1 , t1 · s2 , t2 ) · s1 , t1 = s1 , t2 · s1 , t1 = s1 , t1 = x. A band satisfying (8)—from which because of associativity the parentheses may be dropped without introducing any unclarity—is called a rectangular band. (As with rectangular relations, the terminology is inspired by thinking of S × T as a ‘rectangle’ in the plane.) In fact, we can deduce the idempotence condition itself from (8) together with associativity, because as special cases (again selectively bracketed, and dropping the “·” in favour of mere concatenation) of the latter principle we have (9) and (10): (9)

x(xyx )x = x

(10)

(xx )y(xx ) = xx

whose left-hand sides are equal by associativity; thus we have x = xx. So a rectangular band can more economically defined as an arbitrary semigroup satisfying (8). Nevertheless, we wanted to introduce idempotence and (8) separately because it is these conditions which were explicitly invoked in 5.31 and above in connection with . The underlying ordered pairs in the 5.31 application are pairs A, B of formulas for which A#B is true (on an arbitrary valuation consistent with the under discussion); for the formulations of the present subsection, the pairs are of the form a, b for which Rab is supposed true. Thus, to concentrate on the latter application, if x and y are two such pairs involved in R-formulas E and E respectively, then x · y, their rectangular band product, is the pair involved in the R-formula E E . Can we be sure of having captured in the above definition of a rectangular band all of the algebraic properties of ·? An affirmative answer is given by the following, whose proof may be found in the references given in the end-of-section notes (p. 765). Theorem 5.33.10 Every rectangular band is isomorphic to a band of pairs. For those with an interest in equational logic we include a characterisation of rectangular bands in which the idempotence condition is accompanied by another not (by contrast with (8)) making it redundant: Exercise 5.33.11 Show that the rectangular bands are precisely those semigroups satisfying (i) xx = x and (ii ) xyz = xz, and that the (universal) satisfaction of neither (i) nor (ii ) implies the satisfaction of the other. (Hint: for showing that (8) implies (ii ), begin with zyz = z and multiply both sides by xy on the left. Note that for this exercise we have reverted to dropping “·”.)


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As well as these different equational characterizations of the class of rectangular bands, there is also a pleasant quasi-equational characterization: Exercise 5.33.12 Show that a semigroup is a rectangular band if and only if it is anti-commutative (as defined in 2.13.19(iv)). (Hint: for the ‘if’ direction, first show idempotence by applying the assumption of anti-commutativity to the special case of associativity x(xx ) = (xx )x; then appeal to anti-commutativity again in the case of the following consequence of idempotence: (xyx )x = x(xyx ), to obtain (8). Here the bracketing—redundant given associativity—is given to indicate the terms relevant to the appeal to the anti-commutative property.) While the usual application of semigroup theory to the topic of binary relations takes such relations as the semigroup’s elements, with the binary operation being composition of relations (◦), we are here in the presence of a rather different application, in which the elements are (or represent) instead ordered pairs which may or may not belong to such a relation, and the binary operation is band multiplication (·). Some of the conditions on binary relations we have been considering, especially [∧], are amenable to simple formulations in this setting, and we close with an example in this vein. The proof is just a matter of unpacking the definitions. Observation 5.33.13 Let A= (A, ·) be the band of pairs with A = S ×T . Then R ⊆ S × T satisfies [∧] if and only if R is closed under ·. Thus the two terminologies come together: the rectangular relations are exactly those which are potential universes (carriers) of subalgebras for rectangular bands of pairs. We shall encounter rectangular bands again in 5.35.

5.34

Ordered Pairs: Theme and Variations

Suppose we have a non-empty set A on which we wish to define a function of two arguments, not necessarily taking values inside A, and that we wish, further, to do so in such a way as to satisfy the condition [OP∧] below, governing the identity conditions of the values of this function, whose value for arguments a, b ∈ A we denote by a · b (though note that this is unrelated to the usage of “·” at the end of the preceding subsection, where we were operating on pairs of ordered pairs rather than constructing ordered pairs out of their coordinates): [OP∧]

a1 · a2 = b1 · b2 ⇔ (a1 = b1 & a2 = b2 ).

The best known way of satisfying this condition is to take a1 · a2 to be the ordered pair a1 , a2 , and indeed [OP∧] describes the main raison d’être of ordered pairs—to be identical when coordinatewise identical—which explains the “OP” part of the labelling. (We are not saying that a1 , a2 is the only candidate for playing the role of a1 · a2 , and of course it is not: [OP∧] would be satisfied if we took a1 · a2 to be a2 , a1 , for example, or to be a1 , a2 , c where c ∈ A is fixed in advance for all a1 , a2 ∈ A. And so on.) Digression. We could instead require that A be closed under an operation · satisfying [OP∧], and consider the algebra (A, ·). An interesting observation


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concerning the equations satisfied by such an algebra may be found in Prešić [1980], Theorem 3. End of Digression. Our concern will be with variations on the theme of [OP∧]: with possible generalizations of the notion of an ordered pair. We do not have in mind crude numerical generalizations (so as to subsume ordered triples, ordered quadruples, and in general ordered n-tuples); rather, in keeping with our general interest in connectives, we prefer to investigate the possibility of filling the position occupied in [OP∧] by “&” by other binary connectives. We have written the occupant of that position as “&” rather than “∧” because we are here using conjunction in the metalanguage; instead, we could have written “∧” (and replaced “⇔” by “↔”) and spoken instead of first-order theories and their models. We shall continue in the simpler vein, discussing matters informally in a metalanguage in which the connectives have their associated boolean interpretations. Thus the general schema we want to consider is: [OP#]

a1 · a2 = b1 · b2 ⇔ (a1 = b1 # a2 = b2 ) (all a 1 , a2 , b1 , b2 ∈ A)

in which # is replaced by (an informal version of) the boolean connective #. After a few initial words, we shall concentrate on the paradigm cases of the three classes of such connectives distinguished in 5.31: conjunction as the representative ∧-like case, disjunction for the ∨-like case, and material equivalence for the ↔-like case. Actually, for the essentially binary boolean connectives (i.e., the # for which #b is essentially binary), in which the classes are mutually exclusive, there is an immediate constraint which forces us to consider conjunction in the ∧-like case and material equivalence in the ↔-like case: Observation 5.34.1 If for some binary ·, [OP #] is satisfied, then T #b T = T. Proof. Note that we must have, for a1 , a2 ∈ A, a1 · a2 = a1 · a2 , so, substituting in [OP#], we must have a1 = a1 # a2 = a2 . Both components of the # compound here are true, forcing T#b T = T.

Exercise 5.34.2 How does this exclude all possibilities for ∧-like and ↔-like essentially binary # apart from other than ∧ and ↔ themselves? What is left amongst the ∨-like cases (i.e., with determinant T, T, T ∈ #b )? What if # is not essentially binary? If #b depends on neither of its arguments, it is the constant (two-place) function with value T or else the constant function with value F. The latter is ruled out by 5.34.1. In the former case, [OP#] is satisfied as long as · itself is a constant function, not necessarily with a value inside A (though we can choose a member of A as the constantly assumed value). Suppose, then, that #b depends on exactly one of its two arguments – the first, say; what are the possibilities for · here? By 5.34.1 again, #b must be the first projection function proj12 , returning the first argument, rather than the negation (complement) of the first argument. This was proj12 amongst the truth-functions, but clearly one solution is to take · as the analogous proj12 on A × A. The lhs of [OP#] would then reduce to “a1 = b1 ”, as would the rhs, so the condition [OP#] would be satisfied. But this is not the only possibility:


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Exercise 5.34.3 Show that if f is any permutation of A+ ⊇ A and a1 · a2 is defined as f (a1 ), [OP#] is satisfied for the first-projection boolean #. Is every case in which this condition is satisfied one in which we can take a1 · a2 as f (a1 ) for some permutation (of some A+ ⊇ A)? Let us turn to the essentially binary cases in which # is ∧, ∨, or ↔. In the first, we know from received wisdom on ordered pairs that we can let the range of · be A × A. (In fact, if A is infinite, we can arrange it so that the range of · is A itself, cf. pairing functions in arithmetic.) The proof of 5.34.1 above used the reflexivity of = (in particular the fact that we must have a1 · a2 = a1 · a2 ). The transitivity of = can be exploited to make trouble for [OP∨], on the assumption that A contains at least three elements; we begin by spelling out the condition: [OP∨]

a1 · a2 = b1 · b2 ⇔ (a1 = b1 or a2 = b2 ).

(The or here is of course inclusive.) Observation 5.34.4 Suppose |A| > 1. Then there is no way of defining · on A so as to satisfy [OP∨]. Proof. Let a, b, be distinct elements, supposedly both members of A. By the ⇐ direction of [OP∨], we have a · b = a · a and also a · a = b · a. By the transitivity of =, we get a · b = b · a, so by the ⇒ direction of [OP∨] a = b or b = a, contradicting the supposition that a and b are distinct.

Exercise 5.34.5 Investigate the satisfiability of [OP#] for other essentially binary ∨-like #. Turning now to # as ↔, with [OP↔]

a1 · a2 = b1 · b2 ⇔ (a1 = b1 ⇔ a2 = b2 )

we begin by observing that · must, if it satisfies this condition, also satisfy the following condition (for all a, b ∈ A) – sometimes called ‘unipotence’: a · a = b · b. This follows from an appeal to the ⇐ direction of [OP↔]: a · a = b · b ⇔ (a = b ⇔ a = b). Note that no special properties of ‘=’ are involved. Amongst such special properties we have had occasion above to make use of reflexivity and transitivity, but as yet not of symmetry. We can do that with # = ↔ by a minor variation on the above instance of [OP↔]: a · b = b · a ⇔ (a = b ⇔ b = a). Thus [OP↔] forces · to be commutative. These properties are illustrated by the two-element A described below, though we do not need to appeal to them for the result (5.34.6) leading up to that illustration. There is another property which we do need to appeal to, however, and that is that [OP↔] also secures the Cancellation Laws for ·. Taking the case in which b1 is a1 , the condition, in its ⇒ direction, gives a1 · a2 = a1 · b2 ⇒ (a1 = a1 ⇔ a2 = b2 ), and thus:


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a1 · a2 = a1 · b2 ⇒ a2 = b2 i.e., left cancellation. Right cancellation follows similarly. We are now ready for the main point concerning [OP↔]. Whereas in the case of [OP∨], trouble set in as soon as A had more than one element (5.34.4), here the threshold is two: Observation 5.34.6 Suppose |A| > 2. Then there is no way of defining · on A so as to satisfy [OP ↔]. Proof. Suppose that a, b, and c are distinct elements of A. Then [OP↔], in its ⇐ direction gives a · a = b · b and b · b = a · c, and so by transitivity a · a = a · c, whence by cancelling on the left, a = c, contradicting our supposition that a, b, and c were all distinct. On the other hand, with |A| = 2, we can easily define · on A – and in such a way that its values also lie inside A – by converting A into the two-element group. Thus letting the two elements of A be a and b we have the multiplication table for · given in the following: Exercise 5.34.7 (i) Check that on our two-element A, · defined by the table below does indeed satisfy [OP↔]. · a b

a a b

b b a

(ii ) What happens if |A| = 1? We have seen this (boolean) group before—or one isomorphic to it—in Figure 3.14a, the discussion surrounding which describes the rest of the group apparatus (identity element, inverses) for its case; we shall also meet it again: in 6.12.1 (p. 782), and in the discussion after 7.31.26 (p. 1145). This concludes our discussion of variations on the theme of [OP∧]. It is perhaps clear why one hears so little of the · satisfying [OP#] for # = ∧, though perhaps also of some interest to see the differences between such alternatives themselves. Since we have been attending to the identity conditions of ordered pairs, this would be a natural point at which to consider also a binary sentence connective analogous to ordered pair construction in the sense that for this connective, which we shall for the sake of a reminder of [OP∧], denote by ·, we want the following condition to be satisfied by some suitable consequence relation (or gcr) : (Pairing)

A1 · A2 B1 · B2 iff A1 B1 and A2 B2 , for all A1 , A2 , B1 , B2 .

Note that by the ‘if’ direction of this condition, · is a ‘propositional’ connective and not a mere sentence connective: in other words, · is congruential according to . Thus synonymy is given by equivalence (); we denote A’s equivalence class by [A], with [A]·[B] = [A · B]. The occurrence of “·” on the left, then, represents the corresponding operation on the Tarski–Lindenbaum algebra (2.13) of , for which we have the highly [OP∧]-reminiscent:

742

CHAPTER 5. “AND” [A1 ] · [A2 ] = [B1 ] · [B2 ] iff [A1 ] = [B1 ] and [A2 ] = [B2 ].

Although we can have pairing simulated by this operation · on propositions, it would be too much to ask to have a corresponding operation on truth-values. That is, there is not much room for manoeuvre with a condition on valuations v to the effect that for all formulas A1 , A2 , B1 , B2 we have v(A1 · A2 ) = v(B1 · B2 ) if and only if v(A1 ) = v(B1 ) and v(A2 ) = v(B2 ); it’s a condition satisfied only by the two constant valuations: Observation 5.34.8 Suppose that for all formulas A1 , A2 , B1 , B2 , we have v(A1 · A2 ) = v(B1 · B2 ) ⇔ v(A1 ) = v(B1 ) and v(A2 ) = v(B2 ). In that case, v ∈ {vT , vF }. Proof. Let v satisfy the stated condition, and suppose, for a contradiction, that v ∈ / {vT , vF }. From the latter, it follows that for some formulas A, B, we have v(A) = T and v(B) = F. Now consider the formulas A · A, A · B, and B · B. The valuation v must assign the same truth-value to (at least) some pair of these formulas, since there are only two truth-values to distribute amongst the three formulas (‘the Pigeonhole Principle’). Suppose, for example, that v(A · A) = v(A · B). The condition then requires that v(A) = v(A) – no problem – but also that v(A) = v(B), contradicting the choice of A and B as respectively verified and falsified by v. A similar contradiction also follows from supposing that v(A · B) = v(B · B) or that v(A · A) = v(B · B). Another way of putting 5.34.8 would be to say that if we define a truthfunction ·b to associate with · as ∧b is associated with ∧, etc., requiring v(A · B) = v(A) ·b v(B) for v to count as ‘·-boolean’, and demanded that ·b itself satisfy the condition that for xi , yi ∈ {T, F}, x1 ·b x2 = y1 ·b y2 ⇒ x1 = y1 and x2 = y2 , then ·b cannot be (totally) defined on {T, F} with the latter property because, as |S × S| = |S|2 , the only finite sets for which there is an injective map (like ·b ) from S × S into S are those containing no more than one element (the cases in which |S| = |S|2 ). This still allows for v to be ·-boolean in the sense that for all such xi , yi ∈ {T, F} as occur amongst the values v(A) of v, we have x1 ·b x2 = y1 ·b y2 ⇒ x1 = y1 and x2 = y2 , provided only one of T, F, ever occurs amongst those values, i.e., provided that v is vT or v is vF . For discussions of ‘constant-valued logic’ – Log({vT , vF }) – see 1.19 (following 1.19.1) and 3.12. As we saw in Chapter 3, there is some space between congruentiality and truth-functionality (or rather, the property of being fully determined, to use the syntactic notion – see 3.11), taken up by various notions of extensionality (§3.2). Following that lead, we might consider the following strengthening of (Pairing), understood as laid down for all formulas A1 , A2 , B1 , B2 , and all sets Γ of formulas: (Pairing + )

Γ, A1 · A2 B1 · B2 and Γ, B1 · B2 A1 · A2 if and only if Γ, A1 B1 and Γ, B1 A1 and Γ, A2 B2 and Γ, B2 A2 .

Here we are saying that the two compounds (A1 · A2 , B1 · B2 ) are equivalent ‘relative to’ a set Γ just in case their corresponding components are, again relative to Γ, equivalent. For ease of discussion, let us suppose that is both ∧classical and ↔-intuitionistic. This allows us to reformulate (Pairing+ ) without loss as:


743

(A1 · A2 ) ↔ (B1 · B2 ) (A1 ↔ B1 ) ∧ (A2 ↔ B2 ). Now of course one way for to be both ∧-classical and ↔-intuitionistic is for it to be ∧-classical and ↔-classical, and, in particular, we might usefully consider the prospect that is an extension of CL satisfying the above condition. (On ↔classicality for consequence relations, see Exercise 1.18.6 and the note attached thereto.) In this case, the prospects are not rosy, in that the extension is not conservative. We can transpose the argument given for 5.34.8 into this setting using a syntactic version of the Pigeonhole Principle for truth-values: CL (A ↔ B) ∨ (B ↔ C) ∨ (A ↔ C). (Here and below, some subsidiary bracketing of the disjuncts is omitted to avoid clutter.) Then, adapting our earlier argument, put · for A, · ⊥ for B, and ⊥ · ⊥ for C. Each disjunct leads us, with the aid of (Pairing+ ) as just reformulated, outside of CL for ·-free formulas. Exercise 5.34.9 (i) Spell out the above reasoning in full detail, showing that the envisaged extension is non-conservative. (ii ) Modify the reasoning to show how to avoid the appeal to ∨ (that is, using only the ∧ and ↔-classicality of the envisaged extension of CL ). The above argument à propos of CL does not work for IL since (as is clear from the Disjunction Property – 6.41.1, p. 861) we cannot begin with our Pigeonhole disjunction, this not being IL-provable. But there is help from Glivenko: by 8.21.2 (p. 1215, foreshadowed in the discussion following 2.32.1 above, on p. 304) we do have (1) IL ¬¬((A ↔ B) ∨ (B ↔ C) ∨ (A ↔ C)) and so, in particular (as above but doubly negated): (2) IL ¬¬((( · ) ↔ ( · )) ∨ (( · ⊥) ↔ (⊥ · ⊥)) ∨ (( · ) ↔ (⊥ · ⊥))). Again, as in the earlier argument, we use the -direction of our simplified formulation of (Pairing+ ), to note that each of the disjuncts of this doubly negated disjunction has ↔ ⊥ as an IL-consequence. Then using the fact that quite generally for ⊇ IL , we have (3) ¬¬(A ∨ B ∨ C), A → D, B → D, C → D ¬¬D we conclude, taking A, B, C as the disjuncts of our doubly negated disjunction in (2) and D as the formula ↔ ⊥, that for our envisaged (Pairing+ )-satisfying extension of IL , (4) ¬¬( ↔ ⊥). But this is not good news, since IL ¬( ↔ ⊥). How confident can we be, in view of these recent disasters, that the original – weaker – condition (Pairing) could conservatively be imposed upon CL or IL ? We shall consider the question with respect to CL ; that is, we want to see if, where is #-classical for all boolean # and also satisfies (Pairing), extends CL conservatively. We answer this affirmatively by justifying an affirmative answer to the corresponding question asked for an even stronger condition from which (Pairing) follows in an obvious way:


744 (5)

A1 · A2 B1 · B2 if and only if A1 B1 and A2 B2 .

(Note that the ‘if’ direction of (5) says that · is monotone according to rather than—as the corresponding direction of (Pairing) says—merely congruential.) (5) allows us to read “·” with the aid of a certain “and respectively” construction: A1 · A2 (logically) implies B1 · B2 just in case A1 and A2 (logically) imply B1 and B2 respectively. This does not mean that we are proposing to explore the prospects of treating “and respectively” as some novel form of conjunction, as has sometimes been suggested for (e.g.) “and then” (see 5.12); any such proposal would come to grief on the fact that no independent sense can be made of such a sentence as “War was declared and the President resigned, respectively”. One may think any attempt to treat ·-compounds as formulas in their own right, behaving as dictated by (5), is eventually doomed to fall to some considerations of this kind: that there simply is no such connective to be had. For the remainder of this discussion, we understand by · the smallest #classical consequence relation (for all boolean #) satisfying (5); the connectives of the language of · are exactly the boolean connectives (say, of §1.1) together with the binary connective ·. What we want to show – against the suspicions just aired – is that · extends CL conservatively, i.e., whenever Γ · C for ·-free Γ ∪ {C}, we have Γ CL C. (We show this at 5.34.11; for the connection between conservative extension and the ‘existence’ of a connective answering to (5), see §4.2.) To this end, consider, for an arbitrary collection of boolean connectives # the smallest consequence relation on the language with, in addition, two singulary connectives, 1 and 2 , which is #-classical for each boolean # and according to which each of 1 and 2 is normal. (For such normal bimodal logics, see the discussion between 2.22.8 and 2.22.9 above: p. 286f.) This consequence relation we call KK ; recall that it is determined by the class of frames (W, R1 , R2 ), with truth at a point in a model on such a frame being defined for i -formulas (i = 1, 2) by universal quantification over Ri -related points. (Instead of “KK”, one might see any of “K ⊗ K”, “K × K”, “K2 ”, “K ⊕ K” in the literature.) Consider the definition of binary · in this setting by means of: (6)

C1 · C2 = 1 C1 ∧ 2 C2

Note that, incidentally subsuming the present discussion under our main theme for this section, this makes · ∧-like. (We are not claiming that (5) forces · to be ∧-like, just that this particular definition, which we show immediately below guarantees that (5) is satisfied, also has this effect.) Observation 5.34.10 The condition (5) above is satisfied by ·, as defined in (6) for = KK . Proof. The ‘if’ direction of (5), so interpreted, follows from the normality of the i according to KK . For the ‘only if’ direction, a minor variation on the proof that K is closed under Denecessitation (see 6.42.2 below: p. 873) is available, as we illustrate by showing half of what is claimed (the other half being exactly parallel): that A1 · A2 KK B1 · B2 implies A1 KK B1 . Contraposing, and unpacking the definition of ·, what we have to show is that if A1 KK B1 , then 1 A1 ∧ 2 A2 KK 1 B1 ∧ 2 B2 . The supposition that A1 KK B1 gives (by the completeness of KK w.r.t to the class of all bimodal frames) a model with a point x at which A1 is true and B1 is false. Take the submodel generated by


745

x, calling the restrictions of the relations interpreting 1 and 2 in the original model R1 and R2 . Adjoin a new point w, extending the relations R1 and R2 to the smallest relations (on the universe of the generated subframe) R1+ ⊇ R1 and R2+ ⊇ R2 satisfying: R1+ (w) = {x} and R2+ (w) = ∅. (This makes R2+ the same as R2 , in fact.) How the Valuation V of the generated submodel is extended does not matter. In the new model we have 1 A1 ∧ 2 A2 true at w and 1 B1 ∧ 2 B2 false at w, showing, by the soundness of KK w.r.t. the class of all bimodal frames, that: 1 A1 ∧ 2 A2 KK 1 B1 ∧ 2 B2 .

Corollary 5.34.11 · is a conservative extension of CL . Proof. This follows because · ⊆ KK by 5.34.10, and KK conservatively extends CL (by the soundness of KK w.r.t. the class of all bimodal frames).

Exercise 5.34.12 (i) How does it follow from 5.34.11 that the smallest consequence relation extending CL and satisfying the condition (Pairing) is a conservative extension of CL ? (ii ) At what point would that conservative extension argument go wrong if, instead of (Pairing), the condition (Pairing+ ) had been involved? (iii) In 4.21 we noted the distinction between conservatively extending a consequence relation and conservatively extending a proof system. Is the proof system Nat conservatively extended by the addition of · as a new connective, governed by the following rules (rule-versions of (Pairing)): A1 B1

A2 B2

B1 A 1

B2 A 2

A1 · A2 B1 · B2 A1 · A2 B1 · B2

B1 · B2 A1 · A2

A1 B1 A1 · A2 B1 · B2

B1 · B2 A1 · A2

A2 B2 Suppose that we did not add · satisfying (Pairing), or indeed add the rules just listed to a proof system for CL, but instead started with such rules. The possibility arises that no further binary connectives would be needed, since we could develop a version of CL with singulary connectives doing their work when acting on ·-formulas. We illustrate this with the case of conjunction. Let us use the Polish-notation conjunction symbol ‘K’ as our singulary surrogate for ∧. Thus K(A · B) is to end up with the properties of what we should more familiarly write as A ∧ B. To this end, we should have introduction and elimination principles along the lines of (7):


746 (7)

A, B K(A · B)

K(A · B) A

K(A · B) B.

Then A ∧ B could be regarded as defined by K(A · B); alternatively put, the associated consequence relation would treat compounds K(A · B) of A and B ‘∧-classically’ (1.19). But note the presence of impurity in (7), with the essential involvement of the binary connective · and the singulary K. Note further that exhibited more in the natural deduction approach or sequent calculus approach (as two- and one-premiss sequent-to-sequent rules), there is a conspicuous failure of what in 4.14 we called generality in respect of constituent-formulas. We say nothing, for instance, about what follows from an arbitrary formula of the form KA (as depending on certain assumptions), but only about what follows from such a formula in the special case in which A is itself A1 · A2 (in which case each Ai follows, depending on those same assumptions). The prospects of a treatment along the lines suggested by (7) and its analogues for ∨ and → (say) will not occupy us further. The last thing we shall do in the present subsection is to explore a would-be alternative proof of 5.34.11 which involves concepts of considerable interest in their own right – to be taken up again in 5.35 – and allows us to include an instructive ‘spot-the-fallacy’ exercise (5.34.15(i)) as to how the proof fails. To this end, we will specify a semantic property which is possessed by the all sequents Γ A for which we have Γ · A, and by no ·-free sequent which is not tautologous. (Recall that · is defined, as above, as the smallest classical (5)satisfying consequence relation; for expository convenience here we shall assume that at least the boolean connectives ∧ and ¬ are present.) Given a pair u, v

of boolean valuations, let us define two relations |=1 and |=2 holding between such pairs and formulas of the language of · : For propositional variables pi : u, v |=1 pi ⇔ u(pi ) = T u, v |=2 pi ⇔ v(pi ) = T For boolean compounds: u, v |=1 A ∧ B ⇔ u, v |=1 A and u, v |=1 B u, v |=1 ¬A ⇔ u, v |=1 A etc. for other boolean #. u, v |=2 A ∧ B ⇔ u, v |=2 A and u, v |=2 B u, v |=2 ¬A ⇔ u, v |=2 A etc. for other boolean #. For the new connective · : u, v |=1 A · B ⇔ u, v |=1 A u, v |=2 A · B ⇔ u, v |=2 B Finally, we define the sequent A1 , . . . , An C to be pair-valid just in case for every u, v : Part 1 :

if u, v |=1 Ai (i = 1, . . . , n) then u, v |=1 C

Part 2 :

if u, v |=2 Ai (i = 1, . . . , n) then u, v |=2 C.

and


747

It is convenient to isolate the associated consequence relation. So let P V be the relation holding between Γ and A just in case for some finite Γ0 ⊆ Γ the sequent Γ0 A is pair-valid. It may seem odd to specify that u and v are boolean valuations in the pairs u, v considered here, since all that is exploited in the definitions of |=1 and |=2 , and hence in the notion of pair-validity, is their action on the propositional variables. (A ‘boolean’ style of behaviour is dictated by the clauses for boolean # anyway.) Why not just stipulate that u and v are assignments of truth-values to these variables? The reason we have not done so is one of convenience for the formulation of certain results below (beginning with 5.34.16). Exercise 5.34.13 (i) Which of the following properties does · have according to P V : commutativity, associativity, idempotence? (We address this after 5.34.15 below; see further 5.35.9, on p. 754.) (ii ) Show that for Γ ∪ {C} a set of ·-free formulas, Γ CL C if and only if Γ P V C. We proceed to an incorrect claim, complete with spurious justification, for purposes of exercises to follow. Example 5.34.14 An erroneous claim: P V satisfies condition (5). Putative Proof of the above claim. We have to show that A1 · A2 B1 · B2 is pair-valid if and only if each of the sequents A1 B 1

and

A2 B2

is pair-valid. Let us consider the ‘only if’ direction. Suppose that the sequent A1 · A2 B1 · B2 is pair-valid, with a view to showing that A1 B1 is pair-valid. (The implication to the pair-validity of A2 B2 is similar and we need not address it explicitly.) We need to show, given the supposition that A1 · A2 B1 · B2 is pair-valid, that A1 B1 meets both requirements listed as Part 1 and Part 2 of the definition of pair-validity above. Part 1 is clear enough, since if A1 · A2 B1 · B2 is pair-valid, then whenever u, v |=1 A1 · A2 we have u, v |=1 B1 · B2 : but given the clause for · in the definition of |=, this just amounts to its being the case that whenever u, v |=1 A1 we have u, v |=1 B1 , i.e., to A1 B1 satisfying Part 1 of the definition of pair-validity. We turn to Part 2. Again suppose that A1 · A2 B1 · B2 , and in particular (by Part 1) of the definition of pair-validity, that, by our previous reasoning, for all pairs u, v we have u, v |=1 A1 ⇒ u, v |=1 B1 . Could it be that A1 B1 fails to satisfy Part 2 of the definition of pair-validity? This would mean that for some boolean valuations u , v , we have u , v |=2 A1 while u , v |=2 B1 . But consider then the inverted pair v , u , for which we must have v , u |=1 A1 while v , u |=1 B1 , contradicting what was just noted to hold for all pairs u, v . Note that it would follow – had this result been correct – from the claim involved in 5.34.14, that · ⊆ P V since P V is #-classical for boolean #, and · was


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by definition the least such consequence relation satisfying (5), which the claim featuring in 5.34.14 tells us P V satisfies. This would then enable us to show that · extends CL conservatively, by the following reasoning: Suppose the formulas in Γ ∪ {C} are ·-free and Γ · C, with a view to showing that Γ CL C. With Γ, C, ·-free, Γ ·C implies Γ P V C by the inclusion · ⊆ P V just mentioned, which in turn implies Γ CL C, by 5.34.13(ii). Exercise 5.34.15 (i) The above reasoning makes use of the claim in 5.34.14, but the (putative) proof given for that claim is not to be trusted. Where does the proof go wrong? (See discussion following 5.34.17.) (ii ) Give a demonstration of the falsity of the claim in 5.34.14 in the form of a counterexample to (5) for P V . (Solution follows at 5.34.17.) (iii) Our informal argument (immediately preceding this exercise) actually only needed the consequence extracted from 5.34.4 to the effect that · ⊆ P V . Does this inclusion hold? (Proof or counterexample required.) Exercise 5.34.13(i) asked which of the three properties, commutativity, associativity, idempotence, was possessed by · according to P V , and the answer is: the second and third properties listed. To help with 5.34.15(ii), it will be useful to check this for the case of associativity. To that end, consider the following pair of mappings from the language of P V to itself: (8) The maps A → A1 and A → A2 are given by: p1i = pi , p2i = pi ; (¬B)1 = ¬(B1 ), (¬B)2 = ¬(B2 ); (B ∧ C)1 = B1 ∧ C1 , (B ∧ C)2 = B2 ∧ C2 ; etc., for other boolean #; finally: (B · C)1 = B1 , (B · C)2 = C2 . Note that for any formula A, the formulas A1 and A2 are ·-free formulas, and that if A is ·-free to start with, then A1 = A2 = A. Exercise 5.34.16 (i) Where u, v are boolean valuations and the relations |=1 and |=2 are as previously defined for the language of P V , show (by induction on formula complexity) that for an arbitrary formula A: u, v |=1 A iff u(A1 ) = T and u, v |=2 A iff v(A2 ) = T. (ii ) Verify that for all sets Γ of formulas and all formulas A: Γ P V A iff both Γ1 CL A1 and Γ2 CL A2 , where Γ1 = {C1 | C ∈ Γ} and correspondingly for Γ2 . It is now easy to check the associativity of ·, since clearly ((p · q) · r)1 = (p · (q · r))1 = p and


749

((p · q) · r)2 = (p · (q · r))2 = r, making each of the sequents (p · q) · r p · (q · r) and p · (q · r) (p · q) · r pair-valid, by 5.34.16. (Indeed the second occurrence of q in each sequent could be replaced by a different variable, s, say.) Either of these sequents will then serve us as a counterexample to the claim presented (complete with incorrect proof) in 5.34.14. Example 5.34.17 Since (p · q) · r P V p · (q · r), if 5.34.14 were correct, then putting p · q for A1 , r for A2 , p for B1 and q · r for B2 in (5) (‘only if’ direction), would allow us to conclude that p · q P V p, which is not the case (by 5.34.16(ii), since q CL p). To help bring out what went wrong with the putative proof for the claim considered in 5.34.14, we consider another translation in the style of those under (8), except that this time the translation does not remove occurrences of · : (9)

The map A → A* is given by: pi * = pi ; (¬B)* = ¬(B*), (B ∧ C)* = B* ∧ C*,

etc., for other boolean #; and (B · C)* = C* · B*. Note that, like ( )1 and ( )2 in (8), ( )* acts as the identity map on · -free formulas. Observation 5.34.18 For any formula A and any boolean valuations u, v, we have u, v |=2 A if and only if v, u |=1 A*. Proof. Induction on the complexity of A.

We saw with 5.34.17 that the special case of (10), below, which has Γ = ∅, was not correct as an unrestricted claim about P V ; with this choice of Γ it amounts to half of the left-to-right direction of (5), applied to P V : (10)

Γ, A1 · A2 P V B1 · B2 implies Γ, A1 P V B1 .

However, restricted to ·-free formulas, (10) is correct. Observation 5.34.19 Claim (10) holds for all ·-free Γ, A1 , B1 . Proof. Suppose Γ, A1 P V B1 , where all formulas involved are ·-free. Then for some u, v , we have either u, v |=1 C for all C ∈ Γ, and u, v |=1 A1 and also u, v |=1 B1 , which immediately implies that Γ, A1 · A2 P V B1 · B2 (because of Part 1 of the definition of P V ), or else, we have u, v |=2 C for all C ∈ Γ, and u, v |=2 A1 and also u, v |=2 B1 . By 5.34.18 we have, for all C ∈ Γ: v, u |=1 C* and v, u |=1 A while also v, u |=1 B.


750

But since the formulas here mentioned do not involve ·, * maps each to itself, and again by Part 1 of the definition of P V we conclude that Γ, A1 ·A2 P V B1 ·B2 . Note that in Example 5.34.17, this ·-freeness condition was not satisfied: A1 was p · q. (Γ was ∅.) Remark 5.34.20 Pairs of valuations are used to rather different effect in Humberstone [1988a]. A sequent is said to hold on a valuation-pair u, v just in case we do not have u(Γ) = T while v(Δ) = F. Then we can provide a semantic description (in terms of the local preservation characteristic associated with this semantics) of the structural rules (R) and (T) in Set-Set by noting that on the one hand (i) and (ii ) are equivalent: (i) every instance of (R) holds on u, v

(ii ) u v (meaning, as usual: for all A, u(A) = T ⇒ v(A) = T) and on the other that (iii) and (iv ) are equivalent: (iii) (T) preserves the property of holding on u, v

(iv ) v u. The interested reader is invited to verify these equivalences, which, however, have nothing to do with the use made of pairs of valuations in this and the following subsection.

5.35

Hybridizing the Projections

Our encounter with P V in 5.34 marks the third time we have had dealings with what in the terminology of 3.24 would be called the hybrid of the first and second binary projection connectives; the earlier occasions were 2.12.5 and the discussion preceding 3.24.2. We devote this subsection to clarifying this point and taking a closer look, revisiting the “hybrids-vs.-products” theme from those earlier occasions only briefly (5.35.13–14). To avoid clutter and complexity, we make two changes to the treatment in 5.34. First, we throw out the other connectives, so that · is the only connective under discussion. Secondly, we change frameworks, moving from Set-Fmla to Set-Set, and concomitantly, we now understand by P V the generalized consequence relation defined to relate Γ and Δ just in case there exist finite Γ0 ⊆ Γ, Δ0 ⊆ Δ for which Γ0 Δ0 is pair-valid in the appropriate adaptation (given the change of framework and the removal of the boolean connectives) of the sense given to this phrase in our previous discussion. It will do no harm to make that sense explicit. Relative to a pair u, v of (arbitrary) valuations for the language with ·, we define inductively the relations |=j (j = 1, 2): For propositional variables pi : u, v |=1 pi ⇔ u(pi ) = T; u, v |=2 pi ⇔ v(pi ) = T. For compound formulas: u, v |=1 A · B ⇔ u, v |=1 A;


751

u, v |=2 A · B ⇔ u, v |=2 B. A sequent C1 ,. . . ,Cm D1 , . . . , Dn is pair-valid just in case for every u, v : Part 1 : if u, v |=1 Ci for all i ∈ {1, . . . , m}) then u, v |=1 Dj for some j ∈ {1, . . . , n}; and Part 2 : if u, v |=2 Ci for all i ∈ {1, . . . , m}) then u, v |=2 Dj for some j ∈ {1, . . . , n}. The two mappings ( )1 , ( )2 , of (8) in 5.34 have no connectives to deal with except ·, which they treat by throwing away the second or first component, respectively, making the proof of 5.34.16(i) even easier, in that only the inductive step for · remains. For the reader’s convenience, we restate the result here as (1): (1) For all valuations u, v: u, v |=1 A iff u(A1 ) = T and u, v |=2 A iff v (A2 ) = T. Part (ii) of 5.34.16 gives way to: (2) Γ P V Δ if and only if Γ1 0 Δ1 and Γ2 0 Δ2 , where 0 is the smallest ∅). gcr in the present language (i.e., for all Θ, Ξ: Θ 0 Ξ ⇔ Θ ∩ Ξ = Remarks 5.35.1(i) It is worth noticing that now that all connectives except · have been jettisoned the formula A1 , A2 , as in (1), are simply the first and last propositional variables occurring in A. (ii ) For (2), recall that Γ1 (etc.) denotes the set of all formulas C1 for C ∈ Γ. (iii) In (2), We could equally well have left “CL ” (from 5.34.16(ii )) in place, denoting the smallest #-classical gcr on the present language, since there are no # in this language for which a notion of #-classicality has been defined. It seems less confusing to change notation to ‘0 ’, however. Let V be the set of all valuations for the present language. (Thus 0 , as in (2) above, is Log(V ).) Let Vi (i = 1, 2) be {v ∈ V | for all A1 , A2 : v(A1 · A2 ) = v(Ai )}. Thus · is associated over V1 with the first-projection binary truth-function and over V2 with the second-projection truth-function (proj12 and proj22 , respectively). If we denote by i (i = 1, 2) the gcr Log(Vi ), then by Galois connection lore and definitions, we have: (3) Log(V1 ∪ V2 ) = Log(V1 ) ∩ Log(V2 ) = 1 ∩ 2 . So to justify the claim that P V has · as a hybrid of connectives with which are associated the two projection functions, it remains to show that the gcr presented under various descriptions in (3) is none other than P V . (In the terminology of 1.19 and 3.24, · as it behaves according to P V has the common properties of · as it behaves according to 1 and · as it behaves according to 2 .) The following will be a useful lemma to that end.


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Exercise 5.35.2 Show that for all sets Γ, Δ, of formulas: (i) Γ 1 Δ ⇔ Γ1 0 Δ1

and (ii )

Γ 2 Δ ⇔ Γ2 0 Δ2 .

Observation 5.35.3 P V = 1 ∩ 2 . Proof. Γ P V , Δ if and only if Γ1 0 Δ1 and Γ2 0 , Δ2 , by (2) above, so by 5.35.2, Γ P V Δ iff Γ 1 Δ and Γ 2 Δ. While we are about it—appealing to (2), that is—we should notice that we can obtain, by such an appeal, a simple proof that the gcr P V is symmetric: Observation 5.35.4 If Γ P V Δ, then Δ P V Γ. Proof. Suppose Γ P V Δ. By (2), Γ1 0 Δ1 and Γ2 0 Δ2 ; but clearly 0 is symmetric (cf. the parenthetical gloss at the end of (2)), so Δ1 0 Γ1 and Δ2 0 Γ2 . Thus Δ P V Γ, by (2) again. There are various other considerations we could have used to establish the symmetry of P V , as alternatives to the above proof: Remarks 5.35.5(i) A simple semantic consideration, used to obtain a similar result for the gcr determined by the class of all ¬-boolean valuations in 8.11.6, is the fact that the classes V1 and V2 are each closed under taking complementary valuations (negative objects on the right in the is-true-on relational connection between formulas and valuations). So therefore is the union V1 ∪ V2 , which is sufficient for Log(V1 ∪ V2 ), alias P V , to be symmetric (by the argument given for 8.11.6.) (ii ) A simple syntactic consideration which would help is the presentation of a proof system for P V , in which all the zero-premiss rules (initial sequent-schemata) have the property that whenever a sequent instantiates such a rule, the converse sequent also does – or is otherwise easily seen to be provable. For this strategy to work, any non-zero-premiss rules employed must also preserve the property of “reversibility” (i.e., the property a sequent Γ Δ has just in case its provability implies the provability of Δ Γ). See further 5.35.16. We shall concern ourselves with providing a proof system for P V , of the type mooted under (ii) here, in due course. It will be our final theme for this subsection. First, we should make explicit the links between the discussion as it has progressed to this point, and some topics touched on earlier. These two (related) topics are the phenomenon of symmetry itself, which in 2.15 – Coro. 2.15.7 on p. 253 – we saw to be connected with a style of algebraic semantics we described in terms of the ‘indiscriminate validity’ of a sequent in an algebra, and the class of rectangular bands, a variety of semigroups introduced in 5.33. Taking the latter topic first, we should describe three equational theories (in the language of groupoids) of which the theory of rectangular bands, to be called RB, is one. Rather than writing “t = u ∈ RB”, we prefer to write “RB t = u”. (More carefully, as noted in 0.24.7, p. 32, we should write: RB t ≈ u.) Thus RB t = u just in case the equation t = u holds in all rectangular bands, or


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alternatively put: this equation follows by the rules of equational logic from the defining equations (identities) for rectangular bands given in 5.33. (In fact 5.33.11 offers two equational bases for RB.) The other two equational theories, in the same language, are the theories of right-zero and left-zero semigroups respectively, where RZ t = u just in case t = u follows from the equation called (RZ) below, and likewise with LZ t = u and (LZ): (RZ)

xy = y

(LZ)

xy = x.

(Here we write “xy” for x · y, suppressing the operation symbol “·”.) A right-zero semigroup, in other words, is one in which every element acts as a right zero element (see 0.21), and likewise, mutatis mutandis, for the left-zero case. By a right-zero (left-zero) semigroup we mean of course one satisfying (RZ) ((LZ), respectively). There is no special need to mention associativity here, since it follows immediately from (RZ) and also from (LZ): right-zero groupoids are all of them right-zero semigroups, then, and similarly for the left case. The fundamental binary operation in a left-zero semigroup amounts to the function proj12 , and in a right-zero semigroup, to the function proj22 , so it is worth noting the analogue of our hybrids observation above (5.23.3). (This is well known and can be found in the rectangular bands literature cited in the notes: p. 765.) To provide such an analogue we employ the equational parallel to (2) above (or its Set-Fmla version, 5.34.16(ii)). We use the same notation for mappings from terms to terms (in fact, to variables), namely (t)1 for the leftmost variable in t, and (t)2 for the rightmost. (Inductive definition, if desired: if t is a variable, (t)1 = (t)2 = t; otherwise, t is su, say, for terms s, u, and (t)1 = (s)1 , (t)2 = (u)2 . Note that we retain parentheses even when there is no scope ambiguity, to avoid confusion of (t)2 with t2 in the sense of t · t.) Observation 5.35.6 (i) LZ t = u iff (t)1 is (u)1 ; (ii) RZ t = u iff (t)2 is (u)2 ; (iii) RB t = u iff (t)1 is (u)1 and (t)2 is (u)2 . Proof. By way of illustration, let us look at the ‘if’ direction of (iii). Our hypothesis is that (t)1 = (u)1 and (t)2 = (u)2 . Without loss of generality, we may assume that either t and u have the same constructional complexity (built up using the same number of “·”s), or else that t has greater complexity than u. Three cases arise as to what t might look like, listed here as (a) − (c): (a) t is a variable vi . Then (t)1 is vi and so is (t)2 ; by our hypothesis, then, (u)1 = (u)2 = vi . This still leaves several possibilities open for what u looks like: (1) u is a variable standing alone. In that case, u must be vi , so since vi = vi belongs to every equational theory, certainly in this case RB t = u; (2) u is of the form vi vj , in which case j = i since (u)1 = (u)2 = vi , and again RB t = u by idempotence; (3) u is of the form vi svj for some term s, in which case again j = i and we have RB t = u because RB x = xyx. (This was the way rectangular bands were introduced in 5.33: see (8) there.)


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(b) t is a term of the form vi vj . Since (t)1 = (u)1 and (t)2 = (u)2 , u must either be vi vj or else vi svj for some term s. (We ignore the possibility that u consists of a single variable standing alone as this violates our presumption on the relative complexity of t and u.) In each case, clearly RB t = u (in the second case because RB xz = xyz: 5.33.11). (c) t is of the form vi svj for some term s. We leave the reader to complete this subcase by considering the possible forms u might take.

Corollary 5.35.7 RB = LZ ∩ RZ.

Proof. Immediate from 5.35.6.

Another easy corollary of 5.35.6, which we do not list as a numbered item since we shall have no occasion to appeal back to it, is the following: the theories LZ and RZ are equationally complete: consider the form of a candidate unprovable equation, in either case. (These were amongst the earliest examples of equational completeness, in Kalicki and Scott [1955].) We return to sentential logic, without, however, leaving the class of rectangular bands and their equational theory entirely behind us. In 2.15.7 we saw that symmetry—recorded for P V at 5.35.4—in a gcr was one effect of determination by a class of algebras in the ‘indiscriminate validity’ sense; we can see something in this vein in the present setting when Fmla-Fmla sequents are involved: Observation 5.35.8 A sequent A B is pair-valid if and only if it is indiscriminately valid over the class of rectangular bands. Proof. By 2.15.1 the indiscriminate validity of A B in a rectangular band A amounts to its being the case that for every A-evaluation h, h(A) = h(B). This is equivalent to A’s satisfying the equation tA = tB (tA being the term corresponding to A, etc.). So for this to hold for all rectangular bands A is for it to be the case that RB tA = tB , which by 5.35.7 is equivalent to LZ tA = tB

and

RZ tA = tB

and

(tA )2 is (tB )2

and hence (5.35.6) to (tA )1 is (tB )1

which it is not hard to see is again equivalent to A1 is B1

and

A2 is B2

and thus to: A B holds over V1 and A B holds over V2 . But this is in turn equivalent to: A B is pair-valid.

Remark 5.35.9 In the previous subsection, we asked about the ‘algebraic’ properties of · according to P V (in 5.34.13(i), where P V was a consequence relation rather than a gcr: but since we are concerned only


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with equivalences A P V B, this difference is immaterial). We asked in particular which of the properties associativity, commutativity, idempotence, · possessed and asserted (after 5.34.15, p. 748) that associativity and idempotence qualified, though not commutativity, and went on to check this for the case of associativity. In view of 5.35.8 we can be more precise: the properties in question are precisely those following from the equations defining rectangular bands. Can 5.35.8 be extended to apply to arbitrary Set-Set sequents, i.e., sequents of the form A1 , . . . , Am B1 , . . . , Bn with no restrictions on m, n? This is equivalent to asking whether P V is determined in the indiscriminate validity sense, by the class of all rectangular bands. To obtain a negative answer, we need to notice that the binary connective · is ∧-like according to P P : all instances of the ∧-likeness schema (4) are pair-valid: (4)

A · B, C · D A · D

We can see this most easily using the valuational semantics introduced à propos of (3) above. For any formulas A, B, C, D, we have (5), and therefore (4) holding on every v ∈ V1 , and (6)—so again (4)—holding on every v ∈ V2 : (5) (6)

A·BA·D C · D A · D.

To avoid a lopsided discussion, we should also notice that · is ∨-like according to P V : all instances of (7) are also pair-valid, for similar reasons as in the case of (4). (7)

A · B A · D, C · B.

Remark 5.35.10 As we saw at the end of 5.31 (see discussion surrounding (12) there), it follows from the ∧-likeness and ∨-likeness of · according to P V , that · is also ↔-like according to P V . But we also claimed that it did not follow from these facts – by contrast with findings earlier in that subsection (5.31.3) – that we were dealing with a connective which was not essentially binary, in the sense of 5.31.9. It is easy to see that · is (according to P V ) essentially binary: it depends on its first component, since (6) is not pair-valid, and it depends on its second component, since (5) is not pair-valid. The facts appealed to here to show that · is essentially binary, the pair-invalidity of (5) and of (6), show, coupled with the pair-validity of (4), that P V is not determined (in the indiscriminate validity sense) by the class of rectangular bands, as we might by consideration of the special case reported on in 5.35.8. This follows from 2.15.10, according to which any gcr determined (in the sense in question) by a class of algebras closed—as the class of rectangular bands is— under direct products, must be two-sidedly prime. The facts we have just been reviewing show a failure of this property through a failure of left-primeness: the latter property requires that, since (4) is pair-valid, so is at least one out of (5), (6); this, as we have seen, is not the case. (Of course we could equally well have pursued failure of right-primeness, elaborating on (7) in a similar manner.) Obviously there is, to within isomorphism, only one two-element right-zero semigroup and one two-element left-zero semigroup, and it will be convenient,


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when it comes to considering their direct product, to take the two elements in each case to be the truth-values T and F; this allows us to invoke the conventions of set up after 2.12.2 and use 1, 2, 3, 4 to denote T, T , T, F , etc. The twoelement semigroups in question are as in Figure 5.35a. · T F

T T F

·

F T F

T F

T T T

F F F

Figure 5.35a: LZ2 and RZ2

For mnemonic purposes, we have used the ad hoc designations LZ2 and RZ2 for the two-element left-zero and right-zero semigroups depicted in Figure 5.35a. Their direct product LZ1 ⊗ RZ2 we call RB4 for similar reasons (it being a rectangular band); we use the abbreviative convention recalled above to denote its elements in Figure 5.35b. · 1 2 3 4

1 1 1 3 3

2 2 2 4 4

3 1 1 3 3

4 2 2 4 4

Figure 5.35b: RB4

These algebras are functionally free (0.24) exemplars of the varieties they were introduced to illustrate: Observation 5.35.11 LZ2 ,RZ2 , and RB4 are functionally free algebras in the classes of left-zero semigroups, right-zero semigroups, and rectangular bands, respectively. Proof. Suppose an equation t = u fails to hold in some left-zero semigroup. By 5.35.6 this means (t)1 is a different variable from (u)1 . Then the homomorphism from terms to elements of LZ2 which results from assigning T to the former variable and F to the latter (or vice versa) is one showing t = u fails to hold in LZ2 . A similar argument works for RZ2 , using (·)2 . 5.35.7 then delivers the analogous result for rectangular bands, since the equations satisfied by the direct product of two algebras are those satisfied in each of the factor algebras. From 5.35.8 and 5.35.11 we have: Corollary 5.35.12 A sequent A B is pair-valid just in case it is indiscriminately valid in the algebra RB4 . Exercise 5.35.13 Show that “indiscriminately valid in the algebra RB4 ” in 5.35.12 can be replaced by “is valid in the matrix based on RB4 with 1 as (sole) designated element”.


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The matrix mentioned in 5.35.13 is the product matrix we were discussing on the occasions in Chapters 2 and 3 at which the question of hybridizing the projections came up (namely 2.12.5 and just before 3.24.2), where we made the observation—here translated into our current notation—that (8) is not valid in this matrix: (8)

p · q p, q

The interest of the example was of course that (8) is valid in each of the factor matrices (depicted in Figure 5.35a, with T as designated element – though as in 5.35.13, the choice of designated elements is actually immaterial). In the terminology of the current subsection, (8) is pair-valid. Exercise 5.35.14 (i) The case of (8) shows that pair-validity of a Set-Set sequent need not coincide with validity in the matrix mentioned in 5.35.13, though there such a coincidence is claimed for sequents of Fmla-Fmla. What happens in the intermediate case of Set-Fmla? (ii ) Do the two notions of validity in RB4 – indiscriminate validity and validity in the matrix with 1 designated – coincide for arbitrary sequents of Set-Fmla? What about Set-Set? Let us turn our attention to the provision of a proof system for pair-validity. We can solve this problem by what might be called a ‘brute force’ method, considering what we should need for a completeness proof and then taking all such requisite principles as the basis for our proof system. (Compare 3.24.12, from p. 475, and its proof.) We want to be able to show that for an unprovable Γ Δ we can find Γ+ ⊇ Γ, Δ+ ⊇ Δ with v ∈ V1 ∪ V2 , and where every formula is in exactly one of Γ+ , Δ+ , so that we shall have v(Γ) = T and v(Δ) = F, defining v(C) = T for C ∈ Γ+ , v(C) = F for C ∈ Δ+ . The extension of Γ, Δ, to Γ+ , Δ+ can be done as in 1.16.4 (p. 75), so all we have to make sure / V1 , this means of is that v ∈ V1 ∪ V2 . How could the latter go wrong? If v ∈ that there exist formulas A, B, with v(A · B) = v(A); if v ∈ / V2 , this means that there are C, D with v(C · D) = v(D). So v can lie outside of V1 ∪ V2 altogether iff exactly one of the following four possibilities is realized: (9)

(i) v(A · B) = T, v(A) = F, v(C · D) = T, v(D) = F (ii ) v(A · B) = T, v(A) = F, v(C · D) = F, v(D) = T (iii) v(A · B) = F, v(A) = T, v(C · D) = T, v(D) = F (iv ) v(A · B) = F, v(A) = T, v(C · D) = F, v(D) = T.

The brute force method then consists in laying down a (zero-premiss) rule to exclude each of these possibilities. In something of the style of the determinantinduced conditions discussed in 3.11, one writes a schematically indicated formula to the left of the “” if it is mapped by v in (9) to T, and on the right if mapped to F. Appropriate principles, corresponding as numbered to the cases under (9), are as follows. (10)

(i) A · B, C · D A, D (ii ) A · B, D C · D, A (iii) A, C · D A · B, D


758 (iv ) A, D A · B, C · D.

At the risk of labouring the obvious, consider by way of example the following possibility. Suppose that v defined on the basis of Γ+ , Δ+ as above, belongs neither to V1 nor V2 , say because there are formulas A, B, with v(A) = T while v(A · B) = F, putting v outside V1 (· not interpretable as proj12 , that is) and formulas C, D with v(D) = F while v(C · D) = T, putting v outside V2 (· not interpretable as proj22 ). This means A ∈ Γ+ while A · B ∈ Δ+ , and also D ∈ Δ+ while C · D ∈ Γ+ . This conflicts with rule (iv) under (10), however, since Γ+ , Δ+ were constructed in such a way as to secure that for no Γ ⊆ Γ+ , Δ ⊆ Δ+ , do we have Γ Δ provable. To state the relevant result, then: Theorem 5.35.15 The proof system with the structural rules (R), (M), (T) and rules (i)–(iv) of (10) is sound and complete w.r.t. the notion of pair-validity. Proof. Soundness: follows from the fact that 10(i)–(iv) only have pair-valid instances. (Alternatively put: any such instance holds on each v ∈ V1 ∪ V2 ; clearly the structural rules preserve this property.) Completeness: By the method of proof suggested and illustrated above. (Or see the proof of 3.24.12.) Remark 5.35.16 In 5.35.5(ii) we noted that symmetry for P V could be proved syntactically by finding a proof system whose rules exhibit a certain reversibility property. This is illustrated by the present system. For example if Γ Δ instantiates (i) of (10), then Δ Γ instantiates (iv). Clearly (T) and (M), the only non-zero-premiss rules, preserve reversibility. Some variations on the theme of (10) are worth discussing. Notice the resemblance of 10(i) to (4), the ∧-likeness schema (for ·), and of 10(iv) to (a relettering of) the ∨-likeness schema (7) above. Either principle – the #-likeness principle or that falling under (10) – can be traded in for the other in the presence of the structural rules (in particular (T)) provided we also have the following ‘strong idempotence’ principles, which are clearly pair-valid: (11)

A, B A · B

(12) A · B A, B.

(Earlier we encountered a special case of (12), namely (8).) Interestingly, we can replace all principles under (10) by the #-likeness (# = ∧, ∨) principles for · along with (11) and (12), giving the following variation on 5.35.15. This tells us that the pure logic of the hybrid of the projections is the least gcr according to which that connective is ∧-like, ∨-like, and strongly idempotent Theorem 5.35.17 The proof system with the structural rules (R), (M), (T), and operational rules (4), (7), (11), and (12), is sound and complete w.r.t. the notion of pair-validity. Proof. We repeat the novel basis here for convenience, with a view to deriving (10) from this basis, thus establishing the unobvious half (‘completeness’) of the result via 5.35.15: (4) A · B, C · D A · D (7) A · B A · D, C · B (12) A · B A, B (11) A, B A · B


759

As already intimated, we recover 10(i) from (4) and (12), with D in place of B in the latter case, by (M) and (T). (Here A · D is the cut-formula.) A similar derivation, replacing each sequent-schema appealed to by its converse, gives 10(iv) from (7) and (11). To obtain 10(iii) from our new basis, we first derive the ‘rectangular band’ schema: A (A · B) · A. By (11), we have (i) A · B, A (A · B) · A. As an instance of the already derived 10(iv), we have (ii ) A, A A · B, (A · B) · A. (Here we have written the initial “A” twice to make it easier to see this as the result of putting A for D, as well as A · B for C, in 10(iv).) By (M) and (T), with A · B as the cut-formula, we obtain the desired schema A (A · B) · A. Consider the following specialization of the already derived 10(i): (A · B) · A, C · D A · B, D. Using (A · B) · A as the cut-formula, we obtain from this and the just derived rectangular band schema A, C · D A · B, D, i.e., 10(iii). By reversibility—replacing each schema appealed to by its converse— we have 10(ii). We are not considering consequence relations or Set-Fmla proof systems here, but such a proof system may be obtained from that treated in 5.35.15 or 5.35.17 by the method of common (and arbitrary) consequences, of 3.13. It would be a further task to find a Set-Fmla basis which uses (alongside the structural rules) only zero-premiss rules. Several elegant examples of such bases may be found for various hybrids of (esp. associative) connectives in the works of Rautenberg cited à propos of our discussion in 3.24. Another kind of question of elegance arises as to whether a more economical presentation than either of those we have treated can be obtained: for instance, one in which fewer basic schemata are employed, or in which schemata involving fewer schematic letters. But we leave the further consideration of such questions to the interested reader. Exercise 5.35.18 (i) Prove, either from (10) or from the basis described in 5.35.17, the two associativity sequents: p · (q · r) (p · q) · r

and

(p · q) · r p · (q · r).

(ii) Show that every idempotent truth-function is associative. (Alternatively put: that every idempotent two-element groupoid is a band.) (iii) Does (ii) here mean that the only part of the basis given in 5.35.17 that is needed for completing part (i) of this exercise is (11) and (12), since strong idempotence suffices for idempotence? (Explain why or why not.)


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Conditions (11) and (12) were described as ‘strong idempotence’ principles, as earlier in 3.24, where the point was made that while they have (13) and (14) as consequences – indeed, since we are in Set-Set rather than Mset-Mset, as special cases – respectively: (13)

AA·A

(14)

A · A A,

they do not follow from (13) and (14) without the help of an extensionality principle such as the lhs-extensionality condition (15)

A, B, C(A) C(B).

As a special case of (15) we have A, B, A · A A · B, which with the help of (13) (and (M), (T), with A · A as the cut formula), we get (11). (In the discussion in 3.24 – pp. 475–478 – “·” appeared as “◦”, (11), (12), (13) and (14) as (I), (II), (I) and (II) , and in place of (15) we appealed to a principle called the TTx Rule.) Thus further avenues of variation on the theme of 5.35.17 are opened up: we could simplify the idempotence-related parts of the basis by explicitly positing extensionality principles. Another line of variation that might be pursued involves enriching the language of P V , with constants indicating membership in V1 or V2 . Let us use “1” and “2” as nullary connectives to serve as ‘markers’ for the two interpretations of ·. In the notation of 3.33, HV (1) = V1 and HV (2) = V2 . (In longhand, this means: v(1) = T iff v ∈ V1 and v(2) = T iff v ∈ V2 .) If we had boolean connectives and also propositional quantifiers available, we could define 1 and 2 as in (16) (16) 1 = ∀p∀q((p · q) ↔ p), 2 = ∀p∀q((p · q) ↔ q) but since we don’t, we adopt 1 and 2 as primitive, to be governed by some simple principles spelling out their role as marker constants; these say that given 1 (given 2) a ·-compound is equivalent to its first (resp. second) component: (17) (19)

1, A · B A 2, A · B B

(18) (20)

1, A A · B 2, B A · B.

Now in fact, we may as well exploit the fact that we are in Set-Set, and that either 1 or 2 is true on every valuation in the class of interest (V1 ∪ V2 ) and drop “2” from the language, rewriting (19) and (20) by putting “1” on the right rather than “2” on the left: (21)

A · B B, 1

(22)

B A · B, 1.

(If we retain “2” in the language, then we should acknowledge our interest in V1 ∪ V2 with an endorsement of the sequent 1, 2.) ‘Pair-validity’ can continue to be understood as before – amounting to the property of holding over V1 ∪ V2 , but with the restriction that v(1) = T iff v ∈ V1 ; it is clear enough that all instances of (17), (18), (21), and (22) are pair-valid in this sense. Somewhat less obvious is the fact that all of our earlier principles governing · can be derived from them: Example 5.35.19 A proof of A · B, C · D A · D ( = (4)). (Here we use two lines numbers with an annotation such as “cut A” to mean that the two lines yield, by an application of (T), preceded


761

if necessary by two applications of (M), the given line, A being the cut-formula.) (i) (ii ) (iii) (iv ) (v) (vi ) (vii)

1, A · B A 1, A A · D 1, A · B A · D C · D D, 1 D A · D, 1 C · D A · D, 1 A · B, C · D A · D

(17) (18) (i), (ii ) cut A (21) (22) (iv ), (v) cut D (ii ), (vi ) cut 1

Exercise 5.35.20 (i) Provide a proof, in the above style, of the sequent schema: A · B A, B ( = (12)). (ii ) Show that the reversibility property remarked on for the pure · proof systems is lacking here, and suggest a reformulation which would be correct for the system with (alongside the structural rules) (17), (18), (21), (22). Remark 5.35.21 Since the other principles of the proof system described in 5.35.17, namely (7) and (11), are similarly forthcoming, all pair-valid sequents involving only · can be proved from the basis described in 5.35.20(ii). Redescribing this as the set of sequents holding over V1 ∪ V2 , we have the further question whether such a result extends to the case of sequents in which 1 appears. For instance, given a valuation v defined on the basis of an unprovable sequent (as for the proof of 5.35.15) we have v ∈ V1 or v ∈ V2 as there, but do we also have, as desired, v(1) = T iff v ∈ V1 ? Unless further modifications are made, there is certainly a problem about the ‘only if’ direction here, but we shall not consider the matter further. A final question about the material in this subsection arises over the special status that the particular hybrid we have been considering has over other hybrids. It may seem odd that we have used the term ‘pair-valid’ specifically for (what amounts to) the property of holding over V1 as well as V2 , for our particular choice of V1 and V2 . After all, we could just as well have considered a collection of valuations over which · (or whatever) was associated with the truth-function ∧b and another collection over which it was associated with →b , and described as pair-valid (perhaps introducing this term via valuation-pairs in the style of 5.34, with a first coordinate interpreting the connective as ∧ and a second interpreting it as →). It would be nice to be able to hold that whatever the merits of this or that terminological proposal, there is something special about hybridizing the projections, which we might describe as their role as ‘master hybrids’. By this would be meant that every case of hybridizing of a pair of connectives can be achieved in a simple uniform manner, as explained below. We will present the idea – which appears (in a different logical framework) in Setlur [1970b] – in a fairly informal way, rather than introducing all the machinery that might be required for a proof (of what will appear as (26) below). Let us use the following notation, introduced in 3.24. For a formula A built up with the aid of a binary connective ∗, we denote by A(#1 ) and A(#2 ) the results of replacing all occurrences of ∗ in A by #1 , #2 , respectively, and write


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A as A(∗) to emphasize the connective replaced. If we have a set of formulas Γ, built using A then we may write Γ(∗) to indicate this, and Γ(#1 ), etc., for {A(#1 ) | A ∈ Γ}. To recall the example of the preceding paragraph, ∗ would be the hybrid according to a gcr , of classical conjunction and implication just in case: (23)

Γ(∗) Δ(∗) if and only if Γ(∧) CL Δ(∧) and Γ(→) CL Δ(→).

If we allow the formulas in Γ(∗), Δ(∗), to be constructed with the aid of connectives other than the proposed ∗, we could actually obtain from CL with the aid of our projection hybrid. The idea is to define ∗ by (24)

A ∗ B = (A ∧ B) · (A → B).

Notice that there is no reason for the compounds hybridized in this way to be constructed from binary connectives; for #1 , #2 , both n-ary, we could put, in the style of (24): (25)

∗(A1 , . . . , An ) = #1 (A1 , . . . , An ) · #2 (A1 ,. . . ,An ).

Here we see, in full generality, the idea that we can use · as a master key which at once unlocks all questions of hybridization. For the definition in (25) to make sense, we need to assume that #1 and #2 are (not necessarily primitive) connectives of the same arity (here n), and in fact that they both belong to the language of a single gcr, 0 say. First, we extend the language of 0 to include our special projection hybrid · at the same time boosting 0 to the least gcr for which 0 ⊆ and satisfies all the principles in (10) or those given in 5.35.17, with “” replaced by “”. It is in the context of that we envisage (25) being given as a definition of ∗, the candidate hybrid of #1 and #2 (as they behave according to 0 ), which to say, we want the general case illustrated by (24) above to obtain: (26) Γ(∗) Δ(∗) if and only if Γ(#1 ) 0 Δ(#1 ) and Γ(#2 ) 0 Δ(#2 ). So much for explaining the idea. (Compare (Hyb.) in 3.24, p. 468.) As already indicated, we shall not show rigorously that (26) is in fact guaranteed to hold by this procedure, entrusting several loose ends to the reader. With reference to the ‘if’ direction of (26), let us consider the example set of (23) and (24), and take Γ(∧) as {p, p ∧ q} with Δ(∧) as {q}; this turns (23) into: (27)

p, p ∗ q q if and only if p, p ∧ q CL q and p, p → q CL q

and so, unpacking the left-hand side, into: (28)

p, (p ∧ q) · (p → q) q iff p, p ∧ q CL q and p, p → q CL q.

What would guarantee the ‘if’ direction of (26) in general, here illustrated by the case of (28) in particular, would be the following property of : (29)

A1 , . . . , Am B1 , . . . , Bn and C1 , . . . , Cm D1 , . . . , Dn imply: A1 · C1 , . . . , Am · Cm B1 · D1 , . . . , Bn · Dn .

For those formulas in Γ(#1 ), here listed as A1 , . . . , Am or Δ(#1 ), appearing here as B1 , . . . , Bn , with no occurrence of #1 , we can exploit the idempotence


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of · according to , to obtain ·-free equivalent formulas. For example, let us take the case of (28), with A1 = p, A2 = p ∧ q (#1 and #2 being ∧ and →); we can choose C1 and C2 as p and p → q respectively so that the subscripts on “A” and “B” match up for corresponding formulas, and similarly for “C”, “D”. (We can always achieve such a matching up by suitable re-ordering, by 5.31.12(iii).) Thus A1 · C1 is p · p, which is equivalent to p, while A2 · C2 is (p ∧ q) · (p → q), alias p ∗ q. In these two cases, we have a #i -free (i = 1, 2) formula and a formula with exactly one occurrence of #i , that occurrence being as the main connective. Of course, these do not exhaust the possibilities. Perhaps #i occurs several times, perhaps within the scope of another occurrence of #i or of some other connective. A fully explicit treatment would require the re-introduction of the translations (·)1 and (·)2 of (8) of 5.34; we content ourselves with a couple of illustrations, and say no more about how to get the ‘if’ direction of (26) from (29). Examples 5.35.22(i) Here we consider hybridizing the identity connective with (classical) negation, with the assistance of ·; thus we have a case in which (25) above is realized, with n = 1, as: ∗A = A · ¬A. Consider ∗∗p p, a typical ‘common property’ of the two connectives under consideration. We have, for ∗∗p = (p · ¬p) · ¬(p · ¬p). ((p · ¬p) · ¬(p · ¬p))1 = (p · ¬p)1 = p1 = p ((p · ¬p) · ¬(p · ¬p))2 = (¬(p · ¬p))2 = ¬¬p2 = ¬¬p Thus in this case (∗∗p)1 p and (∗∗p)2 p, as expected (cf. 5.35.3). A general feature – emphasized in Humberstone [1986a] – of these hybridizing logics, which emerges in the additional presence of further (non-hybridized) boolean connectives, is Halldén-incompleteness, as defined in 6.41 below (and whose emergence here is explained by a minor variation on Theorem 6.41.4), which is illustrated by the fact that (∗p ↔ p) ∨ (∗q ↔ ¬q). (ii) Return to the case raised above, of hybridizing (classical) conjunction and implication, with A ∗ B = (A ∧ B) · (A → B), but this time examining a case involving (as in (i) above) embedding of ∗ in the scope of ∗, namely the ‘contraction’ principle: p ∗ (p ∗ q) p ∗ q. Unpacking the formula on the left of the , we have the subformula p ∗ q = (p ∧ q) · (p → q), which is therefore also what we have on the right. to this we get: Applying p ∗ (p ∧ ((p ∧ q) · (p → q))) · (p → ((p ∧ q) · (p → q))), which, discarding all right-hand components of any ·-subformulas – as we may describe applying the translation ( )1 , or equivalently, evaluated by interpreting · as proj12 – gives just p ∧ (p ∧ q), as desired, while discarding all left-hand ·-components (applying ( )2 , or interpreting · as proj22 ) gives p → (p → q), as desired. (And the succedent formula evaluates to p ∧ q, p → q, respectively, in the two cases, also as desired.)


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As for (29) itself, for as the gcr associated with a proof system supplying · with the powers conferred by the rules appealed to in 5.35.15, 5.35.17, a syntactic demonstration would be a proof that the following rule is derivable: (30)

A1 , . . . , Am B1 , . . . , Bn

C1 , . . . , Cm D1 , . . . , Dn

A1 · C1 , . . . , Am · Cm B1 · D1 , . . . , Bn · Dn

We shall just consider the simple m = n = 1 case of (30), however, deriving a rule to the effect that · is monotone (see 3.32) according to our : (31)

AB

CD

A·CB·D

First: a proof of A · C, D B · D, given the provability of A B; proof annotation conventions as in 5.35.19: (i) A · C, D B · D, A 10(ii) (ii ) B, D B · D (11) (iii) A B Given (iv ) A, D B · D (ii ), (iii) cut B (v) A · C, D B · D (i), (iv ) cut A (Note that line although the justification given for the schema at line (i) is “10(ii)” what actually appears is a relettering of the latter; similarly elsewhere.) Next, a proof of A · C B · D, D given the provability of C D and A B: (i) (ii ) (iii) (iv ) (v) (vi ) (vii)

A · C, B B · D, C A · C A, C AB A · C B, C A · C B · D, C CD A · C B · D, D

10(iii) (12) Given (ii ), (iii) cut A (i), (iv ) cut B Given (v), (vi ) cut C

Thus on the hypothesis that A B and C D are both provable, we have proofs of A · C, D B · D and of A · C B · D, D; from these by a cut on D, we obtain a proof of A · C B · D, completing the demonstration that (31) is derivable. With all this discussion of (29), it must be acknowledged that nothing has been said about the converse, also required for a vindication of (26). Exercise 5.35.23 Discuss the converse of (29) insofar as it bears on (26). Is it enough to establish that this holds for ·-free Ai , Ci , Bj , Dj ? (Compare 5.34.19, p. 749; define appropriate classes of valuations to play the role of V1 and V2 in our discussion above following 5.35.1, p. 751.) Exercise 5.35.24 What if we wanted to look at the hybrid of three, as opposed to two, connectives (as they behave according to CL)? For example, suppose we wanted to take up Example 5.35.22(ii) above, in which classical implication and conjunction we hybridized by setting A ∗ B = (A ∧ B) · (A → B), but this time we wanted to look at the hybrid of (classical) ∧, → and ↔. Would we succeed if we took ∗ as just defined and then defined (say), the proposed hybrid, by A B = (A ∗ B) · (A ↔ B), where ·, as before, hybridizes the two binary


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projection connectives? If not, what prevents success here and what alternative route along roughly similar lines might be pursued?

Notes and References for §5.3 The notion of extractibility featuring in 5.32.11 (and the discussion preceding it) is taken from note 8 of Humberstone [1998d ]. A condition much weaker than the condition that pi should be extractible from a formula A in the sense of that discussion would involve extracting a positive and a negative occurrence of pi : in classical logic, any formula A is equivalent to a formula (pi ∧ B) ∨ (¬pi ∧ C), for some pi -free formulas B and C, as one sees by a reflection on the truthtable for A. (The disjunction just cited is classically equivalent to (pi → B) ∧ (¬pi → C).) To cite a more representative illustration of the phenomenon under consideration, suppose that A(p, q, r, s) is a formula in the variables indicated and we choose to ‘extract’, in the current weaker sense than that of 5.32, the variables p and q, we can provide as an equivalent formula: ((p ∧ q) ∧ A(, , r, s)) ∨ ((p ∧ ¬q) ∧ A(, ⊥, r, s)) ∨ ((¬p ∧ q) ∧ A(⊥, , r, s)) ∨ ((¬p ∧ ¬q) ∧ A(⊥, ⊥, r, s)). Much interesting use is made of similar equivalences in Zolin [2000]. #-representable relations (5.33), for boolean #, are discussed in Humberstone [1984b], [1991b], [1995b] (the second of these papers supplying a list of errata for the first). The conditions we have, because of our emphasis on the role and behaviour of various connectives, called [∧] and [↔]1 , are (as was remarked) more often known as rectangularity and regularity respectively; see, for example, Riguet [1948] and Jaoua et al. [1991]. The term ‘regular’ is used here in a slight variation of the sense it has for semigroup elements in general (an element a being regular if it is of the form axa for some element x – parentheses and operation symbol suppressed here). Our 5.33.5 corresponds to Proposition 4.12 (p. 335) of the latter paper; the former paper calls regular relations di-functional (‘relations difonctionelles’) so presumably Riguet had a similar characterization in mind, though the present author was not able to detect an explicit formulation along these lines in Riguet [1948]. The information about rectangular bands exploited in 5.33 and 5.35 can mostly be found in McLean [1954] and Kimura [1958]; for their location in the lattice of varieties of all bands (idempotent semigroups) see Fennemore [1970] or Gerhard [1970]. For more elaborate versions of the structure-of-terms considerations (identity of leftmost variable, of rightmost variable, etc.) that feature in 5.35.6, see Ježek and McNulty [1989].

Chapter 6

“Or” He’s making a list, He’s checking it twice; He’s gonna find out Who’s naughty or nice. Santa Claus is Coming to Town (traditional Christmas song)

§6.1 DISTINCTIONS AMONG DISJUNCTIONS 6.11

Introduction

Before beginning with an introduction proper, giving a brief overview of this section and of the rest of the chapter, we offer, by way of an hors d’oeuvres – an “Or” d’oeuvres, we might say – a glimpse at a problem besetting studies of this most intriguing of the sentence connectives, the word or. The problem is that it is often not clear when a feature of the usage of this word constitutes a genuine datum for a semantic account to accommodate, and when it represents a confusion on the part of speakers (and writers) – a semantic (rather than syntactic) ‘performance error’. On p.216f. of Banfield [1982] in a discussion of the expectation that a literary text will be consistent, we read: Examples are easily found of inconsistencies, which are, however, judged unacceptable by readers. Heintz [1979] points to one from Tolstoy: “according to Anthony Savile, a careful reading of Anna Karenina reveals that it begins on a Monday or a Tuesday. Since nothing in the novel turns on this inconsistency, critics treat this as a slip on the author’s part and they do not treat the world of Anna Karenina as inconsistent on that account” (p. 92).

One can hardly avoid reacting to this passage in the following terms. So what, if according to the story, the action begins on a Monday or a Tuesday? That’s not inconsistent by any stretch of the imagination: the date of the action is underdetermined rather than overdetermined. Indeed, even if the story is quite clear that the action begins on a Monday – so that there is no underdetermination on this score after all – a report that according to the story it begins 767

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CHAPTER 6. “OR”

on a Monday or a Tuesday would still seem correct (though less informative than it might usefully be). What is meant by the remark attributed to Heintz is presumably something along the following lines: according to (some parts of) the story, the action begins on a Monday, while according to (other parts of) the story, the (same) action begins on a Tuesday. This would actually be what is predicted by one theory of how the word “or” functions: to mark a wide scope conjunction, which was just expressed using “while” in place of “and” for purely stylistic reasons. (We will see versions of this theory advocated by E. Stenius and R. Jennings in 6.14 and the appendix to this section.) Unfortunately the theory delivers a reading for the sentence about Anna Karenina which that sentence surely does not have – though the theory does have the interesting merit of explaining why Banfield should have written it. If one looks at p.92 of Heintz [1979] (to which there is a footnote crediting the inconsistency observation to Anthony Savile, in discussion) one finds, not very surprisingly perhaps, that Banfield has misquoted Heintz at the crucial point: “a Monday or a Tuesday” should be “a Monday and a Tuesday”! The question remains of course as to why Banfield did not notice that she had mistranscribed Heintz’s remarks: presumably she thought that the whole thing made good sense with “or”. It is as an explanation of how that might have come about that the ‘wide-scope and ’ suggestion, which works very well for a good many otherwise puzzling occurrences of or, may do some work. (A more mundane explanation is possible, too: unconscious interference from the title, “Monday or Tuesday,” of a 1921 Virginia Woolf prose poem.) Another area in which performance errors in semantics are apt to arise is that in which there are several scope-bearing elements, and conspicuously when among these are negation and modal auxiliaries. An example from the transcript of a radio interview (the details of which are given in the end-of-section notes: p. 816) is the following remark from the interviewee: Well, surely in terms of resistance exercise, you know, one can start by walking up stairs just to increase the strength of the thighs, some gardening where you’re lifting, pulling, pushing, shoving, all of those activities have a sufficient resistant component that that’s going to begin to you know increase strength. One can then begin to do, you know, callisthenics, push-ups, if you have the strength, pull-ups, all of those activities, you don’t need to go to a gym or you don’t even need to buy any special equipment.

Italics – and a few commas – have been added here. The italicized portion of the quotation is our current focus of interest, of course. The speaker is trying to say that neither going to a gym nor buying any special exercise equipment is required, and could have done so satisfactorily by saying “You don’t need to go to a gym or even buy any special equipment” or by keeping the longer form as quoted but changing the “or” to an “and”. As it stands, though, the speaker has simply failed to express the intended thought. It is of course perfectly common to produce such crossed constructions in everyday unrehearsed speech. We don’t need a special semantic account of the ingredients involved according to which what is said is in perfectly good semantic order as it stands. (Of course the sentence in the inset quotation is syntactically fine as it stands in this case, though one can get crossed wires in such cases which equally well result in syntactically ill-formed sentences.)

6.1. DISTINCTIONS AMONG DISJUNCTIONS

769

A less clear case is one discussed in Simons [2005b], in which it is claimed that the following sentence is ambiguous: Jane must not sing or dance. As well as the obvious reading, on which this means that neither singing nor dancing is permitted for Jane, Simons holds that there is also a reading according to which it means that Jane must not sing or Jane must not dance, a reading which is ‘made salient’ by adding “. . . but I don’t know which”. (The paper then proceeds to give a compositional semantics which would show how the sentence can manage to mean this.) But does the sentence actually bear this second interpretation? It would certainly be taken that way if someone added the don’tknow-which rider, but that may be not so much because the rider disambiguates an ambiguous sentence, as because it forces the hearer to reconstrue what has been said as something else in the vicinity (another sentence, that is), appended to which the rider makes sense. This is not to suggest that here is anything questionable about the vast majority of the examples in Simons [2005a], [2005b], used to motivate a very promising semantic theory of disjunction in terms of ‘supercovers’, described in 6.14 below (p. 810). By the time we get there, we will have seen some moves made (especially by Erik Stenius and Ray Jennings) concerning or as having the special role of a wide scope conjunction rather than its traditional role as expressing disjunction in situ (with more than a hint that this is actually the source of disjunction proper). It is with such suggestions in mind that the opening quotation from ‘Santa Claus is coming to town’ was chosen to head this Chapter; it should be admitted that the or of interest, in “He’s going to find out who’s naughty or nice” (meaning something like “He’s going to find out who’s naughty and he’s going to find out who’s nice”) appears in some printed versions of the lyrics as “and” – though one can see why this isn’t entirely the ideal word. These cautionary preliminaries out of the way – and we will return to the issues they raise at more than one point in what follows – we turn to the introduction proper. This section and §6.2 are the analogues for or of §5.1 for and ; collectively, they address disjunction in natural language with special reference to its treatment in formal logic. As in the case of conjunction, so here too we can usefully organize the material around objections to the familiar introduction and elimination rules. The case of the elimination rule (∨E) is reserved for §6.2; in the present section we concentrate on two worries to the effect that or does not always obey the informal analogue of (∨I). Few would deny that there is at least one sense for which that introduction rule is appropriate. There is the thought that if one were in a position to assert S1 one would typically not assert the disjunction S1 or S2 : but this is obviously a matter of conversational implicature rather than of what follows from what. Thus, in practice, opposition to (∨I) goes hand in hand with the claim that ‘or’ has more than one sense, one sense allowing (∨I), and one (or more) not allowing inference in accordance with this rule. We will discuss two such ambiguity claims, the first (6.12) alleging an inclusive/exclusive ambiguity, and the second (6.13) an intensional/extensional ambiguity. Before broaching the subject of the elimination rule in §6.2, we also include a discussion of contexts in which or appears to behave rather like and in disguise (6.14). (Several of these themes are treated at length in Jennings [1994b], which contains much additional information on uses of or.) A treatment of ∨-Elimination leads naturally to a consideration of multiple right-hand

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sides in Set-Set (§6.3). We conclude the chapter in §6.4 with a look at some semantic accounts designed to capture the logical behaviour of disjunction in intuitionistic logic (6.43), relevant logic (6.45) and quantum logic (6.47); the other subsections contain related material not summarized here. But first, we must back up and correct a couple of oversimplifications in what has already been said, as well as devoting a word to the rationale for discussing disjunction in such detail given that conjunction has already been treated: why not replace this Chapter with the words: “dualize Chapter 5”? The first potentially misleading suggestion made above was that one in a position to assert S1 will typically not assert the needlessly weaker disjunction of S1 with something else (S2 , say). The presumption here is that the speaker is being as informative as is appropriate to the occasion, so that to give the diluted information “Either S1 or S2 ” is to implicate (conversationally) that one is not in a position to assert S1 outright. An inchoate appreciation of this fact sometimes causes students to feel that there is something especially ‘pointless’ about the rule (∨I), and it is usual to put their minds at rest on this score by reminding them of contexts in which an inference of just this form is naturally made, such as in answering a questionnaire requesting a tick in a box if the respondent has three or more children: a respondent with three children who accordingly ticks that box is in effect reasoning “I have three children, therefore I have three or more children”. In this case the conversational implicature noted above does not set in, no opportunity for giving the more specific information having arisen. But even, as in ordinary conversation, when there is such an opportunity, the maxim “be informative” does not always dictate that when one’s evidence would justify asserting S1 it would be misleading to assert instead “S1 or S2 ” (hence the “typically” in the opening sentence of this paragraph). Jackson [1987], §2.2, gives an example in which one would do so without impropriety, thereby conveying that this disjunction had a high (subjective) conditional probability given the negation of each of the disjuncts; he calls this property (two-sided) robustness. (One-sided robustness arises with the special phrases “or anyway”, “or at any rate”, etc., as in “She is over forty, or at any rate over thirty-five”, in which the disjunction is taken as highly probable given the negation of its first disjunct only.) We will not summarise the example here, but refer the reader to Jackson’s discussion, with the suggestion that it seems a delicate matter whether what is thus conveyed is so as a matter of conversational implicature (as Jackson argues) or instead conventional implicature. Nor will any further consideration be given here to ‘tonal’ (conventional-implicature-signalling) aspects of or. Digression. As remarked in 5.11, we shall not consider disjunction as it appears in non-declarative contexts (questions, commands) though the interested reader will find that the distinctive or of (direct and indirect) ‘alternative’ questions is related to ordinary disjunction on p. 204 of Lewis [1982] (as well as Groenendijk and Stokhof [1982]) – see also the end of 4.22 above in the Digression on p. 560, where the label ‘whether -disjunction’ is used for or explicitly embedded under whether. Groenendijk [2009] presents a novel account of the relation between the or of alternative questions and regular declarative or. We pause here to note some interesting observations made by Samuel Fillenbaum on mixed imperative/declarative disjunctions such as “Get out of my way or I’ll hit you”. (See Fillenbaum [1974] for data, and Fillenbaum [1977], [1978], esp. p. 204f., [1986] for discussion, as well as Lakoff [1971], p. 144, and Clark [1993], which


771

provides many further references.) There is a very striking non-commutativity phenomenon here, since reversing the disjuncts appears to turn these constructions ungrammatical. Further, they are always interpreted as threats (or perhaps warnings) whereas the form instantiated by “If you don’t get out of my way, I’ll hit you” can also be used for conditional offers or promises – ‘positive’ inducements – as in “If you don’t mention this to the police, I’ll give you half the money”. Thus, this last example cannot be rephrased, retaining its character as an inducement to silence, as “Mention this to the police or I’ll give you half the money”. In fact, there seems to be some spill-over of this negative inducement effect into the case of pure declaratives, as in the example “Mary gets home by midnight or her parents are furious”, which is grammatical, but bizarre, with disjuncts reversed, and a suitable name/pronoun switch. (This example is from Sweetser [1990], p. 99, though – following Lakoff [1971], p. 143, which contains a similar example – she restricts herself to suggesting that the non-commutativity reflects the direction of causation. In fact, though, some kind of generalized Fillenbaum effect would appear to be operating here: for compare “Mary gets home after midnight or her parents are pleased”. We return to Sweetser in 6.13.) A general guide to the psychological literature on processing disjunctions is provided by Newstead and Griggs [1983]. Non-commuting or sentences already arise with Jackson’s “or anyway”, from the preceding paragraph, and various other constructions noted on p. 291 of Jennings [1994b], as well as the quasi-counterfactual or sentences noted on p. 154f. of that work, like “I didn’t know you were coming, or I would have offered you a lift”. (See p. 152f. for a possible explanation of the construction in question.) As in some of the earlier cases, reversing the order of the disjuncts here destroys grammaticality. Note or _ _ also for many of these cases the unavailability of the fuller “either _” form. For an informative discussion and a review of other publications not mentioned here, see also Gillon [1997]. Gillon gives an interesting case in which commutativity is putatively problematic, much as in the declarative–imperative cases, but which are imperatival on both sides “Give me liberty or give me death”. By contrast with the mixed cases, however, it is not obvious that we have more than a difference in poetic quality between the two orderings, that given, on the one hand and “Give me death or give me liberty”, on the other. (Here I am indebted to Steve Gardner.) End of Digression. On the more straightforward conversational implicature front, it is worth re-quoting (with emphasis added) from 5.11 the point that such implicatures arise from the maxim of informativeness on the basis of “the presumption that one does not without good reason say something less informative on a given topic than one might have said at little or no extra cost in sentence length” that the speaker is inferred not to have been in a position to make a stronger statement. Thus, if you tell someone, “I’ve just bought a kitten”, you will not be taken as conveying ignorance of its sex, or of the existence of any other special reason for not saying (e.g., that the hearer is now supposed to guess) what sex it was, whereas if you say “I’ve just bought a male kitten or a female kitten” just such implicatures and expectations will be triggered. This second, disjunctive sentence – for we may take it to amount to no more than “I have just bought a male kitten or I have just bought a female kitten” – is longer than the original, and in particular, it is longer than two sentences the original is not longer than, namely (i) and (ii):

CHAPTER 6. “OR”

772 (i) (ii )

I have just bought a male kitten. I have just bought a female kitten.

So the question arises as to why, rather than coming out with the disjunction, you didn’t come out with (i) or with (ii), to which the answer that perhaps for some reason you were ignorant of the sex of the kitten suggests itself to the hearer. Clearly no such question is raised by simply saying “I’ve bought a kitten”. So it would appear that some such consideration as sentence length (in words) is operative. (No doubt a more accurate description would replace talk of S1 ’s being longer than S2 by talk of S1 ’s having S2 as a proper part, though the precise explication of ‘proper part’ would take some work.) The phenomenon here illustrated has been noted by Jackson ([1987], p. 27) who also observes that it serves as a counterexample to the claim of Grice [1967] that conversational implicature is ‘undetachable’ in the sense that logically equivalent sentences – as we may reasonably if not quite accurately take “I’ve bought a kitten” and the longer disjunction to be – cannot be uttered to different effect in respect of conversational implicature. (However, Jackson stresses the robustness implicature mentioned in the preceding paragraph rather than the present length-related considerations; the latter are emphasized in §3 of Gasking [1996] – a paper written, according to the editors of the volume in which it appears, in the late 1950s or early 1960s. An even earlier anticipation of Grice’s own views on the relation between disjunctions and their disjuncts may be found in Ryle [1929], from which we shall have occasion to quote below.) The second oversimplification in our opening comments was the claim of a consensus as to the existence of at least – if not exactly – one sense of or for which the rule (∨I) was appropriate. It should be mentioned that there is a certain notion of implication, analytic implication, pioneered by W. T. Parry, according to which disjunctions are not (in general) implied by their disjuncts. We regard this as a specialized notion of implication rather than an eccentric view about implication, for reasons of charity. However, it is sometimes taken the other way, and denied that there is any sense in which a disjunction follows from its disjuncts. (For example, so taken, we would have a view that blocked the ‘Lewis Argument’ given as 6.13.1 on p. 789 below, at line 3. This view is called conceptivism, in §2.5 of Routley et al. [1982], in which work such a blocking effect is noted, because of the informal idea that when one thing analytically implies another, every concept required for grasping the latter should figure in the former. For further variations on this them, see also §2.2 of Makinson [1973a], Paoli [2007], and §5.2 of Burgess [2009].) Finally, there is the question of duality. Obviously several topics arising for conjunction do not arise for disjunction and vice versa. In the latter category falls the inclusive/exclusive issue, mentioned at the end of this subsection. Into the former falls the case of temporal conjunction (5.12): there is no corresponding “or next” or “or then” construction (for fairly obvious reasons: there is no time parameter to pick up on). Such divergences notwithstanding, for many logical purposes, as perhaps 1.16 illustrates, conjunction and disjunction are attractively treated in a symmetrical fashion. Inherent asymmetries in the informal conceptual apparatus we bring to bear on logic often make duality an inappropriate consideration to bring in for philosophical purposes, however. To illustrate this, recall that in 5.13 we considered the question ‘What is the point of having conjunction in a language? Instead of saying S1 and S2 , after all, one


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might content oneself with saying S1 and saying S2 . (Our answer to this question turned on noting that not all saying is asserting, and in particular for certain sentences featuring embedded occurrences of conjunction, no conjunction-free set of alternatives was to be had.) Consider the analogous question for disjunction. ‘What is the point of having disjunction in a language? Instead of saying S1 or S2 , after all, one might just as well come out and either say S1 or say S2 .’ Here the inclination is to reply immediately (and without even appealing to the phenomenon of embedding) that one is often in a position to assert a disjunction without being in a position to assert either disjunct, since one can believe that p or q without either believing that p or believing that q. (Such a response is even possible for an intuitionist, given a distinction between possessing a proof of one of the disjuncts and possessing a method which, if applied, would provide such a proof.) A reflection of the way these considerations disallow a treatment of or by replacing the present discussion with the instruction “Dualize Chapter 5” arises from the notion of collective equivalence in play there, and in particular in the discussion after 5.13.3, in which the question was raised as to whether, in view of the fact that for every formula in the language with connectives ¬, ∨, →, ∧ there is a set of ∧-free formulas collectively equivalent to that formula according to IL , we should regard the addition of ∧ with its accustomed properties to the {¬, ∨, →}-fragment of IL as representing an increase in expressive power. Favouring a negative answer to this question is the consideration that whatever can be said with the aid of ∧ can be said without it, provided that we allow the ‘saying’ to consist in committing oneself to a (finite) set of formulas. No such once-and-for all answer was returned in 5.13 on the grounds that the question can be made precise in more than one natural way, but the present point is that on at least one such precisification, a negative answer is plausible. For a finite set Γ of formulas to be collectively equivalent to a formula A, in the case of a language containing ∧ and for a consequence relation which is ∧-classical, is for the conjunction of the formulas in Γ to be equivalent to A. One could isolate a notion dual to this with its being the disjunction of formulas in Γ that is required to be equivalent to A, but there is little temptation to think of Γ as being in any sense collectively equivalent to A when this relation obtains, or to think of the presence of ∨ as being expressively redundant in the case in which for every A there is a set Γ of ∨-free formulas standing in this relation to A. (A closely related point is the difference in utility remarked on in Example 1.11.3, which began on p. 50, between what are there called deductive disjunction and deductive conjunction.) Digression. For the case of assertion, it would be still further from the truth to say that someone asserting a disjunction is asserting one of the disjuncts, than to say that someone asserting a disjunction represents themselves as being in a position to assert one of the disjuncts. The even less plausible claim has been made, however, as in the following passage from James [1985b], p. 468f. (in which “conjunction” is used in its traditional grammatical sense, for binary connectives in general): A sentence with or always asserts at least one of the independent clauses joined by the conjunction, although which clause is to be asserted is left in doubt.

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Perhaps the author’s realization that the initial claim is not quite right—a misformulation of the thought that what is asserted is (of course) that one of the clauses is true—is what prompted the insertion of the words “to be” in “to be asserted”, words which certainly don’t fit with the thought whose expression precedes the although: Asserting that one of two things is true is not the same as asserting one of those things. End of Digression. Here the sticking point is the concept of belief. A full dualization would invoke a concept, say belief *, thought of as like , where belief is thought of as like : to believe* that p is to fail to believe that not-p (these being the dual doxastic operators of Hintikka [1962]. It is true that one will believe* a disjunction only if one believes* at least one of its disjuncts; but now we have lost our moorings: it is beliefs rather than beliefs* that the premises of arguments we endorse are supposed to express, beliefs rather than beliefs* that our assertions are supposed to reflect, and so on. (Similar considerations have been urged against the use of multiple-conclusion frameworks such as Set-Set in §7 of Rumfitt [2008].) At a more superficial level, the pragmatic role – mentioned above – played by considerations of sentence length itself injects an asymmetry into the discussion of conjunction and disjunction, since the logically stronger element in the former case is the longer (S1 and S2 having each conjunct as a proper part, the conjunction is longer that the implied conjuncts) whereas in the latter case it is the shorter (S1 , S2 , each being shorter than S1 or S2 ). In what way is this observation superficial? The dependence on considerations about length shows that it relies on the concrete conception of languages and connectives described in 1.11. There is no reason for the disjunction of two statements—for what counts as S1 ∨ S2 according to an ∨-classical consequence relation (say)—to be physically constructed out of them with the aid of an additional symbol between (or in front) of them, as the discussion of (i) and (ii) above was intended to show. Less superficially: the very propriety of a dual treatment of and and or is itself not something to be taken for granted, and nor is the question of what precisely a dual treatment would consist in (e.g.: take Set-Set as our paradigm framework?) We opened our discussion of conjunction in 5.11 with some remarks about the equivalence between the proof-theoretic perspective, considering the syntactically described condition of ∧-classicality on consequence relations, and the perspective of valuational semantics, with the notion of ∧-boolean valuation. Taking the latter notion first, recall the point that for any collection V of ∧-boolean valuations, we have HV (A ∧ B) = HV (A) ∩ HV (B), for all formulas A and B. (The “H” notation is from the discussion surrounding 3.33.1, p. 495.) This dualizes readily to the case of disjunction: a collection V of ∨-boolean valuations must satisfy: HV (A ∨ B) = HV (A) ∪ HV (B). Turning to the former – inferential – characterization, we noted that for ∧-classical consequence operations, we had Cn({A ∧ B}) = Cn({A, B}), for all A, B. The most direct analogue of this for disjunction would be Cn({A ∨ B}) = Cn({A}) ∩ Cn({B}). Rather than capturing ∨-classicality for consequence operations, this condition corresponds only to a restricted version thereof, combining the effects, not of (∨I) and (∨E), but of (∨I) and the rule (∨E)res given in 2.31 (p. 299, more on which will be found in 6.47). To capture the effect of (∨E) we have to allow side formulas (ancillary assumptions), here represented by “Γ”:


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Cn(Γ ∪ {A ∨ B}) = Cn(Γ ∪ {A}) ∩ Cn(Γ ∪ {B}) In the case of conjunction, we could also have used a formulation with side formulas, though, by contrast with the present case, it would have made no difference there: Exercise 6.11.1 Show that the conditions (understood as prefixed by ‘for all formulas A, B, and all sets of formulas Γ’) (i) and (ii ) are equivalent conditions on a consequence operations Cn: (i) Cn({A ∧ B}) = Cn({A, B}) (ii ) Cn(Γ ∪ {A ∧ B}) = Cn(Γ ∪ {A, B}). (See Grzegorczyk [1972], p. 34, for a formulation along the lines of (ii ), though using a slightly different notion of consequence operation, as well as for remarks about the disanalogies between the appropriate conditions for ∧ and ∨. All such formulations go back to a 1930 paper by Tarski – chapter V in Tarski [1956] and are often called Tarskistyle conditions as a result). In Pogorzelski and Wojtylak [2001] the phrase “Cn-definitions (of propositional connectives)” is used instead, and some effort is made to investigate which connectives in which logics are amenable to characterization in terms of such conditions. See also Chapter 4 of Pogorzelski and Wojtylak [2008]. We have had occasion to remark that when (f, g) is a Galois connection then f ◦ g and g ◦ f are closure operations (0.13.4(ii), p. 9), that every closure operation can be represented as being of this form (0.13.10, p. 11), and that in particular the closure operation obtained as f ◦ g for the Galois connection (f, g) between the set of formulas of a language and any class V of valuations for that language induced by the relation ‘is true on’ is none other than the consequence operation determined by V (1.26). Our discussion in the latter subsection used the neutral notation (f, g) with f mapping a set of formulas to the set of valuations verifying all of them and g mapping a set of valuations to the set of formulas true on all of them. Let us use the more memorable abbreviations H and Tr for this f and g respectively. That is, keeping as fixed in the background some set L of formulas and V of valuations, (H, Tr) is the Galois connection between L and V which arises by putting, for Γ ⊆ L, U ⊆ V : H(Γ) = {v ∈ V | v(A) = T for all A ∈ Γ} Tr (U ) = {A ∈ L | v(A) = T for all v ∈ U }. If we agree to write H(A) for H({A}), as well as Tr (v) for Tr ({v}), then this use of H agrees with that above, as applied to single formulas; indeed we could have defined the present use as an extension of the earlier use by setting H(Γ) = {H(A) | A ∈ Γ}. Now, if we put Cn = Tr ◦ H, we can rewrite the ∧-classicality condition given in 6.11.1(i) above as the first line of a derivation as follows: (1) Tr (H(A ∧ B)) = Tr(H({A, B}) (2) H(Tr (H(A ∧ B))) = H(Tr(H(H({A, B}) (3) H(A ∧ B) = H({A, B})

CHAPTER 6. “OR”

776 (4) H(A ∧ B) = H({A} ∪ {B}) (5) H(A ∧ B) = H(A) ∩ H(B).

In passing from line (1) to (2), we have applied H to both sides, then obtaining (3) by 0.12.2 (p. 4), (4) by rewriting the rhs of (3), and, finally, (5) by 0.12.3. Thus from the ∧-classicality of Cn, taken as Tr ◦ H, the distinctive ∧-booleanness condition, here formulated as the ∩-principle (5), follows. Since all steps are reversible, ∧-booleanness condition similarly yields ∧-classicality for Cn: a familiar fact in a perhaps less familiar notational setting. (To go in reverse from (2) to (1), apply Tr to both sides and invoke 0.12.2 again.) For disjunction, this last point also applies, for if we assume the corresponding ∪-principle – that H(A ∨ B) = H(A) ∪ H(B) – as our formulation of ∨booleanness (of all valuations in V ), then we can derive the ∨-classicality condition Cn(Γ ∪ {A ∨ B}) = Cn(Γ ∪ {A}) ∩ Cn(Γ ∪ {B}), for Cn = Tr ◦ H, thus: (6) H(Γ)∩ (H(A) ∪H(B)) = (H(Γ) ∩ H(A)) ∪ (H(Γ) ∩ H(B)) (7) H(Γ) ∩ H(A ∨ B) = (H(Γ) ∩ H(A)) ∪ (H(Γ) ∩ H(B)) (8) H(Γ ∪ {A ∨ B}) = H(Γ ∪ {A}) ∪ H(Γ ∪ {B}) (9)

Tr (H(Γ ∪ {A ∨ B})) = Tr (H(Γ ∪ {A}) ∪ H(Γ ∪ {B}))

(10) Tr (H(Γ ∪ {A ∨ B}) = Tr (H(Γ ∪ {A}) ∩ Tr (H(Γ ∪ {B})). Here (7) derives from (6), the distributive law, by appeal to the ∪-principle, and (8) follows by using 0.12.3 on both sides, with Tr then applied to each side to obtain (9), with a final appeal to 0.12.3 again on the right to yield (10). This derivation corresponds to the reverse derivation mentioned earlier, from (5) to (1). But a derivation in the direction from (1) to (5) for the present case, showing that the ∪-principle follows for H from the ∨-classicality condition on Tr ◦ H, is not available. Dividing the ∨-classicality condition into the two inclusions, we find that we can work our way back to one half of the ∪-principle from one of them, that corresponding to (∨I), while we cannot similarly work back from the other, corresponding to (∨E), to the remaining half of the ∪-principle. First we have a simplification to note in the case of the inclusion for which the derivation is available: Exercise 6.11.2 Show that the conditions (understood as prefixed by ‘for all formulas A, B, and all sets of formulas Γ’) (i) and (ii ) are equivalent conditions on a consequence operations Cn: (i) Cn({A ∨ B}) ⊆ Cn({A}) ∩ Cn({B}) (ii ) Cn(Γ ∪ {A ∨ B}) ⊆ Cn(Γ ∪ {A}) ∩ Cn(Γ ∪ {B}). It is because of the equivalence of (i) and (ii) here that there is no need for a distinction between (∨I) and (∨I)res analogous to the corresponding distinction for (∨E). Now, since conditions 6.11.2(i) and (ii) are equivalent, we work with the simpler form (i), rather than (ii) which directly states one of the two inclusions embodied in the current formulation of ∨-classicality, in order to derive the corresponding half of the ∪-principle. As usual, we begin by rewriting the


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condition in terms of Tr ◦ H, and in fact we rewrite in this form only part of the content of (i), namely Cn({A ∨ B}) ⊆ Cn({A}): (11)

Tr (H(A ∨ B)) ⊆ Tr (H(A))

(12)

H(Tr (H(A))) ⊆ H(Tr (H(A ∨ B))

(13)

H(A) ⊆ H(A ∨ B).

Here (12) comes from (11) by ‘Galois-flipping’ (condition (G1) or (G2) from 0.12), and (13) then by appeal to 0.12.2 (p. 4). Similar reasoning from another consequence of (i), Cn({A ∨ B}) ⊆ Cn({A}), gives H(B) ⊆ H(A ∨ B). Thus putting both conclusions together, we obtain as promised, one of the two inclusions involved in the ∪-principle: (14)

H(A) ∪ H(B) ⊆ H(A ∨ B).

If we attempt a similar derivation of the converse of (14) from the converse of 6.11.2(ii), we find our route blocked; the starting point would be: (15)

Tr (H(Γ ∪ {A})) ∩ Tr (H(Γ ∪ {B})) ⊆ Tr (H(Γ ∪ {A ∨ B})).

We cannot in this case proceed to consider A and B separately, since (15) does not imply Tr (H(Γ ∪ {A})) ⊆ Tr (H(Γ ∪ {A ∨ B})) or the corresponding inclusion with B in place of A on the left. Note that the appropriate starting point, (15), is the converse of 6.11.2(ii) rather than that of 6.11.2(i), since the latter—cf. (∨E) vs. (∨E)res —is a weaker condition. However, the further difficulties in giving anything like the derivation (11)–(13) in this case are illustrated more simply and with no essential loss by considering the weaker condition, here stated as (16): (16)

Tr (H(A)) ∩ Tr (H(B)) ⊆ Tr (H(A ∨ B))

(17)

H(Tr (H(A ∨ B))) ⊆ H(Tr (H(A)) ∩ Tr (H(B)))

(18)

H(A ∨ B) ⊆ H(Tr (H(A)) ∩ Tr (H(B)))

Here we have obtained (17) from (16) by ‘Galois-flipping’, and passed to (18) by collapsing H ◦Tr ◦ H to H in accordance with 0.12.2; on the rhs, however, we do not find the H ◦Tr ◦ H arrangement for a similar simplification – eliminating the Tr there. We could appeal to 0.12.3, replacing Tr (H(A)) ∩ Tr (H(B))

by

Tr (H(A) ∪ H(B))

on the right to obtain (19)

H(A ∨ B) ⊆ H(Tr (H(A)) ∪ (H(B)))

which brings in the desired occurrence of “∪” (for the ∪-principle), but again does not present “Tr ” in an eliminable position. Remark 6.11.3 Something (19) does draw to our attention, however, is the following: If we had, for all A, B, some formula or other ϕ(A,B) with H(ϕ(A,B)) = H(A) ∪ H(B), then we could show that our ∨-classically behaving disjunction would be such a formula, since we could replace the rhs of (19) by H(Tr (H(ϕ(A,B)), reduce the H ◦Tr ◦ H to H, and then reinstate H(A) ∪ H(B).

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Of course what we should dearly like to be able to do is to appeal to an equality H(Γ ∩ Δ) = H(Γ) ∪ H(Δ), which, with Γ and Δ as Tr (H(A)) and Tr (H(B)) respectively, would take us from (18) to: H(A ∨ B) ⊆ H(Tr (H(A)) ∪ H(Tr (H(B)))) and thus, by 0.12.2, to the desired H(A ∨ B) ⊆ H(A) ∪ H(B). Unfortunately the needed equality, H(Γ ∩ Δ) = H(Γ) ∪ H(Δ), is in general only available in perfect Galois connections, and the present connection is perfect neither on the left nor on the right. (We return to this point at the start of 6.24.) The failure of the attempts described above, to derive the ∪-principle from the ∨-classicality of H ◦ Tr, is a manifestation of the fact that what in 1.14 (p. 67) we called the Strong Claim for ∨ is false: ∨-classicality, by contrast with the corresponding property for conjunction, does not guarantee that a consequence relation – or, for the present discussion, the associated consequence operation – possessing this property has amongst its consistent valuations only those which are ∨-boolean. Our discussion has suppressed the reference, in a subscript on “H” to the class of valuations under consideration, it being understood as arbitrary but fixed throughout (6)–(19). According to Thm. 1.14.9 (p. 70), the valuations consistent with a consequence relation determined by some class V are not only those in V but also those which can be obtained therefrom by conjunctive combination. Let us denote the set comprising the latter by V . As we recall from the proof of Thm. 1.14.6 (p. 69), not all conjunctive combinations of ∨-boolean valuations are ∨-boolean. Thus, where BV ∨ is the set of ∨-boolean valuations, while Tr ◦ H(BV∨ ) = Tr ◦ HBV∨ is ∨-classical, in the case of the left-hand designation of this consequence operation, the ∪-principle fails; more specifically, the inclusion H(A ∨ B) ⊆ H(A) ∪ H(B) does not hold when H is taken as H(BV∨ ) . In several subsections of §6.4, we shall exploit the possibility of non-∨-boolean valuations in the semantics of disjunction. For the present, we hope that the ‘take conjunction and dualize’ recipe for thinking about disjunction has lost some of whatever appeal it may have had, in the light of the above discussion. What follows should further convince the reader that disjunction has a life of its own and raises numerous distinctive issues not profitably treated by attempts simply to dualize remarks about conjunction, either those made in Chapter 5 or those which (as the reader may feel) deserved to be made but weren’t. Although we are concerned in this section with possible objections to the rule of ∨-introduction thought of as applying in the case of natural language or, there is one such objection deserving only of passing mention, and no more will be said about it than what follows. In the course of a discussion of negation, Mabbott [1929] has occasion to a consider a certain railway signal, of such a type as to display either red or green. He writes (p.73) of the claim that at some particular time the signal round the next corner is either red or green that this is “definitely false, for at this moment the signal round the corner is not ‘either red or green’; it is green”. Thus Mabbott rejects ∨-introduction for or, holding that although the disjunct “it is green” is true, the disjunction “it is either red or green” (which we may take to be no different from “either it is red or it is


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green”) is false. A closely associated view would be that the disjunction, unlike the disjunct, is somehow more subjective in character, a report more of the speaker’s ignorance than of the state of the signal. Against all such reactions, Ryle, a co-symposiast of Mabbott’s, sounds the appropriate—and in retrospect we may say, Gricean—note: Take another case. I judge at Reading, say, “That train is going either to Swindon or to Oxford”; and I do so without necessarily implying that the engine-driver, the passengers, the signal-man or even I myself are in ignorance or doubt which its route actually is. Ordinarily, of course, I would not bother to make the statement if I was not in some doubt, since if I could identify its route it would be superfluous to mention such nonindividuating facts about it. But facts do not cease to be facts or cease to be known when it becomes superfluous to mention them. Ryle [1929], p. 92f.

These passages from Mabbott and Ryle are quoted in Chapter 1 of Jennings [1994b], along with many others in the former vein from the earlier years of the twentieth century attesting to similar confusions. Not wholly dissimilar sentiments have been expressed in more recent years, as we shall see in 6.13; compare Mabbott’s “definitely false” with the “certainly false” of the passage from Fisk [1965] quoted before 6.13.2 below. Digression. The idea behind Mabbott’s remarks about the signal is that it cannot be correctly described in such disjunctive terms as ‘red or green’ since this would suggest falsely that the signal is indeterminate in respect of colour between the alternatives offered, whereas it is quite determinate in colour. One is reminded of the discussion in Geach [1979], p. 47, of the metaphysical fiction of the bare particular: To be sure, some people are so hypnotized by mere words that even without jargon they will think to avoid contradiction by change of stress: ‘A substance can have no QUALITIES because IT is WHAT HAS the qualities’.

(Here Geach’s italics have been replaced by capitals to suggest the fortissimo stress needed even to appear to bring off the attempted feat.) In the case of disjunction, the corresponding move would be to claim that the signal cannot be EITHER red OR green, because it is either RED, or GREEN. End of Digression. We turn, then, to the inclusive/exclusive distinction. That such a topic comes up at all for discussion, incidentally, illustrates the claim that disjunction has something of a life of its own, raising its own issues: no ambiguity claim dual to the inclusive/exclusive ambiguity claim for or has ever been suggested for and. The Post dual of exclusive disjunction is material equivalence, not normally something one thinks of and as expressing. It is of course also perfectly possible to introduce a connective of (as we might somewhat facetiously put it) exclusive conjunction @ with the explanation that, relative to any valuation, A @ B is true iff A is true and B is true, but not both. This associates with @ the binary constant truth-function with value F. More seriously, conjunction does raise to an issue about distinctness which is reminiscent the exclusivity issue for disjunction:

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Example 6.11.4 In a broadcast of the television quiz show The Rich List in Australia (known elsewhere under the names The Money List and Who Dares Wins) in 2008, a contestant challenged to list prime numbers under 100 lost out at the first entry proffered, which was “1”. And indeed on the usual conventions, 1 is regarded as neither prime nor composite. But of present interest is the compère’s explanation as to why 1 did not constitute a prime number, which was that “a prime number is a number divisible only by 1 AND itself” – capitals here for reporting emphasis. Thus evidently the compère took it that the “and” implied the distinctness of the “1” and the “itself” (putting the point loosely). This raises the usual question as to whether we have implication proper (i.e. entailment) here, or simply implicature. The latter seems more likely (which makes the compère’s explanation faulty). If someone says, “Betty and my sister walked into the room”, they manage to convey the impression that two people entered, by means of this ‘distinctness implicature’, but presumably could defend themselves against a charge of having spoken falsely by saying, “Betty is my sister, so when Betty walked into the room, it was true that Betty walked in and my sister walked in, and hence true that Betty and my sister walked in”. Harder to settle would be the question of what happens, truthconditionally, if the word both is inserted into such sentences. Presumably there is no problem with the fully sentential version: “It is the case both that Betty walked in and that my sister walked in”, but in the subsentential case a verdict seems less clear: “Both Betty and my sister walked in”. Here the distinctness of the parties referred to does seem required for the truth (and not just the non-misleadingness) of the remark.

6.12

Inclusive/Exclusive

Let be a binary connective (as syntactic operation) distinct from all those mentioned to date in this book. In the style of §1.1, call a valuation v -boolean if for all formulas A and B, v(A B) = T iff v(A) = T or v(B) = T, but not both. The last three words of this definition ‘exclude’ as one of the verifying cases for A B, the case in which both components are true, and when attention is restricted to such -boolean valuations, it is usual to refer to A B as the exclusive disjunction of A and B. Interestingly, in certain early writings on sentential logic, this was part of what was meant by “disjunction” simpliciter, as is indeed suggested by the etymological connection with disjointness – on which more below – with a separate term (e.g., “alternation”) for what we now call disjunction. More recently, Curry [1963] remarks (p. 154) that only the exclusive disjunction of two formulas is ‘properly called’ their disjunction. Indeed the term was even used for the negation of the conjunction of two formulas, with the exclusivity remaining but the – as we would now put it – disjunctiveness altogether lost. (See for example Swenson [1932]. This is the Sheffer stroke of example 3.14.4, p. 405.) However, our interest here is not on what may or may not have been meant by “disjunction” but on what may or may not be meant by “or”, and in particular on the meaning of this expression as it connects declarative sentences (a restriction imposed on our study of all connectives in 5.11). According to the inclusive/exclusive ambiguity thesis, while such occur-


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rences of ‘or’ are sometimes best represented by “∨”—these being the cases of inclusive disjunction—sometimes the more faithful representation is given by “ ”, the “but not both” being part of the very meaning of the word or in these cases. Since sequents of the form A A B (and likewise with B on the left of the “”) are obviously not in general tautologous—do not, that is, hold on all boolean valuations—such a claim involves denying that a rule analogous to (∨I) appropriately governs all occurrences of “or”. But what can be said for the claim? Obviously the last place to look for substantiation of the claim would be in disjunctions S1 or S2 in which the disjuncts themselves represent ‘disjoint’ (in the sense of mutually exclusive) possibilities. For (A ∨ B) ↔ (AB) is a tautological consequence of ¬(A ∧ B), so these cases furnish no evidence for a difference between an inclusive and an exclusive reading of the or. (See Exercise 6.12.9 below, with the contrast between exclusive disjunction and exclusionary disjunctions, the latter being inclusive disjunctions with mutually exclusive or ‘contrary’ disjuncts.) Yet just such examples were widely cited in the heyday of the inclusive/exclusive ambiguity thesis: “Either Fiona is in Rome or she’s in Paris”, etc., being held to display the exclusive or in its most obvious form, since Fiona evidently cannot be in both places. This would be like arguing that and must sometimes mean and then (cf. 5.11) on the basis of examples like “At 8 o’clock she got up and at 8.10 she had breakfast”. Citations from the offending texts and articles may be found in the references by Barrett and Stenner, Dudman, Richards, and Pelletier listed in the notes to this section, which begin on p. 816. The present remarks draw heavily on these as well as Jennings [1986]. (The latter paper also puts paid to the story that the inclusive/exclusive distinction was alive and well in Latin, being lexically marked there as the distinction between vel and aut. It turns out that the latter distinction does a rather different job. See also Jennings [1994b], Chapter 9. Chapter 3 of that work examines the claim that English or is sometimes exclusive in meaning.) Other considerations urged to favour the ambiguity claim currently under discussion tend similarly to miss the point. Strikingly abundant amongst the evidence urged on its behalf are occurrences of “or” which do not link (whole) declarative sentences, such as the famous “You may have coffee or tea”, “You can go to the cinema or to the beach”, and so on. The fact that even in these cases “or both” can be added without forcing a retrospective reinterpretation of the earlier “or” tells against the exclusive “or” interpretation, with its not being added counting as evidence for a pragmatically derived (i.e., conversationally implicated) exclusivity presumption, to explicitly cancel which we have evolved the specially emphatic and/or. One popular route via ‘scalar implicatures’ suggests a “but not both” rider is implicated on the grounds that any speaker attempting to say the strongest informative thing would have asserted the conjunction had that been available for assertion, so an asserted disjunction implicates its unavailability. (See Sauerland [2004] for discussion and references.) Unavailability here would be a matter of not being known to the speaker, however, rather than falsity, so we don’t really recover exclusive disjunction by these Gricean manoeuvres. An excellent discussion of the pragmatics of disjunctive assertions in general is supplied by Simons [2001] (or Chapter 2 of Simons [2000]); for discussion specifically of an exclusivity implicature, see Horsten [2005] (cf. Verhoeven and Horsten [2005]).

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For the (coffee/tea, cinema/beach) examples mentioned above, there is a complication from the notorious ‘conjunctive’ behaviour of the or in question; thus “You may have coffee or tea” amounts to “You may have coffee and you may have tea”, which of course does not entail “You may have coffee and tea”. (Think of p ∧ q vs. (p ∧ q).) Nor is the negation of the latter sentence entailed, as the possible “or both” rider just noted shows, and which in turn shows that this phenomenon – which will occupy us in 6.14 – has nothing especially to do with exclusive disjunction. (It is surprisingly common to see the waters muddied by these modal elements. Compare Clark [1989], p. 102: “Exclusive ‘or’ is true just in case either A or B is true but not both A and B are true. You can have your cake or eat it, but not both.”) Rebutting arguments for the ambiguity claim does not refute the claim, though its attractiveness will be reduced for those sympathetic to what has been called Grice’s Razor: the principle that senses are not to be multiplied beyond necessity (Grice [1967], p. 47). We content ourselves with one consideration— from Gazdar [1977] and Dudman [1978]—that comes as close as one could wish to a definitive refutation. A statement S1 or S2 is always negated by the corresponding Neither S1 nor S2 . If or were sometimes, in virtue of its meaning, exclusive, then the latter would in those cases be true when both S1 and S2 were true: but in fact, Neither S1 nor S2 always requires for its truth the falsity of S1 and of S2 . (Complication: it might be objected that we can negate without passing to the “neither” formulation; thus someone could say with heavy stress on the “or”, “She didn’t visit her sister or her father: she visited both”, apparently denying the exclusivity. By analogy with Grice’s terminology in a passage to be quoted in 6.13 below, we might call this a “substitutive” – also called “metalinguistic” – use of negation: it has the flavour of “That’s not the best thing to say” rather than “That’s not the case”. Cf. “She doesn’t think her car has been stolen, she knows it has” or “He didn’t die, he was killed.”) Having disposed of the suggestion that “or” sometimes has an exclusive meaning, we are of course free to investigate the logic of exclusive disjunction in its own right. (Cf. the discussion of temporal conjunction in 5.12.) One point of interest already emerged in response to Exercise 5.23.2: in contrast to inclusive disjunction (and conjunction) the associated truth-function here has a first- and second-argument reciprocal function (namely: itself); this is closely related to the parenthetical comment after 6.12.1 below. Here we discuss two further topics. The first concerns iterated exclusive disjunction, and the second, the question of what becomes of outside of the confines of classical logic. For illustrative purposes, we will consider the setting of intuitionistic logic. But first, the matter of iteration. Exercise 6.12.1 Check that the truth-function, associated with over the b class of -boolean valuations is associative and commutative. (Indeed the truth-values T and F form a group—in the sense of 0.21—with this function as the binary operation, F as the identity element, and each of T, F, being its own inverse. Groups in which each element is its own inverse are sometimes called boolean groups, as we remarked on encountering an isomorphic copy of this group in Figure 3.14a.) In view of the associativity noted in this exercise, one can without dangerous ambiguity write A1 . . . An for the n-termed exclusive disjunction of


783

the Ai . But whereas the analogous expression for inclusive disjunction can be interpreted as saying that at least one of the Ai is true, this n-fold iterated exclusive disjunction does not, as one might perhaps expect, amount to: exactly one of the Ai is true. Rather, as the discussion in Reichenbach [1947] of the n = 3 case suggests (see also Barnes [2007], p. 221), what we have is the wellknown general situation described in Observation 6.12.2. (Reichenbach’s point about the discrepancy between n-fold iterated binary exclusive disjunction and the “exactly one of the n cases” disjunction was rediscovered in Pelletier and Hartline [2007].) Observation 6.12.2 For any -boolean valuation v and formulas A1 , . . . , An , we have v(A1 . . . An ) = T if and only if the set {i | v(Ai ) = T} contains an odd number of elements. Proof. By induction on n, using 6.12.1 for the induction step. (The inductive proof here alluded to can be found conveniently on p. 6f. of Jennings [1994b].)

Remark 6.12.3 A rough and ready way of summarizing 6.12.2 would be to say that an exclusive disjunction of n formulas is true iff an odd number of those formulas are true. This ignores the possibility that Ai = Aj even when i = j. (To deal with this, interpret “formulas” in the rough and ready formulation, as formula-occurrences.) Note that this is reflected by the absence of idempotence alongside associativity and commutativity in 6.12.1. In response to this situation, it has been suggested that perspicuous representations of natural language iterated exclusive disjunction (supposing such a thing exists) call for something other than a binary connective: for example a multigrade sentence connective, which is to say – recalling 1.11.5 (p. 53) – a connective of no fixed arity (Hendry and Massey [1969], p. 288ff.). Such a ‘multigrade’ connective, #, would attach to any number of formulas to make a formula, so that both #(p, q) and also #(p, q, r), for instance, would be formulas, the former true on any of the valuations we are interested in just in case exactly one out of p, q, is true, and the latter if exactly one out of p, q, r is true. (Note that this means no category label in the sense of 5.11.2, p. 636, can be given to #). A variation on this suggestion is to think of the operator # envisaged here as attaching to the name of a set – or perhaps better, in the light of 6.12.3 – a multiset of formulas. (See McCawley [1972], and §8.5 of McCawley [1993]. Multisets of formulas will be given some attention in 7.25 below – as they were in 2.33.) We will say no more about multigrade connectives here, save to refer the reader to Halbasch [1975] (with some comments thereon in Borowski [1976], Gazdar and Pullum [1976]), in which a point like McCawley’s is made with respect to neither –nor rather than exclusive disjunction, and to mention that still within the familiar ‘unigrade’ domain, one can simulate the behaviour of the multigrade exclusive disjunction by using a family of n-ary connectives, one to do the job for each n (as in Chytil [1974]). Turning now to the second of the topics mentioned above, namely exclusive disjunction in intuitionistic logic, we begin with the observation that grafting

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the above -booleanness condition onto the Kripke semantics for IL (2.32) by saying that, relative to a model M and a point x []Direct

M |=x AB iff M |=x A or M |=x B, but not both M |=x A and M |=x B.

yields compounds which are not guaranteed to be persistent, since with A true and B false at x passage to a ‘later’ y at which B is true would turn A B from true to false. This ‘direct’ transcription of a truth-functional account into the apparatus of the Kripke semantics is therefore not to be regarded as promising (a semantic account for) an intuitionistic version of exclusive disjunction. (Cf. 4.34.2, p. 593.) Let us take a somewhat different tack, beginning with a review of how things stand in classical logic for this connective, and asking if that might be how things stood for any intuitionistically intelligible connective. The terminology to be used was introduced in §3.1. The conditions on a gcr induced by the four determinants of the truthfunction associated with over the class of -boolean valuations are: b

(3)

A B, A, B A A B, B B A B, A

(4)

A B A, B

(1) (2)

Exercise 6.12.4 (i) Using the method of common consequences, give conditions on a consequence relation derived from (1)–(4) above, from which it follows that for any consequence relation on a language with as one of its connectives, if satisfies the conditions you give, then if Γ C, there exists an -boolean valuation consistent with verifying all of Γ but not C. (Cf. 1.13.4 on p. 66) (ii ) Replacing “” in the conditions provided for part (i) by “”, give Set-Fmla sequent-to-sequent rules for the language whose sole connective is which serve as a sound and complete proof system w.r.t. tautologousness. (Cf. 1.25.1, p. 127.) (iii) Adding the rules (∧I) and (∧E) to those in (ii), provide a proof in the resulting system of the sequent p ∧ (q r) (p ∧ q) (p ∧ r) and its converse. Note that, by contrast with ∨, the result of interchanging ∧ and in the sequent displayed is not tautologous. Part (iii) here gives a distribution law for ∧ over ; this together with the parenthetical comment in 6.12.1 leads to an algebraic treatment of classical propositional logic in terms, not of boolean algebras (as described in 2.13) but of algebras called boolean rings, not defined in this book. The interested reader is referred to Church [1956], note 185, and, for the algebraic background, p. 24 and pp. 122–126 of Burris and Sankappanavar [1981]. So much for the classical setting. Passing to IL, we assume the presence of the connectives considered in 2.32, and also and ⊥. (The reference to INat in


785

6.12.5 below is to the system of that name in 2.32, supplemented by rules—e.g., the zero-premiss rules and ⊥ A—for these connectives.) Since we include here ⊥ and ∨, we will not bother with pure rules, transcribing (1)–(4) above into sequent-schemata directly: (1) (3)

A B, A, B ⊥ A (A B) ∨ B B (A B) ∨ A

(4)

ABA∨B

(2)

A first question we can ask about exclusive disjunction in IL is whether there is any formula constructed out of formulas A and B which behaves in the way A B is behaving here. A negative answer to this question follows from the fact that the answer to the more liberal ‘existence’ question (see 4.21) of whether INat can be conservatively extended by adding the schemata (1) –(4) for the new connective is itself negative: Observation 6.12.5 The extension of INat by (1) and (2) is non-conservative. Proof. Putting for A and p for B in (2) gives ( p) ∨ p, and hence ( p) ∨ p. By (1) , by the same substitutions, and omitting the resulting occurrence of on the left, we get ( p), p ⊥, which gives p ¬p. Thus by (∨E) from ( p) ∨ p we obtain ¬p ∨ p. Note that although the above proof does not simply copy the proof from 3.18, this observation is just a special case of 3.18.2, where we saw trouble (from an intuitionistic point of view) from any pair of determinants for an n-ary connective comprising (i): Tn , F and (ii): any T-final determinant. For the current case n = 2 and we have determinants T, T, F – inducing the condition (1) – and T, F, T , inducing (2) . Thus if one took the existence of exclusive disjunction to hang on the satisfaction of (1) –(4) above, then from the point of view of intuitionistic logic, no such connective could be acknowledged to exist. (This implies that the pure rules produced in response to 6.12.4(ii) non-conservatively extend INat.) But since similar reasoning would lead to the unpalatable conclusion that (for example) nothing in IL deserved to be called negation, or implication, we should do better not to set too much store by such considerations. Let us look with a more open mind at some candidates for the title of ‘intuitionistic exclusive disjunction’, beginning with some compositionally derived connectives in ¬, ∨ and ∧ to begin with (turning to → and ↔ later). Exercise 6.12.6 A B is classically equivalent to each of the following four formulas: (i) (A ∨ B) ∧ ¬(A ∧ B) (ii ) (A ∨ B) ∧ (¬A ∨ ¬B) (iii) (A ∧ ¬B) ∨ (B ∧ ¬A)

CHAPTER 6. “OR”

786 (iv ) ¬(¬A ∧ ¬B) ∧ ¬(A ∧ B).

What are the deductive relations between (i), (ii), (iii) and (iv) in IL? (Solution follows immediately.) The answer to the question in 6.12.6 is that (i),(ii) and (iii) are intuitionistically equivalent, and stronger than (iv). That is, any instance of (i),(ii) or (iii) has the corresponding instance of (iv) as a consequence by IL , though not (in general) conversely. This illustrates a quite general phenomenon: the weaker the logic is, the more distinctions it makes. (General, though not universal: see Humberstone [2005a].) In QL (2.31), by contrast, without the distribution laws at our disposal, (iii) would no longer be equivalent to (i) and (ii), though because of the more ‘classical’ behaviour of negation, (iv) would now fall into the same equivalence class as (i) and (ii). Which, if either, of (i)/(ii)/(iii) and (iv) has the stronger claim to be the intuitionistic incarnation of classical exclusive disjunction? (Given 6.12.5, we are not looking for a ‘perfect match’.) The proposal, to count A B true at a point in a Kripke model iff exactly one of A, B, is true there – embodied in our clause []Direct above – was rejected as giving a non-persistent compound (even for components A, B, assumed persistent). The natural way of fixing up the proposal would be to secure persistence by requiring for the truth of A B at a point that exactly one of A, B, be true (there and) at all later points. (i), (ii), and (iii) of 6.12.6 have precisely these truth-conditions, and they are perhaps what would first come to mind to associate with the description ‘intuitionistic exclusive disjunction’. There is a point of interest to note à propos of the equivalence of (i) and (ii): although in general ¬(A ∧ B) is intuitionistically weaker than ¬A ∨ ¬B, the difference between them is nullified in the presence of the added conjunct A ∨ B. 6.12.6(iv) is included as something of a direct transcription of the idea that the exclusive disjunction of A and B is something which says that A and B are not both false and not both true. Now, the question of which, out of this and (i)/(ii)/(iii), is more appropriately associated with that description is a question best considered with a wider field of contestants in view; at this point we may usefully bring in the conditional and especially the biconditional. The latter connective has an obvious right to enter any discussion of exclusive disjunction. For, returning to our introduction of the notion of an -boolean valuation as one verifying A B iff it verifies one but not both of A, B, we notice that this can alternatively be imposed as a condition on v by saying that v(A B) = T iff v(A) = v(B). This reformulation reminds us that for a {, ↔}boolean valuation v, v(A B) = T iff v(A ↔ B) = F. Thus, had we intended to take as a defined connective, the definition of A B as ¬(A ↔ B) would have been suitable (for CL). We can repeat the preceding exercise with variations on this theme: Exercise 6.12.7 A B is classically equivalent to each of the following three formulas: (i)

¬(A ↔ B),

(ii )

A ↔ ¬B,

(iii)

¬A ↔ B.

What are the deductive relations between (i), (ii), and (iii) in IL? How are these formulas related to those in 6.12.6? And which of (1) –(4)


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are IL-unprovable with A B replaced by the various formulas of this exercise and 6.12.6? We now have several examples, provided by 6.12.6–7, of formulas which are not intuitionistically equivalent but each of which is classically equivalent to A B. Of course further examples could be added to the list. For example, we could rewrite (i) in 6.12.7 as ¬((A → B) ∧ (B → A)) and ‘De Morgan’ it into ¬(A → B) ∨ ¬(B → A). What should we do, faced with this embarras de richesses? In 4.38 a similar question was considered in some detail: what is ‘the’ intuitionistic analogue of the (classical) Sheffer stroke connective? The discussion in and around 4.38.6 (p. 620) suggested an answer for connectives # for which a notion of #-booleanness was introduced, which includes therefore the present case of . Such notions associate truth-functions with connectives (#b with #, in the notation of 3.14), and the preferred transfer to the setting of the Kripke model theory for IL urged in that discussion was in terms of what we called the topoboolean condition induced by that truth-function. Applying this to the present case, in which (ξ, ζ) = T iff ξ = ζ (for ξ, ζ ∈ {T, F}) we obtain the b condition []tb below. (We avoid our usual variables x, y,. . . for truth-values in this context because here these are used for frame-elements.) []tb

(W, R, V ) |=x A B iff for all y ∈ R(x): (vy (A), vy (B)) = T b

where vy is the (small “v”) valuation which assigns T to precisely those formulas C for which (W, R, V ) |=y C. Thus the induced topoboolean condition amounts to deeming A B true at x just in case every R-related y verifies exactly one of A, B. The case of inclusive disjunction was reviewed from this perspective in 4.38.6(ii), where in place of “exactly one”, we had “at least one”. Reformulating that discussion, we can think of this clause for ∨ as an ∀∃ condition (for every Rrelated y there exists at least one of A, B, such that y verifies A or y verifies B), and then, exploiting reflexivity and persistence, we found that the corresponding (as typically stronger ) ∃∀ condition was implied by this: there is at least one of A, B, such that for all R-related y, that formula is true at y. In the case of the topoboolean condition is an ∀∃! condition (“!” for uniqueness), which again implies the corresponding ∃!∀ condition: that there is exactly one of A, B, such that for all R-related y, that formula is true at y. Though the reasoning follows that given in 4.38.6(ii), it will do no harm to go through it. The implication holds because since R is reflexive, if every point R-related to x verifies exactly one of A, B, then this applies in the case of x itself: exactly one of A, B, is true at x. Let the formula in question be C (so C is one of A, B). By persistence – which is automatically secured by the form of these topoboolean conditions (and the transitivity of R) – the formula C will be true at all y R-related to x. So there is exactly one of A, B, such that for all y ∈ R(x), that formula is true at y. This does not, however, in turn imply that each y ∈ R(x) verifies exactly one of A, B. In general, while an ∃∀ statement entails the corresponding ∀∃ statement, an ∃!∀ statement does not in general entail the corresponding ∀∃! statement. (From “Exactly one of my friends has visited every European capital”, there does not follow “Every European capital has been visited by exactly one of my friends.”) And in the particular case we are

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considering, it is easy to see that the converse entailment does not hold, or else like the topoboolean ∀∃!-condition with which it would then be equivalent, this ∃!∀-condition would yield (from persistent components) persistent compounds: which it does not for the same reasons as afflicted the clause [∨]Direct . Rewriting [∨]tb so that its import is clearer, we have, where M = (W, R, V ) and x ∈ W: []tb

M |=x A B iff for all y ∈ R(x), either M |=y A and M |=y B or else M |=y B and M |=y A.

Exercise 6.12.8 Show that if the clause []tb is employed, then for all A, B, M, x, we have M |=x A B iff M |=x (A ∧¬B) ∨ (B ∧¬A). The moral we draw from this is that if the topoboolean condition is deemed to give the preferred intuitionistic analogue for classical A B, then any of (i)/(ii)/(iii) from 6.12.6 gives us intuitionistic exclusive disjunction, while (e.g.) (iv) of that Exercise is too weak. In 6.44 we will have occasion to consider a clause for disjunction we shall call [∨]∀∃ , and remark here in advance that the labelling in that discussion is an allusion to a pair of quantifiers over elements in Kripke-style models for modal logic (thought of as ‘possibilities’ rather than ‘possible worlds’), and has nothing to do with the above discussion, in which one of the quantifiers ranged over formulas (specifically, the disjuncts) and the other over elements of Kripke models (for intuitionistic logic). We close with an exercise relating IL to an earlier topic of discussion in this subsection. Exercise 6.12.9 Given a logic with ∧, ∨ and ¬ amongst its connectives, let us say that a disjunction A ∨ B is exclusionary, relative to the logic, when ¬(A∧B) is provable in the logic. (In cases of interest, this will amount to saying that A and B are contraries according to the logic concerned; see 8.11. Note that the is no question of suggesting that there is a special “exclusionary” meaning of or – though Goddard [1960] makes exactly this mistake.) In classical logic, every disjunction is equivalent to an exclusionary disjunction, since A ∨ B is CL-equivalent to A ∨ (¬A ∧ B). Obviously the equivalence invoked here does not hold intuitionistically, but this still leaves us able to ask the following question: Is every disjunction equivalent to some exclusionary disjunction in IL? Give a proof or a counterexample, as appropriate. One can imagine a special partially defined operation of exclusionary disjunction, defined only for the case in which the disjuncts were contraries according to the logic under consideration, though this would be tiresome to work with. It would be like disjoint union, understood as set union of sets but defined only for the case where the sets concerned were disjoint. According to Badesa [2004], p. 3, this is what Boole understood by the sum of two sets. (The usual notion of disjoint unions, as defined on p. 285 above, for example, is defined for arbitrary pairs of sets, taking disjoint isomorphic copies and taking the ordinary union of these. This leaves open which copies are to taken; one common convention is to take the disjoint union of X and Y to be (X × {0}) ∪ (Y × {1}), with a similar convention for unions of arbitrary families of sets, though in our discussion of disjoint unions of frames in 2.22 we did not need go into such details since only the isomorphism-type of the frame mattered.)


6.13

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Intensional/Extensional

In 5.16 the description of fusion (in relevant logic) as ‘intensional conjunction’ was mentioned; indeed in the more general setting of linear logic we encountered such a connective (under the name of multiplicative conjunction) back in 2.33. We acknowledged this as a close relative of “∧” but were not persuaded (in 5.16) that it embodied a sense of and. A similar verdict seems appropriate for the case of disjunction – cf. multiplicative disjunction as in 2.33 – but before coming to that conclusion, we should review the considerations which have led people to make the ambiguity claim in the first place. (A similar ambiguity claim has also been made for the case of the existential quantifier: see Paoli [2005].) The historical setting for this debate is C. I. Lewis’ argument that Ex Falso Quodlibet is a correct principle for deducibility. To give a version of this argument, we supplement the ∧ and ∨ rules of Nat (1.23) by the rule of Disjunctive Syllogism (‘(DS)’ here): Γ A∨B

(DS)

Δ ¬A

Γ, Δ B

In terms of this rule the ‘Lewis proof’ (which actually goes back several centuries: see Read [1988] p.33) is the following proof of a representative Ex Falso Quodlibet sequent. Example 6.13.1 A proof using (∨I) and (DS) of p ∧ ¬p q: 1 1 1 1 1

(1) (2) (3) (4) (5)

p ∧¬p p p∨q ¬p q

Assumption 1 ∧E 2 ∨I 1 ∧E 3, 4 DS

((DS) is of course a derivable rule of the full system Nat.) By contrast with the proof of a close relative of this sequent (with a comma instead of an “ ∧” on the left) at 1.23.4 on p. 119, the ‘assumption rigging’ that was so conspicuous there ((∧I) followed by (∧E)) is absent here. However, as is usually pointed out by advocates of relevant logic, any demonstration that (DS) is a derivable rule of Nat itself passes through assumption-rigging steps: in effect, through precisely the rule form (EFQ) – from 1.23 – of the sequent here proved. One could take the rule (DS) as primitive, but its being an impure rule (in the sense of 4.14) speaks against such a policy: what is it, one would prefer to know, about disjunction and negation, with the rules which ‘give their meanings’ separately, that results in their interacting according to (DS)? However, our purpose here is not to rake over the debate about (EFQ) and relevance, but to introduce Anderson and Belnap’s response to 6.13.1 (conveniently accessible in §16.1 of their [1975]). Their suggestion is that or has an extensional meaning – for which (∨E) and, more to the point here, (∨I) – are appropriate rules, and an intensional meaning, with A-or -B more appropriately paraphrased in terms of some (preferably relevant) conditional → as ¬A → B; from 2.33 we recall that this can also be written using the single fission connective +, A + B being a kind of intensional disjunction and will return to this

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briefly at the end of the present subsection. However we choose to represent it, this is supposed to involve some connection between A and B. The term “extensional” here is meant to exclude this (as part of what is meant, that is), suggesting the suitability of a purely truth-functional treatment; cf. 5.15–16. Note that this use of “extensional” is not quite that of §3.2. “Intensional” in the present context just means “not extensional”.) Then the claim is that while the transition from line (2) to line (3) in 6.13.1 is correct for “∨” read extensionally (though not for “∨” read intensionally), the subsequent transition from line (3) to line (5), with the help of line (4), is correct for “∨” read intensionally (though not, this time, for the extensional reading). Thus on neither reading is the whole proof satisfactory: a fallacy of equivocation is involved. In particular, the rule (∨I) is held not to be suitable for the intensional reading. The merely truth-functional ground—that one of the disjuncts is true—does not suffice for the stronger claim that the falsity of either disjunct would in some appropriate sense imply the truth of the other. Anderson and Belnap did not just claim that in the interests of good reasoning, we should never argue by (DS) from a disjunction which has been inferred from one of its disjuncts, but that in all ordinary contexts, including everyday reasoning and mathematical argument (but excluding, of course, trying to make points in logical theory, as in the Lewis proof) we never in fact do this. More accurately, they claimed that the distinction between extensional ((∨I)-allowing, (DS)-disallowing) and intensional ((DS)-allowing, (∨I)-disallowing) disjunction is already marked in general inferential practice. This bold claim was far too bold, and is no longer believed by those in broad sympathy with the aims of Anderson and Belnap [1975]; decisive counterexamples and abundant references may be found in Burgess [1983]. (Some further informal remarks appear in Burgess [2005]; see also Chapter 5 of Burgess [2009].) Grice [1967] uses the phrase “strong disjunction” for any putative sense of or forming compounds which imply not only that one of their components is true but also that there is some non-truth-functional reason for accepting that this is so. The reader is referred to Grice’s presentation (pp. 44–47) of the case against supposing that there is any such sense of or, which is therefore a case against the alleged intensional reading of or posited by Anderson and Belnap. Predictably, the argument proceeds by explaining away apparent counterexamples by appeal to the notion of conversational implicature. One positive consideration exactly parallels that urged by Gazdar and Dudman, and cited in 6.12, against an exclusive reading of or : namely that for Neither S1 nor S2 there is no temptation to think that anything matters other than the falsity of each of S1 , S2 . That is, one never denies a disjunction on the grounds that the ‘connection’ between the disjuncts (allegedly required for the stronger or intensional sense) does not obtain. Something that may initially appear as a counterexample to this claim is so well treated by Grice that we cannot do better than quote his words ([1967], p. 64): Suppose you say “Either Wilson or Heath will be the next Prime Minister.” I can disagree with you in either of two ways: (1) I can say “That’s not so; it won’t be either, it will be Thorpe”. Here I am contradicting your statement, and I shall call this a case of ‘contradictory disagreement’. (2) I can say “I disagree, it will either be Wilson or Thorpe”. I am not now contradicting what you say (I am certainly not denying that Wilson


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will be Prime Minister). It is rather that I wish not to assert what you have asserted, but instead to substitute a different statement which I regard as preferable in the circumstances. I shall call this ‘substitutive disagreement’.

Jackson [1987], p. 34, discusses this kind of case. The phrase “metalinguistic negation” has been used for similar phenomena by Horn [1989]; cf. further McCawley [1991]. (It is not entirely clear how metalinguistic disjunction, if that is what we want to call it, is related to the Sweetser-style ‘speech act’ interpretation of or, mentioned after 6.13.3 and elsewhere below.) Fisk [1965] puts the case for a strong or intensional reading of or (alongside what he calls the ‘material disjunction’ sense) by stressing the exhaustiveness of the alternatives to be presented (a theme to which we return in the Digression on ‘Dyirbal disjunction’, starting on p. 795 below): Suppose that White attempts to list the alternatives exhaustively when he says that Black, who is considered only as a faculty member of a university, is either an assistant or a full professor. Even though Black is, let us say, an assistant professor, the truth of White’s statement does not follow. For, since there are further alternatives, his statement is certainly false. Fisk [1965], p. 39.

Exercise 6.13.2 Discuss the above passage; when Fisk says “does not follow”, what does he have in mind as that from which the truth of White’s statement does not follow? Another author who likewise objects to (∨I) for ordinary English or – and also describes this as an attack on ‘material disjunction’ is Hartley Slater. Beginning on solid ground with the observation that we distinguish between (a) and (b): (a)

They agreed either that they would go to the pictures or that they would go out for a meal.

(b)

They agreed that either they would go to the pictures or they would go out for a meal.

on the grounds that (a) does not tell us what the ‘they’ in question agreed on, while (b) does – which is really a matter of the scope distinction between, in the case of (a): O(p ∨ q) and, for (b): O(p ∨ q), where the operator “O” represents “they agreed that it should be the case that”, Slater [1976], p. 227, proceeds as follows (quotation marks as in the original, bracketed word a suggested typographical correction): It is sometimes said that where ‘p’ may be inferred, or ‘q’ may be inferred – ‘p’ and ‘q’, naturally, being propositions – there ‘p or q’ may be inferred, or at least certainly ‘p ∨ q’ may be inferred, where ‘∨’ is the material disjunction – it being essentially the definition of ‘p ∨ q’ that it should follow in such cases. The study above reveals not only that in such circumstances ‘p or q’ certainly may not be inferred, but that ‘p ∨ q’ is grammatically malformed if it is defined invariably to follow. For there can be no proposition ‘r’, generally, which may be inferred from two propositions ‘p’ and ‘q’. What follows [from] ‘p’ and ‘q’ is either ‘p’ or ‘q’, not ‘either p or q’; what follows in the two cases has no identity, it can only be delimited.

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The final sentence here is to remind us that in (a) what the parties in question have agreed on is not identified but only said to lie in a certain range (‘delimited’) – a range of two, as it happens. Slater continues with a diagnosis (here quoted with a relabelling of the examples): What seems to have happened is that the distinction between, say, (a ) it is implied either that p or that q, and (b ) it is implied that either p or q, has not been kept clear. The fact that the latter says what is implied appears to have been taken as a fact about the former, so that cases where it is implied either that p or that q have come to be taken as cases where the same one, certain, thing is implied – that either p or q, in fact. But cases where it is implied that either p or q are quite distinct from cases where it is implied either that p or that q. (. . . ) Cases where it is implied either that p or that q are just not cases where invariably the same one thing is implied.

For this to support the claim that, in Slater’s words, “there can be no proposition ‘r’, generally, which may be inferred from [presumably: each of] two propositions ‘p’ and ‘q’ ”, the phrase “what is implied” in this last passage must be interpreted as meaning “what follows logically from” or “what may legitimately be inferred” (from some given premiss or set of premisses). Now of course there is in general no such thing as ‘what logically follows from’ this or that premiss or set of premisses, if this is supposed to denote a single statement (or proposition). When someone replies to a remark with the question “And what follows from that?”, they intend to be asking for some especially salient consequence to be singled out, rather than to be presupposing that there is a single logical consequence and asking for that to be specified. So the phrase “what follows” should not lead us to think that just because one of the things that follows from p is p itself, it can’t be that another of the things that follow from p is something that also follows from q. And indeed their disjunction (understood as p ∨ q for ∨-classically behaving ∨) not just any old common consequence of p and q, but specifically their (logically) strongest common consequence – to recall the terminology of 4.14 above and elsewhere. In terms of (a ) and (b ) understood with a fixed statement doing the implying, the existence of a disjunction for every pair of statements, answering to the ∨-classical characterization just given we have an entailment from (a ) to (b ). The distinction between them, Slater says, “has not been kept clear”: but this would be so only if the existence of such disjunctions also required an entailment from (b ) to (a ), which it does not. (And indeed there would be trouble for any connective ∗ with the property that for all Γ, A, B, we had Γ A ∗ B iff Γ A or Γ B, as was noted in 4.22.2; fortunately classically behaving ∨ is not such a connective. Indeed, if the “Op ∨ Oq” gloss above on statements like (a) and (a ) is correct, Slater himself deploys the allegedly problematic ‘material disjunction’ both connective as the main connective in considering them, as well as in embedded position in (b) and (b ). Further discussion of Slater [1976] will be found in Peetz [1978].) Some authors have been prepared to discern even more than one ‘nonextensional’ (or ‘non-material’) sense of or. For example, according to Woods and Walton [1975], p. 111, there is room for an intensional and also a ‘weakly


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intensional’ sense, distinguishing these on the grounds that whereas the latter is merely consistent with the disclaimer “I don’t know which”, the former, as they say, ‘virtually implies’ this disclaimer. The latter term (from Hintikka [1962]) amounts roughly to the provability of a material implication in a suitable system of epistemic logic. The fact that for reasons obvious from the (Gricean) theory of conversation and already rehearsed in 6.11 above, disjunctions are often asserted because the speaker doesn’t know which disjunct to assert, does not, however, provide any grounds for importing into the meaning of “or” any allusion to that ignorance. (Let alone supposing that there are two distinct such meanings involving such ignorance in two different ways.) Similar excesses may be found in Van der Auwera [1985a], where four ‘readings’ of (inclusive) or are discerned, which he calls “dubitative” (the ignorance-conveying reading), “material” (the familiar ∨), as well as two further readings dubbed ‘contingency’ and ‘enumerative’ or. The present discussion is aimed specifically against the so-called dubitative reading – a point already made especially forcefully in the passage quoted near the end of 6.11 from Ryle [1929]. Not only would the postulation of such ignorance-related ingredients of meaning be unmotivated: it would lead immediately to awkward results. We can show this with an example from Grice [1967], in which the host at a birthday party announces: “The prize is either in the garden or in the attic”. Grice’s point is that the implicature of ignorance is explicitly cancelled by adding “but I’m not saying which”. However, in the absence of such an addition, we can consider the host’s words and the suggestion that the condition of the speaker’s ignorance – as against Grice’s (conversational implicature) treatment – is actually part of the meaning of or. Suppose that, of the children at the party, Sandra hears the announcement but Samuel doesn’t. He asks her where the prize is, and she repeats what she was told: it’s either in the garden or in the attic. Surely her words do indeed mean just what the same words as uttered by the host meant: she is indeed merely relaying the message. This entails that matters of whether the disjunction is accepted because of one disjunct’s being accepted, on the one hand, or instead in the face of ignorance as to which disjunct is correct, on the other, have nothing to do with the meaning of or. (For another example, consider the following passage from p. 38 of the memoir Driberg [1956]: Only a few days before the war broke out, Putlitz dined with Ribbentrop and either Göring or Goebbels (Guy cannot remember which); he passed on to his British friends a full account of their conversation.

Note the parenthetical explanation, playing a role somewhat analogous to that of “but I’m not saying which”, as to why a disjunctive construction is being used.) Such considerations notwithstanding, Zimmermann [2000] proposes that the ignorance-relocated aspects of disjunction be relocated from pragmatics to semantics, with a modal – more explicitly epistemic – theory of what “or” means. A disjunction amounts to roughly the conjunction of the results of prefixing “it is epistemically possible that” to each of the disjuncts. (This is actually the ‘open’ as opposed to ‘closed’ reading of disjunctions, in terms explained in the Digression on Dyirbal Disjunction below – p. 795.) This has the consequence that the rule (∨I) is in danger, since from S1 it certainly doesn’t follow that

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it might be that S1 and it might be that S2 , for any old S2 . Here is Zimmermann reflecting on the validity of this inference (though formulated somewhat distractingly in terms of the truth of a conditional): Mathematically speaking, “If 2 is prime, then in particular 2 is odd or prime” is just as true as “If 2 is an even prime number, then 2 is even”, I think this reflects a difference between the mathematical and the ordinary use of ‘or’. Zimmermann [2000], p. 274, n. 29

The word “use” here has to be understood as “meaning” – rather than “(conversational) usage” – in order for this to bear on the point about or -introduction, and on this interpretation it has no plausibility. When exactly are children taught this special meaning of or ? Why does there turn out to be an exactly matching lexical ambiguity between the ordinary and the mathematical sense of French ou? And so on. (A more promising modal theory of the semantics of disjunction, due to Mandy Simons, will be described at the end of 6.14.) Exercise 6.13.3 Zimmermann [2000], p. 275, tries to show that the would-be ignorance implicature is really part of the meaning of disjunctions with an example with an embedded disjunction (slightly reworded here to avoid a grammatical difficulty): (*) If Mr. X is either in Regent’s Park or in Bloomsbury, we may as well give up. The “we” in question are trying to catch Mr. X but don’t have the resources to mount searches in areas of London as far apart as Regent’s Park and Bloomsbury, so it is because we can’t narrow down the search area – since we are ignorant as to whether the location is Regent’s Park or Bloomsbury – that we “may as well give up”. Does the example support Zimmermann’s claim about the epistemic analysis of or, or can the acceptability of an utterance of (∗), assuming it to be acceptable, be explained in a different way? Sweetser [1990] presents an account of our understanding of sentence connectives which stresses the systematic role of a three-way distinction amongst their interpretations, which she labels as content domain interpretation, epistemic interpretation, and speech act interpretations. (We had occasion to allude to this in the notes – on but – at the end of §5.1: p. 674.) These are illustrated in the case of because by the following examples (p. 77): (1) (2)

John came back because he loved her. John loved her, because he came back.

(3)

What are you doing tonight, because there’s a good movie on?

In (1) – content domain interpretation – it is John’s coming back which is putatively explained by what follows the because; in (2) – epistemic interpretation – it is the speaker’s reason for believing that John loved the woman in question which is putatively explained by what follows the because; and in (3) – speech act interpretation – it is the speaker’s asking the question for which what follows the because is offered as an explanation. Probably “cognitive” would be a better term than “epistemic”, but for simplicity we stick with Sweetser’s terminology


795

here and below, in 7.11, when we mention its application to conditionals. Our present concern is with Sweetser’s application of this taxonomy, informative as it undoubtedly is for because, to the case of or. She gives (4) and (5) as illustrations of content domain and epistemic interpretations of disjunction (Sweetser [1990], examples (33) and (34) on pages 93 and 94, complete with parenthetical glosses): (4) Mary will go to the grocery store this evening, or John will go tomorrow. (Sweetser’s gloss: Either “Mary will go” or “John will go” truthfully describes the future state of affairs.) (5) John is home or somebody is picking up his newspapers. (Sweetser’s gloss: The only possible conclusions I can reach from the evidence are (a) that John is home to pick up the newspapers, or (b) that somebody is picking them up for him.) The trouble with the so-called epistemic interpretation here is that what the speaker’s envisaged evidence supports is precisely the truth of the disjunction, interpreted in the content-domain manner. Suppose the speaker, having somewhat stronger evidence, had been in a position to assert simply: (6)

John is home.

Considerations parallel to those given by Sweetser’s glosses would have us register a content domain interpretation (“John is home” truly – or as Sweetser puts it ‘truthfully’ – describes the real state of affairs), and alongside it an epistemic interpretation (the only possible conclusion I can reach from the evidence is that John is home): one too many interpretations, surely. The case is simply not parallel to that of because. (This was the line we saw Sweetser taking with “but” in the notes to §5.1: p. 674.) Since we are concentrating here on questions of knowledge and ignorance as they pertain to disjunction, we defer consideration of here Sweetser’s speech act interpretations of or -sentences to the following subsection, but something falling loosely under this heading deserves to be mentioned – if only to be put to one side – and that is what we might call the or of reformulation, as when one says or writes (7) (7)

We were, as usual, hunting for the giant puffball, or Calvatia gigantea.

There may but need not be an element of self-correction in the reformulation: (8)

I then introduced myself to Miss – or rather to Ms. – Henderson.

A striking illustration of this kind of construction appears in the correspondence of Philip Roth (see Bloom [1996], p. 149), in which he is describing a walk in the countryside: (9)

A skunk crossed my path; or, as the skunk would say, a Jew crossed my path.

The following Digression returns us to matters epistemic. Digression. (‘Dyirbal Disjunction’) From time to time, reports emerge of this or that language lacking any direct means for expressing disjunction. It is difficult for a non-native speaker of the languages in question to assess these

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reports, and native speakers’ own judgments are hardly to be taken at face value. Particularly interesting, nonetheless, is a well-known and thoughtful discussion (‘Appendix A: Dyirbal Logic’) in Dixon [1972]. Apparently, the speakers of Dyirbal, an Australian aboriginal language of northern Queensland, would say what is said in English by (1)

I saw a fish – it was either a barramundi or a red bream.

using a Dyirbal sentence whose verbatim translation into English runs: (2)

I saw a fish – perhaps it was a barramundi, perhaps it was a red bream.

Dixon’s main interest in this case concerns a contrast between what he calls closed disjunction, in which alternatives are exhaustively listed, and open disjunction, in which likely alternatives are listed, but without the presumption that the listing exhausts the possibilities. The closed/open distinction is illustrated by (1)/(2): the latter clearly ‘leaves open’ the possibility that the fish sighted was of another kind, while this is not left open by (1). Dixon says that, by contrast with English, “Dyirbal deals mainly with open disjunction”, adding (p. 363f.): Although Dyirbal could add a rider stating that it could not have been anything else, i.e., closing the disjunction, just as English can create an open disjunction by including ‘or something else’.

Interestingly enough, Jennings [1994b] (esp. Chapter 11) sees the history of English or -constructions as having evolved to recognise the contemporary disjunctive interpretation (not, even now, their dominant use in his view) from a stage at which “even with no occurrences of the weakening adverb “perhaps”, successions of principal clauses separated by or can demand to be understood as a succession of mooted possibilities” (p.295) via a two-stage transition, one stage of which is the addition of an implicature of exhaustivity, to the effect that (at least) one of the speaker’s attempts at ‘getting it right’ (the possibilities mooted) has succeeded (p. 299): It is the commitment to the exclusion of further possibilities, or, to put the matter positively, it is the element of commitment to the sufficiency of the slate of candidates and the understanding of that commitment on the part of the audience.

The other stage involved in the genealogy of disjunction, as Jennings tells the story, is the idea of seeing the succession of multiple stabs at ‘getting it right’ as a single stab at getting something else (the disjunctive claim, we might say) right. We return to this theme of Jennings below, for the moment noting only that the transition to the exhaustive interpretation, whether seen as a matter of implicature or as a matter of what is explicitly said, corresponds to what in Dixon’s terminology, is a passage from open to closed disjunctions. From the point of view of our own current discussion, however, it is not the closed/open distinction – interesting though it is – which is the most salient point of contrast between (1) and (2). It is, rather, the explicit intrusion of an epistemic element in the content of (2) which is absent from the content of (1) and comes to be associated with utterances of (1) via the pragmatic overlay of Gricean considerations. This talk of an ‘epistemic element’ is deliberately


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vague; the intention is to allow not only the inclusion of material which makes what is said false unless the speaker is in some specific epistemic state, but also of material which conventionally implicates the presence of such a state, making what is said inappropriate or unassertable in its absence. (The previous reference to a pragmatic overlay was of course to conversational implicature.) Presumably the latter is the correct account of the English word “perhaps”, as well as for “(it) might (be)” – an alternative gloss suggested by Dixon for the Dyirbal particle (yamba) used in the sentence translated as (2). (Cf. also p.51 of Hutchins [1980] in which amongst the English renderings of a single Trobriand islander word, kaina – or, in the transcription of the reference cited next, kena – are to be found “or” and “perhaps”. The language concerned has had several names; at the time of writing that favoured is apparently “Kilivila”; see Senft [1986], which contains in pp. 94–99 a discussion of various connectives. See also 3.3.6 of Wälchli [2005], for an interesting discussion, citing inter alia a language of Papua New Guinea (Toaripi) with a similar treatment of disjunction to that attributed to Dyirbal by Dixon.) Either way, an inference by the closest analogue of (∨I) that the above discussion gives us would look very odd: from S1 one would be inferring Perhaps S1 , (and) perhaps S2 for any sentence S2 , including those whose falsity is apparent to the speaker. (Such a conjunction of possibility statements is part of Van der Auwera [1985a]’s definition of dubitative disjunctions, mentioned above. We will see something similar in the semantics for disjunction in 6.45, 6.46, where a conjunctive clause called [∨]Plus is used, whose conjuncts can be thought of as roughly analogous to those of the conjunction inset above. See also Hinton [1973], §7(v). The existential quantifiers in the formulation of [∨]Plus conveniently prevent the apparently ∧-like form of the gloss inset above from satisfying – something that would be disastrous for any candidate for disjunction – the condition (∧-like) given in 5.31.) Dixon ([1972], p. 361) remarks that the absence of anything exactly corresponding to English or does not prevent Dyirbal from being “quite as capable as English” of expressing alternatives. As we have seen, the best that can be said for the relation between (1) and (2) is that of amounting to the same thing when the distinction is set aside between what is conversationally implicated and what is explicitly said by an utterance: (2) making explicit in its content part of what (in normal conditions) of is implicated by an utterance of (1). This leaves the question of embedded occurrences of or untouched, and it would be interesting to know how they would be translated in Dyirbal. The most popular context into which such embedding is considered is that of the antecedent of a conditional (cf. 5.13), and since according to Dixon, Dyirbal also lacks any direct equivalent of English if, investigation of this particular context would appear to be anything but straightforward. (We consider problems about English subjunctive conditionals with disjunctive antecedents in some detail in 7.17. Since Dyirbal is rapidly dying out, the chances of ever settling some of these matters for that language are not good; ideally one would want a native speaker with a reasonable grasp of the distinction between semantic and pragmatic considerations.) Some of the above considerations were rediscovered in Zimmermann [2000], mentioned earlier, which proposes a solution to the problem of free choice permission – see 6.14 below – using the idea that “disjunctions are conjunctive lists

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of epistemic possibilities”. (Compare Jennings [1994b], p. 295, already quoted above: “successions of principal clauses separated by ‘or’ can demand to be understood as a succession of mooted possibilities”.) While the most obvious line of attack on this issue would see it as involving an (unexpected) equivalence between (A ∨ B) and A ∧ B, whether for an epistemic or for a deontic , Zimmermann implausibly tries to explain the phenomenon with a special account of what happens when we have a deontic (which, at p. 285, he writes as “Δ”, rendering the crucial parallel invisible) in the scope of an epistemic . The account would predict, wrongly, that just as “You may eat or drink” can mean “You may eat and you may eat”, “You must eat or drink” can mean “You must eat and you must drink”. (See also the problem raised for his account in the second paragraph of note 47 of Zimmermann [2000].) End of Digression. If one believes (e.g., with Read [1988], §2.7) in both an exclusive/inclusive and an extensional/intensional ambiguity for or, it will be hard to resist the further suggestion that these distinctions cut across each other, and make for (at least) four senses (though Read’s discussion is headed ‘Three Senses of Disjunction’: either way, a highly implausible ambiguity claim). On the intensional front, for example, whether characterized epistemically (as with Woods and Walton) or in terms of more objective connections between the propositional contents of the disjuncts (as with Fisk), there is the possibility of an ‘intensional’ exclusion of the joint truth of the disjuncts. For example, Anderson and Belnap think of the intensional disjunction of A and B, which they write as A + B, as having amongst its consequences ¬A → B, where the “→” is a favoured conditional connective (for entailment or, better, relevant implication), which admits of an intensional exclusive variant in the form of (¬A → B) ∧ (A → ¬B). We could imagine further variations (whence the above reference to ‘at least’ four senses), one taking the disjunction extensionally and the exclusivity intensionally: (A ∨ B) ∧ (A → ¬B) or vice versa: (A + B) ∧ ¬(A ∧ B) [alias (¬A → B) ∧ ¬(A ∧ B)]. However weak the case for an intensional disjunction expressed by or may be, we can certainly consider on its own merits the connective Anderson and Belnap have in mind under this name. It is usually written (as above, and also in 2.33) “+” and called fission, to suggest a kind of duality w.r.t. fusion (◦), one possible definition of + in terms of ◦ being: A + B = ¬(¬A◦¬B). In the relevant logic R (2.33) it turns out that, so defined, A + B is equivalent to ¬A → B – as we saw in the more general setting of Linear Logic in 2.33, in which context – but we shall not here go into as much detail as in the case of fusion (5.16) on the proof-theoretic behaviour of this connective. On the other hand, (ordinary) disjunction ∨ as it fares in R, and more especially the Urquhart semantics for that system, will occupy us below, in 6.45. We encountered the contrast between ∨ and + in 2.33 in its linear-logical incarnation as the distinction between, respectively, additive and multiplicative disjunction (the latter also called par ).


6.14

799

Conjunctive-Seeming Occurrences of Or

A considerable literature attests to the recurrent interest logicians have taken in what at first seems like a mere quirk of idiom: the tendency of or to appear as a kind of displaced and. But the phenomenon seems too systematic to dismiss in any such terms, and the problem has become: to articulate the system in question. Samples from the literature alluded to include Kamp [1973], Hilpinen [1982], Makinson [1984], and Jennings [1986], [1994a], [1994b] (Chapter 3). The present subsection reviews some of their contributions. Although the discussion will not issue in a definitive conclusion, several points of interest will be raised and the reader who hopes to pursue the matter further will have somewhere to start from. In the references cited, such examples of our phenomenon as the following are to be found; in each cases the (a) sentence, with or, seems to amount to much the same as the (b) sentence, with and : (1a) Susan is a faster runner than John or Jane. (1b) Susan is a faster runner than John and Susan is a faster runner than Jane. (2a) Susan may speak to John or (speak to) Jane. (2b) Susan may speak to John and Susan may speak to Jane. (3a) If John comes alone or Susan brings Paul, Jane will be pleased. (3b) If John comes alone Jane will be pleased, and if Susan brings Paul, Jane will be pleased. The pair (1) is included for the sake of completeness. Though it will come in passing below, we will not discuss this case in any detail, since it does not (obviously) feature the interaction of or with another sentence connective, as the remaining cases do. (However, see Seuren [1973] for an argument to the contrary.) There are indeed some connectives with a similar comparative feel to them, such as before when used to connect sentences – though, interestingly, while we get the displaced and occurrences of or here, we do not get them with after ; likewise only before – and not after – creates a negative polarity environment, hospitable to words like any and ever. Similarly in the case of preferability, sometimes taken as a dyadic connective in deontic logic. (See Jennings [1966] for references and commentary on this point.) Finally – another point from Jennings – (1a) might indeed be used to abbreviate not (1b) but the corresponding disjunction, even if inserting “than” before “Jane” in (1a) is perhaps better. We return to Jennings’s explanation for the availability of the conjunctive reading (i.e., as amounting to (1b)) of this example below. We also remark here that sentences like (2b) and (3b) are also heard, with much the same meaning, but in which or replaces and, and thus at least apparently has the modal or conditional connective in its scope. Discussion of this phenomenon is reserved for an Appendix to the present section (p. 812). Let us turn to (2). The “may” might be taken as a “may” of permission or as a “may” of (epistemic) possibility; in both cases (2b) paraphrases (2a). (In fact the epistemic reading is unnatural for this example. To bring it out, change “may speak” to “may have spoken”.) The former case subdivides according as

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we think of the speaker as giving permission or as reporting that something is permitted. Focus on the first of these senses. Although not connecting, as it stands, two sentences, this or can be massaged into such a position if we shift to the somewhat stilted style customary in direct renderings of the symbolism of deontic logic: (2a ) It is permissible that Susan speak(s) to John or Susan speak(s) to Jane. (2b ) It is permissible that Susan speak(s) to John and it is permissible that Susan speak(s) to Jane. The parenthetical “s” is for the benefit of readers who find this kind of subjunctive construction archaic. (Alternatively, insert the word “should” before “speak”, but note that this is not a deontic should : it cannot, for instance, be replaced by “ought to”.) Now, recalling the deontic interpretations of and mentioned in 2.11, “it ought to be that” and “it is permissible that”, we note that whereas in any normal modal logic, such as the system KD often regarded as a minimal deontic logic what we have is (i), rather than, as suggested by the case of (2a ) and (2b ), (ii): (i) (A ∨ B) ↔ (A ∨ B)

(ii ) (A ∨ B) ↔ (A ∧ B).

And of course we cannot simply graft on (say as an additional axiom-schema) the equivalence (ii) here, since the ← direction of (i) together with the → direction of (ii) give: A → B (in effect: “If anything is permissible, then everything is permissible”.) The problem of how to respond to this difficulty is called the problem of free choice permission in the literature. One style of solution has been to acknowledge, quite separately from the permissibility notion ( in (i)) which is dual to that of obligatoriness (‘oughtness’), a new deontic operator, P say, for ‘free choice permission’ which is stipulated to obey a principle like (ii). The idea behind this label is that (2) or (2 ) gives Susan a free choice as to whether to speak to John or to Jane. Digression. In fact the form of (2) doesn’t actually specify who is being permitted to do what, only that a situation in which Susan speaks to John is permissible. This is the same as a situation in which John is spoken to by Susan, but is one thing to permit Susan to speak to John and another to permit John to be spoken to by Susan. References to the literature on agent-relativized deontic logic, in which such matters are made explicit, may be found in Humberstone [1991]. We ignore these complications here, however. End of Digression. There are two reasons for not being satisfied with a suggestion that we simply express free choice permission with the new operator P , however. In the first place, its relation to the existing operators and is unclear. In the second (noted by Hilpinen), as long as P is congruential, trouble results from (ii) rewritten with its aid, since putting p ∧ q for A and p ∧ ¬q for B, its → direction gives (iii)

P ((p ∧ q) ∨ (p ∧¬q)) → (P (p ∧ q) ∧P (p ∧ ¬q)).

Thus, assuming our underlying non-modal logic is classical, so that p is equivalent to (p ∧ q) ∨ (p ∧ ¬q), congruentiality yields (iv )

P p → (P (p ∧ q) ∧P (p ∧¬q))


801

and thus in particular P p → P (p ∧ q).

(v)

Yet there seems no notion of permissibility (‘free-choice’ or otherwise) for which (v) is a satisfactory principle (pace van Benthem [1979b], p. 36, and Humberstone [1983a], p. 347). One could of course abandon congruentiality, but that would be tantamount to giving up on the task of making semantic sense of the relationship between pairs like (2a) and (2b). We return to sketch Hilpinen’s solution to the problem of free choice permission after looking at the problem raised by examples (3), repeated here for convenience: (3a) If John comes alone or Susan brings Paul, Jane will be pleased. (3b) If John comes alone Jane will be pleased, and if Susan brings Paul, Jane will be pleased. This time, what we have is known as the problem of disjunctive antecedents. But what – it may well be wondered – is the problem here? The examples seem directly to have the forms (A ∨ B) → C and (A → C) ∧ (A → C), and these are indeed equivalent in CL (or IL, even): so, unlike the case of (2a) and (2b), no special manoeuvres seem called for, familiar logical principles affording us all the explanation we need for why in this case an or should behave like a displaced and. And in fact, (3a) and (3b) do not present what is usually called the problem of disjunctive antecedents, which arises rather with counterfactual or subjunctive conditional analogues, presented as (4) below, to the indicative conditional construction in (3). But before getting to that, let us note that one way of accounting for the otherwise puzzling appearance of or as a displaced and might be to reduce it to a case, such as that in (3) in which a logic-based explanation is already available. This was the inspiration, van Benthem [1979b] remarks, for his semantic treatment of P , mentioned parenthetically above. A more straightforward version of this strategy was employed in Makinson [1984], at p. 145 of which we read “You may work or relax” may be seen as reflecting the thought “For all x, if x is the (generic) activity of work or x is the state of relaxation, then x is permitted for you”.

Applied to our examples (2), the “or” in (2a)

Susan may speak to John or (speak to) Jane.

would, on Makinson’s account, be cast in the role of a trace of the “or” in (2c)

For all x, if x is John or x is Jane, then Susan may speak to x

and Makinson explicitly applies the suggestion to non-deontic cases – cf. the above reference to an epistemic reading of examples (2) – as on p. 144: For instance, “Either Brazil or Argentina could become the dominant power in Latin America” may be seen as representable along the lines “For all x, if x is Brazil or x is Argentina then . . . ”.

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Unfortunately, Makinson’s universally quantified conditionals with their disjunctive antecedents – ‘checklist’ conditionals, as he calls them – are available as equivalents in more cases than actually present the phenomenon they are invoked to explain. For instance, to vary the theme of the example in the preceding quotation, we can express the fact that Brazil and Argentina both participate in the World Cup soccer competition by saying “For all x, if x is Brazil or x is Argentina, then x participates in the World Cup”, but in this case the or -for-displaced-and formulation which such a paraphrase is supposed to explain in the cases quoted, simply doesn’t exist: “Brazil or Argentina participates in the World Cup” isn’t a way of reporting the fact in question, and has only a straightforward disjunctive reading. Similarly, in the deontic case, it is only with “may” and “permissible”, and not with “ought” or “obligatory”, that the phenomenon under discussion arises, even though the availability of checklist conditional paraphrases is the same for both cases. (A variation on Makinson’s account, which may not be vulnerable to this objection, appears in Higginbotham [1988], p. 226.) The same objection applies to Makinson’s attempt offer the above explanation in the case of disjunctive antecedents—the topic we take up immediately below—his suggestion (p. 146f.) being that ‘when we say “If John or Mary had come to the party, it would have livened up”, what we want to convey is that for all x, if x is John or x is Mary, then if x had come to the party it would have livened up’. In this case, the problem is that such a checklist paraphrase is available also for the consequent position: wanting to convey that for all x if x is John or x is Mary, if Sally had come to the party then she would have spoken to x, we precisely cannot do so by saying “If Sally had come to the party then she would have spoken to John or Mary”. There is no ‘problem of disjunctive consequents’. Returning, then, to the problem of disjunctive antecedents in the context in which it is usually given that name, let us rewrite (3) replacing the indicative conditionals by subjunctives. (More will be said on the distinction between these two kinds of conditionals in 7.11 and 7.18.) We have also transposed the examples into the past tense: (4a) If John had come alone or Susan had brought Paul, Jane would have been pleased. (4b) If John had come alone Jane would have been pleased, and if Susan had brought Paul, Jane would have been pleased. One promising treatment of the problem of free choice permission, that of Hilpinen [1982], consists in solving the problem of disjunctive antecedents, as illustrated by the pair (4), and then reducing the former problem to the latter. We cannot give the details of the latter solution here, since that would require some familiarity with a certain approach to the semantics of subjunctive conditionals; accordingly the task is deferred to 7.17. But we can say something about how there is a problem of disjunctive antecedents for (4) where there wasn’t for (3), and we can outline the reduction of the one problem to the other. To this end, we adopt (as in 7.17) the symbolization of Lewis [1973] for the conditional construction in question, with “→ ” as a binary connective. Read “A → B” as “If it were the case that A then it would be the case that B”. Then (4a) and (4b) appear to be of the forms (vi) and (vii) respectively:

6.1. DISTINCTIONS AMONG DISJUNCTIONS (vi )

(A ∨ B) → C

(vii)

803

(A → C) ∧ (B → C)

and, on the semantics for → alluded to above, these two are by no means equivalent. Further, if we make whatever changes might be called for in order to force them to be equivalent, we should be in trouble (as Hilpinen notes) since a notoriously invalid schema: (Strengthening the Antecedent)

(A → C) → ((A ∧ B) → C)

would then be forced upon us. (The – at least apparent – invalidity of this principle is explained in the discussion between 7.15.6 and 7.15.7, pp. 990–994.) We argue as in (iii)–(v) above, and again assuming congruentiality: (viii) (ix ) (x)

(((p ∧ q) ∨ (p ∧¬q)) → r) ↔ (((p ∧ q) → r) ∧ ((p ∧ ¬q) → r) (p → r) ↔ (((p ∧ q) → r) ∧ ((p ∧¬q) → r) (p → r) → ((p ∧ q) → r).

Here (viii) is an instance of the schema claiming (vi) and (vii) are equivalent; (ix) follows by replacing (p ∧ q) ∨ (p ∧ ¬q) with its equivalent p; and (x) is a tautological consequence of (ix). Yet (x) is just a representative instance of the unwanted principle of Strengthening the Antecedent. Thus replacing → by → in passing from consideration of (3a, b) to (4a, b) deprives us of a straightforwardly logic-based account of the displaced conjunctive “or” in (4a). Hilpinen’s solution (as in Humberstone [1978], inspired by Åqvist [1973]) lies in discerning a structural ambiguity in subjunctive conditionals with disjunctive antecedents, such as (4a). In describing the ambiguity as structural the intended contrast is with a lexical ambiguity. That is, no word in (4a) is claimed to be ambiguous, though those unambiguous words can be thought of as put together in either of two different ways, corresponding to the two senses discerned (somewhat as with the familiar “Old men and women walked in the park”). Only one of the these two readings is captured by the → notation in (vi), a more fine-grained symbolism being required to contrast the two readings. (See 7.17: p. 1020.) Hilpinen adopts a notation according to which the distinction emerges as one between (xi) and (xii) (xi ) (xii)

If (A ∨ B) then C (If A ∨ If B) then C.

In both cases, the “If ” is intended subjunctively rather than indicatively; (xi) is what would usually be written as (vi) above, and (xii) is not directly representable in the → notation, though, unlike (vi), it is equivalent to (vii). By an understandable ellipsis, the second “If ” in (xii) may be omitted, giving rise to the above-mentioned structural ambiguity in counterfactuals with disjunctive antecedents, as between sense (xi) and sense (xii). As for permission statements, we think of “It is permissible that A” as amounting roughly to “If A were the case, things would be all right”. Abbreviating the latter consequent to “O.K.”, the ordinary permissibility claim that it is permissible that A or B, which follows from the claim that it is permissible that A, is represented by (xiii), and the corresponding free choice permissibility claim, which entails that A is permissible and so is B, is represented by (xiv): (xiii) (xiv )

If (A ∨ B) then O.K. (If A ∨ If B) then O.K.

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Although Hilpinen’s treatment of the displaced “and” occurrences of “or” in the antecedents of subjunctive conditionals itself has considerable plausibility, there is a worry about trying to reduce the free choice permission problem to the problem of disjunctive antecedents. The worry is that other (non-deontic) examples in which essentially the latter phenomenon arises, may not be amenable to a parallel treatment. Take Makinson’s example, “Either Brazil or Argentina could become the dominant power in Latin America”. Is there an analogue for what was written as “O.K.” above, for this case also? None comes to mind. A similar non-generalizability worry arises for some other solutions of the free choice permission problem. Kamp [1973] toys with a solution which, rather than reducing this problem to another (e.g., that of disjunctive antecedents), pays close attention to what is done with the aid of permissibility statements. (See also Kamp [1978], Lewis [1979a], Merin [1992].) If we think of them as rescinding prohibitions, what such a statement does is to enlarge the class of permissible worlds for the addressee. So the net effect of issuing two such statements will be to end up rendering permissible a class larger than each taken separately would render permissible, indeed perhaps “something like” their union, as it is put on p. 65 of Kamp [1973], which continues: This is precisely what we should expect, given the meaning of the word ‘or’. For, as logicians have realized with regard to assertions for at least a century, ‘or’ stands essentially for set-theoretic union: The set of possible situations in which a disjunctive assertion is true is the union of the sets of possible situations which realize its disjuncts. ‘Or’, I maintain, always stands for set theoretic union. Previously we could not see that it does have this function also in disjunctive permission statements because we lacked the appropriate understanding of what permission statements do. And so we saw ourselves forced into the implausible and totally ad hoc doctrine, that in such statements ‘or’—usually at least—plays the same role that ‘and’ plays in assertions.

In the later parts of his paper, Kamp recognizes that if a uniform account is to be provided of the various and -reminiscent or s we have seen, then the contrast between permission statements and assertions is overplayed here. For in the case of conditionals, can-statements of ability, may-statements of epistemic possibility, etc., we seem to be over on the assertion side of any such boundary. (Kamp’s paper can be consulted to assess the attractiveness of his suggestions for these cases.) Indeed, that is even more clearly where we are with statements which report rather than give permission, as in “Susan was permitted to speak to John or to Jane”. One argument to the conclusion that the phenomenon of free choice permission is pragmatic rather than semantic is worthy of note. Jayez and Tovena [2005b] write (p. 131) that “Kamp [1978] and Zimmermann [2000] have shown that FC [ = free choice] implicatures are cancellable, as expected for implicatures”, meaning by this conversational implicatures, and they use the following example to make the point: You may reach the island by boat or by plane, but I don’t remember which. The last Gricean rider “but I don’t remember which” does not really show that there is no sense of the sentence before it according to which free choice permission is being expressed. Taking without the rider, the sentence could simply be


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ambiguous, with the rider serving to disambiguate it in favour of the non-‘free choice’ reading; cf. the qualms aired about a similar rider at the start of 6.11. This is not to suggest a lexical ambiguity in the word or. (A range of structural – or syntactic – ambiguities in or -constructions is treated plausibly in Simons [2000a], [2005b], by treating disjunctions semantically in terms of sets of propositions which can be collapsed into single propositions (by taking unions) at any of various stages in the calculation of their truth-conditions, giving rise to what are essentially scope ambiguities. The ‘set of propositions’ idea she traces back through work of Angelika Kratzer to a treatment of the semantics of questions by C. L. Hamblin in the 1970s.) We opened this survey of reactions to the ‘displaced conjunction’ occurrences of or with a reference to the suggestion that such occurrences represent a mere quirk of idiom, and the remark that the systematicity of the phenomenon seemed to call for more than such a suggestion offers. Let us note that there are two ways of taking the suggestion. Any claim that a linguistic construction is idiomatic is a claim that its constituents and modes of composition are not doing their usual semantic work in the construction in question. The fact that the whole construction means what it does mean is not to be explained by citing the meanings of the parts: rather, that fact is to be treated from a semantic point of view as accidental (which does not mean that there is no—for example, historical—explanation for the emergence of the construction). In the present case the idiomaticity suggestion could amount either (i) to the suggestion that nothing unifies the occurrences of or in the problematic constructions with the usual disjunctive role of or, it being merely accidental that the same word is used in both cases, or (ii) to the more extreme suggestion that even amongst the different classes of problematic occurrences (free choice permission, disjunctive antecedents, etc.), it is purely accidental that the same word is used from case to case. By making the ‘quirk of idiom’ move in accordance with construal (i) rather than construal (ii), there is indeed room for a systematic account of what the problematic occurrences of or have in common. Just such a position is taken in Stenius [1982]. Particularly useful for our purposes here is the following characterization of that position, from Makinson [1984], p. 142: Stenius points out that there is in ordinary language a strong tendency to contract compound sentences that involve repetition of some element, such as subject, object, principal verb, or other ingredient. In particular, if a compound sentence formed by “and” has elements repeated on each side, and can be simply telescoped leaving the and intact without creating confusion, then it will tend to be so. In this way, “Anne-Lise is telling a story and Anne-Lise is laughing” telescopes into “Anne-Lise is telling a story and laughing” (. . . ) But if such a telescoping gives rise to a sentence which evidently says something more than the original, or could easily be confused with one saying something more, then in the telescoping process some more serious modification may be made, such as replacing the and by an or. It is for this reason, Stenius suggests, that “You may play baroque music and you may play New Orleans jazz” is contracted to “You may play baroque music or New Orleans jazz”. The or does not serve as a sign of disjunction at all, but is rather a ‘dummy connective’, whose only purpose is to record the trace of the contraction.

CHAPTER 6. “OR”

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A legislator for language might have preferred to see a quite different word here, say “uh”; but natural language has seized on the word or for the purpose, just as it appropriated the temporal indicator then in order to express the distinct notion of conditionality.

The “uh” suggestion here indicates that the proposal described would see any relation between disjunctive or and the problematic occurrences of this word as purely accidental, but the proposal at the same time gestures toward a unified account of those problematic occurrences: all are traces of a reduced conjunction where a dangerous ambiguity would result if and were to appear in the reduced form. (Some references to the linguistic literature on, inter alia, such conjunction-reduction processes were given in the notes to §5.1, p. 674.) One of the most striking examples of this phenomenon comes from Simons [2005b], where it appears as example number (27): If Jane is singing or dancing, then so is Jennifer. As Simons notes, while this sentence can mean other things as well, one thing it certainly means is that if Jane is singing, then Jennifer is singing, and if Jane is dancing, then Jennifer is dancing. A somewhat more general version of the Stenius suggestion is what Jennings [1986], [1994b], calls the ‘punctuationist’ account of or. To see a further application of the idea, made by Jennings in these works, recall the earlier example of (1a) as most naturally interpreted to mean what (1b) means: (1a) Susan is a faster runner than John or Jane. (1b) Susan is a faster runner than John and Susan is a faster runner than Jane. Jennings points out that with a comparative such as “heavier” in place of “a faster runner” in a sentence otherwise like (1a), if an and had appeared where the or appears, there arises a reading according to which Susan is said to be heavier than John and Jane together: for her weight to exceed their combined weights. Since this would not be equivalent to the claim made by the longer sentence, its reduced equivalent cannot safely use and to link the names in the than-phrase, and the punctuationist account predicts that or will replace it accordingly. He speculates that this use of or after comparatives then becomes established as the standardized idiom, even in cases (such as perhaps (1a), or where – Jennings’s own example – the comparative is “is politer than”) in which the unwanted ‘combinative’ reading is not available. This suggestion is subject to an ‘overgeneralizing’ objection of the kind to which Makinson’s proposals were seen to be vulnerable. The trouble is that the combinative reading is just as available for names linked by “and” before the comparative as for names in the than-phrase. Yet the reduced form of “John is heavier than Susan and Jane is heavier than Susan” remains steadfastly “John and Jane are heavier than Susan” rather than, as punctuationism predicts, “John or Jane is heavier than Susan”. So the account’s explanation for the genuine phenomenon of a conjunctive-seeming or after comparatives overextends itself, incorrectly predicting such an or before comparatives. The distinctive behaviour of or in than-contexts – well known to be a negative polarity environment, a consideration completely orthogonal to Jennings’s concerns


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– remains unexplained after all. (See the Digression below for some references to the negative polarity literature.) It might be objected that one can give a conjunctive reading to the sentence just cited. Someone says, “I’m looking for someone who is heavier than Susan”, and is told, “Well, John or Jane is heavier than Susan: take your pick”. However, whatever is at issue here, it is quite a separate phenomenon from the routine use of ‘or’ after comparatives. Jennings [1994b], p. 86, notes the difference between (a) and (b): (a)

Kimberley is heavier than Jack or Bob.

(b)

Kimberley is related to Jack or Bob.

While the most natural reading of (a) is as a conjunction of heavier than claims, the most natural reading of (b) is as a disjunction of related to claims. And he notes (p. 87) that one could, though less naturally, say “Kimberley is related to either Jack or Bob, so if, as you claim, any relative will do, it doesn’t matter which you choose”. Compare our earlier rider “take your pick”. What explains the unusual appearance of “or” in such contexts of picking and choosing? One answer might be: simple confusion – a line of response foreshadowed at the start of 6.11. (This would be an example of what in the Appendix to the present section, which begins on p. 812, we call the ‘explaining away’ response, as opposed to the ‘justifying’ response.) Consider the widespread use of such sentences as (c) in place of—what is perhaps better expressed by—(d), a usage one is tempted to explain as resulting from interference in the speaker’s mind from (e): (c)

* Sarah had to choose between a full-time job as a nurse or a part-time position in a department store.

(d)

Sarah had to choose between a full-time job as a nurse and a part-time position in a department store.

(e)

Sarah had to choose whether to take a full-time job as a nurse or a parttime position in a department store.

Of course, we have said nothing about the or that occurs with whether, as in (e), or in reduced forms thereof (“A simple choice: X or Y ”), and refer the interested reader to the sources cited in the Digression near the start of 6.11 for this. (Choosing whether to F or to G is a matter of choosing to F or choosing to G, just as knowing whether p or q is a matter of knowing that p or knowing that q.) The present point is only to air the possibility that the “or” after “between” results from a simple confusion on the speaker’s part – a possibility which can be acknowledged even by someone wanting to go on and say that this is the historical source of what is by now a perfectly acceptable special idiom in its own right, just as much as by someone wanting to see it as a persistent misuse. (Jennings [1994b] notes the anomalous “between” constructions at p.92.) Whether confusion – at some stage in the past or on each current occasion – is held to be the explanation for the logically unexpected ‘choicy’ or sentences, there is another possibility that deserves to be considered in connection with (b), and that is that we are in the speech act domain of Sweetser [1990]’s threefold division into content, epistemic and speech act domains. (See our discussion

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following 6.13.3 above, where the case for distinguishing the first two for or sentences was found wanting.) In illustration of the speech act interpretation for or, she cites (f ) and (g): (f )

Have an apple turnover, or would you like a strawberry tart?

(g)

King Tsin has great mu shu pork, or China First has good dim sum, or there’s always the Szechwan place just around the corner.

(Numerous examples along the lines of (f ) appear in §3.5 of van Dijk [1979], where they are described as uses of ‘pragmatic or ’.) The occurrences of or here seem to link alternative suggestions—proposed as solutions to a “Where shall we go for a Chinese meal?” problem; one is strongly reminded of the ‘sequence of mooted possibilities’ gloss offered in Jennings [1994a], [1994b] – see the Digression on Dyirbal Disjunction in 6.13 (p. 795) – in the case of somewhat similar examples. Whether the emphasis on a speech act of suggestion is the best way, ultimately, to respond to such examples, for the present point it suffices to note their similarity to (b) in its conjunctive interpretation: the hearer is looking for a relative of Kim, and speaker proceeds with a couple of alternative suggestions. The talk of a conjunctive interpretation is a quick but misleading way of describing the case. What is meant is that this is a case in which the speaker would be happy to accept the conjunction “Kimberley is related to Jack and she is related to Bob” as an accurate description of the facts of the case. What is misleading is the idea that the speaker must accordingly be seen simply as reporting these facts conjunctively in uttering (b). Similarly, any discussion of (g) which described it as providing a context in which the occurrences of or were to be interpreted as and, would manifestly fail to reveal the parallel with (f ), in which such a replacement would distort the meaning. (The words “there’s always” – indicative, as Sweetser notes, of the successive suggestions construction – would probably be best replaced by “there’s also” in any such conjunctive restatement.) The situation with (a) is quite different. In this case it seems not at all misleading to speak of a conjunctive interpretation— even if, as Jennings [1994b] has emphasized, this leaves incomplete the project of accounting for how an or in this position succeeds in contributing to such interpretation. With this we return to the case of (a), and more generally that of (1a), (2a) and (3a) with which we began the present subsection. Digression. The range of positions in which the word “any” can occur in English shows a surprising similarity to the range of positions in which or puts in an appearance of the sort we have described as problematic. (Kamp [1973] gives an extended discussion of this fact; see also Ladusaw [1980], and the references cited therein to the early literature on negative polarity items in general and ‘affective or ’ in particular. There is an excellent discussion of these matters in Krifka [1991] and in various contributions to the collection Horn and Kato [2000]; see also Ladusaw [1996] and, on the connection with ‘free choice’ constructions, Jayez and Tovena [2005a].) For example, it can appear in the antecedents, though not the consequents (unless further negative polarity items occur there), of conditionals; with can and may though not with must; after comparatives. . . Hintikka has conjectured that “any” only occurs where its replacement by “every” would yield something non-equivalent. (This is the anythesis, discussed for example in Chapter 9 of Hintikka and Kulas [1983], q.v.


809

for further references, as well as Horn [2005].) Now Stenius’s proposal may be thought of as analogous to this, since the ambiguities described above as dangerous are cases in which putting and for or (in the reduced forms: the “uh” of the linguistic reformer envisaged in the above quotation) gives something non-equivalent. The analogy in question is given by any:every::uh:and. The relationship between or and any is also explored in Jennings [1994b], esp. Chapters 6 and 7. End of Digression. Two possible views were distinguished above as to the amenability of the various occurrences of or to a unified treatment. The less unificatory view might try to treat the various types of cases all differently, with the more unified approach being that suggested by Stenius, of distinguishing the genuinely disjunctive or from the dummy connective “uh”. An even more radically unifying suggestion put was forward in a somewhat speculative manner by Jennings [1986] (with Jennings [1966] as an earlier foray into the area). The suggestion is that a Stenius-style account of the displaced-and occurrences of or is right for every occurrence of or. As Jennings puts it ([1986], p. 254), “the invention of disjunction came after and not before the abbreviation of ‘It is false that p and it is false that q’ by ‘It is false that p or q’.” Recall that we provisionally distinguished the cases of disjunctive antecedents with “ →” from those with “→ ” on the grounds that the equivalence with a corresponding conjunction in the former case but not the latter could be given a ‘logic-based explanation’. Such an explanation would traditionally be given for the equivalence of not-(S1 or S2 ) with not-S1 and not-S2 , the explanatory work being done by De Morgan’s laws (and thus ultimately by ‘pure’ principles governing the connectives involved). But what Jennings is suggesting (under the name of ‘punctuationism’) is that or appears here, for exactly the same reason as it appears in the free choice permission cases, as a replacement for and whose appearance in the reduced sentence would confusingly suggest something nonequivalent (not-(S1 and S2 )). This is what Jennings refers to as the ‘separative conjunctive function’ of or in the following passage in which he takes up the theme in Jennings [1994b], p. 314f.: The deliberate architectonic of logical theory belies the rough and ready free-market economy, tricking us into seeing symmetries where none are. No feature illustrates the fact better than the equivalences noted by Petrus Hispanus and named after De Morgan. It is true that something like these equivalences hold for English constructions in ‘and’ and ‘or’. But the conjunctive distribution of negation over ‘or’ is an instance of a fundamental separative conjunctive function of ‘or’, not a consequence of an allegedly primary disjunctive meaning. The disjunctive distribution of negation over ‘and’, by contrast, is a discoverable consequence of its primary combinative use. We learn the equivalence of ‘. . . not F or G’ with ‘. . . not F and not G’ as a use of ‘or’, along with a great diversity of other instances of the same idiomatic practice. The one is one of the ways of conveying the other. We learn the equivalence of ‘. . . not F and G’ with ‘. . . not F or not G’ by having it drawn to our attention. The one is not merely the form in which we idiomatically express the other.

Digression. While Jennings is clearly thinking mainly of linking predicates with and and or in this passage, we can think of the point as applied also—

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and indeed primarily in view of the categorial grammatical explanation of how sentence connectives manage to apply to predicates (5.11.3, p. 637)—in the domain of sentential logic. Someone agreeing with his observation that the one De Morgan equivalence (viz. ¬(A ∨ B) ¬A ∧ ¬B) seems more to flow from the meanings of the connectives involved than the other (¬(A ∧ B) ¬A ∨ ¬B) might attempt a different explanation of the fact observed. This alternative reaction begins by noting that while both equivalences hold for = CL , only the first holds for = IL . In view of the fact that the natural deduction rules for the connectives in classical logic do more than uniquely characterize these connectives – are ‘stronger than needed’ for this purpose, as it was put in 4.32, it might be suggested that it is the intuitionistically acceptable rules which are to be seen, in the style of §4.2, §4.3, as giving the meanings of those connectives. Such a position might even be adopted by an adherent of classical logic, if coupled with the view that the excesses of CL over IL represent the incorporation of a metaphysico-logical assumption of bivalence which goes beyond anything given in the meanings of the connectives but nonetheless deserves to be regarded as part of logic. (See Kuhn [1981], note 9, for such a possibility; also the end of the first paragraph in Humberstone and Lock [1986]; both of these sources use a terminology according to which the stronger system is a theory rather than a logic, because the logic/theory distinction is drawn in a different place than we have found convenient in the present work.) Though such a position would require delicate development—since one could not think of the correct logical principles as all being ‘analytic’ in the narrow sense of following from the meaning of the logical vocabulary employed—it would certainly afford an explanation of the different status of the two De Morgan laws noted by Jennings. End of Digression. Whether Jennings’s punctuationist account could be elaborated to cover all occurrences of or however (e.g., unembedded occurrences, occurrences in the scope of and itself, or of it is necessarily the case that,. . . ) must be left as something for the reader to ponder, with assistance from Jennings [1994b]. It is no simple matter, in particular, to reconcile the ‘punctuationist’ account with the ‘sequence of mooted possibilities’ suggestion we have more than once had occasion to cite from the latter work (and will again in the Appendix following: p. 812). One particularly promising suggested reconciliation between the straightforward boolean account and some of the ‘displaced conjunction’ proposals does deserve summary, and that is the hypothesis developed in Simons [2005a]. (See also Simons [2005b].) To formulate the hypothesis, we need Simons’s notion of a supercover. Fix some background set U and suppose S ⊆ U and P ⊂ ℘(U ). Then we say P is a supercover of S just in case: (i) X ⊆ P (ii) for all X ∈ P, X ∩ S = ∅. Here we have use the letter “P” for the supercover, to remind the reader of partitions, though, as Simons remarks, even disregarding the elements of elements of P which lie outside of X, what we are left with is not a partition, as overlap is permitted between the would-be cells or blocks of the partition. (In fact, we are left with what is often called a cover of S, whence Simons’s terminology.) However, as with a partition, when P is a supercover of S, part (ii) of the above definition entails that ∅ ∈ / P (and also that S = ∅). Let us think of the propositions expressed by sentences A, B, as the sets of possible worlds at which they are true (the set of all possible worlds serving as


811

the background universe U in the sense of the preceding discussion), and use !A! and !B!, as usual, to denote these sets. Simons’s proposal is that !A or B!, comprises all w ∈ U for which the following holds: for some S ⊆ U , w ∈ S and {!A!, !B!} is a supercover of S. Let’s come back to how close this comes to the usual boolean story (i.e., !A or B! = !A ∨ B! = !A! ∪ !B!) after seeing why these supercovers are entering the picture. The motivation is from the free choice permission phenomenon and from an analogous phenomenon concerning disjunctions (a term we use for or compounds without prejudice to any given proposed semantic account of them) in the scope of the strong modal – for definiteness, deontic modal – must. Let us use R for a binary deontic accessibility relation and as usual abbreviate “{x ∈ W | wRx}” to “R(w)”. This is the set of worlds in which all obligations are met. Simons proposes that a compositional account of the meanings of must and or should have the upshot that: w ∈ !Must(A or B)! iff {!A!, !B!} is a supercover of R(w), while for the permissibility case we want to end up with: w ∈ !May(A or B)! iff {!A!, !B!} is a supercover of some S ⊆ R(w). The second of these gives the free choice equivalence between May(A or B) and May(A) and May(B), while the first has the interesting consequence that Must(A), taken for deontic-free A to be true at w when R(w) ⊆ !A!, no longer implies Must(A or B), since the latter requires inter alia that B be true at at least one x ∈ R(w). The latter non-implication is independently argued for in Cariani [2007] on the basis of examples such as that of an examination which could not be reported as requiring students to answer one long essay question or three short ones, when it simply demands that one long essay be written. Cariani makes a persuasive case against a purely Gricean response to the effect that the under the circumstances such a report would be misleading, as a violation of the maxim that one give the most informative description rather than gratuitously weakening it. Coincidentally, Simons [2005b] makes use of a similar examination case in her §4.1 against a similar pragmatic attempt in the case of May rather than Must with a disjunction of option-descriptions in its scope. The account offered by Cariani draws on Jackson’s semantics for deontic logic (see for instance Jackson and Pargetter [1986]) rather than Simons’s supercover semantics, and we do not go into further details here. Nor has any attempt been made in this summary to explain how a systematic derivation of the truth-conditions offered by Simons can proceed along compositional lines. (See the papers cited for this.) Simons ([2005a], p. 292) very sensibly asks whether the supercover semantics which seems to do so well for embedding under modals delivers the results we would like to see – the standard boolean truth-conditions – for unembedded disjunctions, and notes that the above account for the non-embedded case delivers the right result for “John sang or Jane danced”. For this to be true at w, we need w to be an element of some set S of worlds (whose deontic accessibility is not now to the point) for which {!John sang!, !Jane danced!} is a supercover. Since this requires every world in S to lie in the first or the second set (or both) of the supercover, at least one of the disjuncts must be true at

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w. But, as Simons remarks (thanking Graeme Forbes for the observation) in the other direction matters are more problematic in the general case. In the current instance, if one of the disjuncts of our sentence is true at w, then we can take the union of !John sang! with !Jane danced! as the desired S, since this set is ‘supercovered’ by {!John sang!, !Jane danced!}: it is included in – indeed coincides with – the union of the sets in this supercover, and further each of those sets has a non-empty intersection with our current choice of S. So we must have !Jane danced! ∩ (!John sang! ∪ !Jane danced!) = ∅ and !John sang! ∩ (!John sang! ∪ !Jane danced!) = ∅. By the absorption laws, the left-hand sides here reduce to !Jane danced! and !John sang! respectively, which is fine here since each of the sentences expresses a logical possibility. But evidently any disjunction with an impossible disjunct is going to be false on the supercover semantics, and if A is true (at a world) while B is impossible, we will have an unwanted counterexample to ∨-introduction in the passage from A to A or B. The question then becomes: whether it is possible to tinker with the semantics to avoid this result while still retaining the benefits for the cases of interaction with deontic (and other) modals. Further work is needed, evidently. But Simons has come very close to the needed reconciliation between the simple boolean story and the modal account designed to make sense of what she begins Simons [2005a] by calling “the variably bad behavior of the word or ”.

Appendix to §6.1: Conjunctive-Seeming Or with Apparent Wide Scope As was mentioned at the start of 6.14 after the introduction of examples (2b) and (3b), apparently disjunctive forms of these sentences are sometimes encountered with much the same meanings as they have. Unlike the or of the reduced forms (2a) and (3a), this or appears to occupy the position of a sentence connective, connecting complete clauses, and to have the modal and conditional material in its scope. This amounts to examples (2) and (3) in the following list. We have added (0) to show a natural example of the phenomenon in question, adapting (1) from Jennings [1994b], p. 297; (4) and (5) are from Cooper [1968] and Barker [1995], respectively; the sources for (6) and (7) are given in the notes (p. 818), along with further examples. As in 6.14, our purpose here is to survey reactions to these examples and direct the reader to relevant literature, rather than to argue in favour of a specific reaction. (0)

You can stay here and starve, or you can come with me and search for food.

(1)

It might indicate envy or it might indicate contempt.

(2)

Susan may speak to John or she may speak to Jane.

(3)

If John comes alone Jane will be pleased, or if Susan brings Paul, Jane will be pleased.


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(4)

If the water supply is off then she cannot cook dinner, or if the gas supply is off she cannot cook dinner.

(5)

Either the cheque will arrive today, if George has put it in the mail, or it will come with him tomorrow, if he hasn’t.

(6)

Man either revels in his misery or, if he reveals it, he prides himself on being aware of it.

(7)

If the milk is above 49◦ when the yoghurt starter is added, it will curdle, or below 43◦ , nothing will happen.

While the customary practice of logicians would be to replace these “or”s with “and”s to correct for something presumed to be either an error of formulation or a quirk of syntax, Cooper ([1968], p. 298) asks, with his case (4) primarily in mind, “If the speaker meant ‘and’, why didn’t he say ‘and’ ?”. This covers the misformulation reaction; for the ‘quirk of syntax’ reaction, there remains the analogous question as to why an apparently disjunctive construction should be the natural vehicle for the expression of an apparently conjunctive thought. Cooper’s question was actually rhetorical in intent. Called on to provide an explanation for the occurrence of conjunctive-seeming or in such examples as (0)–(7), one may offer either an attempt at justification – showing that the formulations exhibited are exactly what a correct pragmatic, semantic or logical theory would call for – or an attempt at ‘explaining away’ the phenomenon: showing why its existence should not be deemed troubling for such a theory because it represents a mistake or confusion of some kind. In Cooper’s case, the question was rhetorical because he wanted to offer an alternative to classical logic in which – recall the focus on (4) – a disjunction of conditionals was equivalent to the corresponding conjunction, the general non-equivalence of conjunctions and disjunctions notwithstanding. This alternative logical account is thus part of a justificatory response to the phenomenon; we will review the logic Cooper proposes in detail in 7.19, saying no more about it here. Another reaction which can perhaps best be classified as justificatory is that of Jennings [1994b]. We have already, in 6.13 and 6.14, had occasion to cite his suggestive description of (at least some) or -sentences as demanding to be “understood as a succession of mooted possibilities”, and there are many examples in the same vein as (0)–(7) above to be found, with commentary, in that work. The suggestion seems especially apposite in connection with the modal examples (0)–(2), but because it does not seem to lend itself to a precise formulation, we pass over it in silence here, referring the reader to the very full discussion Jennings [1994b], esp. Chapter 11. We return to a third justificatory response – that of Barker [1995] – after mentioning some instances of the ‘explaining away’ stance. Suppose one had a satisfactory account of the conjunctive-seeming occurrences of or discussed in 6.14, and in particular for such occurrences of or as appear in (8)

It might indicate envy or contempt.

Such an account would explain why this is understood as meaning that it (whatever it is) might indicate envy and it might indicate contempt. As in 6.14, though there we concentrated on may rather than might, we take (8) to involve

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or in the scope of the weak (i.e., -like rather than -like) modal operator might. (In speaking of -likeness, we assume a normal modal logic for ; the weak/strong contrast is so described on the presumption that underlying logic extends KD.) With such an account to hand, we might explain away examples like (1) above as mere variants on (8) in which the modal element is copied over in much the same way as some language (including some dialects of English) copy negation onto suitable sites in its scope (“I didn’t see nobody take nothing” for “I didn’t see anybody take anything”); there are many similar reduplicative phenomena (sequence of tenses, etc) in natural languages, and we do not regard their products as challenging otherwise plausible logical doctrines (the law of double negation, for instance). Even ‘strong’ modal elements—a proponent of this particular style of ‘explaining away’ might continue—seem subject to such semantically gratuitous (and therefore potentially misleading) reduplication: (9)

You have to drink your coffee standing up or else you have to stand in line and wait for a seat.

(10) I want that money on my desk by two o’clock or I want your resignation. (Jennings [1994b], p. 236, mentions (10) to illustrate a somewhat different point. Simons [2005a], pp. 274, 282, and elsewhere, also discusses these cases with a repeated modal.) A second response which falls into our ‘explaining away’ category would be one according to which examples like (0)–(7) simply do not have the conjunctive meanings which might provide a challenge (‘the problem of disjunctive antecedents’ or whatever). Rather, people confuse them with the corresponding conjunctive sentences for reasons which themselves deserve empirical examination. In particular, Johnson-Laird and Barres [1994] take it that a ‘bottle-neck in inferential processing’ (p. 478) occasioned by limitations in working memory underlies errors in performance on reasoning tasks which reveal these confusions, and they also claim that the ‘theory of mental models’ developed by JohnsonLaird and collaborators (see Johnson-Laird and Byrne [1993]) offers a level of representation of the content of the confused sentences which precisely explains the confusions actually encountered in the responses of logically untrained experimental subjects. However, the theory has by no means been set out with a level of clarity which would enable such claims to be assessed. (Complaints on this score may be found, for instance, in Hodges [1993b].) Fortunately we do not have to assess them in order to offer Johnson-Laird and Byrne [1993] as a clear illustration of the ‘explaining away’ reaction arising from the combination of an unquestioned regard for a particular logic (here CL) as determining which of the subjects’ responses are erroneous, with a particular hypothesis in cognitive psychology (here the mental models theory) to explain the errors. (The ‘errors’ are genuinely explained ; what is explained away is any appearance of a need to change the favoured logic.) Digression. For the record, it should be mentioned that Johnson-Laird and Barres [1994] actually provide representations of the content of a sentence at three different levels of refinement, with errors accordingly to be expected at greater frequency with decreasing refinement. Apart from the confusion of (A → B) ∨ (C → D) with (A → B) ∧ (C → D), on which we have been concentrating, the authors found a tendency to conflate the latter with another form (another


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disjunction, as it happens): (A∧B)∨(C∧D). Though their preferred explanation is again in terms of the mental models hypothesis, it is worth noting that in the presence of the additional assumption—sometimes suggested by context, or explicitly enforced by choosing C as ¬A—that A and C are contradictory (i.e., incompatible – ‘contraries’ – and also ‘subcontrary’), (A → B) ∧ (C → D) and (A ∧ B) ∨ (C ∧ D) are equivalent by truth-tables. End of Digression. We close our discussion with a return to the justificatory stance. In the case of Barker [1995] it takes the form of a pragmatic theory of conditionals, and applies to the conditional examples (3)–(8) in our initial list. (“Pragmatic” rather than “semantic” because no conditional propositions are assigned as the content of conditional sentences; what propositions are assigned, and what becomes of the conditionality, will be clear presently.) The general category is that of the ‘conditional assertion’ (more generally, conditional question, conditional command, etc.) approach to conditionals—for more information about which, see below. The particular form of Barker’s proposal, within this genre, is that someone saying ‘If S1 then S2 ’, for the particular case in which S2 (like S1 ) is a declarative sentence is thereby asserting S2 , but only on the supposition that S1 , which is to be understood as not committing the speaker to the truth of S2 but only committing him/her to its truth should S1 turn out to be true, and in addition as conveying by means of a conventional implicature that this conditional commitment is warranted by other suppositions and background beliefs which are in force. When the whole conditional occurs embedded in a context in which it is not asserted—such as one disjunct of a disjunction—its propositional content remains simply that of its consequent (S2 ). As to what happens to the conventional implicature just mentioned, Barker suggests that we follow the lead of theorists of presuppositions, who have interested themselves in the conditions under which the presuppositions of components are inherited by the compounds in which they are components. (A helpful survey of this literature may be found in Kay [1992].) Oversimplifying somewhat for present purposes, this amounts, in the case of disjunctions, to saying that the implicatures/presuppositions (to use Barker’s formulation) of a disjunct are inherited by the disjunction unless they are explicitly negated by the other disjunct. Thus, while (11) presupposes that George has been beating his children, the disjunction (12) does not; on the other hand, (13) does have this presupposition because the second disjunct does nothing to ‘cancel’ it. (These examples are from Barker [1995], p. 203.) (11) George has stopped beating his children. (12) Either George has stopped beating his children or he never started. (13) Either George has stopped beating his children or has gone on holiday. Applying this to the case of Barker’s (5), repeated here for convenience: (5)

Either the cheque will arrive today, if George has put it in the mail, or it will come with him tomorrow, if he hasn’t.

we have the result that, since neither disjunct cancels the ‘conditional implicature’ (as we might call it) of the other, the disjunction inherits both implicatures. Its propositional content, however, is obtained by disjoining the contents of the two disjuncts, and is thus is simply that of

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(14) Either the cheque will arrive today or it will come with him ( = George) tomorrow. The apparent conjunctivity of (5) is accordingly explained on Barker’s account by the fact that both conditionality implicatures survive in the disjunction; nevertheless, the correct representation is indeed disjunctive (so this is a justificatory explanation), though shorn of its implicatures, all that is disjoined is given in (14).

Notes and References for §6.1 The monographs devoted to the syntax, semantics and logic of or in our bibliography are Jennings [1994b] and Simons [2000]; of course there are many individual articles on the topic, referred to in the main body of this chapter and in these and other end-of-section notes. The 2004 French–Israeli movie entitled Or (directed by Keren Yedaya) turned out not to be the definitive documentary on disjunction the world had been waiting for, but just another drama, with the title being the name of one of the characters – perhaps the biggest disappointment for cinema-goers since 1968, with the appearance of Lindsay Anderson’s If. The quotation from a radio interview near the start of 6.11 is from the August 20, 2007, episode of the Health Report broadcast on Australia’s ABC Radio National network, with transcript available as of the time of this writing at http://www.abc.net.au/rn/healthreport/stories/2007/2010834.htm The interview subject, whose words are quoted, is Bill Haskell (and the interviewer, Norman Swan). Here is another example, this one taken from the Melbourne newspaper The Age of December 17, 2009 (p. 13, article headed ‘Abuse remarks doubtful: lawyer’, by Carol Nader, the remarks here quoted being there attributed to a lawyer, Howard Draper): A court has to decide if it is proven or not. No one knows what is true or not.

The first “or” is a whether “or”; the second is problematic: should it be dignified with a wide-scope “and” defence? (“No one knows what is true and what is false.”) In 6.12, it was suggested that views according to which or has an exclusive sense will tend to be views that also acknowledge an inclusive sense, and hence espouse an (implausible) ambiguity claim. The tendency is not universal. For example Bradley [1922], Vol. 1, Chapter IV, argues that “or” strictly speaking always expresses exclusive disjunction. The first paper to turn against the ambiguity claim, which at the time of its publication had something of the status of a received view, was Barrett and Stenner [1971]. Subsequent works in the same vein include Gazdar [1977], Pelletier [1977], Dudman [1978], Richards [1989] (originally published 1978), and Jennings [1986]. Richards’s paper contains a justified criticism of Goddard [1960] on this matter. See also Horn [1989], p. 225 and elsewhere, Gazdar [1979], pp. 78–83, and Browne [1986], as well as Noveck, Chierchia et al. [2002] for a reasonably sophisticated discussion of considerations favourable to the exclusivity implicature. Rearguard actions


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in defence of the inclusive/exclusive ambiguity thesis include Hurford [1974], and Girle [1989] (originally published 1979). An early attempt to assess the ambiguity thesis with the aid of questionnaires is reported in Naess [1961]; naturally the results of any such survey have to be handled carefully because of pragmatic interference with semantic judgments on the part of naive subjects. One result of the survey is that ‘S1 or S2 ’ is more likely to receive an exclusive ‘interpretation’ if preceded by Either. Indeed Rautenberg [1989a] distinguishes ∨ and as or and either/or respectively – and he is not alone in doing so. A stance more consilient with 6.12 would have it that the use of either, just like the placing of heavy stress on “or” (without the addition of “either”) is just a way of drawing attention to the fact that disjunction rather than conjunction has been used, with the consequential implicature that the speaker would not be prepared to assent to the conjunctive sentence. An even better suggestion (Zimmermann [2000], top p. 268) – best put of open and closed disjunctions, as in the Digression (on Dyirbal Disjunction) from p. 795 – would be that the role of “either” is to make it explicit that the disjunction is to be taken as closed (exhaustive). Questions about inclusive vs. exclusive interpretations of “or” are amongst the topics discussed in Newstead and Griggs [1983]. An early and somewhat confused discussion of exclusive disjunction – under the name ‘the (binary) contradictory function’ – may be found in Rose [1957]. We do not pretend to have covered in our discussion every proposed distinction amongst disjunctions; nothing is said, for example, about the alleged ‘generically interpreted’ disjunctions of Kratzer [1989], p. 638. The same goes for some other roughly situational-semantic discussions of disjunction, such as those in Restall [1996], Read [2000b], (on the notion of truth-making) and also Wolniewicz [1980] and references therein (on early Wittgensteinian metaphysics). (See also van Fraassen [1969b] for references to the ‘disjunctive facts’ issue, and Wolniewicz [2000] for Wolniewicz’s more recent thoughts.) A propos of our discussion of Dixon’s example (2) in our Digression on Dyirbal disjunction at the end of 6.13 (p. 795): (2)

I saw a fish – perhaps it was a barramundi, perhaps it was a red bream.

we note that a related example concerning the or in so-called alternative questions, and how to phrase such questions for the benefit of speakers of Aboriginal English, is cited in Humberstone [1998b], note 30. Apart from the papers cited in 6.14, the survey paper Hilpinen [1981b] gives an extensive bibliography on work (prior to its date of publication) on the problems of disjunctive antecedents and free choice permission. Hilpinen’s use of a counterfactual construction to explicate the latter concept is a refinement of a suggestion of von Wright [1971] to use (approximately) strict implication in a similar capacity. In the quotations in 6.14 from Makinson [1984] the spelling “Brasil” has been altered to (the usual English spelling) “Brazil”. Makinson has observed to me that the wording after examples (3a) and (3b) in 6.14: “The examples seem directly to have the forms (A ∨ B) → C and (A → C) ∧ (A → C), and these are indeed equivalent in CL (or IL, even)” features in this last parenthetical remark the kind of conjunctive-seeming occurrence of or that is officially under discussion in this subsection. (This was not intentional.) In a further personal communication, Makinson notes that as well as the occurrences of or that seem to demand interpretation as and, we also meet the reverse situation – not touched on in 6.14 (or elsewhere) – as in (his examples): “Most

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aircraft accidents occur on take-off and landing”, and “Most of our bookshop’s sales are of thrillers and biographies”. We have certainly not given comprehensive coverage to all the qualms (∨I) has raised. Question 1 on p.111 of Goldstein [1990] asks the reader to consider its role in the celebrated argument of E. L. Gettier that knowledge is not analyzable as justified true belief. Another aspect of the behaviour of disjunction that has caught the attention of epistemologists is the way it figures in the definition of Nelson Goodman’s predicate grue (from Goodman [1965]); Roxbee Cox [1986], for instance, bears the title ‘Induction and Disjunction’. In this case the term induction refers to the empirical inductive inferences whose legitimacy has been so much discussed by philosophers. The same title suggests also a somewhat different topic, taking ‘induction’ as ‘mathematical induction’. Disjunction is used in casual formulations of what are intended to be inductive (or recursive) definitions, making them look like (circular) explicit definitions, as in “For all x, x is a number iff x is 0 or x is the successor of a natural number”. Such a would-be definition fails to single out a determinate set as the extension of “(natural) number”, the intention being to single out specifically the least set satisfying the stipulation. (See further p. 287f. in Keefe [2002].) Example (6) from the Appendix to this section is from Pascal [1670], Remark 119 on p. 142 of the translation cited in our bibliography. (The interplay of if and or in the English translation mirrors that of the French original.) Example (7) is adapted from a remark on p. 36 of the June–July 1986 issue of a British vegetarian lifestyle and cooking magazine, Lean Living. The full text is given below. The author (Celia Storey) is discussing yoghurt thermometers, for use in making yoghurt at home: It is marked with the maximum and minimum temperatures of 43◦ and 49◦ . The milk should be somewhere between those two before adding the starter. If it is above, it will curdle or below, nothing will happen.

As reproduced in the Appendix (i.e., as (7)), a comma has been inserted after ‘curdle’, for extra readability, and the reference to the particular temperatures from the earlier sentence inserted into an accordingly expanded version of the last sentence. From another magazine – the Australian women’s magazine New Idea of September 4, 2004, on the second page of an article, pp. 62–3, “The Arthritis Epidemic”, by Bronwyn Marquardt – comes the following example (into which the second comma has been inserted for this quotation): The items are available at most health food stores, Chinese grocers or markets, or ask your natural therapist for help.

In the same vein, here are two examples from Stephanie Alexander’s column in the ‘Epicure’ section of the Melbourne Age newspaper (June 14, 2005, p. 4, this particular instalment being headed ‘Walnut wonders’): Some growers sell cracked kernel, or you must do it yourself.

Later on the same page: Walnuts and grapes or fresh pears and goat’s cheese or blue cheese tossed with soft leaves and a dressing that includes a little roasted walnut oil is a lovely starter.


819

It is an interesting question whether one should have an account of “or” which makes sense of such examples or instead hold them all to involve confusion. Here, still within the arena of food choices, and with the element of choice apparently exerting its influence (pernicious or otherwise, depending on how the question just raised is handled) on the form of the connective, is a most striking example from Lehrer [1997], p. 81. It concerns the American philosopher Sidney Morgenbesser, the subject of many anecdotes: morgenbesser: I would like a roll or something. What have you got? waitress: I have a sweet roll or a bagel. morgenbesser: I’ll have a sweet roll. waitress: I forgot. We also have onion rolls. morgenbesser: Oh, in that case I’ll have a bagel instead. Lehrer mentions the example à propos of the principle of the independence of irrelevant alternatives, from the theory of rational choice, whereas for our purposes the interest lies in the or in the waitress’s first line. (As to the rationality issue: of course in practice there need be nothing irrational about Morgenbesser’s reported behaviour, since the new information that onion rolls are available may provide crucial evidence about how the sweet rolls and bagels on offer are likely to have been prepared, evidence affecting their relative desirability. This was not the kind of consideration the decision-theoretic principle mentioned above was intended to rule out, however.) Still in the gastronomic domain, consider the following passage from Tobie Puttock (‘Easy Italian’, The Sunday Age (Melbourne), Sunday Life Magazine, May 20, 2007 pp. 36–7), which precedes a recipe for spaghetti with olive oil, croutons and chilli: This is my “nothing in the fridge” favourite. I can have dinner on the table in 15 minutes. You can add practically anything to it. But rocket or prosciutto are especially good.

Our interest of course is in the “or” of the last sentence – but note the plural form, “are”, of the verb. A similarly plural verb is used (surely ungrammatically, whatever one thinks of the preceding case, though not marked as such by the authors) in an example on p. 303 of Noveck, Chierchia et al. [2002]: If Jules or Jim come, the party will be a success.

Here is another example, from the opening page of Halmos [1960]: A pack of wolves, a bunch of grapes, or a flock of pigeons are all examples of sets of things.

Here we are mainly interested in the appearance of or where and might be expected, and also in the choice of plural are, rather than the correctness of the sentiment expressed. (But, for the record, note that what Halmos says in trying to explain the mathematical concept of a set is in fact false. When is the last time a set – in that sense – of wolves roamed the woods? The locus classicus for this objection is Black [1971].) Again, we have the following program summary for an episode (‘Natural Causes’) of the British television series The Inspector Lynley Mysteries, variations on which appeared in several Australian newspaper TV guides in March 2008:

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CHAPTER 6. “OR” Lynley is suspended from duty awaiting the outcome of his disciplinary hearing. Havers is working with another Detective Inspector, Fiona Knight, investigating a woman drowned in her car. The prime suspects are the woman’s husband or boyfriend, until Havers persuades Lynley to take an interest in the case.

With less pressure on space, no doubt, “or boyfriend” would have been written out in full as “or her boyfriend”, to avoid an unintended reading, but note again the plural “prime suspects are”: in the situation described each of the two men mentioned is a suspect, and the summary is putting or for and to avoid the suggestion that the two men are suspected of having both committed the crime. Nevertheless, the result is surely not English. (Some further remarks on unexpected plurals with or -constructions may be found in LeGrand [1974], p. 398f.)

§6.2 ARGUMENT BY CASES IN THEORY AND IN PRACTICE 6.21

Problematic Applications of ∨-Elimination

The rule (∨E), or its informal analogue, is often called ‘argument by cases’, since in reasoning that C follows whichever of A, B, is assumed, we have covered all the cases in which A ∨ B could be true, and so are justified in inferring C from the supposition that it is true. In §6.1 we considered the suggestion that for certain senses of or (in particular, exclusive and ‘intensional’ senses), the usual introduction rule (∨I) was compromised, and here we shall survey some examples in which it may seem that all is not well with the elimination rule. Note that there is no threat here from exclusive or intensional disjunction (whether or not one thinks of these as representing a sense of or ): even if both cases obtain, or there is more to the truth of a disjunction than at least one of them’s obtaining, it remains legitimate to ‘argue by cases’ to any conclusion that follows from each disjunct. And in fact, the examples we shall mainly consider in this connection do not come from theorists trying to interest us in new senses of or. Our final example does come from an area of logical theory, however, namely quantum logic (2.31). Even here, though, as we saw in 4.32.2 (p. 587) and the surrounding discussion, there is no room for an ambiguity claim (classical vs. quantum disjunction senses of “or”, as it might be put). A quite different style of objection to (∨E) can be found in Girard et al., [1989], p. 74, in which offence is taken at the appearance of the extraneous “C” (the formula which in any application of this rule has to be derived from A and from B so that the rule allows us to derive it from A ∨ B), which is not (in general) a subformula of the latter disjunction. This feature is alleged to make for awkwardness in Prawitz-style proof theory for natural deduction (normalization, as touched on in §4.12). Nothing will be said about such objections here. (For some discussion, see Ferreira and Ferreira [2009], and references there cited.) The first example to consider is a simplified version of a widely known story about prisoners; the usual version may be found in Bar-Hillel and Falk [1982].

6.2. ARGUMENT BY CASES IN THEORY AND IN PRACTICE War

Peace

Remain armed

−100

0

Disarm

−50

50

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Figure 6.21a

Example 6.21.1 Suppose there are three prisoners a, b, and c, one of whom will be picked at random to be released, with the other two being executed. Prisoner c reasons: (i) Either a or b will be executed. (ii) Suppose that a is executed; then the probability that I will be executed is 1/2 (since each of b and I would be equally likely as a candidate to be executed alongside a). (iii) Suppose that b is executed; then the probability that I will be executed is again 1/2. (iv) Therefore, in view of (i), the probability that I will be executed is 1/2. Clearly something has gone wrong with c’s reasoning here, since, with only one of the three to be released, and each of the three an equally likely candidate to be that one, c’s chances of being executed are 2/3 rather than 1/2. But what? In particular: is the attempt to ‘argue by cases’ on the basis of (i) at fault? The reader who has not encountered this puzzle before may care to ponder it before proceeding. In any case, we postpone discussion of the examples until the following subsection. Our second example also involves probability, though in a somewhat different way. The example is taken from Jeffrey [1965], §§1.1, 1.5. The background is as follows: an agent with control over whether the United States is to retain or abandon its nuclear arsenal, is trying to work out which option is preferable. The agent attaches a probability of .1 to the proposition that there will be war given continued armament (and therefore .9 to the probability of peace under that condition) and .8 to the proposition that there will be war given U.S. disarmament (and therefore .2 to the probability of peace under that condition). This talk of probabilities is to be taken subjectively (i.e., as recording our agent’s degrees of belief). The low probability of war in the former case reflects a confidence in the policy of deterrence, the high probability in the latter case, the estimated vulnerability to attack of an unarmed nation. Now, the agent is to reason in accordance with the canons of Bayesian decision theory, and therefore to choose the option with the greater expected utility, so we need some utilities to feed into the calculation. As Jeffrey describes the case, these are as in Figure 6.21a. Thus the expected utility of remaining armed is the sum of two things: first, the product of the utility of War while armed (−100: the lower figure because of the destructiveness of a bilaterally nuclear war) and the probability of war given the ‘remain armed’ option (.1), which is to say −10, and second, the product of the utility of Peace while remaining armed (0: this is the status quo) and the

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probability of Peace given the ‘remain armed’ option (.9), which is to say 0. Thus the expected utility of remaining armed is −10. Corresponding calculations for the option of disarming give (−50 × .8) + (50 × .2) = −40 + 10 = −30. So the two options emerge with expected utility −10 and −30 respectively, and the rational choice would therefore be for the first option (remain armed). This much said, we can present: Example 6.21.2 A fallacious (?) argument that the second option (‘disarm’) is to be preferred in the above situation: (i) Either there will be war or there will be peace. But looking at Figure 6.21a, we see the following: (ii) Suppose there will be war. Then the utility of disarming exceeds the utility of remaining armed. (iii) Suppose there will be peace. Then the utility of disarming again exceeds the utility of remaining armed. (iv) Therefore, since in either case, disarming proves to have the greater utility, the rational choice is to disarm. An option which would have greater utility in any of the conditions considered (such as War, Peace, here) than some other option is said to dominate that other option, and an option which dominates all other options is called a dominant option. The reasoning of 6.21.2 generalizes to any decision situation in which there is a dominant option, giving the conclusion that this dominant option should be selected. Such reasoning is described by Jeffrey (p. 8) as fallacious, since it does not accord with the (in this case, at least, reasonable enough) prescriptions of Bayesian decision theory. He remarks (p. 9, emphasis added), “The assumption that the dominant act is the better is correct if an extra premise is introduced: namely, that the probabilities of the conditions are the same, no matter which act is performed”. This of course is not the case in the present example, since the agent’s subjective probabilities reflect a confidence in the policy of deterrence: our agent thinks that the chances of war are enormously reduced by remaining armed. But it is one thing to say all this, and another to show how, in the light of these considerations, the reasoning of 6.21.2 goes wrong: after all, we seem to have a straightforward application of (∨E). If either A or B, and option X would be better given A and also better given B, then why doesn’t that show that option X would be better tout court? Since when did ∨-Elimination need an ‘extra premise’ ? Our next example retains a practical flavour, though the (fatalistic) advice it issues in is much less likely to be taken seriously than that of the previous case. The argument involved goes back to antiquity, though the present version is taken with minor adaptations from Dummett [1964]. Example 6.21.3 Imagine yourself in London at the time of an air raid in the Second World War, hearing the sirens and making for a shelter. Someone tries to argue you out of taking cover, saying: (i) Either you are going to be killed by a bomb or your aren’t. (ii) Suppose you are. Then any precautions you take will be ineffective. (iii) Suppose you are not. Then any precautions you take will be superfluous.

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(iv) Therefore, either way, it is pointless to take precautions. The reader has perhaps encountered (as curiosities) other arguments of this form; for example, against taking swimming lessons (“Either you will die by drowning or you won’t” etc.). Presumably, those persuaded by such arguments have a higher mortality rate than those who remain sceptical. Should ∨-Elimination, then, carry a government health warning? Finally, we illustrate the kind of case which leads some—the proponents of Quantum Logic—to urge the replacement of (∨E) by the restricted rule (∨E)res described in 2.31 (p. 299). Recall that the intention behind this restriction is to block the proof of the distribution laws. Example 6.21.4 According to quantum mechanics, it is not possible to assign to an electron simultaneously a determinate position and momentum. For a given position, however, a range of possible momenta can be assigned. So let P and M1 , . . . , Mn be statements to the effect that a certain electron has a certain position, and that it has a corresponding momentum from a range of n possibilities, and further, that P ∧ (M1 ∨ . . . ∨ Mn ) is true. Clearly, use of the unrestricted (∨E) rule enables us to prove P ∧ (M1 ∨ . . . ∨ Mn ) (P ∧ M1 ) ∨ . . . ∨ (P ∧ Mn ). But whereas the left-hand formula is, ex hypothesi, true, the right-hand formula is a disjunction of conjunctions each attributing a particular position and momentum, and none of these can be true: therefore the disjunction as a whole is not true, and we should reject—or restrict—the rule (∨E) which led to the provability of the above sequent. Note that a proponent of the above line of reasoning (e.g., Putnam [1969]) has to mean something non-epistemic by the claim that it is not possible ‘to assign’ a determinate position and momentum. This can’t just mean it is not possible to ascertain both the position and the momentum of a given electron at a given time, or it would not follow that no conjunction P ∧ Mi is true. It must mean that an electron cannot simultaneously have a determinate position and momentum.

6.22

Commentary on the Examples

In especially the first three examples of 6.21, an argument by cases seemed to lead to an anomalous conclusion. These arguments involved four steps. Step (i) presented the disjunction in question; steps (ii) and (iii) derived that conclusion from the first and second disjuncts respectively; and step (iv) presented that conclusion as depending on the disjunction. By way of reminder, those steps in the case of Example 6.21.1, in which prisoner a was contemplating the chances of being executed, were: (i) Either a or b will be executed. (ii) Suppose that a is executed; then the probability that I will be executed is 1/2 (since b and I are equally likely to be executed alongside a).

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CHAPTER 6. “OR” (iii) Suppose that b is executed; then the probability that I will be executed is again 1/2. (iv) Therefore, in view of (i), the probability that I will be executed is 1/2.

An assumption of the example is that two of the three prisoners (the choice to be made at random) will be executed. This certainly guarantees step (i), which is just a reduced form of the (inclusive!) disjunction “Either a will be executed or b will be executed”. The trouble comes with steps (ii) and (iii). The (completely standard) concepts to be used in describing this trouble were introduced in 5.14. What makes step (ii) look reasonable is the fact that the probability that c will be executed given that a is executed is indeed 1/2; likewise in (iii) for the probability that c will be executed given that b is. Thus starting with a probability function Pr, the two probability functions got by conditionalizing Pr on what follows the “given that” in these two cases—call them Pr a and Pr b respectively—do indeed assign the value 1/2 to the statement that c will be executed. But since in (ii) we have the conclusion that Pr a (c will be executed) = 1/2, and in (iii), we have instead the conclusion that Pr b (c will be executed) = 1/2, no common conclusion has been derived from each of the disjuncts in 1, and (∨E) does not apply. We can usefully put this point in terms of the original probability function Pr, and a rewriting of (ii) and (iii) in conditional terms, say as (ii) (a is executed) → Pr (c will be executed) = 1/2. (iii) (b is executed) → Pr (c will be executed) = 1/2. From these it would indeed follow that Pr (c will be executed) = 1/2, by (i) and the rule, sometimes called ‘Constructive Dilemma’: to pass from A → C and B → C to (A ∨ B) → C; think of this as a kind of conditional version of (∨E). But, to rephrase the point of the preceding paragraph, what we are given by the details of the example are not (ii) and (iii) but rather: (ii) Pr (c will be executed/a is executed) = 1/2 (iii) Pr (c will be executed/b is executed) = 1/2. If from these, it followed that Pr (c will be executed/a or b is executed) = 1/2, then the conclusion (iv) of Example 6.21.1 would in turn follow, since Pr (a or b is executed) = 1. So what the example shows is that a certain kind of (∨E)-like move is not available for conditional probabilities; it establishes, to be precise: Observation 6.22.1 It is not in general the case for a probability function Pr and statements A, B, and C that (∗)

If Pr(C/A) = r and Pr(C/B) = r, then Pr(C/A ∨ B) = r.

It is interesting to note that with the additional hypothesis on (*) that A and B are mutually exclusive (in the sense that Pr (A ∧ B) = 0), we get something true. This is worth showing. We begin with a simple lemma from real arithmetic using the concept of betweenness, taken inclusively in respect of end-points (i.e., we count j as between i and k if i j k or i j k).


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Lemma 6.22.2 For positive real numbers a, b, x, y: (x + y)/(a + b) lies between x/a and y/b. Proof. Note that x(a + b) − a(x + y) bx − ay x x+y − = = a a+b a(a + b) a(a + b) Similarly,

b(x + y) − y(a + b) bx − ay x+y y − = = a+b b b(a + b) b(a + b)

Since the denominators here, a(a+b) and b(a+b), are in both cases positive, and the numerators are the same, either (bx − ay)/(a(a + b)) and (bx − ay)/(b(a + b)) are both positive, or both negative, or both zero, from which the Lemma follows. (If (x + y)/(a + b) is not between x/a and y/b, one of the above subtractions has to yield a positive result while the other yields a negative result.)

Theorem 6.22.3 Let Pr be a probability function with P r(A ∧ B) = 0. Then for any C, P r(C/A ∨ B) lies between Pr(C/A) and Pr(C/B). Proof. The latter two conditional probabilities are by definition P r(C ∧ A)/P r(A)

and

P r(C ∧ B)/P r(B)

respectively, while the former is: P r(C ∧ (A ∨ B)) P r((C ∧ A) ∨ (C ∧ B)) P r(C ∧ A) + P r(C ∧ B) = = P r(A ∨ B) P r(A ∨ B) P r(A) + P r(B) (We use the assumption that Pr (A ∧ B) = 0 to simplify the denominator in the second equality here.) Now the Theorem follows, taking x = Pr (C ∧ A), y = Pr (C ∧ B), a = Pr (A) and b = Pr (B) in Lemma 6.22.2. Correcting the ‘(*)’ part of 6.22.1, we have the following immediate consequence of 6.22.3: Corollary 6.22.4 For any probability function Pr with P r(A ∧ B) = 0, if Pr(C/A) = r and Pr(C/B) = r, then P r(C/A ∨ B) = r. The incompatibility or ‘mutual exclusiveness’ condition here (Pr (A ∧ B) = 0) may—cf. 6.12—erroneously suggest that we can delete the “not” from 6.22.1 and replace “∨” by “ ” to obtain a truth. Add to the conditions on probability functions in 5.14 the condition that Pr (A B) = Pr ((A ∨ B) ∧ ¬(A ∧ B)) for the sake of:

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Exercise 6.22.5 Give an example to show that it is not the case that for any probability function Pr, if Pr (C/A) = r and Pr (C/B) = r, then Pr (C/A B) = r(for all A, B, C). Before wrapping up our treatment of the first example from the preceding subsection, we make a further connection with 5.14. Say that (binary) # is a probabilistic disjunction connective if for all A, B, and all Pr: Pr (A # B) = max (Pr (A), Pr (B)). In 5.14.3 (p. 654) we saw the untoward effects of postulating the existence of a connective for probabilistic conjunction (as here, but with “min” for “max ”). Exercise 6.22.6 (i) Show that if # is a probabilistic disjunction connective, then for any probability function Pr and any A, B, C, if Pr (C/A) r and Pr (C/B) r then Pr (C/A # B) r. (ii ) Does the above conditional hold for # as ∨? (iii) Can you derive any unacceptable consequences from the supposition that there is a connective of probabilistic disjunction? This concludes our discussion of Example 6.21.1; the failure of probabilistic ∨-Elimination (Observation 6.22.1) can be applied to diagnose what happens in the ‘Twin Ace’ paradox which has puzzled many writers. See the references given in the end-of-section notes (p. 842). The other example from 6.21 we intend to discuss in some detail is the ‘fatalistic’ 6.21.3 (p. 822). For discussion of 6.21.4 (p. 823), the reader is referred to Dummett [1976], and for 6.21.2, to supply his or her own resolution. (Many of the papers in Campbell and Sowden [1985] are of interest in connection with the notion of dominance in decision theory and game theory.) Those readers familiar with the notion of conditional obligation may care to consider also the acceptability of the (∨E)-like inference in dyadic deontic logic (e.g., as in Lewis [1974]) from O(C/A) and O(C/B) to O(C/A ∨ B). And to illustrate the natural occurrence of informal argument by cases in everyday practical reasoning, we include here a quotation from Zimmer [1986], in which a female warder is reflecting on her physical security in a male prison: No inmate wants to go after a woman. What does it prove if he beats her? You see, if he beats a woman in a fight, it’s no big deal—any man could do that. On the other hand, if he should lose in a fight with a woman, his reputation is completely ruined in here. So all around, it’s just to his advantage to avoid physical confrontations with women.

Turning, then, to the air-raid example 6.21.3, we begin by noting that the label “fatalistic”, used above, is actually somewhat contentious, since in philosophical parlance fatalism is more commonly taken to be some version of the view that we are powerless to act in any other way than the way we in fact act (especially when this view is urged on the basis of purely logical considerations, without reference to causation or natural laws), rather than of the view that our actions do not themselves affect the course of history, including such aspects of that course as who gets killed in a given bombing raid (Cahn [1967].) We will move to a consideration of fatalism in that more proper sense after reviewing 6.21.3.


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To recall the example: someone trying to talk you out of taking cover in an air raid reasons as follows. (i) Either you are going to be killed by a bomb or your aren’t. (ii ) Suppose you are. Then any precautions you take will be ineffective. (iii) Suppose you are not. Then any precautions you take will be superfluous. (iv ) Therefore, either way, it is pointless to take precautions. From a formal point of view, we don’t quite have the structure of an (∨E) argument here, but, to fix that, let us grant that precautions count as pointless when they are either ineffective or superfluous. (Or else: simply rewrite (iv) as a disjunction.) Of course, to reject the above reasoning, we might, instead of abandoning argument by cases, dissent from step (i), step (ii), or step (iii). These reactions are not all on a par. To reject step (i) would, no less than rejecting the pattern of inference now under discussion, involve a repudiation of classical sentential logic. Just this move was made, for essentially the reasons now before us, by Łukasiewicz, as we saw in 2.11. But there is a more obvious and more logically conservative response available: namely to reject (iii). (Ayer [1963], Dummett [1964]; the less conservative, (∨E)-rejecting response is described in 6.23. For a different response, see Section V of Stalnaker [1975].) Someone as yet unpersuaded of the pointlessness of taking cover will of course be someone who thinks that taking cover will—or may well (there being some uncertainty as to exactly where the bombs will fall)—result in being saved. Taking this to be your position, as the addressee of the above reasoning, you will precisely deny that, on the assumption that you are not killed in the raid, such precautions as you take will be superfluous: it will be because you take precautions that you aren’t killed! Thus any plausibility (iii) initially seemed to have, it owed to be being interpreted along the lines of (iii) Suppose you are bound not to be killed, regardless of what you do. Then any precautions you take will be superfluous. This step would be unobjectionable, but then to reconstruct the reasoning in accordance with (∨E), (i) would have to be re-written as: (i) Either you are going to be killed by a bomb, or your aren’t going to be killed regardless of what you do. This of course is no longer an instance of the law of excluded middle, and it is not something anyone inclined to place any reliance on taking cover as a way of avoiding getting hit has been given reason to accept. If the motivation to avoid fatalism in this popular sense is insufficient to cast doubt on (∨E), we may ask after a similar motivation in respect of fatalism in the more philosophical sense distinguished above. To formulate this doctrine, we need a necessity-like notion, to be written ‘’, and for which a suitable informal reading (from Prior [1967], on which the present discussion draws) is “it is now unpreventable that”. What is unavoidably the case at a given time for a given agent is whatever ends up being the case regardless of what that agent does as of the time in question. Let the agent be you, and suppose you are considering whether to accept the offer of a friend to look after (and have the use of) her house while she is on vacation next month. Abbreviate the statement that you

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do accept this offer (tomorrow, say, when you are due to say yes or no) to: A. The fatalist claims that you are not free in respect of whether or not to accept the offer; more formally, the claim is that either A or ¬A is now true. The reasoning is given by: Example 6.22.7 (i) Either A or ¬A. (ii) Suppose A. In that case it was true yesterday that A. Abbreviate the last statement as B. Since B pertains to the past, and you cannot now do anything about the past, B. (iii) But as well as being entailed by A, B entails A. So (assuming has a normal modal logic), B entails A. Therefore A (from (ii)), and hence A or ¬A. (iv ) Alternatively, suppose ¬A. Then, reasoning as in (ii), we have: ¬B. (v) From (iv), in the same way as we obtained (iii) from (ii), we obtain ¬A, and hence A or ¬A. (vi ) Thus, since we have our conclusion A or ¬A on the assumption that A and also on the assumption that ¬A, we conclude that A or ¬A. Does this argument establish that it is not up to you now whether or not to accept the offer tomorrow? Its conclusion certainly rules out its now being up to you, since if A is now true then nothing you do can make it the case that ¬A and if ¬A is true, there is nothing you can do to make it the case that A. Rather than querying the role of (∨E), we may note that this reasoning seems problematic enough in alleging that A follows from A, and ¬A from ¬A: never mind what happens once these concessions have been made. In 6.23 we will consider a different response, according to which this is not taken as problematic, and the pressure is placed back on (∨E). But for the moment, let us go along with the idea that A simply does not follow from the assumption that A (and likewise for ¬A and the assumption that ¬A). This also makes redundant Łukasiewicz-style worries about step (i) in the above argument. It concentrates our attention on (ii) and (iii) (or, in the case of ¬A, (iv) and (v)). Let us consider these steps, then. Although -statements are prone to variation in truth-value with the passage of time, since what was not unpreventable yesterday morning (e.g., that you would sleep last night on a camp bed) may be so this morning (if you are now awakening after a night on just such a bed), the A and B of 6.22.7 are not intended to interpreted as thus temporally variable. Imagine these spelt out in English with specific dates rather than words like tomorrow and yesterday. In this case, A has the form “You accept the offer at time t2 ” for the specific time t2 twenty-four hours from now, with t1 for the present moment, and t0 for the time twenty-fours hours before t1 . Thus B is the statement that at t0 it was (or “is”, understood tenselessly) true that you (would) accept the offer at t2 . We imagine the argument of Example 6.22.7 presented at t1 , so that in one rather obvious sense the claim in step (ii) that “B pertains to the past” is correct, since in spelling out the content of B as above – at t0 it was the case that A – explicit reference is made to a time which is in the past, as of t1 . The simplest way of defusing the above argument is to reject the subsequent inference in (ii)


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to the conclusion that from the truth and the ‘past-pertainingness’ in this sense of B, it follows that B. The strategy (which goes back to William of Ockham) is to claim that such an inference is only legitimate for a much stronger sense of “pertaining to the past”, which we could baptize as “being genuinely about the past”. The distinction just drawn can be illustrated with the following example (White [1970]): while the claim that a certain person, a, was stabbed yesterday, and the claim that a was fatally stabbed yesterday, both pertain to the past, only the former is genuinely about the past, since whether or not yesterday’s stabbing (supposing there was one) counts as fatal depends on whether or not a subsequently dies from it. If a is even now losing blood while waiting for medical attention, a doctor can make it the case that the stabbing was non-fatal by giving the needed attention, and can make it the case that the stabbing was fatal (we may suppose) by withholding that attention. Say the doctor does not help, and a dies. In that case it is true that a was fatally stabbed yesterday, but is it is not true that it is now-unpreventably the case that a was fatally stabbed yesterday, since the doctor could (although ex hypothesi she won’t) save a’s life: and doing this would prevent its being the case that a was yesterday fatally stabbed. What really is ‘over and done with’ or ‘now unpreventably the case’ is the genuinely past fact that a was stabbed: unless we were suppose that backward causation is possible, there is nothing anyone could now do which would make this not be the case. With this distinction at hand, it does not take much to see that B in the above argument – the statement that it was the case at t0 that you would accept the offer at t2 – is like the case of a’s having been fatally stabbed, and not like the case of a’s having been stabbed. Whether B was true at the time t0 to which it ‘pertains’ depends on what you later freely do when it comes to t2 , in the same way that whether or not the stabbing was fatal depends on what the doctor later freely does. Although, as of t1 , B is true and pertains to the past, B is not genuinely about the past, and its unpreventability does not (even with the help of the assumption that there is no backward causation) follow from its truth. Step (iv) errs in the same way, the diagnosis proceeding by substitution of ¬B for B in the above. This (entirely standard) response to Example 6.22.7 shows that there is really no need to question either the law of excluded middle or the principle of inference by (∨E). Of course a fuller exposition would say more about the matter of being genuinely about a time (or set of times); for an explication of this notion in terms of possible worlds, see Section V of Lewis [1988]. Sometimes in the literature, truths genuinely about past times are described as expressing ‘hard facts about the past’. (Adams [1967], Wideker [1989].) This description combines the element of genuine pastness with the element of unalterability consequent thereon according to the view that there is no backward causation (or: that the past is fixed – to put it more colloquially). One aspect of the conception of time involved in this response to 6.22.7 has given some thinkers pause. An alternative conception, that of ‘branching time’, has arisen in reaction to the misgivings in question, and (whether or not we share those misgivings) an exploration of this conception will provide us with an opportunity to examine an important technique: the use of supervaluations. This idea is closely bound up with a repudiation of (∨E), and it has applications ranging far beyond that most immediately prompting its inclusion here.

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6.23

Supervaluations

Valuations, in Chapter 1, are (total) functions from sets of formulas to the set {T, F} of truth-values. By a partial valuation we mean a partial function (from the set of all formulas of whatever language is under consideration) to {T, F}. Certain three-valued logics – arising out of the Kleene matrices from 2.11 – may naturally be thought of in these terms, with (what we called) evaluations into the three-element set of values as partial valuations, the third value representing the ‘undefined’ case. (See Blamey [1986]/[2002].) Here we make a rather different use of the idea. Following van Fraassen (and others: see notes), we say that a partial valuation S is the supervaluation induced by V, for some set V of (total) valuations when: ⎧ ⎪ ⎨T S(A) = F ⎪ ⎩ undef ined

if v (A) = T for all v ∈ V if v (A) = F for all v ∈ V otherwise.

To indicate in the notation that S is the supervaluation induced by V in the above manner, we write S as SV . Van Fraassen’s motivation for introducing supervaluations was as a way of dealing with ‘truth-value gaps’ (failures of bivalence, in the sense of cases in which a statement is neither true nor false). If we start with a partial valuation, we can consider all possible total valuations which extend it, in the sense of agreeing with it in the ‘defined’ cases, and filling in (with T or F) the ‘undefined’ cases. Call this set of valuations V (S), where S is the partial valuation concerned. (This double use of “V ” should not give trouble.) Exercise 6.23.1 Show that for any partial valuation S, S = SV (S) . From 6.23.1, it follows that we can recover S from V (S), in the sense that if V (S) = V (S ), then S = S . The mapping from V to SV is not similarly injective: Exercise 6.23.2 (i) Given an example of V , V with V = V but SV = SV . (ii ) Show that if S = SV then V ⊆ V (S). A famous example of the application of this idea of supervaluations is to vagueness (Lewis [1972], Fine [1975], Kamp [1975]). If we think of a vague predicate such as “is tall” and (the name of) an individual, a, which is a borderline case of that predicate, then we may want to say that the statement “a is tall” is neither true nor false, and that the same goes for its negation. Thus our assignment of truth-values is only partial. But corresponding to each possible way of making “(is) tall” precise we obtain – pretending for a moment that this is the only vague (atomic) predicate in the language – a total valuation. “Possible” here means: possible compatibly with what tall means, rather than: logically (or combinatorially) possible. For instance if a is taller than b, a precisifying total valuation should not count “b is tall” true unless it counts “a is tall” true. Thus if S is our original partial valuation, we are not interested in V (S) but in a subset thereof, usually described as comprising the admissible precisifications.


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(See the references cited for more detail.) Still, we can make sense of this smaller set of total valuations, and calling it V , note that our original S is SV . What becomes of the logic of vagueness on this proposal? For partial valuations, no less than for valuations proper, the definition of what it is for a sequent to hold, given in 1.25, continues to make sense: Γ A holds on S iff: if S(C) = T for all C ∈ Γ, then S(A) = T. Here we have treated the Set-Fmla case; for Set-Set, where the sequent concerned is Γ Δ, we add to “S(A) = T” the words, “for some A ∈ Δ”. Although traditionally (e.g., with Łukasiewicz) a rejection of the principle of bivalence and a renunciation of the law of excluded middle were felt to go together, as long as the valuations in V are all boolean, then any sequent of the form A ∨ ¬A turns out to hold on SV . For this just means that every v ∈ V assigns the value T to A ∨ ¬A. In terms of the application to vagueness, this means that although we count neither “a is tall” nor “a is not tall” as true, we count the disjunction “Either a is tall or a is not tall” as not only true, but logically true, since wherever we draw the line between the tall and the not tall (however we precisify the predicate tall ) a falls on one or other side of that line. More generally, not only the law of excluded middle, but any tautologous sequent of Set-Fmla, holds on every supervaluation SV induced by a class V of boolean valuations. The easiest way to see this is to note explicitly at this point—as the reader has probably already noticed—that for a class of valuations V , there is an intimate connection between the supervaluation SV and what was introduced in 1.14 as the conjunctive combination (V ) of all valuations in V . Unlike SV , the latter is an ordinary total valuation, assigning T to formulas to which every valuation in V assigns T, and F to the others. So the only difference between (V ) and SV arises for formulas A with SV (A) undefined: these are assigned F by (V ), along with those for which SV (A) = F. But the above definition of what it is for a sequent to hold on a partial valuation is in terms of when various formulas receive the value T. Thus any sequent holds on SV iff it holds on (V ). The above claim that for Set-Fmla, any tautologous sequent holds on SV for V ⊆ BV therefore follows from Theorem 1.14.5, stating that consistency with a consequence relation is passed from a collection of valuations to their conjunctive combination. As this reminder of material from Chapter 1 forewarns us, even if classical logic in Set-Fmla remains intact when we turn our semantic attention from the property of holding on every boolean valuation to the property of holding on every supervaluation induced by a class of boolean valuations, the same will not be so for Set-Set. The sequent p, ¬p does not enjoy the latter property, for example, and nor, in general, do sequents of the form A ∨ B A, B. We can easily have SV (A ∨ B) = T without SV (A) = T or SV (B) = T, since the valuations in V not verifying A might all verify B. The point just made about Set-Set translates into Set-Fmla as a point— not about sequents—but about sequent-to-sequent rules, and in particular the rule (∨E) with which this whole section is concerned. For whereas this rule has what in 1.25 we called the ‘local’ preservation characteristic of preserving the property of holding on an arbitrary boolean valuation, it does not preserve the property of holding on the supervaluation SV induced by a class V of boolean valuations. (Of course, it is more specifically the ∨-booleanness of valuations that is at issue here.) When we turn to the more ‘global’ preservation characteristic afforded by the supervaluational semantics, how things turn out depends

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on what else is in the language. The global characteristic we have in mind here is that of preserving validity in the following sense – which for explicitness we baptise as ‘supervalidity’: a sequent σ is supervalid if σ holds on every supervaluation SV for V ⊆ BV. Since, as already noted, the supervalid sequents are precisely the tautologous sequents, a rule preserves supervalidity iff it preserves tautologousness. But it is natural to consider additional connectives forming compounds whose truth-values on the underlying bivalent valuations depend – very much in the style of modal logic – on what is happening at other bivalent valuations. For example, relative to a consideration of acceptable precisifications of vagueness as given by the class V of valuations, we might introduce an operator for “it is determinately the case that” with the stipulation that v(A) = T iff for all v ∈ V , v (A) = T. (The “” notation is used because the consequence relation with which we shall be concerned is -normal in the sense of 2.23, p. 291.) Understanding by supervalidity the property of holding on every supervaluation SV with this last constraint (as well as that V ⊆ BV ) in force, we have the following: (1) All sequents of the form A A are supervalid. Thus in particular, each of p p and ¬p ¬p is supervalid, as of course is p ∨ ¬p, whereas (2) The sequent p ∨ ¬p is not supervalid. Thus the rule (∨E) does not preserve supervalidity in the present language (and with the above stipulation concerning in force). Nor, though it is our special concern here, is (∨E) the only rule of Nat so affected: Exercise 6.23.3 Show that the rule (→I) does not preserve supervalidity, as currently understood. (Hint: put p for A in the schema appearing in (1) above, and consider the result of applying (→I).) We return now to the example (6.22.7) of the argument for fatalism from the preceding subsection. A supervaluational treatment of the idea of the future’s being, unlike the past, ‘open’, was given in Thomason [1970], and it is summarised here. Think of time as branching into the future with different branches representing alternative ways the future might turn out to be, where this “might” reflects an indeterminacy in reality rather than an ignorance on our part as to which future will be the actual future: indeed, the present picture is undermotivated by any conception that there is, from the point of view of the present moment, one actual future. Such structures could of course be regarded as frames for tense logic in the ordinary way, but—after noting some anomalies in so proceeding—Thomason instead relativizes truth to not only a moment of time but a path through the tree (a maximal chain of points). GA is then true at t w.r.t. a path containing t if every later point on that path has A true at it (again, w.r.t. the path in question), and likewise for HA with earlier replacing later. For the ‘weak’ tense operators F and P, change every to some. The boolean connectives are treated in the obvious way: for example, ¬A is true at a point w.r.t. a path iff A is not true at that point w.r.t. that path. This doubly relativized notion of truth gives us our underlying bivalent valuations, and we take the supervaluation induced by the class of such valuations for a given moment t to give us the singly relativized notion of truth at t: truth


833

at t, in other words, is a matter of truth at t relative to all paths through t. The supervalid sequents are those which hold at t in this path-non-relative sense, for each point t in all of the structures concerned. As before, this notion of validity coincides, in Set-Fmla though not Set-Set, with that we should have arrived at by a more conventional route: in this case, by considering only linear frames in the first place (with the usual tense-logical semantics). But as before, the picture changes if an operator such as above is introduced. With the same stipulation as before, which now amounts to counting A true at t w.r.t. a given path iff A is true at t w.r.t every path through t, we can think of as representing the notion “it is now unpreventably the case that”, exactly as in 6.22. With the present understanding of what it takes for a statement to be true at a time, the flaw in the argument (of Example 6.22.7) for fatalism is relocated. On this understanding it is not fallacious—as was charged the discussion of 6.22—to move from something’s being true at a past time to its expressing a genuine fact about the past (and hence a ‘hard fact’ about the past—nothing anyone can do anything about now), since the sequent A A is indeed supervalid. But of course the argument now falters in its appeal to (∨E), the first casualty (as we have seen) of supervaluational semantics. Which diagnosis of the troubles with 6.22.7 one prefers will depend on one’s willingness to apply the principle of bivalence to statements about the future. The diagnosis of 6.22 took this as unproblematic, assuming it to be a determinate (if unknown) fact whether things would turn out one way or another in the future. The Thomason-style treatment presented here is suitable for a view on which no such assumption is made. From the perspective of a time t2 later than the time t0 , one may wish to say that a statement about what would happen at an intermediate moment t1 was either true or false, depending on how things went. This is because the various paths passing through t2 all agree on such details. But, as the view underlying the supervaluational semantics would have it, this is only a kind of ex post facto or ‘valedictory’ truth (Lucas [1989], p. 66): as of t0 , considered in itself and without reference to the particular paths made salient by subsequent developments (those on which t2 lies), there need be no question of either the predictive statement or its negation being true. The comparative merits of this position and the more simple-minded linear time + bivalence position underlying our earlier (6.22) treatment make for an interesting metaphysical discussion, but one it would take us too far away from the ups and downs of (∨E) to enter into. Supervaluations will be mentioned again in 6.46. A clearer formulation of some themes on the truth of statements about the future from Lucas [1989] appears in MacFarlane [2003].

6.24

Distributive Lattices of Theories

In the discussion surrounding (6)–(19) in 6.11, we were concerned with a Galois connection H, Tr between formulas and valuations which we had occasion to point out was perfect neither on the left nor on the right. On the left, specifically, we do not have Γ = H(Tr (Γ)), because Γ is—or, for that discussion was—an arbitrary set of formulas, and not necessarily a Cn-theory, where Cn is the consequence operation H ◦ Tr . Ignoring for the moment the decomposition of Cn into H and Tr, we address the structure of the lattice of Cn-theories in a fixed language (with ∩ and ∪˙ as meet and join, the latter forming the

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closure under Cn of the union), for a given consequence operation Cn on that language. The main feature of interest here is the distributivity of that lattice, which follows for the case of ∨-classical Cn. (The distributivity of the lattice of consequence operations on a given language is another matter, and was touched on in 1.17.2.) Recall that a consequence operation and a consequence relation are mutually associated by [Def. ] and [Def. Cn] (1.12); the first point to be established is most easily put with the “” notation. Exercise 6.24.1 Show that if is an ∨-classical consequence relation, then for any sets Γ1 , Γ2 of formulas and any formula A of the language of : Γ1 , Γ2 A ⇒ Γ1 , {C ∨A | C ∈ Γ2 } A. The solution to the above exercise requires (T)+ and what in the style of 1.19 we could call ∨E -classicality. The condition inset above may be replaced by one in which Γ2 is restricted to being {C} for some C: compare here Exercise 3.32.4(iii) (on p. 492) and the Digression following that Exercise. This simplification is not exploited here since it relies on the set Γ1 of ‘side formulas’, and below (with the rule (ρ)) we shall be requiring instead that Γ1 = ∅. (In fact, while on the subject of 3.32, we should note that the condition in Exercise 6.24.1 is equivalent to saying that ∨ is monotone with side formulas in its first position and also such that A ∨ A A for all A; we return to the latter point in 6.24.3(i).) Recalling from 0.13 that the half of the distributive law: a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) is automatically satisfied by all lattices, our attention here is on what corresponds to the -direction, which is to say, on: Γ ∩ (Δ ∪˙ Θ) ⊆ (Γ ∩ Δ) ∪˙ (Γ ∩ Θ). We wish to show that this inclusion holds for all Cn-theories Γ, Δ, Θ, provided that Cn (equivalently: the associated consequence relation ) is ∨-classical. To that end, suppose that A ∈ Γ ∩ (Δ ∪˙ Θ), i.e., (*)

A ∈ Γ and (**)

Δ, Θ A.

Putting Γ1 = ∅ and Γ2 = Δ ∪ Θ in 6.24.1, from (**) we infer, on the assumption that is ∨-classical: {C ∨ A | C ∈ Δ}, {C ∨ A | C ∈ Θ} A But since Γ and Δ are both -theories for ∨-classical (this time by ‘∨I classicality’), {C ∨A | C ∈ Δ} ⊆ Γ ∩ Δ: for A ∈ Γ (our supposition (*)) and the C’s in question were all in Δ. Similarly, {C ∨A | C ∈ Θ} ⊆ Γ ∩ Θ. Thus A ∈ (Γ ∩ Δ) ∪˙ (Γ ∪ Θ), completing the proof of: Theorem 6.24.2 The lattice of -theories for an ∨-classical consequence relation is distributive.


835

Since this argument uses only a restricted form of the principle cited in 6.24.1 (the restriction being that Γ1 = ∅), we pause to examine the combined effect of this (restricted) principle, taken in rule form, and with “∨” replaced by “⊕” to avoid confusion: (ρ)

A1 , . . . , An B A1 ⊕ B, . . . , An ⊕ B B

alongside (∨I), which for present purposes we may take as the pair of schemata, again replacing “∨” by “⊕”: (⊕I)1

AA⊕B

(⊕I)2

BA⊕B

along with the Set-Set structural rules (R) (M) and (T). (Because of our conventions with Set-Set, we have only finitely many formulas to the left of the “” in the rule (ρ); this restriction is not essential to the point of the discussion to follow, which could be reconstructed in a framework allowing arbitrarily many formulas to the left, or alternatively, couched entirely in terms of conditions on consequence relations, as in the formulation in 6.24.1.) In view of the argument given above for the distributivity of the lattice of -theories for ∨-classical , the only properties of ∨ which the reference to ∨classicality appeals to are the properties ascribed to ⊕ by the rules (ρ), (⊕I)1 , and (⊕I)2 , together with the structural rules. (This discussion will occupy us up to 6.24.12 – p. 839 – below.) The first question that naturally arises is whether the consequence relation associated with any proof system closed under those rules treats ⊕ ∨-classically. If the answer to this question is negative, then this shows that ∨-classicality was a stronger condition than we needed to employ, not only for the distributivity property recorded in 6.24.2 to hold, but even for the very argument given for that result to go through. We shall show that this negative answer is indeed correct, by showing (6.24.8) that the sequent p ⊕ q q ⊕ p is not provable in the proof system presented by those rules. (Thus the full effect of (∨E), for ⊕ rather than ∨, is not made available by (ρ), even when taken in conjunction with the introduction rules.) We call the proof system in question the ⊕-system. For familiarization: Exercise 6.24.3 Where is the consequence relation associated with the ⊕system, show that: (i) p ⊕ p p

(ii ) is left-prime.

(Hint for (ii ): Extend the style of argument suggested in the Hint to 1.21.5(i), from p. 110.) Observation 6.24.4 If we add ∨ to the language, and (∨I), (∨E) to the above basis for the ⊕-system, then A ∨ B A ⊕ B for all formulas A, B, where is the consequence relation associated with the extended proof system. Proof. By (∨E) and (⊕I)1 , (⊕I)2 , we get A ∨ B A ⊕ B. Using the n = 1 form of the rule (ρ), which allows passage from A B to A ⊕ B B, we obtain A ⊕ B A ∨ B from A A ∨ B, which is itself provable by (∨I); therefore A ⊕ B A ∨ B. (Note that only the (∨E)res force of (∨E) has been exploited

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here, and only (∨I) from the first disjunct.)

We obtain the promised result (unprovability of p ⊕ q q ⊕ p) by a consideration of the poset whose Hasse diagram appears as Figure 6.24a.

1 ? ??? ?? ?? 2 3 OOO o OOO ooo o Oo oooOOOOO o o O oo 4

5

Figure 6.24a

This poset is not a lattice. More to the point, since the only connective to be interpreted is disjunction-like rather than conjunction-like, it is not even a join (or ‘upper’) semilattice, in that not every pair of elements have a least upper bound. (From a purely algebraic point of view there is no difference between join semilattices and the dually defined meet – or ‘lower’ – semilattices: the equations satisfied by either class are just those defining semilattices in 0.21.) Although every pair of elements have an upper bound, for the pair 4, 5 there is no least upper bound, each of 2 and 3 qualifying only as a ‘minimal’ upper bound (an element a to which no other upper bound of 2 and 3 bears the partial order relation ). 6.24.4 explains why we need to have at least one pair of elements without an l.u.b.: otherwise the operation we introduce to interpret ⊕ will collapse into the join operation, and we certainly won’t be able to invalidate p ⊕ q q ⊕ p. We denote the operation which will be used to interpret the connective ⊕ by “⊕” again, for simplicity. Whenever elements a, b, of the poset depicted in Figure 6.24a have a least upper bound, we stipulate that a ⊕ b is to be that l.u.b.; this leaves out, as already remarked, only the pair 4, 5, and for these we stipulate that 4 ⊕ 5 = 2, while 5 ⊕ 4 = 3. A tabular representation of the operation thus defined appears in Figure 6.24b. ⊕ 1 2 3 4 5

1 1 1 1 1 1

2 1 2 1 2 2

3 1 1 3 3 3

4 1 2 3 4 3

5 1 2 3 2 5

Figure 6.24b

This table is not a matrix in the sense of many-valued logic, even though there are no other connectives to deal with (in the language of the ⊕-system),


837

because no indication – as by asterisking – has been given of designated values. This is because we have in mind the -based algebraic semantics of 2.14. In that discussion, we defined a sequent A B of Fmla-Fmla to hold on an evaluation or homomorphism h just in case h(A) h(B); since the only connective involved at present is ⊕, the condition that a mapping from the language to the algebra just defined is that h(C ⊕ D) = h(C)⊕h(D) for all formulas C, D. (Here ⊕ on the left is the connective, on the right, the operation defined in Figure 6.24b.) Our current framework is Set-Fmla, however, so we need a more general account of holding: A1 , . . . , An B holds on h just in case for all elements a of the algebra, if a h(Ai ) for each Ai (i = 1, . . . , n) then a h(B). Note that when n = 1, this works out to be equivalent to the Fmla-Fmla definition – a point of some importance since although we are in Set-Fmla, the difference from Fmla-Fmla is not actually exploited in view of 6.24.3(ii). As in 2.14, a sequent is valid (in an algebra) just in case it holds on every homomorphism (into that algebra). We want to show that every sequent provable in the ⊕-system is valid in the five element algebra described above. Lemma 6.24.5 For any elements a, b of the algebra described: (i)

a a ⊕ b and b a ⊕ b.

(ii) If a b then a ⊕ b b, and hence, by (i), a ⊕ b = b. Proof. Immediate from the way ⊕ was defined. In particular, for (ii), note that ⊕ gives the l.u.b. wherever this exists, as it always does for a, b, with a b. Let the reduced ⊕-system be the system whose basis is as for the ⊕-system above, except that only the rule (ρ) is restricted to the n= 1 case. Exercise 6.24.6 Show that the same sequents are provable in the ⊕-system as in the reduced ⊕-system. (Hint: Use 6.24.3(ii ).) Theorem 6.24.7 Every sequent provable in the ⊕-system is valid in the five element algebra with one binary operation as in Figure 6.24b, using the relation of Figure 6.24a to judge -based validity. Proof. By 6.24.6, it suffices to give the demonstration (by induction on the length of the shortest proof of the sequent in question) for the reduced ⊕system, as presented above. (⊕I)1 and (⊕I)2 have only formulas valid on this algebra amongst their instances, by 6.24.5(i), and (ρ) preserves the property of holding on a given homomorphism (and a fortiori preserves validity in the algebra) by 6.24.5(ii)): note that since we are in the restricted proof system, we are speaking of the n = 1 form of (ρ) here.

Corollary 6.24.8 The sequent p ⊕ q q ⊕ p is not provable in the ⊕-system. Proof. Putting h(p) = 4, h(q) = 5, gives h(p ⊕ q) = 2 while h(q ⊕ p) = 3; since we do not have 2 3, for our chosen relation , the sequent does not hold on h and so by 6.24.7 is not provable.

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Recall that our interest in the sequent whose unprovability is noted at 6.24.8 was in showing that the full effects of (∨E) are not made available by having, alongside (∨I), a rule for ∨ like the rule (ρ) above; thus an assumption weaker than the ∨-classicality of a consequence relation is actually required for the proof of the distributivity of the lattice of -theories. We could have used associativity, rather than commutativity, as an example of something underivable in the ⊕-system, since (4 ⊕ 5) ⊕ 3 = 2 ⊕ 3 = 1, which does not stand in the relation to 4 ⊕ (5 ⊕ 3) = 4 ⊕ 3 = 3: hence, the sequent (p ⊕ q) ⊕ r p ⊕ (q ⊕ r) is also unprovable, by 6.24.7. Before we leave it entirely behind us, there is a little more philosophical interest that we can extract from this discussion of the ⊕-system – specifically on the matter of Observation 6.24.4 above, which tells us that when disjunction is present (with the familiar inferential properties appealed to in the proof of that Observation), then ⊕ collapses into it. In the first place, this gives us a rather interesting case of non-conservative extension: Example 6.24.9 The rules (∨I) and (∨E) give a non-conservative extension of the ⊕-system. For in the unextended system, we cannot prove the ∨-free sequent mentioned in 6.24.8, though with them (via 6.24.4) we can. We have already encountered cases in which a connective with certain inferential powers collapsed into a connective with stronger inferential powers, namely in the discussion (4.32) of cases in which the former inferential powers were already sufficient to characterize a connective uniquely, and the latter accordingly fell into the ‘stronger than needed’ category. This brings us to the second point of interest: the case of the ⊕ rules is not like those cases, in that the weaker collection of rules (the ⊕ rules) does not suffice for unique characterization, so the stronger collection (those for ∨) do not—on this score—qualify as stronger than needed. (See further the final parenthetical comment in the proof of 6.24.4.) Observation 6.24.10 The rules (⊕I)1 , (⊕I)2 , and (ρ) do not characterize ⊕ uniquely. Proof. We must show that in the proof system for a language with two binary connectives ⊕ and ⊕ , satisfying (alongside the structural rules for Set-Fmla) respectively {(⊕I)1 , (⊕I)2 , (ρ)} and reduplicated forms thereof for ⊕ , {{(⊕ I)1 , (⊕ I)2 , (ρ )}, the sequent p ⊕ q p ⊕ q is not provable. Obtain a table for an operation ⊕ to associate with (the connective) ⊕ from Figure 6.24b by duplicating the given table for ⊕ except for the entries for the l.u.b.-less pair 4, 5; pairs 4, 5 and 5, 4 ; instead, for these cases, put 4 ⊕ 5 = 3, 5 ⊕ 4 = 2 (reversing, in other words, the arbitrary decision made in the case of ⊕). For the remainder of the argument, proceed as with 6.24.7 and 6.24.8. (Note that the partial order remains as before, and that since a ⊕ b = b ⊕ a, the h given in 6.24.8 invalidates p ⊕ q p ⊕ q.) Recalling a notation introduced in 4.22.10 (p. 555), let us say that a con¨ (primitive or otherwise) in its sequence relation with a binary connective ∨


839

¨ -intuitionistic if A is a consequence of Γ according to that relation language is ∨ whenever A is a consequence of Γ according to IL , where A is obtained from ¨C A, and the formulas of Γ from those of Γ, by replacing any subformulas B ∨ by (B → C) → C. ¨ , the rules (ρ), (⊕I)1 , Exercise 6.24.11 Check that when “⊕” is interpreted as ∨ (⊕I)2 all preserve IL-provability. ¨ is sufficiently disjunctionThis shows that our intuitionistic pseudo-disjunction ∨ like for an analogue of 6.24.2 to go through, via our analysis above of which features of disjunction are actually exploited in the proof; we have added a restriction to finitary (sometimes called ‘Tarski’) consequence relations to avoid some complications: this way the following is an immediate consequence of 6.24.11. ¨ -intuitionistic finitary consequence relaTheorem 6.24.12 The lattice of all ∨ tions is distributive. It is clear that from 6.24.2 the lattice of all →-classical consequence relations is distributive, since we can define A ∨ B as (A → B) → B; in an intuitionistic ¨ B. But we have just seen context, this definition leads only to the weaker A ∨ that this still allows the distributivity argument to go through; hence: Corollary 6.24.13 The lattice of all →-intuitionistic finitary consequence relations is distributive. (Recall from 2.33 that is →-intuitionistic just in case for all Γ, A, B, we have Γ, A B iff Γ A → B.) This concludes our exploration of the features of disjunction used in establishing 6.24.2. An example of a consequence relation for which the lattice of -theories is not distributive—by contrast with what 6.24.2 above shows for the case of every ∨-classical consequence relation—is given in van Fraassen [1971], p.176, where it is remarked that for what van Fraassen calls ‘bivalent propositional languages’, the distributivity of the corresponding lattice is established by a proof of Beth [1959], on p. 549. (We capture the intent of van Fraassen’s phrase here with a formulation in terms of classicality at the start of the following paragraph.) Beth’s proof is in several ways instructive and we devote the remainder of the present subsection to it. What Beth is proving is that for any -theories Γ, Δ, Θ, where is {∧, ∨, ¬}classical, we have Γ ∩ (Δ ∪˙ Θ) = (Γ ∩ Δ) ∪˙ (Γ ∪ Θ) (Obviously we could replace the reference to {∧, ∨, ¬} by one to {∨, ¬} or {∧, ¬} in the above statement, which is in any case overconditioned in view of 6.24.2.) This is our notation rather than Beth’s. As before, we use “Δ ∪˙ Θ” (etc.) to denote Cn(Δ ∪ Θ), where Cn is the consequence operation associated with . Beth’s proof makes heavy use of the Galois connection (H, Tr ) with which we were concerned in 6.11, except that we now restrict our attention to the closed sets on the left (Cn-theories, for Cn = Tr ◦ H) and on the right (classes of valuations closed under the operation H ◦ Tr ). Translated into this notation, Beth exploits the fact that this is a perfect Galois connection, to establish the equality inset above by showing that

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H(Γ ∩ (Δ ∪˙ Θ)) = H((Γ ∩ Δ) ∪˙ (Γ ∪ Θ)). The steps in Beth’s proof – justified below – are as follows; to reduce parentheses, we omit them when a single Greek letter is acted on by “H”, writing “HΓ” (etc.) for “H(Γ)”: (1) (2) (3) (4) (5) (6) (7)

H(Γ ∩ (Δ ∪˙ Θ))

= = = = = = =

H(Γ ∩ Tr (H((Δ ∪ Θ))) H(Tr (HΓ ∪ H(Δ ∪ Θ)) H(Tr (HΓ ∪ (HΔ ∩ HΘ))) H(Tr (HΓ ∪ HΔ)) ∩ H(Tr (HΓ ∪ HΘ)) H(Γ ∩ Δ) ∩ H(Γ ∩ Θ) H(Tr (H((Γ ∩ Δ) ∪ (Γ ∩ Θ)))) H((Γ ∩ Δ) ∪˙ (Γ ∪ Θ)).

While line (1) and the transition from (6) to (7) simply unpack the definition ˙ some of the other steps are less obvious, making it—in particular—hard of “ ∪”, to see where the ∨-classicality of Tr ◦ H comes into Beth’s argument. Spelling out in more detail the move from (1) to (2), we have: (1) (1a) (2)

H(Γ ∩ (Δ ∪˙ Θ))

= H(Γ ∩ Tr (H((Δ ∪ Θ))) = H(Tr (HΓ) ∩ Tr (H((Δ ∪ Θ))) = H(Tr (HΓ) ∪ H(Δ ∪ Θ)).

in which the passage from (1) to (1a) is justified by the fact that Γ = Tr (HΓ), and from (1a) to (2) by the general Galois identity 0.12.3 (p. 4), as is the step from (2) to (3). This identity also justifies the passage from (4) to (5); in more detail: (4) H(Γ ∩ (Δ ∪˙ Θ)) = H(Tr (HΓ ∪ HΔ)) ∩ H(Tr (HΓ ∪ HΘ)) (4a) = H(TrHΓ ∩ TrHΔ) ∩ H(TrH Γ ∩ TrH Δ) (5) = H(Γ ∩ Δ) ∩ H(Γ ∩ Θ). Here 0.12.3 applies in the move from (4) to (4a), with (5) then following from the identity of Γ with Tr (H(Γ)), and similarly for Δ. We amplify the transition from (5) to (6) thus: (5) H(Γ ∩ (Δ ∪˙ Θ)) (5a) (6)

= H(Γ ∩ Δ) ∩ H(Γ ∩ Θ) = H(Tr(H(Γ ∩ Δ) ∩ H(Γ ∩ Θ))) = H(Tr(H((Γ ∩ Δ) ∪ (Γ ∩ Θ)))).

(5a) follows from (5) by 0.12.10 (p. 6); the passage to (6) is then via 0.12.3 again. A slight variation on this part of the argument is available thus, where (5b) is obtained by 0.12.3, whence (6) follows by 0.12.2: (5) H(Γ ∩ (Δ ∪˙ Θ)) = H(Γ ∩ Δ) ∩ H(Γ ∩ Θ) (5b) (6)

= H((Γ ∩ Δ) ∪ (Γ ∩ Θ)) = H(Tr (H((Γ ∩ Δ) ∪ (Γ ∩ Θ)))).

All transitions in Beth’s derivation have now been justified by general principles concerning perfect Galois connections except for one: that from (3) to (4). It is here, accordingly, that we expect to have to appeal to the hypothesis that Tr ◦H is ∨-classical. Under magnification, we represent the transition in question thus:


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(3) H(Γ ∩ (Δ ∪˙ Θ)) = H(Tr (HΓ ∪ (HΔ ∩ HΘ))) (3a) = H(Tr ((HΓ ∪ HΔ) ∩ (HΓ ∪ HΘ))) (4) = H(Tr (HΓ ∪ HΔ)) ∩ H(Tr (HΓ ∪ HΘ)). (3a) rewrites (3) using the distribution law for the boolean lattice of sets, and we could pass from (3a) to (4) by 0.12.10 if only we could write HΓ ∪ HΔ in the form HΞ (i.e., H(Ξ)) for some set Ξ of formulas, and likewise for HΓ ∪ HΘ (though it is not suggested that we need the same set Ξ for this case). Let us concentrate on the case of HΓ ∪ HΔ. It would be a mistake to appeal to 0.12.5 (p. 5) to argue that we must have HΓ ∪ HΔ = H(Γ ∩ Δ) simply on the grounds that the present Galois connection is perfect. For that relates to the Galois connection (H, Tr ) between closed classes of formulas (alias theories) and closed classes of valuations, under the respective closure operations Tr ◦ H and H ◦ Tr . (See the Warning following 0.12.11.) We have as yet no guarantee that HΓ ∪ HΔ is appropriately closed. Note that a class V of valuations being closed in the sense of being equal to H ◦ Tr (V ), i.e., to H(Tr (V )), is equivalent to V ’s being of the form H(Ξ) for some set Ξ of formulas. (Why?) To justify the transition from (3a) to (4), then, we must show that in our Galois connection (H, Tr ), unions of closed sets on the right (closed sets of valuations) are themselves closed. If HΓ ∪ HΔ is of the form HΞ for some Ξ, then, as suggested by the premature proposal of the preceding paragraph, we can always take Ξ to be of the form Γ ∩ Δ; cf. Remark 6.11.3. (Here we continue to assume Γ, Δ, and Ξ are closed sets from the left of this Galois connection.) For suppose (i) HΓ ∪ HΔ = HΞ; in that case: (ii) Tr (HΓ ∪ HΔ) = TrH Ξ = Ξ. So: (iii) H(Tr (HΓ ∪ HΔ)) = HΞ, and therefore: (iv) H(TrH Γ ∩ TrH Δ)) = HΞ, i.e., (v) H(Γ ∩ Δ) = HΞ; but then by (i): (vi) HΓ ∪ HΔ = H(Γ ∩ Δ). So the one lacuna in our annotated version of Beth’s proof remaining to be filled can be filled if and only if we can establish (vi). We have already given the result whose proof we are now considering, as 6.24.2, with a different proof, and the relevant condition on Cn = Tr ◦ H is of course that this consequence relation is ∨-classical. We want to show (again) that from this hypothesis, it follows that the lattice of Cn-theories is distributive. Now we should recall that the notation “H” is really elliptical for something of the form ‘HV ’, and attend to this subscript position for a moment. A mild reformulation of part of the content of Theorem 1.13.4 (from p. 66) might read: if is an ∨-classical consequence relation then is determined by the class of all ∨-boolean valuations consistent with . Thus even if we are given Cn as Cn = Tr ◦ HV for some V containing non-∨-boolean valuations, we can, given the ∨-classicality of Cn, replace V by U ⊆ BV ∨ , and continue to reason about Cn, this time presented as Cn = Tr ◦ HU . Let us read into all the occurrences of “H” above, then, such a restriction to delivering classes of ∨-boolean verifying valuations. Returning to (vi) with this in mind, let us first observe that the ⊆direction of (vi) is trivial in the sense of holding in arbitrary Galois connections, so that all we need to establish is the ⊇-direction.

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To that end, suppose we have v ∈ H(Γ ∩ Δ) while v ∈ / H(Γ) and v ∈ / H(Δ). Since v ∈ / H(Γ), there is some C ∈ Γ with v(C) = F, and since v ∈ / H(Δ), there is some D ∈ Δ with v(D) = F. As v is ∨-boolean, we therefore have v(C ∨ D) = F. Since Γ and Δ are both Tr ◦H-theories for ∨-classical Tr ◦H, C ∨ D ∈ Γ and C ∨ D ∈ Δ; thus C ∨ D ∈ Γ ∩ Δ and we have a contradiction with the supposition that v ∈ H(Γ ∩ Δ).

Notes and References for §6.2 I am grateful to Frank Jackson for discussions many years ago on problems such as that concerning the prisoners (Example 6.21.1), and to Judi Humberstone for suggesting the proof of Lemma 6.22.2. Theorem 6.22.3 overlaps observations made in some 1940 papers by Bernard Koopman, references to which may be found in Makinson [2010], along with discussion of related issues concerning disjunction and probability. 6.22 also mentions the kinship between that problem and the ‘twin ace’ problem; see Faber [1976], Goldberg [1976], Bar-Hillel and Falk [1982]. The last mentioned paper gives some history of the problem and further references. (Subsequent discussions include Weintraub [1988], Shimojo and Ichikawa [1989], and pp. 165–169 of the popular survey Piatelli-Palmarini [1994].) A fuller discussion than 6.22 provides would address the relation between Pr (C/A) and the degree to which one would believe C on coming to learn that A; one’s account of this relation needs to be sensitive to the morals of Example 6.21.1. Again the Bar-Hillel and Falk paper is a good reference; see also Spielman [1969], Friedman [1975], and §3.3 of Isaac [1995]. An example of fallacious apparently ∨-eliminative reasoning in a probabilistic context even more puzzling than the three prisoners/twin ace problem is provided by what is called ‘Simpson’s Paradox’; see Gardner [1976] for discussion and references. For an apparent failure of dominance-related reasoning (as in the decision-theoretic examples in 6.21), see the discussion in Tversky and Shafir [1992], or the account of the ‘Monty Hall Paradox’ in pp. 162–165 in Piatelli-Palmarini [1994]. This problem is also discussed in Isaac [1995], pp. 1–2, 8–10. For a more recent and more logically motivated discussion of dominance reasoning, see Cantwell [2006]. As well as being of logical and philosophical interest, Fine [1975] provides information about the history of (the use of) supervaluations, with Fine [2008] voicing reservations about the application of this apparatus to vagueness in that earlier paper. Herzberger [1982] is a later technical study, while variations on the theme – with special attention to the application to vagueness – are discussed in Varzi [2007]. See also Belnap [2009]. (Williamson [1994a] was already mentioned in this connection in the notes to §2.1, p. 268.) See also van Fraassen [1971], Chapter III §6, and Chapter 4, esp. §4, of Haack [1974]. The latter discussion concerns the issue of fatalism broached in our 6.22 and 6.23; for more on the treatment of truth as it figures in 6.22’s suggested response to that issue, see White [1970]. White is especially clear on the distinction between (as he would put it) a prediction’s not being true unless what is predicted comes to pass, and its not being true until what is predicted comes to pass. An interesting discussion of the relation between the supervaluational approach and Łukasiewicz’s three-valued response to the issue of future indeterminacies is to be found in Restall [2005a].

6.3. COMMAS ON THE RIGHT

843

The rule of ∨-elimination with which we have been concerned in this section (and will be again through much of §6.4) is (∨E) as formulated in our presentation of Nat in 1.23; namely as: ΓA∨B

Δ, A C

Θ, B C

Γ, Δ, Θ C though we have also had occasion to consider (and will again in 6.47) the quantum-logical ‘restricted’ form (∨E)res of this rule, with Δ and Θ required to be empty. The verbal description of (∨E) in Lemmon [1965a], however, is open to an unintended construal in which the set of right-hand formulas of the conclusion-sequent is to be, not Γ ∪ Δ ∪ Θ (as above), but rather Γ ∪ Δ ∪ Θ {A, B}. (Compare the rule (→I)P ra from 4.12.) So construed, the rule would allow for the proof of non-tautologous sequents, whenever applied in cases in which A or B occurs as an element of Γ. This was pointed out in the final paragraph of Coburn and Miller [1977], and, at greater length, in Pfeiffer [1990], both of which draw attention to possible misinterpretations of Lemmon’s verbal formulation of (∨E); compare also the example given in 7.19.11 below. (Actually, our standard formulation of (∨E) is itself not quite faithful to Lemmon’s conception of the rule, since he thinks of assumptions to be discharged not as formula-types—formulas, that is, in the sense in which a single formula may occur several times in a proof, but as formula-tokens—these very occurrences themselves; this involves a subtle interplay between the frameworks Set-Fmla and Mset-Fmla which we have not attempted to capture in our formulation of Nat; Remark 7.13.2, p. 976, addresses this same point as it arises in a different context.) Investigation of the properties of lattices of -theories (or Cn-theories) on a given language, such as are included in 6.24, was initiated by Tarski in the 1930’s; see Paper XII in Tarski [1956]. Our discussion draws on Chapter 19 of Beth [1959], and §6 of Chapter 2 of van Fraassen [1971]; see also Appendix I there. The discussion of Beth’s argument ([1959], p. 548) as the deduction (1)– (7) is approximate; in place of H as interpreted in our discussion (assigning a set of verifying valuations) H there assigns to a set of formulas the set of models (verifying interpretations) in the sense of first-order logic.

§6.3 COMMAS ON THE RIGHT 6.31

GCRs Agreeing with a Given Consequence Relation

The title of this section refers to the use of commas on the right of the “” in sequents of the logical framework Set-Set, as well as to the commas which appear on the right of the “” in statements of such forms as “Γ A, B” where is a generalized consequence relation. (The intimate association between gcr’s and collections of sequents of Set-Set was given in 1.22.2 (p. 113); we will not be considering “commas on the right” in substructural logics such as the in the sequent calculus for classical linear logic from 2.33.) It is because of the informal advice that such commas should be thought of ‘disjunctively’ that we include this discussion in the present chapter, though as we saw in 6.23 there are perfectly well-motivated gcr’s – in that instance, the gcr associated on the supervaluational semantics with collections of ∨-boolean valuations – in

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which claims of the form Γ A, B and of the form Γ A ∨ B do not amount to the same thing. A further example of considerable independent interest in the same vein will be presented in 6.33, after some background (from modal logic) has been supplied in 6.32. The present subsection picks up the topic of ‘agreement’ which we considered à propos of the relation between Set-Fmla and the less general framework Fmla in 1.29; in fact we put the points in terms of consequence relations and collections of formulas. Here we consider instead the relation between Set-Set, as the more general, and Set-Fmla, as the less general framework. The sense of comparative generality is given by the fact that while every sequent of the more general framework is a sequent of the less general, the converse is not the case. And as before, rather than putting things in terms of these logical frameworks, we prefer to put them in terms of gcr’s and consequence relations. We will sometimes use “” in place of to indicate that it is figuring as a variable ranging over gcr’s rather than consequence relations. The following discussion is based on Scott [1974b]. There is an extended treatment in Shoesmith and Smiley [1978], Chapters 5 and 16. The key concept, already encountered in the discussion after 3.13.9 (p. 391) above, is that of a gcr agreeing with a consequence relation , which we be the case when the language of and is the same and for every set of formulas Γ and formula A of that language Γ A iff Γ A. (We will be equally happy to say, conversely, that agrees with , or simply that and agree, when this is the case.) Thus with a given consequence relation there will typically be various agreeing gcr’s, and their agreeing with that consequence relation is a matter of them agreeing on the question of when a single formula – A above, recalling that we do not bother to write this as “{A}” on the right of a gcr-statement – follows from a set of formulas. (By contast any gcr induces a unique consequence relation as the consequence relation agreeing with it.) The variation between such gcr’s agreeing with might in principle arise over the case of what happens when there are more than one and also when there are fewer than one (i.e., zero) formulas in the sets denoted on the right. In fact the latter type of variation arises only in a rather unusual case, namely when the gcr’s concerned lack the property that L ∅, where L is the language of : Exercise 6.31.1 (i) Suppose that 1 and 2 are both gcr’s agreeing with a consequence relation , with L the language of 1 , 2 , , and that L i ∅ for i = 1, 2. Show that for all Γ ⊆ L: Γ 1 ∅ iff Γ 2 ∅. (ii) Show that if and are respectively the gcr and the consequence relation determined by some fixed class of valuations, then agrees with . Now amongst the various gcr’s agreeing with a consequence relation we may distinguish two of special importance, to be called − and + . (We avoid the notation here to preserve the link to the given .) The former, already encountered briefly before the formulation and in the proof of 3.13.13 (p. 394), we define by: Γ − Δ iff for some A ∈ Δ: Γ A.


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Note that this gcr falls into the ‘unusual’ category just described, since we never have Γ − ∅, even for Γ = L. It does, all the same, agree with : Exercise 6.31.2 (i) Check that − , as just defined, is a gcr. (ii ) Check that − agrees with . (iii) Check that − is right-prime (1.16.9, p. 79). In fact, amongst gcr’s agreeing with , − occupies a special place: Lemma 6.31.3 If is any gcr agreeing with the consequence relation , then − ⊆ . Proof. Suppose Γ − Δ. Then for some A ∈ Δ, Γ A, so, if agrees with , Γ A. The conclusion that Γ Δ follows by (M). For a consequence relation , the following definition Γ + Δ iff for all C and all Γ ⊇ Γ: if Γ , D C for every D ∈ Δ, then Γ C. is reminiscent of the method of common consequences in 3.13, except that there we were concerned to derive conditions on a consequence relation from determinant-induced conditions on a gcr. Exercise 6.31.4 Verify that if is a finitary consequence relation then + is a finitary gcr and that it agrees with . The finitariness restriction in 6.31.4 is essential: the result does not hold if we delete each occurrence of the word “finitary”; an ingenious counterexample is given in Shoesmith and Smiley [1978], p. 75. The problem is that although, on the weaker hypothesis that is a consequence relation, + meets the condition (T), it need not meet the condition (T+ )—these being the conditions mentioned under those names in 1.12. The notation we have used will have led the reader to anticipate what comes next, in spite of the failure of symmetry with 6.31.3: Lemma 6.31.5 If is any gcr agreeing with the finitary consequence relation , then ⊆ + . Proof. Suppose that for agreeing with , Γ Δ and Γ, D C for all D ∈ Δ. We need to show that Γ C. Since agrees with , we have Γ, D C for all D ∈ Δ. From this and the fact that Γ Δ, it follows by (T+ ) that Γ C. Therefore, appealing again to the fact that agrees with , we have Γ C. We sum up the results 6.31.2–5 in: Theorem 6.31.6 For any finitary consequence relation , the relations − and + are, respectively, the weakest and the strongest gcr’s agreeing with .

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As noted in 1.21 (e.g., 1.21.1(i), p. 106), there is a certain natural resistance to regarding the framework Set-Set as belonging to logic proper, on the grounds that argumentation in natural language proceeds from premisses to a single conclusion. Some philosophical discussion of this resistance is given in pp. 4–6 of Shoesmith and Smiley [1978]; Restall [2005b] provides an interesting and sophisticated defence (though arguably vulnerable to objections like those against an earlier proposal of Brian Ellis urged in pp. 65–70 of Sorensen [1988]); see also §7 of Rumfitt [2008], in which note 2 makes the point just alluded to). A brief outline of Restall’s idea can also be found in Beall and Restall [2006], p. 46f.) Theorem 6.31.6 indicates how a consideration of multiple conclusions can arise as an adjunct the study of the single-conclusion case. (In 1.16 and 1.26 we motivated their consideration out of concerns with symmetry vis-à-vis the semantics of boolean valuations; the following vertical/horizontal terminology is from the latter subsection.) The maximal gcr + agreeing with a (finitary) consequence relation is especially interesting in this connection, since it allows for the ‘horizontalizing’ of such ‘vertical’ conditions on consequence relations as ∨-classicality. In particular, recall that aspect of the latter condition requiring that if Γ, A C and Γ, B C then Γ, A ∨ B C. This, understood as universally quantified w.r.t. the variables Γ, A, B, C, is completely equivalent to the -formulation of (∨E) in the form: if Γ A ∨ B and Δ, A C and Θ, B C, then Γ, Δ, Θ C. The horizontal (alias unconditional) gcr condition A ∨ B A, B is equivalent to this condition in the sense given by (i) of the following Exercise. Parts (ii) and (iii) are included for the sake of a concrete example of the phenomena with which we have been dealing; it concerns classical logic. The relation of intuitionistic logic to the distinction between − and + will be commented upon below, in 6.43.8 (p. 899). Part (iv) brings in for subsequent use the matter of consistent valuations; note that (as with “Log”) above, “Val ” here means something different according as whether consequence relations or gcr’s are involved. Exercise 6.31.7 (i) Show that a consequence relation meets the (∨E) condition just described iff A ∨ B + A, B for all A, B. (ii ) Show that if = CL (the tautological consequence relation) then + is the tautological gcr; assume here that the language of contains all boolean connectives. (iii) With as in (ii), show that Γ − Δ if and only if for some finite Γ0 ⊆ Γ, Δ0 ⊆ Δ, the sequent Γ0 Δ0 is supervalid in the sense of 6.23. (iv ) Show that if is any gcr agreeing with a consequence relation , then we have Val() ⊆ Val(). Parts (ii) and (iii) have interesting analogues (discovered by D. Gabbay) for as IL , reported below at 6.43.8 (p. 899). As for part (iv), it is not hard to see that we can reverse the inclusion here for = − : Observation 6.31.8 For any consequence relation , Val(− ) = Val(). Since not only Val (− ), but also Val (+ ), is uniquely fixed given Val (), we can consider the possibility of obtaining an analogue of 6.31.8 for the latter case also (assuming finitary). Parts (i) and (ii) of 6.31.7 tell us, however, that there


847

is no such analogue, since valuations consistent with a gcr such as + CL , satisfying A ∨ B + CL A, B all assign T to at least one disjunct of any disjunction to which they assign T, whereas this is not so for the underlying consequence relation CL itself: taking conjunctive combinations of CL -consistent valuations gives an endless supply of counterexamples to the suggestion that Val (CL ) comprises only valuations with this property. (See 1.14.5, 6.) Let us call valuations with the property in question prime; in other words, prime valuations are those respecting the determinant F, F, F for ∨ (Cf. 3.13.7, p. 390) and a prime set of formulas in the sense of 2.32 is one whose characteristic function is a prime valuation. We will have a look at this property for a moment in connection with those conditions on consequence relations which can be formulated as sequentto-sequent rules in Set-Fmla. The following material will be useful for §6.4. To recall a distinction introduced in 1.25, with a rule ρ and a class of valuations V in mind, we contrast ρ’s having the local preservation characteristic associated with V with ρ’s having the global preservation characteristic associated with V . Supposing ρ to be an n-premiss rule (and understanding rules as in §1.2 rather than with the refinements of 4.33), the former is a matter of satisfying: For all σ1 , . . . , σn ∈ ρ, for all v ∈ V: if σ1 , . . . , σn−1 all hold on v, then σn holds on v. while the latter is a matter of satisfying: For all σ1 , . . . , σn ∈ ρ, if σ1 , . . . , σn−1 all hold on every v ∈ V then σn holds on every v ∈ V . We will treat the latter, global preservation characteristic for ρ = (∨E) in 6.47, here remarking on the local characteristic: Observation 6.31.9 (i) The rule (∨E) preserves the property of holding on a valuation if and only if that valuation is prime. (ii) The same is true for the restricted rule (∨E)res of 2.31. Proof. (i) ‘If’: Obvious. (Compare the proof of 1.25.1, p. 127.) ‘Only if’: Suppose v is not prime, so that for some A, B, we have v(A ∨ B) = T, v(A) = v(B) = F. Then an application of (∨E) of the form A∨B

AC

B C

C in which C is chosen as any formula false on v (e.g., C = A) provides a case in which all premiss-sequents hold on v but the conclusion-sequent does not. (ii) ‘If’: Any application of (∨E)res is an application of (∨E). ‘Only if’: The above application A ∨ B; A C; B C; C of (∨E) is itself an application of (∨E)res . Now, (∨E) is a three-premiss rule, so some reminders are in order about the prospects for zero-premiss rules in this context. For such rules (in any logical

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framework) there is no distinction between the global and local preservation characteristics (associated with a given V ). No zero-premiss rule in Set-Fmla, or collection of such rules, can replace (∨E) in a result such as 6.39.1(i); this follows from 1.26.3, together with the fact that the class of valuations satisfying the (∨I)-securing condition that v(A) = T or v(B) = T implies v(A ∨ B) = T is sequent-definable in that framework. The opening topic for this subsection was the multiplicity of gcr’s agreeing with any given consequence relation – a situation we noted was similar to the fact that there are in general many consequence relations agreeing with a given logic in Fmla in the sense of having as the consequences of ∅ precisely the formulas in that logic, while differing amongst themselves as to the consequences of other sets. Suppose we start with a consequence relation and just consider its deliverances for the empty set Cn (∅). In certain unfavourable cases – the cases of atheorematic – Cn (∅) = ∅ and we have nothing left of our logic as far as the Fmla framework is concerned. The analogous situation can’t happen as we move from Set-Set to Set-Fmla, at least if the familiar structural rules are in place. That is, given a generalized consequence relation , the unique consequence relation agreeing with is never empty, since consequence relations never are: but we would still have lost everything distinctive about , as in the case of passing from an atheorematic consequence relation to a Fmla logic, if the resulting consequence relation, , say, was minimal as a consequence relation on its language; in other words if for all sets Γ of formulas of that language and all formulas A thereof, we had: Γ A if and only if A ∈ Γ. In saying that in this eventuality we are losing everything distinctive about , a presumption is being made to the effect that there is something distinctive here, i.e., that is not itself minimal as a gcr on its language, where this time minimality amounts to: Γ Δ if and only if Γ ∩ Δ = ∅. Let us present an example (6.31.11) to show that this can happen: we can have a non-minimal gcr whose agreeing consequence relation is minimal. Some background will be supplied in an exercise. Exercise 6.31.10 Say that formulas A, B, in the language of a gcr are subcontraries according to that gcr just in case A, B, and similarly, that they are subcontraries w.r.t. a class of valuations V for the language, just in case for each v ∈ V , either v(A) = T or v(B) = T. Show that if # is a 1-ary connective in the language of then for every formula A, the formulas A and #A are subcontraries according to if and only if they are subcontraries w.r.t. to the class Val() of all valuations consistent with . (We encountered the special case in which V = BV in 3.34 – see for example 3.34.9 on p. 506.) Example 6.31.11 For definiteness let us suppose that the only primitive connective in the language of , from Exercise 6.31.10, is # itself, and that is the smallest gcr according to which A and #A are subcontraries for all formulas A. Letting be the unique consequence relation agreeing with , we claim that although is not minimal as a gcr (for instance,


849

since p, #p), satisfies the minimality condition above. Suppose otherwise, and we have Γ A (equivalently, Γ A) for some Γ, A, with A∈ / Γ. Consider the valuation v defined by: v(C) = T iff C = A. We have v(Γ) = T while v(A) = F, yet v is consistent with : a contradiction. (See further 8.11.2(vii) – p. 1166 – in the discussion preceding which a notion of subcontrariety relative to a consequence relation is introduced.) Exercise 6.31.12 What is the justification for the claim at the end of 6.31.11 that the valuation v there under consideration is consistent with ? Our main business is now over, since we have seen an example in which the consequence relation induced by the of 6.31.11 says nothing at all about the relation between A and #A that it doesn’t say about an arbitrary pair of formulas. (It says ‘nothing special’, we might put it, anticipating the discussion of Chapter 9 below about this relation.) But let us include a further puzzle here in view of its continuity with the preceding discussion. Actually, it is most conveniently introduced by means of the notion of contrariety rather than subcontrariety. (For more details on both notions, see 8.11 and 8.22.) This is to be understood as dual to subcontrariety, in the case of classes of valuations: A and B are subcontraries relative to V when no v ∈ V assigns T to each of them. For gcr’s, we can also dualize the above definition: A and B are contraries according to . We have A, B ∅, whereas for consequence relation we need a variation: A, B C for all formulas C. (In what follows we drop the special “” notation for gcr’s.) Examples 6.31.13(Fallacious arguments.) (i) Suppose #A and A are contraries according to a consequence relation or gcr , for any formula A. Informally, while this means that while A and #A and are incompatible according to , they need not be jointly exhaustive: #A could be logically stronger than (classically behaving) ¬A. Privately, whatever the resources of , we could think of #A as the conjunction of ¬A with something else – call it D. So, since ##A is the result of applying # to #A, we can think of this applying # to ¬A ∧ D, and thus as being equivalent to ¬(¬A ∧ D) ∧ D. The latter can be reduced by De Morgan, Double Negation and Distribution equivalences to (A ∧ D) ∨ (¬D ∧ D) and hence to A ∧ D, which should imply A. Thus with # and as here, we should expect ##A A. (ii) Suppose that #A and A are subcontraries according to a gcr , for any formula A. Again informally, this means that A and #A are jointly exhaustive though they may not be mutually exclusive: #A could be logically weaker than ¬A. So whatever resources of makes available, we could think of #A as the disjunction of ¬A with something else – call it (as before) D. ##A then amounts to ¬(¬A ∨ D) ∨ D, which reduces to A ∨ D. So, since disjunctions follow from their disjuncts, we expect A ##A. These obscure arguments are multiply fallacious if considered as arguments for the claims (i) that if # is a contrary-forming operator according to , then we must have ##A A for all A in the language of and (ii) that if # is a

CHAPTER 6. “OR”

850

subcontrary-forming operator according to , then A ##A for all A. It is conclusion (ii) that makes them topical here, since this contradicts the findings of 6.31.11: A ##A would be a case of the single formula on the right not being amongst those on the left. Conclusion (i) is even more obviously wrong, since it predicts that in IL, we have the double negation elimination principle ¬¬A A (as intuitionistic negation is a contrary-forming operator). Exercise 6.31.14 Indicate some points at which the arguments of 6.31.13, to the extent that they can be understood, fail to establish the conclusions (i) and (ii) just mentioned. Let us close with something more straightforward; for more on contraries and subcontraries in this vein see 8.11 below (p. 1163) and, for even more, Humberstone [2005d], from some of which the following exercise is taken. (The use of “σ” here is strictly temporary. Otherwise it would clash with our use of the same symbol as a variable over sequents. An unrelated use of “κ” will be made on p. 1112.) Exercise 6.31.15 (i) Let be a gcr in whose language we have subcontrary and contrary forming operators (1-ary connectives) σ and κ respectively, by which description is meant that σA, A and κA, A , for all formulas A of that language. Show that the following claims are correct for all A: κA A

κσA A

A σκA.

(ii) Suppose now that is a consequence relation with subcontrary and contrary forming operators σ and κ, where this means that for all formulas Γ ∪ {A, C} of the language of : A, κA C and: if Γ, A C and Γ, σA C then Γ C. Are the three claims inset above (in part (i) of this exercise) correct for this ? (Justify your answer.)

6.32

The Model-Consequence Relation in Modal Logic

The present subsection is included by way of preparation for 6.33, in which gcr’s (or logics in Set-Set) are considered which generalize the consequence relations (or logics in Set-Fmla) described here. Among the consequence relations arising from the Kripke semantics for modal logic which we distinguished in 2.23 from the inferential consequence relations, were the model-consequence consequence relations, and it is these in which an interest will be taken here. By way of remainder, where C is a class of frames, we call a formula B a C-model-consequence of a set Γ when any model on a frame in C verifying all of Γ verifies B. This is often called the global consequence relation determined by C – though that terminology, as used in 2.33, covered in addition the distinct consequence(s) relation holding between Γ and A when A is valid on any frame (in C) validating all formulas in Γ: the corresponding frame-consequence relation, that is. Since this a consequence relation, one naturally reaches for the framework Set-Fmla, but to avoid confusion with the corresponding concepts under an inferential construal, we introduce the terminology with which to discuss the current relation with an initially superscripted “m” (for model ). In particular, then, let us say that a sequent Γ B m holds


851

in M = (W, R, V ) unless M |= A for all A ∈ Γ, while M |= B. (There is no notion of a sequent’s m holding at a point.) A sequent is m valid on a frame if it m holds in every model on that frame. A collection of sequents is m sound w.r.t./ m complete w.r.t./ m determined by a class of frames when that collection comprises (respectively) only/all/precisely the sequents m valid on each frame in the class. We proceed to a system m K which will be shown to be m determined by the class of all frames. For definiteness, we assume that all connectives # are present for which in Chapter 1 a notion of #-booleanness was defined, as well as . Actually, this is more than a convention for avoiding the kind of relativity to Φ (Φ being a set of such connectives) which featured in 2.33. We will make heavy use of the results of 2.32, and the smoothest way of doing so is to work in a language not suffering from ‘boolean impoverishment’. The first point at which this reliance appears is in our even-lazier-than usual presentation of m K. For, instead of just helping ourselves to classical truth-functional logic at one fell swoop in the basis, we here help ourselves to anything provable in (the Fmla system) K: as a zero-premiss rule we take K0

A

for any formula A provable in (Fmla) K.

This of course includes all tautologous formulas, and their substitution instances in the present language. If it were desirable to render the specifically modal reasoning visible, we could use a proof system with K0 restricted to such A, and a ‘horizontal version’ of the rule mentioned in Exercise 2.21.1 (p. 277): (A1 ∧ . . . ∧ An ) → B (A1 ∧ . . . ∧ An ) → B. We need the n = 0 case of this schema as our second primitive rule anyway: (Nec0 )

A A

Thirdly, we take a variation on Modus Ponens: (Imp.)

Γ A→B Γ, A B

The structural rules (R) and (M) are derivable from (K0 ) and (Imp.), but we need to take as primitive the rule (T). The second and third of our four primitive rules call for some comment. Of course (Nec0 ) here is a horizontal version of the rule of Necessitation, which would be very much out of place in a minimal system for the inferential interpretation of “” as in 2.23. Since that formula-to-formula rule preserves truth in (i.e., truth throughout) an arbitrary model any sequent instantiating the present schema (Nec0 ) is m valid on every frame. It thus poses no threat to the m soundness, w.r.t. to the class of all frames, of the system m K. The reader will be able to check that the same goes for the implication rule (Imp.), and the relevant comment to make is that an inverted (‘Conditional Proof’ or (→I)) form of this rule would not be similarly favoured. For example, it would deliver from (Nec0 ) all sequents A → A, which are not m valid on every frame. This is the same situation as obtains for the supervaluational semantics of 6.23: see

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852

6.23.3 on p. 832. (Notice that the m validity of a sequent Γ B on a frame, when Γ = ∅, is precisely the same as the validity of the sequent on that frame in the sense of 2.23, and amounts to the validity of the formula B on the frame in the sense of 2.22.) Taking it, then, that the m soundness of m K w.r.t. the class of all frames is clear enough, we pass to a lemma which will assist us in establishing the m of the system w.r.t. that class. We use the notation Γ for completeness n n∈ω { Γ}: in other words for the set of all formulas got by applying some number (including zero) of times to formulas in Γ. Lemma 6.32.1 If a sequent ΓB is not provable in m K, then the set Γ ∪ {¬B} is K-consistent (in the sense of 2.21). Proof. Suppose that Γ ∪ {¬B} is not K-consistent. Then amongst the elements of Γ are formulas C1 , . . . , Ck with n1 C1 → (n2 C2 . . . → (nk Ck → B) . . .) provable in (Fmla) K, for some values of n1 , . . . , nk . Then by (K0 ) and (Imp.), we have provable in m K: n1 C1 , n2 C2 , . . . , nk Ck B whence, by (Nec.) and (T), the sequent C1 , C2 , . . . , Ck B is provable in and thus by (M), likewise the sequent Γ B.

m

K,

The above Lemma does most of the work needed for the completeness theorem. The remainder is done by Lemma 2.22.4 (on generated submodels: p. 285), as well as the ‘Fundamental Theorem’ for normal modal logics – Lemma 2.21.4 in our presentation. Theorem 6.32.2

m

K is

m

determined by the class of all frames.

Proof. All that needs to be shown is the m completeness half of this claim, i.e., that if Γ B is not provable in m K, then there is a model (on some frame or other) verifying Γ but not B. For unprovable Γ B, 6.32.1 tells us that the set Γ ∪ {¬B} is K-consistent, and so by Lemma 2.21.4, all its elements are true at some point x in the canonical model for K. Consider the submodel thereof generated by the point x. All these formulas are true at x in this submodel by 2.22.4, so B is not true at x and hence not true in the submodel. But by the definition of Γ , the truth at x in the submodel of all formulas in this set guarantees that all formulas in Γ itself are true throughout the submodel (each point in the submodel being some number of ‘R-steps’ from x). We pause here to recall the example from our discussion of ‘validity at a point’ in 2.23 of the formula ⊥, which, as we noted there, can be valid at a point in a frame (and will be iff that point bears the accessibility relation to some point which in turn bears it to none), but cannot be valid on a frame. Since this formula contains no propositional variables, it is true in a model just in case it is valid on the frame of that model, and we can revive the example for present purposes. For any formula B, the sequent ⊥ B is m valid on every frame, since no model on any frame verifies ⊥. By 6.32.3, therefore, each such sequent is provable in m K. Choosing B = ¬⊥, we have, writing


853

for the consequence relation associated with m K, ⊥ ¬⊥, and also of course, ¬⊥ ¬⊥. But ¬⊥, since the formula on the right is not valid on every frame. Thus, since we have a case in which A B and ¬A B but not B, the present consequence relation fails to be ¬-classical. This of course is only to be expected, since in general we do not expect either a formula or its negation to be true throughout a model. Before leaving the example, we press it into service to show that our consequence relation is not congruential either (3.31). We have ⊥ ⊥, but not, for example ¬⊥ ¬⊥. There is a striking contrast, on both scores, with the inferential consequence relations of 2.23. There has been little specific to the Fmla system K in our discussion. We could equally well consider, for example, m KB, given by the above basis for m K supplemented by the zero-premiss rule A → A, and show this system to be m determined by the class of all symmetric frames. We describe the general situation: Theorem 6.32.3 If the Fmla logic K + Γ is sound w.r.t. a class C of frames to which all point-generated subframes of its canonical frame belong, then the Set-Fmla logic with basis as above for m K supplemented by A for each A ∈ Γ is m -determined by C. By considering the logic m KT covered by this general statement (Γ being the set of instances of the schema T), we have another simple example of noncongruentiality, since, by (Imp.) from A → A we have A A for any A, and the converse sequent is given by (Nec0 ): but in general A and A are not synonymous in this logic (e.g., choose A as p). The sequent-schema just derived in m KT is of considerable interest in its own right, since just as (Nec0 ) horizontalizes the rule of Necessitation, this schema, call it (Denec0 )

A A

gives a horizontal Set-Fmla sequent version of the rule of Denecessitation mentioned below in 6.42. Adding it as a zero-premiss rule to those above for m K gives a system we may call m KDenec , for which the following result may be obtained, by combining the above considerations with 6.42.8; a converse-serial frame is a frame (W, R) in which for all x ∈ W there exists y ∈ W with xRy: Observation 6.32.4 serial frames.

m

KDenec is

m

determined by the class of all converse-

By the considerations of 6.42 below, not only is the Fmla system K determined both by the class of all frames and by the class of converse serial frames, but the same goes for its Set-Fmla (and Set-Set) analogue under the inferential interpretation. Such double determination does not apply here, with the model-consequence interpretation: 6.32.4 could not survive with “ m KDenec ” replaced by “ m K”: the latter system does not contain, for example, p p, so the m completeness claim would be false. Nor could Theorem 6.32.2 above have “ m K” replaced by “ m KDenec ”, since then (using the same example, we see that) the m soundness claim would fail. More distinctions are possible, then, with

854

CHAPTER 6. “OR”

the current apparatus than were available in Fmla or, for that matter, in SetFmla and Set-Set treated, as in 2.23, with the inferential consequence relation in mind. Distinct classes of frames which determined in those other settings the same logic here determine – or rather, m determine – different logics. We can bring out this greater discriminatory power more directly by appropriating some of the concepts of modal definability theory, and in particular (we retain the “m” superscript to avoid confusion) saying that a collection of sequents Σ modally m defines a class C of frames when for all (W, R), every sequent in Σ is m valid on (W, R) iff (W, R) ∈ C. If Σ = {σ}, we will say that the sequent σ itself modally m defines C. It is an immediate consequence of Theorem 2.22.5 that the class of converse-serial frames is not modally definable in Fmla: for this class of frames is not closed under generated subframes. The same applies in Set-Fmla, understanding the notion in terms of the inferential consequence relation, with the above definition (minus the superscripted occurrences of “ m”) in place. However, in the present setting, by contrast: Observation 6.32.5 The class of converse-serial frames is modally by p p.

m

defined

Proof. The sequent cited is obviously m valid on every converse-serial frame. On the other hand, if (W, R) is not converse-serial, so that, where W is the set of elements of W lacking R-predecessors, W = ∅. In that case, put V (p) = W W , and verify that (W, R, V ) |= p while (W, R, V ) |= p. Thus the ‘Generation Lemma’ (2.22.4, p. 285) is robbed of its effects in the current context, there being, as was mentioned above, no ‘local’ notion of m holding at a point in terms of which m holding in a model is defined by universal quantification.

6.33

Generalizing the Model-Consequence Relation

In 2.23, we passed from a consideration of inferential consequence relations to the corresponding gcr’s. The same passage brings somewhat more dramatic effects when it is the model-consequence relation (associated with a class of frames) that is to be generalized. We are thinking of the C-model-gcr as relating Γ to Δ when any model on a frame in C which verifies all formulas in Γ verifies at least one formula in Δ. A Set-Set sequent Γ Δ m holds in a model M provided that if M verifies each formula in Γ, then M verifies at least one formula in Δ. The other ‘m’-superscripted vocabulary introduced for Set-Fmla is to be understood as defined in terms of m holding exactly as in §2.2. We can introduce a Set-Set version of m K, which we shall not separately baptize, with a basis as for its Set-Fmla namesake, except that now we should include (M) as a primitive rule, since thinning-on-the-right would otherwise not be derivable. Theorem 6.32.2 (p. 852), now read as speaking of this Set-Set system, is again correct. Nothing very exciting so far, then: our provable Set-Set sequents are precisely those m valid on every frame. The interest of the new framework will emerge when we exploit the opportunity it provides for multiplicity on the right (of “”) to define some extensions of (Set-Set) m K. Before doing so, we sketch the proof of the Set-Set modification of 6.32.2. Suppose that Γ Δ is unprovable in this system, where Δ = {B1 , . . . , Bn }. Then each of the sets


855

Γ ∪ {¬Bi } for 1 i n is K-consistent, giving (as in the proof of the original Theorem) points x1 , . . . , xn in the canonical model for (Fmla) K verifying Γ but not B1 , . . . , Bn respectively. The submodel of that model generated by {x1 , . . . , xn } is then a model verifying every formula in Γ but no formula in Δ. Before proceeding to an illustration of what can happen when more than one formula appears on the right, we might pause to ask whether any gain in expressive power is after all achieved by this transition to Set-Set. Suppose we have a sequent of the form A B, C. This m holds in a model verifying A iff either B or C is true in ( = thoughout) that model. But perhaps there is some formula which will serve as the ‘model-disjunction’ of A and B – some derived connective # of the language of modal logic with M |= #(B, C) iff either M |= B or M |= C. Now of course for particular cases of B and C such a formula may be found: but in general, we know from 2.22.8(ii) – see p. 286 – that there is no such formula to be had; for its existence would mean that there were disjunctive combinations on the left of the connection (is-true-in, formulas, models). So we really do need to move to Set-Set to proceed. Turning now to the promised exploitation of ‘commas on the right’, we begin by noting that the model-gcr determined by a class of frames is not in general ∨-classical: it is only too easy for a sequent A ∨ B A, B to fail to m hold in a model, since the model need only verify A ∨ B in virtue of different points verifying the disjuncts, neither of which need be true throughout the model. Thus the m validity of a sequent Γ A, B on a frame in general implies but is not implied by the m validity of the sequent Γ A ∨ B. To illustrate this greater strength, let us recall three modal principles from 2.21, in the first and third cases, changing the implicational form they had there so that all schemata have the form of disjunctions. The original labels are retained: .2 A ∨¬A

.3 (A → B) ∨ (B → A)

5 A ∨¬A

Such schemata are best known for their roles in axiomatizing S4.2, S4.3, and S5. The first two systems, as well as K.2 and K.3 are the subject of the following limbering-up exercise. Exercise 6.33.1 (i) As in 2.32, a frame (W, R) is convergent if for all x, y ∈ W there exists z ∈ W such that xRz and yRz, and is piecewise convergent when for all w, x, y ∈ W , if wRx and wRy then there exists z ∈ W such that xRz and yRz. Show that the frames on which .2 is valid are the piecewise convergent frames, and that the canonical frame of any normal modal logic containing .2 is piecewise convergent. Conclude that K.2 is determined by the class of all piecewise convergent frames and that S4.2 is determined by the class of all reflexive transitive piecewise convergent frames. Show that the point-generated subframes of reflexive transitive piecewise convergent frames are convergent, and hence that S4.2 is also determined by the class of all reflexive transitive convergent frames. Axiomatize the logic determined by the class of all convergent frames. (Solution below.) (ii ) Call a frame (W, R) connected if for all x, y ∈ W either xRy or yRx, and piecewise connected when for all w, x, y ∈ W , if wRx and wRy then either xRy or yRx. Show that the frames on which .3 is valid are the piecewise connected frames, and that the canonical frame of any normal modal logic containing .3 is piecewise connected. Conclude that K.3 is

856

CHAPTER 6. “OR” determined by the class of all piecewise connected frames and that S4.3 is determined by the class of all reflexive transitive piecewise connected frames. Show that the point-generated subframes of reflexive transitive piecewise connected frames are connected, and hence that S4.3 is also determined by the class of all reflexive transitive connected frames. (See Segerberg [1971] for a proof that this system is also determined by the class of all linear (or linearly ordered) frames in the sense of: reflexive, transitive, antisymmetric frames which are connected.) Axiomatize the class of all connected frames. (Again, solution follows.) (iii) The notion of linear ordering mentioned parenthetically at the end of (ii) is appropriate for connected partial orderings in (chains) of 0.13. If instead we have irreflexive ‘strict’ orderings in mind, we should consider the following condition of weak connectedness. A frame (W, R) is weakly connected if for all x, y ∈ W either xRy or yRx or x = y; similarly we may call (W, R) piecewise weakly connected when for all w, x, y ∈ W , if wRx and wRy then either xRy or yRx or x = y. Show that the latter condition is modally defined by the formula: (.3w ) (p → q) ∨ ((q ∧ q) → p) and that the canonical frame of any normal modal logic containing this formula is piecewise weakly connected. Show that the completeness result this provides for K4.3w can be supplemented by one to the effect that this system is determined by the class of transitive weakly connected frames. (iv ) Show that the formula (p ∧ q) → ((p ∧ q) ∨ (p ∧ q)) is valid on the same frames as .3; add another disjunct to its consequent to obtain a formula valid on the same frames as .3w . Show that (p ∧ q) → (p ∧ q) is K-interdeducible with .2. (v) Analogously with the notion of weak connectedness, define a frame (W, R) to be piecewise weakly convergent when for all w, x, y ∈ W , if wRx and wRy and x = y, then there exists z ∈ W such that xRz and yRz. Find a modal formula defining (the class of frames satisfying) this condition. (vi ) Along the lines of 2.22.6 (p. 286), show that none of the following classes of frames is modally definable: the class of convergent frames, the class of connected frames, the class of weakly connected frames, the class of asymmetric frames, the class of antisymmetric frames.

The above Exercise asks, inter alia, for axiomatizations of the logics (in Fmla) determined by the classes of all convergent and all connected frames. To set the stage for our discussions of extensions of m K in Set-Set, we need to clear up these matters. Let us say that a frame (W, R) is generalized piecewise convergent if for all w, x, y ∈ W such that for some m, n ∈ N, wRm x, wR n y, there exists z ∈ W such that xRz and yRz. Note that we could equally well have said simply that for w, x, y with wR*x, wR*y, there exists z. . . . Here R* is, as usual, the ancestral (or reflexive transitive closure) of R. However, the formulation with numerical superscripts for iteration helps make sense of the ‘generalized’ version of .2 given below (as .2g). Likewise, though with reference to a generalization of .3, for the


857

following definition. A frame (W, R) is generalized piecewise connected if for all w, x, y ∈ W such that for some m, n ∈ N, wR m x, wR n y, either xRy or yRx. Let us now abbreviate generalized piecewise to “gpw”; the sense in which the notions introduced here are more general than those in the above Exercise (6.33.1) is of course that the ancestral of the accessibility relation figures instead of that relation itself. (Alternatively put: that for the original notions we require that the values of m and n be 1, whereas we now allow them to be greater, or lower. R0 is the identity relation.) We can make a similar generalization in the case of the condition of euclideanness. The original condition (2.21.8, p. 281) is that if wRx and wRy then xRy; we say that (W, R) is generalized euclidean when for all w, x, y ∈ W , if wR m x for some m, and wR n y for some n, then xRy. We now introduce the three schemata with which to axiomatize the logics determined by the three classes of frames just introduced. More strictly speaking, each ‘schema’ is really an infinite class of schemata (each of which schemata in turn represents infinitely many formulas), depending on how the parameters m and n are realised: .2g m A ∨ n ¬A

.3g m (A → B) ∨ n (B → A) 5g m A ∨ n ¬A

The normal modal logic in Fmla we call K.2g is the smallest such logic containing, for any choice of m, n, A, the formula m A ∨ n ¬A. We understand K.3g and K5g analogously. Without the “g”, of course, the labels apply to the smallest normal modal logics containing the m = n = 1 instances of the three schemata. To tie up the matter of Exercise 6.33.1, we note that the canonical frames of K.2g and K.3g can easily be seen to be, respectively, gpw-convergent and gpw-connected, and that such frames are all frames for these logics (again, respectively). Further, the point-generated subframes of gpw-convergent frames are convergent, while for gpw-connected frames, the point-generated subframes are connected. Thus, as a solution to 6.33.1, one may take the logic determined by the class of convergent frames (of connected frames) to be given by adding by all instances of .2g (of .3g) to the axiomatization of K in 6.22. Notice (also à propos of 6.33.1) that S4.2g = S4.2 and S4.3g = S4.3, since given the m = n = 1 case of the schema, one can raise the value of m or of n by appeal to 4 and reduce it by appeal to T. Putting the point semantically, gpw-convergence and convergence (or gpw-connectedness and connectedness) are equivalent conditions on a frame with a reflexive and transitive accessibility relation, such a relation being its own ancestral. We have said nothing of K5g. In this case, the canonical frame is generalized euclidean, and so we obtain a completeness result for this logic w.r.t. the class of all such frames. The point-generated subframes of generalized euclidean frames are universal, since if (W, R) is such a subframe, generated by w ∈ W , say, then given any x, y ∈ W we must have xRy since for some m, n, wR m x and wR n y, whence that conclusion follows by the property of generalized euclideanness (which is inherited by generated subframes of frames possessing it). This modeltheoretic reasoning, which shows that K5g = S5, alias KT5, can be mirrored syntactically by noting that putting m = n = 0 in 5g gives T while putting m = n = 1 in 5g gives 5. (If you prefer to think of S5 under the description KTB4, note that B results from setting m = 0, n = 1, while to obtain 4, put m = 2, n = 0.) To draw this excursion back into the Fmla territory of 2.21–22 to a close, we point out – so as to reassure the reader who has already become

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suspicious on this score – that there was really no need to introduce the concept of a generalized euclidean frame: such frames are just the equivalence-relational frames under another description. We have now reviewed several examples, in Fmla, though the point is the same for Set-Fmla and Set-Set under the inferential interpretation of 2.23, of the determination of one modal logic by more than one class of frames. K.3g is determined, for example, both by the class of all connected frames, and by the much more extensive class of all gpw-connected frames. (For an example of a gpw-convergent frame which is not convergent, consider the disjoint union of any two gpw-convergent frames.) Thus to the extent that one wants to describe the ‘modal theory’ of one rather than another of such classes, this preference is one that cannot surface as a preference for one rather than another modal logic. For a case in which one might naturally have such a preference—on which the K.3g example obviously bears—consider tense logic (2.22): the hypothesis that time has this rather than that sort of structure will frequently not be reflected in the selection of one tense logic rather than another as the weakest system compatible with the hypothesis. (Of course the examples in our discussion have been—and will continue to be—monomodal, for the sake of simplicity.) The Fmla system K is determined both by the class of all frames, and – as will follow from 6.42.2 (p. 873), 6.42.6 (p. 876) – by the class of all converse-serial frames, though we saw in 6.32 that this particular instance of multiple determination vanishes in the Set-Fmla systems for the model-consequence relation (taking ‘determined’ as ‘m determined’). We were able to prise apart the weaker system m K, giving the logic of all frames from the stronger system m KDenec , giving the logic of the converse-serial frames. One can to some extent give these semantic distinctions their proof-theoretic due by distinguishing between modal logics not solely on the basis of what is provable in them, but in terms of which rules are derivable in them. (Individuating at level (2) of the hierarchy (0)–(3) from p. 186 in the Appendix to §1.2.) But the treatment of such distinctions in this subsection allows us to construe modal logics as collections of sequents (though not sequents of Fmla), retaining the original simplicity of identifying logics containing the same sequents regardless of the derivability of rules of proof. Further, the treatment of m KDenec in Set-Fmla ‘horizontalized’, as we put it, the rule of Denecessitation (6.42), whereas the cases we now consider in Set-Set use multiple right-hand sides and so do not correspond to rules of proof in Fmla (with their single conclusions) in this way, giving us something more general. Our illustration concerns three systems which could equally well have been presented in Set-Fmla, namely m K.2g, m K.3g and m K5g ( = m S5), and three other systems which could not have been dealt with in Set-Fmla. With the three cases just mentioned, the intention here is to single out the systems, in Set-Set, presented by extending the basis for Set-Fmla m K given in 6.32 with the Set-Set rules (M) and (T), and every sequent A in which A instantiates the schema in question (.2g, .3g, 5g, respectively). If we leave out this last addition ( A) we have a Set-Set version of m K, about which we remark that this system is m determined by the class of all frames, and that its associated gcr is the relation − (as in 6.31) where is the consequence relation associated with Set-Fmla m K. The other three systems we call m K.2+ , m K.3+ , and m K5+ , bases for which are obtained from that for (Set-Set) m K by adjunction of the following sequent-schemata, respectively:

6.3. COMMAS ON THE RIGHT .2+ A, ¬A

.3+ A → B, B → A

859 5+ A, ¬A.

Note that for any m, n, A, the sequent m A, n ¬A is provable in m K.2+ , and similarly for -prefixed versions of the other schemata in their respective systems (by Nec0 and (Imp)). These commas can then be turned into occurrences of ∨ by the K0 principles C C ∨ D and D C ∨ D (together with (T)). So the ‘+’ systems are extensions of the ‘g’ systems. Further, these extensions are proper, as the following semantic argument shows. (We stick with the .2 case for illustration.) m K.2g is m sound w.r.t. the class of all gpw-convergent frames, since any sequent instantiating the schema .2g is m valid on any such frame. But the sequent (an instance of .2+ ) p, ¬p is m invalid on some such frames: in fact on any such frame which is not actually convergent. (For, given x, y with R(x) ∩ R(y) = ∅, put V(p) = R(x); then p is false at y and ¬p is false at x, so neither is true throughout the model.) Therefore p, ¬p is not provable in K.2g, which is accordingly properly extended by m K.2+ . Note in passing that this shows that the sequent p, ¬p modally defines—or rather modally m defines—the formerly undefinable (because not closed under disjoint unions) class of convergent frames. (Compare 6.32.5 at p. 854 above, and the discussion immediately following it.) More information on modal m definability may be found in Kapron [1987]. We return to the topic of completeness, to conclude our illustration of the greater discriminatory power of the ‘m K-framework’ we summarize the situation. The proof of the following Theorem is omitted. With our earlier explanation as to how to transfer 6.32.2 to Set-Set, together with the remarks in the present paragraph, this should present no difficulty. Theorem 6.33.2 (i)m K.2g is determined by the class of gpw-convergent frames, while the stronger system m K.2+ is determined by the more restrictive class of convergent frames. (ii) m K.3g is determined by the class of gpw-connected frames, while the stronger system m K.3+ is determined by the more restrictive class of connected frames. (iii) m K5g(=m S5) is determined by the class of all equivalence-relational frames, while the stronger system m K5+ is determined by the more restrictive class of universal frames. This concludes our illustration of how differences between classes of frames that go unmatched by differences in the logics determined in Fmla or in Set-Fmla and Set-Set with the inferential interpretation, are here recorded as issuing in suitably distinct logics. On the inferential interpretation, the consequence relations and gcr’s associated with a class C of frames are, to use the terms of §1.2, Log(V ) for V = {vx | x ∈ W for some (W, R, V ) with (W, R) ∈ C}, where vx (A) = T ⇔ M |=x A. These valuations vx are boolean; our usual understanding of truth-preservation is of preserving truth on such valuations. The term inferential for the (generalized) consequence relations requires such preservation because to infer one statement from another (in an interpreted – e.g., natural – language) is tantamount to inferring the latter’s truth in the possible world in which that inference is made from the truth of the former in that world. In this case of model-consequence relative to a class C, we have instead Log(V ) with V = {vM | for some M = (W, R, V ) with (W, R) ∈ C}, where

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860 vM (A) = T ⇔ M |= A.

Such valuations are #-boolean for # ∈ {∧, , ⊥} though not for # ∈ {∨, →, ¬}, and indeed, more generally, it is readily seen that none of the latter connectives are truth-functional w.r.t. the class of all vM for M a Kripke model. (This terminology is from 3.11.) The non-truth-functionality, just noted, of connectives one expects to be described as truth-functional (though of course, our official understanding of either phrase is always relative to a class of valuations), offers a disincentive to taking the valuations vM seriously, and concomitantly, to horizontalizing the model-consequence relation. While the technical interest of such a move has perhaps been made evident by the material in this and the preceding subsection, it must also be conceded that in introducing “” in 1.21 with the suggestion (following Lemmon [1965a]) that at least in Set-Fmla when the left-hand side is not empty, this is a formal analogue of ‘therefore’, we have built an inferential interpretation of sequents into the motivation for considering sequents at all. No-one would accept “S, therefore necessarily S” (and likewise for many other readings of ) as a good form of argument, while (Nec0 ) announces as a fundamental principle of m K: A A. It may be that some such considerations as have just been aired were amongst those that led Scott ([1974a], pp. 149f.) to complain at those – he names H. B. Curry – who would allude to the rule of Necessitation in terms of the condition that A A. “Thus”, writes Scott, “we argue here that the Curry view is not only inconvenient but misleading – though of course it is possible to be consistent about it”. As our discussion shows, not only is it possible to be “consistent about it”, but such a perspective is actually very convenient after all for certain purposes. As for misleadingness, the danger is over confusion with the more natural (‘local’) inferential interpretation, and to reap the benefits of the model-consequence (‘global’) interpretation while avoiding this danger, one could develop a two-tier structure of inferentially interpreted sequents themselves playing the role formulas normally play in the construction of (this time, model-consequential) sequents. Thus we could retain “ ” to form the lower-level sequents, and use (e.g.) “'”, with a second-level sequent such as Γ1 Δ1 ; Γ2 Δ2 ' Γ3 Δ3 understood to m hold in a model M unless Γ1 Δ1 and Γ2 Δ2 hold in M while Γ3 Δ3 does not. Higher-level sequents of (roughly) this sort have been deployed in modal logic in Mints [1971] and Došen [1985]. They will not be considered further here, for the same reasons as we set aside ‘hypersequents’; see 1.21.8 (p. 111). An extended treatment of various alternative logical frameworks for modal logic may be found in the anthology Wansing [1996b] and in the monograph Wansing [1998]; see also §2.2 of Wansing [2000] (not specifically on modal logic). There are some interesting comparative remarks on various treatments in Restall [2007] (see also §4 of Restall [2009]) and in Brünnler [2006], esp. pp. 107–8.

Notes and References for §6.3 The discussion in 6.32 and 6.33 is based on the paper abstracted as Humberstone [1986b]; Kapron [1987] is an independent venture in a similar area.

6.4. DISJUNCTION IN VARIOUS LOGICS

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§6.4 DISJUNCTION IN VARIOUS LOGICS 6.41

The Disjunction Property and Halldén-Completeness

In classical and intuitionistic logic, a conjunction is provable if and only if each conjunct is provable; this is a special case (the case where no formulas lie to the left of the “”) of the fact that both of the associated consequence relations are what in Chapter 1 we called ∧-classical. From the ∨-classicality of both consequence relations, we cannot similarly infer that a disjunction is provable if and only if at least one disjunct is provable. In classical logic, the ‘if’ but not the ‘only if’ part of this claim is correct, the most conspicuous illustration of which is the fact that we have CL p ∨ ¬p without CL p or CL ¬p. This missing converse is called the Disjunction Property (for a given logic). That is, a logic (not necessarily one in Fmla) is said to have the Disjunction Property if whenever B ∨ C is provable in the logic, so is either B or C. In every such counterexample to the Disjunction Property for classical logic, the disjuncts B and C have a propositional variable in common, which leads to the isolation of a weaker property called Halldén-completeness: every provable disjunction whose disjuncts share no propositional variable has at least one of those disjuncts provable (6.41.2). (Another modification in the case of CL involves considering modal extensions thereof, and a modalized variation on the Disjunction Property: see 6.42.) For intuitionistic logic no such weakening is required; it possesses the Disjunction Property unadulterated by any such qualifications: Observation 6.41.1 For all formulas B, C, in the language of INat: If IL B ∨ C then IL B or IL C. Proof. Suppose IL B and IL C. We show that in that case IL B ∨ C. Our supposition implies, by the ‘completeness’ half of 2.32.8 (p. 311), that for each of B, C, there can be found a point in a Kripke model for IL (as in 2.32) at which that formula is false. Without loss of generality, we take the models, say M1 = (W1 , R1 , V1 ) and M2 = (W2 , R2 , V2 ) respectively, to be generated by the points in question, x1 and x2 , say, and to have W1 ∩ W2 = ∅. Define a model M = (W, R, V ) by adjoining a new point x (“new” meaning x ∈ / W1 ∪ W2 ) thus: W = W1 ∪ W2 ∪ {x}; R = R1 ∪ R2 ∪ { x, u | u ∈ W }; V (pi ) = V1 (pi ) ∪ V2 (pi ). The condition (Persistence) is satisfied because x never belongs to V (pi ). Further, M |=x B ∨ C, since we have neither M |=x B (as B is not true at x1 in M) nor M |=x C (as C is not true at x2 in M): here we are appealing to 2.32.2 (p. 306) and the obvious analogue of 2.22.4 (p. 285), on persistence of formulas and generated submodels respectively (as in the proof of 2.32.4, p. 309). By 2.32.8 again (this time the soundness half) therefore, IL B ∨ C. As was pointed out in 2.32, an easy proof of this result is also available on the basis of the sequent calculus formalization of IL (IGen, as it was in our discussion), once one has a proof of Cut Elimination, since a cut-free proof (in

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this system) of a sequent with nothing on the left and a disjunction on the right must have had an application of (∨ Right) as its final step, in which case cutting off the last step yields already a proof of one or the other disjunct. (Since we have not provided a proof of Cut Elimination here, however, the above proof keeps things self-contained.) Observation 6.41.1 can be generalized. It is the Γ = ∅ case of the assertion that Γ IL B ∨ C ⇒ Γ IL B or Γ IL C, an assertion that clearly fails for arbitrary Γ (e.g., consider Γ = {B ∨ C} where B is p and C is q). Harrop [1960], however, shows that if no formula in Γ contains an occurrence of ∨ not in the scope of some occurrence of ¬, the result does hold. (Actually Harrop’s condition on Γ is somewhat more liberal than this, and Kleene [1962] shows that it can be liberalized even further.) We will state simple versions of these generalizations as 6.42.10, 6.42.13, below. Note also that 6.41.1 shows the set of provable formulas of IL is itself an element of WIL of the canonical model MIL used in 2.32 to obtain the completeness result exploited in the above proof (since this set is ‘prime’). Further, this particular element generates the whole of MIL , since all elements of WIL are deductively closed. (The canonical model for the modal logic K is similarly point-generated, as noted after 6.42.5 below – p. 875 – though not only by one point, as here; by contrast with the present case, indeed, the set of provable formulas of K is not itself a point in the canonical frame. Intermediate logics – in Fmla – with the Disjunction Property are precisely the substitution-invariant elements of WIL .) The Disjunction Property for IL is to be expected on the ‘constructive’ account of disjunction: if one is, in general, in a position to assert a disjunction only when in a position to assert one of the disjuncts, this should hold also in particular when on the basis of logic alone a disjunction is asserted. The fate of certain intermediate logics, in respect of the Disjunction Property, will be touched on in 6.42: see the Digression on p. 875 and the discussion following 6.42.9 (which is itself to be found on p. 877). A restricted version of the Disjunction Property holds also in relevant logic. More precisely, and by way of example, Meyer [1972] shows that if B ∨ C is provable in R (with connectives ∧, ∨, ¬, →) then if neither B nor C contains ¬, one or other of these disjuncts is provable in R. For classical sentential logic, we have, as already intimated, the following – Halldén-completeness – property: Theorem 6.41.2 For any variable-disjoint formulas A, B, if CL A ∨ B then CL A or CL B. Proof. We argue contrapositively. If neither CL A nor CL B, then there are boolean valuations u, v, with u(A) = F and v(B) = F. Then since A and B have no common propositional variables, by 2.11.7(i), there is a boolean valuation w with w(A) = w(B) = F. For any such w, w(A ∨ B) = F, showing that CL A ∨ B. There is some work on Halldén-completeness amongst intermediate logics – since those lacking the Disjunction Property may or may not be Halldéncomplete – as well as in various other areas, which we shall not be reviewing here.


863

(For discussion and references, see pp. 201–205 of Chagrov and Zakharyaschev [1991].) Digression. To sidestep issues about the Disjunction Property, one may consider any of various ∨-free analogues of Halldén-completeness: for example the property that A → B implies, for variable-disjoint A, B, either ¬A or else B: a property which Komori [1978b] shows not only IL but every intermediate logic possesses. Another more general variation is the ‘separation of variables’ property considered in Maksimova [1976], [1995], the general definition of which we avoid here, mentioning only a special case: If A0 and A1 , taken together, share no propositional variables with B0 and B1 , and (A0 ∧ B0 ) → (A1 ∨ B1 ) then either A0 → A1 or B0 → B1 is provable. Note the resemblance to the Shoesmith–Smiley Cancellation Condition – see 2.11.8 on p. 205, and surrounding discussion (especially that below 2.11.8 giving the analogous condition for gcr’s). The explanation for the appearance of this condition, raised in that earlier discussion à propos of determination by a single matrix, lies in 6.41.5 below, which tells that the modal logic (not, in general, normal) comprising the set of formulas valid at any given point in a frame is bound to be Halldéncomplete. For a matrix formulation, we take the modal algebra ma(W, R) obtained (as explained in the Digression on p. 290) from the frame (W, R), with a view to isolating the set of formulas valid at a particular x ∈ W , and consider the matrix based on this algebra with its set of designated elements being {a ∈ ℘(W ) |x ∈ a}. The Set-Set sequents – or the corresponding conjunctionto-disjunction implicational formulas – valid in this matrix are precisely those valid at x in (W, R). Note also the equivalence of the above condition (from Maksimova) to Halldén-completeness, in any extension (with additional connectives) of CL, as we can always rewrite a counterexample of the form inset above, in the form (A0 → A1 ) ∨ (B0 → B1 ), when it will appear explicitly as a ‘Halldén-unreasonable’ disjunction. (The converse direction, of transforming any such disjunction into a counterexample to the above separation-of-variables condition, is left as an exercise.) End of Digression. We pursue the topic instead in the area of modal logic, for which Halldén [1951] originally proposed it as a desideratum. (Actually, he emphasized a particular kind of failure of the property, that in which each disjunct contained exactly one propositional variable and the variables in the two disjuncts were distinct. The more general definition given above is that which has been subsequently employed. For some differences between logics to which the current version and the original conception of Halldén-completeness apply, see the survey in Schumm [1993].) More specifically, Halldén suggested that provability in a logic lacking the property could not be equivalent to any notion of validity (or “truth” as he puts it). It is easy to see that a Halldén-incomplete logic cannot be determined by a matrix (in the style of 2.11) – cf. the Digression above – in which a disjunction receives a designated value precisely when at least one disjunct does. But, as we shall presently note, many complete normal modal logics are Halldén-incomplete, so such a negative verdict was premature: provability

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here does indeed coincide with validity over a class of frames. First, we consider the matter in the more general setting of modal logics (in Fmla) without the requirement of normality: collections of formulas of the language of §2.2 containing all tautologies and closed under Uniform Substitution and tautological consequence. The latter closure condition could equivalently be replaced by: closed under Modus Ponens. This is the more traditional characterization, used for example in Kripke [1965a] and Lemmon [1966b], from which the following two results come. For a formula A and a modal logic S, we denote by SA the smallest modal logic extending S and containing A. Lemma 6.41.3 For any modal logic S and variable-disjoint formulas A, B, if S A ∨ B, then S = SA ∩ SB . For a proof, see Theorem 2 of Kripke [1965a]; in Lemmon’s paper, the proof is absorbed into that of the following Theorem, which can be expressed in terms of the lattice of all modal logics thus: A modal logic is Halldén-complete if and only if it is meet-irreducible in the lattice of all modal logics. (The meet of this lattice is intersection, and the join is the operation taking two modal logics to the smallest modal logic including each of them. Note that we are working with logics in Fmla here and considering arbitrary modal logics, not specifically normal modal logics. For the notion of meet-irreducibility, see 0.13.9(ii), p. 11.) The main ingredient in the proof 6.41.3 is the fact that a formula C is provable in SA if and only if there are substitution instances A1 , . . . , Am of A for which S (A1 ∧ . . . ∧ Am ) → C together with the analogous fact in the case of SB . Here we rely on the fact that substitutions can be pushed to the top of the proof of C in S, preceding any applications of Modus Ponens (early 1.29), together with the Deduction Theorem (late 1.29), which yields the above conclusion in an alternative formulation: S A1 → (A2 → . . . (Am → C). . . ). See further the remarks preceding 6.41.8 below. Theorem 6.41.4 A modal logic is Halldén-incomplete if and only if it is the intersection of two modal logics neither of which is included in the other. Proof. ‘Only if’: Suppose S is a Halldén-incomplete modal logic, so that for some variable-disjoint A, B, we have (i) S A ∨ B, (ii) S A, and (iii) S B. By (i) and 6.41.3, S = SA ∩ SB ; by (ii) S = SA ; and by (iii) S = SB . The reader is left to verify that we have neither SA ⊆ SB nor SB ⊆ SA , completing the proof. ‘If’: Suppose S = S ∩ S , with neither S ⊆ S , nor S ⊆ S . Select A ∈ S S and B ∈ S S , in the latter case choosing B so as to share no propositional variables with A. (This can be done because if B’s variables overlap those of A, B can by replaced by the result of substituting for such shared variables, new variables not in A or B, and the result belongs to S S iff the original does.) Then since S A, S A ∨ B, and since S B, S A ∨ B. Therefore, as S = S ∩ S , we have S A ∨ B, even though S A and S B. Thus S is not Halldéncomplete.


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Schumm [1993] has a simpler formulation: instead of speaking of a logic’s being the intersection of two modal logics neither of which is included in the other, he speaks of its being the intersection of two of its proper extensions. A moment’s thought will reveal the equivalence of the two formulations. (The terminology in Lemmon [1966b] is most unfortunate: he calls two logics neither of which is included in the other “disjoint” – as though this meant they had no theorems in common.) One aspect of the situation revealed by 6.41.4 deserves special mention: if you are interested in various modal logics for a particular application, but unsure as to which of two different axioms A and B to add to some system S, though sure that one or other represents a correct modal principle, you can—making sure B has been relettered to be variable-disjoint from A (as in the parenthetical comment in the above proof)—view with favour the Halldén-incomplete system SA∨B , since, being the intersection of SA and SB , have as theorems only such formulas as represent correct modal principles whichever choice were to be made between A and B. For more information on, and examples of, this kind of ‘common ground’ motivation for Halldén-incompleteness, see Hughes and Cresswell [1968], pp. 270–272, as well as Humberstone [2007a] for further applications. When we turn from modal logics in general to normal modal logics, the situation is not so pleasant: the ‘if’ part of 6.41.4 survives intact, with “normal” inserted before “modal”, but the “only if” part fails, depending as it does on 6.41.3 which is not correct for normal modal logics (i.e., taking SA now to be the smallest normal modal logic extending S and containing A – though in fact we later prefer to denote this by S + A). We will consider two ways of reacting to this circumstance below, the first introducing a concept of Halldén-normality, and the second introducing instead a concept of global Halldén-completeness). Here, however, we illustrate the failure of the ‘if’ half of 6.41.3 for the lattice of normal modal logics. We can do so by reference to the zero of that lattice, the system K. This smallest of all normal modal logics is itself Halldén-incomplete, since its theorem ⊥ ∨ is a variable-disjoint disjunction: since there are no propositional variables in the disjuncts, there are certainly no shared propositional variables. If we do not have available the constants ⊥, , we can rewrite the example as (p ∧¬p) ∨ (q ∨ ¬q) neither of whose variable-disjoint disjuncts belongs to K. (Note that the first disjunct could be further simplified to p.) In spite of this, K is not the intersection of any pair of its ⊆-incomparable normal extensions. To see this, it helps to have Schumm’s reformulation in mind, mentioned above: K would then be the intersection of two of its proper normal extensions (i.e., of two normal modal logics properly extending K). If there were such extensions, call them S and S , then we could find A ∈ S K and B ∈ S K, in which case, since A ∈ S and B ∈ S , A ∨ B ∈ S ∩ S , so A ∨ B ∈ K. But by a result (6.42.1) from the following subsection, this implies A ∈ K or B ∈ K, contra hypothesi. Thus in particular, the disjunction displayed above notwithstanding, K is not the intersection of KVer with KD. (Not that there is any difficulty in describing this intersection in simple axiomatic terms: it is the normal extension of K by all formulas m (p ∧¬p) ∨ n (q ∨ ¬q) for each m, n.

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We shall have occasion to mention this system again below, in 6.41.10.) We can distinguish two ways, at the level of informal motivation, in which Halldén-incompleteness can arise from semantic considerations for normal modal logics. Let us call a class of frames heterogeneous if not all frames in the class validate the same formulas, and let us call a single frame (again) heterogeneous if not all points in the frame validate the same formulas. (For the notion of validity at a point in a frame, see 2.23.) We apply the term homogeneous to a frame or to a class of frames, meaning “not heterogeneous”. (Warning: in the discussion of Halldén-completeness to be found in van Benthem and Humberstone [1983], a homogeneous frame is defined to be one in which for any points x, y, in the frame, there is an automorphism of the frame which maps x to y. This condition is sufficient but not necessary for the frame to be homogeneous in the current more ‘linguistic’ sense.) Then the two types of source we have in mind for Halldénincompleteness are as follows: (i) the aim might be to give the logic determined by a single frame which is heterogenous; or (ii), the aim might be to give the logic determined by a heterogeneous class of (possibly homogeneous) frames. For an example falling under (i), suppose that the frame we have in mind has for its elements the natural numbers and for its accessibility relation the relation greater than. (Perhaps we are interested in the tense logic of ‘beginning time’, with a discrete ordering of moments of time. , with this interpretation is the past tense operator that would be written as H in A. N. Prior’s notation, as explained in the discussion between 2.22.8 and 2.22.9 above: p. 286f.) Then the element 0 validates the formula ⊥, while all the other elements (the positive integers) validate ⊥. Since we can insert (as above) different variables in these two cases, we get the formula (p ∧ ¬p) ∨ (q ∧ ¬q) valid on the frame, showing the logic it determines to be Halldén-incomplete. Indeed without such an insertion the disjunction ⊥ ∨ ⊥ is as its stands a disjunction with no propositional variables in common between the disjuncts. (The second disjunct here is an example of the kind most recently mentioned after the proof 6.32.2: a formula which can be valid at a point in a frame but cannot be valid on any – even one-element – frame.) And for a formula without constants we have the choice of a simpler candidate than that given before, namely p ∨ q. For an example falling under (ii), modally describing a heterogeneous class of frames, we can again consider a tense-logical system. The notation introduced in this connection toward the end of 2.22 will be used. Suppose we have a tense logic for strict (i.e., irreflexive) linear time – for the details of which any text in tense logic may be consulted (6.33.1(iii) being very much to the point here also) – and become taken with the thought that time is either a discrete or else a dense linear ordering (density as in 2.21.8(vi), p. 281). The former characterization may be taken to amount to the existence, for any point x, of an ‘immediate’ successor in the sense of: a point y later than (R-related to) x, everything earlier than which is either x or something earlier than x; the latter property is a matter of having, whenever x is earlier than z, some point y intermediate between x and z. This leads us to conclude that either GGp → Gp, a version of the 4c K4c@K4c (modal logic) (with G for ), or else (p ∧ Hp) → FHp, a formula due to C. L. Hamblin, is an acceptable tense-logical principle, and hence that, relettering the latter away from the former, the disjunction: (GGp → Gp) ∨ ((q ∧ Hq) → FHq)


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should be added to our system for linear time, even though we do not wish to commit ourselves to one disjunct or to the other. (Strictly, we should also consider a past-tense version of Hamblin’s axiom also, putting “G” for “H” and “P” for “F” in the above formula, since each point can have an immediate successor without each point’s having an immediate predecessor.) This is of course the matter of common ground compromises mentioned above as a source of Halldénincompleteness, though our present interest is in seeing the phenomenon as arising for a logic determined by the union of two classes of frames—the dense and the discrete linear orderings—which union is itself a heterogeneous class in the sense introduced above. These are two aspects of the same situation, in view of the Galois-connection equation (0.12.3: p. 4): ML(C1 ∪ C2 ) = ML(C1 ) ∩ ML(C2 ) Although from the point of view of informal motivation, we can distinguish the above two sources of Halldén-incompleteness, heterogenous frames vs. heterogeneous classes of frames, the latter topic can be subsumed under the former since any logic determined by a class of frames (any complete normal modal logic, in other words) is determined by a single frame – though it will typically not be a point-generated frame, an issue to which we return at the end of this subsection. To obtain the frame in question from the determining class, select an invalidating frame for each non-theorem, and form the disjoint union of the selected frames. (Note: it would not do say “Consider the disjoint union of all frames for the logic”, since the supposition that such a disjoint union exists would lead to a contradiction – it would be a frame for the logic, and therefore a proper subframe of itself. Nor can one simply point to the frame of the canonical model for the logic, since this may not be a frame for the logic.) If the original class of frames was a heterogeneous class of homogeneous frames, then the disjoint union is a single heterogeneous frame. So let us concentrate for a moment on individual heterogenous frames. This directs our attention to what is and what is not valid at this or that point in a frame. Here the property Halldén thought of as required for semantic tractability is very much in evidence: the set of formulas valid at any given point in a frame is a Halldéncomplete modal logic. (Though containing all theorems of K and closed under Uniform Substitution and tautological consequence, such a set of formulas is not typically closed under Necessitation, and so does not in general constitute a normal modal logic.) We record this fact—leaving a proof to the reader—as Observation 6.41.5 For any frame F = (W, R), any x ∈ W , and formulas A, B, having no propositional variables in common: F |=x A ∨ B iff F |=x A or F |=x B. An immediate consequence of this gives the following connection with heterogeneity: Corollary 6.41.6 If F is a homogeneous frame, then ML(F) is Halldéncomplete. S4 and S5 and several other normal modal logics were shown in Kripke [1963] to be Halldén-complete by exhibiting for each of the logics concerned a homogeneous characteristic frame and using the fact recorded in 6.41.6 (though the

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result itself for S4 and S5 goes back to McKinsey [1953], where it is proved by algebraic/matrix-theoretic methods). We cannot say, conversely, that if F is not homogeneous, then ML(F) is Halldén-incomplete. Consider, for example, the two-element frame (W, R) where W = {w1 , w2 } and R = { w1 , w1 , w1 , w2 , w2 , w2 }. This frame is heterogeneous because p ↔ p is valid at w2 though not at w1 . However, every formula valid at w1 is valid at w2 , as can be demonstrated with the aid of a concept— that of a ‘p-morphism’—not covered in our survey (§2.2) of modal logic (though described in most of the references given for that section). Thus there is no way of finding a formula A valid at w1 but not w2 and assumed relettered so as not to contain the variable p, for which we can claim that (W, R) |= A ∨ (p ↔ p) while (W, R) |= A and (W, R) |= p ↔ p. This suggests the following definition. Say that a frame (W, R) is strongly heterogeneous if W has two proper subsets W1 , W2 , with W = W1 ∪ W2 , for which there are formulas A, B, where A is valid at every element of W1 but not at every element of W2 , and B is valid at every element of W2 but not at every element of W1 . (Note that we do not require W1 ∩ W2 = ∅.) Using this concept we can strengthen 6.41.6 to a biconditional formulation, as is noted in part (i) of the following exercise; the remaining parts are included for practice with Halldén-completeness in normal modal logic. Exercise 6.41.7 (i) Show that for any frame F, ML(F) is Halldén-incomplete iff F is strongly heterogeneous. (ii ) Show that the normal modal logic KTc is Halldén-incomplete. (iii) Show that KD and KVer are Halldén-complete. (The references on Halldén-completeness given at the end of this section may need to be consulted in case of difficulties.) We return to the failure of 6.41.3, 4 in the lattice of all normal modal logics. The situation can be fixed by tampering with the notion of normality. (After 6.41.11, we consider a different reaction, replacing Halldén-completeness with global Halldén-completeness.) Keeping all the parts of the definition of what it takes to be a normal modal logic in Fmla (2.21) except the requirement of closure under Necessitation, we strengthen the latter requirement to closure under the rule we call Halldén-necessitation, and call a logic meeting this revised set of conditions Halldén-normal : (Halldén-necessitation)

A∨B A ∨ B

Provided that A and B share no propositional variables. Note that Necessitation can be recovered by taking B as pi ∧ ¬pi , where pi is some variable not occurring in A (or just take B = ⊥). For this choice of B, the conclusion A ∨ B yields A by truth-functional reasoning. Thus every Halldén-normal modal logic is a normal modal logic. Note also that Halldénnecessitation preserves validity on any homogeneous frame (indeed on any frame meeting the weaker condition that if y is accessible to x then any formula valid at y is valid at x). The latter claim follows from 6.41.5. The Lemmon–Kripke proof of 6.41.3 adapts straightforwardly to provide a proof of the ‘only if’ half of 6.41.8 below, in which, given any formula A, S + A


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is the smallest normal modal logic extending S and containing A. (A more common notation for this is S ⊕ A or S ⊕ {A}.) For the adaptation, one should note that for any formula C, S+A C if and only if for some substitution instances A1 ,. . . ,Am of A and some k1 ,. . . km (ki 0), we have (since all applications of Necessitation can be pushed up the proof of C so as to precede any applications of Modus Ponens – see also the remarks after 6.41.3 above): S (k1 A1 ∧ . . . ∧ km Am ) → C and similarly S+B C iff for some substitution instances B1 ,. . . ,Bn of B, and “degrees of iteration” l1 ,. . . ,ln , S (l1 B1 ∧ . . . ∧ ln Bn ) → C. It follows that the conjunction of the disjunctions ki Ai ∨ lj Bj provably implies C in S; but if S A ∨ B for variable-disjoint A, B, and Halldén-normal S, then each such disjunction is provable by Halldén-necessitation (interlaced with interchanging the disjuncts to comply with the letter of our formulation of that rule above), giving ki A ∨ lj B, whence we get ki Ai ∨ lj Bj by Uniform Substitution. Lemma 6.41.8 For any Halldén-normal modal logic S and variable-disjoint formulas A, B, if S A ∨ B then S = (S + A ∩ (S + B). This Lemma serves in the proof of the ‘if’ half of the following Theorem as 6.41.3 served for 6.41.4; the ‘only if’ half goes through exactly as before. Theorem 6.41.9 A Halldén-normal modal logic is Halldén-incomplete iff it is the intersection of two normal modal logics neither of which is included in the other. Note that the intersected logics promised here are claimed only to be normal, not to be Halldén-normal, because the logic S + A (S + B) of 6.41.8 is the smallest normal, not the smallest Halldén-normal, modal logic extending S and containing A (respectively, B). The formulation of 6.41.9 can be somewhat simplified by the observation, which the reader is invited to check, that any normal modal logic which is Halldén-complete is a Halldén-normal modal logic. Thus the concepts of normality and Halldén-normality coincide for those modal logics which are Halldéncomplete. This fact is also of use in pinning down the zero of the lattice of Halldén-normal modal logics. For it implies, given 6.41.7(iii), that KD and KVer, and hence also their intersection KD ∩ KVer, are Halldén-normal. (This because being closed under a rule, such as Halldén-necessitation, is a Horn property, and thus is inherited by intersections. More generally, although the intersection of two Halldén-complete logics is not guaranteed to be Halldéncomplete, it is guaranteed to be Halldén-normal.) As we had occasion to notice above, this intersection can be axiomatized as K + m (p ∧¬p) ∨ n (q∨¬q) for all m, n 0. (In fact, we can make do with just m = 0: see Exercise 6.41.10(i).) But by our earlier considerations, we have that if S is normal (and therefore, if S is Halldén-normal) then S (p ∧ ¬p) ∨ (q ∨ ¬q), whence by Halldén-necessitation, preceded and followed, as called for, by permuting the

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disjuncts, and Uniform Substitution, S m (p ∧¬p) ∨n (q ∨¬q) for all m, n. So every Halldén-normal system extends the Halldén-normal (though not Halldén-complete) system KD ∩ KVer, which is accordingly the bottom element of the lattice of all such systems. Exercise 6.41.10 (i) Show that the normal modal logic mentioned above, K + m (p ∧ ¬p) ∨ n (q ∨¬q)

(for all m, n)

is equally well described as K + (p ∧ ¬p) ∨ n (q ∨¬q) (for all n). (ii ) Show that the logic mentioned under (i) is also K + . (Note. These points are made on p. 75 of Chellas and Segerberg [1994], though the topic of Halldén-completeness is not there under discussion.) Example 6.41.11 Van Benthem [1979a] describes a fairly simple incomplete modal logic, namely the smallest normal modal logic containing the formula A = → ((p → p) → p) every frame validating which also validates B = ⊥ ∨ ⊥ while B is not provable (as he shows using general frames) in this logic. The simplicity of the example resides not so much in the low complexity of the formulas involved as in the fact that the argument showing B to be a frame-consequence of A can be formalized in a very weak version of second-order logic. As van Benthem explains, if we just take the consequent of A, and remove the initial “”, we can deduce ⊥ from the result, using only the deductive resources – including in particular necessitation – of K. (See (1)–(4) at the top of p. 73 of van Benthem [1979a].) It is not hard to see that leaving the initial “” intact on the consequent, we have a formula from which we can deduce ⊥ using those same resources. Let us transform A into the more convenient form A = ((p → p) → p) ∨ ⊥. We can put the problem of deducing B from A like this: one can no longer apply necessitation to one disjunct of a disjunction, so the fact that we can deduce ⊥ from the first disjunct alone does not mean that we can deduce B from A (which the incompleteness proof demonstrates is indeed impossible). However, for whatever interest it may have, we note that because the second disjunct of A contains no propositional variables, extending the usual deductive resources of normal modal logic to include Halldén-necessitation means that we can do precisely this in the present case. Whether this is a coincidence – applying only to some but not all cases of Kripke incompleteness in which only weak second-order logic is needed to construct a derivation that cannot be matched by a modal deduction – the author has no idea. We add that the present role of Halldén-necessitation can be thought of as emerging from the characterization of the smallest Halldén-normal modal logic given in 6.41.10(ii), in which the antecedent of A appears as an axiom,


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allowing us to detach the consequent and thence deduce ⊥ in the manner already indicated. (Van Benthem’s example is also presented in pp. 57–62 of Hughes and Cresswell [1984], where A is called VB. A brief summary may be found in Hughes and Cresswell [1996], p. 185.) Of more general interest in connection with this example is that in the minimal tense-logical extension Kt VB of this monomodal logic KVB – which adjoins axioms securing an additional operator, here written as −1 (with dual −1 ) is normal as well as the bridging axioms. (See the discussion after 2.22.8: p. 286f.) p → −1 p and p → −1 p – the formula B above can be deduced from A (or A ). This observation is due to F. Wolter and is explained (though without an explicit syntactic deduction) at p. 163f. of Litak [2005]; for related themes (though not this issue of non-conservative tense-logical extension) see further Wolter [1996]. We conclude with an issue left over from our earlier discussion: the fact that since to obtain a single characteristic frame for a modal logic we have to use a disjoint union construction – a kind of cheating, it might be felt, if what we wanted was a tree and what we got was a forest – with the resulting frame not being point-generated. Any logic determined by a point-generated frame is not only Halldén-complete, as Coro. 6.41.6 assures us, but has a stronger property in this vicinity, which we shall call global Halldén-completeness, in view of the role played by Necessitation in the definition, a rule with the global preservation characteristic of passing the property of being true throughout a given model from premiss to conclusion – as opposed to the local analogue of preserving truth at an arbitrary point in a model. (It should be noted, however, that the following terminology is used with a different meaning in Kracht [1999], p. 142. The present discussion follows Humberstone [2007a]. See the end-of-section notes, p. 922.) We call a modal logic S globally Halldén-complete just in case, for any formulas A, B, having no propositional variables in common, if m A ∨ n B ∈ S for all m, n 0, then either A ∈ S or B ∈ S. S is globally Halldén-incomplete otherwise. Thus global Halldén-incompletenes implies Halldén-completeness simpliciter (which we could for contrast have called “local Halldén-completeness”), since the existence of A, B with m A ∨ n B ∈ S for all m, n 0, despite the S -unprovability of either of A, B, means in particular for m = n = 0 we have a provable variable-disjoint disjunction with neither disjunct provable. In the style of 6.41.3–4 above, according to which the Halldén-incompleteness of an arbitrary modal logic amounted to being the intersection of two ⊆-incomparable modal logics, here we have the following development, with proofs left to the reader; as above S + A is the normal extension of S by A. Lemma 6.41.12 For any normal modal logic S and variable-disjoint formulas A, B, if S m A ∨ n B for all m, n ∈ N, then S = (S + A) ∩ (S + B). Theorem 6.41.13 A normal modal logic is globally Halldén-incomplete iff it is the intersection of two normal modal logics neither of which is included in the other.

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Thus while Halldén-completeness is meet-irreducibility in the lattice of all modal logics, global Halldén-completeness is meet-irreducibility in the lattice of all normal modal logics. Exercise 6.41.14 Suppose that S is a Halldén-normal modal logic. (i) Show that S is Halldén-complete if and only if S is globally Halldéncomplete. (ii) Which, if either direction of (i) – the ‘if’ direction or the ‘only if’ direction, does not require the hypothesis that S is Halldén-normal? In the style of Coro. 6.41.6 above we have here the following, whose proof again is left to the interested reader. Theorem 6.41.15 If F is a point-generated frame, then ML(F) is globally Halldén-complete.

6.42

The ‘Rule of Disjunction’ in Modal Logic – and More on the Disjunction Property

For this subsection we consider normal modal logics in Fmla and a phenomenon closely related to the Disjunction Property which some of them manifest. We shall also have occasion to make a few remarks not specifically concerned with disjunction but arising out of a special case (‘Denecessitation’) of the phenomenon in question; these remarks concern the status of rules of proof in modal logic. Towards the end of the discussion we turn back to the Disjunction Property itself, with special attention to its status in intermediate logics, and to a generalization of the property which holds for intuitionistic logic itself (“Harrop’s Rule”). The notion of rule in play in this section should be taken to be the simple-minded one from Chapter 1; in terms of the refined conception offered in 4.33, what we are talking about as rules here are language-specific application-sets of rules. An interesting theme in Lemmon and Scott [1977] is what they call the ‘rule of disjunction’, which we here recapitulate for a special case (treated already in §5.3 of Kripke [1963]). We can show that if K A ∨ B then either K A or K B, using essentially the same kind of argument as was offered in the proof of the Disjunction Property for IL (6.41.1, p. 861): Suppose that neither A nor B is provable in K; then by the completeness of K w.r.t. the class of all frames, there exist frames (W1 , R1 ) and (W2 , R2 ) and Valuations V1 and V2 with (W1 , R1 , V1 ) |=x A and (W2 , R2 , V2 ) |=y B, for some points x ∈ W1 and y ∈ W2 , which we may, without loss of generality (in view of 2.22.4: p. 285) take to generate their respective models, whose sets W1 and W2 we may again assume to be disjoint. Now append a new point z (“new” meaning: z∈ / W1 ∪ W2 ), to obtain a new model (W, R, V ) with W = W1 ∪ W2 ∪ {z}, R = R1 ∪ R2 ∪ { z, x , z, y }, and V any Valuation such that for all pi , V (pi ) ∩ W1 = V1 (pi ) and V (pi ) ∩ W2 = V2 (pi ). Since (W, R, V ) |=x A and (W, R, V ) |=y B, we have (W, R, V ) |=z A and (W, R, V ) |=z B (by 2.22.4 again), so (W, R, V ) |=z A ∨ B and hence K A ∨ B. In fact, more generally, we have (by the same argument for arbitrary n in place of 2):


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Theorem 6.42.1 For any n, and any formulas B1 , . . . , Bn , if K B1 ∨ . . . ∨ Bn then for some i (1 i n) we have K Bi . As already remarked, Lemmon and Scott call the property 6.42.1 ascribes to K, the property of admitting the ‘rule of disjunction’. This terminology is of course unfortunate, since only for the case of n = 1 does such an implicational metatheorem indicate the admissibility of a rule. The rule in question (for this case) we call Denecessitation: (Denec)

From A to A.

Note that we considered a ‘horizontalized’ version of this rule in §6.3, calling it (Denec0 ) in the discussion following 6.32.3. Corollary 6.42.2 The rule of Denecessitation is admissible for K. Proof. Apply 6.42.1 for the case of n = 1.

More will be said about this rule below. With some variations on the construction of the new model from the n countermodels to B1 , . . . , Bn , we could give a proof of the analogue of 6.42.1 for various other normal modal logics. For example, in the case of K4, we need to change the way R is constructed from R1 ,. . . , Rn in order to make the frame of the model with its appended point z transitive. Without this, we could not conclude from the falsity of the disjunction of the Bi at z in that model, that this disjunction was not provable in K4. But this is easily done, since we can put into R each pair z, w for w in any of the original models. A similar variation is possible for the case of KT and S4. In the latter case, the argument is strikingly like the proof in the preceding section that intuitionistic logic has the Disjunction Property, since the frames are required to be both reflexive and transitive. The presence of the ‘’s in the modal case can be thought of as compensating for the fact that here we do not require arbitrary formulas to be persistent (though all formulas of the form A are persistent in models on transitive reflexive frames) in the sense of being true at points accessible to any points at which they are true. This connection between IL and S4 is made explicit in: Exercise 6.42.3 Let τ map the formulas of the language of INat to formulas of the language of modal logic in the following way: τ (pi ) = pi ; τ (A ∧ B) = τ (A) ∧ τ (B); τ (A ∨ B) = τ (A) ∨ τ (B); τ (¬A) = ¬τ (A); τ (A → B) = (τ (A) → τ (B)). Where S4 is the inferential consequence relation associated (2.23) with S4, show that for any set Γ of formulas, and formula A, drawn from the language of INat: Γ IL A iff τ (Γ) S4 τ (A), where τ (Γ) = {τ (C) | C ∈ Γ}.

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CHAPTER 6. “OR” Hint: use the soundness and completeness results for IL and S4 w.r.t. the appropriate classes of Kripke frames, showing how a countermodel to the claim that Γ IL A can be converted into one to the claim that τ (Γ) S4 τ (A), and vice versa. Notice that the “”s appear in the translation for the cases of ¬ and →, the connectives for which the Kripke semantics for intuitionistic logic bring in a universal quantification over R-related points, as well as for the propositional variables, where the semantics demands persistence. The translation τ was one of several considered in McKinsey and Tarski [1948]. The others yielded the above result only for Γ = ∅, and so are suited to work for Fmla rather than Set-Fmla; cf. Smiley [1963b], p. 238. Gödel [1933c] also considered an S4-translation falling into this category. Modal translations of IL are also discussed in Chapter 3 of Rescher [1968], Došen [1986b], and Hazen [1990a]. An extension S of S4 is called a modal companion of the intermediate logic S (in Fmla, for simplicity) when A ∈ S if and only if T (A) ∈ S; since T is not surjective (even to within S-synonymy) a given intermediate logic has in general many modal companions, even when attention is restricted to normal modal logics as companions. S4.2 and S4.3 are modal companions of the intermediate logics KC and LC (Dummett and Lemmon [1959] – see also the discussion below). General discussions of this companionship relation may be found in Chagrov and Zakharyaschev [1992] and Wolter [1997]; modal translations for substructural logics are studied in Došen [1992b]. A more general discussion of translational embeddings may be found in Chapter X of Epstein [1990]; an elegant treatment with special attention to modal logic is provided by Zolin [2000] – some of which we quoted in 3.24. (On this topic see also Humberstone [2006d].) We present further examples of such embeddings at 7.22.29 on p. 1086 and 8.13.8 on p. 1191 (due to C. A. Meredith and R. Goldblatt, respectively). The ‘abstract algebraic logic’ approach reviewed in 2.16 can also be thought of along these lines, though with the one of the two languages being equational and the other sentential. For further treatment of translations and related matters, often concentrating on the relation between intuitionistic and classical (non-modal) logic, see §81 of Kleene [1952], Prawitz and Malmnäs [1968], Tokarz and Wójcicki [1971], Prucnal [1974], Wojtylak [1981], Furmanowski [1982], Chapin [1974], Wójcicki [1988] (see especially the discussion surrounding Theorems 2.6.8 and 2.6.9) and Caleiro and Gonçalves [2005], where further references may be found. A less demanding kind of embedding is discussed in Humberstone [2000a]. Some penetrating questions about relative strength and weakness of logics in the face of translational embeddings are raised in Béziau [1999a]; further discussion of the issues raised may be found in Humberstone [2005c], which also supplies further references.

In view of the misleadingness, noted above, of talk of the ‘rule’ of disjunction, let us say instead that when for a given n, the provability of B1 ∨ . . . ∨ Bn in a system, for any B1 , . . . , Bn , implies that at least one of the formulas B1 ,. . . ,Bn is provable, the system has the n-ary -disjunction property. A simple illustration of how the above style of argument for the possession of this property, even for just the case of n = 2, would break down is the case of S5; in order


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to get the new model with its additional point to have an equivalence relation (or more generally, for the accessibility relation to be euclidean), needed for the passage from invalidity on the frame of the model to unprovability in the logic, we would have to have R relating points in the various original countermodels to the disjuncts, which would then make them not generated submodels of the new model (blocking an earlier part of the argument). And this is as one would expect, since the property under investigation fails for S5 (indeed for K5), as witness the n = 2 case: ¬p ∨ p (or p ∨ ¬p). Likewise the other modal principles (most recently) considered in 6.33, at which point we re-wrote the implicational form of .2 given in 2.21 into the present disjunctive form: .2 A ∨ ¬A

and

.3 (A → B) ∨ (B → A).

Substituting p for A and q for B give instances of these schemata which serve as counterexamples to the 2-ary -Disjunction Property for systems such as K.2, K.3, S4.2 and S4.3. The latter pair of systems are related by the translation of Exercise 6.42.3 to the intermediate logics KC and LC (from 2.32) as S4 is to IL, leading us to expect a failure of the Disjunction Property for those intermediate logics. Such a failure is indeed evident from the facts that KC ¬p ∨¬¬p and LC (p → q) ∨ (q → p). Digression. With such examples as KC and LC in mind, one might conjecture (Łukasiewicz [1952], final paragraph of §7) that no proper consistent extension of IL had the Disjunction Property. In 1957, Kreisel and Putnam refuted this conjecture with the example of an intermediate logic, since called KP, which we shall be looking at later on in the present subsection (after Exercise 6.42.9, on p. 877). Gabbay and de Jongh [1974] give a countable chain of further counterexamples to the conjecture — all of them intermediate logics with the disjunction property. ‘Corrected’ versions of Łukasiewicz’s conjecture may be found in de Jongh [1970] and Skura [1989]. Intermediate propositional logics with the Disjunction Property are sometimes called constructive logics – with an analogous property for existential formulas also required in the case of predicate logic. (A general discussion of what might reasonably required of a logic for it to count as constructive may be found in Wansing [2008].) Faced with the fact that IL has such logics amongst its proper extensions, one may naturally be curious as to whether there is a greatest such extension, including all the others; Kirk [1982] shows that there is not. This still leaves open the possibility of constructive logics which are maximal in the sense of having no consistent extensions with the Disjunction Property. The existence of such logics is mentioned by Kirk, and they have been studied in Maksimova [1986] and Bertolotti, Migioli and Silvestrini [1996], q.v. for further references. End of Digression. Returning to the -disjunction property in modal logic, we note the following result, whose converse can be equally easily verified, also from proved by Lemmon and Scott [1977]: Theorem 6.42.4 If a consistent normal modal logic has the n-ary -disjunction property then its canonical frame (W, R) satisfies the condition: ∀x1 , . . . , ∀xn ∃z(zRx1 & . . . & zRxn ).

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Corollary 6.42.5 The canonical frame for K, (WK , RK ) satisfies the condition: For all finite X ⊆ WK , there exists z ∈ WK such that zRx, for each x ∈ X. Proof. By Theorems 6.42.1 and 6.42.4.

The word “finite” can be dropped from 6.42.5; it follows from this, taking X = WK , that there are points in the canonical frame for K which bear the relation RK to all points in the frame. Corollary 6.42.6 If a consistent normal modal logic S is closed under the rule (Denec) then (WS , RS ) is converse-serial, in the sense that for all x ∈ WS , there exists z ∈ WS with zRS x. Proof. This is just the special case of 6.42.4 with n = 1.

Again, the converse of this Corollary also holds. We continue with the special n = 1 case of the n-ary -disjunction property: the rule of Denecessitation. Observation 6.42.7 For all formulas B and sets of formulas Γ, B is a Ccs model-consequence of Γ if and only if B is deducible from Γ using arbitrary theorems of K and the rules Necessitation, Denecessitation, and Modus Ponens. Proof. ‘If’: It is necessary only to check that Denecessitation preserves truth in any model on a converse-serial frame. ‘Only if’: By 6.42.6. This last result obviously bears on the rule-soundness and rule-completeness of a modal logic understood not as just a collection of theorems but as a collection of (theorems and) derivable rules. Here we are thinking of rule-soundness w.r.t. a class of models (such as those on frames in Ccs ) as amounting to the derivability only of such rules as preserve truth in the models in question, with rule-completeness taken as the converse property. Rather different notions suggest themselves w.r.t. classes of frames, with rule-soundness w.r.t. C as having only such rules derivable as preserve validity on each F ∈ C and rulecompleteness again understood conversely. Both types of notions have their interest, though we shall not pursue these matters here. (Attention is paid to rules, rather than just formulas, in the area of normal modal logic, by Porte [1981], [1982b], though not in quite the way suggested by the above remarks.) However, we include some exercises for the benefit of interested readers. Exercise 6.42.8 (Williamson [1988a].) (i) Show that, on the basis of the axiomatization if K provided in 2.21, the following rules are admissible but not derivable: A → B A A A → B Call these rules, respectively, Depossibilitation and Inverse Monotony. (ii ) Show that the canonical frame (W, R) for any normal modal logic closed under Depossibilitation satisfies the condition that for all x ∈ W


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there exists z ∈ W such that z bears the relation R to at most x, and that this rule preserves truth in any model on a frame meeting that condition. (iii) Show that the canonical frame (W, R) for any normal modal logic closed under Inverse Monotony satisfies the condition that for all x ∈ W there exists z ∈ W such that z bears the relation R to x and only to x. (iv ) Finally, show that closure under Inverse Monotony is equivalent to closure under the rule (‘Inverse Congruentiality’) we get by replacing the occurrences of “→” in the formulation of that rule by “↔”. The rule labelling parenthetically mentioned under part (iv) here invokes the terminology according to which a modal logic (in Fmla) is said to be congruential just in case it contains A ↔ B whenever it contains A ↔ B (for arbitrary A, B). Note that all normal modal logics (as studied in §2.2) are congruential in this sense, and that it is an obvious adaptation to logics in Fmla of the notion of ↔-congruentiality start of 3.31. In a monomodal logic we do not need to distinguish between calling the logic congruential and calling the sole non-boolean connective congruential, and nor, à propos of the latter, do we need to add “in its ith position”, since only one position is at issue ( being 1-ary). Note that the Lindenbaum algebra of a congruential modal logic has for its elements the equivalence classes of formulas under the relation ‘is provably equivalent to’, and that in view of the behaviour of the operation corresponding to in these algebras, the inverse congruentiality rule for could equally aptly be called an injectivity rule for . For more information on these matters, see Humberstone and Williamson [1997]. Exercise 6.42.9 Say that a (formula-to-formula) rule modally defines a class of frames if the frames in the class comprise all and only those frames on which the rule preserves validity. (For a set of formulas, conceived as a zero-premiss rule, this amounts to the same as the class of frames being modally defined in the sense of 2.22 by that set of formulas.) (i) Find a modal formula which defines the same class of frames as the rule of ‘Possibilitation’: from A to A. (ii ) Show that neither the rule of Denecessitation nor any other rule modally defines the class of converse-serial frames (cf. 6.32.5), and likewise for the classes of frames singled out in connection with Depossibilitation and Inverse Monotony under (ii) and (iii) of Exercise 6.42.8. For the remainder of this subsection, we take up the topic foreshadowed in the Digression on p. 875. As we had occasion to observe there, the best known examples of intermediate logics, such as those – KC and LC – mentioned in 2.32 conspicuously lack the Disjunction Property, so a natural enough conjecture (Łukasiewicz [1952]) would be that perhaps IL is itself the strongest intermediate logic with the Disjunction Property. The conjecture was refuted five years later by G. Kreisel and H. Putnam – see the notes to this section (under ‘Intermediate Logics and Admissible Rules’, p. 923) – with the aid of an intermediate logic, usually called the Kreisel–Putnam logic, or KP for short, presented in Fmla with the following axiom-schema: (KP)

(¬A → (B ∨ C)) → ((¬A → B) ∨ (¬A → C))

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Since we concentrated on Set-Fmla in 2.32, we will treat (KP) as a label indifferent for the above formula schema and for the sequent schema we get by replacing the main “→” by a “”; when speaking of the logic KP in Set-Fmla, we, mean the logic obtained by adding the latter schema as a zero-premiss rule to the rules of INat. Aside from its bearing on Łukasiewicz’s conjecture about the Disjunction Property, (KP) has been of interest because of its status as a ‘horizontalized’ form of the rule, sometimes called Harrop’s Rule: (Harrop)

¬A → (B ∨ C) (¬A → B) ∨ (¬A → C)

(As was mentioned at the start of 6.41, Harrop actually provided for a greater generalization of the Disjunction Property than the version – “¬A IL B ∨ C implies ¬A IL B or ¬A IL C” – we can obtain from the above rule together with the form of that Property given in our discussion.) For the proof system INat, and any other conventional system for IL in Set-Fmla, (Harrop) is an admissible rule which is not derivable. As we have recently had occasion to recall à propos of modal logic, to say a rule is admissible in a proof system is to say that the set of provable sequents is closed under the rule, which is to say in the present instance that for any A, B, C for which we have IL ¬A → (B ∨ C), we also have IL (¬A → B) ∨ (¬A → C). The non-derivability claim is established by the same method as will be used below for a different rule (due to Mints): an appeal to what we call the Horizontalization Property (6.42.14). Admissibility is not a proof system-dependent notion, since it is simply a matter of closure of the class of provable sequents under a rule, regardless of what rules were used to obtain that class of sequents in the first place. Especially convenient for the present question is the fact that IGen, our Gentzen-style SetFmla0 sequent calculus from 2.32 enables us to reason that the last step of a cut-free proof of ¬A B ∨ C must have inserted either the exhibited occurrence of ¬ (on the left) or that of ∨ (on the right). But the rule (¬ Left) cannot have applied since this requires the rhs to be empty. So it must be the rule (∨ Right) that was last applied, in which case the second last line of the proof must have been either ¬A B or else ¬A C, giving us our conclusion that ¬A IL B or ¬A IL C. Notice that an even more straightforward application of the same kind of reasoning establishes the Disjunction Property itself for IL. In order for our discussion of such points to be self-contained, however, since we did not prove the completeness of the cut-free version of IGen (IGen − , as it might be called by analogy with Gen − in 1.27), we gave, in treating the Disjunction Property for IL at 6.41.1, an argument in terms of the Kripke semantics for IL with respect to (the class of all frames supplied by) which the natural deduction system INat had been shown to be sound and complete in 2.32.8 (p. 311). Using the same construction as in the proof of 6.41.1 (p. 861), we can show that (Harrop) is an admissible rule of IL. We actually give the formulation without implication here, suited to IL in Set-Fmla. (The admissibility of (Harrop) is equivalent to this, thanks to the →-rules of INat and the Disjunction Property for IL.) Observation 6.42.10 For any formulas A, B, C, in the language of IL : if ¬A IL B ∨ C, then either ¬A IL B or ¬A IL C.


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Proof. Suppose ¬A IL B and ¬A IL C. We want to conclude that ¬A IL B ∨ C. As in the proof of 6.41.1, we then have disjoint models, M1 = (W1 , R1 , V1 ) and M2 = (W2 , R2 , V2 ) generated by respectively by x1 and x2 , at which points the sequents ¬A B and ¬A C respectively fail. Define the new model M = (W, R, V ) exactly as in that proof by adjunction of an additional point x. We claim that ¬A B ∨ C fails at x in this new model. By the earlier argument, we already know that M |=x B ∨ C, so it suffices to show that M |=x ¬A. Since R(x) = R1 (x1 ) ∪ R2 (x2 ) ∪ {x} and ¬A is true at each of x1 , x2 , the only way for ¬A not to be true at x is to have M |=x A: but this is incompatible with having ¬A true at either of x’s successors x1 , x2 (let alone both), by the fact that formulas are persistent. Note that in the absence of the Disjunction Property, the formulation in 6.42.10 is a very different matter from the admissibility of Harrop’s Rule. For example, for the intermediate logic KC, where the Disjunction Property is (of course) lacking, the analogue of 6.42.10 would be false: take A = p ∧ q, B = ¬p, C = ¬q (cf. 2.32.10(iii), p. 318). Yet, surprisingly enough, not only KC but every intermediate logic is closed under (Harrop), as Prucnal [1979] shows (by an argument considerably more complex than ours, via 6.42.10, for the simple special case of IL); further developments appear in Minari and Wroński [1988]. A more general discussion of admissibility and derivability of rules for intermediate logics is provided by Iemhoff [2006]. Digression. We have before us the materials for something of a puzzle. If, as just claimed, (Harrop) is admissible in every intermediate logic, then it is admissible in every logic extending IL (since any rule is admissible for the inconsistent logic, the only logic in the connectives of IL extending IL and not intermediate between IL and CL). But doesn’t admissibility in all extensions of a logic amount to derivability in the logic, contradicting the earlier claim that (Harrop) is not a derivable rule of INat? The apparent puzzle is resolved by noting that intermediate logics are required to be substitution-invariant, whereas Uniform Substitution is not a derivable rule of INat. (Substitutions in fact play a crucial role in the argument of Prucnal [1979].) End of Digression. A more general form of 6.42.10 can be obtained with the help of the following: Lemma 6.42.11 Let M1 = (W1 , R1 , V1 ) and M2 = (W2 , R2 , V2 ) be Kripke models for intuitionistic logic, with W1 ∩ W2 = ∅, generated respectively by points x1 , x2 . Then there is a model M on the frame (W, R) where W = W1 ∪ W2 ∪ {x}, for some x ∈ / W1 ∪ W2 , and R = R1 ∪ R2 ∪ { x, u | u ∈ W }, such that for all formulas A built up from propositional variables using only ∧ and ¬, M |=x A if and only if M1 |=x1 A and M2 |=x2 A. Proof. Since we have described the frame of M, it remains to specify V for which M = (W, R, V ). Put u ∈ V (pi ) for u ∈ W1 just in case u ∈ V1 (p) and put u ∈ V (pi ) for u ∈ W2 just in case u ∈ V2 (p); this leaves out the case of the ‘new point’ x. Here put x ∈ V (pi ) just in case both x ∈ V1 (pi ) and x ∈ V2 (pi ). This gives the basis case, in a proof by induction on the complexity of A, of what we are trying to show: that M |=x A if and only if M1 |=x1 A and M2 |=x2 A, provided A is a {¬, ∧}-formula. Because of the latter restriction, there are only two cases to consider for the inductive step: A = ¬B for some {¬, ∧}-formula

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B, and A = B ∧ C for some {¬, ∧}-formulas B, C. We treat only the former case explicitly. We have to show that M |=x ¬B ⇔ [M1 |=x1 ¬B & M2 |=x2 ¬B,] on the hypothesis that such a relationship holds for B itself. Since xRx i (i = 1, 2) the ⇒ direction follows from the persistence of the formula ¬B (Lemma 2.32.3, p. 308). As to the ⇐ direction, suppose ¬B fails to be true at x in M even though it is true at xi in Mi (i = 1, 2). Then for some y ∈ R(x), M |=y B. But R(x) = R(x1 ) ∪ R(x2 ) ∪ {x}. Since ¬B is true at x1 and x2 in their respective generated submodels (and therefore in M), this leaves only x itself as a possible choice of y: and this contradicts the inductive hypothesis, which requires, in that case, that B is true at each of x1 , x2 (whereas it is in fact true at neither). The way we dealt with x in specifying V in the above proof guarantees, it should be noted, that the condition (Persistence) on models is satisfied. That specification can be put in terms of conjunctive combinations of valuations by observing that if for a given model M, vu (more explicitly, we should include a superscript “M”, as in 2.21, p. 279, and write “vuM ”) is defined by vu (A) = T ⇔ M |=u A, then vu is vx1 vx2 . We have already had occasion to observe (cf. the proof of 1.14.6, p. 69) that conjunctive combinations of ∨boolean valuations are typically not ∨-boolean, which in the present context amounts to observing that Lemma 6.42.11 does not extend to the case of A also built up using disjunction. For example, if A is p ∨ q and p (but not q) is true at x1 while q (but not p) is true at x2 , there is no way to arrange to have the disjuncts and the disjunction all satisfy the condition that they are true at x iff true at each of x1 , x2 . Exercise 6.42.12 Can Lemma 6.42.11 be extended to cover implication? (I.e., to allow for the case in which A may be constructed using not only ∧ and ¬ but also →.) We are now in a position to return to the generalized version of 6.42.10. Theorem 6.42.13 For any formulas A1 , . . . , Am , B, C, at all, and any formulas D1 , . . . , Dn constructed using only the connectives ∧ and ¬: if D1 , . . . , Dn , ¬A1 , . . . , ¬Am IL B ∨ C, then either D1 , . . . , Dn , ¬A1 , . . . , ¬Am IL B or D1 , . . . , Dn , ¬A1 , . . . , ¬Am IL C. Proof. Suppose D1 , . . . , Dn , ¬A1 , . . . , ¬Am IL B ∨ C with D1 , . . . , Dn constructed using only ∧, ¬. Then D1 , . . . , Dn , ¬(A1 ∨ . . . ∨ Am ) IL B ∨ C, and since for A1 ∨ . . . ∨ Am there is a CL-equivalent formula, A, say, built using only ∧, ¬, by 8.21.2 (p. 1215), we have D1 , . . . , Dn , ¬A IL B ∨ C. Thus we need only show that D1 ∧ . . . ∧ Dn ∧ ¬A IL B ∨ C implies that either D1 ∧ . . . ∧ Dn ∧ ¬A IL B or D1 ∧ . . . ∧ Dn ∧ ¬A IL C. So suppose not. Thus there are points in Kripke models verifying the formula D1 ∧ . . . ∧ Dn ∧ ¬A but falsifying, in the one case, B, and in the other case, C. Since D1 ∧ . . . ∧ Dn ∧ ¬A is a {¬, ∧}-formula, the construction described in the proof of 6.42.11 gives a model with this formula true at the new point, at which B ∨ C is false, showing


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that D1 ∧ . . . ∧ Dn ∧ ¬A IL B ∨ C, and thereby establishing the Theorem. Note that the reference in this proof to 8.21.2 is to Glivenko’s Theorem, proved at that point (though described above in the discussion following 2.32.1, p. 304). For a formulation more in the style of (Harrop), we could take 6.42.13 as showing the admissibility of the following rule of Set-Fmla: (Harrop + )

Γ ¬A → (B ∨ C) Γ (¬A → B) ∨ (¬A → C) Provided that Γ contains only {¬, ∧}-formulas.

However, notice that because of the proviso on Γ, this is not a substitutioninvariant rule – a consideration whose interest will become apparent in the following paragraph. Let us return to the case of the rule (Harrop) itself. What we have seen is that this is a ‘merely admissible’ rule of any conventional proof system (such as INat or IGen) for IL – admissible though not derivable, that is. Since the rule is substitution-invariant this shows that such systems are not structurally complete, by contrast with analogous systems for CL (see for example 1.25.9, p. 131). Mints [1976] gives the following example to show that the presence of negation is not required to make this point: (A → C) → (A ∨ B) ((A → C) → A) ∨ ((A → C) → B) For the admissibility of this rule, Mints gives a proof-theoretic argument like those sketched above (“the only way to have had a cut-free sequent calculus proof for the premiss sequent would yield such a proof for the conclusion”). In more detail, suppose we had a cut-free proof in IGen of the premisssequent, in which case the last rule to apply must have been (→ Right) and the sequent to which this rule was applied was A → C A ∨ B, in which case the preceding rule to apply was either (∨ Right) or (→ Left). In the former case we had either A → C A or else A → C B provable, from which it is clear how to proceed to obtain the conclusion sequent of Mints’s Rule. In the latter case, the preceding lines must have been A → C A (or perhaps just A) and C A ∨ B, from the first of which again we get the desired conclusion. Note that this last part of the argument would not have gone through if we had been trying to show the admissibility of a ‘variegated’ form of Mints’s Rule: From (A → C) → (D ∨ B) to ((A → C) → D) ∨ ((A → C) → B). (We omit “”.) Indeed the latter rule is not IL-admissible, as one sees by taking A to be any provable formula C to be p ∨ q, and D and B to be respectively p and q. As to the non-derivability of Mints’s Rule (in a system such as INat or IGen), the argument of Mints [1976] was that if it were derivable, then all sequents of the horizontalized form (A → C) → (A ∨ B) ((A → C) → A) ∨ ((A → C) → B)

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would have to be IL-provable, which they are not. (By contrast, Mints shows that the {∧, →}-fragment of IL is structurally complete; our version of this result appeared in a slightly more general form as Theorem 1.29.19) This involves an appeal to the ‘Horizontalization Property’ (for formulas) established for Nat as 1.25.11; the following reads identically except that references to Nat are replaced by references to INat (though of necessity the proof is somewhat different): Observation 6.42.14 If A1 , . . . , An−1 , An is an application of a derivable rule of INat, then the sequent A1 , . . . , An−1 An is provable in INat. Proof. Since all the primitive rules of INat preserve the property of holding in an arbitrary (Kripke) model, the hypothesis implies that An is true throughout any model in which all of A1 , . . . , An−1 are true. Thus by 2.32.4 (p. 309) the sequent A1 , . . . , An−1 An holds in every model, and by 2.32.8 (p. 311) this sequent is then provable in INat. Digression. Several asides on structural completeness: (1) Dummett ([1973], p. 436, [1977], p. 169) cites IL-admissible rules in the style of Harrop’s and Mints’s them as showing the failure of what he calls smoothness, which seems to be intended to mean the same as structural completeness. A vague initial characterization from the first of these sources has it that a logic is smooth if for that logic every rule of proof is a rule of inference. The vagueness of such a characterization arises over what is meant by the rules of proof of a logic: we have met this distinction already (1.29), and it is clear that, though he thinks of a logic as a (substitution-invariant, Tarski) consequence relation, Dummett has in mind formula-to-formula rather than sequent-to-sequent rules; leaving the only an unclarity as to whether we are concerned with the admissible rules of proof, or the derivable rules of proof. In his more precise definition, Dummett makes it clear that it is the former interpretation that is appropriate. (2) Further, and as already remarked in 1.29, it is necessary to distinguish structural completeness as a property of proof systems (“every substitution-invariant rule is derivable”) from structural completeness as a property of consequence relations (“If sAi for i = 1, . . . , n − 1 implies sAn for every substitution s, then A1 , . . . , An−1 An ”). The two notions come to be related via the association of a consequence relation with a proof system in Fmla in the way this is done for the sake of the Deduction Theorem. But such facts as the failure of structural completeness for a sequent calculus in which the cut rule is admissible but not derivable (the typical cut-free calculi) cannot be reformulated in terms of structural incompleteness as this notion applies to consequence relations. (This distinction cross-cuts a finitary/infinitary distinction for structural completeness depending on whether non-finitary rules or consequence relations are considered. See Makinson [1976].) (3) We note that the conjunction–negation fragment of IL is structurally incomplete in 8.21.7(ii), or more accurately ask for a proof of this, with a hint that the double negation rule is a substitution-invariant rule which is admissible but not derivable in the fragment in question. Alternatively, in terms of consequence relations, and understanding by IL here the restriction of the usual consequence relation of that name to the language with only ∧, ¬, as connectives, while ¬¬p IL p, there is no formula A for which IL ¬¬A while IL A, by 8.21.5 on


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p. 1216. (Thus there is no substitution s with IL s(¬¬p) and IL s(p).) Of course these claims would be incorrect if we gave the usual unrestricted interpretation to “IL ”, since we could take A as any formula classically but not intuitionistically provable – e.g., Peirce’s Law or the Law of Excluded Middle: but all such counterexamples lie outside the fragment under consideration. This may seem to conflict with Theorem 1 in Mints [1976], which reads, after adjusting the notation and terminology to match ours (“⊃” to “→”, “IPC” to “IL” – by which, understand IL – and “derived” to “derivable”) as follows: “Every rule admissible in a corresponding fragment of IL not including → is derivable in IL.” This is because Mints understands the possible substitutions which are relevant to the admissibility of a rule to extend beyond the confines of the fragment concerned, whereas we have been understanding the target language for the substitutions relevant to the structural completeness of any to be simply the language of . (Mints’s policy means that one and the same consequence relation could count as structurally complete qua such-&-such fragment of one logic but not qua such-&-such fragment of another logic.) Further discussion of items (2) and (3) in this Digression can be found in Humberstone [2006a]. End of Digression. Let us now return to the schema which ‘horizontalizes’ (Harrop), namely (KP). For definiteness we take this in the Set-Fmla form and discuss KP as the corresponding intermediate logic in Set-Fmla (proof system IL + (KP), for definiteness). Kreisel and Putnam gave a proof-theoretic argument of the type we have had occasion several times now to allude to, for the conclusion that KP has the disjunction property. (Hence the adverse bearing of KP on Łukasiewicz’s conjecture, mentioned earlier.) In this case, by contrast with those of the Disjunction Property and the mild generalization thereof—admissibility of (Harrop)—we have just been addressing à propos of IL, a simple modeltheoretic argument to this conclusion does not seem to be available, because the class of frames for which the most direct completeness proof we can give (Thm. 6.42.18) is a class of frames without the crucial feature we need for such an argument: the feature that any two disjoint point-generated frames of the class can be linked by the adjunction of a new point to which each of those generating points is accessible, to yield a new frame in the class. The class in question is the class of all (W, R) satisfying the following condition, which we shall call van Benthem’s condition for KP, in which all quantifiers range over W: ∀x∀y, z ∈ R(x) ∃u ∈ R(x)[uRy & uRz & ∀w(uRw ⇒ ∃t(wRt & yRt) ∨∃t(wRt & zRt))]. Something very like this formulation is credited to van Benthem under Example 6.5 in Rodenburg [1986], where the condition is cited as defining the class of all frames for KP (a correspondence-theoretic notion – the intuitionistic analogue of modal definability) whereas what we shall be doing is showing that KP is determined by this class of frames (a completeness-theoretic point). The exact condition mentioned by Rodenburg is like that above, except with the added hypothesis concerning y, z ∈ R(x) that neither yRz nor zRy: but this addition makes no difference. (Why?) For the completeness proof just alluded to, we shall need to begin with a notion of more general applicability.

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Elements x and y of the canonical frame (for an intermediate logic, defining such frames as in 2.32) are compatible if there exists no formula A with A ∈ x, ¬A ∈ y; this, as we shall see, coincides with their having a common successor (i.e., some z such that xRz and yRz ). Since the latter is evidently a symmetric relation, it will help to check that the former is also. So suppose that x and y are compatible, with a view to showing that y and x are compatible. Suppose the latter not to be the case. Thus for some formula B, B ∈ y and ¬B ∈ x. Then taking A in the above definition of compatibility as ¬B, we have A ∈ x and ¬A ∈ y (since B IL ¬¬B and B ∈ y), contradicting the supposition that x and y are compatible. Thus we feel free to speak, concerning any two points, of their compatibility or otherwise, without specifying the points in any particular order. Exercise 6.42.15 (i) Show that two points in the canonical frame for any intermediate logic are compatible if and only if they have a common successor. (ii ) Show, using (i), that any two points in a point-generated submodel of the canonical model for the intermediate logic KC have a common successor, and conclude that KC is determined not only by the class of ‘piecewise convergent’ Kripke frames (those frames (W, R) satisfying: for all w ∈ W , if x, y ∈ R(w), then there exists z ∈ W with xRz and yRz ), but also by the class of ‘convergent’ frames (satisfying: for all x, y ∈ W there exists z ∈ W with xRz and yRz ). Remark 6.42.16 With respect to the class of convergent (also called ‘directed’) frames mentioned in 6.42.15(ii), KC has the finite model property, from which given the transitivity of the frames concerned, it follows that KC is also determined by the class of frames (W, R) such that for some y ∈ W , for every x ∈ W , xRy. (A logic for which a semantics in terms of frames and models has been provided has the finite model property w.r.t. a class C of frames if it is determined by the class of models on finite frames belonging to C. The strengthening of convergence mentioned in this Remark may be found in Gabbay [1981], p. 67. The idea of the finite model property emerged originally in matrix semantics, as noted in the discussion leading up to 2.13.5 on p. 228. above; see Harrop [1958], [1965], Ulrich [1986].) As it happens, for canonical model arguments arising with certain intermediate logics (and KP is one of them) in Set-Fmla, it is simpler to conduct the argument as though one were in Set-Set, with maximal pairs Γ, Δ rather than, as in the case sketched in 6.42.15(ii) for KC, deductively closed prime consistent theories. (This device, used throughout Gabbay [1981], allows one to ensure that a point x in the canonical model bears the accessibility relation to one or more given points by putting their Δ-formulas into its Δ-set; this precise effect cannot be achieved by putting those formulas’ negations into x’s Γ-set because this would do more: it would prevent all of x’s successors from bearing the accessibility relation to the given points.) We concentrate on giving the relevant definitions just for the case at hand: the consequence relation KP associated with the Set-Fmla logic KP (for which a presentation is given by subjoining the schema (KP) to the basis for INat). To say that Γ, Δ is


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KP-consistent is to say that for no A1 , . . . , Am ∈ Γ, B1 , . . . , Bn ∈ Δ do we have A1 , . . . , Am KP B1 ∨ . . . ∨ Bn ; a KP-consistent Γ, Δ is maximally so (or ‘maximal’ for short) if the KP-consistency of Γ , Δ with Γ ⊇ Γ, Δ ⊇ Δ, implies Γ , Δ = Γ, Δ . Note that any consistent pair can be coordinatewise extended to a maximal pair (essentially as in 1.16.4, p. 75). For any maximal pair Γ, Δ , the set Γ is a deductively closed prime consistent KP -theory, and given any such theory, the pair which had it in the first position, and its complement (relative to the set of all formulas) in the second, is a maximal pair. If x is such a pair, we denote its first component (its “Γ-set”, as we put it above) by Γx , and its second by Δx . (Thus x = Γx , Δx , y = Γy , Δy , etc.) Corresponding to claims of the form “A ∈ x” in canonical model arguments of the type given in 2.32, we have instead “A ∈ Γx ”. 6.42.15(i) above, for example, continues to apply provided that we understand x and y to be compatible just in case for no formula A do we have A ∈ Γx , ¬A ∈ Γy (or equivalently, as we saw, for no formula A do we have ¬A ∈ Γx , A ∈ Γy ). Let us define, then, the canonical model MKP for KP to be the structure (WKP , RKP , VKP ) in which WKP is the set of all maximal pairs (relative to KP), xRy just in case Γx ⊆ Γy (which in turn holds iff Δy ⊆ Δx ), and VKP (pi ) = {x ∈ WKP | pi ∈ Γx }. We need, as usual, to show that truth (at x ∈ WKP ) and membership (in Γx ) coincide. But this requires only reformulation of 2.32.7 (p. 311): Lemma 6.42.17 For any formula A, any x ∈ WKP , M |=x A if and only if A ∈ Γx . Thus to show the completeness of KP w.r.t. the class of frames satisfying van Benthem’s condition for KP, it will accordingly suffice to show that the frame of MKP satisfies the condition. To that end, suppose that we have x, y, z ∈ WKP with xR KP y and xR KP z. We must find u ∈ RKP (x), for which (1) uR KP y, (2) uR KP z, and (3) for all w ∈ RKP (u) either w and y have a common RKP -successor, or else w and z have a common RKP -successor. Given x, y, z related as above, we claim that the pair Γx ∪ {¬(A ∧ B) | ¬A ∈ Γy , ¬B ∈ Γz }, Δy ∪ Δz

is KP-consistent and that if we let u = Γu , Δu be a maximal pair coordinatewise extending this pair, than u satisfies the condition placed on it in the above formulation of the condition. u ∈ RKP (x) because Γx ⊆ Γu , and conditions (1) and (2) are met because Δy ⊆ Δu and Δz ⊆ Δu ; so it remains only to check (3). For a contradiction, suppose that u has a successor w which lacks a common successor with y and also lacks one with z. By 6.42.15(i), this means that for some A, B, we have ¬A ∈ Γy , A ∈ Γw , and ¬B ∈ Γz , B ∈ Γw . But for every A and B answering to this description (the first with its negation in Γy , the second with its negation in Γz ), we put ¬(A ∧ B) into Γu , so since uR KP w, we cannot have A ∧ B ∈ Γw , which gives a contradiction with the fact that A ∈ Γw , B ∈ Γw . Thus as long as our claim that the pair Γx ∪ {¬(A ∧ B) | ¬A ∈ Γy , ¬B ∈ Γz }, Δy ∪ Δz

is KP-consistent is correct, we are done (with showing that (WKP , VKP ) meets van Benthem’s condition for KP). Suppose, to establish this claim, that the

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pair in question is not KP-consistent, so that for some A1 , . . . , An ,B1 , . . . , Bn , we have Γx , ¬(A1 ∧ B1 ), . . . , ¬(An ∧ Bn ) KP C ∨ D where C is some disjunction of formulas in Δy and D is some disjunction of formulas in Δz . Then, replacing the commas on the left by ∧ and applying the intuitionistically acceptable de Morgan equivalence which replaces conjunctions of negations by negations of disjunctions: Γx , ¬((A1 ∧ B1 ) ∨ . . . ∨ (An ∧ Bn )) KP C ∨ D Then by (→I) Γx KP ¬((A1 ∧ B1 ) ∨ . . . ∨ (An ∧ Bn )) → (C ∨ D) so by (T) and (KP) Γx KP ¬((A1 ∧ B1 ) ∨ . . . ∨ (An ∧ Bn )) → C) ∨ ¬((A1 ∧ B1 ) ∨ . . . ∨ (An ∧ Bn )) → (C ∨ D). From this we get (by IL-properties of ∧ and ¬): Γx KP ((¬A1 ∧ . . . ∧ ¬An ) → C) ∨ ((¬B1 ∧ . . . ∧ ¬Bn ) → D)) Thus one or other of the disjuncts here must belong to Γx . But the first disjunct cannot, since xR KP y and each ¬Ai (i = 1, . . . , n) belongs to Γy , while C belongs to Δy ; a similar consideration applies for the second disjunct, in view of the provenance of the ¬Bi and D in relation to z. This then gives us the ‘if’ (completeness) half of our soundness and completeness theorem for KP. Theorem 6.42.18 A sequent is provable in KP if and only if it is valid on every frame satisfying van Benthem’s condition. Proof. ‘If’: Suppose Γ A is not provable in KP. Extend the KP-consistent pair Γ, {A} to a maximal pair x = Γx , Δx , which, as we have seen, is an element of a frame (WKP , RKP ) satisfying van Benthem’s condition for KP, and on which the model MKP verifies all of Γ but not A, at the element x (by Lemma 6.42.17). Γ A is not valid on the frame (WKP , RKP ), then. ‘Only if’: It suffices to show that any sequent instantiating the schema (KP) is valid every frame meeting the cited condition. Suppose otherwise; i.e., that for some A, B, C, we have ¬A → (B ∨ C) true at a point x in some model M on a frame (W, R) meeting the condition, with neither ¬A → B nor ¬A → C true at x. Then there are points y, z both verifying ¬A, with B false at y, C false at z. The condition then gives us a point u each of whose successors shares a successor with either y or z, and amongst whose successors are y and z themselves. Since uRy, B must be false at u (all formulas being ‘persistent’); since uRz, C must be false at u. So M |=u ¬A, because M |=x ¬A → (B ∨ C) and M |=u B ∨ C (and xRu). Thus some successor of u must verify A: but this is impossible, since each of u’s successors shares a successor with y or with z, at which ¬A is true.


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Exercise 6.42.19 Draw a diagram indicating the inclusion relationships amongst IL, KC, LC, and KP. (Suggestion: To place KP in relation to LC, it may help to consult 2.32.10(i), p. 318.) Digression. (‘Weak Disjunction’). Let us recall the condition on Kripke frames (W, R) for IL that we have been calling van Benthem’s condition for KP: ∀x∀y, z ∈ R(x)∃u ∈ R(x) uRy &uRz &∀w(uRw ⇒ ∃t(wRt & yRt) ∨∃t(wRt & zRt)). Following Chagrov and Zakharyaschev [1997], we abbreviate “∃t(wRt & yRt)” to “wCy” (informally: “w and y are compatible” – see 6.42.15 and surrounding discussion). ∀x∀y, z ∈ R(x)∃u ∈ R(x) uRy & uRz & ∀w(uRw ⇒ (wCy ∨ wCz )) Especially in respect of its final conjunct, this condition is reminiscent of the following clause in the definition of truth at a point for a new kind of compound, A ( B of formulas A, B, called the weak disjunction of those formulas. The following is given by Chagrov and Zakharyaschev [1997], p. 59, as the condition for such a formula to be true at x in a model M = (W, R, V ), it being understood that x ∈ W and that all quantifiers range over W : ∀y ∈ W [ M |=y A or M |=y B] or ∃u, v(yRu & yRv & |=u A & |=v B) & ∀w(yRw ⇒ (wCu ∨ wCv ))]. We quote from the same page of Chagrov and Zakharyaschev [1997], adjusting their formal notation to our own: Let us understand points in frames as “reasons” or “arguments” in a controversy. (. . . ) A point x in a model is a reason for A if M |=x A. Say that A is intuitionistically established at a point y by reason x if M |=x A and xRy; A is classically established at a point y by reason x if M |=x A and xCy. The intuitionistic disjunction A ∨ B is true at x if either A or B is intuitionistically established at x. The classical disjunction (. . . ) ¬¬(A ∨ B) is true at x, if for every initial development of the controversy, either A or B remains classically established (by some reasons) at any point. The relation M |=x A B can be represented in the form ∀y(xRy ⇒ M |=y A B), where the restricted quantifier “∀y(xRy ⇒” corresponds to an arbitrary initial development of the controversy and “M |=y A B” means that either the intuitionistic disjunction A ∨ B has been already established at y or (above x) there are two reasons for A and B, respectively, which alone establish the classical disjunction at x. Roughly speaking, the weak disjunction A B reduces to the problem of classically establishing A or B by some unique, concrete reasons for them and not on the ground of the general distribution of truth-values of A and B.

The source of the above Kripke-semantical clause for ( is a 1983 paper by D. P. Skvortsov which appeared in Russian and is not accessible to the present author, leaving us with only the above (somewhat embryonic) remarks to rely

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on. Weak disjunction was originally proposed (with this same notation) at p.859 of Medvedev [1966], within a different semantical framework. Some interesting connections between Medvedev’s intermediate logic (‘logic of finite problems’) and the logic KP may be discerned by following up the references to Medvedev’s logic given in the notes to §4.3 (p. 630). End of Digression. The final topic we shall cover here deserves to be regarded as a corollary to the Disjunction Property for IL, though we shall approach it somewhat indirectly to begin with, the connection with that property emerging in due course. This topic is the argument of Gödel [1932] to the conclusion that IL is not tabular (“has no finite characteristic matrix” as it is often put). For our discussion we need to recall that while our once-and-for all enumeration of the propositional variables in any sentential language is p1 ,. . . pn ,. . . we sometimes resort to using “q1 ,. . . ,qn ” to list n occurrences of propositional variables in a formula, allowing repetitions, in the sense that we allow qi = qj even when i = j (whereas for i = j, pi = pj ). Define a (possibly derived) n-ary connective G of some language to be an n-ary Gödel connective for a set of formulas S of that language, which we shall assume to be closed under Uniform Substitution, when, writing “S A” for “A ∈ S”, for any propositional variables q1 ,. . . ,qn we have (∗)

S G(q1 ,. . . ,qn ) ⇔ qi = qj for some i, j, 1 i < j n.

Theorem 6.42.20 If there is an n-ary Gödel connective for the language of S then S is not sound and complete w.r.t. any m-element matrix where m < n. Proof. Since unless S is closed under Uniform Substitution, S is not determined by any matrix, we may as well suppose that S is so closed. Let G be an n-ary Gödel connective for S and suppose that S is determined by M = (A, D) with |A| = m < n, with GA the n-ary operation on A induced by G. First note that GA (a1 , . . . , an ) ∈ D for any a1 , . . . , an ∈ A, since, with A having m < n elements, some pair ai , aj , with i < j must be equal. We justify this claim with a representative illustration of the possibility envisaged: an−1 = an . For this case note that S G(p1 ,. . . ,pn−1 , pn−1 ) by the ⇐ direction of (∗), so we must have GA (a1 , . . . , an ) ∈ D when an−1 = an , or else an M-evaluation h with h(pi ) = ai would invalidate the formula G(p1 , . . . , pn−1 , pn−1 ), contradicting the hypothesis that S is sound w.r.t. M. Now since GA (a1 , . . . , an ) ∈ D for any a1 , . . . , an ∈ A, the formula G(p1 , . . . , pn−1 , pn ) holds on each M-evaluation h, as for each such h we must have GA (h(p1 ), . . . , h(pn−1 ), h(pn )) ∈ D. So since S is supposed to be complete w.r.t. M, and G(p1 , . . . , pn−1 , pn ) is as we have just seen, valid in M, we must have S G(p1 , . . . , pn−1 , pn ). But this contradicts the ⇒ direction of (∗), since all the indicated variables are distinct.


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Corollary 6.42.21 IL in the framework Fmla, and hence IL in Set-Fmla, is not tabular. Proof. Suppose IL is determined by an n-element matrix. In view of 6.42.20, this supposition is contradicted by the fact that G(B1 ,. . . ,Bn+1 ) defined as the disjunction of formulas all Bi ↔ Bj with 1 i < j n makes G an (n + 1)-ary Gödel connective for IL. To verify that (∗) is indeed satisfied for this choice of G, we note that if amongst the variables q1 ,. . . ,qn+1 we have qi = qj for some i, j, 1 i < j n, then one of the disjuncts of G(q1 , . . . , qn+1 ) as currently defined has the form pk ↔ pk for some k, which is IL-provable, whence (by appropriate appeals to (∨I) or the equivalent) so is G(q1 ,. . . ,qn+1 ), giving the ⇐ direction of (∗). For the ⇒ direction, we use the Disjunction Property. If G(q1 ,. . . ,qn+1 ) is IL-provable then so is one of the disjuncts qi ↔ qj , where i < j. But evidently no such equivalence of variables is provable in IL (or even in CL) unless qi and qj are occurrences of the same variable. We have used here the definition of G used in Gödel [1932] – the original source for this result – but many variants are possible. Let us begin with looking at alternative occupants of the role played in the above proof by ↔. Clearly we could equally well have used → in this role, for example, since for variables q1 , q2 , we have q1 → q2 provable if and only if q1 = q2 . At an intermediate level of abstraction between that at which Theorem 6.42.20 is formulated and that employed in Coro. 6.42.21, we will work with the ideas of ‘variable-identifiers’ and ‘alternators’, abstracting from the roles of ↔ and (iterated) ∨ in the specific G defined for the proof of 6.42.21. We say that a binary connective (again not necessarily primitive) I is a variable-identifier for a logic S in Fmla, presumed closed under Uniform Substitution if for any variables q1 , q2 , we have (∗∗)

S I(q1 ,q2 ) ⇔ q1 = q2

We have already observed that ↔ and → are variable-identifiers for IL. The case of → shows that a variable identifier need not satisfy the condition that S I(B1 ,B2 ) iff S I(B2 ,B1 ) for arbitrary formulas B1 , B2 (e.g. take B1 as p and B2 as q → p) even though this might initially seem to follow by appeal to Uniform Substitution from the fact (indeed implied by (∗∗) that S I(q1 ,q2 ) iff S I(q2 ,q1 ) – substituting Bi for qi . Exercise 6.42.22 (i) Explain exactly how the above appeal to Uniform Substitution is mistaken. (ii ) Show that putting I(A, B) = ¬(A ∧¬B) makes I a variable-identifier for IL (i.e., (∗∗) is satisfied), from which we conclude that a variation on the proof of Coro. 6.42.21 is available appealing only to properties (in IL) of ∨, ∧, and ¬. (This choice of I appears in Rose [1953].) Where Φ ⊆ {→, ∧, ∨, ¬, ↔} we have so far in effect observed that if {∨, ↔} ⊆ Φ, the Φ-fragment of IL is non-tabular (the original proof of Coro. 6.42.21), and likewise if {∨, →} ⊆ Φ or if {∨, ∧, ¬} ⊆ Φ, by suitable choices of variableidentifier (in the latter case from 6.42.22(ii), the present point being noted in McKay [1967]). Both of these last two preliminary findings can be improved.

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For the second, note that taking I(A, B) = ¬¬(A ∨ ¬B) gives us a variableidentifier for IL, so we can reduce {∨, ∧, ¬} to {∨, ¬}. The first case can also be reduced by discarding ∨, as reported in the following, the proof for which case appears below. Observation 6.42.23 If {∨, ↔} ⊆ Φ or → ∈ Φ or {∨, ¬} ⊆ Φ, then the Φ-fragment of IL is non-tabular. Before tackling the “→ ∈ Φ” part of this observation, we address the question of whether the provisionally isolated {∨, ∧, ¬} can be reduced by discarding ∨ rather than ∧. The answer here is negative. In Fmla the {∧, ¬} fragment of IL is two-valued and in Set-Fmla it is three-valued. See 8.21.5 (p. 1216) and 8.21.12 (p. 1221), respectively. The project of abstracting from the roles played by ↔ and (iterated) ∨ in the specific proof of Coro. 6.42.21 has so far concentrated on the former role, with the introduction of the concept of a variable-identifying connective. In the latter case, the most obvious concept to introduce is that of what we shall call an n-ary alternator (for a logic S) by which for the moment we shall mean an n-ary connective A in the language of S such that for any n formulas B1 ,. . . ,Bn : (∗∗∗)

S A(B1 , . . . , Bn ) ⇔ for some Bi (1 i n) we have S Bi .

The n-ary alternator chosen for the proof of 6.42.21 was of course B1 ∨ . . . ∨ Bn (which, if definiteness is desired, may be thought of as bracketed to the left), and at this intermediate level of abstraction we can describe the n-ary Gödel connective used in the proof of Coro. 6.42.21 thus: G(q1 ,. . . ,qn ) = A((I(q1 ,q2 ), I(q1 ,q3 ),. . . ,I(q1 ,qn ), I(q2 ,q3 ), I(q2 ,q4 ),. . . ,I(qn−1 ,qn )) in which A is an n(n − 1)/2-ary alternator. Thus we may extract the moral that any logic for which there is such an alternator and also a variable-identifier possesses an n-ary Gödel connective. With a view to justifying the “→ ∈ Φ” part of Observation 6.42.23 above we shall be liberalizing the notion of an n-ary alternator, but for the moment we stick with the definition given. Any logic S in Fmla has a 1-ary alternator since when A(B) is taken as B itself, (∗∗∗) is satisfied. For n > 1, different choices of n don’t come to much: Observation 6.42.24 For any logic S and any n 2, S has an n-ary alternator if and only if S has a binary alternator. Proof. Given a binary alternator A for S we can move to higher n using the fact that if A is (n − 1)-ary then A defined thus is n-ary: A (B1 ,. . . , Bn ) = A(A (B1 ,. . . , Bn−1 ), Bn ); conversely, we can move down since if A is n-ary then A is (n−1)-ary when defined by: A (B1 ,. . . , Bn−1 ) = A (B1 , . . . , Bn−1 , Bn−1 ). Note that the “going up” part of the above proof amounts to using iterated A while the “coming down” part uses something which in the binary case is reminiscent of idempotence. One should not jump to the conclusion that if (for a given S) A is a binary alternator then it is idempotent or that it is associative, however – understanding this, to illustrate with idempotence, as meaning that


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formulas B and A(B, B) are synonymous in S for any B. All that follows is that for any B, B and A(B, B) are equi-provable in S (either both provable or neither provable, that is), and in the case of associativity, that A(A(B, C), D) and A(B, A(C, D) are equi-provable, for any B, C, D. By way of illustration: Example 6.42.25 Where S is any normal modal logic with the Rule of Disjunction (i.e., for which we have the analogue of 6.42.1, there stated for K), B ∨ C is a binary alternator satisfying, normality here securing (via Necessitation) the ⇐ direction of (∗ ∗ ∗). One may verify that in K, the formulas A(A(B, C), D) and A(B, A(C, D) are not in general synonymous (i.e., for this logic, provably equivalent) amounting respectively to (B ∨ C) ∨ D and B ∨ (C ∨ D). (To see this most easily, note that formulas of these forms need not be provably equivalent even in the stronger logic KD, and in particular putting p, ⊥, q for B, C, D, the two sides of the would-be equivalence reduce to p ∨ q and p ∨ q respectively, evidently not KD-equivalent. (We choose KD here so that we can “throw away” the KD-refutable subformula ⊥.) The (idempotence-relevant) non-synonymy in K of B ∨ B and B, in general, is even more obvious. No doubt the more natural way of using 6.42.1 would be to obtain ternary and further alternators not (as here) by iteration of the binary case, but by directly invoking the general form with variable n, i.e., using B1 ∨ . . . ∨ Bn as an n-ary compound A(B1 , . . . , Bn ) satisfying (∗∗∗). The above example shows us that although there is no binary alternator for CL in Fmla, as one may check – if necessary – by considering separately each of the 16 binary connectives associated over BV with some boolean truthfunction, the addition of such an alternator yields a conservative extension. (Of course one cannot just “add” such an alternator for the same reason that the ‘rule of disjunction’ isn’t really a rule, so more circumspectly formulated: the existence of such an alternator in an extension does not imply that the extension is non-conservative.) Exercise 6.42.26 Combine the proposed alternator(s) from 6.42.25 with any suitable variable-identifier to show how 6.42.20 yields the conclusion that, e.g., S4 is not tabular. (See the discussion before 6.42.3, p. 873, on S4 and the Rule of Disjunction.) In fact the proof of the non-tabularity of the Lewis modal logics S1–S5 in Dugundji [1940] (S1–S3 being non-normal modal logics and S4 and S5 being the normal modal logics S4 and S5, alias KT4 and KT5 respectively), though inspired by Gödel [1932] does not proceed along the lines suggested by 6.42.26, since it makes no use of the Rule of Disjunction – which in any case is not possessed by S5, as remarked in our earlier discussion. Instead Dugundji considers the disjunctions (using “ ” for iterated binary disjunction): (Dugn )

1i<jn

(pi ↔ pj )

Dugundji then notes that (Dugn ) is valid in any matrix with fewer than nelements, as in the first part of the proof of 6.42.21 above, and then gives a separate proof that for no n is any of the formulas (Dugn ) provable in any of

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the logics to be shown non-tabular (by exhibiting a suitably large matrix to the logic invalidating the formula in question). So there is no need to appeal to the missing Rule of Disjunction. Similarly, a Gödel-inspired non-tabularity can be given for intermediate logics lacking the Disjunction Property, which was an essential part of our treatment above (at any rate in the isolation of alternators), by a judicious choice of matrices to show unprovability. This is the procedure followed in Dummett [1959b] in showing that the intermediate logic LC is not tabular. Dummett remarks (p. 100) à propos of the result in question (Theorem 6 of that paper) that it is obtained “by using exactly the proof given by Gödel [1932]”, in which the “exactly” seems something of an exaggeration, since Gödel makes essential use of the fact that IL has the Disjunction Property, which LC obviously does not. (We have inserted our bibliographical reference into the quotation just given to replace Dummett’s reference to the same work.) Information on the maximal non-tabular – to use the customary term – pretabular ) modal and intermediate logics may be found at the end of 2.32 and in the references there cited. We return to the question of the “→ ∈ Φ” part of 6.42.23, and to the possibility of liberalizing the above definition of an alternator. The idea is that we can drop the requirement that to be an n-ary alternator for S, A should be a (not necessarily primitive) connective in the language of S which forms from any n formulas B1 ,. . . ,Bn , a compound A(B1 ,. . . ,Bn ) which is S-provable just in case at least one of the Bi is ( = (∗) above). In fact all we need for the argument to work is that for each n formulas B1 ,. . . ,Bn , there should be some formula A(B1 ,. . . ,Bn ) satisfying this condition: not that such a formula needs to have been formed by applying a (possibly derived) connective to B1 ,. . . ,Bn . We recall in the first place that arbitrary algebraic functions and not just term functions in the algebra of formulas are of occasional use: we generally refer to them as contexts (e.g., 3.16.1, p. 424), with the application of the latter functions being only a special case. Thus, we could allow A(B1 ,. . . ,Bn ) to be a formula containing propositional variables not occurring in any of the Bi , for instance, something not possible on the narrow definition requiring A to be an n-ary connective. For the present case, we shall require that there be one such additional variable involved. But since we cannot specify the variable in advance of being given the B1 ,. . . ,Bn , we are not here even dealing with a fixed n-ary context. In the case of n = 2, the operation we have in mind takes formulas B1 and B2 to their deductive disjunction B1 B2 , in the sense of 1.11.3 (p. 50), where this was defined to be the formula (B1 → pk ) → ((B2 → pk ) → pk ) in which pk is the first propositional variable (in the enumeration p1 ,. . . pn ,. . . ) not occurring in B1 , B2 . Because of this dependence on the variables occurring in B1 , B2 , we cannot think of _ _ _ as a context with two gaps for the formulas B1 , B2 , which is filled instead by two other formulas C1 , C2 , to yield C1 C2 , since the pk required will typically change. For present purposes we need to generalize this binary operation and will just indicate the n-ary form (for arbitrary n) by iteration, without the parentheses of the above notation. Thus the n-fold deductive disjunction of formulas B1 , . . . , Bn is the formula B1 . . . Bn = (B1 → pk ) → ((B2 → pk ) → . . . → ((Bn → pk ) → pk ). . . )


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in which pk is the first propositional variable not occurring in any Bi (i = 1,. . . ,n). Exercise 6.42.27 Show that for any formulas B1 ,. . . ,Bn in the language of S = the Φ-fragment of IL in Fmla for any Φ with →∈ Φ, we have S B1 . . . Bn iff for some Bi (1 i n), S Bi . Thus, even though disjunction (i.e., ∨) is not definable in IL in terms of →, we have what is to all intents and purposes a ‘Disjunction Property’ even in just the →-fragment, which justifies the “→ ∈ Φ” part of Observation 6.42.23 above. In more detail, one applies the earlier reasoning with G(q1 ,. . . ,qn ) = A((I(q1 ,q2 ), I(q1 ,q3 ),. . . ,I(q1 ,qn ), I(q2 ,q3 ), I(q2 ,q4 ),. . . ,I(qn−1 ,qn )) where I is a variable identifier for IL (such as → itself, ex hypothesi available), but now taking as our alternator A the corresponding n(n − 1)/2-fold deductive disjunction of the exhibited alternands. (This device is used in Rautenberg (1986) – see p. 132 for this and for a slightly different way of extracting a general moral from Gödel [1932]; cf. also the ‘modified disjunction property’ in Section 2 of Humberstone [2003a].)

6.43

Disjunction in the Beth Semantics for Intuitionistic Logic

For the remainder of §6.4, we shall discuss semantic treatments of disjunction in three areas of logic which object-linguistic disjunction (“ ∨”) is not explained by (our classical) metalinguistic disjunction (“or”). Specifically, we are concerned with model-theoretic semantics rather than (e.g.) algebraic semantics; that is, with accounts of validity which proceed via a definition of the truth of a formula (or the holding of a sequent) in a model, itself a notion defined in terms of the truth of formulas at various points in the model. The common feature of the cases we shall consider is that the valuations vx corresponding to such points x are not in general ∨-boolean valuations. While in every case the truth of at least one of the disjuncts at a point will be sufficient for the truth at that point of the disjunction, it will not be necessary. We have already encountered such a situation in our discussion of supervaluational semantics (6.23) as well as in the treatment of non-∨-boolean valuations consistent with the relation of tautological consequence (1.14). In 6.44 and 6.45, we will see this pattern emerging for the cases of classical and relevant logics, while in the present subsection it will be intuitionistic logic that is used to illustrate the phenomenon. 6.46 extracts the nub of the semantics of 6.45 for more general application, as well as making some remarks about the valuational semantics of (∨E); in 6.47 we look at model-theoretic semantics for the quantum-logically motivated restricted version (∨E)res of this rule (from 2.31: p. 299). Our presentation of the Brouwer–Heyting–Kolmogorov (or ‘BHK’) explanation of the meanings of the connectives given in 2.32 was vague as to whether specifically a disjunction was assertable on the basis of possession of a proof of at least one of the disjuncts, or on the weaker basis of possession of an effective method which would yield such a proof. The latter explanation allows the possibility that one’s state of information could warrant a disjunctive assertion without, as it currently stands, warranting the assertion of either disjunct. The

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Kripke semantics for IL accords with the former style of explanation, since for a disjunction to be true at a point in a Kripke model, it is required that one or other disjunct be true there. Corresponding to the latter – more liberal – explanation we have the semantics of E. W. Beth. (See end-of-section references, under ‘Beth Semantics for IL’: p. 924.) We present the Beth semantics for IL in as closely parallel a manner as possible to the way the Kripke semantics was presented in 2.32 (p. 302). Here, as elsewhere, three ingredients are usefully distinguished: the notion of frame employed, the notion of model employed, and the (inductive) definition of truth of a formula at a point in a model. Validity of sequents is defined in terms of the last concept exactly as in 2.32, and, as there, our framework will be SetFmla. Thus, the notion of validity afforded by the present semantics will be one w.r.t. which the proof system INat of 2.32 is sound and complete, exactly as was the case for the Kripke semantics. (We could equally well mention the sequent calculus IGen described there, filling any gap on the right – since the framework was Set-Fmla0 rather than Set-Fmla – with ⊥.) For the first ingredient, the notion of a frame, there is no change from the notion of Kripke frame from 2.32. Beth frames are frequently taken to be not arbitrary posets, but specifically trees; there is no need to make such a restriction, however. (As in 2.32, the condition of antisymmetry is also optional: what is essential is reflexivity and transitivity; but we will take it that the frames are antisymmetric also, since this allows for convenient formulations of various notions, such as that of a ‘path’, introduced below.) Given a frame (W, R), a model on (W, R) will be defined almost as before, but not quite: in addition to the condition of persistence, we shall impose another condition, to be called the ‘barring’ condition below. Finally, the truth-definition proceeds as in 2.32 for atomic formulas, conjunctions, negations, and implications, but diverges – accounting for the location of the present discussion – for disjunctions. In accordance with the idea that a disjunction may be asserted by one not yet in a position to assert either disjunct, what we demand is that at certain later (viz. R-related) points, one or other disjunct is true. Which later points? No particular later points: but we insist that sooner or later, every foreseeable extension of our state of knowledge puts us in a position to assert one or other of the disjuncts. This idea can be made precise with the help of the notion of barring. We say that a set X ⊆ W bars a point x ∈ W in the frame (W, R) if every path through x leads into X. The latter phrase means that every maximal linearly ordered subset of W (linearly ordered by R, that is) which contains x has a non-empty intersection with X. Since the X’s we shall be interested in having bar a point x will themselves be closed under the relation R (0.13.4(iii), p. 9), this will amount to its being inevitable that one eventually, whichever possible future expansion of knowledge is considered, becomes stuck within the subset X. Note the following immediate consequences of the definition of barring: Observation 6.43.1 Let (W, R, V ) be a Beth model, with X, Y ⊆ W, x ∈ W . Then (i) if x ∈ X, then X bars x (ii) If X bars x and X ⊆ Y then Y bars X. (iii) X bars x iff X ∩ R(x) bars x.


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We use this (second order) notion of a set’s barring a point to give the clause for disjunction in the Beth semantics; here !A! (etc.) denotes the truth-set of the formula A in whatever model, (W, R, V ), say, is under consideration, i.e. {y ∈ W | (W, R, V ) |=y A}: [∨]Beth (W, R, V ) |=x A ∨ B iff the set !A! ∪ !B! bars the point x. Understood relative to any model, each of !A!, !B!, is a subset of !A! ∪ !B!; it therefore follows by 6.43.1(ii) that, since (by 6.43.1(i)) if either of A, B is true at a point x the truth-set of that formula bars x, if either of A, B, is true at x so is their disjunction. This shows that the rule (∨I) preserves truth at a point in a Beth model, and a fortiori that it preserves truth in such a model. The situation for (∨E) in the former (‘local’) respect is different, since a disjunction can be true at a point in a Beth model without either disjunct being true at that point. To have this rule nonetheless preserve truth in a model (and therefore to preserve validity on the frame of that model), we require the additional condition alluded to above, namely, the following condition on models: (Barring) For any Beth model (W, R, V ), if V (pi ) bars x ∈ W, then x ∈ V (pi ). As with the condition (Persistence) from 2.32, we can check that the condition here laid down for propositional variables holds for arbitrary formulas: Lemma 6.43.2 For any Beth model (W, R, V ), any formula A and any point x ∈ W , if !A! bars x then x ∈ !A!. Proof. By induction on the complexity of A; the case where A is a disjunction is given by [∨]Beth . We can use this result to show that (∨E) preserves validity on a frame, since it preserves the property of holding in any given Beth model: Lemma 6.43.3 For any Beth model M = (W, R, V ), if Γ A ∨ B and Δ, A C and Θ, B C all hold in M, then so does Γ, Δ, Θ C. Proof. Suppose that the three premiss-sequents hold in M, and that x ∈ W verifies all formulas in Γ ∪ Δ ∪ Θ; we need to show that M |=x C. This supposition implies (1) and (2): (1) M |=x A ∨ B (2) R(x) ∩ !A! ⊆ !C! and R(x) ∩ !B! ⊆ !C!. By [∨]Beth and (1), !A! ∪ !B! bars x.So, using 6.43.1(iii): (!A! ∪ !B!) ∩ R(x) = (!A! ∩ R(x)) ∪ (!B! ∩ R(x)) bars x. Therefore, by (2) and 6.43.1(ii), !C! bars x, and so by 6.43.2, M |=x C. Lemma 6.43.3 supplies the crucial step in the soundness (‘only if’) half of the following result; the completeness half, we do not give: Theorem 6.43.4 A sequent is provable in INat iff it holds in every Beth model.

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Though we have not given a proof of the completeness half of this result, let it be noted only that one does not require primeness on the part of the analogue of the elements of the canonical Kripke model for IL: disjunctions have to ‘wait’ for one or other of their disjuncts to become true at later R-related points. Sooner or later we get ‘stuck’ inside the union of the sets of points verifying the separate disjuncts, as demanded by [∨]Beth . Above, we described this in terms of inevitability, thinking of an R-closed subset X of elements of a Beth frame (or of a formula A for which X= !A!) as strongly inevitable at an element x when X bars x. The clause [∨]Beth says that a necessary and sufficient condition for a disjunction to be true at a point is that the union of the disjunct’s truth-sets be strongly inevitable at the point in question. There is an interestingly weaker notion of inevitability with which to contrast this: not that we must eventually become stuck inside X, but that it will always remain open that we should enter X. More precisely, let us say that an R-closed subset X of W , for a frame (W, R) is weakly inevitable at x ∈ W when for all y ∈ W such that xRy, there exists z R-related to y with z ∈ X. This concept appears in the consequent of (i) of the following Observation, which we state without proof. Observation 6.43.5 Given a frame (W, R) and X an R-closed subset of W and x any element of W: (i) X bars x ⇒ ∀y(xRy ⇒ ∃z(yRz & z ∈ X)) (ii) If W is finite, then the converse of (i) also holds. (In fact, all that is required under (ii) is that all paths are finite.) Whereas [∨]Beth requires the union of !A! and !B! to be strongly inevitable at a point verifying A ∨ B, the notion of weak inevitability for this same union emerges in the Kripke semantics – as the reader is left to check – as a necessary and sufficient condition for the truth at the original point of, not the disjunction of A with B, but rather the formula ¬¬(A ∨ B). Since Kripke’s and Beth’s semantics do not diverge on the treatment of negation, we can also say that if ¬¬(A ∨ B) is true at a point in a Beth model, then !A ∨ B! (even though this is no longer equal to !A! ∪ !B!) is weakly inevitable at that point. So by 6.43.5(ii), if the model is finite, this set is strongly inevitable at the point, and hence (6.43.2(i)) true there. Thus any sequent of the form ¬¬(A ∨ B) A ∨ B holds in all finite Beth models, even though – by considerations of soundness, since not all such sequents are IL-provable (e.g., put p and q for A and B) – such sequents do not all of them hold in every Beth model. With respect to a modeltheoretic semantics offering a notion of validity co-extensive with provability in some logic, the logic is said to have the finite model property when every unprovable sequent fails in some finite model, as was mentioned in 6.42.16. The foregoing example shows that IL does not have the finite model property w.r.t. the Beth semantics. Though we do not prove it here, there is in this respect a contrast with the Kripke semantics for IL. Usually the point is illustrated with the case of the law of excluded middle, obtained by putting A = B = p in the above sequent-schema, and noting that IL ¬¬(p ∨ ¬p); see 6.43.7 below. In fact – though again we shall not go further into this here – every classically valid but intuitionistically invalid sequent can only be got to fail in an infinite Beth model. Some remarks as to how to think about this, and on the relation between Kripke models and Beth models, may be found in Dummett [1977],


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Chapter 5. We shall encounter the notion of weak inevitability again in the following subsection. To conclude the present discussion, we consider, first, the prospects for mixing a Kripke-style and a Beth-style disjunction in the same language, and then, what becomes of the Kripke-Beth contrast in Set-Set. Suppose we work in a language like that of IL, but instead of containing the connective ∨, containing two connectives ∨Kr and ∨Be , interpreted respectively by Kripke’s and by Beth’s clauses for disjunction. This would be the natural setting for exploring the hypothesis that there are here two notions of disjunction. The situation is comparable to that with ¬c and ¬i considered in (especially) 4.34, except that the polemical background (IL ‘versus’ CL) is absent. In that case, we noted that (RAA) for ¬i could only be counted on to preserve validity if the formulas which would remain on the left of after an application of the rule (collectively, Γ, in our discussion there) were all persistent, but that formulas built with the aid of ¬c were not themselves guaranteed to have this property. An examination of the case of (∨E) for ∨Be , as treated in Lemma 6.43.3, reveals a similar situation, though this time with regard to the condition (Barring), rather than (Persistence), and for the formula on the right, rather than those on the left. That is, where C is the common consequence of the disjuncts A and B, in the presence of ancillary assumptions Δ and Θ, we concluded the proof of that Lemma by inferring that M |=x C from the fact that !C! barred x. This is an appeal to Lemma 6.43.2, stating that all formulas behave as C is required to behave here. But the inductive proof of that Lemma would break down – even supposing its basis case is secured by retaining the condition (Barring) for propositional variables – for ∨Kr -disjunctions. That part of the inductive step would go through only if we had in general that if !A! ∪ !B! bars a point, then either !A! or else !B! bars that point: and this is manifestly not the case. Thus in a proof system for the envisaged combined (∨Be + ∨Kr ) logic, (∨E) for ∨Be would need to be restricted to formulas with the Barring property, i.e., formulas for which 6.43.2 holds. For example—though we are not concerned here with obtaining a complete proof system for the combined logic—we could secure this by requiring that ∨Kr not appear in the formula C in an application of (∨E) for ∨Be . (Note that this blocks that part of a uniqueness argument which would establish A ∨Be B A ∨Kr B.) As long as we are imposing the condition (Barring) for propositional variables, this restriction will clearly suffice. And to contemplate rescinding that condition would leave hardly any scope for the application of (∨E) for ∨Be . Retaining the condition, we note that the set of valid sequents of the combined language—as in the case of ¬i with ¬c when (Persistence) is retained for the variables—is not closed under Uniform Substitution, in view of Example 6.43.6 The sequent p ∨Be p p holds in every Beth model, but the sequent (p ∨Kr q) ∨Be (p ∨Kr q) (p ∨Kr q) does not. The left-hand formula in the second sequent cited here is true at a point x in a model iff !(p ∨Kr q)! ∪ !(p ∨Kr q)! bars x, by [∨]Beth , which is to say, iff !(p ∨Kr q)! bars x, which is to say, iff !p! ∪ !q! bars x. As, we have already remarked, this does not imply that either !p! bars x or !q! bars x, which allows us to falsify each of p, q, and hence p ∨Kr q (the right-hand formula in our sequent). For example we can take x as the point 0 in the following infinite frame partially depicted in Figure 6.43a, in which the accessibility relation is the ancestral of

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the relation indicated by the arrows. (Thus the diagram is upside-down, if we are thinking of elements y accessible to x as elements y x.) { 0 CCC CC {{ { CC { { CC }{{ ! −1 1 CC { CC { { CC {{ CC { { C! }{ −2 2@ {{ @@@ { { @@ {{ @@ }{{ −3 3? ~~ ??? ~ ~ ?? ~~ ?? ~~~ −4 4

etc. Figure 6.43a

A model on this frame with V (p) taken as the set of negative integers and V (q) the set of all integers other than 0 and −1, has !p! ∪ !q! barring 0 without either !p! or !q! barring 0, so that while the left-hand formula of the second sequent in Example 6.43.6 is true at 0, the right-hand formula is not. Remark 6.43.7 Though there is no need to consider the infinite frame of Figure 6.43a for the purpose of illustrating the invalidity of the sequent mentioned—we could have made do with just the points 0, −1, 1—we include it here to show that just making use of the above assignment V(p), we get a countermodel to p ∨ ¬p. In this case, as already noted, we cannot avoid the infinite frame. For example, stopping with 0, −1, and 1 would leave {−1, 1} barring 0 and so, since this is a union of a p-verifying and a ¬p-verifying singleton, we should have p ∨ ¬p true at 0. (A similar comment applies in the case of the ∨-free sequent ¬¬p p. The simple 2-element Kripke countermodel – originally described à propos of Figure 2.32a, p. 309 – violates the Barring condition on V .) Instead of considering models with the Barring condition in force, we can introduce the notion of barring into the setting of ordinary Kripke models and have a singulary connective # for (as we would put it) ‘strong inevitability’: We count #A as true at a point in such a model iff !A! bars that point. The idea of introducing a connective so interpreted as a ‘new intuitionistic connective’ (cf. 4.37) is due to M. Kaminski. (Compare also Hazen [1990a].) Suitable axioms for the extension of IL in Fmla by such a connective may be found under Remark 30 of Gabbay [1977]; a sequent calculus presentation in Set-Fmla0 may be found in §4 of Kaminski [1988].


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The final topic on our agenda for the present subsection is the more general framework Set-Set. We return to the language with just one disjunction connective ∨, and the Beth semantics. The contrast with Kripke’s treatment of disjunction is evident from a consideration of the schema A ∨ B A, B. Since this is valid on the Kripke semantics but not on the Beth semantics, it is clear that the two associated gcr’s are quite different, even though the associated consequence relations are identical (and coinciding with the consequence relation IL ). Letting Kr and Be be the gcr’s given by Kripke’s and Beth’s semantics for IL respectively, then we can state the following observation (due to Gabbay) succinctly. The “+” and “−” superscripts denote, as in 6.31, the maximal and minimal gcr’s agreeing with the consequence relation to whose name they are appended. − Theorem 6.43.8 Kr = + IL and Be = IL .

A proof may be found in Chapter 3 of Gabbay [1981], Thm. 6 of §1 (for the first assertion) and Thm. 5 of §2 (for the second).

6.44

Disjunction in the ‘Possibilities’ Semantics for Modal Logic

In Kripke’s semantics for modal logic (with classical logic for the boolean connectives), no less than in his semantics for intuitionistic logic, a disjunction counts as true at a point just in case at least one disjunct is true at that point. This seems reasonable because for many applications we think of the points as possible worlds, fully determinate in every respect. Possible worlds are sometimes thought of as ‘points’ in logical space. We can similarly think of ‘regions’ of logical space, either as sets of such points, or as primitive in their own right, their interrelations being construed in mereological (part/whole) terms. The latter perspective is quite interesting as the articulation of a philosophical dissatisfaction with the idea of fully determinate non-actual worlds (see Humberstone [1981a]). The thought would be that what is possible is what is true throughout some non-empty such region – or ‘possibility’ – rather than true at some point. Dually, what is necessary is what is true throughout every such region. This is analogous to explaining necessity in the possible worlds semantics in terms of a universal accessibility relation. Corresponding to the general case in the latter setting, in which we allow arbitrary accessibility relations, an elegant treatment (from Fine [1974]) in the possibilities framework is available, using—as in the algebraic semantics for modal logic—a singulary operation instead. If we denote this operation by f , then the clause for would run (relative to a model): |=x A iff |=f (x) A. We encountered such a clause in our discussion of modal logic (2.22.9, p. 288), but were in that context thinking of the x as a point rather than as a region. This makes a difference to what treatment the boolean connectives should receive, and it is on these, and specifically on ∨, that we shall concentrate in what follows. After reviewing the possibilities semantics, we will close this subsection with some discussion of the distinctive treatment given to disjunction in Fine [1974]. Although we do not have to consider an accessibility relation as in §2.2, or even—since we shall say no more about and —its functional surrogate f just introduced, there is another binary relation between possibilities that does

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need to be attended to. A very general possibility, such as the possibility that grass is green, has various refinements or further specifications, for example, that grass is green and is eaten by cows. If we think of the spatial metaphor, the former possibility represents a region of logical space which has the latter as a proper subregion. As we add more specifications, we single out a smaller region. This results in a tension over whether to represent the fact that y is a refinement of x by a notation such as “y x” (or “y ⊆ x”), thinking of the part-whole partial ordering of regions, or on the other hand as “x y” (or “x ⊆ y”), thinking instead of increase in specificity as we pass from x to y. Here we adopt the latter choice, and write “x y” to mean that either x = y or else y is a refinement of x, and use “y x”, when convenient, for the same thing. (To emphasize the part/whole way of thinking, the following subsection adopts the reverse convention.) Since in passing from x to y x things become more specific, whatever is true over x should remain true over y. In other words, for this interpretation of the machinery we have been informally setting up, a persistence condition is appropriate. We now present the apparatus more formally, with a view to studying the boolean connectives in terms of it. (Reminder : boolean connectives are simply those connectives # for which we have defined the notion of a #-boolean valuation. It is not assumed that such connectives will exhibit the logical behaviour which a restriction to those valuations would induce.) Actually, we shall only consider ∧, ¬, and ∨. The models we consider are of the form (P, , V ) in which (P, ) is a poset (the frame of the model) and V assigns subsets of P to propositional variables, subject to two conditions, the first of which is familiar from 2.32; unless otherwise stated, individual variables range over elements of P : (Persistence)

For all x, y ∈ P, if x ∈ V (pi ) and x y, then y ∈ V (pi )

(Refinability)

For all x, if ∀y x ∃z y z ∈ V (pi ), then x ∈ V (pi ).

In the terminology of 6.43, this second condition says that if pi (or, strictly, V (pi )) is weakly inevitable at a possibility x, then pi is true at x. It is easier to get a feel for this condition if we have negation at our disposal; although the following clause for ¬ is exactly as in the Kripke semantics for IL, (Refinability) will exert a ‘classicizing’ influence over subsequent proceedings. We also give the clauses for ∧ and ∨, and comment on them in due course. Let M = (P , , V ) be a model; then for any variable pi , and any formulas A, B, and any x ∈ P : [Vbls]

M |=x pi iff x ∈ V (pi );

[¬]

M |=x ¬A iff ∀y x, M |=y A;

[∧]

M |=x A ∧ B iff M |=x A and M |=x B;

[∨]∀∃

M |=x A ∨ B iff ∀y x ∃z y M |=z A or M |=z B.

The clause for ¬ says that ¬A is true w.r.t. a possibility if no further refinement thereof makes A true. In terms of the spatial analogy: ¬A is true throughout a region just in case no subregion verifies A. It would not have been appropriate instead to have x verify ¬A just when x does not verify A, since we want to allow that x simply leaves open the question of whether or not A: it is not so


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specific as either answer this question affirmatively or negatively. There is also a temporal analogy to bear in mind, giving rise to a semantics for tense logic – mentioned above in 5.12 – in which formulas are evaluated w.r.t. to intervals rather than instants, and when considering whether “Tess is not at home” is true over a certain interval, we require that Tess not be at home over any subinterval thereof, rather than that she fails to be at-home-over-the-whole-period. We can usefully re-express the condition (Refinability) with this understanding of negation in mind. If we contrapose the above formulation of the condition, and make use of the clause [¬], then we get: if pi is not true over x, then there is some y x over which ¬pi is true. We shall see below that (Refinability), like (Persistence), applies to formulas in general and not just to the propositional variables. Thus the general import of the condition is that any possibility which fails to verify a formula has a refinement verifying its negation. Every case of indeterminacy or non-specificity comes eventually, on this picture, to be resolved positively or negatively at some later stage of refinement. This is not to say that there is some single later stage of refinement at which every such indeterminacy gets resolved: that would be the possible worlds account. If we think of the poset (P, ) as arising out of a boolean lattice, what is not required for the present treatment is that this lattice is atomic. Such atoms would be (in one-to-one correspondence with) possible worlds. (The zero of the lattice would represent the one region which is not a possibility—the empty region. Here the tension, remarked on above, as to how to notate the refinement relation, is particularly acute, since the associated poset is really not (P , ), on our notational conventions, but (P , ).) Passing over the (presumably) uncontroversial case of [∧], we turn to the distinctive [∨]∀∃ , so named for the quantifier-pattern on its rhs. It is clear that a clause deeming a possibility to verify a disjunction just in case it verifies at least one disjunct would not be appropriate, since a region of logical space might be sufficiently circumscribed so as to allow that A ∨ B without determinately requiring the one or the other disjunct. (Think of the complex, but not fully specific, possibility described by a consistent story.) This consideration by no means dictates the clause [∨]∀∃ as appropriate – for example, the more demanding [∨]Beth would deserve our attention in a more extended treatment – but here we content ourselves with an exposition of Humberstone [1981a], in which (roughly) the current semantic treatment is seen to issue in the tautological consequence relation CL , despite the various similarities with Kripke’s semantics for IL. The emergence of classical logic (for the connectives under consideration) in this rather uncustomary setting would be no surprise if the logical framework we were working in was Fmla, in view of a result of Gödel [1933b], given in Chapter 8 (p. 1216) as 8.21.5, according to which the provable formulas of IL whose only connectives are ¬ and ∧ are precisely those provable in CL (the tautologies, then). For the truth-conditions given by [∨]∀∃ to disjunctions A ∨ B are precisely those the Kripke semantics (as well as the present semantics) gives to the corresponding formulas ¬(¬A ∧ ¬B). Thus we are dealing essentially with just those formulas to which Gödel’s result applies. However, as is noted in the discussion after 2.32.1 (p. 304), this result most emphatically does not extend from Fmla to Set-Fmla. What allows this extension for the present semantic proposal is the condition (Refinability). First, we need to show that the constraint this condition imposes on propositional variables spreads upward to formulas of arbitrary complexity, just as (Persistence) does – exactly as in

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2.32.3 (p. 308) – and as with (Barring) in the Beth semantics (6.43.2). We do this in a ‘Refinability Lemma’, 6.44.2. Remark 6.44.1 According to the present semantics, A ∨ B is true at a point in a model when !A! ∪ !B! is weakly inevitable at that point, whereas on the Beth semantics we require instead that this union be, as it was put in 6.43, strongly inevitable at the point. (And of course for the Kripke semantics, we demand, even more strongly, that the point in question belong to !A! ∪ !B!.) We noted in 6.43 that the weak inevitability requirement amounted to treating A ∨ B the way that ¬¬(A ∨ B) is treated by the Kripke semantics. Yet in the discussion above of Gödel’s [1933b] result, we cited instead the formula ¬(¬A ∧ ¬B). There is no inconsistency here, since ¬¬(A ∨ B) and ¬(¬A ∧¬B) are IL-equivalent. Lemma 6.44.2 In any possibility model M = (P, , V ), for all x ∈ P and all formulas A, if ∀y x ∃z y M |=z A, then M |=x A. Proof. By induction on the complexity of A. (Recall that the only connectives are ∧, ∨, and ¬.) As remarked above, we note that a more convenient formulation may be given in contraposed terms: any point at which a formula which is false (= not true) has a later point at all of whose extensions the formula is false. Exercise 6.44.3 Show that the rules (∨E) and (¬¬E) of Nat preserve the property of holding in any possibility model. (Hint: 6.44.2 will be needed.) The soundness of Nat w.r.t the present semantics is easily established; the rules (∨E) and (¬¬E) were explicitly mentioned in the above Exercise because of the novelty of [∨]∀∃ , in the first case, and the failure, in the second case, of the rule in the Beth and Kripke semantics for IL. Thus by the completeness theorem for Nat (1.25.1, p. 127: though here we rely on the fact, extractable from the proof of that result, that the →-rules are not needed for the proof of valid sequents in ∧, ∨, ¬), all tautologous sequents of the present language are valid on the possibility semantics. The converse follows from a consideration of the one-element frame. Thus we have: Theorem 6.44.4 A sequent of Set-Fmla, all of whose connectives are among ∧, ∨, ¬, is tautologous if and only if it holds in every possibility model. Exercise 6.44.5 Explain what goes wrong with 6.44.4 if “Set-Fmla” is replaced by “Set-Set”. The question arises as to whether, giving the possibility models of the present subsection an epistemic construal like that suggested for the Beth models of 6.43 (or the Kripke models of 2.32), we have provided a justification for classical logic—at least in Set-Fmla, and for the connectives mentioned in 6.44.4—in terms of proof and assertibility: a justification of a type generally assumed to be available only for intuitionistic logic. Let us compare the present semantics with that of Beth. There are two differences: the use of (Refinability) in place of (Barring) as a condition on models, and the use of [∨]∀∃ in place of


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[∨]Beth in the definition of truth. The two differences are not independent. If we retain (Barring) from the Beth semantics and combine it with [∨]∀∃ , then we cannot establish the inductive case for ∨ in the proof that arbitrary formulas have the Barring property—so that the analogue of Lemma 6.43.2 for the envisaged hybrid semantics does not go through. (We leave the reader to ponder the combination of [∨]Beth from the Beth semantics with condition (Barring) with (Refinability) from the present semantics.) Now these two differences between the two semantic treatments are related in that (Barring) and [∨]Beth both employ the notion of strong inevitability precisely where (Refinability) and [∨]∀∃ employ instead weak inevitability; both differences are accordingly nullified when attention is restricted to finite models, by 6.43.5(ii). (And indeed, we already remarked in 6.43 that the logic determined by the class of finite Beth models is CL.) Thus the question of whether the kind of epistemic considerations generally held to motivate IL rather than CL might well be reopened with the present semantics in mind; we do not go into this here, however. The spatial metaphor deployed above, in which y’s being a refinement of x is thought of in terms of region y’s being a subregion of region x, can be given an epistemic flavour, if we think of a region of logical space as representing the information that the actual world lies in that region. This works well for the case of empirical information, though not so well for the application to mathematical discourse with which IL is traditionally associated, since in this case the acquisition of information reveals a previously open epistemic possibility not to have been a real possibility after all. However, we shall ignore this complication. The condition (Persistence) is a ‘forward-looking’ condition on models in the sense that it constrains truth to propagate along R-chains, to recall the notation for Kripke models from 2.32, or down -chains in the present notation: from truth at a possibility, there follows truth at any refinement. By contrast, both the conditions (Barring) and (Refinability) are ‘backward-looking’ conditions, propagating truth in the reverse direction. We conclude this subsection with another approach to disjunction, from Fine [1974], which also involves a backward-looking condition on models. Although devised for the case of relevant logic, we can consider its application more generally, and so considered, it affords a useful point of contrast with the above [∨]∀∃ treatment. The latter shares with the Beth semantics the feature that not only are the models not required to consist exclusively of elements which verify at least one disjunct of disjunctions they verify: they are not even required to contain any such ‘prime’ elements. (Some philosophical mileage is extracted from this feature of the possibilities semantics in Humberstone [1981a]: the semantic account is compatible with the view that fully determinate possible worlds are a fiction we can do without, holding that any specification of a way things could have been is capable of further refinement. That is, we could impose a condition on our frames to the effect that for every x there is a y > x, without altering the logic determined.) Fine’s treatment of disjunction is not like this. Although not all elements are ‘prime’, or, as we shall say, saturated, the truth of disjunctions at all points depends on the behaviour of the disjuncts at refinements which do possess this feature. (It follows from the clause [∨]Fine introduced below that the existence of some such saturated points is a necessary condition for the falsity of a disjunction at any point.) We consider as the frames for this approach certain expansions of those

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underlying possibility models, namely structures (P, , S) in which (P, ) is a poset and S ⊆ P will contain the ‘saturated’ elements. S is required to be closed under . When we form models on these frames, we require that V (pi ) is also so closed: in other words, we impose the condition (Persistence). The ‘backward’ looking condition on models alluded to above will be introduced after the definition of truth is presented. We shall suppose that disjunction is the only connective present so as not to be distracted by other matters (such as how to treat negation). As before, we have the usual clause for propositional variables in our models M = (P, , S, V ): [Vbls]

M |=x pi iff x ∈ V (pi ).

The distinguished subset S enters into the clause for disjunction: [∨]Fine

M |=x A ∨ B iff ∀y x if y ∈ S then either M |=z A or M |=z B.

Exercise 6.44.6 (i) Check that disjunctions are persistent in the present models. (ii) Verify that the set of formulas true at a saturated element is indeed prime (i.e., that for all x ∈ S, for all A, B: M |=x A ∨ B only if M |=x A or M |=x B). Without further ado we can see that (∨I) preserves the property of holding at a point in a model, and accordingly that of holding in a model. This requires 6.44.6(i). By 6.44.6(ii), (∨E) preserves the property of holding at a saturated point. But for a soundness proof relating the (∨I), (∨E) proof system to validity over the class of frames currently in play, we should like to show that (∨E) preserves the property of holding in a model. By 6.31.9(i), p. 847, we cannot do this in the present context by showing the rule to preserve the property of holding at a point, since the valuation associated with a given point (in a model) is not in general ‘prime’ in the sense of 6.31. A look at the proof of 6.43.3, doing the same thing in the Beth semantics for IL, where again primeness was lacking, suggests an approach. This motivates the second condition on models that needs to be imposed – analogous to (Barring) for 6.43.3. For consider the premiss-sequents for an application of this rule: Γ A ∨ B

Δ, A C

Θ, B C.

Suppose that the conclusion-sequent Γ, Δ, Θ C fails at x in M. Then in particular, x verifies Γ, and so, assuming the first premiss sequent holds in M, M |=x A ∨ B. Thus every saturated refinement of x verifies A or else B. So all such refinements verify C, on the assumption that the second and third premiss-sequents hold in M, since x itself verifies Δ ∪ Θ. How does this help to show that x itself verifies C? What we need is a condition which will ensure that whenever a formula is true at all of the saturated refinements of a point, the formula is already true at that point. Let us impose such a condition, as usual, on the Valuations of our models (P, , S, V ): (Saturated Refinability)

For all x, if ∀y x such that y ∈ S, y ∈ V (pi ), then x ∈ V (pi ).

From now on, we require models to satisfy both (Persistence) and this new condition. The condition extends to arbitrary formulas:


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Lemma 6.44.7 For any model M = (P , , S, V ) and any x ∈ P, for all formulas A, if M |=y A for every y x such that y ∈ S, then M |=x A. Proof. The basis case for an induction on complexity is settled by (Saturated Refinability). The only inductive case is that of A = B ∨ C. Suppose M |=x A. Then by [∨]Fine , there exists y ∈ S, y x, M |=y B and M |=y C. So by 6.44.6(ii) M |=y B ∨ C. Thus not every saturated refinement of x verifies A. (Note that the inductive hypothesis is not actually required here; matters would be different if we had ∧ present, interpreted by the earlier [∧].) Thus (∨E) preserves the property of holding in a model, securing the soundness (‘only if’) half of: Theorem 6.44.8 A sequent is provable from (R), (∨I), and (∨E) if and only if it is valid on every frame (P, , S). Proof. The completeness (‘if’) part of the Theorem remains to be established. A suitable canonical model is obtained by taking P as the set of deductively closed sets of formulas (relative to the proof system specified in the Theorem), S comprising the prime deductively closed sets, and x y iff x ⊆ y. It is left to the reader to verify that, when V (pi ) = {x ∈ P | pi ∈ x}, the condition (Saturated Refinability) is satisfied and that membership in x and truth at x in this model coincide for all formulas, for arbitrary x ∈ P .

6.45

Disjunction in Urquhart-style Semantics for Relevant Logic

We shall discuss disjunction in the Fmla system R, axiomatized in 2.33 by subjoining to a basis consisting of the pure implicational schemata B, C, I, W (alias Pref, Perm, Id, Contrac) and the conjunction schemata (∧E)1 (∧I)

(A ∧ B) → A

(A ∧ B) → B

(∧E)2

((A → B) ∧ (A → C)) → (A → (B ∧ C))

the following schemata for disjunction: (∨I)1

A → (A ∨ B)

(∨E)

((A → C) ∧ (B → C)) → ((A ∨ B) → C)

(Distrib)

(∨I)2

B → (A ∨ B)

(A ∧ (B ∨ C)) → ((A ∧ B) ∨ (A ∧ C)).

The (non-zero-premiss) rules are Modus Ponens and: Adjunction

A A ∧ B

B

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The use of the labels (∨E) and (∧I) is of course intended to be reminiscent of their use for the natural deduction rules of Nat; since the latter will not be under discussion here, no confusion should result. Axioms governing negation can be omitted for the sake of the present discussion, which concerns therefore the system sometimes called ‘Positive R’. Interestingly this is not so—even though our primary interest is in disjunction—for the above principles concerning conjunction. (Of course, since we are considering a Fmla system, the ever-present → is not to be avoided.) The reason for this is not only the above mixing principle (Distrib), but also the fact that “∧” puts in an appearance in the schema (∨E). A proof of, for example (p ∨ q) → (q ∨ p) on the above basis requires not only appeal to that schema but also to the ∧-rule of Adjunction. Exercise 6.45.1 What would go wrong if (∨E) were replaced by the ∧-free schema: (A → C) → ((B → C) → ((A ∨ B) → C))? Remark 6.45.2 The involvement of ∧ noted above in the proof of principles pertaining to (→ and) ∨ creates an asymmetry between conjunction and disjunction. To remedy this, we could think of the ‘deductive glue’ role of conjunction taken over by a separate connective—call it (like the above rule) adjunction. Let us symbolize this by “&”. The envisaged new basis for Positive R (with &) would state the rule of Adjunction as passing from A and B to A & B, and replace the first occurrence of ∧ in (∧I) and (∨E) by “&”. For a relevant logic in which the rule of Adjunction (for ∧) is replaced by a ‘rule form’ of (∧I), justifying the passage from A → B and A → C to A → (B ∧ C), see Semenenko [1979]. Exercise 6.45.3 Add the axioms governing t given in 2.33 to the basis suggested in 6.45.2 and show that the old Adjunction rule, licensing the transition from A and B to A ∧ B, is derivable. After this digression we return to Positive R as axiomatized initially (without “&”), and remind the reader of the Urquhart semilattice semantics from 2.33, which uses models (in our notation) (S, ·, 1, V ) whose frames (S, ·, 1) are semilattices with identity element 1. If M is such a model, then we have in the definition of truth, the following clauses for the connectives → and ∧: M, x |= A → B iff for all y ∈ S such that M, y |= A, we have M, xy |= B; M, x |= A ∧ B iff M, x |= A and M, y |= B. (As in 2.33, we denote x · y by xy, and we write “y |= A” in place of “|=y A”.) The formulas valid on every semilattice frame ( = true at the identity element 1 in every model on the frame) in which only ∧ and → appear are then, as can be gleaned from Urquhart [1972], precisely those formulas in these connectives which are provable from the above axioms and rules governing ∧ and → in R. This coincidence no longer holds for the {∧, ∨, →} fragment of R when disjunction is treated, as suggested by Urquhart, with the following clause:


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M, x |= A ∨ B iff M, x |= A or M, x |= B. Rather, the desired soundness theorem for positive R goes through, but not the completeness theorem; examples – discovered by Dunn and by Meyer – of formulas valid on every frame but not provable in R may be found in Urquhart’s paper. (Or see 8.13.21 below, p. 1210.) Such a completeness theorem can be obtained, however, by modifying the semantics. We will indicate the kind of modification called for, referring the reader to Humberstone [1988b] for the completeness proof; however, to illustrate the machinery, some remarks pertinent to establishing the soundness of the above axiomatization will be made. (Note: we are considering in this discussion what is sometimes called ‘extensional disjunction’ in the literature on relevant logic, and usually symbolized by “∨”, not the ‘intensional disjunction’ or fission briefly mentioned at the end of 6.13.) In the preceding subsection, the distinction was made between regions and points of logical space. Now the elements of a semilattice frame are supposed to be thought of as representing ‘pieces of information’. Generally, information has the form: you are in such-and-such region of logical space, rather than the form: you are at such-and-such point of logical space. That is, in general, the propositional content of a piece of information is not so specific as to narrow down the class of possible worlds (compatible with the information) to one. So we should ask what the appropriate way to think of disjunction is when formulas are evaluated w.r.t. regions of logical space. To keep this image in mind, we will reverse the practice of 6.44 and write “” for the subregion relation; renotated in these terms, the semantic clause for ∨ we offered there in the possibilities semantics—possibilities being thought of as non-empty regions—becomes: [∨]∀∃

M, x |= A ∨ B iff for all y x, there exists z y such that M, z |= A or M, z |= B.

(We have also raised the subscripted “x”, “y”, “z” to conform with present conventions.) Rather than adopting [∨]∀∃ for our current purposes, we make a somewhat different-looking proposal (closer in spirit in fact to [∨]Beth , though we shall not make a detailed comparison). More in keeping with Urquhart’s semantics would be the introduction of a new binary operation to combine pieces of information to form their ‘common content’, or alternatively, thinking now in terms of the logical space metaphor, for aggregating two regions of logical space into the smallest region including each of them. We denote this operation by “+”. Just as the operation · is used in the interpretation of →, so this operation will enter into the interpretation of ∨. For the machinery to work smoothly, we shall also need to introduce a special zero element for this operation, which will be written as 0. We call the structures to be employed ‘expanded frames’, though they are not all expansions of Urquhart’s frames, since the (S, ·, 1) reduct is not required to be a semilattice (with unit element 1). We shall be requiring commutativity and associativity here, but not idempotence; this last property would end up— taken in conjunction with the other demands we need to make for showing the soundness of R—validating the Mingle schema A → (A → A) and providing a semantics for the Fmla logic RM mentioned in 2.33 (e.g., on p. 334). To be precise, then, an expanded frame is an algebra (S, ·, +, 1, 0) similarity type 2, 2, 0, 0 whose reducts (S, ·, 1) and (S, +, 0) are respectively a commu-

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tative semigroup with unit element 1 and a semilattice with zero element 0, in which four conditions are satisfied for all x, y, z ∈ S, the first two being: (y + z) = xy + xz

Ring Distribution x Zeroing

0x = 0.

For the other two conditions, we introduce the partial ordering associated with the (S, +, 0) semilattice (thinking of + as forming joins—0.13): xy ⇔ x+y =y (The definition x y ⇔ ∃z ∈ S. x + z = y would be equivalent; we have in effect recovered the earlier subrelation by this definition.) We write x2 for xx (alias x · x) in the first of the two remaining conditions: Pseudo-idempotence Decomposition

x x2

x u + v ⇒ ∃u u, v v.x = u + v .

Before proceeding to describe models on these expanded frames, we make a few remarks on the four conditions. Ring Distribution is a form of distributivity found in a class of algebras called rings, not reviewed in §0.1. Pseudoidempotence is incorporated for validating the schema Contrac; as already noted, we have “” rather than “=” to avoid validating the converse schema (equivalent, giving the other axioms, to Mingle). The Decomposition condition is really just a condition on the additive semilattice (S, +); semilattices satisfying this condition have been called ‘distributive semilattices’ in the literature. (See notes.) It is not hard to verify that a (join-)semilattice reduct of any distributive lattice must satisfy the condition. (We could equip our expanded frames with a lattice operation dual to +, giving a distributive lattice, and using this operation in the clause for ∧ in the definition of truth, but this will not be not necessary.) In 2.33 we saw that Urquhart imposed special conditions on (the Valuations of) models in order to obtain a semantic account of RM0 and IL, but no such conditions for the case of (implicational) R itself. (In fact 2.33.4(ii), from p. 339, relabelled with proof as 7.23.5(ii) below – p. 1090 – touches only in the Mingle side of the story and not on IL; but the reader will be able to reconstruct the appropriate condition on models for the latter case, or else find it in Urquhart [1971], [1972].) For our account of positive R, we shall need some such conditions on models, constraining the way Valuations treat the additive semilattice structure. We say that a model on an expanded frame is a 6-tuple (S, ·, +, 1, 0, V ) in which for each propositional variable pi , V (pi ) ⊆ S and for all x, y ∈ S: (Plus Condition)

x + y ∈ V (pi ) ⇔ x ∈ V (pi ) & y ∈ V (pi );

(Zero Condition)

0 ∈ V (pi ).

The latter condition gives us some idea of how the element 0 will be behaving in our models: as a maximally undifferentiating element, verifying all formulas. Here we have laid this down for the atomic formulas of the present language;


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the general result is given as 6.45.4(ii) below. Thinking in terms of regions of logical space, 0 is the only such region as not to count as a ‘possibility’ in the terminology of 6.44. The former condition is a combination of what, again in the terminology of that discussion, we called a forward-looking and a backwardlooking condition. Its ⇒-direction is just (Persistence) in a new notation, and its ⇐-direction is a ‘backward’ condition reminiscent of (Barring), (Refinability), and (Saturated Refinability). To define truth in a model M = (S, ·, +, 1, 0, V ), we employ the same clauses for ∧ and → as were given at the start of this subsection, and for propositional variables pi of course we say that M, x |= pi iff x ∈ V (pi ). The novelty is the case of disjunction, for which we say [∨]Plus

M, x |= A ∨ B iff there exist y, z ∈ S with x = y+z such that M, y |= A and M, z |= B.

The idea is that a disjunction counts as true over a region if that region can be decomposed into subregions verifying the separate disjuncts; note that if A is already true over the whole of x, we can choose x as the required y and the element 0 as the z. As noted above, though not yet shown, we shall always be assured that M, 0 |= B. This will secure the validity of (∨I)1 , with a similar result in the case of (∨I)2 . The notion of validity on an expanded frame here employed is as for the original frames: truth at the element 1 in every model on the frame. We should now note that the properties conferred by the Plus Condition and the Zero Condition on the propositional variables: Lemma 6.45.4 For any formula A and any model M = (S, ·, +, 1, 0, V ): (i) M, x + y |= A if and only if M, x |= A and M, y |= A, for all x, y ∈ S. (ii) M, 0 |= A. Proof. In both cases, by induction on the complexity of A, the basis being given by the Plus and Zero Conditions respectively. The reader is invited to work through the inductive part of the proof here, to see how the various conditions on expanded frames pull their weight in the argument. (The Decomposition condition is needed for the case of A = B → C, for example.) Since the novelties of the present account pertain to disjunction, and we have already indicated how to show that the schemes (∨I)1 and (∨I)2 are valid on every expanded frame, we will content ourselves here with showing the same for (∨E). We must show that for any model M = (S, ·, +, 1, 0, V ), we have M, 1 |= ((A → C) ∧ (B → C)) → ((A ∨ B) → C) whatever formulas A, B, and C may be. So suppose M, x |= (A → C)∧(B → C), with a view to showing that M1x(= x) |= (A ∨ B) → C. Then – dropping the reference to M, for brevity – we have x |= A → C and x |= B → C, and we must show that if y |= A ∨ B then xy |= C (for arbitrary y ∈ S). Accordingly, suppose y |= A ∨ B, and hence, by [∨]Plus , that there are z1 , z2 with y = z1 + z2 and z1 |= A, z2 |= B. Thus xz 1 |= C and xz 2 |= C, so by 6.45.4(i), xz1 +xz2 |= C. But this last mentioned element is x(z1 + z2 ) by Ring Distribution, and since

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y = z1 + z2 , we have our desired conclusion that xy |= C. The reader may care also to check the schema (Distrib), which also involves ∨, to say nothing of the ∨-free schemas. The result will be a soundness theorem for positive R w.r.t. the present semantics, given as the ‘only if’ half of the following; for the ‘if’ half, Humberstone [1988b] may be consulted. (The above argument that (∨E) is valid on all expanded frames is an improvement on the corresponding argument, which appears to have got somewhat scrambled, at p. 70 of that paper.) Theorem 6.45.5 A formula is provable in positive R if and only if it is valid on every expanded frame. Remark 6.45.6 The sentential constants t and T , with axioms as in 2.33, are readily accommodated in the present semantics (in the sense of permitting 6.45.5 to be demonstrated for this extension) by supplying the clauses: x |= t ⇔ x 1, and x |= T (for every x). In fact the same goes also for F (from 2.33), with: x |= F ⇔ x = 0. The treatment of disjunction in R afforded by models on expanded frames with truth defined (inter alia) using [∨]Plus is certainly not the only kind of treatment possible. For example, Fine [1974] uses the clause [∨]Fine from 6.44 in a certain elaboration of the models of Routley and Meyer [1973] , while those authors themselves use a direct (‘classical’, ‘boolean’) clause to the effect that a disjunction is true at a point iff at least one disjunct is. Rather, the [∨]Plus treatment represents a ‘minimal mutilation’ of Urquhart’s semantics to bring validity into line with provability in positive R. (Urquhart [1972] is actually concerned to criticize this system in the light of his semantics; but perhaps it will be agreed that the ‘pieces of information’ idea naturally suggests the above deviation from a direct boolean clause for disjunction.) In the following subsection, we shall disentangle the machinery for dealing with disjunction from other parts of the expanded frame apparatus in order to make connections with the Strong/Weak Claim distinction from 1.14 as that distinction bears on disjunction.

6.46

‘Plus’ Semantics and Valuational Semantics for Disjunction

We have seen several approaches to the semantics of disjunction which allow disjunctions to be true at points in models at which neither disjunct is true, while at the same time inducing ∨-classical consequence relations; in particular (∨E), which might have seemed a likely casualty of such approaches, manages in each case to preserve validity on the frames of these models. The treatments incorporated [∨]Beth , [∨]∀∃ , and [∨]Plus , studied especially in 6.43, 6.44, and 6.45, respectively, provide for this feature without requiring that any point in a model resolve every disjunction, in the sense of verifying one or other disjunct of all disjunctions verified there, while the treatment incorporated [∨]Fine (also mentioned in 6.44) does require some such points (the ‘saturated’ elements). In this subsection, we will extract the ‘Plus’ semantics for disjunction from 6.45, and present it in isolation from that part of the machinery of expanded (Urquhartstyle) frames employed for the semantic treatment of relevant implication. Those expanded frames were the structures (S, ·, +, 1, 0) of type 2, 2, 0, 0 meeting various conditions mentioned in 6.45, so our interest now is in their semilattice (with


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zero) reducts (S, +, 0). The three conditions of Ring Distribution, Zeroing, and Pseudo-idempotence all involve the now discarded operation and lapse in the present setting; Decomposition is an exclusively additive condition and remains in force. We call these structures simply semilattice frames; a model on such a frame is a semilattice model, it supplements the frame with a Valuation as in 6.45, satisfying for each propositional variable pi , V (pi ) ⊆ S and all x, y ∈ S: (Plus Condition)

x + y ∈ V (pi ) ⇔ x ∈ V (pi ) & y ∈ V (pi );

(Zero Condition)

0 ∈ V (pi ).

We are mainly interested in disjunction, but include conjunction also in the language for added zest; negation will not be considered, and, in contrast with 6.45, nor will (any kind of) implication, for which reason we move from Fmla back to Set-Fmla. As usual, a sequent holds in a semilattice model if every point in the model at which all the formulas on the left of the “” are true is a point at which the formula on the right is true, and the sequent is valid a semilattice frame if it holds in every semilattice model on that frame. We repeat the clauses for ∧ and ∨ to be used in the definition of truth at a point x ∈ S in a model M = (S, +, 0, V ): [∧] [∨]P lus

M, x |= A ∧ B iff M, x |= A and M, x |= B. M, x |= A ∨ B iff there exist y, z ∈ S with x = y+z such that M, y |= A and M, z |= B.

As in 6.45.4, the Plus and Zero Conditions are satisfied by arbitrary formulas: Lemma 6.46.1 For any formula A and any model M = (S, +, 0, V ): (i) M, x + y |= A if and only if M, x |= A and M, y |= A, for all x, y ∈ S; (ii) M, 0 |= A. The proof is by induction on the complexity of A; the Decomposition condition is needed for the ‘only if’ direction of (i) in the case of A = B ∨ C. 6.46.1 is a Lemma for the ‘only if’ direction of 6.46.2, which is the analogue of 6.45.5 (p. 910) in the present setting. The references cited there give details sufficient for a proof of the ‘if’ direction. A sequent Γ A is valid on a semilattice frame (S, +, 0) if it holds in every model M on that frame in the (usual) sense that for any x ∈ S with M, x |= C for each C ∈ Γ, we have M, x |= A. Theorem 6.46.2 A sequent is provable using (∧I), (∧E), (∨I), (∨E), alongside the standard structural rules, if and only if it is valid on every semilattice frame. This concludes our discussion of the semilattice semantics for (conjunction and) disjunction. Note that, in contrast with (say) 2.14, which would yield – applied to the disjunction fragment of the above proof system – something also deserving the name ‘semilattice semantics’, we have here, as throughout this section, been interested in model-theoretic semantics. For the remainder of the present subsection, we discuss, with special attention to (∨E), valuational semantics for disjunction in the style of §1.1.

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The relational connection between formulas and (arbitrary) valuations has, we noted in 1.14, conjunctive combinations on the right. We denote such a combination of valuations v1 and v2 by v1 v2 . There is an intimate relationship between conjunctive combination and the + operation of this and the preceding subsection, which we can spell out with the aid of the valuation associated with a point of one of the models M of that discussion, namely the valuation vx which assigns T to A iff M, x |= A: vx+y = vx vy In the terminology of 6.23, this makes the valuation vx+y behave in all logically relevant ways like the supervaluation SV where V = {vx , vy }. (As noted in 6.23, there is a difference, in that the supervaluation is not defined for the cases in which the elements of V assign different values to a formula, whereas the conjunctive combination returns the value F in this case: but this makes no difference to logic, since truth-preservation is not affected by any internal subdivision—such as false vs. undefined —amongst the non-truths. Indeed even this formulation takes the notion of falsity, when this is distinct from failure of truth, far too seriously.) For the semilattice models of our recent discussion, the equation above is given by 6.46.1(i). The conjunctive combination of the empty set of valuations assigns T to all formulas to which every valuation in that set assigns T; thus, there being no valuations in ∅, every such valuations assigns T to all formulas, so the conjunctive combination of the empty set of valuations is the valuation vT verifying every formula. Thus by 6.46.1(ii): v0 = v T . This suggests a purely valuational semantic reworking of the Plus semantics for disjunction, in terms of the following condition on V : [∨]

For all v ∈ V, all A,B, v(A ∨ B) = T iff ∃v1 , v2 ∈ V v1 (A) = v2 (B) = T and v = v 1 v2 .

Theorem 6.46.3 Let V be a collection of valuations for the language whose sole connective is ∨, such that V is closed under (the binary operation) , contains vT , and satisfies the condition [∨] . Then (∨I) and (∨E) have the global preservation characteristic associated with V. Proof. Since for v ∈ V , v = v vT = vT v, if v(A) = T or v(B) = T, then v(A ∨ B) = T. This shows that (∨I) has the local preservation characteristic associated with V (i.e., preserves holding on any given v ∈ V ) and a fortiori the global preservation characteristic. For (∨E), suppose that Γ A ∨ B and Δ, A C and Θ, B C all hold on every v ∈ V , and that v(Γ ∪ Δ ∪ Θ) = T, with a view to showing that v(C) = T. Instantiating the universally quantified variable v as v, we get v(A ∨ B) = T. Thus by [∨] , there are v1 , v2 in V with v1 v2 = v, v1 (A) = T, v2 (B) = T. Now instantiating v as v1 and then as v2 we get v1 (C) = v2 (C) = T, so since v1 v2 = v we have v(C) = T. Note that the multiple instantiations of “v ” in the above proof—which is just a reformulation of the results in 6.45—reflect the fact (6.31.9(i)) that (∨E)


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has only the global and not the local preservation characteristic. Since (∨I) is equivalent (given the structural rules) to the zero-premiss rules A A ∨ B and B A ∨ B, the local and global preservation characteristics for any V coincide, and are easily seen to amount to the following condition: [∨]Intro

For all v ∈ V, and all formulas A and B, if v (A) = T or v (B) = T, then v (A ∨ B) = T

The converse of this condition is satisfied precisely when all valuations in V are prime, in the sense of the discussion surrounding 6.31.9 (p. 847): [∨]Prime

For all v ∈ V, and all formulas A and B, if v (A ∨ B) = T then v(A) = T or v(B) = T.

Thus V ⊆ BV ∨ iff V satisfies both [∨]Intro and [∨]Prime . In the case of [∨]Intro we have the result that (∨I) possesses the global preservation characteristic associated with V iff V satisfies [∨]Intro . Some interest attaches to the question of an analogous result for (∨E). (Such questions are raised especially in Garson [1990], [2001]; see also Garson [2010].) We cannot – by contrast with the local preservation characteristic – offer [∨]Prime as a corresponding condition for (∨E), since although V ’s satisfying this condition is sufficient for (∨E) to preserve the property of holding-over-V (holding on every valuation in V ), it is not necessary. For example, from the proof of 6.46.3, we extract the result that the condition [∨] is also sufficient, but this condition on V does not imply [∨]Prime . Indeed the whole point of the semantic clauses [∨]Beth , [∨]∀∃ , [∨]Fine , and [∨]Plus – the last of which we have transposed from the model-theoretic setting to become [∨] – was to have (∨E) preserve the property of holding in a model even though disjunctions could be true at points therein without either disjunct being true. One condition which is not only sufficient but also necessary for (∨E) to preserve holding over V is the following, in which v v means: for all formulas A, v(A) = T ⇒ v (A) = T. [∨]Elim

For all formulas A and B, and all valuations v ∈ V :

v(A ∨ B) = T ⇒ for all formulas C such that v (C) = F, there exists v ∈ V such that v v , v (C) = F and either v (A) = T or v (B) = T. For the ‘necessity’ part of our argument, we shall need to make one assumption on V , namely that Log(V ) – the consequence relation determined by V – is a finitary consequence relation (1.12); recall (1.22.1, p. 113) that this condition is always satisfied when Log(V ) coincides with the consequence relation associated with a proof system in Set-Fmla. Observation 6.46.4 Let V be any class of valuations for which Log(V ) is finitary. Then (∨E) has the global preservation characteristic associated with V if and only if V satisfies [∨]Elim . Proof. ‘If’: immediate from the definition of [∨]Elim ; the finitariness condition is not required. ‘Only if’: Suppose V does not satisfy [∨]Elim . Then there exist formulas A and B such that for some v ∈ V , v(A ∨ B) = T while for some formula C, v(C) =

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F and (1) for every v ∈ V with v v and v (A) = T we have v (C) = T and (2) for every v ∈ V with v v and v (A) = T we have v (C) = T. Let Ξ be the set of all formulas true on v; since these include A ∨ B we have Ξ A ∨ B, where = Log(V ); by (1) and (2), we also have Ξ, A C and Ξ, B C, but since v(C) = F, Ξ C. Now, since is finitary, there exist finite subsets Γ, Δ, Θ ⊆ Ξ with the sequents Γ A ∨ B; Δ, A C; and Θ, B C all holding on every valuation in V , from which (∨E) would deliver the sequent Γ, Δ, Θ C, which, however, does not hold over V , as Γ ∪ Δ ∪ Θ ⊆ Ξ and Ξ C. Thus (∨E) lacks the global preservation characteristic associated with V . Unlike the conditions [∨]Prime , [∨] , etc., [∨]Elim is not suited to incorporation as a clause in an inductive definition of truth. The less significant point to make here is that we have “⇒” rather than “⇔”; one can imagine this fixed. Indeed in 1.26 (see p. 138) the biconditional version appeared, under the name [∨]Garson . Exercise 6.46.5 Show that the condition obtained from [∨]Elim by replacing “⇒” by “⇐” is equivalent to the condition [∨]Intro . (Note: it would be an instance of the ‘converse proposition’ fallacy—1.15—to conclude that [∨]Elim is itself equivalent to the converse of [∨]Intro , viz. [∨]Prime .) The more significant difficulty is the universal quantification “for all formulas C” in [∨]Elim ; this ranges over formulas of arbitrary complexity, including those of complexity greater than A ∨ B itself. So there is no chance of reconstruing that condition as it stands – even with “⇔” in place of “⇒”, as in the condition we called [∨]Garson in the discussion leading up to 1.26.13 (p. 138) – as part of the inductive definition of truth at a point in a model, supporting some relation corresponding to , with the valuations v ∈ V being those induced as vx for points x in the model. (A different worry for an inductive compositional semantics for disjunction – which spreads to other connectives also – is raised at p. 126 of Garson [2001], over the use of such -style persistence relations, on the grounds that they themselves conceal a universal quantifier over arbitrary formulas.) But we noted in 1.26.13 that this feature of the envisaged condition can be avoided, by using instead: [∨]∃∃ For all v ∈ V , v(A ∨ B) = T iff ∃U0 , U1 ⊆ V : U0 ⊆ !A!, U1 ⊆ !B!, and v = (U0 ) (U1 ). Note that this (i.e., [∨]∃∃ ) is close to the reformulation of the ‘Plus’ semantics, except that we have not required that (U0 ), (U1 ) ∈ V . Digression. This would not be fully satisfactory by the lights of Garson [2001], however, for the following reason. If there is something objectionable about using in a clause in the inductive definition of truth, for the reason given parenthetically before the statement of [∨]∃∃ here that there is a problem with in this role, then there is a problem about appealing to conjunctive combinations, since we could just define u v as: u v = u. On the other hand, there may be ways of avoiding the apparent circularity in defining by following the lead of the Kripke (or Beth) semantics for IL and working with relations stipulated to have atomic formulas persistent w.r.t. them and then demonstrably extending this privilege to arbitrary formulas and ‘canonically’ coinciding with the relation in play here. (That is, for the canonical model for IL from 2.32.8,


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MIL = (WIL , RIL , VIL ), the consequence relation IL is determined by the class of all valuations vxMIL with x ∈ WIL , where vxMIL (A) = T iff MIL |=x A. Dropping the superscript, we have for any such vx , vy : vx vy iff xRIL y iff x ⊆ y.) Garson [2001] is interested in what in 1.26 we called the global range of a collection of (pure) rules governing a connectives # as a way of deriving a semantic account of # implicit in the rules, which he calls a natural semantics for # (as governed by those rules): this is what we get when the V in the global range of the rules are precisely those for which every v ∈ V satisfies some condition apt for treatment as a clause in the inductive definition of truth on v (or more accurately, v and other elements of V , over which we may quantify). The issue we have just been airing is whether the condition [∨]∃∃ imposes on v ∈ V is in this sense ‘apt’. See Garson [2001], [2010], for further discussion of similar issues and the general program of extracting natural semantics. End of Digression. We can usefully subsume our earlier clauses for disjunction under [∨]Elim in such a way as to see how they secure the preservation characteristic for (∨E) in which we have been interested. In the case of [∨]Prime , we can see that the consequent of [∨]Prime implies that of [∨]Elim , for a given choice of V , v ∈ V , A, B, thus: for any C with v(C) = F, we may take the promised v v as v itself. For a similar subsumption of [∨] , it is useful to reformulate the consequent of [∨]Elim in a ‘distributed’ form: For all formulas C such that v (C) = F, either (i) there exists v ∈ V such that v v , v (C) = F and v (A) = T or (ii ) there exists v ∈ V such that v v , v (C) = F and v (B) = T Now suppose that the rhs of [∨] is satisfied, i.e., that there exist v1 , v2 ∈ V with v1 (A) = v2 (B) = T and v = v1 v2 . Then the italicized condition above must be met because v1 v and v2 v and if neither (i) nor (ii) is satisfied by (respectively) v1 or v2 this can only be because v1 (C) = T and v2 (C) = T, which contradicts the choice of C as a formula for which v(C) = T, since v = v1 v2 . The reader may undertake to subsume the clauses [∨]Beth , [∨]∀∃ , and [∨]Fine from 6.43,4 in a similar way under [∨]Elim , after first converting then into conditions on collections of valuations in the way [∨]Plus was converted into [∨] . We have two further topics to broach before leaving the valuational semantics of disjunction. For the first, we recall the status of ‘non-prime’ valuations in terms of the discussion preceding 6.31.9 (p. 847). Such valuations illustrate the existence of non-boolean valuations consistent with the consequence relation Log(BV Φ ) where Φ is a collection of connectives (for each of which a notion of booleanness has been defined) containing ∨. In general, these non-boolean valuations fail to respect some determinant for a boolean connective #, and in the present instance of course, that determinant (taking # as ∨) is F, F, F . In Figure 3.13a (p. 390) this is indicated by the use of parentheses around the output-value in the bottom row of the truth-table. In the terminology of Belnap and Massey [1990], such a determinant represents a ‘semantic indeterminacy’ for the connective concerned. An interesting question considered in that paper (and originally in Carnap [1943]), and on which we report here, is the question of whether the removal of one such indeterminacy results in the removal of all.

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Let us restrict our attention, for example, to the valuations consistent with CL understood as the relation of tautological consequence in the language containing every connective considered in §1.1, which also respect the determinant F, F, F for ∨. Then these valuations also respect the determinant F, T for ¬, since CL A ∨ ¬A for any formula A. We need also to address the other determinant for ¬, namely T, F . As we noted in 1.14.2, the valuation vT is consistent with every consequence relation, CL included, and this valuation does not respect T, F . So to remove all indeterminacies, we shall certainly have to restrict attention to valuations other than vT . Thus we are led to consider imposing both restrictions, and to consider (Val (CL ) ∩ Pr ){vT }, where Pr denotes the class of prime valuations (i.e., those respecting F, F, F for ∨). Then, where Φ comprises all connectives considered in §1.1 (including → and V ), we have Observation 6.46.6 BVΦ = (Val(CL ) ∩ Pr) {vT }. This result is proved in Belnap and Massey [1990]. In fact, they give the stronger result that the same equation holds for the result of replacing Pr here by the class of valuations satisfying any given determinant which is not enforced by CL (resolving any indeterminacy, that is, for any connective, in the prescribed boolean manner)—such as for example F, T for ¬ (see also Massey [1981]); and they prove this for a somewhat more extensive class of connectives than the present choice of Φ: in fact, for all connectives of arity 2. The latter restriction is, however, essential, as Belnap and Massey show with an example we shall repeat here. For the example, we extend the language of CL as lately conceived (namely, as Log(BV Φ )) by the addition of a connective maj for the ternary ‘majority’ truth function maj defined by: maj (x1 , x2 , x3 ) = T iff for at least two choices of i ∈ {1, 2, 3}, xi = T. A valuation v is maj-boolean when for all formulas A, B, C, we have v(maj(A,B,C)) = T iff maj (v(A), v(B), v(C)) = T. Note that where Ψ = Φ ∪ {maj}, maj(A,B,C) is synonymous according to Log(BV Ψ ) with (A ∧ B) ∨ (A ∧ C) ∨ (B ∧ C). (The truth-function this stipulation associates with maj on the maj-boolean valuations serves, incidentally, to answer Exercise 3.14.10(ii) from p. 411 in the negative: although essentially ternary, it is self-dual.) Observation 6.46.7 With BVΨ as above, and = Log(BVΨ ), we do not have BVΨ = (Val() ∩ U ) {vT }, where U is the class of valuations satisfying the determinant F, F, F, F for maj. Proof. Letting v1 and v2 be the unique Ψ-boolean valuations satisfying v1 (pi ) = F for all i 1, and v2 (pi ) = F iff i > 1, / BVΨ . To show we have v1 v2 ∈ (Val () ∩ U ) {vT }, even though v1 v2 ∈ this latter part of the claim first, we note that v1 v2 is not ¬-boolean, since v1 v2 (p) = v1 v2 (¬p) = F. (Recall that p1 is p.) Next, we note that these same equations show v1 v2 = vT . It remains to show that v1 v2 ∈ Val () and v ∈ U . The first claim follows from the fact that v1 , v2 ∈ Val () by 1.14.5.


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For the second, suppose that v1 v2 (A) = v1 v2 (B) = v1 v2 (C) = F, and yet v1 v2 (maj(A, B, C)) = T. Then v1 (maj(A,B,C)) = T, so at least two out of A, B, C, are true on v1 (this being Ψ-boolean), and likewise for v2 . It follows that at least one of A, B, C must be true on both valuations v1 and v2 , contradicting the supposition that each of A, B, C is false on v1 v2 . For further philosophical discussion on the significance of such examples, see Belnap and Massey [1990], Massey [1990]. (Warning: the terminology and notation of these writers is quite different from our own. For example, they call a valuation consistent with Log(BV ) sound, and generally mean by a valuation a valuation other than vT meeting this condition.) Our final topic in valuational semantics is something we shall need for the following subsection, as well as being of some interest in its own right. It involves us in a return to the language whose sole connective is ∨, and in a change from considering conjunctive combinations of valuations to considering disjunctive combinations. Rather than just considering disjunctive combinations of pairs of valuations, we need to consider the disjunctive combination (V ) of an arbitrary collection of valuations V ; this is defined to be the valuation which assigns T to a formula A just in case for some v ∈ V , v(A) = T. We shall connect closure under disjunctive combination with left-primeness of a consequence relation, i.e., with the property that Γ A only if for some C ∈ Γ, C A. (This notion was introduced just before 1.21.5, p. 110.) Lemma 6.46.8 If V is any collection of valuations such that for all V ⊆ V , we have (V ) ∈ V , then Log(V ) is left-prime. Proof. Suppose that for each C ∈ Γ, there exists vC ∈ V such that vC (C) = T but vC (A) = F. We must show that in that case Γ, A ∈ / Log(V ), on the assumption that V satisfies the condition in the Lemma. Thus we must find v ∈ V assigning T to each formula in Γ but F to A. It is easily verified that taking V as {vC | C ∈ Γ} gives (V ) as such a v. Now let us consider the smallest ∨-classical consequence relation ∨ on the language whose only connective is ∨. (In fact, any language with ∨ amongst its connectives would do as well, provided we take ∨ as the least ∨-classical consequence relation on that language.) Theorem 6.46.9 ∨ is left-prime. Proof. By 1.13.7, p. 67, ∨ = Log(BV ∨ ), so the result follows by appeal to 6.46.8 once we have checked that V ⊆ BV ∨ implies (V ) ∈ BV∨ . Supposing V ⊆ BV ∨ , this requires that we check that (V ) assigns T to A ∨ B iff it assigns T to at least one of A, B. This is a routine matter, left to the reader.

Exercise 6.46.10 Show that if A ∨ pi then pi ∨ A. (Suggestion: induction on the complexity of A.) From 6.46.9, 6.46.10, we draw an interesting conclusion for atomic consequences in the pure logic of disjunction:

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Corollary 6.46.11 Γ ∨ pi if and only if for some A ∈ Γ, A ∨ pi .

6.47

Quantum Disjunction

Our simple Set-Fmla natural deduction system, QNat in 2.31, for the {∧, ∨}fragment of quantum logic, was presented there with the familiar (∧I), (∧E), and (∨I) rules, and the restricted ∨-Elimination rule (∨E)res – see p. 299 – alongside the usual structural rules. The motivation for restricting (∨E) by disallowing side-formulas in the derivation of the common conclusion from each disjunct is explained in Example 6.21.4 (p. 823). The title of the present subsection indicates our intention to focus on the behaviour of disjunction in this setting. A more accurate title might therefore be ‘ortho-disjunction’, given that we are considering the logical analogue of ortholattices rather than of orthomodular lattices. See the discussion following 6.47.4. Since we are considering only conjunction and – especially – disjunction, this amounts to arbitrary lattices and the proof system Lat treated in 2.14 gives us a foretaste, though with based algebraic semantics in mind, of the current topic. However, there we worked in Fmla-Fmla rather than Set-Fmla, so that the distinction between (∨E) and (∨E)res does not emerge. First let us consider disjunction alone. Perhaps surprisingly, although the combined set of rules just described constitutes a weaker collection (4.32) of rules than if we had the unrestricted rule (∨E), when it comes to just the rules governing disjunction, there is no loss of provable sequents when the restricted elimination rule is used. We state this as 6.47.3. Exercise 6.47.1 If C and D are formulas whose only connective is ∨, and every propositional variable occurring in C occurs in D, we call C a subdisjunction of D. Show that for such formulas, the sequent C D is tautologous iff C is a subdisjunction of D. Lemma 6.47.2 If C is a subdisjunction of D, then C D is provable using (R), (T), (∨I) and (∨E)res . Proof. By induction on the complexity of C. Basis case. The complexity of C = 0, so C is pi for some i; then assuming pi occurs in D, we can prove C D by (∨I) and the structural rules. Inductive step. Suppose C = C1 ∨ C2 is of complexity n > 0, and C is a subdisjunction of D. Then so are C1 , C2 , and the inductive hypothesis gives us the provability of C1 D and C2 D. Then (R) yields C1 ∨ C2 C1 ∨ C2 , so from this and the two sequents just mentioned, we obtain C D by (∨E)res .

Observation 6.47.3 If a Set-Fmla sequent σ in the language whose only connective is ∨ is tautologous then σ is provable using (R), (M), (T), (∨I) and (∨E)res . Proof. Suppose σ = Γ D is a tautologous sequent of the language concerned. Then by 6.46.9 there is some C ∈ Γ such that C D is tautologous, in which case C is a subdisjunction of D (6.47.1), and accordingly the sequent C D is provable using the rules listed, by 6.47.2. (M) then yields Γ C.


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Remark 6.47.4 The above observation gives an example of what Garson [1990] calls ‘non-modularity of completeness’. Alongside the standard SetFmla structural rules, the restricted ∨-rules are sound and complete w.r.t. tautologousness (the property of holding on every ∨-boolean valuation); likewise for the usual ∧-rules. A combined proof system (QNat) with all these rules is not complete, however, since it does not prove the distribution sequent p ∧ (q ∨ r) (p ∧ q) ∨ (p ∧ r), which holds on all {∧, ∨}-boolean valuations. As the above Remark indicates, the interest of (∨E)res emerges only when we consider the interaction between disjunction and other connectives (such as conjunction); accordingly, for the remainder of our discussion, we concentrate on the system QNat of 2.31. As in 6.43–5, we are going to describe a semantic account in terms of which a completeness theorem for this proof system can be given. The semantics is based on Goldblatt [1974a], who treats also negation, roughly along the lines of the systems of orthologic (OL) and orthomodular logic (OML) mentioned in 2.31. In fact, Goldblatt takes disjunction as a defined connective, with A ∨ B = ¬(¬A ∧ ¬B), but we can extricate the derived semantic account of disjunction for a treatment on which that connective is primitive. (See 8.13 for ¬.) Unlike traditional semantic treatments of these logics, the present semantics is model-theoretic rather than algebraic (cf. 2.31.2), p. 300. Following our usual procedure, we take the models to be triples (W, R, V ) in which W is a non-empty set on which R is a binary relation, and V assigns subsets of W to propositional variables, subject to a certain condition to be described presently. Goldblatt describes an informal interpretation of these structures, in which W is thought of as comprising a set of possible experimental outcomes and R is a certain relation (‘orthogonality’) between such outcomes – roughly: a kind of incompatibility. He insists that R is symmetric and irreflexive, conditions which are important for the correct treatment of negation, but which we need not impose for the present ¬-free language. However, the condition on Valuations alluded to above will be important, and it is something of a mouthful; the condition is that for all pi , and for all x ∈ W : [∀y ∈ W ((V (pi ) ⊆ R(y) ⇒ x ∈ R(y))] ⇒ x ∈ V (pi ). To put this condition more succinctly, we introduce a piece of notation “R ” for a relation between subsets of the set W (of some model (W, R, V )) and elements of W . The definition is as follows. For any X ⊆ W , x ∈ W : [Def. R ]

X R x iff ∀y ∈ W (X ⊆ R(y) ⇒ x ∈ R(y)).

The notation here is suggestive of that for consequence relations, even though we are relating, not sets of formulas and formulas, but sets of points in a model, and individual points. However, it is not hard to see that the three conditions (R), (M) and (T+ ) on consequence relations are satisfied (when the “Γ”, “A”, variables are given this non-linguistic interpretation); such relations – like consequence relations but not restricted to what we are prepared to consider as languages – are called ‘closure relations’ in Humberstone [1996a]. Let us say that a set X ⊆ W is closed under R , or (for brevity) is R -closed when for all x such

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that X R x, we have x ∈ X. Then we can restate the above condition on Valuations as: (R -Closure)

For any model (W, R, V ) and propositional variable pi , V (pi ) is a R -closed subset of W.

Truth at a point in a model is defined in the usual way for propositional variables and conjunctions, while for disjunctions there is a considerable novelty: [∨]Gldb

(W, R, V ) |=x A ∨ B iff ∀y ∈ W : !A! ∪ !B! ⊆ R(y) ⇒ x ∈ R(y).

As already remarked, this clause (whose labelling is intended to abbreviate ‘Goldblatt’) is the result of transcribing the truth-conditions for ¬(¬A ∧ ¬B) from the semantics of Goldblatt [1974a], which paper should be consulted for motivation and for connections with the algebraic quantum-logical tradition; “!A!”, etc., stands for the set of points in the model at which A is true. Lemma 6.47.5 For any model (W, R, V ) and any formula A, the set !A! is R -closed. Proof. By induction on the complexity of A, the basis case being given by (R Closure), and the inductive case for A = B ∧ C being straightforward. Suppose, then, that A = B ∨ C. We have to show that if !B ∨ C! R x then x ∈ !B ∨ C!, i.e., M |=x B ∨ C, where M = (W, R, V ) is the given model. Accordingly, assume (for a contradiction) that (1) !B ∨ C! R x but (2) M |=x B ∨ C. By [∨]Gldb , (2) means that there exists y0 ∈ W with !B! ∪ !C! ⊆ R(y0 ) but x ∈ / R(y0 ). By [Def. R ], what (1) means is that for all y ∈ W such that !B! ∪ !C! ⊆ R(y), we have x ∈ R(y). But the point y0 is a counterinstance to this claim. The above Lemma evidently generalizes the condition on models (R -Closure) from propositional variables to arbitrary formulas, as in the case of our earlier conditions (Persistence), (Barring), etc. Theorem 6.47.6 Any sequent provable in QNat holds in every model. Proof. We content ourselves with showing that the rules (∨I) and (∨E)res preserve, for an arbitrary model M = (W, R, V ), the property of holding in M. For (∨I), it suffices to show that M |=x A implies M |=x A ∨ B (the case of M |=x B ⇒ M |=x A ∨ B being similar). So suppose that M |=x A, and that we are given y with !A! ∪ !B! ⊆ R(y). We must show that x ∈ R(y). Since x ∈ !A! and !A! ⊆ !A! ∪ !B!, this conclusion is immediate. Suppose now, for (∨E)res , that Γ A ∨ B, and A C and B C, hold in M, and take x ∈ W with M |=x D for all D ∈ Γ. From our supposition, we have M |=x A ∨ B. Thus for all y ∈ W : (*) if !A! ∪ !B! ⊆ R(y) then x ∈ R(y). But from what we have supposed about A C and B C, we have !A! ⊆ !C! and !B! ⊆ !C!, and so !A! ∪ !B! ⊆ !C!. Therefore, by (*), !C! ⊆ R(y), then


921

x ∈ R(y), for any y ∈ W , which is to say !C! R x. So by 6.47.5, M |=x C. The reader may care to attempt a soundness proof but with (∨E) in place of (∨E)res ; where does the argument break down? We will now show that QNat is complete, in the sense that the converse of 6.47.6 holds. The elegant completeness proof in Goldblatt trades heavily on the presence of negation with orthocomplementation properties and cannot be mimicked here in its absence. The complication which this forces (as far as the author can see) is that the canonical model we need for the proof has to have for its elements not certain sets of formulas, but certain pairs, the first element being such a set, and the second being a formula not in the set. We call the model to be constructed MQL = (WQL , RQL , VQL ). Where QL is the consequence relation associated with the proof system QNat of 2.31, the elements of WQL are to be all pairs Γ, C in which Γ is a deductively closed set of formulas—closed under QL , that is—and C is a formula not in Γ. Where x = Γ, C ∈ WQL we put In(x) = Γ and Out(x) = C. The definition of RQL is as follows: For x, y ∈ WQL : yR QL x ⇔ Out(y) ∈ In(x). As above, we will write “x ∈ RQL (y)” in place of “yR QL x”. Finally: VQL (pi ) = {x ∈ WQL | pi ∈ In(x)}. Presently, we shall show (6.47.8) that membership in In(x) and truth at the point x in MQL coincide, from which the completeness of QNat will follow (6.37.9). First we need a piece of notation and a Lemma. For any formula A, we use the notation |A| to denote the set {x ∈ WQL | A ∈ In(x)}. Lemma 6.47.7 For all y ∈ WQL and all formulas A: |A| ⊆ RQL (y) if and only if A QL Out(y). Proof. ‘If’: If A QL Out(y), then every z ∈ |A|, since it has A ∈ In(z) and In(z) is deductively closed, has Out(y) in In(z), so z ∈ RQL (y). ‘Only if’: Suppose |A| ⊆ RQL (y). That is, for every z ∈ |A|, Out(y) ∈ In(z). One such z is got by setting In(z) to be the deductive closure of {A}, and Out(z) to be any formula C such that A QL C. (For example, C can be chosen as some propositional variable not occurring in A; C is not then a tautological consequence of A, and so a fortiori not a consequence of A by QL .) So our supposition implies that the deductive closure of {A} contains Out(y), i.e., A QL Out(y).

Theorem 6.47.8 For any formula A and any x ∈ WQL , M |=x A if and only if A ∈ In(x). Proof. As usual, by induction on the complexity of A, the interesting case being that of A = B ∨ C. Note that the inductive hypothesis implies that |B| = !B! and |C| = !C!. Thus our task is reduced to showing that B ∨ C ∈ In(x) iff for all y ∈ WQL such that |B| ∪ |C| ⊆ RQL (y), we have x ∈ RQL (y).

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‘Only if’: Suppose B ∨ C ∈ In(x) and we have y ∈ WQL with |B| ∪ |C| ⊆ RQL (y). We want to show that x ∈ RQL (y). Since |B| ∪ |C| ⊆ RQL (y), we have |B| ⊆ RQL (y) and |C| ⊆ RQL (y), so by 6.47.7, B QL Out(y) and C QL Out(y). Then, in virtue of (∨E)Res , we have B ∨ C QL Out(y), and so, since In(x) is deductively closed and contains B ∨ C, In(x) contains Out(y), i.e., x ∈ RQL (y). ‘If’: Suppose B ∨ C ∈ / In(x). We must find y ∈ WQL with |B| ∪ |C| ⊆ RQL (y) while x ∈ / RQL (y). As in the ‘Only if’ part of the proof, we can appeal to 6.47.7 and rewrite this inclusion as (i)

B QL Out(y)

and (ii) C QL Out(y).

Since we also want x ∈ / RQL (y), we need (iii)

Out(y) ∈ / In(x).

To obtain y satisfying (i), (ii), (iii), we may choose y = In(x), B ∨ C . Since B∨C ∈ / x, this y ∈ WQL , and (iii) is satisfied. So also are (i) and (ii), in virtue of (∨I).

Corollary 6.47.9 Any sequent which holds in every model is provable in QNat. Proof. We have to check, first, that MQL is a model, and the only part of this which is not immediate from the way WQL , RQL , and VQL were defined is that the condition (R -Closure) is satisfied. We are only required to show for A = pi that, for all x ∈ WQL , if for every y ∈ WQL , !A! ⊆ RQL (y) implies x ∈ RQL (y), then x ∈ !A!. But we can easily show this for arbitrary A. By 6.47.7, 8 and the preceding discussion, what has to be shown can be reformulated thus: for all x ∈ WQL , if for every y ∈ WQL such that A QL Out(y), we have Out(y) ∈ In(x), then A ∈ In(x). Contrapositively, we must show how, given x with A ∈ / In(x), how to find y such that (i) A QL Out(y)

and

(ii) Out(y) ∈ / In(x).

But since A ∈ / In(x), we can choose y as In(x), A , and (i) and (ii) are obviously satisfied. Now that we have established that MQL is a model, it remains only to check that no unprovable sequent of QNat holds in MQL . If the sequent Γ C is unprovable, then Γ+ , C is an element of WQL , where Γ+ is the deductive closure of Γ. By 6.47.8, all formulas in Γ are true at this element, while C is not. So the unprovable sequent does not hold in the model MQL .

Notes and References for §6.4 Disjunction Property and Halldén Completeness. The Disjunction Property of 6.41 was claimed to hold for intuitionistic logic in Gödel [1933c], and proved in McKinsey and Tarski [1948]. For more information on the property see Meyer [1971], [1976b], and (with special reference to intermediate logics) Wroński [1973], Maksimova [1986], Zakharyaschev [1987], Chagrov and Zakharyaschev [1993]; there is an excellent review of results obtained on the Disjunction Property (and some related conditions, including Halldén-completeness)


923

for intermediate logics in Chagrov and Zakharyaschev [1991]. (See also the references in the Digression before 6.42.4, on p. 875.) Apart from the works cited in 6.41 bearing on Halldén completeness, the following may be cited: McKinsey [1953], Wroński [1976], van Benthem and Humberstone [1983]. Techniques similar to those of this last paper are applied to certain relevant logics in Routley et al. [1982], pp. 392ff. and 416f. Halldén-incomplete logics are also called ‘Halldén-unreasonable’ in the literature, an epithet used here to apply to disjunctions witnessing Halldén-incompleteness; Schumm [1993] provides a thoughtful overview of all these matters. Halldén-completeness is introduced (apparently independently of Halldén) under the name “quasi-completeness” at p. 188 of Łoś [1954]. The related topic of intersections of logics in Fmla is discussed in Miura [1966], Wroński [1977] and (for modal logic) Schumm [1975]; there is a very readable discussion in pp. 80–85 of Meyer [1976a]. A different definition of Halldén completeness is given in Kracht [1999], p. 29, according to which a consequence relation is said to have this property just in case for any variable-disjoint formulas A, B, if A B and A is consistent, then B (where “A is consistent” means that for some C, A C). We prefer the more standard definition in terms of the provability of a disjunction, so that Halldén completeness emerges as a weakened form of the Disjunction Property. (Kracht’s notion is a special case of the Shoesmith–Smiley Cancellation Condition (**) given on p. 205: the special case in which Γ is empty and Θ is {A}. Also here we are taking as the appropriate inferential or truth-local consequence relation, giving what Kracht calls local Halldén completeness. If we take as the truth-global or modal consequence relation we get what he calls global Halldén completeness. As remarked before 6.41.12, this is not the ‘global Halldén completeness’ of our discussion.) The notion of a Halldén-normal modal logic is new, though a rule closely analogous to what we call Halldén-necessitation may be found (under the name RLC) on p. 191 of Zeman [1973], the connection with Halldén-completeness being noted on p. 192. The notions of homogeneity, heterogeneity and strong heterogeneity employed in the discussion are linguistic, in that they are defined in terms of the validity (at various points) of formulas. For a purely structural notion of homogeneity, defined in terms of automorphisms of frames, see p. 39ff. of van Benthem [1983]. The paragraph leading up to 6.41.5 owes much to (private communication from) van Benthem. The original source for the discussion in Lemmon and Scott [1977] of the ‘Rule of Disjunction’ (our 6.42) is based is Kripke [1963]; Williamson [1992] should be mentioned as a source of variations on the theme, as well as Williamson [1994b] (in which, inter alia, Denecessitation is discussed) and Chapter 15 of Chagrov and Zakharyaschev [1997] (which also treats Halldén-completeness), for a textbook presentation.

Intermediate Logics and Admissible Rules. Since the 1957 paper by Kreisel and Putnam, introducing the intermediate logic which came to be known as KP, appeared in German and no English translation has appeared, it is not included in our bibliography, but a summary, and references to reviews and abstracts, may be found under entry 72 in Minari [1983]. A completeness proof, w.r.t. a class of frames defined by a second-order condition on frames, together with references, may be found in Gabbay [1970]. (The condition in the text of 6.42 – referred to as van Benthem’s KP condition – is, by contrast, a first-order condition: there is no quantification over sets of points, only points themselves.)

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More information on Mints’s Rule, also mentioned in 6.42, may be found in Tsitkin [1977]. A comprehensive discussion of the admissibility of various rules in intuitionistic and intermediate (as well as normal modal) logics is provided by Rybakov [1997]; Mints’s rule (with variations) is treated there in §5.2. See also pp. 50–51 of Chagrov and Zakharyaschev [1997] and Iemhoff [2006] as well as references there cited. Beth Semantics for IL. Beth’s somewhat opaque presentation of his semantics for IL (6.43) is conveniently accessible in §145 of Beth [1959]. An extensive discussion of the philosophical issues it raises may be found in Dummett [1977], Chapter 5, and §3 of Chapter 7, which also presents the semantics (as in Kripke [1965b], Gabbay [1981]) in more or less the same form as 6.43. Dummett’s discussion includes an argument (p. 208) that the Beth semantics is preferable to the Kripke semantics as a “representation of the intended meanings of the intuitionistic logical constants”; this argument – which concerns the sequent mentioned under 4.21.4 – hangs on the two semantics as applied to sequents of intuitionistic predicate logic, not covered in our exposition. See also the works cited in the notes to 2.33 (‘Intuitionistic and Intermediate Logics’, p. 370) on comparative discussions of various alternative semantic approaches to IL. Hazen [1990a], mentioned briefly in 6.43, contains an interesting application of barring to normal modal logic, as well as some remarks on disjunction in IL. A somewhat garbled account of Beth’s treatment of disjunction appears in Garson [2001], p. 121, which overlooks the role of the condition of Barring on models (see the proof of our Lemma 6.43.3). References for the Later Subsections. 6.44 and 6.45 present material discussed in Humberstone [1981a] and [1988b]; see further van Benthem [1986b] on the former material. Some of that material also appears in Mares [1999]; another venture close to the possibility semantics of 6.44 is reported in Cresswell [2004]. As was mentioned in 6.45, the condition there called Decomposition has been used to provide a notion of distributivity for semilattices; see Grätzer [1968] p. 329, [1978], p. 99f., and for further discussion and additional references, Hickman [1978]. The discussion in 6.45 and 6.46 of the ‘plus’ semantics for disjunction is based on Humberstone [1988b]. Similar treatments may be found in Bell [1986], esp. p. 91; and in Ono [1985], Ono and Komori [1985]: see the clause labeled (d) on pp. 189 and 177 of these papers, respectively; and, finally, in Orłowska [1985], p. 455. (There is also a plus-like component in the clause for disjunction on p. 323 in Terwijn [2007].) See further p. 419 and note 10 of Humberstone [1988a]. In fact Beth’s own presentation of the semantics of disjunction (Beth [1959], p. 447, clause (iv)) is formulated along these lines. The condition [∨]Elim from 6.46 may be found in note 6 on p. 167 of Garson [1990].

Chapter 7

“If” The material conditional has an ingratiating perspicuity: we know just where we are with it. But where we are with it seems, on the whole, not where we ought to be. Nerlich [1971], p. 162.

§7.1 CONDITIONALS 7.11

Issues and Distinctions

This chapter addresses not only some of logical and semantic issues raised by ‘if then. . . ’, but also the behaviour of related connectives which have been proposed to formalize various special relations of implication and entailment. We associate the label conditionals with the former project – to be pursued in much of the present section – and tend to use implication for the latter. Such a division is not without problems, but will do well enough for imposing structure on the material in terms of motivating considerations. When these amount to an interest in how “if ” behaves in English, they issue in a logic of conditionals, while attempts at providing a connectival treatment of what in the first instance is most naturally thought of as a relation between proposiimplies tions or sentences, we have an implicational logic: a logic for ‘that that. . . ’. As it happens, material implication as well as strict implication will, their names notwithstanding, receive attention in the present section (7.13, 7.15, respectively). Intuitionistic (or ‘positive’), relevant, and ‘contractionless’ forms of implication will occupy us in §7.2, the first also being touched on in 7.14 here. Many important topics under the general heading of implication will receive scant attention or no attention at all. These include connexive implication, mentioned only in passing in 7.19, and analytic implication (see Parry [1968], [1986], and Smiley [1962b]). (For comparative remarks on these – and the associated doctrines of ‘connexivism’connexive implication and ‘conceptivism’ – and relevant implication, see Routley, Plumwood, Meyer and Brady [1982].) Implicational statements and conditionals have converses; conjoining one with its converse gives an equivalence statement or biconditional, respectively, and these we shall discuss in §7.3. (“Statement”, here, means the linguistic expression of a proposition; as noted in 1.15, propositions themselves do not have converses.) 925

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“IF”

The main stimulus for much of the formal work on conditionals has undoubtedly been over the claim that the inferential behaviour of at least indicative conditionals (to use a term whose sense will be explained presently) is adequately captured by the behaviour of → according to an →-classical consequence relation, to put the matter proof-theoretically. (A good deal of the opposition to a treatment of indicative conditions by means of → construed in terms of the broader class of →-intuitionistic consequence relations – as defined on p. 329 – would go through similarly; we will not take a special interest in IL as opposed to CL on this matter.) Speaking more semantically: that much if not then. . . is given by the all there is to be said about the interpretation of if familiar truth-table account (i.e., by the condition of →-booleanness on valuations). These formulations are not quite equivalent but either would do as a way of making precise the contested claim – see the opening quotation above, from Nerlich, as well as that from Haiman below – that material implication gives the logic of the indicative conditional. The ‘paradoxes of material implication’ are much appealed to in opposition to such a claim, or indeed to a variant in which “→-classical” is replaced in the above formulation by “→-intuitionistic”. (These appear as (P + ) and (P − ) in 7.13 below.) This replacement would lead us to a pattern of discussion of if along the lines of the discussion of and and or in the preceding chapters: looking at the natural deduction introduction and elimination rules for →. We do not pursue this strategy, however, because it turns out that many differences between distinct proposals hang not on the rules as such so much as on what the appropriate logical framework is taken to be, how careful one is with assumption-dependencies, and so on. (See 7.13 for more on this.) But let us recall that Modus Ponens has come under fire, as mentioned in the Digression on p. 528, as well as Conditional Proof in the relatively framework independent simple form that if one can derive B from A, one is entitled to assert “If A then B” outright (or as depending on such any additional assumptions used in that derivation). See 7.18.2, p. 1036, for one example of this. Many problematic inference patterns slightly less obvious than the paradoxes of material implication and not, this time, all holding for intuitionistic implication, have been enlisted in the campaign against classically behaving → as the indicative conditional; a selection may be found in 7.19.8. The hostility, especially on the part of non-logicians, to the claim just formulated is evidenced by such remarks as the following, from Haiman [1993], p. 923: The yawning chasm between ordinary language conditionals and conditionals defined by “material implication” is of course well known.

Others have been put of the whole subject of logic by the same (apparent) chasm, as we see from the following quotation from Waugh [1992], p. 128, in which the term hook is a reminder of the use of the notation “A ⊃ B” in place of our “A → B” (and there is a little incidental confusion on the word “therefore”): There are no circumstances in which ‘A therefore B’ or ‘If A then B’ automatically become true statements when A is shown to be false. If the hook does not cover either the consequence or the conditional relationship, but one of its own invention, then its only applied function must be either as a parlour game or as a liars’ charter. Formal logic, like so many other things studied in the universities, must surely be a colossal waste of time.

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927

No doubt, with reactions like these fuelling them, more pages have been written about conditional and implicational connectives than about any others, and the present chapter looks only at a few of the of the topics that have come up in this literature. While no particular account of (especially, indicative) conditionals is defended here, the discussion should provide a reasonably comprehensive set of pointers to the literature, so that any issues of special interest can be followed up. (Further references and a lively discussion of many topics not covered here may be found in the excellent survey Edgington [1995]. See also Edgington [2001] and Bennett [2003].) Despite the adverse comments here recalled, psychologists studying adult reasoning, as well as children’s acquisition of reasoning skills, often accept without question the deliverances of the usual truth-table account of indicative conditionals as material implications, regarding experimental subjects’ verdicts not in accord with that account as errors on their part. (See for example Paris [1973].) Theorists of conditionals are in no consensus as to even the most fundamental questions, and we will begin by citing one such contentious question: the subjunctive vs. indicative distinction amongst conditionals. Since the ‘material implication’ claim mentioned above is usually only advanced with the indicative conditionals in mind, it is important to see what kind of restriction this is. The distinction, we shall see, is connected with a second problem concerning ‘modified’ conditional constructions, such as involve even if and only if, rather than if by itself (7.12). Though much of the present subsection addresses the indicative/subjunctive distinction and matters arising in its clarification, at the end we shall also mention – though they will turn out to be of lesser significance – two further distinctions suggested in various parts of the literature on conditionals, namely (i) ‘consequential’ (or ‘causal’ or ‘resultative’) vs. ‘non-consequential’ conditionals, and (ii) what we shall call the distinction between ‘helping’ and ‘non-hindering’ conditionals. Topics arising out of this material on indicative conditionals will occupy us until 7.15, at which point we take up the issue of subjunctive conditionals, through till 7.17, returning to the subjunctive/indicative contrast itself in 7.18. Suppose, for the first of many illustrations, that you have a friend, Pennie, who, as far as you know, never wears a hat. You can see a group of people waiting at a bus stop in the distance, their backs to you, so that none is recognizable. But one thing you can see is that all of them happen to be wearing hats. Here is something which, under the circumstances, you cannot but agree to: (1)

If Pennie is at that bus stop, she is wearing a hat.

And here is something with no claim whatever on your assent: (2)

If Pennie were at that bus stop, she would be wearing a hat.

(1) and (2) are described respectively, as an indicative and a subjunctive conditional; likewise with (3) and (4) (again, respectively): (3) (4)

If there was pepper in the tea, I didn’t taste it. If there had been pepper in the tea, I wouldn’t have tasted it.

Having just enjoyed a cup of tea, and then been told that it may have been the cup into which a child had sprinkled some pepper, a speaker might well respond with (3), but not with (4): instead of (4), our speaker would instead probably assert with confidence, “If there had been pepper in the tea, I would have tasted

CHAPTER 7.

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“IF”

it”. The same distinction as (3) and (4) illustrate for the past, arises for the present, with (5)

If there is a kitten in the room, it is concealed from view.

(6)

If there were a kitten in the room, it would be concealed from view.

A brief glance round the room disclosing no visible kitten might prompt one to come out with (5), but it would be no grounds for asserting (6); such grounds might come instead from information to the effect that kittens tend to curl up in nooks and crannies, rather than to stay out in the open. We will give some further illustrations of the difference (first emphasized in Adams [1970]) between indicative and subjunctive conditionals; the Oswald– Kennedy example Adams made famous will appear in 7.18.4. The following example is inspired by one given in Curley [1972]. Suppose you know no more than that one of Gilbert and Sullivan wrote the music, and the other the words, for the operettas with which their names are associated. Then you can reasonably say (7)

If Sullivan didn’t write the music then Gilbert did.

Would you, however, be prepared to assert (8)? (8)

If Sullivan hadn’t written the music, then Gilbert would have.

Presumably not: it is compatible with what you know that Sullivan wrote the music and that Gilbert was a librettist with no ability at musical composition whatever, and in that case, had Sullivan not written the music in question, it would never have been written, and certainly wouldn’t have been written by Gilbert. Remark 7.11.1 In a reply to Curley [1972], Barker [1975] draws attention to a special subjunctive conditional construction which he calls the extrinsic counterfactual, as opposed to the ‘intrinsic’ counterfactual of (8). The labels are not especially suggestive, but a variant on (8), modelled on the example discussed by Barker (p. 371), is more likely to have what he calls the extrinsic interpretation, namely: “If it hadn’t been Sullivan who wrote the music, then it would have been Gilbert”. This special interpretation is all but forced by a slight rephrasing: “If it had turned out that Sullivan didn’t write the music, then it would have to have been Gilbert who did”. This is indeed a reasonable thing for someone knowing that one or other of them wrote the music to say. Note the epistemic flavour conveyed by the phrase “turned out”: a possible change in one’s state of knowledge is alluded to without being explicitly mentioned. (See also Jackson and Humberstone [1982]. The dialectical setting of this paper, as well as those of Curley and Barker, is provided by 6.13.1 on p. 789.) The extrinsic/intrinsic distinction appears not to have received much subsequent attention, and we shall be concentrating on the intrinsic counterfactuals in those parts of the present chapter in which subjunctive conditionals are at issue. Finally, suppose that you have been expecting a phone call from your sister at 4p.m. When the phone fails to ring at 4, you say:

7.1. CONDITIONALS (9)

929

If she dialled my number at 4 o’clock, then there is a problem with the line.

What you don’t say is: (10) If she had dialled my number at 4 o’clock, then there would have been a problem with the line. Instead, there being no reason, other than the hypothesis that she did dial and your phone didn’t ring, to suppose that there was a problem with the line, you will say that if she had dialled the number, then your phone would have rung in the normal way. As will be evident from 7.11.1, instead of being called ‘subjunctive’, conditionals such as (2), (4), (6), (8), and (10), are often called ‘counterfactual’ conditionals, or counterfactuals, for short (older form: ‘contra-factual(s)’). Notoriously, neither label is completely apt. We consider the term ‘subjunctive’ first. It might be held that other constructions, to the extent that any trace of the subjunctive is conceded to survive in contemporary English, have a better claim to this title: for example, the (archaic) construction in “If there be a kitten,. . . ” or that in “If I should die before tomorrow,. . . ”. We take up this latter example in 7.11.2 (and the comment which follows that Exercise). Historically, the former is a present tense subjunctive (such as survives more robustly in “The officer demanded that the hut be cleared completely”); while the form in (6) is morphologically a past tense subjunctive, even though it is interpreted semantically as present tense. (Compare “I wish I were in Paris now”.) A similar tense backshift has occurred in (4), of course, with a pluperfect form having a simple past interpretation. As has been emphasized by V. H. Dudman – in several of the papers listed in the end-of-section references (which begin on p. 1053) – the past, and even pluperfect (or ‘past perfect’), forms may arise with a future interpretation, as in (11) and (12) respectively: (11)

If war broke out tomorrow, I would enlist.

(12)

If war had broken out tomorrow, I would have enlisted.

The subtle but perceptible semantic difference between (11) and (12) is well described in Dudman [1991], in particular. (Similar pairs in which the distinction is anything but subtle have been used to striking effect in Ippolito [2007] and other papers by the same author.) As we shall see presently, however, Dudman by no means agrees that the subjunctive/indicative contrast we are now drawing is as significant as another distinction which cuts across it (projective vs. non-projective if -sentences). It would seem then, that although some confusion is possible, the description of conditionals such as (1), (3), (5), (7) and (9) as indicative, and (2), (4), (6), (8), and (10) (as well as (11), (12)), as subjunctive, is not wholly misleading. Matters are slightly worse with the label counterfactual, which may convey the impression—endorsed by much of the literature (especially the early literature)—that assent to the conditionals in question requires that one believe their antecedents to be ‘contrary to fact’ (to be false, that is). In its more modern usage – that followed here – as simply an alternative to subjunctive, this is a misimpression, as is shown by each of the following examples (from p. 158 of Davies [1979], Comrie [1986]):

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(13) If John had been at the scene of the crime at the time when the murder was committed, Mary would have seen him leaving. So we must get hold of her to find out if she did see him. (14) If the butler had done it, we would have found just the clues that we did in fact find. The point made by these examples—that one may perfectly well use a subjunctive conditional without commitment to the falsity of the antecedent (let alone commitment to the falsity of both antecedent and consequent) was also mentioned in Strawson [1952], p. 84n. and von Wright [1957], p. 134 (in von Wright’s case to show that not all subjunctive conditionals are counterfactual conditionals, using the latter term in its older ‘contrary-to-fact’ sense). Probably the first occurrence in print of this subsequently much overlooked point is a brief remark in Chisholm [1946], p. 291. Examples (13) and (14) have a forensic flavour characteristic of those generally given in the literature to make this point, prompting Chisholm to remark: “It is said that detectives talk in this manner”. The point may be found again, further developed, in Anderson [1951]. There is a full and sensible discussion of this matter in the first five pages of Harrison [1968], from which we shall have occasion to quote extensively in 7.17; see also 7.11.3 below. The need for a qualification has, however, been noted by Moro [1989]: Dudman (‘future had ’) constructions such as that exhibited in (12) above do not seem amenable to this neutrality, but rather to convey, as part of their meaning, that the antecedent does indeed run contrary to fact. We have cited (14), rather than just (13), for the sake of drawing attention to the bizarre results (as indicated by the prefixed question mark) of replacing the subjunctive construction with its indicative analogue: (15)

?

If the butler did it, we found just the clues that we did in fact find.

Contrasts such as those between (14) and (15) have been used by Frank Jackson to show that whatever the merits of other distinctions, the traditional indicative/subjective distinction is at least a distinction of some semantic significance. (As already noted, Dudman tends to play down this distinction, and some go further still—cf. the following remark from the opening page of Lowe [1995]: “Nor do I think that the subjunctive/indicative distinction, as applied to conditionals, is semantically or logically important”.) Rather than working with “in fact”, Jackson uses “actually” to make the point, with such pairs as (16) and (17) (adapted from Jackson [1987], p. 75): (16) If it had rained, then things would have been different from the way they actually were. (17)

?

If it rained, then things were different from the way they actually were.

We should note that the antecedent, too, provides a hospitable environment for “actually” in (only) the subjunctive case: (18) If it had rained for longer than it actually did yesterday, the race would have been called off. (19)

?

If it rained longer than it actually did yesterday, the race was called off.

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Now, as we saw in the discussion surrounding 3.31.8, the role of the word “actually” is to achieve a back-reference from non-actual possible worlds to the actual world. This is not the only role this word plays (consider “Actually, it was the Smiths, not the Browns, who said they had to leave early”: more on this rhetorical or ‘corrective’ use in 7.18), but it is surely plausible to suppose that this is the role it is playing here. Jackson concludes from this that some kind of possible worlds account would be correct for the semantics of subjunctive conditionals but incorrect for the case of indicative conditionals. This line of thought we refer to as Jackson’s “actually” argument. Its significance is not just in showing that an indicative conditional does not mean the same as the corresponding subjunctive – a moral we already drew from (1) and (2), etc. – but rather: that an account of the meanings concerned should, in the case of the subjunctive, and should not, in the case of the indicative, invoke worlds other than the world at which the conditional is being evaluated. (In particular, then, it would not be appropriate to respond – as in Stalnaker [1975] – to the conceded difference in meaning by simply selecting sets of worlds for the indicative and subjunctive on a different basis.) The natural understanding of “evaluated” here is as “evaluated for truth”, though one particularly extreme form of the claim that indicative and subjunctive conditionals exhibit radically different semantic behaviour has been the denial (e.g., in Adams [1975]) that indicative conditionals themselves have truth-values. (Some writers have even made such a claim for the case of subjunctive conditionals. Further exposition sympathetic to Adams’s position maybe found in Edgington [1995], [2001], [2006].) We should notice that, even without the further suggestion that an account in which alternative possible worlds are invoked by the antecedent of a subjunctive conditional but not by the antecedent of an indicative conditional, the difference illustrated by (16) as against (17), or (18) as against (19), already shows the subjunctive/indicative distinction to mark a genuinely significant contrast. (In fact, the point can be made without even bringing in the word “actually”, in terms of the presence or absence of ambiguity in the sentences resulting from (16)–(19) after this word is deleted, as was observed in Postal [1974], p. 391; we return to Postal in 7.18 below.) The fourfold classification of Dudman [1991] of if -sentences is as follows. The first category comprises what he elsewhere calls hypotheticals and the second category habitual or generic if -sentences. They are illustrated respectively by “If the door was locked, Grannie leapt in through the window”, understood as pertaining to a specific occasion, and by “If the door is locked, Grannie leaps in through the window”, understood as reporting a habitual practice. (Several authors seem to think these cases deserve special consideration, for example Barwise [1986].) The third category comprises all of what we are calling subjunctive conditionals, together with indicative conditionals such as (20) and (23) below, in which the antecedent, though exhibiting no “will” or “shall”, is understood as relating to a future time. What these have in common is that all manifest the tense backshift we have noted for subjunctive conditionals proper. They are the only category to which Dudman is happy to call conditionals, when he is not using the phrase “projective if -sentences” to refer to them. (The fourth category contains some odd cases, some of which we shall have occasion to mention later on in this subsection.) That will be sufficient background to render intelligible the following quotation from Dudman [1991], p. 206, which appears in a footnote to the presentation of this fourfold classification of conditionals:

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Received logical thought about ‘if’-sentences rejects my four categories. It favours a dichotomous taxonomy which draws a line smack dab through my third category so as to separate two ‘moods’. (. . . ) How to respond? Polite incredulity gives way to gathering outrage when investigation reveals that the sole authority for this improbable taxonomy resides with a hallowed pair of antique terms of art. Take away ‘indicative’ and ‘subjunctive’ and the whole idea collapses.

As we have seen with the aid of Jackson’s “actually” argument (to the conclusion that subjunctive but not indicative conditionals involve a world-shift), however, this is an overstatement on Dudman’s part: the indicative/subjunctive division does mark a semantically significant distinction – whether or not one wishes to use the terminology of moods. (Here we assume also the result of Exercise 7.11.2(i) below.) This is not to say that there need (in principle) be anything wrong with other classifications, including those which cross-cut this one. There have been several objections raised in the literature to Dudman’s hypothetical/conditional distinction with its reclassification of some indicative conditionals alongside all subjunctives: for example Dale [1985b] and Palmer [1986], p. 190. The latter is somewhat surprising since Palmer [1974], §5.5.3 isolates as ‘predictive conditionals’ precisely the Dudman-conditionals, also called by Dudman projective if -sentences. From now on we refer to these as projective conditionals. (So as to minimize misunderstanding, it seems better to speak of projective conditionals rather than of conditionals tout court for this class, since most writers use “conditional” and “if-sentence” more or less interchangeably.) See also §§7–10 of Bennett [1988] for further phenomena which would have us classify the (Bennett’s term:) ‘relocated’ conditionals—such as (20), (23)—along with subjunctive conditionals, and §11 for suspicions that further relocation may be called for. Digression. Bennett [1995] voices second thoughts about all this. His revised opinion there is that the traditional account draw the line in the right place after all; for his subsequent thoughts, see Bennett [2003]. (Jackson [2009] also reconsiders some aspects of Jackson’s position in the material cited in this section, but not in ways that affect the points here under discussion.) What is most seriously in need of questioning is any idea of “the line” to be drawn. Why should only a single distinction be expected to be helpful for all purposes? For example, consider the distribution of the sentence pro-form so. In the combination “if so”, this seems to demand a hypothetical rather than a projective interpretation. This does not undermine the distinction between subjunctive and indicative conditionals whose semantic significance the “actually” argument reveals. End of Digression. Some of the data appealed to by Dudman à propos of this projective vs. hypothetical distinction are in any case well worth further consideration, including in particular the cases in which (what we would call) an indicative conditional can take a future (will or shall ) tensed if -clause. Compare (20) with (21): (20) If the ambulance doesn’t get here soon, Nicky will die. (21) If the bus won’t be here for another hour, we can afford to go and have a cup of coffee.

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Impressionistically we may say (roughly following Dudman) that in the first (projective) case, we project ourselves into the future, whereas in the second (hypothetical) case, we consider, from the perspective of the present, a hypothesis concerning the future. Similarly with (22) (taken from Dudman [1991] and elsewhere) and (23); note how putting is for will be in (22), or making the reverse substitution in (23), changes the flavour of the example (in the former case giving something anomalous): (22) If Grannie will be dead by sundown, then we can start selling her clothes now. (23) If Grannie is dead by sundown, then we can bury her tonight. References to published discussions of these matters may be found in the notes to this section. Digression. We have seen that Dudman questions the significance of the traditional subjunctive/indicative distinction in particular because it cuts a line through the middle of the category of projectives grouping the future-directed projectives with hypotheticals (these together comprising the indicative conditionals), and leaving behind their past-tense cousins to comprise the category of subjunctives. More than this, he even questions he wisdom of discerning antecedents and consequents in projective conditionals. It is not uncommon to hear the objection that the conditional should not be treated as a binary sentential connective on a par with and and or (e.g., Gazdar and Pullum [1976], Lycan [1984], Haiman [1993]), on the grounds that such a treatment would ignore the syntactic distinction between subordinating and coordinating binary connectives. (The subordinate or if -clause in much traditional grammatical literature is referred to as the protasis, while the main clause – which may or may not be introduced, should it follow rather than precede the protasis, by the word “then” – is similarly called the apodosis. We touch briefly on this matter of the presence or absence of then early in 7.12.) As explained in 5.11.2 on p. 636, however, this perfectly genuine syntactic distinction (between subordinating and coordination) would appear to be of no semantic or logical importance. Note that the correspondence between the grammatical terminology of protasis and apodosis and the logical terminology of antecedent and consequent is only rough. If one is thinking of a representation of “A only if B” in terms of a conditional connective “⇒” of whatever kind, as “A ⇒ B” then one will regard A as the antecedent of the conditional construction, despite its being the apodosis rather than the protasis (the if -clause) in terms of the grammatical distinction. (This is not to urge the correctness of an A ⇒ B style of treatment for “A only if B”sentences. As we shall see in the following subsection, the appropriate account of such sentences is a controversial and unresolved matter.) There are in any case many syntactic and semantic distinctions arising within the class of subordinating connectives. Consider the case of if and when for example, the latter participating in cleft-constructions (as in (a) here), the former not: (a) (b)

It was when the bell rang that the victim awoke. *It was if the bell rang that the victim awoke.

(Amongst other subordinators, because behaves like when in this respect, while although behaves like if. Cross-cutting this division, we have the fact that

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because, when, and if can be modified—see 7.12 for our discussion in the case of if —by only, whereas although cannot. The asterisk on (b) should arguably be reduced to a question mark, indicating borderline acceptability; a similar sentence, numbered (60a) in Iatridou (1993), p. 192, is rated fully grammatical – by contrast with one attempting to insert a then after the that.) Or again, for a more semantics-related consideration, from Geis [1985], consider the fact that (c) is ambiguous—will I leave when you make your announcement, or at your announced time of departure?—while (d) is not: (c)

I will leave when you say you’ll leave.

(d)

I will leave if you say you’ll leave.

Returning to the coordinating vs. subordinating issue: David Lewis offers a routine formulation of his semantic account of subjunctive conditionals tailored to a syntax which discerns the if -clause as a constituent attaching to the rest to make a sentence at p.60 of Lewis [1973]; for somewhat less routine manoeuvres in this vicinity on the part of Risto Hilpinen, see 7.17 below (p. 1020). A different view, tying the subordinating role of if to theories—touched on later in this subsection—in the ‘conditional assertion’ camp, may be found in James [1986b]. Dudman goes further in, as it were, subordinating the if -clause, suggesting that “Sir Jasper will be killed if he falls” is best understood as a simple subjectpredicate sentence along the lines of “Sir Jasper will-if-he-falls be killed”. (The example is from Dudman [1986], p. 179.) But what of “It will hail tomorrow if it rains tonight”, or other cases in which only in the most superficial syntactic sense is there anything deserving to be called a subject? And what of the fact that simply taking out the words between the if and the then (supposing both occur, and in that order) we do not arrive—because of the temporal backshift and other oddities—at something which means what it meant when in that conditional context? Should this stop us from speaking of antecedent (and consequent)? No. The fact that the sequences of words “Pennie is at that bus stop” and “Pennie is wearing a hat” do not bodily appear in (2) above, for example, need not prevent us from describing them as respectively the antecedent and consequent of (2), which is after all understood as saying that if the first were true then the second would be. This is no different from thinking of “Joe didn’t go” as the negation of “Joe went”, the non-appearance of the latter as a part of the former notwithstanding. It is with the above understanding of the antecedent/consequent distinction that we may say that Modus Ponens is the principle licensing the inference of the consequent of a conditional (in our broad sense, not Dudman’s restricted sense) from the conditional together with its antecedent. The projective (20), and the hypothetical (21), are alike in respect of their antecedent in this sense, the relevant applications of Modus Ponens being respectively:

(20) If the ambulance doesn’t get here soon, Nicky will die. (20 ) The ambulance won’t get here soon. Therefore: (20 ) Nicky will die.

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and (21) If the bus won’t be here for another hour, we can afford to go and have a cup of coffee. (21 ) The bus won’t be here for another hour. Therefore: (21 ) We can afford to go and have a cup of coffee. End of Digression. Exercise 7.11.2 (i) Find examples to show that the contrasts in intelligibility like those between (16) and (17) or (18) and (19) show up also for subjunctives and indicatives whose antecedent and consequent pertain to future times. (For the indicative case, give examples like (19) as well as examples like (20), to verify that it is the subjunctive/indicative contrast that is at work, rather than Dudman’s conditional/hypothetical distinction.) Note: It seems to be necessary to use the “future had ” type of construction for this purpose, rather than the more ‘open’ subjunctive construction – a point presumably connected to that mentioned in connection with Moro [1989] earlier. (ii ) What happens with “actually” when conditionals of the form “If it should be that A then it would be that C” are considered? What is the difference in meaning between “If anyone should see us, we’ll be in trouble” and “If anyone sees us, we’ll be in trouble”? (iii) Explain the ambiguity in the following example (of Dudman’s): “If Grannie missed the bus, she would walk home.” (This shows that the form of a conditional alone does not always fix whether it is subjunctive or indicative – though of course this is not how Dudman would put it.) A propos of part (ii) of the above, the moral of which is that for present purposes the should -construction featuring there belongs with the indicative rather than subjunctive conditionals, the interested reader is referred to Nieuwint [1989]. We take this opportunity also to quote from Cantrall [1977], p. 23, where the contrast in acceptability between (24) and (25) is noted, with the remark that “you can’t simultaneously deny and hypothesize about a proposition”: (24) (25)

If Hitler, dead as he undoubtedly is, were alive. . . ? If Hitler, dead as he undoubtedly is, should be alive. . .

Remark 7.11.3 Subjunctive conditionals such as (18) are guaranteed by the meaning of their antecedents to be contrary-to-fact, in the sense that one can know a priori that their antecedents are false. (See Davies and Humberstone [1980].) This is not part of the meaning of the subjunctive conditional construction, though, as we saw with (13) and (14). In many cases, the falsity of the antecedent is merely conversationally implicated (Lewis [1973], p.3, and – though this Gricean terminology is not used by him – Harrison [1968]), though the difference between what is implicated by an utterance and what is implied by the content of the utterance is lost on many commentators, as the following illustration shows.

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Example 7.11.4 Leech [1971], §168, considers the sentence: I won’t tell Susan about my problems: (even) if she were my closest friend, I (still) wouldn’t want to tell her about them. He writes: “Clearly the implication of the if -clause is ‘Susan is not my closest friend’.” But, as with (13) and (14) above, there is no tendency to draw such a conclusion in the case of a third-person variant of this example: having no idea about how close Simon and Susan are, but knowing that Simon’s problems are acutely embarrassing and that he is very reticent, I could say, “Even if Simon were Sheila’s closest friend, he wouldn’t want to tell her about them”. Thus it is the fact that one is taken not to be ignorant as to the closeness of one’s own friends, that discourages a similarly neutral interpretation of the original example. Semantic accounts of indicative and subjunctive conditionals meeting the demands of Jackson’s “actually” argument – the latter directing us away from, and the former keeping our feet planted firmly in, the actual world – will be considered in 7.13 (indicatives) and 7.15, 7.17 (subjunctives). Before getting there, we shall note (in 7.12) some facts which, inter alia, threaten to embarrass this ‘dualist’ strategy. How best to respond to the threat is left as an open problem. Let us pause for a moment to say a bit about what the “actually” argument does not show. A rough approximation to the semantics for subjunctive conditionals of Lewis [1973] is that such a conditional, with antecedent A and consequent C is true in a world w just in case in each of a range of worlds meeting some condition (determined by A and w), C is true. (See 7.17 for the condition in question, and 7.17.3 for a correct statement of Lewis’s own preferred account.) Now, it is specifically the employment here of quantification over worlds that the “actually” argument shows would be inappropriate in the case of indicative conditionals. Jackson’s own proposal for indicative conditionals is—as we shall see in more detail in 7.13—that while the truth conditions for (indicative) If A then C (in w) are that either A is false (in w) or C is true (in w), this apparently austere ‘material implication’ account of truth conditions needs to be supplemented by something else to provide an account of the meaning of indicative conditionals. Nothing in Jackson’s “actually” argument prevents a less austere account of the truth conditions of If A then C. Although quantification over worlds is ruled out, the argument does not touch universal quantification over epistemic states, as in Veltman [1986], or over situations or cases, as in Lycan [1984] (inspired by Geis [1973], [1985]; see also Lycan [2001]). In the present brief survey, we cannot discuss these proposals, though they are touched on in 7.12 below. What is undermined, however, is any attempt to use quantification over entities of the same kind, for given the semantics of both indicative and subjunctive conditionals, and in Stalnaker [1968], [1975], Davis [1979], (in all of which we have possible worlds for both) or Ellis [1978], Gärdenfors [1986], [1988] (belief systems for both). So much for what the “actually” argument does not show. On a more positive note, let us consider the following fact. Turning on the seven o’clock news, one thing you would be guaranteed not to hear is this: today more troops entered region X than actually did. Why would no such news item be broadcast? After all, it is perfectly possible for there to have been a number of troops greater than the actual number of troops which entered the region—so why shouldn’t

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reporters have discovered, and relayed through the newsreader, the information that precisely this possibility was realized? The reason is clear enough: the actual number of troops just is whatever number of troops did enter the region. The newsreader is purporting to inform you of what happened in the actual world after all. Contrast the absurdity of (26) with the perfect intelligibility of (27): (26)

?

Let me inform you that twice as many troops entered region X as actually did.

(27) Let me ask you to imagine that twice as many troops entered region X as actually did. What follows the “that” in the two cases shows itself to be a fit propositional content for imagining or supposing, suitable for following up with speculation as to what would have happened in the situation as (counterfactually) envisaged, but not a fit propositional content for adding to one’s stock of beliefs. More generally, in the latter case, we should say, not “adding to one’s stock of beliefs” but rather, “adjusting one’s stock of beliefs to accommodate”, since the new (putative) information may contradict what one previously believed. A convenient briefer description with some currency in the belief revision literature for this process is updating (one’s beliefs). The moral of (26)/(27), then, is that a statement which, though capable of being true, announces its own falsity in the actual world – as with what follows the “that” in our examples – is something one can reasonably suppose in terms of, but not something one can reasonably update with respect to. (A preference for this use of “suppose”, requiring that one suspend all relevant disbelief, is somewhat stipulative. Someone can easily be imagined to say, with a different usage, “But I can’t suppose that—I know first-hand that it isn’t so!”. For the present usage, see Skyrms [1987] and Collins [1989], the latter paper making appropriate connections with the theory of conditionals. Notice that the currently preferred notion of supposition is that expressed by suppose with a subjunctive rather than an indicative construction: “Suppose Harry had been here yesterday” vs. “Suppose Harry was here yesterday”. A slightly confusing presentation of the contrast, to be found in Lance and White [2007], speaks of “two modes of updating” instead of distinguishing updating from supposing.) The above correction from talk of addition of beliefs to talk of adjustment and accommodation is Stalnaker’s variation on a suggestion of F. P. Ramsey’s, to the effect that “If two people are arguing ‘If p will [it be the case that] q?’, and are both in doubt as to p, they are adding p hypothetically to their stock of knowledge and arguing on that basis about q”. (Ramsey [1929], p. 247n.; Stalnaker [1968].) This suggestion has been enthusiastically endorsed in subsequent theorizing about conditionals, in the form of the ‘Ramsey Test’ for seeing what conditionals one accepts: these should be precisely those whose consequents express a belief one would possess on minimally revising one’s state of belief so as to include the antecedent. The content of the previous paragraph, taken together with the difference between (18) and (19) (only the latter being anomalous), observed earlier – or any other subjunctive/indicative pairs of conditionals with ‘contra-actual’ antecedents – strongly suggests that Ramsey’s proposal is apposite more for the assessment of indicative conditionals than for subjunctives. In the case of the former conditionals, the antecedent is to be

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thought of as a potential update, while for the latter, as a supposition. This underlies our impatience (also stressed by Ramsey) with indicative conditionals whose antecedents we are so certain are false that we cannot envisage coming to discover their truth. We have no time for an interlocutor who says “If the Earth has more than one moon,. . . ”, but are prepared to listen when a counterfactual supposition is on offer: “If the Earth (had) had more than one moon, . . . ”. With (18) and (19) above, this contrast is presented in an especially extreme form. The Ramsey Test has been used as a basis for an ‘epistemic’ semantics (in terms of belief states rather than truth) for conditionals by Gärdenfors, elaborating the kind of account mentioned in 5.12, but we shall not discuss this here. See Gärdenfors [1986], [1988] (Chapter 7), where some interesting connections are forged between the logic of conditionals and the process of belief revision. (See also Ellis [1978] and Levi [1988].) A latter-day descendant of the Ramsey Test is a thesis associated especially with E. W. Adams, identifying the acceptability (to a given subject) of an indicative conditional with the conditional probability of its consequent given its antecedent, where the probability in question is subjective probability (for the given subject). We take up this idea in 7.13. Another thought about conditionals is suggested in embryonic form in Ramsey [1929], and like the Ramsey Test, it has attracted numerous defenders in succeeding years. This is the thought that asserting a conditional is not to be viewed as the assertion of a conditional proposition, but rather as the conditional assertion of the consequent, the condition in question being given by the antecedent. If the latter condition is not satisfied (i.e., if the antecedent is false), then it is as though no assertion had been made. The parallel is with conditional bets, which are void – in that no money changes hands – unless the condition they are conditional upon obtains. Considering a certain conditional, which Ramsey calls a hypothetical, and whose antecedent (using the terminology mentioned in the Digression beginning on p. 933 above) he calls its protasis) that “it asserts something for the case when its protasis is true” (Ramsey [1929], p. 240). This suggests – rather than clearly expresses – the conditional assertion idea just described, since Ramsey does not add “and expresses nothing otherwise”. The idea appears more explicitly in Quine [1950], §3: An affirmation of the form If p then q is commonly felt less as an affirmation of a conditional than as a conditional affirmation of the consequent. If, after we have made such an affirmation, the antecedent turns out true, then we consider ourselves committed to the consequent, and are ready to acknowledge error if it proves false. If on the other hand the other hand the antecedent turns out to have been false, our conditional affirmation is as if it had never been made.

Given this conception of conditional assertion (or ‘affirmation’), we need to know what it is for an argument to be valid, in which conditionals figure as premisses or conclusions to be valid. (Here argument is used in what was called the premisses-&-conclusions sense, in 1.23) One obvious suggestion is to supply a ‘defective’ truth-table, with only two determinants: T, T, T and T, F, F . Such partially determined connectives were discussed in 3.11. But this does not capture an important part of the conditional assertion idea, namely that when its antecedent is not true, a conditional affirmation is as though it ‘had never been made’. We want to exclude, that is, rather than merely fail to

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include, determinants whose first element is F and whose final determinant is either T or F. The psychologist Peter Wason suggested that for these cases, the appropriate description of the conditional is not as true or as false, but as ‘irrelevant’. Another way of putting this is that unless the antecedent is true, the question of the truth or falsity of the consequent ‘does not arise’. As in §2.1, we prefer not to use “T” and “F” as labels for truth-values in any but a bivalent setting. This allows us to keep a clear distinction between the values h(A) – a matrix-element – and vh (A) – a genuine truth-value recording the designation status of h(A) – for a formula A. Accordingly, following Wason [1968], let us notate our trichotomous classification using “1”, “2”, and “3” for what he would call true, irrelevant, and false, respectively. Thus the earlier determinants become 1, 1, 1 and 1, 3, 3 , and we can now add in 3, 1, 2 and 3, 3, 2 . (We stretch the term determinant to cover these three-valued analogues.) Note that we are far from having here specified a three-element matrix. In the first place, even setting aside the question of what the tables should look like for connectives other than the conditional, determinants with the ‘new’ value, 2, in the either of the first two positions have not been provided (so the underlying three-element algebra has not been described.) And secondly, we have said not which values are to be designated (so no passage from algebra to matrix has been settled). The first difficulty need not detain us, since there is no reason not to consider the three-valued analogue of what in the two-valued case (3.11) we called partially determined connectives. (For consistency with the terminology of §2.1, we should describe such connectives as partially determined w.r.t. a class of evaluations rather than of valuations.) This will still allow us to assess the validity of sequents, as long as we make good the second lack, and come clean on the matter of designation. The obvious choices here are (i) the generous choice, counting 1 and 2 as designated, and (ii) the ungenerous choice, counting only 1 as designated. Wason’s motivation (in Wason [1966], Wason [1968]) for introducing 2 in the first place was to provide a hypothesis for the fact that the fallacy of affirming the consequent – arguing from a conditional together with its consequent to its consequent – is more common (under experimental conditions designed to elicit inferences) than the fallacy of denying the antecedent – arguing from a conditional together with the negation of its antecedent. This motivation would appear to require choice (ii). But choice (ii) does more than ‘validate’ affirming the consequent: Exercise 7.11.5 Check that on the ungenerous choice, all sequents of the form A ⇒ B A are valid, as well as all those of the form A ⇒ B B. (Here and at many places in this section, we use “⇒”, elsewhere our metalinguistic implication connective, for a non-specific conditional connective of the object language, to be interpreted indicatively or subjunctively according to the context of the discussion. The point made by the present exercise appears in van Fraassen [1975], p. 53.) Presumably these represent patterns of inference which subjects are rather less likely to exhibit than either of the fallacies described above; there are in any case reasons culled from Dummett [1959a] and deployed in Jeffrey [1963] for considering each of 1 and 2 as a designated value. Jeffrey’s paper contains further suggestions for the form of the three-valued tables that might be suitable, though rather than go further into this question here, we defer the

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discussion until 7.19, where a fully worked out proposal along these lines (from Cooper [1968]) will be presented in some detail: see especially the discussion between 7.19.10 and 7.19.11 – p. 1052f. – for the connection between that and an elaborate semantic account of conditional assertion found in Belnap [1973] and various papers (some of them mentioned the 7.19 discussion just alluded to) elaborating Belnap’s ideas. One way of describing this suggestion is in terms of two-dimensional modal logic – on which see 3.31.9, p. 489, and (for a different application in connection with conditionals) the discussion between 7.17.4 and 7.17.5 below (pp. 1015–1021): what proposition the conditional assertion of B conditional on A expresses in world w depends on whether A is true in w. If so, then the proposition expressed is the set of worlds at which B is true, and if not, then no proposition is expressed. In the terminology of 3.31.9, we are thinking of the conditional If A then B as being evaluated from the perspective of w (as reference world) in x (the world of evaluation), and provided A is true at w, as being true with respect to the pair w, x just in case B is true at x. (Here for simplicity we assume there is no two-dimensionality called for by A and B themselves, and say nothing about the case in which A is not true at w.) Again, we return briefly to this matter in 7.19, for the moment noting some qualms aired by Dummett [1959a]. Dummett casts serious doubt on the whole idea of conditional assertion, whether intended as an account of some special conditional construction, or as an account of all conditionals. In Dummett [1959a], the objection is based on a general and potentially controversial (cf. Holdcroft [1971]) doctrine about assertion and truth; but the following formulation, from Dummett [1991] (p. 115) avoids reliance on that doctrine while successfully undermining the ‘conditional assertion’ project: Someone who asserts a conditional sentence is not asserting the antecedent: it is senseless to think of assertoric force—or force of any other kind—as attaching to the antecedent clause. Nor is it a good description to say that he asserted the consequent conditionally, as if he had handed his hearers a sealed envelope marked “Open only in the event that. . . ”: someone who believes the speaker, and knows the consequent to be false, may infer the falsity of the antecedent.

(It is the second sentence which makes the point; the first is quoted here to make the “Nor” intelligible. The ‘sealed envelope’ analogy, incidentally, appeared in Jeffrey [1963].) We shall have no more to say here on the topic of conditional assertion, a full discussion of which requires consideration of the distinction—this one much having a much greater claim to be recognised—between questions with conditional content on the one hand, and conditional questions on the other, as well as the rather more controversial (alleged) distinction between commands to make a conditional true and conditional commands. Although such nonassertoric contexts are outside the scope of our survey (as announced in 5.11), the reader will find food for thought on these matters in Dummett [1959a], Holdcroft [1971], and Van der Auwera [1986]. The ‘sealed envelope’ analogy is subjected to particularly telling criticism in Edgington [1995], p. 289. See further Barker [1995] and the discussion thereof in the Appendix to §6.1, p. 812 above. In the interests of completeness, mention must here be made of a use of if which is reminiscent of what people say under the heading of conditional

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assertion. These are sometimes called ‘sideboard’ conditionals, after a famous example from Austin [1961], p. 212: (28)

There are biscuits on the sideboard if you want them.

in which the if -clause gives a condition for the appropriateness of saying something rather than a condition for the truth of what is said. Other labels include “pragmatic conditionals”, “speech act conditionals”. (See Haegeman [1984], and, for a more recent foray, Predelli [2009], who misquotes the example with “in the cupboard” for “on the sideboard”; what kind of host would that be, expecting the visitor to ferret around for – and presumably unpack – any biscuits to be consumed?) Note that the “speech act conditionals” terminology is part of the general project of Sweetser [1990] according to which connectives in general can be expected to participate at a ‘content domain’ level, at an epistemic level, and at a speech-act level. (We had occasion to dip into this analysis for our discussion of disjunction in 6.13, as well as in the notes – on but – to §5.1: p. 674.) A perusal of Chapter 5 of Sweetser [1990] reveals her content domain conditionals and epistemic level conditionals to correspond rather closely, as one would perhaps expect, to the projective and hypothetical conditionals (respectively) of Dudman’s classification. (We are speaking here of indicative conditionals. For the subjunctive case there is no speech-act interpretation possible, and there may or may not be an epistemic interpretation – depending on whether one thinks of the ‘extrinsic’ conditionals of 7.11.1 as supplying examples of precisely this.) Digression. Suggestive of Sweetser’s epistemic level conditionals is the following passage from Zepp [1982], p. 211, discussion the southern Bantu language Sesotho; the colon has been introduced for grammaticality, and the final sentence is included for its likely interest to many readers (negative polarity and all that): The literal translation of “ if . . . then” is roughly, “if it is that . . . you should know (subjunctive) . . . ” Thus implication in Sesotho, which differs from English, might be expected to cause some differences between English and Sesotho groups. This is especially the case since the word for “if” is “ha”: the very same word which is used to introduce negative sentences.

End of Digression. Returning specifically to the content level vs. speech-act interpretation, we have the distinction well brought out by the following pair, from Fauconnier [1985], p. 126: (29) (30)

If you’re hungry, we’re having dinner at six If you’re hungry we’ll have dinner at six.

The first of these is a ‘sideboard’ conditional, while the second is not, though it is still ambiguous between “if you’re hungry now,. . . ” and ‘if you’re hungry at six,. . . ’. Fauconnier also gives the following example, in which there is a ‘sideboard’/non-‘sideboard’ ambiguity: (31)

If you’re interested, Prof. Murgatroyd will speak tomorrow at five.

Digression. It is interesting to note that some of these recent “if”s (those in (28) and (31), at least) could be replaced without much change in meaning by the

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usually far from synonymous “in case”. (See also Wakker [1996], p. 179.) When this replacement is made in (28), a comma needs to be inserted after “sideboard” to avoid giving an unrelated reading—in fact a content-level rather than a speech-act interpretation, according to which what is given is an explanation as to why the biscuits have been placed there. A pleasant (language-teaching) discussion of the contrast between if and in case may be found on p. 226f. of Murphy [1994]. At an earlier stage of English these two subordinators seem to have been closer in meaning. The if -like meaning of in case seems to survive in the complex preposition “in case of” as in “Do not use in case of fire”, which appearing on the door of a lift, certainly does not advise the reader not to use the lift in case there should be a fire. The somewhat technical use of “just in case” to mean “if and only if” presumably also attests to this earlier usage. End of Digression. Most discussions of conditionals in the logical literature—and the present survey will be no exception—set aside as peripheral, marginal, or just plain ‘odd’ (Veltman [1986]) not only these sideboard conditionals but a number of other constructions with if, sometimes having the status of special idioms. (Some of these are included in the ‘fourth category’ of Dudman’s [1991] classification, sketched in our discussion of the subjunctive/indicative distinction above.) By way of reminder to the reader, we recall the so-called ‘Dutchman’ or ‘monkey’s uncle’ conditionals having such as (32), and what have been called asseverative conditionals such as (33); these cases (understood idiomatically) are interpreted as rhetorically embellished ways of respectively denying their antecedents and asserting their consequents, presumably through an invited Modus Ponens or Modus Tollens (again, respectively—for the latter rule, see 1.23.6(i)): (32)

If he’s a competent lawyer then I’m a Dutchman.

(33)

If I’ve told you once I’ve told you a thousand times. . .

Some discussion of these rhetorical phenomena may be found in Sadock [1977], Carden [1977], Sørensen [1978]. (The term asseverative is taken from the last of these. A related phenomenon, arising for the construction “If such-and-such then why so-and-so?” as used to cast doubt on whether such-and-such is the case, is the subject of Kay [1991].) We have not included amongst our numbered examples ‘verbless’ protases of rather restricted location in the sentence and with a distinctly idiomatic feel to them. There are those of the “Ida was her best – if not her only – friend”, for discussion of which the references given in Brée [1985], p. 314, may be consulted, as well as §5 of Yamanashi [1975]. The following passage, from a work on culinary uses of plants, illustrates a nonnegative version of construction with two occurrences. The author opens his entry on (lemon) balm with the words: A good, if vigorous, plant for the garden with bright green, golden or variegated leaves and a nice, if unsubtle, lemony smell. (Stuart [1984], p. 74.)

Our final topic for the present subsection is a further pair of distinctions which have been proposed as taxonomically useful for the study of conditionals. They will not detain us to anything like the extent to which the subjunctive/indicative distinction did earlier. These are, first, the distinction between

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‘consequential’ and ‘non-consequential’ conditionals, and secondly the distinction between (as we shall put it) ‘helping’ and ‘non-hindering’ conditionals. The consequential/non-consequential terminology is from Dancygier and Mioduszewska [1984]; closely related distinctions are presented in Funk [1985], Comrie [1986], and elsewhere. (Indeed the distinction seems close to Dudman’s projective/hypothetical distinction, discussed above.) The idea will be clear if we cite some of the examples offered by Dancygier and Mioduszewska. The following they list as consequential: (34)

If I catch the train I will come on time.

(35)

If I tell him the truth, he’ll get angry.

(36)

If John hadn’t been ill yesterday, he would have passed the exam.

(37)

If I knew how it worked, I would tell you what to do.

Note that the first pair are indicative and the second pair are subjunctive, and that in all cases the words “as a result” can be appended to the consequent without too much unnaturalness. The authors give the following as examples of non-consequential conditionals: (38)

If Susie is at the door, she is breathing quietly.

(39)

If he told you that last night, he was lying.

(40)

If John didn’t pass the exam, then he was ill.

(41)

If my son is a genius, I’ve underestimated him.

(Example (41) had appeared in §160 of Leach [1971].) Here, given plausible settings for the examples, the words “as a result” would be out of place in the consequents. Dancygier and Mioduszewska suggest instead as a diagnostic test the insertability of “it means that” or “that means that” before the consequent. In the same spirit, it might be suggested that “means” here is sometimes better replaced by “shows”. At any rate, the idea is clear enough: in (40), for example, it is not suggested that if John didn’t pass the exam, then he was ill as a result (of not passing), but that John’s not passing indicates (means, shows,. . . ) that he was ill (at the time of sitting the exam). Further, as Dancygier and Mioduszewska observe, the non-consequential cases do not admit of subjunctive conditional formulation (except, we might add, with the ‘extrinsic’ construction of 7.11.1). Two points suggest that this distinction between consequential and nonconsequential conditionals is not so much a difference between kinds of conditionals as a difference in the kinds of grounds one has for assenting to them, or perhaps in the kinds of thing that make them true. The first is that some (and arguably all, though not with equal naturalness) of the consequential cases will allow the “it/that means that”; best from the above list is (35), which becomes: (42)

If I tell him the truth, that means that he’ll get angry.

The point here is that if the antecedent’s being true would causally contribute to the consequent’s being the true, then this is one way for the antecedent’s truth to serve as an indicator of the consequent’s truth. Secondly, there is no difficulty in ‘amalgamating’ paradigmatically consequential and paradigmatically non-consequential conditionals into a single (‘mixed’ ?) conditional, which again

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suggests we are not dealing with a fundamental distinction here. (Such amalgamations are clearly not available for subjunctive and indicative conditionals, for example.) We may illustrate this possibility with the non-consequential (41), and the consequential (43) If my son is a genius, then he will do well in business. which we may amalgamate to obtain: (44) If my son is a genius, then I’ve underestimated him and he will do well in business after all. The final distinction between conditionals – the ‘helping’ vs. ‘non-hindering’ distinction – which deserves some attention is that between those in which, as we might put it, the truth of the antecedent is claimed to help make true the consequent (perhaps causally, perhaps constitutively), or more generally, to ‘favour’ (e.g., confirm or bear some other favourable evidential relation to) the consequent, and those in which the truth of the antecedent merely fails to hinder the consequent’s being true, or more generally, fails to ‘disfavour’ the consequent. Arguably, all of the Dancygier and Mioduszewska examples, (34)–(41) are of the former type. Restricting his attention to subjunctive conditionals, Pollock [1976], p. 42, describes two ways that what he calls ‘simple subjunctives’ with antecedent P and consequent Q can be made true: “Either Q is made true by P , or Q is already true and P would not disrupt this”. The former disjunct, Pollock represents as a special ‘necessitation conditional’, and the latter as an even if conditional. According to Pollock’s preferred logic of these constructions, a ‘simple subjunctive’ is provably equivalent to the disjunction with these two disjuncts. In our deliberately vague terminology, that is to say, a subjunctive conditional is equivalent to the disjunction of a ‘helping’ with a ‘non-hindering’ conditional. (Versions of Pollock’s equivalence thesis, weakened to the claim that the subjunctive conditional entails the disjunction in question may be found in the opening paragraphs of Barker [1991], and, adapted to the case of indicative conditionals, in Barker [1979], p. 144. The former paper does not, however, accept the “even if” formulation of the non-hindering disjunct. Readers not consulting the bibliography ambulando should note that the authors here are different Barkers. We return to even if in 7.12.) The fact that not all conditionals (subjunctive or indicative) are ‘helping’ conditionals – to go along for the moment with a terminology we shall shortly be arguing incorrectly suggests a division amongst the conditionals themselves – can be brought to bear against the suggestion that conditionals always purport to express some kind of connection between the antecedent and consequent. The misleadingness of this suggestion was noticed already in Chisholm [1946], with the remark (p. 298) that we affirm many subjunctive conditionals in order to show that there is no relevant connection between antecedent and consequent; e.g., “Even if you were to sleep all morning, you would still be tired”.

In this case, one might expect sleeping all morning to be a recipe for eliminating tiredness, and Chisholm’s assertor is countering this expectation for the particular case, without denying the general grounds for the expectation. The

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envisaged morning-long sleep has the status of what we shall call, below, a ‘potential obstacle’ to the addressee’s subsequent tiredness, giving perhaps a ‘connection’ of sorts between antecedent and consequent for this kind of nonhindering example. But Chisholm’s point can be made with other examples. For suppose the speaker is trying to correct the hearer’s belief that having avoided walking under a ladder last Friday would have averted her child’s subsequent attack of meningitis, by saying: (45) If you had not walked under the ladder, your child would still have become ill. Here no concession is involved to the effect that a general ground for expecting bad luck to follow upon walking under ladders is in this particular instance not operative. For an indicative case, imagine the following addressed to this same mother, whose superstitious beliefs also include the belief that ill fortune is warded off by visiting a shrine: (46) Whether or not you visit the shrine, your child will need medical attention for at least a further week. This, which is precisely a way of denying that there is any connection between the mother’s shrine-visiting and her child’s recovery can be more or less paraphrased by (47) If you visit the shrine your child will need medical attention for at least a further week, and if you don’t visit the shrine your child will need medical attention for at least a further week. Digression. We say that (46) is “more or less” paraphrased by (47). The need for some such qualification is made clear by the following examples from Zaefferer [1991b], p. 232: (a)

If you take the plane to Antwerp, the trip will take three hours; if you take the car or go by train, it will take ten hours.

(b)

Whether you take the car or go by train to Antwerp, the trip will take ten hours; if you take the plane, it will take three hours.

(c)

If you take the plane to Antwerp, the trip will take three hours; whether you take the car or go by train, it will take ten hours.

The difference in acceptability between (b) and (c), as Zaefferer suggests, results from the fact that the use of the whether -construction implies that the ‘disjuncts’ exhaust the possibilities under consideration – a presumption it is easier to sustain in advance of an as yet uncontemplated further option (as in (b), where what follows the semicolon has the status of an afterthought) than in the wake of an already articulated additional option. However, in view of the semantic similarity between (46) and (47) it may not be wise to create – as Zaefferer proposes – a special category of ‘unconditionals’ for the likes of (46). End of Digression. With these last examples we have tried to avoid formulations involving even if (postponed until 7.12), though some theorists of conditionals want to carefully segregate out conditionals with and without “even”, giving quite different

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accounts of the two, and going so far as to say that the absence of an explicit even is no guarantee that we are not really dealing with an even if conditional. Hunter [1983], p.7, says, for example, that some conditionals that do not require a connection between antecedent and consequent “are covert ‘even ifs’ ”, giving (48) as an example, where the background is that a certain switch is not wired into the circuit: (48) If (i.e., even if) you turn that switch to “Up” the light doesn’t go on; I’ve tried it. The parenthetical insertion is of course Hunter’s. We will return to this example, or rather to the following variation on it, presently: (49) If you turn that switch to “Up” the light won’t go on. Discussing indicative conditionals, Rundle [1983b] addresses what we are calling the ‘non-hindering’ cases, distinguishing them from the ‘helping’ cases by saying (p. 34f.) “In the standard case, the antecedent of a conditional will specify a condition whose fulfilment is thought to favour that of the consequent, whereas here it is a matter of a condition in the face of which the latter is affirmed”. The phrase “in the face of which” is actually rather more specific than the general idea of not hindering, since the latter includes the case in which the antecedent’s truth is neither here nor there as far as the consequent is concerned, as well as the case in which the possibility is made salient – though held not to be realized – of the antecedent’s truth ruling out that of the consequent. Thus Rundle is considering in particular the idea of the fulfillment of the condition specified by the antecedent as a potential obstacle which the conditional statement itself claims to be/have been overcome, and contrasts this with the ‘helping’ case with the pair of examples: (50) (51)

I shall buy that clock if it takes every penny I have. I shall buy that clock if I can afford it.

Rundle remarks that “this may lead us to suppose a difference of substance” (cf. Hunter’s views, as quoted above) between the conditional constructions exemplified by (50) and (51), raising the question of an ambiguity, with if meaning even if in (50), and provided (that) in (51). He argues wisely against any such ambiguity claim, and notes – recalling the “in the event that” paraphrase of “if” which was especially emphasized in Geis [1973] – that with both [(50) and (51)] it can be said that the speaker’s claim is that he will buy the clock in that event. Whether the event is an aid or an obstacle is something to which the ‘if’ is indifferent.

We should add, as a reminder that ‘not hindering’ includes not only the case of the obstacle overcome but also the case where there is no obstacle to be overcome (as in (45)), that whether the event is an aid or an obstacle or neither is something to which the if is indifferent. (We return to Geis [1973] in 7.12, especially a propos of what will there – p. 964 – be called the Geis–Fillenbaum Equivalence.) It is one thing to assert such ‘indifference’, and other to justify this assertion. We proceed, as with the citation of (43)/(44) in the case of the

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consequential/non-consequential distinction, to dispose of the suggestion that the helping/non-hindering distinction marks a semantic difference between two classes of conditionals with a point about amalgamability. The point is simply that if, in such cases as (49) above, repeated here for convenience, the conditional construction has one semantic interpretation (whether characterized as an even if or as a non-hindering interpretation), while that in (52) has another (‘standard’, ‘helping’, ‘connection-indicating’. . . ) interpretation: (49) If you turn that switch to “Up” the light won’t go on. (52) If you turn that switch to “Up” you’ll make a loud noise (the switch being very stiff). then there ought to be no way of making sense of (53) If you turn that switch to “Up”, you’ll make a loud noise and/but the light won’t go on. In fact, of course, there is no difficulty (or even any hint of syllepsis) in interpreting (53). So there cannot be two different senses of if in (49) and (52), requiring an impossible double construal of “if” in (53) to accommodate the double consequent. (Note also that in the circumstances envisaged, though “still” could appear after “the light”, “even” could not appear before “if”.) Exercise 7.11.6 Wierzbicka is discussing one form of the widespread view that there is a link between if and because, in the course of which discussion she writes (Wierzbicka [1997], p. 19f – example numbering adjusted for this quotation): I claim, that the ‘if’ connection is sui generis, and cannot be reduced to anything else; and that a link with ‘because’ is not always present. For example the sentence (i) If he insults me, I will forgive him. does not imply that I will forgive him BECAUSE he has insulted me: it is true that I can forgive him only if he has done something bad to me (e.g., if he has insulted me), but it is not true that the insult will be the “cause” of my forgiveness. Similarly, the sentence (ii) If he invites me for dinner I will not go. does not mean that I will not go because he has invited me: if he doesn’t invite me, I will not go either; and the sentence: (iii) If he is asleep, I will not wake him up. does not mean that I will not wake him up because he is asleep: on the contrary, I could wake him up only if he were in fact asleep.

Discuss Wierzbicka’s comments on her examples (i)–(iii) here.

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7.12

“IF”

Even If, Only If, and Unless

We consider some constructions with an obviously conditional flavour, other than the straightforward if -sentences which featured in 7.11. The discussion will be more in the genre of (informal) natural language semantics rather than logical in any stricter sense. Some of these constructions involve what we might loosely call a modified if, of which the two most important are the even if and the only if constructions. (‘Loosely’, because it is not being suggested that even if or only if has the status of a syntactic constituent of sentences containing them.) We shall also attend to unless. Of course there are many other ways conditionality is expressed, but those just listed will be our main concern here. For example, there is the whether construction of (46) of 7.11, and analogous constructions as in “Wherever you go, you will be followed”, an auxiliary pre-posing construction mentioned under 7.12.1(ii), different lexical items replacing if, such as provided that, monoclausal conditionals (as they are called in Fujii [1995]) like “Without a grant we wouldn’t have been able to manage”, and so on; see further 7.17.1, p. 1010. (See also Anscombe [1981], and Wen-li [1983], in which such examples are listed amongst many others as ‘sentences of implied condition’.) An interesting case is mentioned in Steward [1997], p. 181 (though she does not remark on the feature of current interest): “If it hadn’t been for the dictator, the war wouldn’t have happened”. This does have an explicit “if” and looks like a subjunctive conditional. But what, in that case, is its antecedent? And what would the corresponding indicative conditional be? Additional references to the extensive literature on much else in our discussion – which does not pretend to be comprehensive – will be found in the notes (which begin on p. 1053). Digression. The labelling of even if and only if as modified conditional elements is not intended to suggested that they have more in common with each other than either of them does with unless. For example, Taylor [1997] notes that in answer to a yes/no question like “Are you going to that conference?”, “Only if my paper gets accepted” is fine, whereas “Even if my paper doesn’t get accepted” is not – without a preceding Yes to append this clause to. (The change of “gets” to “doesn’t get” is made in the interests of plausibility given the content.) The reader is invited to consider which of even if and only if the word unless more closely resembles in this respect – or see the discussion in Taylor [1997], 295–297. Also worth considering is whether the list of modified “if”s should include “as if”, assumed by Shuford [1967] to function as a binary connective. (Pizzi [1981] is more careful.) The assumption seems unwise. Unlike “even if” and “only if”, this would give bizarre indicative/subjunctive mismatches: “Sarah is trembling as if she were afraid”. What this means is not that one thing is the case “as if” something else were the case – to use a form of words which is itself nonsense, in order to make the point – but rather, that Sarah is trembling in the way she would (or might) be trembling if she were afraid. Here the relevant antecedent and consequent are explicitly displayed within a construction itself functioning as a complex manner adverbial, modifying “trembling”. End of Digression. It would be nice to be able to begin with two simple observations about what the three examples we have selected for closer inspection have in common, namely (i) that even if, only if, and unless are alike in not allowing (by contrast with unmodified if ) a “then” in the paired clause, and (ii) that all

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three items are as comfortable introducing subjunctive as indicative conditionals. But the second of these points has been often been denied for the case of unless, and this may well be a matter of dialectal or idiolectal variation. Thus Geis [1973] appends an asterisk to the following example, which is, however, perfectly grammatical for the present author, as also for Brée [1985], who takes this as a weakness of Geis’s discussion rather than a point of idiolectal/dialectal difference: (1)

Unless you had helped me, I would never have been able to finish on time.

Quirk and Greenbaum [1973] (p. 325) express a similar sentiment, declaring “Unless I had arrived” to be impossible as an unless-clause. (They presumably have in mind a subjunctive construction rather than an indicative pluperfect.) Dancygier [1985] even attempts a detailed semantic explanation of the nonoccurrence of subjunctive unless. (For a more considered position, see Dancygier [1998], §6.2, and especially, on this point, pp. 175–177.) The status of commonality (ii), then, is somewhat unclear. As for (i) – absence of “then” – there seems to be no corresponding disagreement, though of course there is a question as to what might be behind this, especially in the case of the modified-if phrases since without the modification an accompanying then is permitted. One possibility (suggested in conversation by Brian Weatherson) is that the presence of then is a signpost to the ‘helping’ side of the helping/nonhindering distinction we were at pains in 7.11 to down-value, but no further speculation will be entered into here. Instead, the reader is referred to the useful discussion (as well as the bibliographical references there given) in Dancygier and Sweetser [1997]; see also Geis [1985], esp. pp. 146–148, Schiffrin [1992] and Dancygier [1998], §6.3; a general discussion of the cases in which then may occur appears in §4.1 of Bhatt and Pancheva [2005]. Some of the intuitions voiced by Dancygier and Sweetser are fairly subtle, and we close these remarks about then with one such example, a variation on Austin’s biscuits example (28) from 7.11: (That example had the if -clause second rather than first, so there is no grammatical possibility of inserting a then into it.) (2) (3)

If you’re hungry, there are biscuits on the sideboard. If you’re hungry, then there are biscuits on the sideboard.

According to Dancygier and Sweetser [1997], p. 129, (2) is less rude than (3) as a way of offering biscuits, in that (3) seems to “insist on the specific circumstances under which the biscuits are offered”. (See Iatridou [1993] for a detailed discussion, and Hegarty [1996] for some criticism, and also Predelli [2009].) Let us turn to the modification of if by only and even, beginning by pointing out what not to say about them. The wrong approach to even if and only if, as has been widely recognized in theorizing about conditionals, is to take them as semantically unstructured idioms, related to unmodified if by nothing more than historical connections. (The need for compositional treatments of the constructions were stressed, perhaps first, in Fraser [1969] for even if and in Geis [1973] and McCawley [1974] for only if. Contrast the case of the ‘optative’ if only, which is clearly a special idiom.) Obviously, only is doing the same job – however that job is best described – in each of (4)–(6): (4)

These flowers need watering only in the summer.

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(5)

That kind of epidemic takes hold only after an earthquake strikes a populous region.

(6)

Trudi will come with us only if we decide on a vegetarian restaurant.

Presumably no-one will suggest that there is a special “only after” temporal connective involved in (5). There should be the same resistance to an analogous claim for a special “only if” conditional connective in (6). Note also the same mobility for “only” in the two cases: (7)

That kind of epidemic only takes hold after an earthquake strikes a populous region.

(8)

Trudi will only come with us if we decide on a vegetarian restaurant.

as well as the subject-auxiliary inversion when the only-modified material is pre-posed: (9)

Only after an earthquake strikes a populous region does that kind of epidemic take hold.

(10) Only if we decide on a vegetarian restaurant will Trudi come with us. (Similarly, in the case of (4), we have “Only in the summer do these flowers need watering”.) Remarks 7.12.1(i) Dancygier and Mioduszewska [1984] claim (pp. 125, 127) that “Only if he loves you, he will come” and “Only if she called yesterday, I was out” are, respectively, grammatical and ungrammatical. Of course, neither is grammatical, because of the failure to invert. This illustrates the especially great risk taken by non-native speakers of a language in commentary on the syntax and semantics of conditionals in that language – not that native speakers are above this kind of oversight: “Only if you don’t give me your money, I will kill you” can be found unasterisked at p. 184 of Fillenbaum [1986], for instance. (In this section, the subject is conditionals in English; references to some comparative linguistic literature are given in the notes – p. 1053 onwards – as well as in occasional asides in this subsection.) (ii ) Subject-auxiliary inversion appears in quite another place in conditional constructions also: as an if -less alternative for subjunctive antecedents, as in “Had the candidate not been late for the interview, her chances would have been considerably better”. (The same applies to the should construction of 7.11.2(ii).) (iii) The inversion phenomenon alluded to under (ii) is reminiscent of subject-auxiliary inversion in the formation of yes/no questions in English, and is amongst the grounds occasionally urged for semantically relating such questions and conditionals. Another is the close relationship between if and whether ; both can introduce conditionals—cf. (46) in 7.11—and both can introduce indirect yes/no questions. §5.2 of Chapter 10 in Huddleston and Pullum [2002] provides some interesting details on the possibility of using if in place of whether in indirect questions

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(or embedded interrogatives, as the authors prefer to say) in contemporary English. (Various similarities between conditionals and questions are noted in Mayo [1970], Bell [1974] and Harman [1979]. According to Jespersen [1924] (p. 305), the inversion in conditionals developed historically from that in questions; Johnston [1996], p. 105, suggests instead a ‘common cause’ explanation: inversion is used in both cases as an assertion-cancelling device. While for English the relevant conditionals are subjunctive, a similar inverted construction is available in the indicative case; for a German example, the parallel with questions being explicitly noted, see (40)/(41) on p. 222 of Zaefferer [1991b]. For examples in the English of the Polish authors of a paper we have elsewhere had occasion to cite for its content, see the second and third sentences of the second paragraph on p. 179 of Łoś and Suszko [1958]. Semantic differences, some of them rather subtle, between if and whether in indirect questions, are emphasized in Bolinger [1978]. A more recent attempt to explain the why conditionals and interrogatives make use in many languages of common lexical and syntactical devices appears in 3.2.2 of Bhatt and Pancheva [2005].) (iv ) The details of subject-auxiliary inversion for subjunctive conditionals and questions are none the less somewhat different; for the latter a reduced “not” in the form of an -n’t suffix counts as part of the (first) auxiliary element to be permuted, while this is not so for the former. Compare: (a) Had we not already eaten by that time? (b) Hadn’t we already eaten by that time? (c) Had we not already eaten, we would be happy to accept. (d) *Hadn’t we already eaten, we would be happy to accept. The inversion triggered by initial “only if” appears to follow the (c)/(d) pattern rather than the (a)/(b) pattern – pace Fillenbaum [1986], at the base of p. 184 in which we find (cited as grammatical) “Only if you don’t give me a ticket won’t I give you $20”. Having decided that only if should not be taken as an unanalyzable idiomatic chunk, but should rather be given whatever account flows from combining independently justified accounts of only and if, we face the problem of supplying those accounts. The whole of the present section is devoted to the question of what form the latter account should take, but quite apart from the details of that account, we can notice a general problem for this strategy. (This difficulty is raised in Van der Auwera [1985b].) Consider (11) and (12): (11) (12)

Only Harry survived the accident. Harry survived the accident.

Does (11) entail (12)? We might say no, and try to explain away the appearance of an entailment here by an appeal to the idea of conversational implicature, invoking the following reconstruction of a hearer’s reasoning: Why am I being told that only Harry survived if in fact nobody survived? In that case, the speaker would have given me the stronger pertinent information that nobody survived, so it must be that somebody survived, and since—the suggestion would

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go—(11) says literally that nobody other than Harry survived—it follows that Harry survived. The objection to this suggestion is that it represents the literal content of (11) as being equally well provided by (13) (13)

At most Harry survived the accident.

and (13) has no tendency conversationally to implicate (12). (Indeed it has, if anything, an opposite implicature.) So the meanings of (11) and (13) are not the same. All is not lost for the suggestion that there is an implicature, rather than an entailment, from (11) to (12), however, since this might be claimed to be a conventional rather than a conversational implicature. Let us return to this possibility after considering the result of returning the more obvious answer yes, instead of no, to the question of whether (11) entails (12). The difficulty in this case is that if deleting only from (11) gives something (namely (12)) which follows from (11), and we wish to treat the only in only if constructions in the same way as other occurrences of only, then (6) (or, equally well, (8)) ought to entail (14): (6)

Trudi will come with us only if we decide on a vegetarian restaurant.

(14)

Trudi will come with us if we decide on a vegetarian restaurant.

But there does not seem to be an entailment here. Perhaps Trudi has imposed several conditions on dining with us, one of which is that we choose a vegetarian restaurant, and the another of which is that it be located within walking distance of where we all now are. Knowing this much we will assent to (6) while dissenting from (13). The point here is simply the familiar distinction between necessary conditions and sufficient conditions: (6) says that our deciding on a vegetarian restaurant is a necessary condition for Trudi to accompany us, while (13) represents this condition as sufficient. There is, then, a prima facie difficulty facing any account which would unify the only in (6) with the only in (11). In the case of (11), deleting the only yields something apparently entailed by (11), while in the case of (6) deleting the only yields something apparently not entailed by (6). So one way of surmounting this difficulty would consist in showing that in at least one of these cases, the appearance is deceptive. The second case looks strong, given the distinction between necessary and sufficient conditions (on which, more below), and we have seen that for the first case, the prospects are not good for the project of undermining the apparent entailment by appeal to conversational implicature. We now take up—briefly and inconclusively—the possibility, mentioned above, of utilising the notion of conventional implicature to this end. The paradigm for this manoeuvre is the case of but (described in 5.11 and in the notes to §5.1: p. 674). The conditions under which S1 but S2 is true are the same as those under which S1 and S2 is true, but there is a difference in the appropriateness of asserting the two, because of (something we may inaccurately describe as) a conventional implicature of contrast. We might maintain that (11) and (13), (11)

Only Harry survived the accident.

(13)

At most Harry survived the accident.

although having the same truth-conditions, differed in that (11) conventionally implicated (12):


953

Harry survived the accident.

There is obviously considerable pressure in that case to maintain, correspondingly, that (3) and (14), or any other pair consisting of an “only if” conditional and the result of deleting the “only” are alike in respect of truth-conditions and differ only over what is conventionally implicated. An earlier proposal (Horn [1969]) located the unstraightforward part of the story not in conventional implicature but in presupposition. According to Horn, (11) above presupposes (12) and asserts (13). Thus for example—without going too deeply into the murky area of presupposition—someone asking “Did only Harry survive?” would be representing themselves as taking it for granted that Harry survived, and wondering whether anyone else did. A negative reply to the question would be taken as a claim that others also survived, and any response denying that Harry survived would be taken as correcting a misconception involved in the question rather than as simply answering it. We may at this point bring in “even” by noting that of correcting the misconception here would be by saying: “Not even Harry survived”. And with even, as with only, treatments in terms of conventional implicature and in terms of presupposition have been offered. To take the latter first, we have Horn [1969], whose account we give as (16), next to the account of only, given as (15): (15)

(16)

“Only a is F ” presupposes: a is F ; asserts: nothing other than a is F “Even a is F ” presupposes: something other than a is F asserts: a is F .

(according to Horn [1969])

(according to Horn [1969])

While there is no doubt a symmetry of sorts here, most other writers on “even” have, beginning with Fraser [1969], have thought differently. They have held that the presupposition under (16)—whether we want to consider it as a presupposition, or as a conventional implicature, or something else—should have the upshot not just that a fails to be alone amongst the things which are F , but that a should be, compared with other F things, less likely than they are to be F . (The conventional implicature view has been the commonest, dating back at least to Karttunen and Peters [1979].) Indeed, it is not completely clear that we need other things to be F for the sentence with even to be felicitously uttered as an assertion: Exercise 7.12.2 Suppose that the least gymnastically able student, Pam, in a gym class, is selected by the teacher to be the only one to try and climb a rope, and she is successful. One of the other students says to a third, “I’m not sure I would have been able to manage that”, and gets the reply, “Don’t be silly! Even Pam climbed it – of course you could have.” Is this a counterexample to the claim that a felicitous “Even a is F ” requires that at least one other individual is F , on the grounds that here Pam is the only one to have climbed the rope (on the occasion in question)? A ‘universal’ variant of the existential condition described as a presupposition under (16)—though not this description of it—has also enjoyed considerable

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currency, as in Bennett [1982], where “Even a is F ” is taken to require that everyone, a included, in a contextually specified class, is F . The original version of this view, in Fauconnier [1975a,b], derives it by thinking of the context as supplying also a scale (such as gymnastic ability, in 7.12.2), whose low point is occupied by a (Pam), and a pragmatically generated implication that anyone higher on the scale is than someone who is F (here slightly varying our example: is able to climb the rope) is in turn F . Thus if the low-point individual is F , everyone ranked on the scale will be, and we get the universal reading in consequence. (This proposal is obviously incorrect, however, as has been pointed out in Francescotti [1995], where various ‘universal quantifier’ accounts of even are criticized – Bennett [1982], Lycan [1991] – see also Lycan [2001], Barker [1992]; we return to the case of Pam climbing the rope in 7.12.3(ii).) What happens if we stick with something like Horn’s (15) and (16), with “asserts” understood to mean that what follows gives the truth-conditions of what precedes, and “presupposes” replaced by “implicates”? (For discussion of these options for only, without specific reference to the only if construction, see Ippolito [2008].) There is still the problem raised before, about the fact that dropping “only” from an only if conditional gives something which seems to stand to the original in a very different semantic relation which results from dropping an “only” from a non-conditional sentence such as (11), to obtain (12). Weatherson [1997] makes the interesting suggestion that the very strong apparent entailment from (11) to (12) is the exception rather than the rule amongst non-conditional sentences with “only”, as we see if we consider a case in which names are replaced by predicates: (17) Only American-trained and qualified pilots are allowed to fly these planes. (18) American-trained and qualified pilots are allowed to fly these planes. (18) is rather hard to assess because of the absence of an explicit quantifier. Obviously a universal reading of (18) does not follow from (17), since it may be clear to all parties that an additional precondition, aside from U.S. training and qualification, is passing a pre-flight drug and alcohol test. But if even an existentially quantified version of (18) – to the effect that some American-trained and qualified pilots (under some conditions) are allowed to fly commercial planes – were false, there would be a question mark as to the point of mentioning such pilots in the first place. This appears to be a straightforward Gricean conversational implicature, explicitly cancellable at will: (19) Only patients who can keep their calorie intake to below 800 a day for a year—and I’m not sure that such patients exist—will be able to benefit from this treatment. Now Weatherson’s suggestion is that in the case of only + name constructions (like (11)) this general conversational implicature has become conventionalized (making it different from other conventional implicatures in respects we shall not go into here, but which make it look more like an entailment than a standard conventional implicature: see Morgan [1978] for a discussion of this general phenomenon). With “only if”, we have the same situation as for the “only” in (17) or (19): a cancellable conversational implicature.

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Digression. There is a similar Gricean argument for a conversational implicature in the reverse direction, from “S1 if S2 ” to “S1 only if S2 ”, which was given a laborious and arguably somewhat confused airing in Geis and Zwicky [1971] encouraging subsequent papers on the would-be topic of ‘invited inferences’ by other linguists unfamiliar with Grice’s work. For early corrective discussions, see Boër and Lycan [1973] and Fillenbaum [1975], [1976]. See further de Cornulier [1983] and Van der Auwera [1997]. End of Digression. Before proceeding further, we should note that (15) and (16) above are really special cases of an idea whose general formulation requires us to remark that “only” and “even” are both what are called focus particles (or ‘focusing adverbs’) with the focus in (15) and (16) being the name a. A general account of such items may be found in various sources listed in our end-of-section notes (which begin at p. 1053), and we propose here simply to reminder the reader of how difference of focus matters to the meanings; the focus is emboldened in each case, and should be read with greater emphasis accordingly: (20a)

Arthur only visits the Ritz with Mary.

(20b)


(20c)


In each case the effect of the focusing could be elaborated: (a) . . . not with anyone else. (b) . . . not any other establishment. (c) . . . he doesn’t stay there with her. If we replace only in (20a–c) with even we get a similar contrast; for instance (21a) and (21b) differ in that the implicature in the first is that Mary is a rather surprising person for Arthur to visit the Ritz with, while in the second, it is that the Ritz is a rather surprising place for Arthur to visit with Mary. (21a)

Arthur even visits the Ritz with Mary.

(21b)

Arthur even visits the Ritz with Mary.

There are many such focus particles in English for which similar distinctions arise (also, at least, especially,. . . ) but our interest here just on even and only. The account for these, updating and generalizing (15) and (16), would be something along the following lines, with the expression Efocus being the focus of the occurrence of “only” (which is here represented initially though that may not be its actual position in the English sentence whose form is given): (22)

Only (—Efocus —) has for its truth-condition: that for nothing E (in some contextually relevant class) apart from E itself, do we have —E — true; implicates: that —E— is true

(23)

Even (—Efocus —) has for its truth-condition: same as —E—; implicates: amongst a range of truths —E —, —E —, etc., the truth of —E— is somewhat surprising, or is in some other way a comparatively extreme case.

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The qualification added hereafter the words “somewhat surprising” is an acknowledgement that it may not be that the scale in the background for uses of even is always one of likelihood. This was urged in Kay [1990], p. 84, though the point is not clear – see Francescotti [1995] – and we ignore the qualification from now on. We shall make five comments on (22) and (23), the first of which is that as formulated the proposals don’t quite makes sense because of a use/mention confusion, which we can illustrate with the case of (22). What do “E” and “E ” stand for here? Linguistic expressions? Presumably, since we obtain —E — from —E— by a substitution of one expression for another. But then what is meant by saying that E is something apart from (roughly speaking, other than) E ? Literally, this would mean simply that they are different linguistic expressions (words or phrases, that is). But that is obviously not right. Suppose Mary is Tim’s sister. Evidently, the truth of (25) does not make (24) false: (24) (25)

Only Mary carried the flowers. Tim’s sister carried the flowers.

Let us, however, not attend to the question of how to correct this aspect of the casual formulation of (22)–(23) – a somewhat delicate matter in view of the fact that the focussed expressions can be syntactically (and semantically) of very diverse categories. Roughly, it is the semantic values of the expressions that need to be what we have described as ‘apart’. Secondly, there is the choice of this apartness formulation in (22) to explain. A first approximation would be to say that the relation between (the semantic values of) E and E should simply be the relation of distinctness, i.e., nonidentity. But this is not quite right, since, using an example given in Weatherson [1997] to make this point, we want to allow it to be true that only the lounge suite has been moved, even though it is also true for something not identical with the lounge suite, namely the sofa, that it has been moved. (Here E is “the lounge suite” and E is “the sofa”.) We don’t hold the moving of sofa against only the lounge suite’s having been moved because though different, the sofa is not something apart from—mereologically disjoint from—the lounge suite. (Weatherson actually has a different way of putting this point, in terms of a standing principle that the domain of quantification is required to be taken as a partition, so that any distinct objects quantified over are disjoint. A somewhat different proposal appeared in Kratzer [1989], p. 608, whose informal exposition uses the notion of something’s not being a thing apart from something else, and whose formal explication is in terms of a certain relation of lumping, which we do not go into here.) Thirdly, (22) and (23) simplify in supposing that the focussed elements are always single (albeit possibly composite) linguistic expressions. This need not be so. Imagine a television documentary about the success rate of ‘blind date’ introductions, which follows fate of three couples who thus introduced over the course of a year. Albert and Alice seem especially well matched, Brian and Bronwyn somewhat less so, and Casper and Cassandra an unlikely pair indeed. Mid-documentary, one might hear: (26) Casper and Cassandra no longer see each other, Bronwyn is on the verge of breaking up with Brian, and even Albert has started to argue with Alice.

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The third conjunct has the pair Albert, Alice as the focus of even, rather than any single expression, and the contrast is with the other pairs. Fourthly, notice that because we have put under the ‘implicature’ part of (23), that there should be a range of truths (—E —) etc., amongst which the truth of —E— is relative surprising, this formulation goes against the spirit of Exercise 7.12.2, in the case of “Even a is F ” (which unambiguously focusses “a”) requiring that at least one other way of filling the blank in “Even ___ is F ” should be less surprising and also true. Remarks 7.12.3(i) It hardly seems appropriate that some alternative filling of is F ” should be obtained by another expression the blank in “Even with the same denotation as the “a” replaced, even if it might be less surprising that the resulting sentence is true. (This observation is of a piece with our first comment, above, on (22) and (23), about the casual use/mention confusion involved in their formulation.) (ii ) What about the case of Pam climbing the rope, from 7.12.2? One response: the student who said “Don’t be silly! Even Pam climbed it – of course you could have”, in fact misspoke, meaning to have said “Don’t be silly! Even Pam was able to climb it – of course you could have.” Fifthly and finally, there is a case for introducing a scalar element into (22) to play a role in the truth condition, somewhat like that played by the surprisingness scale in the felicity condition (i.e., the characterization of the implicature) in (23). Someone uttering (24) for may have in mind a contrast with the case in which someone else as well as Mary carried the flowers, or they may have in mind a contrast with the case in which someone else instead of Mary carried the flowers. If the latter, then there is a tacit scale in the background, and a scale on which Mary occupies a lowly position. Compare (27) and (28): (27) Did the President attend the funeral? — No, only the Vice-President. (28) Did the President and Vice-President attend the funeral? — No, only the Vice President. (Equally good : No, only the President.) Similarly with (20c) above, where “visits” is the focus of only: to interpret the sentence, we have to come up with a scale of relations that Arthur (along with Mary) can bear to the Ritz, in which visiting ranks low, or at least lower than some contextually salient alternative relation. Whether we should say that only is ambiguous – as with the ‘exceptive’ vs. ‘limiting’ distinction of Taglicht [1984], pp. 86–93 – or we can subsume both usages under a simple unifying account, we should expect some such distinction to surface when the compound only if is present, and indeed it does. Let us turn, then, to the upshot of proposals such as (22)/(23) for even and only for even if and only if. As mentioned in the notes (on “but”) at the end of §5.1 (p. 674), conditionals introduced by even if have traditionally been known as concessive conditionals – by analogy with the (also traditionally thus described) concessive clauses (introduced by though, even though, and although) – but this phrase makes all too much of a concession for the likes of some, who would deny that there is anything conditional about the even if construction. (See Fraser [1969], or Mackie [1973], p. 72.) The idea is that the consequent is asserted outright – so there is nothing conditional going on – when such a construction is used (assertively),

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or at least that “Even if S1 , S2 ” has S2 as a logical consequence (for any sentences S1 , S2 ). (This point is neutral as between subjunctive and indicative constructions.) A certain type of counterexample (mentioned but dismissed as elliptical for a longer doubly conditional construction in Pollock [1976], taken up again in Hazen and Slote [1979]) goes as follows: (29) If you drink this bottle of peroxide, you’ll be ill. In fact, even if you drink half the bottle, you’ll be ill. Clearly, one could come out with (29) and not regard oneself as committed, in view of the second sentence, to “You’ll be ill”. The obvious reply, made in not quite this terminology in Bennett [1982], to the use of examples like (29) against the claim that consequents are entailed by even if conditionals, is to clarify the claim using the notion of focus (which Bennett confusingly calls ‘scope’). The claim was intended to cover the case in which the focus was the whole of the if clause, whereas in the second sentence of (29), the focussed expression is “half the bottle” (or perhaps just “half”). On the other hand, proposal (23) gives something of an explanation of the speaker’s commitment to the truth of the consequent when the focus of even is the whole of the antecedent: for what is the range of truths differing from the “Even if S1 , S2 ” under consideration in respect of the focussed “ if S1 ” (or, though this seems slightly less satisfactory, the focussed S1 ). The main alternative that comes to mind would be one in which “if S1 ” is replaced by “if not-S1 ”, and if that is less surprisingly implicated as true, the speaker is committed implicitly to (30), and explicitly to (31), from which S2 follows as longs as the connectives exhibited are governed by the familiar classical principles (for → and ¬, and for ∨ too if this is brought in to appeal to the Law of Excluded Middle with disjuncts S1 and not-S1 ): (30) If not-S1 , S2 . (31) If S1 , S2 . On this line, although someone asserting “Even if S1 , S2 ” is committed to propositions from which the truth of S2 follows, we don’t quite have a vindication that S2 is entailed by that concessive conditional, since the reasoning went in part through an implicature which was not an entailment. (Note also that we rely on the aspect of (23) to which our fourth comment on (22)–(23) above was addressed.) The most important point to note is that a corresponding line of reasoning to the conclusion that S1 is implicated by an assertion of “Even if S 1 , S2 ” is available only on the hypothesis that the focus of even is if S1 , and so would not be available in the case of (29) under the interpretation in which it is intended. We turn to a parallel observation in the case of “only if”, in which apparent counterexamples to a plausible thesis suggest that the thesis be maintained in a form which pays attention to questions of focus. The plausible thesis—defended for example in McCawley [1974], and in a qualified form in McCawley [1993]—is that (32) and (33) will be true under the same conditions: (32) S2 only if S 1 (33) If not-S 1 , then not-S 2 . There are various difficulties – comprehensively reviewed in von Fintel [1997] – in the way of obtaining such an equivalence from an account along the lines

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of (22), and thereby displaying only if constructions (assuming we want this equivalence) as amenable to a compositional semantic account. We mention only one: if, as in the above discussion of even if, we can subsume the salient alternatives to if S 1 under the umbrella of if not-S 1 , we will be in a position to equate (32) with (34)

Not(if not-S 1 then S 2 )

and we are left with the problem that the initial negation is not in the position – attached to S2 – that we need it to be in to have (33). While the implication from (34) to (33) requires only a principle (for the if concerned) called the Conditional Law of Excluded Middle (see 7.17), the converse implication requires the much more controversial ‘connexivist’ principle sometimes (touched on in 7.19 below) called Boethius’s Law. Alternatively, we might try a mixed semantic/pragmatic account of conditionals which directly places the negation where we want it because of some special aspect of conditionality (such as the conditional assertion account employed to this effect in Barker [1993], [1994]). Any such manoeuvering is too specific to the case of if to be appropriate, since we want the same equivalence as is schematically indicated with (32)/(33) in other cases too, such as with where in (35)

It snows only where it hails.

(36)

Where it doesn’t hail, it doesn’t snow.

Thus probably the best response is one which places negations signs more carefully than (22) does, attending fairly carefully to what might otherwise seem to be relatively superficial syntactic matters, such as the notion main verb of the main (as opposed to subordinate) clause: such an account may be found in Appiah [1993]; we do not go into further detail here. (It should be noted that (22), as formulated, does not make any play with negation. But instead of speaking of no —E — being true (E replacing E), we could have put this in terms of the negations of the various —E — being true.) Let us repeat (32) and (33) alongside another contender with far less claim to equivalence with them than they have of equivalence to each other, here given as (37): (32)

S2 only if S 1 .

(33)

If not-S 1 , then not-S 2 .

(37)

If S 2 , then S 1 .

The considerable difference in meaning between (32) and (37) has often been observed (McCawley [1974], [1993], Sharvy [1979]) and is generally linked to the intuitive distinction between something’s being a necessary condition for something else ((32), (33)) and the latter’s being a sufficient condition for the former ((37): see Wertheimer [1968], Sanford [1976], Chapter 11 of Sanford [1989], and – for an early recognition of the point – p. 200 of Roëlofs [1930]). Quite what account to give of this matter of ‘direction of conditionality’ remains unclear, as is the question of whether the notion of sufficiency can plausibly be applied in the case of concessive or other ‘non-hindering’ (to use terminology from 7.11) conditionals, or applied in spite of the unacceptability for the conditionals in question of the principle (discussed below, after 7.15.6 on p. 990) of Strengthening the Antecedent. For a slightly different angle on the issue of directionality,

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we note that the discussion of the clearly non-synonymous pair (38)/(39) elicits talk of priority from McCawley [1993], p.537; the (33)–style counterpart of (39), namely (40), is included for good measure: (38)

If you touch me, I’ll scream.

(39)

You’ll touch me only if I scream.

(40)

If I don’t scream, you won’t touch me.

As McCawley comments: The if -clause must be temporally/causally/epistemologically prior even if it is modified by only.

Thus in (38) represents the envisaged touching as prompting the screaming, while (39) and (40) represent the touching as prompted by the screaming – presumably a matter of causal (and temporal) priority in this case. These last examples are, in Dudman’s terminology (reviewed in 7.11) projective examples—and the same point could equally have been made about subjunctive members of that category (though Dudman would not put it like that). Perhaps the best place to look for McCawley’s ‘epistemological priority’ would be amongst the Dudman hypotheticals, if we were right in 7.11 to think of these as amounting to conditionals with (in Sweetser [1990]’s terms) an epistemic level interpretation. So let us consider a trio corresponding to (38)–(40) from that arena: (41) If the structure we’re looking at is the Eiffel Tower, then we’re in Paris. (42) The structure we’re looking at is the Eiffel Tower only if we’re in Paris. (43) If we’re not in Paris, then the structure we’re looking at is not the Eiffel Tower. (41) would appear to be the most natural thing to come out with for a busy tourist whose companion has just asked what city they are in—or more specifically whether they are in Paris – whereas (42) and (43) are more natural if the question has been about what structure they are looking at – or more specifically whether the structure before them is the Eiffel Tower. And this difference seems to align itself with an epistemic difference: information about what’s in the if -clause is presented for its potential role as evidence on the topic of the then-clause. As McCawley [1993], p.83f., notes in criticism of Braine [1978], this may explain differences in subjects’ abilities to draw conclusions correctly from a conditional and the negation of its antecedent or consequent, and it also calls for some care in deciding what to count as Modus Tollens when the conditional is an only if conditional. (Modus Tollens was introduced for → in 1.23.6(i); here we intend the corresponding rule for the English connectives concerned. Some references to discussions of this rule have been collected in the end-of-section notes, under ‘Modus Tollens’, p. 1055.) See also Castañeda [1975], p. 67, for sentiments somewhat similar to those expressed here about (42)/(43).) Admittedly, this seems considerably less like a difference in truth-conditions than does the distinction between (38) on the one hand and (39), (40), on the other, making it a somewhat delicate matter whether or not to speak of contraposition – the passage from If S1 then S2 to If not-S2 then not-S1 – as a fallacious inference pattern. More explicitly, since we have already spoken, e.g. in 2.11.9 (p. 207), of

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(various forms of) contraposition and had in mind various principles not involving a conditional connective (of all of which the principle that A B implies ¬B ¬A is a special case), we should say ‘contraposition for conditionals’ here. An interesting illustration of these inferential considerations comes from a sign on the back of many trucks for motorists behind them to read: If you can’t see my mirrors, I can’t see you. We need not decide whether this has the same truth-conditions as the (“decontraposed”) version with unnegated antecedent and consequent exchanging positions. That version begins “If I can see you,. . . ” and presents the following motorist with a minor premiss for a Modus Ponens, but a premiss the motorist is not in an evidential position to assent to: how does the motorist know whether the truck driver can see the motorist’s vehicle? But in the actually encountered form, as inset above, the truth of the antecedent is readily accessible and the message succeeds where the de-contraposed version would not. Remarks 7.12.4(i) As well as the causal, temporal and epistemic priority alluded to by McCawley, there seems to be a matter of purely logical priority which influences in a similar way the relative naturalness of S1 only if S2 , on the one hand, and If S1 then S2 on the other. It seems more natural to say that a conjunction A ∧ B can be true only if its first conjunct, A, is true, than to say that a disjunct A can be true only if the disjunction A ∨ B is true. We seem to think of the truth of the conjuncts of a conjunction as a pre-requisite for its truth – A’s truth and B’s truth each being a necessary condition, and their joint truth being a sufficient condition, for the truth of A ∧ B. But we do not think of the truth of A ∨ B as a necessary condition or ‘prerequisite’ for the truth of A – even though from the information that A ∨ B was not true, we could deduce that A was not – preferring instead to regard A’s truth as a sufficient condition of the truth of A ∨ B. It is though in each case, the compound is inferentially ‘downstream’ from the components, regardless of which direction the entailment relation (here CL for definiteness) runs in. The same phenomenon arises with quantifiers: we say that everyone in the boat died only if the captain died, more readily than we say that the captain died only if someone in the boat died. It seems unnatural to say, with Nerlich [1971], p. 161, in illustration of logical sufficiency, that bachelorhood is a sufficient condition for being unmarried – even though the information that someone is a bachelor is sufficient for the conclusion that he is unmarried. What we would normally say, rather, would be that being unmarried is a necessary condition for being a bachelor. (To be fair, Nerlich is not actually considering the relationship between necessary and sufficient conditionality; for this, see Wertheimer [1968], and the other references given after (37) above.) (ii ) As McCawley noted, if the compound connective “if and only if” is treated in accordance with its internal structure, we get results diverging from the usual treatment of S1 if and only if S 2 as the conjunction of (a) If S 1 then S 2 and (b) If S 2 then S 1 , since (a) should be replaced by “S1 only if S 2 ”, or – for simplicity continuing to assume that this amounts to much the same – by:

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If not-S 2 then not–S1 . If the conditional or implicational connective involved does not form conditionals which are equivalent to their contraposed versions, however, the question as to what is more conveniently read as if and only if (or iff ) may sensibly be resolved in a way that does not respect this compositional consideration. For example, in intuitionistic logic, A ↔ B is usefully read as “A if and only if B”, even though by the letter of compositional semantics, the latter should be reserved as a reading of, say, A ≡ B, where we temporarily introduce ≡ in accordance with the definition of A ≡ B as (A → B) ∧ (¬A → ¬B). (Alternatively we could take ≡ as primitive and provide rules securing the equivalence with that definiens; note that a different usage of “≡” as symbolizing a binary connective will be made in 7.32.) Although → satisfies contraposition according to IL , the two are not equivalent because the converse of contraposition – “Decontraposition” as it was called in 1.29.22 – is not satisfied. What we generally want from an equivalential connective are properties analogous to those of an equivalence relation (such as commutativity, by analogy with symmetry – see 3.34: p. 497 onward) together with various replacement properties (cf. §§3.2–3), and these are not forthcoming with ≡ in IL, as 7.12.5(ii) will indicate, whereas they are with ↔, which therefore deserves the handier verbal rendering (as “if and only if”). Put in the terminology introduced after 1.19.9 above, {p ≡ q} does not constitute a set of equivalence formulas for IL . (iii) Lycan [1984], p. 446, says that an “analysis of subjunctive conditionals such as Lewis’s offers no way of distinguishing these”, referring to the pair (a) and (b)—and for more detail on the account of Lewis [1973], already summarised in 7.11, see 7.17: (a) If Joe were to leave, then he would have a headache. (b) Joe would leave only if he were to have a headache. However, on Lewis’s account, a subjunctive conditional (represented A → B) is not equivalent to its contraposed form—cf. 7.18—so it is not clear why Lycan thinks this account cannot distinguish (a) and (b) thus: for (a): for (b):

“Joe leaves → Joe has a headache” “¬(Joe has a headache) → ¬(Joe leaves)”.

Exercise 7.12.5 (i) As was mentioned under 7.12.4(ii), we could take the connective ≡ there described as a new primitive. Show how to extend the proof system INat of 2.32 to accommodate such an additional primitive, keeping the rules as close as possible to the ‘pure and simple’ paradigm. (ii ) Find formulas A and B for which A ≡ B IL B ≡ A. (Hint: use 2.32.1, from p. 304.) (iii) Do there exist formulas A and B, and a context C(·), for which we have: A ≡ B, C(A) IL C(B)? Digression. Although we have more or less assumed that (32) and (33) above are equivalent, this is questioned in Weatherson [1997], with the aid of an example in which there are two urns, each containing 1000 balls. One is a red urn,

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999 of the balls in which are red, the remaining ball being black; the other is a black urn, 999 of the balls in which are black, the remaining ball being red. The speaker for the following examples is imagined to be about to draw a marble at random from one or other urn. According to Weatherson, we think that (a) is probably true, while (b) is certainly false: (a) (b)

If I choose from the black urn, I will choose a black marble. I’ll only choose a red marble if I choose the red urn.

For this to bear on the equivalence of (33) with (32), we have to take it that under the hypotheses of the example, not choosing from the black urn amounts to choosing from the red urn, and not choosing a black marble amounts to choosing a red marble. Weatherson’s idea is evidently that the only in (b) amounts to denying the existence of a possible outcome which has non-zero probability, whereas in the case of (a) we simply overlook a small possibility of error; thus only if conditionals have a modal force—or perhaps a stronger modal force—than the simple if conditionals. In effect we read (b) as something along the lines of “I can only choose a red marble if. . . ”. Certainly the non-equivalence is clear with the latter formulation, but as to whether this is what is literally meant by (b) is less clear. We shall at any rate continue to ignore this worry about the equivalence of (32) with (33). End of Digression. Before passing to a few closing words on the subject of unless, we redeem our promise from the start of this discussion of only if conditionals to illustrate the need for attention to the focus of only in such constructions. Suppose that children learn the rudiments of cooking in classes stage by stage. To pass the Grade I test, they have to demonstrate their ability on either of two tests: they have to fry an egg or else boil a potato (subject to producing an end-product within some quality range we need not go into). Slightly harder, because of the timing involved, is the task of boiling an egg, and doing so successfully is one way of passing Grade II; the other option for a Grade II pass is to (dice and) fry a potato. One student attempts the Grade II test uncertain of what is expected and proceeds to fry an egg (having, we may suppose, passed Grade I by boiling a potato), only to be told (44c), which we list along with parallel formulations recapitulating the above description: (44a) (44b) (44c) (44d)

You pass Grade I only if you fry an egg. You pass Grade I only if you boil a potato. You pass Grade II only if you boil an egg. You pass Grade II only if you fry a potato.

Here the emphasis (in bold) gives the focus of “only”. Disregarding it, our student might reply, “No, it can’t be right that you pass only if you boil an egg, since Margaret passed and she fried a potato”, and there might seem to be a counterexample to the claim that “S1 only if S 2 ” implies “If not-S 2 then not–S1 ”. (More accurately, we are working within the scope of some quantifiers, and dealing with “Any student passes only if that student boils an egg” on the one hand, and “Any student who does not boil an egg does not pass”.) Such an apparent counterexample would arise because the case of Margaret falsifies the “If not-S 2 then not–S1 ”, since she did not boil an egg and yet she did pass. These anomalies are mere appearances, however, since the focus of only is not the if -clause, as in our earlier discussion of even. (44c) with its subclausal focus

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says not that the only condition under which one passes Grade II is that one boils an egg, but that the only way of passing Grade II by cooking an egg is by boiling it. It should be added that the “If not-S 2 then not–S1 ” format does indeed afford a paraphrase of (44c) as long as the emphasis is again on “boil”: if you don’t boil an egg, you don’t pass Grade II. The notes to this section provide references to published discussions of the far-reaching aspects of focus and contrast (p. 1055, under the heading ‘Focus Constructions’), and very little more will be said on the subject here. In fact the only further reference to focus we need to make is to observe that (44c) seems exactly paraphrased by a sentence with unless, thereby providing us with an opportunity to introduce the last of the conditional constructions with which we are concerned in this subsection: (45)

You don’t pass Grade II unless you boil an egg.

Under the circumstances envisaged, while (45) as it stands would be true, the result of taking the whole of “you boil an egg” as the focus would not be, since, as before, you can pass by frying a potato. We call the equivalence between (44c) and (45), with or without the internal focus, an instance of the following equivalence principle. Geis–Fillenbaum Equivalence. The following pair are paraphrases: S1 only if S 2

and

not-S 1 unless S 2

not-S1 only if S 2

and

not-S 1 unless S 2 .

as are:

We name this equivalence-claim after Geis [1973], since it appears to be a consequence of the proposals—summarised at (49) below—here made for the treatment of if, only if, and unless, and Fillenbaum [1976], in which it is explicitly defended. (See also Fillenbaum [1986], and 7.12.7 below, where we shall also note an anticipation of this equivalence-claim, on the part of S. F. Whitaker. This proposal also appears at p. 206 of Sanford [1989].) The second equivalence is a special case of the first, putting not-S 1 for S1 , assuming that the two occurrences of not cancel. For the case in which the only if clause comes first, we can think of the equivalence as one between only if and not unless. (“Only if the Queen is in residence does the flag fly outside” means “Not unless the Queen is in residence does the flag fly outside”.) As was remarked at the start of this subsection, Geis is amongst those for whom unless cannot occur in a subjunctive construction, so he should only be regarded as committed to a version of the equivalence as restricted to the indicative in force. To introduce the subject of unless in a more historical way, however, we need to back up a little. The question of how best to describe the logical force of unless was raised in Ladd-Franklin [1928], a letter to the editors of a philosophical journal apparently written more to amuse the reader than to stimulate investigation – though Roëlofs [1930], of which more in a moment, did indeed take up the issue. More widespread interest was sparked only considerably later by some remarks from Quine [1950], who raised the question of why unless presents the appearance of being non-commutative, (46) and (47) appearing to make very different claims: (46)

Smith will not sell unless he hears from you.

(47)

Smith will not hear from you unless he sells.

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Quine’s explanation is that we take the unless-clause as pertaining to the earlier of the two times of the envisaged events (his selling or otherwise, and his hearing from you or otherwise), and accordingly we fill out the “hears from you” differently in the two cases, as meaning, for (46) that he hears from you to the effect that he should not sell, and, for (47), that he hears from you to the effect that he should not have sold. Chandler [1982] goes on to investigate the possibility of inserting tense operators into the representations of (46) and (47) to make explicit the temporal relationships involved, so that the resulting representations do not even seem to illustrate a failure of commutativity. (As Chandler concedes, a full account of unless will need to take into account more than such matters of temporal priority. See also the discussion of commutativity for unless in Castañeda [1975], p. 160.) Now why should we even be asking about commutativity here? The expectation that unless is commutative—and notice that this is a perfectly legitimate way of putting matters even if we agree that we are dealing with a subordinating rather than a coordinating connective, from the point of view of natural language syntax—is the product of two distinguishable assumptions made by Quine. The first is that S1 unless S 2 can be treated as equivalent to S1 if notS 2 , or alternatively, in the pre-posed form always available for subordinating connectives, that unless S 2 , S1 can be treated as equivalent to if not-S 2 , S1 . The second assumption is that the contraposing behaviour of the if involved here is such as to render if not-S 2 , S1 equivalent to if not-S 1 , S2 . In fact, in for Quine, the assumption is that the if is the material conditional, so that the reasoning justifying the expectation of commutativity can be summarized thus: (48)

(a) Unless S 2 , S1 : (b) Unless S 1 , S2 : (c)

to be represented as: ¬S1 → S2 to be represented as: ¬S2 → S1

¬S1 → S2 CL ¬S2 → S1

In Quine’s discussion, the separate assumptions are rendered slightly less visible by the observation that the left-hand representations of (48a) and (48b) are CLequivalent to S1 ∨ S2 and S2 ∨ S1 , in which notation (48c) becomes simply the commutativity of disjunction (according to CL ). (One sometimes hears the suggestion that the relevant boolean connective in this context is not inclusive disjunction, ∨, but exclusive disjunction . The suggestion is analogous, at the level of truth-functions, to the mistake diagnosed at 7.12.9 below.) It is worth separating out the two distinguishable assumptions as we have done here, however, because we can also distinguish the objections that might be made to these assumptions. The second matter – contraposition – is something we have already had occasion to touch on. Many have thought it more plausible for indicative than for subjunctive conditionals, and we shall return to it in 7.18. Our current concern will be with the first assumption – to the effect that unless means if not, as we may put it for brevity. In particular, Geis [1973] objects explicitly to this assumption that, offering the following paraphrases of several conditional constructions with the aid of the locution “the event that”: (49) (a) (b) (c)

S1 if S 2 S1 unless S 2 S1 only if S 2

means

S1 in the event that S 2

means means

S1 in any event other than that S 2 S1 in no event other than that S 2

We obtain what was called the Geis–Fillenbaum Equivalence above by taking

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the further step of identifying “S1 in no event other than that S2 ” and “notS1 in any event other than that S2 ”. Although Geis does not explicitly take this step, we attribute it the equivalence to him on the basis of its appearing uncontroversially to follow from his proposed “in the event that” paraphrases. In any case, the equivalence can be found vigorously defended by McCawley [1993], p. 553. (Geis’s own preferred formulation is given at 7.12.7 below.) In the setting of the present discussion, however, it is not so much the Geis– Fillenbaum equivalence that we need to attend to as what we might call Geis’s non-equivalence, namely, the denial – the main point of Geis [1973], though, again, also illustrated in Fillenbaum [1976] – that unless means if not. (Recall that it was the assumption that these two did mean the same that underlay steps (a) and (b) of our reconstruction (48) of the reasoning leading to an expectation of commutativity on the part of unless.) Geis took it to be fairly clear that if his account (49) was correct, then the claim that unless means if not could not be, since when particular sentences are plugged in for the variables S1 , S2 , the resulting “S1 in the event that not-S2 ” and “S1 in any event other than the event that S2 ” have rather different sounding results. A problem with this is that it is not clear what the “event” talk itself means—obviously we are not talking about events in the sense of “things that happen” (since we can say, “In the event that S” for any declarative sentence S, whether or not S is a sentence purporting to report the occurrence of an event in the more familiar sense)—and on any clarification there will be the further question as to what could be meant by talk of “the event that S2 ”, required by (49a). This seems to require that there is only one event which counts as the event that S2 , while (49b), for instance, seems to allow that we can have the sentence S1 true under several distinct conditions, the event that S3 , say, the event that S4 , and so on. How are these latter related to what Geis would call “the event that S1 ”? These questions may not be unanswerable—Lycan [1984], [2001], has some suggestions—but they do suggest that reflection on Geis’s paraphrases alone is unlikely to give much guidance as to whether unless means if not. Fortunately, Geis [1973] presents a welter of more helpful considerations, two of which we mention here, concerning co-occurrence restrictions with polarity items and co-occurrence with modifiers of if. We begin with a reminder that “much” in contemporary English, when serving as a stand-alone adverb, is amongst the so-called negative polarity items, as is illustrated by the ungrammaticality of (50a), by comparison with (50c). (We say “stand-alone adverb” to separate this from ‘ad-adjectival’ uses, as in “It’s much too hot”, “She’s much better now”, etc., and pronominal uses, as in “She has much to commend her”.) (50a) (50b) (50c) (50d)

*Her husband talks much. Her husband talks a lot. Her husband doesn’t talk much Her husband doesn’t talk a lot.

We encountered negative polarity items in the second Digression in 6.14. Now consider: (51a) (51b)

If her husband doesn’t talk much, she will need the company of a female friend. *Unless her husband talks much, she will need the company of a female friend.

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Geis also shows that the mirror image category of positive polarity items can occur in the unless construction but not in the if-not construction. We omit the examples here. Since these polarity phenomena are somewhat complex, there being several syntactically different types of item classed under the overall heading of “negative polarity item”, for instance, and they might in any case not be thought capable of bearing for the present discussion the weight they have variously been taken to have in the multifarious versions of grammatical theory that have held sway since the publication of Geis [1973], we include a more straightforward co-occurrence argument from that paper against the “unless means if not” thesis: (52a)

Even if her husband doesn’t respect her, she will remain faithful.

(52b)

*Even unless her husband doesn’t respect her, she will remain faithful.

By a variation on this last pair, we can find pairs differing in that one has if + not where the other has unless, both are grammatically acceptable, but clearly non-synonymous. Here (53) with continuation (a) is example (47) from 7.11: (53)

If you visit the shrine your child will need medical attention for at least a further week, and . . .

. . . (a) if you don’t visit the shrine your child will need medical attention for at least a further week. versus . . . (b) unless you visit the shrine your child will need medical attention for at least a further week. Continuing in this ‘quasi-concessive’ vein, we have an equally clear contrast between these two: (54a)

If you didn’t enjoy the meal, you certainly didn’t complain at the time.

(54b)

Unless you enjoyed the meal, you certainly didn’t complain at the time.

And again with this pair: (55a)

If Mary doesn’t want to come with us, we’ll ask someone else.

(55b)

Unless Mary wants to come with us, we’ll ask someone else.

the distinction between which can be helped along with a re-phrasing: (56a) (56b)

If Mary doesn’t want to come with us, we can always ask someone else. Unless Mary wants to come with us, we can always ask someone else.

?

Another pair, with a rather different flavour; although (57b) is grammatical, it would hardly be a way of re-expressing (57a): (57a)

If you don’t know the way, you can ask a policeman.

(57b)

Unless you know the way, you can ask a policeman.

We can also consider pairs of pairs. The (a) and (b) of (58) seem closer to being interchangeable—in the context of a teacher addressing a class—than do the (a) and (b) of (59): (58a)

If you don’t know the answer, you shouldn’t put up your hand.

(58b)

Unless you know the answer, you shouldn’t put up your hand.

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If you don’t know the answer, you didn’t read Chapter 7. Unless you know the answer, you didn’t read Chapter 7.

Likewise in the case of (60a, b) and (61a, b): (60a) (60b)

If he’s not promoted, he’ll resign in protest. Unless he’s promoted, he’ll resign in protest.

(61a) (61b)

If he’s not promoted, he’ll decide he’s happy in his present position. Unless he’s promoted, he’ll decide he’s happy in his present position.

Although we are supposed to be concentrating on declaratives, it is hard to resist citing the following imperatival pair from Dancygier [1985]: (62a) (62b)

Consult the dictionary if you can’t do without it. Consult the dictionary unless you can do without it.

The (b) example seems much more of an encouragement to consult the dictionary than the corresponding (a) case. Returning to declaratives, we have the following striking contrast (from Fillenbaum [1976], [1986]); the imagined speaker is a motorist trying to bribe a parking inspector: (63a) If you don’t give me a ticket, I’ll give you $20. (63b) Unless you give me a ticket, I’ll give you $20. Further examples of the non-equivalence of unless with if not may be found in Whitaker [1970], in Brée [1985], and at p. 756 of Huddleston and Pullum [2002]. The second of these sources also mentions (p. 316) that there is an implicature – described as ‘conversational’ implicature but presumably we should interpret this as ‘conventional’, since it is occasioned by the specific presence of the word unless – that “the protasis is an exceptional circumstance”. This interesting aspect of the meaning of unless is occasionally explicitly noted, and is implicit in various discussions in the artificial intelligence literature. (See Sandewall [1972] for an easy example in which something called “Unless” is pressed unto service as, surprisingly enough, a one-place connective for tagging exceptional cases; also in roughly this genre: Michalski and Winston [1986].) Fillenbaum [1976] remarks, of his contrast (63), that unlike (63a), (63b) “seems to suggest that in the ordinary run of things the speaker would give the addressee the $20 and only a ticket could prevent him from doing so” (p. 249), picking up the ‘exceptionality’ theme—and notes also the relative unnaturalness of formulating (conditional) offers and promises with unless, as against warnings and threats. (Cf. also the discussion in Noordman [1985].) Contrasting unless with if – with or without a following not – Whitaker [1970], p. 160, says: “Potentially threatening, gloomy, pessimistic, its implications are quite different”. The easiest way for the reader to detect the evaluative or affective element which seems to be present—and no explanation of this appearance will be attempted here—is to imagine that a non-fiction book has been found with the title “UNLESS. . . ”: on the basis of just that information, consider whether the book is more likely to be about the good things that will happen unless something unspecified is done, or about the bad things that will happen in that eventuality. The curious reader can even locate such a book – and confirm any hunches – by tracking down Jeffrey [1975]. Shields [2002] is a novel with the same title, containing a chapter of which ‘Unless’ is again the title, in which one character muses thus:

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Unless is the worry word of the English language. It flies like a moth around the ear, you hardly hear it, and yet everything depends on its breathy presence. Unless—that’s the little subjunctive material you carry along in your pocket crease. (. . . ) Unless you’re lucky, unless you’re healthy, fertile, unless you’re loved and fed, unless you’re clear about your sexual direction, unless you’re offered what others are offered, you go down in the darkness, down to despair. Unless provides you with a trapdoor, a tunnel into the light, the reverse side of not enough. Unless keeps you from drowning in the presiding arrangements. Ironically, unless, the lever that finally shifts reality into a new perspective, cannot be expressed in French, À moins que doesn’t have quite the heft; sauf is crude. Shields [2002], p. 149.

It would be interesting to know whether other Anglo-French bilinguals would agree with the last sentiment expressed here. Certainly, conditional constructions in some other languages other than English are more sensitive to matters of evaluation and affectivity: see Akatsuka [1997a] for some striking data in this connection from Japanese and Korean. (Akatsuka [1997b] is less convincing about evidence for a similar sensitivity in the case of subjunctive conditionals in English itself. Exercise 7.12.6 Explore the prospects for the Geis–Fillenbaum equivalence (p. 964) as an improvement on the “unless means if not” hypothesis, with reference to the (a)/(b) pairs from (53) to (63). (Recall that the equivalence treats S1 unless S 2 as meaning not-S1 only if S 2 ; this makes for uncomfortably many negations which may have to be simplified out of the way before the results of making the replacement intelligible. Also, the imperatival examples (62) may be expected to cause trouble, and care will be needed over the should in (58).) Is the element of ‘exceptionality’ explicable by some aspect of the meaning of the word only in these replacements? Remark 7.12.7 While the subject of exceptionality is in the air, we should mention that the contribution of Geis [1973] to the Geis–Fillenbaum equivalence is in fact somewhat different from the formulation we have given in terms of only if ; Geis’s version simply replaces unless by except in the event that, or—more colloquially (and as Geis notes, unacceptably for some)—simply by except if. We have not wanted to conduct the discussion in these terms because that would oblige us to enter into the not entirely straightforward question of what except means. (Does “Everyone except John voted” entail “John did not vote”, for example? Certainly, with “Everyone other than John voted” we have at best an implicature here. For discussion and references, see von Fintel [1993], Hoeksema [1996].) Quirk and Greenbaum [1973], p. 324, also offer a gloss of ‘except on condition that’ for unless; indeed they also give the only if. . . not paraphrase we have been associating with Geis and Fillenbaum. The first published airing of these points known to the author is provided by Whitaker [1970], where (i) the failure of the unless = if not equivalence is argued with examples, (ii ) the Geis–Fillenbaum equivalence, as we have been calling it, is suggested (p.159), and (iii) the more explicitly Geisian paraphrase of unless as except if is also proposed (p. 160). The syntactic arguments of Geis [1973] do not appear; instead,

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Whitaker gives a subtle semantic discussion attending to the details of many examples. Perhaps through the indirect influence of Whitaker [1970], it is in many circles simply asserted without any fanfare – or any reference to Geis – that unless means except if. This is done in a pedagogical setting in Unit 114, on unless, as long as, and provided/providing of Murphy [1994], for example. Or again, Adams and Perrin [1995] mention in passing that unless means except in the case that. (They say this for “contemporary American English”, and are puzzled by a particular Pennsylvanian dialect in which unless means in case, with speakers saying things like “Shall I leave your fork unless you want it later?” The authors remark that “(e)ven native English speakers found it difficult to use unless precisely; in the seventeenth century, they frequently confused it with lest (. . . ), a word with similar meanings and similarly formed”. They air various hypotheses, mostly connected with the German linguistic ancestry of the population involved, though they do not seem inclined to suggest that it is the same unless/lest confusion that is involved – and surely the main similarity to blame such a confusion on would be the straightforward phonological similarity. For more on in case, see the Digression following (31) above.) Exercise 7.12.8 (i) With respect to the distinction, touched on in 7.11, of Sweetser [1990], between content domain, epistemic, and speech-act interpretations of connectives, how should the use of unless in the following example be categorized? “I saw Tom going into the Town Hall today – unless it was his twin brother Sam.” (ii ) Berry [1994] considers the following pair of examples with regard to the (as we have seen, far from plausible) hypothesis that unless means if not: (a) (b)

I couldn’t have made it on time unless I’d had an executive jet. I couldn’t have made it on time if I hadn’t had an executive jet.

Berry says (p. 104) that in (a) ‘I’ was successful because of the jet, while in (b) ‘I’ didn’t make it. Are these reactions to the examples correct, and if what explains the difference? (Note: although we cite Berry here, these examples go back, via Celce-Murcia and Larsen-Freeman [1983], p. 351, to Whitaker [1970], p. 155.) Digression. On the same page of Berry [1994] as was just cited, the following pair is presented to illustrate the difference “not just in emphasis” between unless and if not: (a) (b)

We’ll take the car if it rains. We won’t take the car unless it rains.

The difference in more than mere “emphasis” turns out to be a difference in truth-conditions; as Berry says (labelling of examples changed to the above) “In (b) rain is the only condition which could lead to the taking of the car. In (a) other causes are not excluded.” This is a satisfactory contrast between the cases, but closer inspection reveals that the difference does not after all bear on the hypothesis that unless means if not. Replacing the former by the latter in (b), and making the concomitant syntactic adjustments, we get (c):

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971

We won’t take the car if it doesn’t rain.

We can reduce the number of negations if we accept the equivalence, favoured in our discussion of only if, between not-S 1 if not-S 2 and S1 only if S 2 , giving, not (a), as would be required for the example to make the point Berry is using it for, but (d): (d)

We’ll take the car only if it rains.

No-one has ever claimed that (a) and (d) are equivalent, and no such claim follows from the “unless means if not” hypothesis, which requires far more delicate selection of examples for its refutation – such as (53)–(63) above. End of Digression. The exceptionality feature can, incidentally, be illustrated for the unless in 7.12.8(i) by a comparison between (64a) and (64b): (64a) I saw Tom going into the Town Hall today – unless, perhaps, it was his twin brother Sam. (64b)

??

I saw Tom going into the Town Hall today – unless, probably, it was his twin brother Sam.

Our final topics for this subsection will be (i) a mention of yet another type of example which has been used in the literature to illustrate the non-equivalence of unless and if not, (ii ) a type of argument in Geis [1973] that we have avoided citing, and (iii) a problem over combining the line we have taken over only if with the line we have taken over unless. Beginning with topic (i), then, we recall the following examples – from Dancygier [1985], p. 69, in which the (b)-member of each pair is marked as of questionable acceptability: (65a)

I’ll be happy if she doesn’t go to hospital.

(65b)

I’ll be happy unless she goes to hospital.

(66a)

I’ll be glad if he doesn’t fail the exam.

(66b)

I’ll be glad unless he fails the exam.

The acceptability or otherwise of the (b)-examples—and (66b) seems more dubious than (66a)—we can agree that they do not fare well as capturing at least one sense of the corresponding (a) sentence. That is the sense in which the if -clause gives, not so much a condition under which the speaker expects to be happy (glad), as what the speaker will be happy about: namely, about its being the case that such-and-such. The difference between (65) and (66) is that in contemporary English we tend to use “happy” both in free-standing descriptions of emotional or affective state, and also in what we might call affective propositional attitude ascriptions (“happy that such-and-such is the case’), giving what the subject is happy about, whereas “glad” is associated particularly with the latter ascriptions – even if the proposition in question has to be gleaned from the context. (“How do you feel about her victory?” – “Very glad.” Contrast *“She was in a glad mood”.) A discussion of the relation between affective states and affective propositional attitudes may be found in Humberstone [1990b]. We introduce the idea here only to make room for the suggestion that the if -clauses in (65a), (66a) are

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behaving rather like that-clauses in attitude ascriptions. Now Harman [1979], p. 49, summarizes footnote 53 of Chomsky and Lasnik [1977] in the following way: Chomsky and Lasnik “regard if as a variant of that before subjunctives” because of the contrast between “I was surprised that they came” and “I would be surprised if they came”, etc., because this assumption simplifies certain principles of grammar.

This is a good exposition of Chomsky and Lasnik, though the if in question has nothing to do with subjunctivity; one could equally well say, “I will be surprised if they come”, for instance. The idea is really that, sometimes or always, the word if is a complementizer – an element whose job it is to parcel up the ingredients of a clause to make them into a sentential complement for (typically) a verb – like that (in “He says that he gave it back”) or for. . . to (in “For her to leave early would be impolite”), rather than a subordinating conjunction as traditionally understood. Well, sometimes or always? We already know that in some of its occurrences – the indirect question introducing occurrences (7.12.1(iii)) – if behaves as a complementizer, so for these considerations to bear on the case of (65)–(66), it suffices to take the proposal to be that for these cases too, we have if as a complementizer. This gives us one of two possible strategies for treating (65a) and (66a), for which we indicate the proposed role of if as a complementizer by writing “ifComp ”; the second would be to retain the traditional conditional if (“ifCond ”) and regard the complementizing effect as a collapsing of this with that-complementation. We do not take sides as to which of these proposals has the better long-term prospects. Proposal 1 :

“a will be happy if S” should be understood as “a will be happy ifComp S”

Proposal 2 : “a will be happy if S” should be understood as “ifCond S, then a will be happy that S” (“a will be happy that S, ifCond S”). We present both proposals only as addressing the special if of affective propositional attitude ascriptions. (In contrast, Harman [1979] maintains that if is always a complementizer and the conditionality of such if-sentences as are conditional is contributed by invisible indicative or subjunctive element called denoted there by IMP or Would, respectively. A similar proposal appears in Kratzer [1986].) Both are compatible with acknowledging the other “purely conditional prediction” sense of (65a) and the like. Thus suppose Drug A and Drug B can be surreptitiously added to the patient’s drip with the effect of increasing or decreasing, respectively, the patient’s sense of well-being. We can imagine a member of the medical staff coming out with (67)—with the comma (or pause) militating against a complementizer-style reading, or indeed (68a): (67)

The patient will be happy, if she is given Drug A.

(68a)

The patient will be happy, if she is not given Drug B.

(68b)

The patient will be happy, unless she is given Drug B.

For (68a) we have to have an already contented patient (perhaps as a result of administering Drug A), and it may help with (67) to dispel any remaining

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‘complementizer’ feel, to suppose that the patient is opposed to taking moodaltering drugs: she certainly won’t be happy that she is given Drug A. And with this understanding of the if -clauses in place, there is conspicuously less difference—(68b) vs. (68a)—between the unless formulation and the if not formulation than there was in examples (65) and (66). (Not that we are defending the unless = if not hypothesis, of course. We are simply pointing out that there is something special about (65)/(66) type of argument against it.) The second of our final topics is a type of argument we have not been attending to in Geis [1973]. In the course of mounting his attack on the unless = if not hypothesis, Geis has occasion to consider the following three examples, and he appends a “*”, indicating some kind of unacceptability, to (69b); here this has been replaced by a weaker annotation (and of course the numbering is different from that in Geis [1973], p. 233): (69a)

Prof. Arid will pass you in Linguistics 123 if you don’t fail the final exam and if you don’t make less than a C on your term paper.

(69b)

?

(69c)

Prof. Arid will pass you in Linguistics 123 if you don’t fail the final exam and don’t make less than a C on your term paper.

(69d)

Prof. Arid will pass you in Linguistics 123 unless you fail the final exam and make less than a C on your term paper.

Prof. Arid will pass you in Linguistics 123 unless you fail the final exam and unless you make less than a C on your term paper.

The difference in status between (69a) and (69b), in particular, is held to count against the unless = if not hypothesis, and in favour of Geis’s preferred unless = except if account. We do not discuss this argument here because of not wishing to raise the general issue of combinations such as and if, and unless, which the and seems to be linking clauses in the sense of “clause” in which it means subordinator plus sentence. We will have occasion below—in the discussion around examples (12)–(17) in 7.17—to present one line of thought (Risto Hilpinen’s) on such constructions. Here we content ourselves with a few remarks on some other examples in the same vein as (69) but involving only if ; they are taken from Geis [1973], p. 248: (70a) (70b)

?

I will leave only if John leaves today and only if Joe leaves tomorrow.

I will leave only if John leaves today and only if Joe leaves tomorrow.

Again, Geis’s annotation on (70a) is with “*” . But it is not clear that he intends to indicate that the (would-be) sentence is ungrammatical, or instead that it is self-contradictory, since he suggests that the a/b contrast in (70) is parallel to that in (71), where again the ‘*’ has been replaced by a ‘ ?’: (71a) (71b)

?

I like no one other than John and no one other than Joe.

I like no one other than John and Joe.

A closer inspection, however, will reveal that (71a) isn’t even self-contradictory, though it would be hard to envisage circumstances in which it would naturally be asserted. Here, it is being assumed that we make sense of (71a) as meaning simply “I like no one other than John and I like no one other than Joe”, which would be true under either of the following circumstances: (i) I like no one at all, or (ii ) John is the same individual as Joe, and I like precisely that individual.

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With this parallel in mind, we might do well to reconsider (71a). (72a) would be a natural way of expressing (72b): (72a)

I will come and work for you only if you pay me twice what you’re offering and (even then) only if my annual leave is raised to six weeks.

(72b)

I will come and work for you only if you pay me twice what you’re offering and (what’s more) I will come and work for you only if my annual leave is raised to six weeks.

The speaker, with (72b) articulates two necessary conditions for coming to work for the addressee. We can accordingly reformulate this using unless by the judicious insertion of negations: (73)

I won’t come and work for you unless you pay me twice what you’re offering and I won’t come and work for you unless my annual leave is raised to six weeks.

Geis does not consider such ‘unreduced’ analogues of the reduced forms such as (69b) which he marks unacceptable so it is not clear what he (as of 1973) would have made of them. Clearly, an acceptable reduced version of (73), in the style of (69d) would be available: (74)

I won’t come and work for you unless you pay me twice what you’re offering and my annual leave is raised to six weeks.

Remark 7.12.9 On the assumption that (73) is a satisfactory alternative formulation of (72a, b), we can note in passing that the suggestion occasionally encountered (e.g., Noordman [1985], p. 303), to the effect that unless is biconditional rather than conditional in force, is incorrect. The first conjunct of (73) does not imply that the speaker will come to work for the addressee if the pay offer is doubled, as the second conjunct makes clear. Any impression to the contrary for the case in which the assertion stops short at the first conjunct is to be explained in terms of conversational implicature: see the Digression between (19) and (20a) of this subsection. Finally, we come to the problem involved in simultaneously defending what the Geis–Fillenbaum hypothesis says about unless and the treatment of only if we have been favouring, to the effect that S1 only if S 2 amounts to not-S 1 if notS 2 . The difficulty can be seen very easily. The Geis–Fillenbaum hypothesis says that S1 unless S 2 means something other than S1 if not-S2 . On our formulation of the thesis what it means is instead (75): (75) Not-S1 only if S 2 Now the view about only if just described would have (76) paraphrasable as (76), or, on cancelling (if we may) the doubled negation, (77): (76) Not-not-S 1 if not-S 2 (77) S1 if not-S 2 But (77) was precisely what S1 unless S 2 is supposed not to mean, on the Geis– Fillenbaum hypothesis. So something has to give. One possibility is that we rethink our formulation of the Geis–Fillenbaum hypothesis, since as we noted

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in 7.12.7, Geis’s own formulation was different: there were no only-if s to be seen, their role being played by the except in the case that construction (except if, to use the cruder form – speaking of which, it is worth doing an internet search on the surprising phrase unless if to see how common it is as a variant on plain unless). Another response might be to question the account of only if, as indeed McCawley [1993] does, though with some far from clear examples unpersuasively discussed in terms of failures of bivalence – which does not, of course, mean that better reasons for reconsidering might not be available. We leave the reader to decide on a preferred resolution of the problem of relating unless, if, and only if, consistently with the examples presented here.

7.13

Material Implication and the Indicative Conditional

Chapters 5 and 6 used the introduction and elimination rules for ∧ and ∨ as a focus for the discussion of the logical behaviour of and and or, considering inter alia various possible objections to those rules as providing a faithful account of that behaviour. One obvious difficulty for a parallel strategy in the case of if lies in the multiplicity of forms the introduction rule might take, even with attention restricted to the framework Set-Fmla against the background of which most of our discussion has taken place. We begin here by reviewing that multiplicity. The informal idea behind the introduction rule for →, alias the rule of Conditional Proof, is that what will become the consequent after the application of the rule, should have been derived from the formula which will become the antecedent—together perhaps with some other assumptions—and that the application of the rule should be capable of discharging that assumption. This can be made precise by settling two questions. First: must the consequent really depend on the antecedent? And second: must the antecedent be discharged? There are thus four possibilities: Must/Must, Must/May, May/Must, and May/May. The first of these answers both questions affirmatively: the assumption which will become the antecedent must be amongst the assumptions on which the formula which will become the consequent depends, and that assumption must be discharged on application of the rule. In the second case, we again insist that the assumption which will become the antecedent must be amongst the assumptions on which the formula which will become the consequent depends, but we say only that that assumption may be discharged as the rule applies. Thirdly, we have the May/Must form of Conditional Proof: the assumption which will become the antecedent may be amongst the assumptions on which the formula which will become the consequent depends, and if it is, then as the rule applies that assumption must be discharged. Finally, with May/May, we say only that the assumption which will become the antecedent may be amongst the assumptions on which the formula which will become the consequent depends, and if it is, then as the rule applies that assumption may be discharged. All but this last form of the introduction rule for →, we have already encountered: Must/Must This is the rule (→I)d – “d” for (compulsory) discharge – from the relevant implication system RNat of 2.33. Γ, A B ΓA→B

Provided A ∈ / Γ.

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Must/May This is the rule (→I) from Nat originally given in 1.23: Γ, A B ΓA→B May/Must This is the Prawitz version of Conditional Proof, (→I)P ra from 4.12: ΓB Γ {A} A → B The reason (→I) is described as a Must/May form of Conditional Proof is that A, the assumption which is to become the antecedent, must appear on the left of “” for the rule to apply, and it may (but needn’t, since, by contrast with (→I)d , we allow A ∈ Γ) to remain on the left after that application. In the case of (→I)P ra , these permissions and obligations are reversed: since we have Γ {A} on the left after the application of the rule, the assumption A must be discharged, if it appears, though since we did not require A on the left before the rule could apply, all we are saying is that B may depend on A. Exercise 7.13.1 Provide a formulation of the May/May version of Conditional Proof. Remark 7.13.2 Although the proof system Nat (1.23) is our version of the system in Lemmon [1965a], this is an appropriate point at which to notice a significant difference. Lemmon allows the same formula to be assumed more than once in a proof, and his assumption-discharging rules actually discharge particular occurrences (as assumptions) of the formula in question. Thus not only is the following a correct Lemmon proof of the sequent p → (p ∧ p): 1 1

(1) p (2) p ∧ p (3) p → (p ∧ p)

Assumption 1,1 ∧I 1–2 →I

but so is the following a correct Lemmon proof of p p → (p ∧ p): 1 2 1, 2 1

(1) (2) (3) (4)

p p p∧p p → (p ∧ p)

Assumption Assumption 1, 2 ∧I 2–3 →I.

This second proof illustrates the discharging of one ‘assumption occurrence’ of a formula multiply assumed (namely p, assumed twice). Thus, although it is not something Lemmon exploits when it comes to citing sequents proved, the actual proofs of Lemmon’s system, when recast as sequent-to-sequent proofs, are proofs not in the framework Set-Fmla (as is the case for Nat), but in Mset-Fmla. If the above fourfold taxonomy is redrawn for the latter framework, then Lemmon’s version of (→I) – ‘(CP)’ as he calls it (for ‘Conditional Proof’) – would be described as Must/Must, since the consequent must depend on some assumptionoccurrence of the antecedent, and that particular assumption-occurrence must be discharged. (We give this as a rule for Mset-Fmla under the

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name (→I)ms in 7.25 – see discussion preceding 7.25.23, which appears on p. 1116; the same rule appeared as (→ Right)ms in the sequent calculus rules for linear logic, just before 2.33.11.) Even this description of Lemmon’s procedure is not perfectly accurate, since the only use of multisets is in respect of ‘multiply assuming’ a formula, as we put it: while Lemmon allows “1, 2” on the left, as in line (3) of the second proof above, he does not allow “2, 2”—representing a multiset in which a given assumption-occurrence figures twice. A similar practice using numerical labelling of assumptions occurs in Prawitz [1965]: the same label can be used on several occurrences of a given formula, all of which are discharged together by a rule such as (→I), though differently numbered occurrences of the same assumption formula remain undischarged. The topic of such ‘assumption classes’, as like-numbered occurrences are said to comprise, is treated in Leivant [1979], where a suggestion is made as to how sequent-to-sequent presentations of these systems doing justice to this aspect of them can be formulated: in Seq-Fmla, according to Leivant. A minor variant worth considering would use heterogeneous sequents with sets of labelled formulas – pairs A, k with A a formula and k a positive integer – on the left and individual formulas on the right. (Think of this as the line number at which A is assumed in the formula-to-formula presentation, (R) having the form A, k A.) Where Ξ is such a set of labelled formulas, Lemmon’s version of (→I) would be represented as allowing the transition from Ξ, A, k B to Ξ A → B, subject to the condition that A, k ∈ / Ξ – which allows that we may have A, j ∈ Ξ for j = k. An ordinary Set-Fmla sequent Γ A would be regarded as proved by any proof of a heterogenous Ξ A whenever Γ = {B | B, k ∈ Ξ for some k}. The multiplicity of variations on the them (→I), even within Set-Fmla and saying nothing about the further possibilities that emerge once consideration is given to alternative logical frameworks, produces something of a difficulty for considering Gentzen’s proposal (4.11) that the introduction rules for a connective capture the meaning of that connective, since there remains an unclarity as to which particular introduction rule(s) for → should be considered when it is the meaning of if that we are trying to capture. The differences between different forms of the →-introduction rule, even within Set-Fmla, to say nothing of variations across frameworks (such as the (→I)ms rule alluded to under 7.13.2) render problematic the strategy – found in Lemmon [1965a] (see the preface “to the teacher”) or Simons [1965] – of persuading the potential objector to accept the material implication account of indicative conditionals since the introduction and elimination rules for → have a plausible ring to them when the → is read as “if , then . . . ”, and they lead (taken together with other equally plausible rules for the remaining connectives) to a logic for which the standard truth-table account (in our exposition, the notion of →-booleanness for valuations) provides a fitting semantics. There are, all the same, various proposals in the literature, usually going by the name of “suppositional accounts”, which see as central to the meaning of conditionals the idea that they record the derivability (modulo background assumptions) of the consequent from the antecedent, and which might therefore naturally be regarded as giving a primacy to (some kind of) Conditional

978

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“IF”

Proof. References may be found in the notes to this section (under the heading ‘Suppositional Accounts’, p. 1056). Before proceeding, we should note that the use of the term ‘supposition’ in this context is somewhat at odds with the use of that term in 7.11, where we associated it with the antecedent of, specifically, subjunctive conditionals. (A formal development of this theme appears in 7.16.) As we discuss such conditionals in 7.15–7.17, a feature will emerge, namely the fact that they do not seem to obey a rule of ‘Suffixing’ (described there) which indicative conditionals do seem to obey. These claims will be given some reconsideration in 7.19, which is the reason for the inclusion of “seem” at this point. But, taking appearances at face value, the fact that this problematic rule would be derivable from any of the above variations on the theme of (→I), taken together with the structural rules (R), (M) and (T), and the rule (→E), provides a reason for pressing the introduction/elimination idea with special reference to indicative conditionals rather than subjunctive conditionals. In the light of the above (‘embarras de richesses’) difficulties for giving primacy to the rule for introducing →, thought of as providing a formal analogue of the indicative conditional construction, we should recall the possibility (4.14, especially the discussion of Peacocke) of allowing an elimination rule to play a constitutive role in meaning-determination. For the present case, this strategy seems independently promising, since what more better evidence could there be of failure to understand if than failure to acknowledge that B follows from If A then B together with A? (See Hare [1970]; for a ‘vertical’ formulation, we should say more explicitly: that from If A then B, as depending on assumptions Γ, and A, as depending on assumptions Δ, there follows B, as depending on Γ ∪ Δ.) Though this is here intended to be taken as a rhetorical question, we encountered much dissent on this score in the Digression on p. 528. A distinctive variation on this ‘primacy-of-(→E)’ doctrine forms the core of the account of indicative conditionals provided in Jackson [1979], [1987]. According to Jackson, the raison d’être of indicative conditionals is their participation in inferences in accordance with (→E) or, as he more commonly calls it, Modus Ponens. What is distinctive about the account is that this consideration is urged to play a role in the assertibility conditions, rather than in the truth-conditions, of indicative conditionals. As far as truth is concerned, Jackson holds that an indicative conditional, If A then B stands or falls with the material conditional, that is, with → as it behaves when attention is restricted to →-boolean valuations. But for If A then B—which we shall now write as A ⇒ B (asking the reader to bear in mind that this is to be an indicative conditional with A and B as antecedent and consequent)—to be assertible, something more is required, namely that the conditional probability of B given A should be high. The probability in question is the subjective probability for the (potential) assertor of A ⇒ B. We will return to the ‘material implication’ part of the story below, first concentrating on this special assertibility condition, of which the above formulation is something of a distortion. As remarked earlier in this discussion, what has come to be called Adams’s Thesis in the literature on conditionals is the thesis that the assertibility of an indicative conditional varies directly with the subjective probability of the consequent given the antecedent. As remarked in 7.11, this can be seen as a development of the Ramsey Test for the acceptability of an indicative conditional. (See Jackson [1987], p. 14.) Nothing hangs on the choice of terminology

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here – “assertibility” vs. “acceptability” – and in particular, it is probably best not to concentrate specifically on outward assertion, so much as on a subject’s assent (silent or communicated) to the conditional. The reader is invited to confirm Adams’s Thesis by reflecting on the situations in which the indicative conditionals listed in 7.11 – in particular (1), (3), (5), (7), and (9) – would and would not command assent, noting comparative strengths of the corresponding conditional probabilities. Now the respect in which the above description of Jackson’s special assertibility rule for indicative conditionals was a distortion was that this special rule does not simply stipulate that the associated conditional probability should be high, but instead requires that the indicative conditional’s probability, conditional upon the antecedent, should be high. Taken in conjunction with the thesis that the truth-conditions of an indicative conditional are those of the corresponding material implication statement, Jackson shows that the satisfaction of this requirement entails that the conditional probability is high ([1987], 2.4, and 7.13.3 below); thus Adams’s Thesis is not so much stipulated, as explained. At least, the latter term is appropriate if we can motivate the special assertibility requirement just described. It is at this point that the idea of Modus Ponens as the raison d’être for the indicative conditional construction enters. What we want this construction for is to have something—an ‘inference ticket’, as Ryle [1950] puts it—which will allow us to infer the consequent, should we come to learn the antecedent. If all we have is a material conditional, with no more to its meaning than what the familiar two-valued truth-table tells us about its truth-conditions, then learning (coming to attach high probability to) the antecedent may not put us in a position to infer the consequent since the increase in probability for the antecedent may be accompanied by a decrease in the probability of the conditional. After all, if our only evidence for A → B, the material conditional, is evidence for ¬A (recalling that A → B, as currently understood, is equivalent to ¬A ∨ B), then any increase in our subjective probability for A will be a decrease in probability for A → B. In terms of Ryle’s analogy, this would be like having a ticket for a train journey which expires as soon as you board the train, rendering you unable (legitimately) to reach your destination. We want a ticket which is good for the duration of the trip. And we want a conditional which is good for the trip from antecedent to consequent, rather than one which is liable to expire as soon as the antecedent is learnt. So we want a conditional whose assertibility requires not only its own high probability, but one whose probability remains high given its antecedent: and this is precisely Jackson’s special assertibility requirement for indicative conditionals. (There are some oversimplifications here in the assumption that the conditional probability Pr (B/A) always corresponds to the probability one would assign to B on learning A; see Jackson [1987], pp. 12–14 for examples and discussion, as well as the Postscript to Lewis [1976] as it appears in Jackson [1991a].) Exercise 7.13.3 (i) Show that for any probability function (as defined in 5.14, but with the language extended to include →) Pr, and any A for which Pr (A) = 0, we have Pr (A → B/A) = Pr (B/A), for all B. (ii ) Show that under the same conditions as in (i): Pr (B/A) Pr (A → B).

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Remark 7.13.4 The way the ‘material implication’ interpretation of → enters into 7.13.3 is through the instruction to take over, for a language with → amongst its connectives, the definition of definition of ‘probability function’ from 5.14, with the privileged role that definition gives to the tautological consequence relation CL . As has been emphasized by Ellis [1984], however, all that is needed for what makes 7.13.3(i) to go through is that a consequence relation playing that role should be ∧-classical and should satisfy: A ∧ (A → B) A ∧ B, and as Ellis remarks (though we put the point in our terminology), this is by no means sufficient to make an →-classical consequence relation. (Note: we can eliminate the ∧ from this formulation by separating the -condition just given into the joint requirement that A, A → B B

and

A, B A → B,

which we could put by saying that A → B and B are A-equivalent according to . If we think of as a gcr rather than a consequence relation, then in the terminology introduced in the paragraph preceding 3.31.4, this amounts to saying that A → B and B are ({A}, ∅)-equivalent according to . If (consequence relation or gcr) is ↔-classical, then the formulation A (A → B) ↔ B gives an alternative formulation – cf. the discussion following 5.23.1 – and Ellis’s point is that from the fact that A (A # B) ↔ B, it does not follow that is →-classical in its treatment of #.) When the conditional probability Pr(A → B/A) for a subject at a time is high, Jackson describes the conditional A → B as robust w.r.t. its antecedent (for that subject and time); the special rule of assertibility for indicative conditionals is then that one should assert only such conditionals as are robust w.r.t. to their antecedents (for one, at the time of assertion). This amounts to saying (and Jackson does say this) that indicative conditionals have a certain conventional implicature, and it is this aspect of their meaning which distinguishes them from material conditionals. The robustness implicature, as we may call it, imposes a special requirement over and above the general rule of assertibility that one should only make assertions whose truth is highly probable for one at the time of assertion, though as we have seen (7.13.3(ii)), taking the truth-conditions of an indicative conditional to be those of the corresponding material conditional implies that the general rule will be obeyed if the special rule is, in this case. The so-called ‘paradoxes of material implication’, by which is meant specifically such schemata as (P + )

BA→B

and

(P − )

¬A A → B

(or their analogues in Fmla, with the left-hand formulas here appearing as antecedents of implications with the right-hand formula as consequent) and more generally, aspects of material implication which seem to disqualify it as playing an important role in the theory of (indicative) conditionals, led Grice [1967] to entertain the idea that the meaning of If A then B could after all be represented by A → B, with the apparent oddity of (P + ) and (P − ) explained by considerations of conversational implicature: there is a gratuitous weakening of informativeness as we pass from left to right in these schemata. (More accurately, Grice addressed himself to “If A, B” rather than to the construction with

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“then”, discerning some possibility of a semantic distinction here; cf. the start of 7.12 above.) With this to explain why one does not reason from B to If A then B, or from ¬A to If A then B, there is no reason to deny the validity of (P + ) and (P − ), for “→” read as “If_ _ _ then___”. (See Johnson [1921], p. 42f. for an earlier pragmatic diagnosis.) The reasons against accepting an account appealing only to conversational implicature are set out in Jackson [1979] and [1987] (2.1, 2.2) and will not be rehearsed in detail here. We have already seen some of the pertinent data, for example in (47) of 7.11, whose first conjunct we repeat here: (1)

If you visit the shrine your child will need medical attention for at least a further week.

This is a perfectly reasonable thing to say for someone fully confident of the truth of its consequent, and what its assertion conveys is that the assertor’s conditional probability for the consequent given the antecedent (equivalently: of the corresponding material implication, given the antecedent) is high. Imagine instead, that, the speaker’s confidence in the consequent remaining as just envisaged (i.e., very close to 1), that the remark had been: (2)

If a new meningitis treatment is discovered at the hospital this afternoon which results in complete and immediate recovery, then your child will need medical attention for at least a further week.

This time, things have gone badly wrong. Although the subjective probability (for the assertor) of the material conditionals associated with (1) and (2) differ at most negligibly (the antecedent of (2) having probability close to 0), (2) is, unlike (1), not assertible. Since the conspicuous difference between (1) and (2) is the vast difference in the conditional probabilities, such considerations favour Jackson’s account in terms of a special conventional implicature of robustness, over any suggestion that general conversational principles are involved about not asserting something weaker than one is in a position to assert: the latter do not distinguish between (1) and (2). Digression. It is perhaps something of an oversimplification to claim that (1) above is a ‘perfectly reasonable thing to say’, if that means that it is perfectly reasonable when said in isolation out of the blue. As the example appeared in 7.11, (1) was conjoined with a conditional having the same consequent and “(If) you don’t visit the shrine” as antecedent, the whole conjunction being our (47). Outside of such a context, considerations of conversational implicature do indeed intrude over and above those of conventional implicature: why is the speaker indicating robustness w.r.t. this particular antecedent? (A judicious use of even or still forewarns against such a line of thought.) The point made with the aid of (1) could in any case be made by comparing (47) with the analogous conjunction of (2) with the indicative conditional having a negated antecedent and the same consequent as (2). End of Digression. At this point we conclude the presentation of the case for Jackson’s robustness implicature as a part of the meaning of the indicative conditional construction. As far as truth-conditions are concerned, Jackson’s account holds that this part of the semantics of the construction is given identifying them with those of the material conditional. At the beginning of this subsection, we noted that an argument for this identification based on the familiar introduction and elimination rules confronted an obstacle in the form of the variety of

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possible introduction rules; we remark here that even selecting the strongest such rule, namely that we have been simply calling (→I), taken together with (→E), yields intuitionistic rather than material (or ‘classical’) implication: but since in concert with the rules of Nat governing other connectives we obtain material implication (that is, the consequence relation associated with the full system is →-classical), this is not such a significant difference. Further, we cannot use Jackson’s derivation of Adams’s Thesis from his robustness implicature in concert with the material implication interpretation of (indicative) if to motivate that interpretation, since by Ellis’s observation (7.13.4), this is not the only interpretation on which that derivation goes through. One occasionally finds arguments in the literature (e.g. Marcus [1967]) to the effect that, given minimal assumptions, the only truth-table account of conditionals with any plausibility is that given of material implication: but this is not very helpful, since the semantic accounts of rival theories are precisely not truth-functional. Faced with these difficulties a common strategy has been to show how material implication gives the truth-conditions of indicative conditionals by linking such conditionals with constructions involving other connectives. We will follow one version of this strategy here. Symbolizing the indicative conditional by ⇒, consider (3), (3 ) and (4): (3) A ∨ B ¬A ⇒ B (3 ) ¬A ∨ B A ⇒ B (4) A ⇒ B ¬(A ∧ ¬B) Taking as either IL or CL , and taking ⇒ as →, (3 ) and (4) are correct claims; further, specifically taking as CL , but making no further stipulation for ⇒, we can conclude that A ⇒ B A → B, since the formula on the left of (3 ) and that on the right of (4) are equivalent, so that each of them—or again equivalently A → B—must be equivalent to the formula A ⇒ B which is ‘sandwiched’ between them. This last part of the argument does not go through intuitionistically, since we do not have ¬(A ∧ ¬B) IL ¬A ∨ B; but this is in any case not so important since what has been contentious about the relation between indicative conditionals and material implication is already wrapped up in (3 ): the controversy has been about whether something more is required for the truth of A ⇒ B than for the truth of A → B. Accordingly, we now transfer our attention specifically to (3 ) and its close cousin (3). To begin with (3), we note that its plausibility was implicit in the discussion in 7.11 of example (7) there: If Sullivan didn’t write the music then Gilbert did. This was supposed to be a reasonable thing to say on the basis of no more than a knowledge that either Sullivan wrote the music or Gilbert wrote the music, and so would appear to confirm the informal acceptability of (3), taking A as “Sullivan wrote the music” and B as “Gilbert wrote the music”. We have to be careful, however, about the precise status of such informal evidence, because of the contribution of conventional implicature to assertibility. In particular, the fact that something would not be a reasonable thing to say on the basis of a certain body of information may be due (even setting aside considerations

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of conversational implicature) to the failure of a special assertibility condition, such as the robustness implicature postulated by Jackson, rather than to the fact that the content of the would-be assertion does not follow from the body of information possessed. With the case just considered, however, this does not arise, since the case concerned a preparedness, rather than a reluctance, to draw some conclusion (namely that if Sullivan didn’t write the music then Gilbert did) on the basis of information, and probability of truth cannot be greater than what is required for assertibility by the robustness implicature (7.13.3(ii)).

7.14

Positive Implication and Clear Formulas

In view of the widely conceded difficulty of understanding conditionals embedded in the scope of disjunctions, in the antecedents of conditionals, and in the scope of negation, it is worth having a look at formulas containing → in none of these environments; to recall this widespread opinion as to their greater intelligibility, such formulas will be described as clear formulas. We shall find that intuitionistic logic and classical logic agree on Set-Fmla sequents composed out of clear formulas. More accurately, we will set negation to one side and consider Positive Logic. Recall (from 4.12) that this is the {∧, ∨, →}-fragment of intuitionistic logic; thus there is little to choose, for the language currently under consideration, between the subscripts “IL” and “PL”, though here we use the former (so as not to have to change when, below, we bring ¬ into the picture). A precise definition of clarity for the formulas of this fragment will be given below, though first we introduce some ancillary notions. (As mentioned at the start of this chapter, the present section accordingly represents a digression over to the ‘implication’ side of the distinction there drawn between the study of conditionals and the study of implication.) By an {∧, ∨}-formula we mean a formula built up by means of no connectives other than ∧ and ∨. By a Belnap–Thomason formula we mean a formula which is either an {∧, ∨}-formula or else an implicational formula whose antecedent and consequent are both {∧, ∨}-formulas. This terminology is a reminder of Belnap and Thomason [1963]; the result which follows may be extracted from their discussion (see Hazen [1990a]), though we shall give a different proof. Observation 7.14.1 For Belnap–Thomason formulas A1 , . . . , An , B, we have: A1 , . . . , An CL B iff A1 , . . . , An IL B. Proof. Since IL ⊆ CL , it suffices to prove that if A1 , . . . , An IL B, then A1 , . . . , An CL B, on the assumption that all formulas exhibited are Belnap– Thomason formulas. Suppose A1 , . . . , An IL B. By 2.32.8 (p. 311), there is a model (as in 2.32) M = (W, R, V ) with x ∈ W at which each Ai is true but B is not. As a simplifying assumption for the present proof, we will assume that all the formulas exhibited are implicational Belnap–Thomason formulas, leaving the reader to see how to adapt the argument for the case in which some (or all) are {∧, ∨}-formulas. Thus each Ai is of the form Ci → Di , and B can be taken to be the formula B1 → B2 . Since M |=x B, there is y ∈ R(x) with M |=y B1 while M |=y B2 . Let M be the model (W , R , V ) in which W = {y}, R = { y, y }, and V (pi ) = V (pi )∩ W . Since B1 and B2 are {∧, ∨}-formulas, they take the same truth-values at y in M as at y in M, so that M |=y B1 → B2 .

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As for the formulas Ci → Di , if one of them were false at y in M , this could only be because M |=y Ci while M |=y Di ; but again, since Ci and Di are {∧, ∨}-formulas, this would imply M |=y Ci and M |=y Di , contradicting the fact that M |=x Ci → Di . Therefore each Ci → Di (i = 1, . . . , n) is true at y in M , while B1 → B2 is not. However, since M is a one-element Kripke model, it induces a boolean valuation assigning T to precisely the formulas true at its sole element ( = y), showing that A1 , . . . , An CL B.

Exercise 7.14.2 Clearly 7.14.1 would not be correct if B were allowed to be a disjunction of Belnap–Thomason formulas; for example, we have p → (q ∨ r) (p → q) ∨ (p → r) for = CL but not for = IL . At what point would the above proof break down for such cases? We are now in a position to define the class of clear (positive) formulas. A formula A of the language of positive logic is clear iff A contains no occurrence of → in the scope of an occurrence of ∨, and no occurrence of → in any formula B for which B → C is a subformula of A. Examples 7.14.3(i) All Belnap–Thomason formulas are clear formulas. (ii ) The formula (from 7.14.2) (p → q) ∨ (p → r) is not a clear formula, and neither is p ∨ (p → r). (In each case, because we have → in the scope of ∨.) (iii) The formula ((p → q) ∧ r) → p is not a clear formula (because → occurs in a subformula of an antecedent. (iv ) The formulas (p → q) ∧ (r → s) and p → (q ∧ (r → s)) are clear formulas which are not Belnap–Thomason formulas. A clue to relating Belnap–Thomason formulas to clear formulas, thereby allowing us (7.14.5) to apply the result above (7.14.1) to show that CL and IL agree on positive clear-formula sequents, is provided by the first example under 7.14.3(iv). Though not itself a Belnap–Thomason formula, it is the conjunction of two such formulas. The second formula given there is not itself a conjunction of any pair of formulas, but it is intuitionistically equivalent to (p → q) ∧ (p → (r → s)), and hence to the conjunction of the two Belnap–Thomason formulas p → q and (p ∧ r) → s. This suggests that each clear formula is intuitionistically equivalent (i.e., bears the relation IL ) to a conjunction of Belnap–Thomason formulas. Taken literally, we can adapt the first of these examples to show that this is not quite correct: take the case of (p → q) ∧ ((r → s) ∧ (q → r)). Here the two conjuncts are p → q, a Belnap– Thomason formula, and (r → s) ∧ (q → r): not itself a Belnap–Thomason formula, but (again) a conjunction thereof. But let us agree to count a formula B as a conjunction of n formulas A1 , . . . , An if B is the formula A1 ∧ . . . ∧ An , understood as bracketed in some particular way (though any way is as good as any other); for example, by association to the left. The latter convention means that, in the case of n = 4, for instance, A1 ∧ . . . ∧ An is the formula ((A1 ∧ A2 ) ∧ A3 )∧ A4 . It is with this liberal interpretation of one formula’s being a conjunction of some others, that the following Lemma is to understood.

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Lemma 7.14.4 Every clear formula is intuitionistically equivalent to a conjunction of Belnap–Thomason formulas. Proof. We show by induction on the complexity of A that if A is a clear formula, then there are Belnap–Thomason formulas B1 , . . . , Bn with A IL B1 ∧. . .∧Bn . If A is of complexity 0, A is pi for some i, and is itself a Belnap–Thomason formula (so n = 1). Now suppose that A is A1 ∧ A2 , each conjunct being IL-equivalent to a conjunction of Belnap–Thomason formulas. Then A is IL-equivalent to the conjunction of all the conjuncts of these two conjunctions. Next, suppose A is A1 ∨ A2 . Here we do not need the inductive hypothesis (that each disjunct is IL-equivalent to a conjunction of Belnap–Thomason formulas); since A is a clear formula, → does not occur in A1 or in A2 , so that their disjunction is already a Belnap–Thomason formula (in fact, an {∧, ∨}-formula). Finally, suppose that A is A1 → A2 . Again, using the fact that A is clear, we have that A1 is an {∧, ∨}-formula. By the inductive hypothesis, A2 is equivalent to a conjunction of Belnap–Thomason formulas, k of which (say) are {∧, ∨}formulas, and m of which are implicational Belnap–Thomason formulas. Thus A2 is IL-equivalent to the conjunction: B1 ∧ . . . ∧ Bk ∧ (C1 → D1 ) ∧ . . . ∧ (Cm → Dm ) in which the Bi , Ci and Di are all {∧, ∨}-formulas. Then A itself is IL-equivalent to: (A1 → B1 ) ∧ . . . ∧ (A1 → Bk ) ∧ ((A1 ∧ C1 ) → D1 ) ∧ . . . ∧ ((A1 ∧ Cm ) → Dm ) which is a conjunction of Belnap–Thomason formulas.

Theorem 7.14.5 If all of A1 , . . . , An , B, are clear formulas, then A1 , . . . , An CL B if and only if A1 , . . . , An IL B. Proof. As in the case of 7.14.1, it suffices to show that for clear formulas A1 , . . . , An , B, if A1 , . . . , An IL B, then A1 , . . . , An CL B. To avoid clutter, we treat explicitly the case of n = 1. So suppose that for clear formulas A, B, we have A IL B. By 7.14.4, each of these formulas is IL-equivalent to a conjunction of Belnap–Thomason formulas, say C1 , . . . , Ck in the case of A, and D1 , . . . , Dm in the case of B. Thus C1 ∧ . . . ∧ Ck IL D1 ∧ . . . ∧ Dm . So, by the ∧-classicality of IL , for some i (1 i m) we have C1 , . . . , Ck IL Di . The formulas C1 , . . . , Ck ,Di are all Belnap–Thomason formulas, so by 7.14.1 we have C1 , . . . , Ck CL Di , from which it follows (this time by the ∧-classicality of CL ) that C1 ∧ . . . ∧ Ck CL D1 ∧ . . . ∧ Dm and therefore, since IL-equivalent formulas are CL-equivalent, that A CL B. Turning our attention now to the full language of IL, with ¬ included, we might at first expect 7.14.5 to fall to obvious counterexamples. What about, for instance, the fact that we have, for = CL , but not for = IL :

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(1) ¬¬p p (2) ¬p → q ¬q → p (3) p ∨ ¬p? The formulas involved are all clear formulas, by our definition, since none involve a conditional in the scope of a disjunction, or a conditional in the antecedent of a conditional. It is, however, possible to formulate IL with ⊥ rather than ¬ as primitive, defining ¬A as A → ⊥. (The associated consequence relation should be ⊥-classical, and the Kripke semantics is adapted to stipulate that ⊥ is false at every point in a model.) Strictly speaking, this way of describing matters (primitive vs. defined) is not congenial to the approach taken in 3.16; for further details, see 8.32. We can, however, decide to call formulas (of the language with ¬) clear if the results of replacing all of their subformulas ¬A by A → ⊥ count as clear by the original definition. (The presence of ⊥ makes no difference to that definition.) The results of making such replacements in (1)–(3) are: (1)

(p → ⊥) → ⊥ p

(p → ⊥) → q (q → ⊥) → p

p ∨ (p → ⊥)

(2)

(3)

and with these changes, we can see that the original ‘clearness’ conditions are violated. (1) and (2) exhibit → in the antecedent position of →, while in (3) , we have → in the scope of ∨. Exercise 7.14.6 Show that Theorem 7.14.5 continues to hold, on the above understanding of clearness, for arbitrary formulas A1 , . . . , An , B of the language of IL. We should remark that the agreement between CL and IL as to which clear formulas follow from which clear formulas is not to be confused with the matter as to whether these logics agree on the admissibility of sequent-to-sequent rules which are formulable with clear formula-schemata. The rules (Peirce) and (Subc)→ considered in 7.21 are rules falling under the latter description which are CL- but not IL-admissible, for example. In Set-Fmla there are, however, close connections between certain such rules and sequents constructed out of clear formulas, brought out by horizontalizing (à la 1.25.10, p. 131) the rules into sequent-schemata format, as explained in Humberstone and Makinson [forthcoming]. Finally, let us make some connections between the present syntactical notion of clearness and the semantic concept of a topoboolean condition from 4.38. For a boolean connective # the topoboolean condition [#]tb for # (or ‘induced by the boolean function #b ’) was introduced in the discussion leading up to 4.38.6 (p. 620); we need a closely related concept here, and use the same ancillary notation to define it. Specifically, relative to a Kripke model (for IL) M = (W, R, V ) we understand by vx (A) the truth-value (in M) of A at x ∈ W . Suppose A(q1 , . . . , qn ) is a formula in which the only propositional variables to occur are those indicated (and q1 etc. are metavariables for arbitrary propositional variables). Then we shall say that A(q1 , . . . , qn ) is a topoboolean compound of q1 , . . . , qn , or for brevity, a topoboolean formula, when for some n-ary truth-function f , we have for all models M = (W, R, V ) and all x ∈ W :

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M |=x A(q1 , . . . , qn ) ⇔ ∀y ∈ R(x), f (vy (q1 ), . . . , vy (qn )) = T. Examples 7.14.7(i) The formulas ¬p, p ∨ q, p → q, p ↔ q, p ∧ q, p → (q → r) and p → ¬q, are topoboolean (i.e., are topoboolean compounds of their variables). (ii ) The formulas ¬¬p, ¬p ∨ q, ¬p → q, p ∨ (q → r), (p → q) → r are not topoboolean. The formulas listed under 7.14.7(i) are all clear formulas, and it is not hard to see that every clear formula is topoboolean. For instance, taking A as the formula (p → q) ∧ (q → (r ∨ s)), we have (for arbitrary M, x as above) M |=x A ⇔ ∀y ∈ R(x), f (vy (p),vy (q),vy (r),vy (s)) = T where f is the quaternary truth-function which can be ‘read off’ from the (classical) truth-table for A. (That is, taking # as the derived boolean connective with definition #(B, C, D, E) = (B → C) ∧ (C → (D ∨ E)), f is the truthfunction #b associated with # over the class of boolean valuations.) It is not only clear formulas that are topoboolean, however, since we can have an ‘unclear’ formula IL-equivalent to a clear formula, such as (p → p) → q, ¬¬¬(q∧r), p ∨ (q → p) and their respectively clear equivalents q, ¬(q ∧ r), and q → p. But such exceptions are the only ones that need to be considered: Observation 7.14.8 A formula is topoboolean if and only if it is IL-equivalent to a clear formula. A proof can be obtained by following the lead of the examples given and (for the ‘only if’ direction) by adapting Rousseau [1968]. Thus we could, as a further corollary, write out Observation 7.14.5 with clear replaced by topoboolean.

7.15

From Strict to Variably Strict Conditionals

Recalling that the necessitation (A → C) of a material implication with A as antecedent and C as consequent – often written more concisely as A C – is called the strict implication with that antecedent and consequent, we work our way towards the account of subjunctive conditionals developed in (e.g.) Stalnaker [1968] and Lewis [1971], [1973], via a study of the logical features of such strict conditionals. But we depart from the language of modal logic as it featured in §2.2 by taking strict implication as a primitive connective, to be symbolized with ‘’, and without supposing either or → to be present in the language at all. We will allow also the connective ∧, since some of the logical principles which distinguish the strict from what (following Lewis [1973]) we shall later call the ‘variably strict’ conditional are conveniently formulated with its aid. Various principles distinguishing different logics of the latter type of conditional, which as we shall offer promising avenues for the formalization of the subjunctive conditionals we distinguished from their indicative counterparts in 7.11, call for the presence of additional boolean connectives. But we defer any mention of, in particular, negation and disjunction until 7.17, and we shall not be concerned to extend the illustrative soundness and (especially) completeness results of the present subsection into the richer language containing those connectives.

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An initial comment is called for to forestall misunderstanding. A common use of such phrases as “strict implication” and “strict conditional” reserves them for (statements given the formal representation) (A → C) in which has the intended reading “it is logically necessary that”. Subjunctive conditionals are of course frequently deemed true despite not having consequents which follow logically from their antecedents: indeed, the latter are a highly special case. So it is worth emphasizing that no such restricted interpretation is intended for here – or rather, since we are suppressing that connective – for our compounds A C. Just as in §2.2 we distinguished modal logic ‘in the broad sense’ from the narrow sense in which this is the study specifically of logical necessity and possibility, so we may make a parallel distinction between strict implication, narrowly construed, and strict implication broadly construed. The semantic reflection of this broader interest is that when evaluating a strict conditional at a point in a model, we have at our disposal an accessibility relation which directs us to the accessible points at which the antecedent is true, and we count the conditional true at the original point when each of the latter points verifies the consequent. The motivation for pursuing such an investigation is of course that these ‘points’ are thought of as representing possible worlds, and we saw in 7.11 that quantification over possible worlds seemed all but forced for the semantics of subjunctive (as opposed to indicative) conditionals. The strict implication paradigm is the simplest way of implementing this idea. As to how best to construe the accessibility relation, we make no positive suggestion—since we shall see in due course that this implementation turns out to be too simple(minded)—though there are some suggestions in the literature. For example, from Burks [1951] can be extracted the suggestion that the implicit operator stands for some notion of causal necessity, so that one world may be deemed to be accessible from another if the none of the causal laws of the former are violated in the latter (with perhaps also some restriction in terms of common ‘initial conditions’). We begin, then, with the models (W, R, V ) in terms of which the semantic discussion of modal logic proceeded in §2.2, and because of its appealing prooftheoretic presentability, let us consider the strict implication (and conjunction) fragment of S4. The logical framework will be Set-Fmla throughout. Thus in the present instance, we are concerned with the sequents A1 , . . . , An B which are valid on every reflexive transitive frame, with the truth-conditions for propositional variables and conjunctive formulas as in §2.2, and—reflecting the way we intend to construe the “” here—the following clause for -formulas: [] (W, R, V ) |=x A B iff for all y ∈ W such that xRy and (W, R, V ) |=y A, we have (W, R, V ) |=y B. The proof-theoretic appeal just alluded to lies in the fact that we need only supplement the structural rules (R), (M), (T), together with (∧I) and (∧E), with the following variations on (→I) and (→E): (I)

(E)

Γ, A B ΓAB

Provided all formulas in Γ are of the form C D.

ΓAB Γ, Δ B

ΔA

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Of course in the case of this second rule, there is no variation at all (except that we have rewritten “→” as “”). The proviso on (I), however, gives us a seriously restricted conditional proof rule; it blocks the would-be proof of p q p from p, q p, since p is not a strict implication formula. And this is just as well, since a world (in some model) could verify p without all p-verifying worlds R-related to it verifying q. Exercise 7.15.1 (i) Show that (I) and (E) preserve the property of holding in any model on a reflexive transitive frame. (ii ) Show that the following rules are derivable in the present proof system: (Prefixing)

B 1 , . . . , Bn C A B1 , . . . , A Bn A C

(Transitivity)

A B, B C A C

Part (i) of this exercise gives a soundness proof for the present system w.r.t. the class of reflexive transitive frames; Part (ii) is useful for the completeness proof, as well as in isolating some principles we shall need to take as primitive for certain weaker logics below. Observation 7.15.2 The rules listed under 7.15.1(ii) in fact preserve the property of holding in any model. The reader is left to check this. Note, then, that the labelling on the second rule has nothing to do with the transitivity or otherwise of the accessibility relation. The idea for the completeness proof is (as ever) to build a canonical model for the system. We take as elements of the set W for this model the sets of formulas which are deductively closed (w.r.t. the associated consequence relation); primeness and consistency need not be imposed since we do not have disjunction or negation amongst our connectives. We define xRy to hold when for all A, B, with A B ∈ x, if A ∈ y then B ∈ y. Exercise 7.15.3 (i) Show that all sequents of the form (a) A B, A B, as well as all sequents of the form (b) A B (C C) (A B), are provable in the current proof system. (ii ) Conclude from the provability of (a) that the canonical frame (W, R) is reflexive and from (b) that it is transitive. We convert our frame into a model by supplying V with the usual definition, so as to secure (*) (W, R, V ) |=x pi iff pi ∈ x. Lemma 7.15.4 The condition (*) above holds for arbitrary formulas A in place of pi . Proof. The inductive case of (∧) is nothing new, so let us consider A of the form B C. By the inductive hypothesis, what we have to show is that B C ∈ x iff for all y ∈ W such that xRy, if A ∈ y then B ∈ y. The ‘only if’ direction

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of this ‘iff’ is given by the definition of R. For the converse, we need to show that when B C ∈ / x, there exists y ∈ R(x) with B ∈ y, C ∈ / y. We will show that we may take as such a y the set {D | B D ∈ x}. First it must be verified that y ∈ W , i.e., that y is deductively closed. This follows from the derivability of the rule Prefixing in 7.15.1(ii). Next, we show that y ∈ R(x). Suppose, for some formulas A1 , A2 , we have A1 A2 ∈ x and A1 ∈ y. We want to conclude that A2 ∈ y. Since A1 ∈ y, B A1 ∈ x. By Transitivity, since A1 A2 ∈ x and B A1 ∈ x, we have B A2 ∈ x; therefore A2 ∈ y, as desired. Finally, we must check that B ∈ y while C ∈ / y. Clearly we have C ∈ / y, since y was to comprise just those formulas D for which B D ∈ x, and we are supposing that B C ∈ / x. On the other hand B ∈ y because B B ∈ x, since the sequent B B is provable (by (I) from B B: here Γ in the formulation of the rule is empty so the restriction is vacuously satisfied). Thus, from 7.15.1(i) and 7.15.4, we conclude: Theorem 7.15.5 The proof system with the structural rules, the ∧-rules, and the rules (I) and (E), is determined by the class of all reflexive transitive frames. The proof of Lemma 7.15.4 had nothing to do with the reflexivity or transitivity of the canonical accessibility relation, so that we can easily peel off these aspects of our discussion, and formulate proof systems for the (conjunction and) strict implication fragments of KT and K (and K4, too, for that matter, though we shall not treat this case). By 7.15.2, the rules there mentioned do not interfere with any soundness results; clearly ( E) depends on the reflexivity of the accessibility relation for its possessing the preservation property cited in 7.15.1(i), so it must be abandoned for the case of the strict implication system corresponding to K (given under (ii) of the following Theorem, part (i) treating of the KT case). Theorem 7.15.6 (i) The proof system with the structural rules, the ∧-rules, and the rules ( E), (Prefixing), (Transitivity) and the rule A A is determined by the class of all reflexive frames. (ii) Dropping ( E) from the rules listed in (i) gives a proof system determined by the class of all frames. We shall not investigate the strict implicational fragments of normal modal logics stronger than S4. The interested reader will find a discussion (for the framework Fmla) of these in Ulrich [1981], to whose completeness proofs those we have given are very similar. (For an example of an incomplete strict implicational logic – one determined by no class of frames – see Kirk [1981].) A proof-theoretically oriented discussion of strict implication in a range of modal logics including some non-normal systems may be found in Hacking [1963], where the proviso on (I) originates. Our present interest is now in the passage from strict implication to variably strict implication. By way of motivation, recall that we are concerned with providing a setting for the exploration of the logic of subjunctive conditionals. The logics considered to this point all contain every instance of the schema we have been calling Transitivity:

7.1. CONDITIONALS

991 A B, B C A C.

It has been maintained that for the conditional constructions with which we are concerned, this represents an incorrect principle of inference. The kind of direct counterexample people have had in mind will be illustrated in 7.17. All of our proof systems have also had as a derivable rule the following highly restricted form of (I): AB A B In the case of the S4 system of 7.15.5, as we noted in the proof of 7.15.4, this is a special case of (I) in which the set Γ of side-formulas left behind after application of the rule is empty, and so a fortiori comprises at most formulas. For the KT and K systems of 7.15.6, we included the rule A A so that Lemma 7.15.4 would still go through, and using this we derive the above rule by applying Prefixing to A B, obtaining A A A B, from which, together with A A, we arrive at the conclusion-sequent A B by (T). None of this is particularly controversial for the intended application to subjunctive conditionals, but using the above rule together with Transitivity we obtain the following highly controversial principle: (Suffixing)

AB BCAC

Note that the admissibility of (Suffixing) for a proof system is equivalent to the claim that the connective is antitone (3.32) in the first position. Similarly, though this has seldom been found problematic, the admissibility of the n = 1 form of Prefixing is equivalent to the claim that is monotone in the second position. What, then, is controversial about (Suffixing)? We can illustrate this with the help of the other of the two connectives of the language of the present subsection, conjunction. The idea is to exploit the fact that (for all the logics with which we are concerned) conjunctions have their conjuncts as consequences, so we may put A ∧ B for A in (Suffixing) and then apply this rule to conclude: (Strengthening the Antecedent)

B C (A ∧ B) C.

From now on, we will usually call this “(Str. Ant.)” for short. The example most commonly used to illustrate what seems unacceptable about (Str. Ant.) originated with Nelson Goodman in 1946 (see Goodman [1965], Chapter 1); here is the version of it which appears in Stalnaker [1968], One is to imagine the speaker holding a normal dry match in a perfectly ordinary room (as regards such matters as the presence of oxygen): If this match were struck it would light; therefore, if this match had been soaked in water overnight and it were struck, then it would light. We seem to find ourselves (in the situation envisaged) inclined to assent to the premiss but not to the conclusion of this argument. It is one thing to follow out the consequences of making the supposition that the match is struck, and another to follow out the consequences of the logically stronger and (given

992

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the circumstances) more far-fetched supposition that the match is struck after having been soaked in water overnight. It is because (Str. Ant.) seems an incorrect principle for subjunctive conditionals, though it is not something whose formal validity we can escape while treating them as strict conditionals, that the strict conditional approach was described in the second paragraph of this section as being in the end too simpleminded. We should notice that not only does this make such proposals as Burks’s ‘causal implication’ account, mentioned in that paragraph, inappropriate for the logic of subjunctive conditionals, it also takes out of contention the suggestion occasionally voiced that relevant implication (introduced above in 2.33) shows us what that logic should look like: for (Str. Ant.), with “” understood as the relevant “→” (say, as in the system R) is a correct principle. The suggested application of relevant implication here rejected can be found in Bacon [1971], Barker [1969], as well as in certain passages of Anderson and Belnap [1975]. Burgess [2009], p. 114, suggests a conditional reading for relevant implication, with A → B as “If A then B for that reason” (emphasis Burgess’s). This is reminiscent of the ‘helping’ discussed in 7.11 in connection the helping/non-hindering distinction; see the Digression below. The problem for Burks’s account raised by strengthening was noted in Rescher [1968], p. 31 (originally published in 1957). Digression. Burgess’s “for that reason” rider has to be interpreted, in connection with a relevant implicational claim of the form A → B in such a way as not to imply that B would not be the case if A weren’t, in view of the provability (in R, for example, of both p → (p ∨ ¬p) and ¬p → (p ∨ ¬p). An interesting discussion of transitivity inferences for indicative conditionals in Dale [1972] involves several of the issues that have been in play in the discussion here and in earlier subsections, though it is not clear that the notion of relevance about to be invoked has anything to do with relevant logic. Dale’s target is the pair of claims from Strawson [1952]: one the one hand that a ‘correct’ conditional requires that the antecedent be relevant to its consequent, which is understood as meaning that it should be at least partially explanatory of the (supposed) truth of the consequent, and, on the other hand, that the transitivity (or ‘Hypothetical Syllogism’) inference: If p then q; If q then r; therefore: If p then r, should preserve correctness. (Strawson’s claim was that the premisses here entail the conclusion, so for this purpose correctness could perhaps be glossed as truth; presumably Dale is using the term correct so as to sidestep the issue of whether it is appropriate to speak of the truth of indicative conditionals, or replace it with talk of something in the area of assertibility.) We then have the following example from Dale: (i) (ii) (iii) (iv) (v)

If I knock this typewriter off the desk then it will fall. If it falls then it is heavier than air. If I knock this typewriter off the desk then it is heavier than air. If the typewriter is heavier than air then an elephant is heavier than air. If I knock this typewriter off the desk then an elephant is heavier than air.

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The idea is that (i), (ii) and (iv) seem intuitively ‘correct’; in the last case a justification might come from the observation that any elephant is heavier than this typewriter. But from (i) and (ii) the transitivity inference pattern yields – the somewhat peculiar – (iii), which together with (iv) yields by that same inference pattern, the decidedly peculiar (v). If we overlook the peculiarity in question and rule it ‘correct’ after all, we have a counterexample to Strawson’s relevance requirement. But if we rule otherwise, we have to reject the transitivity inference. So we cannot endorse both of Strawson’s claims. Already we notice in the passage from (i) and (ii) to (iii) that (i) and (ii) lie are conditionals of very different types, so much so that in pondering the issue of the transitivity inference (or Hypothetical Syllogism, for those who not prepared to accept the justification for speaking of transitivity in this context offered in and around 3.34.4 above – p. 502), they are not the kind of premiss pairs than immediately spring to mind. In terminology aired in 7.11, (i) is a projective (Dudman), consequential (Dancygier and Mioduszewska) or content domain (Sweetser) conditional, while (ii) is a hypothetical, non-consequential, or epistemic domain conditional. (It is not here assumed that these binary distinctions coincide in all cases, or that any one of them has been made sufficiently clear to invoke with any confidence. Not also that the idea of explanatory relevance seems to be working in opposite directions with (i) and (ii).) This heterogeneity seems to have something to do with why the interim conclusion (iii) is already odd, even before we proceed to (v), but we do not speculate further on this here. It would be good, also, to be get clear on the relation between examples like this and examples of apparent transitivity failure for subjunctive conditionals – see the discussion after 7.18.2 below, p. 1036 – as well as examples with indicatives which fail (on Jackson’s account) to fail to preserve assertibility. Presumably the latter do not only involve sets of conditionals which are heterogeneous in respect of the above distinctions (projective/hypothetical etc.). End of Digression. Not everyone, as we shall see in 7.18, has been persuaded that it is a fallacious move to strengthen the antecedent of a subjunctive conditional in this way, but for the moment, let us treat the prima facie unacceptability of (Str. Ant.) as a reason for seeing what happens if the rule that led to our deriving it, namely Transitivity, is dropped from the proof systems we have been considering. To this end, we must ask what role the latter rule played in the completeness proofs. It figured in the proof of Lemma 7.15.4, showing the equivalence of truth and membership for the canonical model(s). We were trying to show that, given B C ∈ / x, we could find y ∈ R(x) containing B but not C, and Transitivity was invoked in the part of the argument showing that if we chose y as {D | B D ∈ x}, we would indeed have y ∈ R(x). Let us repeat the reasoning here. Suppose, we said, for some formulas A1 , A2 , we have A1 A2 ∈ x and A1 ∈ y. This latter means that B A1 ∈ x, so we could put this together with A1 A2 , now that both are seen to belong to x, and conclude by Transitivity that B A2 ∈ x, giving us A2 ∈ y, as required. How would the semantic apparatus need be to adjusted so that this appeal to Transitivity can be forgone? We will no longer be able to show that x bears R to y defined as {D | B D ∈ x}. But this way of specifying y delivered everything else we needed, so let us stick with it. Rather than requiring that for all formulas A1 A2 ∈ x, if A1 ∈ x then A2 ∈ x—the old definition of (canonical) R—we could restrict

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our attention to the case of A1 = B, the antecedent of the -formula we are supposing not to belong to x. The relation we are interested in holds between x and y = {D | B D ∈ x} alias {A2 | B A2 ∈ x}, but now the relation depends on B itself, so let us call it RB . The idea is to define RB in the canonical model to hold between points w and z when {D | B D ∈ w} ⊆ z. This relation will hold between the x and y in play in the above discussion, and we won’t have to appeal to Transitivity because we no longer worry about conditionals with other antecedents (the full range of A1 , above). Thus the change in the semantics needed to make all this work is to replace the one relation R by a whole family of relations RA for various formulas A, and to replace the ‘strict’ clause [] by the following ‘variably strict’ clause, to emphasize which we now change notation from to ⇒ (and any back-reference to a condition or rule already defined for should be understood in the course of what follows as referring to result of replacing “” by “⇒” in the condition or rule): [⇒]

M |=x A ⇒ B iff for all y ∈ W such that xRA y, M |=y B.

Here M is our model, which can be taken to be of the form (W, R, V ) in which R is now a function assigning to each formula a binary relation on W , and we write RA for R(A). We will call such models variably strict models. (There is no change to the clauses for the propositional variables and conjunctive formulas in the definition of truth at a point in a model.) The reader can check that the rule Prefixing preserves the property of holding in a variably strict model, giving the soundness (‘only if’) part of the following result. The completeness part can be demonstrated using the above suggestion about how RA should relate points (deductively closed sets, again) in the canonical model, namely: xRA y iff {D | A ⇒ D ∈ x} ⊆ y. (See Lewis [1971] and Chapter 3 of Nute [1980] for applications of this idea.) Theorem 7.15.7 A sequent is provable using the structural rules, the ∧-rules, and Prefixing if and only if it holds in every variably strict model. We wished to get rid of Transitivity or anything that led to it, but much of the baby has been thrown out with the bathwater in paring down the ⇒principles to just Prefixing. Before putting back some of what has been lost, we pause to observe (with Chellas [1975]) that the minimal system treated by 7.15.7 can be thought of as a multi-modal logic in which each formula indexes a normal modal operator. Instead of writing A ⇒ B, to bring out this parallel, we write A B. Then Prefixing becomes: B1 , . . . , Bn C A B1 , . . . , A Bn A C which says that A is a normal modal operator according to the proof system (or the associated consequence relation), in the sense of 2.23. Chellas [1975] refers to these operators as sententially indexed modal operators, and takes a greater interest in what he calls instead propositionally indexed modal operators. (The former terminology also appears in Lewis [1971]. Actually Chellas writes

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“[A]B” rather than “A B”. The latter notation is from Gabbay [1972], where a similar point is made. Purists will note that whichever notation is used, the latter is not the result of applying a 1-ary connective – “[A]” or “A ” – B; if it were, the propositional variables occurring in the resulting formula would just be those occurring in B. Rather, sticking with the latter notation, A is a 1-ary context, as explained for 3.16.1, p. 424. (See also 5.11.2, p. 636.) We will make use of the “[A]” notation for a rather different purpose in 7.16.) We return to this sentential/propositional distinction below. Now, to put back some of what has here been jettisoned from the weakest of our strict implication systems, let us note that (⇒E) can be associated with the following condition on variably strict models M = (W, R, V ); in each case understand the condition as prefaced by “For all x, y ∈ W and for all formulas A”: (mp)

If M |=x A then xR A x.

and the rule A ⇒ A with the condition: (id ) If xR A y then M |=y A (The labelling, suggestive of “Modus Ponens” and “Identity”, is adapted from Chellas.) The association alluded to is given by: Theorem 7.15.8 (i) A sequent is provable using the structural rules, the ∧rules, Prefixing and (⇒ E) if and only if it holds in every variably strict model satisfying (mp). (ii) A sequent is provable using the structural rules, the ∧-rules, Prefixing and A ⇒ A if and only if it holds in every variably strict model satisfying (id ). (iii) A sequent is provable using the structural rules, the ∧-rules, Prefixing, (⇒ E) and A ⇒ A if and only if it holds in every variably strict model satisfying (id ) and (mp). The proofs of the completeness halves of these assertions are straightforward applications of the idea given before the formulation of 7.15.7. Such results themselves, however, are far from edifying. In the first place, conditions such as (id ) and (mp) are imposed on models rather than frames, giving them an ad hoc flavour: what is it about the underlying structures (W , R) which justifies such conditions? Further, unlike the conditions on Valuations such as (Persistence) in 2.32, and similar conditions in 2.33, 6.43–6.47, these conditions on models are imposed for all formulas by brute force, rather than being stipulated for atomic formulas and then shown inductively to hold for arbitrary formulas. (The trouble here is that we cannot ‘break into’ the subscript position on “RA ” and relate this to “RB ” for subformulas B of A.) Secondly, we have given no informal indication as to how these varying accessibility relations RA should be thought of. Filling this lacuna will turn out to help us to remedy the first (frames vs. models) difficulty. To obtain an intuitive gloss on the relations RA , it is helpful to see how the principle (Str. Ant.) fails on the present semantics, even with both the conditions of 7.15.8(iii) in force. We have a point x in a model, verifying (say) the formula p ⇒ r. So all the points Rp -related to x verify r. But x needn’t verify the antecedent-strengthened formula (p ∧ q) ⇒ r, since perhaps not all points Rp∧q -related to x verify r. For this to happen, there must be points

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to which x bears Rp∧q but not Rp . In the case of Goodman’s match-striking example, x may be thought of as a world housing Goodman’s dry and unstruck match, and the supposition that the match is struck (p) takes us to worlds in which things are somewhat different from the way they are in x: for a start, the match is struck. But there are various other differences consequential upon this together with things with are not required to suppose different (such as the amount of oxygen in the room, for example, and the physical laws governing combustion). In that example, q and r are respectively the statements that the match has been previously soaked in water and that it lights. The fact that the match lights in the worlds in which only the differences just described as consequential upon the supposition that p is no reason for saying that it will light when such differences as are occasioned by the more far-reaching supposition that the match has also been soaked in water. This gives the following way of thinking about the relations RA , which found expression in Stalnaker [1968] and Lewis [1973]: xR A ywhen y is amongst the most similar A-verifying worlds to x. A picturesque way of speaking of such relations of (overall) similarity is in terms of closeness, so that with the above gloss on the relations RA , our clause [⇒] becomes: A ⇒ B is true at a world x iff at all the closest A-verifying worlds to x, B is true. With this way of thinking of RA in mind, let us assess the two conditions imposed above to recover some of what was lost in the passage from strict to variably strict conditionals; we repeat them here for convenience: (mp) (id )

If M |=x A then xR A x; If xRA y then M |=y A.

For the current gloss on RA , both conditions are clearly satisfied: if y is amongst the closest A-verifying worlds to x, then y is an A-verifying world (at least, as long as there are some A-verifying worlds). And if A is true at x, then x must be amongst the closest A-verifying worlds to x. If we take the superlative “closest” seriously, we may want – following Stalnaker and Lewis – to go further and say that if A is true at x then the set of closest A-verifying worlds to x is precisely {x}, no other world being as similar to x as x itself is. We shall return to this stronger requirement (under the name “Centering”) in 7.16, as well as touching on other conditions which the similarity conception of the RA relations suggests. For present purposes, it will do to notice one other consequence of this conception, which brings back some more of the strength of the {∧, }-fragment of K (7.15.6(ii)) that was sacrificed as we traded in [] for [⇒]. Recall the qualms raised above about the rule: (Suffixing)

AB B ⇒ CA ⇒ C

which led to arbitrary strengthening of the antecedent (though we focussed specifically on conjunct-to-conjunction strengthenings). Even if we do not want the consequence relation associated with a proof system for any proposed formalization of the subjunctive conditional to be antitone in the first position, we may well wish the connective concerned to be at least congruential in that position:

7.1. CONDITIONALS (Ant. Cong.)

AB

997 B A

B ⇒ CA ⇒ C

This rule is forced on us by the present understanding of ⇒. For if A B, then the most similar A-verifying worlds to any given world must also be the most similar B-verifying worlds, there being no difference between the class of A- and B-verifying worlds: thus all of this class, specified the former way, verify C iff all of the class, specified the latter way, verify C, and we must have B ⇒ C A ⇒ C. Closure under the above rule can be thought of as saying that according to a logic so closed, the proposition expressed by a conditional formula depends, insofar as its antecedent is concerned, only on the proposition expressed thereby, rather than on the particular formula chosen to express that proposition. It is for this reason that Chellas [1975] refers to the smallest logic closed under Prefixing and (Ant. Cong.) – his minimal ‘normal conditional logic’ – as a logic of propositionally indexed modalities, rather than (as in the absence of the latter condition) of sententially indexed modalities. Though most theorists are agreed that subjunctive conditionals should be treated as congruential in antecedent position, there has been some dissent; since one of the grounds for dissent concerns conditionals with disjunctive antecedents, and we are postponing the enrichment of the current language by disjunction (and negation) until 7.17, the qualms about congruentiality will be described there. If we do go along with this majority view, then, as was mentioned above, the model-oriented semantics with which we have been working can be given a presentation in terms of frames. First, we should observe that 7.15.8 can be repeated for the case in which (Ant. Cong.) is added to each of the sets of rules mentioned in its parts (i), (ii), and (iii), as long as the following condition is also imposed on variably strict models: that if A B, then RA = RB , for any formulas A and B and any such model (W, R, V ). This condition shares the inelegancies of the earlier conditions (id ) and (mp). So next, we rework the apparatus to eliminate those inelegancies. We regard R not as assigning to each formula a binary relation on W , but rather to each subset of W , a binary relation on W . Where X ⊆ W , we have the ‘similarity’ interpretation of xR X y as: y is one of the worlds in X which are maximally similar to x. More generally, we think of y as accessible to x by an accessibility relation determined on the basis of the proposition X. Then we replace [⇒] by: [⇒]

M |=x A ⇒ B iff for all y ∈ W such that xR A y, M |=y B.

in which !A! denotes {w ∈ W | M |=w A}. It is straightforward to replace the conditions (mp) and (id ) on models by conditions on frames (as now conceived): (mp) (id )

If x ∈ X then xR X x; If xR X y then y ∈ X. (Alternative formulation: RX (x) ⊆ X.)

The analogue of 7.15.7–8 then reads: Theorem 7.15.9 The proof systems with the standard structural rules, the ∧rules, (Ant. Cong.) and in addition (i): Prefixing, (ii): Prefixing and (⇒ E), (iii): Prefixing and A ⇒ A, (iv): all rules listed under (ii) and (iii), are – when the truth-definition uses the clause [⇒] – determined respectively by:

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(i): the class of all frames; (ii): the class of all frames satisfying (mp) ; (iii): the class of all frames satisfying (id) ; (iv): the class of all frames satisfying (mp) and (id) . Canonical model arguments for the ‘completeness’ halves of claims (i)–(iv) here turn out not to be adversely affected by the fact that the models concerned have to supply relations RX even for the case of X = !A! for any formula A. In these cases, the relations can be arbitrary. And for the case of RX when X= !A!, our earlier definition can be used: xR X y iff for all C such that A, ⇒ C ∈ x, C ∈ y. Closure under (Ant. Cong.) guarantees that we will get the same set of C’s even if X is also equal to !B! for a different formula B from A, so that the definition can be consistently employed. (See §§6,7 of Chellas [1975] for a full presentation, and Segerberg [1989] for a variation.) For 7.16, we will revert to the earlier conceptions of model and frame, with sententially rather than propositionally indexed accessibility relations, and not attend to the condition of congruentiality until we come, in 7.17, to discuss the applicability of the logics we have been treating, and some of their extensions, to the subjunctive conditional constructions that motivated their introduction. The reader can pass to that discussion from this point, without loss of continuity. The following subsection will take some steps to make the systems we have been examining look more like logics of conditionals from the point of view of philosophical proof theory (§4.1). In particular, we seek pure and simple rules for ⇒ which display, in addition, some close resemblance to the usual natural deduction rules (→I) and (→E).

7.16

Interlude: Suppositional Proof Systems

Actually, the advertisement at the end of 7.15 notwithstanding, it is more the matter of purity and simplicity that interests us than the natural deduction theme, and we shall pursue the latter idea only briefly and informally, before changing to a somewhat different approach. (Here approach is intended in the sense of approaches to logic, as distinguished from logical frameworks in 1.21.) By way of motivation, let us see how we could exclude a natural deduction proof of the sequent p ⇒ r (p ∧ q) ⇒ r in which the antecedent is (we continue to assume) fallaciously strengthened, using rules for ⇒ as much like the rules (→I) and (→E) as possible. If no change is made to those rules, but for a renotation from “→” to “⇒”, we have, in Lemmon-style format: 1 2 2 1, 2 1

(1) (2) (3) (4) (5)

p⇒r p∧q p r (p ∧ q) ⇒ r

Assumption Assumption 2 ∧E 1, 3 ⇒E 2–4 ⇒I

To block a proof like this in a manner consilient with the idea (7.11) that the antecedent of a subjunctive conditional represents a supposition, we could distinguish between two kinds of assumptions: ‘plain’ assumptions, and suppositions.

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Only the latter qualify as possible minor premisses for (⇒E), and we shall mark them explicitly with a superscripted “s”; similarly, conclusions drawn in the course of following out the consequences of a supposition we will mark with a superscript “c” (for ‘consequence of a supposition’). In particular, then, we reformulate the rules (⇒E) and (⇒I) in the following way: s

(⇒E)

A

A⇒B c

B

(⇒I) Given a proof of c B from assumption s A, pass to A ⇒ B (depending on the remaining assumptions used in the proof ). Since the rule (∧E) figures in the above proof of p ⇒ r (p ∧ q) ⇒ r, we need to say something about how this rule deals with the superscript markings; from a conjunctive premiss with an “s” marking – thinking here of the rule in formulato-formula form (as with (⇒E) above – either conjunct should be inferable with a “c” marking, since we are here following out the consequences of a supposition. Now, with these marking restrictions in place, the above derivation is blocked. To use (the formula on) line (1) as a major premiss for (⇒E) it should be a plain assumption, and since, at the end of the derivation, we want to discharge the assumption p ∧ q and turn it into the antecedent of a conditional, it will have to be assumed as a supposition. By what was just said about the ∧-rules, however, this s-marking on p ∧ q will have to traded in for a c-marking on p at line (3), and then the (⇒E) step at line (4) is no longer possible, since only s-marked minor premisses can be used in applying this rule. Intuitively, what the apparatus is doing here is enforcing the observation that it is one thing to suppose p ∧ q, and another thing to suppose p: what follows (given some background ‘plain’ assumptions) from making the latter supposition need not follow from making the former, even though the former is logically stronger in respect of its content. This possibility is what the rejection of the rule (Suffixing) from 7.15 amounts to. The reader may care to check how the s and c restrictions on the ⇒-rules block a proof of the following instance of Transitivity: p ⇒ q, q ⇒ r p ⇒ r. By contrast with (Suffixing), the rule Prefixing had a favourable ride through 7.15, going into the presentation of every proof system we had occasion to consider. A representative instance of the application of this rule is in the passage from q ∧ r q to p ⇒ (q ∧ r) p ⇒ q. Let us illustrate the way in which the obvious natural deduction proof of the latter sequent is unscathed by the marking restrictions. For this example, we have to note that applying (∧E) to a c-marked formula should yield a c-marked conclusion, since the conjunct’s following from a conjunction which is a consequence (together with background assumptions) of a supposition is itself a consequence of that supposition. This is because the ‘following from’ involved here is logical consequence, unaided (by contrast with the case Transitivity) by such additional background assumptions. This explains the step from line (3) to (4) in the following proof. (Note also that we do not need to write “ c (q ∧ r)” to avoid ambiguity: the markers c and s are not singulary connectives but indications of what role in a derivation is being played by the whole formula to which they are appended.)

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Example 7.16.1 A proof of p ⇒ (q ∧ r) p ⇒ q: 1 2 1, 2 1, 2 1

(1) (2) (3) (4) (5)

p ⇒ (q ∧ r) s p c q∧r c q p⇒q

Assumption Assumption 1, 2 ⇒E 3 ∧E 2–4 ⇒I

As remarked above, we do not intend to give a full and explicit discussion of the natural deduction system sketchily introduced by the above remarks, as this would involve various complications into which we do not wish to enter. The reader is invited to pursue the matter privately, if desired, or to consult Thomason [1970b] for a natural deduction system in very much the same spirit, though cast in the format of Fitch [1952]. Rather, we carry over the basic idea of separating plain assumptions from suppositions by shifting both to a different logical framework and to a different approach. Given the structural rules (R), (M) and (T), then the following rules, in which the double underline indicates a downward rule and an upward rule, are interderivable with the natural deduction rules (→I) and (→E); in fact the downward direction just is (→I): Γ, A B (→)

ΓA→B

(Such rules were mentioned in 1.28). The upward direction of the rule is what takes us outside the natural deduction approach, since it neither introduces nor eliminates an occurrence of → on the right of the . But this change of approach is particularly amenable to a systematic implementation of the ‘supposition’ idea, since we can introduce a special position on the left of the to house formulas playing this role. (In the present treatment, for simplicity, we allow only one formula per sequent in this position.) to We shall not use the “s” and “c” method of marking, but rather write a ‘supposed’ formula in square brackets; thus the above pair of rules become, after we make that change, as well as putting “⇒” for “→”: Γ [A] B (⇒)

Γ A ⇒ B

Note that we do not have anything corresponding to the superscripted “ c”; rather, the logical framework into which we have now moved affords two types of sequents, those in which there is no [ ]-formula on the left—call them plain sequents—and those in which there is such a formula on the left—the suppositional sequents; a formula on the right will be interpreted in the latter sequents, as though it had a “c” superscript. We call this framework the suppositional framework, and emphasize the difference just noted between it and the other logical frameworks with which we have been dealing, namely, that there is no one form which every sequent in the framework can be regarded as instantiating. This makes it necessary to pay particular attention to the structural rules we shall impose on what we will call suppositional systems. For present purposes, we identify such a system with its collection of provable sequents, so that

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1001

in defining the class of such systems, the rules in question have the status of closure conditions. (We will sometimes say “logic” as an alternative to “system”.) In this capacity, we take over intact the Set-Fmla rules (R), (M) and (T), for plain sequents, and we also need a form of (M) for suppositional sequents, as well as ‘mixed’ (plain + suppositional) version of (T). We call the latter pair of structural rules (Suppositional M) and (Mixed Cut): (Suppositional M)

(Mixed Cut)

Γ[A] B Γ, Δ[A] B

Γ[A] B1 . . . Γ[A] Bn

B1 , . . . , Bn C

Γ[A] C

The second rule here cannot be replaced by the special case in which n = 1, because of the mixing of plain and suppositional sequents. (Indeed, if one insists that a rule has always some fixed number of premiss-sequents, then (Mixed Cut) is not one rule, but infinitely many—one for each choice of n.) As in 1.12, 1.16, we follow Scott’s lead here in developing the fundamentals of this framework without regard for exactly what connectives are present in the language or what rules they are stipulated to be governed by. Later the connectives ⇒ and ∧ will enter the picture, but for the moment, we do not assume their presence, emphasizing that a suppositional system is simply required to be closed under the above structural rules and not also (in particular) under the two-way rule (⇒) above. (There is a close parallel with the treatment of modal logic in Blamey and Humberstone [1991], which pays somewhat fuller attention to the advantages and interest of such structurally oriented developments.) For a parallel with the discussion of §1.1, we need something analogous to the distinction between arbitrary valuations and boolean valuations, and to keep the internal constitution of the frames of the variably strict models of the preceding subsection in view, we shall make this be the distinction between models and what we shall call ‘structures’. More specifically, the frames we have in mind are those (W, R) in which R assigns to each formula—not (as at the end of 7.15) each subset of W —a binary relation on W , denoted RA for a given formula A. We depart from our usual practice of having a Valuation V on such a frame assign a truth-set only to the propositional variables, this serving as the basis for an inductive definition of the truth-relation (“|=”). Rather, we shall let V assign a truth-set (a subset of W ) to every formula, writing for x ∈ W , “x ∈ V (A)” instead of “x ∈ !A!” (since the latter notation suggests the inductively defined |=). The difference between structures and models is then that no conditions whatever are imposed on V for (W, R, V ) to count as a structure on the frame (W, R), whereas a model is specifically a structure in which V respects the clauses in the truth-definition of 7.15, i.e., in the present notation, for all formulas A and B: [∧] [⇒]

V (A ∧ B) = V (A) ∩ V (B) V (A ⇒ B) = {x ∈ W | RA (x) ⊆ V (B)}.

Of course, additional such conditions would be appropriately imposed if we were to pay attention to further connectives; even the above conditions (and hence

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the notion of a model) will not concern us for the moment, since we intend to work for a little in abstraction from questions of what connectives are present. A plain sequent Γ B holds at x ∈ W in a structure (W, R, V ) just in case we do not have x ∈ V (C) for each C ∈ Γ without having x ∈ V (B), whereas a suppositional sequent Γ[A] B holds at x iff we do not have x ∈ V (C) for each C ∈ Γ without also having RA (x) ⊆ V (B). Here we are making clear the difference, mentioned above, between the way the formula on the right is interpreted in a plain sequent from the way such a formula is interpreted for the case of a suppositional sequent. In the former case, the sequent to hold at a point, the right-hand formula must be true at that point if the left-hand formulas Γ are true there, whereas in the latter case, the right-hand formula must be true, not at the point verifying those formulas, but at all the points RA -related to that point, where A is the suppositional (square-bracketed) formula. In the case of sequents of either sort, we say that a sequent holds in a structure when it holds at every point thereof, and we say that a structure in which every sequent in some set Σ of (plain and suppositional) sequents holds is a structure for Σ. Recall that we identify a suppositional logic with such a set of sequents. We now show how to find, for any such logic, a canonical structure for the logic, this label being deserved since for the structures (WΣ , RΣ , VΣ ) concerned, we have a sequent σ holding in the structure iff σ ∈ Σ. If Σ is a suppositional logic (a set of sequents, that is, closed under the earlier-mentioned structural rules), then WΣ consists of the deductively closed sets of formulas in the sense of: sets of formulas Γ such that Γ Σ B implies B ∈ Γ. The notation “Γ Σ B” means that for some finite Γ0 ⊆ Γ, we have (the plain sequent) Γ0 B ∈ Σ. We shall also use a similar notation for the case of suppositional sequents, writing ‘Γ [A] Σ B’ when for some finite Γ0 ⊆ Γ, we have Γ0 [A] B ∈ Σ. In fact, we need this right now, as we shall define the relation-assigning function RΣ by stipulating that for any formula A and any Γ, Δ ∈ WΣ : ΓRΣA Δ iff {B | Γ[A] Σ B} ⊆ Δ. Finally, and predictably, we set VΣ (A) = {Γ ∈ WΣ | A ∈ Γ}. Lemma 7.16.2 For any suppositional system Σ, (WΣ , RΣ , VΣ ) is a structure for Σ. Proof. We must show that each σ ∈ Σ holds in (WΣ , RΣ , VΣ ). There are two cases. (1) σ ∈ Σ is a plain sequent Γ B. Suppose we have x ∈ WΣ with x ∈ VΣ (C) for each C ∈ Γ. By the definition of VΣ , this means that Γ ⊆ x, so x Σ B, since Γ B ∈ Σ. But since x is deductively closed, B ∈ x, i.e., x ∈ VΣ (B), which is what had to be shown for us to conclude that σ holds at x. Since x was arbitrary, σ holds in (WΣ , RΣ , VΣ ). (2) σ ∈ Σ is a suppositional sequent Γ [A] B. Again, with a view to showing that σ holds at an arbitrarily chosen x ∈ WΣ , suppose that Γ ⊆ x. Then x[A] Σ B. We must show that for all y ∈ WΣ such that xR ΣA y, we have B ∈ y (alias y ∈ V (B)). But this is immediate from the definition of RΣ . The proof just given does not exploit the assumption that Σ is a suppositional system, so 7.16.2 holds for an arbitrary collection of sequents of the suppositional

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framework. (The definition of (WΣ , RΣ , VΣ ) can be employed for any such Σ.) Not so in the following case, where the structural rules are exploited. Lemma 7.16.3 For any suppositional system Σ, if σ ∈ / Σ then σ does not hold in (WΣ , RΣ , VΣ ). Proof. Again there are two cases: (1) σ = Γ B. If σ ∈ / Σ then clearly σ fails to hold at the deductive closure of Γ (the point x ∈ WΣ such that x = {C | Γ Σ C}). (2) σ = Γ [A] B. Again, we show, assuming σ ∈ / Σ, that σ fails at the deductive closure x of Γ. We have x ∈ V (C) for each C ∈ Γ, so what we need now is some / y (so that y ∈ / V (B)). But (i) and (ii) are y ∈ WΣ with (i) xR ΣA y and (ii) B ∈ clearly satisfied by taking y = {C | x[A] Σ C}. This leaves us with having to show that y ∈ WΣ . For all we know, y may not be deductively closed, and so not belong to WΣ . So suppose that C1 , . . . , Cn ∈ y and that C1 , . . . , Cn D ∈ Σ. We must show that D ∈ y. By the way that y was specified, this means that there are (finite) subsets Γ1 , . . . , Γn of x with Γi [A] Ci ∈ Σ (1 i n). Letting Γ* be the union of these Γi , by (Suppositional M) we have in Σ each of the following: Γ*[A] C1 ;. . . ; Γ* [A] Cn . Now since also C1 , . . . , Cn D ∈ Σ, we get Γ* [A] D ∈ Σ, by (Mixed Cut), and therefore since Γ* ⊆ x, x[A] Σ D, so that D ∈ y, as was to be shown.

Exercise 7.16.4 Where are the plain-sequent structural rules (R), (M), and (T), (tacitly) appealed to in the above proof? A collection of sequents Σ is sound w.r.t. a class of structures if every σ ∈ Σ holds in each structure in the class, complete w.r.t. such a class if every sequent holding in every structure in the class belongs to Σ, and (as usual) is determined by a class if it is both sound and complete w.r.t. the class. Theorem 7.16.5 Any suppositional system Σ is determined by the class of all structures for Σ. Proof. Soundness is automatic, given the notion ‘structure for Σ’. As to completeness, the canonical structure (WΣ , RΣ , VΣ ) is a structure for Σ, by 7.16.2, in which an arbitrary sequent not belonging to Σ fails to hold, by 7.16.3. Given a specification of some class of structures other than directly as ‘structures for’ a suppositional system, the soundness part of a claim of determination will not possess the triviality we see in 7.16.5. We will see some such claims in 7.16.8 below, for which purposes we now record three observations on classes of structures and rules. The first, 7.16.6(i), treats of the structural rules under which all suppositional systems are required to be closed, while the remaining observations pertain to additional structural rules we now introduce, namely (Suppositional R)

[A] A

and

(Desupposition)

Γ[A] B Γ, A B

.

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Note that the first rule could equally well have been formulated as “Γ[A] A” (since we can ‘thin in’ Γ by (Suppositional M), and that the second rule takes us from a premiss-sequent of one type (suppositional) to a conclusion-sequent of the other (plain): it converts a formula playing the role of a supposition into a formula playing the role of an ordinary assumption. To describe the semantic effect of these rules, we need to recall the conditions (mp) and (id ), imposed on (variably strict) models in 7.15, and here reformulated so as to apply to arbitrary structures (mp) (id )

If x ∈ V (A) then xR A x If xR A y then y ∈ V (A).

Then we have Observation 7.16.6 (i) For any structure, the rules (R), (M), and (T), (Suppositional M), and (MixedCut) preserve the property of holding in that structure. (ii) All instances of the schema (Suppositional R) hold in any structure satisfying (mp). (iii) The rule (Desupposition) preserves the property of holding in any structure satisfying (id). (iv) The canonical structure for any suppositional system containing all instances of (Suppositional R) satisfies (mp). (v) The canonical structure for any system closed under (Desupposition) satisfies (id). Proof. By way of example, we offer proofs of (iii) and (v). For (iii), suppose that (W, R, V ) is a structure satisfying (id ), and in which Γ[A] B holds. We want to show that Γ, A B also holds in this structure. So suppose further that we have x ∈ W with x ∈ V (C) for each C ∈ Γ and also x ∈ V (A), with a view to showing that x ∈ V (B). Since x ∈ V (A), and our structure satisfies (id ), xR A x; thus for the sequent Γ[A] B to hold in the structure (as assumed), we must have x ∈ V (B), since x ∈ V (C) for each C ∈ Γ. For (v), suppose that Σ is closed under (Desupposition). We want to show that (WΣ , RΣ VΣ ) satisfies (id ). So suppose for x ∈ WΣ that x ∈ VΣ (A), for some formula A, which means (by the definition of VΣ ) that A ∈ x. We need to show that xR ΣA x, i.e., that for any formula C, if x [A] Σ C, then C ∈ x. So take some C for which x[A] Σ C. Then x ∪ {A} Σ C, by (Desupposition). But A ∈ x and x is deductively closed: therefore C ∈ x. Now let us bring conditionals explicitly back into the picture, and with them, for added interest, conjunction. We call a suppositional system in whose language the connectives ⇒ and ∧ occur, and which contains all instances of the (plain-sequent) schemata A, B A ∧ B; A ∧ B A; A ∧ B B, and is closed under the two-way rule Γ [A] B (⇒)

ΓA⇒B

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a {⇒, ∧}-suppositional system. Recall that a structure whose Valuation V respects the intended meanings of ⇒ and ∧ is called a model. A model which is a structure for Σ is a model for Σ. We leave the reader to check the following: Observation 7.16.7 (i) The ∧-schemata above hold in any model, and the rule (⇒) preserves, for any model, the property of holding in that model. (ii) The canonical structure for any {⇒, ∧}-suppositional system is a model. Note that (ii ) does not say that every structure for a {⇒, ∧}-suppositional system is a model. (Compare 1.14.7, p. 70, and see also Blamey and Humberstone [1991].) However, what it does say suffices for completeness results of particular {⇒, ∧}-suppositional systems which are versions in the present suppositional framework of the systems treated in 7.15.7, 8. Let us baptize these systems, in the interests of a concise formulation of the results in question. Let Σ0 be the smallest {⇒, ∧}-suppositional system, Σmp be the smallest such system closed under (Suppositional R), Σid be the smallest {⇒, ∧}-suppositional system closed under (Desupposition), and Σmp,id be the smallest {⇒, ∧}-suppositional system closed under both of these two rules. Theorem 7.16.8 (i) Σ0 is determined by the class of all models. (ii) Σmp is determined by the class of all models satisfying (mp). (iii) Σid is determined by the class of all models satisfying (id ). (iv) Σmp,id is determined by the class of all models satisfying both (mp) and (id ). Proof. Taking the representative case of Σid , we consider (iii). Soundness: 7.16.6(i), (iii), and 7.16.7(i). Completeness: 7.16.6(v) and 7.16.7(ii). Notice that for plain sequents, the notion of holding in a model deployed in this subsection is the same as that used in 7.15; thus, by 7.15.7 (p. 994) and 7.16.8(i), the plain sequents belonging to Σ0 are precisely the sequents provable using (alongside (R), (M), (T)) the ∧-rules and Prefixing. Similarly, by 7.15.8(i) and 7.16.8(ii), the plain sequents of Σmp are the sequents provable using not only the rules just mentioned, but also the rule (⇒E) from 7.15. And so on for the other combinations of rules covered by 7.15.8 (p. 995) and 7.16.8. The present approach, via the suppositional framework, differs in that only one rule, namely (⇒), involves the connective ⇒, and variations from system to system are obtained by changes in the structural rules rather than by laying down (as in 7.15) special rules for ⇒. We should also observe that this (two-way) rule (⇒) uniquely characterizes the connective it governs (§4.3) and can be regarded as assigning to this connective a constant meaning which does not vary as we pass from system to system. What varies instead is the properties we take the to be possessed by practice of making a supposition. Now, it may occur to the reader that a more direct way of exploiting the ‘suppositional’ position in our sequents might be to introduce a singulary connective – there being, after all, only one formula occupying that position – to express occupancy of the ‘supposition’ role. Like the rule (⇒), a rule for this connective will modulate between suppositional and plain sequents, but unlike the latter rule, it will not also move a formula across the “”. Writing ‘*’ for the envisaged connective, we would have the (two-way) rule:

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Γ [A] B Γ, *A B If we want the lower sequent here to hold at a point x in a model (W, R, V ) precisely when the upper sequent does, we shall have to arrange things so that Γ, ∗A B holds at x iff we do not have each C ∈ Γ true at x unless B is true at each point in RA (x). This singulary connective * will appear again in 7.17. We close the present section with examples and exercises for practice with the suppositional framework. Example 7.16.1 Revisited. A suppositional proof of p ⇒ (q ∧ r) p ⇒ q (1) (2) (3) (4) (5)

p ⇒ (q ∧ p ⇒ (q ∧ q∧rq p ⇒ (q ∧ p ⇒ (q ∧

r) p ⇒ (q ∧ r) r) [p] q ∧ r r) [p] q r) p ⇒ q

(R) 1 (⇒) upwards. Basic ∧-principle 2, 3 (Mixed Cut) 4 (⇒) downwards.

Some slightly more complicated examples arise in (i), (ii), of Exercise 7.16.9 below. For later parts of the exercise, we will need to refer to some additional structural rules for the suppositional framework—both of them ‘Cut-like’—so we display them here. (Suppositional Suffixing)

(Suppositional T)

AB

Γ[B] C Γ[A] C

Γ[A] B

Γ[B] C Γ[A] C

Note that no plausibility (given the intended application to subjunctive conditionals via the rule ( ⇒ )) is claimed for the above rules. Exercise 7.16.9 (i) Using just the rules in terms of which Σ0 was defined, give a proof of: p ⇒ q, p ⇒ r p ⇒ (q ∧ r). (ii ) Using the same rules, give a proof of: s ⇒ (p ⇒ (q ∧ r)) s ⇒ (p ⇒ q). (iii) Using the rules mentioned in (i), together with (Suppositional Suffixing), give a proof of: p ⇒ r (p ∧ q) ⇒ r. (iv ) Using the rules of (i), together with (Suppositional T), prove: p ⇒ q, q ⇒ r p ⇒ r.

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(v) Show that a suppositional system is closed under (i.e., contains all instances of) (Suppositional R) iff it is closed under the rule: from A B to [A] B. (vi ) Show that if a suppositional system is closed under (Suppositional R) and (Suppositional T) then it is closed under (Suppositional Suffixing). (Hint: make use of (v).) (vii) Show that the rule (Suppositional Suffixing) preserves the property of holding in a structure, if that structure (W, R, V ) satisfies the condition: for all A, B, V (A) ⊆ V (B) implies RA ⊆ RB . (viii) Show that the condition on structures mentioned in (vii) is satisfied by the canonical structure for any suppositional system closed under the rule (Suppositional Suffixing).

7.17

Possible Worlds Semantics for Subjunctive Conditionals

In 7.11, we noted that a semantic account in terms of possible worlds seemed called for in the case of subjunctive but not in the case of indicative conditionals. Accordingly in 7.15, we looked at two kinds of possible worlds semantics for “⇒”, intended to be read as a subjunctive conditional: strict and variably strict. It was pointed out that certain principles—such as (Str. Ant.)—appeared not to be correct for subjunctive conditionals, and that since the strict treatment validated these principles, while the variably strict treatment did not, the latter treatment was ahead on points. (In 7.18, we will report on the work of some who are not convinced by these considerations.) Further, as was also mentioned in 7.15, there is a natural way of construing the set RA (w) = {x | wRA x} which arises in the semantics of variably strict models, namely, as the set of worlds most similar to w in which A was true. This idea was associated with the names of Stalnaker and Lewis. (The ‘Lewis’ in question here is David Lewis, not C[larence] I[rving] Lewis, who had much to say in defence of strict implication—though as a formalization of entailment rather than of subjunctive conditionals.) In neither case is this association quite accurate. We will spend some time discussing the respect in which this is inaccurate for Stalnaker, correcting the account for the case of Lewis in 7.17.3. (The clause [⇒] which we tend to associate with Lewis until then is actually due to John Vickers: see Lewis [1973], p. 58.) Reminder : we shall conduct the discussion more informally than in 7.15 (and 7.16) – in which, to make life easy, we restricted attention to the language with a conditional connective and conjunction: a restriction here rescinded. In particular, no attention will be paid to demonstrating soundness and completeness for the various proof systems that can be presented with the aid of the principles to be discussed. (For information on these matters, see Lewis [1971], [1973] (Chapter 6), Krabbe [1978], Nute [1980], Burgess [1981b].) By way of further disclaimer, we note that no discussion will be given to the details of a ‘similarity’ interpretation of the variably strict accessibility relations RA . Such details have been the subject of much debate – for example in Bennett [1974], Pollock [1976], Lewis [1979b], Bowie [1979] – but are ignored here, since we are more interested in the upshot of assumptions on world-similarity for the validation of schemata and rules governing subjunctive conditionals than

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in the possibility of providing a reductive analysis of such conditionals in terms of similarity. (See Nozick [1981], p. 174n., Section III of Stalnaker [1975], and §5 of Forster [2005].) In Stalnaker [1968], a conditional A ⇒ B is counted as true at a world w just in case the closest (i.e., most similar) A-verifying world to w verifies B. It is assumed, then, that there is always exactly one such closest A-verifying world. The assumption has two parts: existence and uniqueness. The first part may seem problematic for the case of impossible antecedents; most dramatically, consider the case in which A is the formula p ∧ ¬p. Stalnaker’s solution here is to include in his models a special ‘absurd’ world at which all formulas come out as true. (Thus the clause for, e.g., negation, in the definition of truth has to be modified. We cannot say that ¬A is true at a world iff A is not, since ¬A and A are both true at this special point. The needed reformulations are routine, though they can require careful handling; see Humberstone [2011a].) The second part – uniqueness – is what Lewis concentrates on, under the label Stalnaker’s assumption, in the virtuoso exposition provided by Lewis [1973]: why should there not be ties for maximal similarity to w amongst the A-verifying worlds? Allowance for such ties is built into the notation “RA (w)” since the function RA takes an element of W to a subset of W , rather than to an element of W . In this terminology, Stalnaker’s assumption becomes the condition that RA (w) is always a singleton, for all formulas A and all points w. The most famous logical repercussion of that condition is that all models meeting it verify every instance of the schema: (A ⇒ B) ∨ (A ⇒ ¬B) which is often called the Law of Conditional Excluded Middle. This is best seen as a special case of the fact that the sequent schema A ⇒ (B ∨ C) (A ⇒ B) ∨ (A ⇒ C) holds in all such models. (Take C = ¬B.) It is because of the presence of disjunction (even if, as in this last schema, negation is not explicitly present) that we have deferred discussion of these matters until now. The preceding two subsections confined themselves to questions arising—and not even all of those questions—for a language containing a (subjunctive) conditional connective alongside conjunction. Naturally, we take the definition of truth at a point in a model to have the usual clause for ∨ (as in 2.21, for example). Once the uniqueness assumption for closest antecedent-verifying worlds is abandoned, it is natural to distinguish a stronger from a weaker subjunctive conditional. Lewis calls the two constructions the would-counterfactual and the might-counterfactual, respectively, and symbolizes them with binary connectives → and →. We read “A → C” as “If it had been (or were) that A, it would have been (or would be) that C”, whereas with “A → C” the “would (have)” in this gloss is replaced by “might (have)”. Provisionally, we should think of Lewis’s “→ ” as corresponding to the use we have been making since 7.15 of “⇒”, with “A → C” being defined (cf. and themselves in 2.21) as “¬(A → ¬C)”. This amounts to saying that (not all but) some of the closest A-verifying verify C. Thus a case in which the Law of Conditional Excluded middle would be held to be false would be one in which some but not all of the closest antecedentverifying worlds verify the consequent. Both the would-counterfactuals are false,

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and both of the might-counterfactuals true. A commonly cited case is one in which the would -counterfactuals concerned are “If we had tossed this (fair) coin, it would have landed heads” and “If we had tossed this coin, it would not have landed heads”. Rather then accepting either of these, what we say instead is that had the coin been tossed, it might have landed heads and it might not have. Digression. Interestingly enough, though the example just given about the coin seems to cast doubt on the Law of Conditional Excluded Middle, and thereby to tell in favour of Lewis’s approach and against Stalnaker’s, the final sentence of the last paragraph points in the other direction. Consider the three forms: (1) (2) (3)

If it had been that A, it might have been that B; and if it had been that A, it might have been that C. If it had been that A, it might have been that B and it might have been that C. If it had been that A, it might have been that B and C.

There are ‘direct’ representations in Lewis’s language of (1) and (3), namely: (1) (3)

(A → B) ∧ (A → C) A → (B ∧ C)

whereas there is no such direct representation in the case of (2), which would instead have to be treated as an idiomatic variant on (1), and accordingly rendered as (1) . Analogous points apply if the might in (1)–(3) is replaced by would. We can also mix the two: (4)

If it had been the case that A, then it would have been the case that B and it might have been the case that C.

Examples such as (2) and (4) raise difficult methodological issues about the extent to which the representation of natural language sentences in a formal language should attempt to mirror the construction of those sentences, and about how explicit an account needs to be given in justification of departures from such mirroring. Usually the latter account remains informal and inchoate, and sometimes goes by the name of ‘translation lore’ (e.g., in McGee [1985]). This issue will arise again, below, as we touch on the problem of disjunctive antecedents. (The distinction between would and might subjunctive conditionals has sometimes been held to arise, though formulated using different modal auxiliaries in English, for indicative conditionals also – for example in Lowe [1983].) Of course we can still represent (4) in Lewis’s notation, with an ungainly repetition of the antecedent. Another kind of example in which the only available formal representation of a natural language conditional construction (“If an American male hostage were to be shot, so would his wife be”), seems to exhibit a similar long-windedness is discussed in Humberstone [1988c], at p. 191 of which this sentence and a situation relative to which its truth is to be assessed are given. End of Digression. Although Lewis [1973] presents the availability of a different semantic treatment for might- and would-counterfactuals as a benefit of renouncing Stalnaker’s uniqueness assumption, this consideration has not been universally found compelling. Stalnaker [1981] presents some reasons for dissatisfaction, one of which is that the might that appears in the consequents of some counterfactuals is

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better treated, if at all possible, in the same way as ordinary unembedded occurrences of might – for example, as a modal operator expressing (some suitable notion of) possibility. There is a striking contrast in this respect with ‘would’, which lacks a self-standing interpretation of a piece with its consequent-marking role. For instance, someone’s saying, “It would rain this afternoon”, naturally receives the query “It would rain this afternoon, if what?”. (Exception: with heavy stress on “would” for the idiomatic construction more fully appearing as: “Oh, it would have to go and rain this afternoon, wouldn’t it?”) There is no analogous problem with interpreting “It might rain this afternoon”. Remark 7.17.1 The “if what? ” request for clarification can be forestalled with, (then) . . . ”, out explicitly using a conditional construction in “if and more particularly, with a mere prepositional phrase specifying the ‘condition’. One can say, for example: “Without antibiotics you would not survive.” “Away from the sunlight, this plant’s leaves would fade.” As just remarked, however, in such cases, the prepositional phrases are still understood as specifying circumstances such that if they were to obtain the would -clause would be true. (See further the opening paragraph of 7.12 and the references there given to monoclausal/‘implied condition’ conditionals.) An interesting question raised by these constructions is whether they allow mixed indicative–subjunctive combinations of a kind impossible (in English) with explicit if -conditionals. For example, I tell you, as we discover that Ralph, a very fit swimmer, has been washed overboard: “In water this cold and rough, Ralph will survive only half an hour, and you or I would survive only five minutes.” Having decided to make use of a possibility operator—let’s write it as “”— to deal with the might in might-counterfactuals, there is still a question as to where to place it. In order to avoid the suggesting that the “might” vs. “would” contrast is a matter of duality, let us revert to the more neutral-looking “⇒” notation for subjunctive conditionals of the preceding two subsections, and ask which of the following is to be preferred as the representation of “If it were that A then it might be that C”: (i) (A ⇒ C) (ii ) A ⇒ C Stalnaker [1981] opts for a representation in the style of (i), and explains how the notion of possibility invoked is to be understood. It is harder to arrange the semantics so as to have (ii) do the job, and we shall not go into this here; though a representation along these lines is more strongly suggested by the material in the Digression above. The niceties of matching English idiom to the structure of our formal representations aside, there is a more pressing difficulty with the uniqueness assumption, namely that, as already noted, this will end up validating the Law of Conditional Excluded Middle, which cases like that of the fair coin suggest should not be regarded as valid. Stalnaker’s response to this worry is to claim that what would be objectionable would be the result that for any choice of

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A and C, either A → C or else A → ¬C is true. We often feel that neither is, and to do justice to this feeling, we may use supervaluations (6.23) in order to keep the disjunction of such ‘opposite’ counterfactuals true, without either disjunct having to be true. The underlying bivalent valuations for this application of the technique are those which select, for each A and w, a unique A-verifying world to be deemed the closest A-verifying world to w. To the extent that such a selection represents the arbitrary choice of one out of a range of equally good candidates for this role (i.e., in the event of a ‘tie’), one can iron out the results of making that choice by counting as true only what comes out true on every way of making the choice. On any given way, the disjunction (A → C) ∨ (A → ¬C) is true (so that this formula is valid), though this cannot be said for either of its disjuncts. (This strategy is foreshadowed in Thomason [1970b]; for further discussion, see Stalnaker [1981], van Fraassen [1974]; for more on the Law of Conditional Excluded Middle more generally, see Harrison [1968] (esp. p. 380) and Section V of Hilpinen [1981].) This general disagreement over the cardinality of RA (w) notwithstanding, there is one case in which Lewis and Stalnaker are united in the view that it should equal 1: the case in which A is true at w. In this case, there is a unique closest A-verifying world to w, namely w itself. Recall the condition (mp)

If M |=x A then xR A x

from 7.15, which we can reformulate as saying that if M |=x A then x belongs to RA (x). The current point of agreement would strengthen this consequent to: then x, and nothing else, belongs to RA (x). In other words: (Centering)

If M |=x A then R A (x) = {x}.

Instead of this ‘sentential’ condition, we could use a ‘propositional’ formulation; where M = (W, R, V ), we say, for any X ⊆ W : (Centering)

If x ∈ X then R X (x) = {x}

which makes it clear that we are dealing here with a condition on frames rather than on models. (Of course, this raises again the question from 7.15 as to congruentiality in antecedent position, to which we shall return below.) The label “(Centering)” is taken from Lewis [1973], which, for the most part, works with a differently formulated semantics, in terms of ‘spheres’ of similarity comprising all worlds containing, for some world x, all worlds at least as similar to x as any world they contain. Such a system is described as centered on x if {x} is itself one of these spheres. The most obvious repercussion of (Centering) is the validation of the sequent schema (Cent.)

A ∧ C A → C

though since (Centering) implies (mp), its full effect is only registered by combining (Cent.) with A, A → C C, as in A (A → C) ↔ C.

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This, we should observe, gives the subjunctive conditional those features which Ellis emphasized an indicative conditional needed to conditional probabilities to be guaranteed not to exceed the probabilities of the corresponding conditionals: see the end of 7.13.4. Although (Cent.) contains only the connectives (with ⇒ rewritten as → ) available to us in 7.15 (and of course the ∧ on the left could be replaced with a comma, leaving only “→ ”), we remark that a completeness proof along the lines of that subsection is not possible. We would wish to show that for the canonical model, as defined in that subsection, of a logic containing A → A and (Cent.), the condition (Centering) is satisfied. The schema A → A (alias A ⇒ A) is included because that condition subsumes the requirement that if M |=x A then x ∈ RA (x). What we expect appealing to (Cent.) to deliver is the rest of (Centering), namely that if y = x, then y ∈ / RA (x), assuming that M |=x A. So suppose that y ∈ RA (x) in the canonical model in question, where A ∈ x. We need to show that x ⊆ y and y ⊆ x. The first inclusion presents no problem, since if B ∈ x, A ∧ B ∈ x, and so by (Cent.) A → B ∈ x, so B ∈ y, as y ∈ RA (x). But the converse inclusion cannot be established in the expressively austere setting of 7.15. With negation in the language, and behaving classically in the logic, the imposition of maximal consistency on the elements of the canonical model takes care of the problem, since for maximal consistent x, y, x ⊆ y implies y ⊆ x (essentially because of 0.14.4). (Other examples of this phenomenon may be found in Humberstone [1990a].) (Cent.) has had a mixed reaction from theorists of conditionals, being accepted by Stalnaker and Lewis, as well as by Harrison [1968], but rejected (or at least questioned) in, for example, Clark [1973], Bennett [1974], Nute [1975], Bigelow [1976], Nozick [1981] (p. 680f.), Uchii [1990]. Without taking sides on this issue, we can at least draw attention to a good point made by Lewis ([1973], p. 27) that in any conversational setting in which it was appropriate to assert “A and C”, it would usually be inappropriate (for Gricean reasons) to assert “If it were that A it would be that C” without further ado—such as is provided in (13), (14), of 7.11. As Lewis writes: Our principal response will be not that the conditional is true or that it is false, but that it is mistaken and misleading because of its true antecedent. So it is, but that is not at issue. The false information conveyed by using a counterfactual construction with a true antecedent eclipses the falsity or truth of the conditional itself.

Two adjustments suggests themselves to this diagnosis. Firstly, the eclipsing effect spoken of results not from the truth of the antecedent, but from the fact that the antecedent is accepted as true. Secondly, some role should be attached to the fact that this is also the status of the consequent C, when we imagine someone inferring from A ∧ C to A → C, since there is less resistance to the Modus Ponens inference from A → C and A to the conclusion C than to (Cent.) and the joint acceptance of the premisses here differs from the case of accepting a premiss of (Cent.) only in that, prior to drawing the conclusion, the consequent is not explicitly accepted. Lewis [1973], p. 27f., can be consulted for Lewis’s way of arguing informally for the validity of (Cent.) in the face of the above difficulty for the direct approach of showing that people do in fact reason from its premiss to its conclusion, and Bennett [1974] for some adverse commentary on the course this

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argument takes. We will quote instead from Harrison [1968], who considers a case in which Tom and Mary are visiting their grandmother Jane: Just before they arrive Tom says to Mary ‘If she had known we were coming, I’ll bet she would have baked a cake’. When they eventually do arrive, they discover that Jane did know that they were coming, and had in fact baked a cake. Tom (. . . ) will probably regard the fact that she both did know they were coming, and did bake a cake, as showing conclusively that he was right in what he said. What better evidence, after all, could he have that baking a cake is what she would have done, had she known that they were coming, than that when she did know they were coming, that is just what she did? Harrison [1968], p. 372.

A potentially confusing feature of this example is that in the situation as described, one may attach some weight to the hypothesis that Jane baked a cake because she knew Tom and Mary were coming; Harrison makes it clear that no such weight is to be attached in writing (p. 373) of the joint truth of its antecedent and consequent as sufficient to establish a subjunctive conditional (called by Harrison an ‘unfulfilled’ conditional – somewhat infelicitously, given this very point). Now, in the above example, Tom retrospectively endorses what he had said with the words “If she had known we were coming, she would have baked a cake”; it is not that he would now – after learning of the truth of the antecedent and consequent – be happy to repeat them. As Harrison puts it (p. 373): If Tom had known that Jane knew that he and Mary were coming and had baked a cake, and had said to Mary ‘If she had known we were coming, she would have baked a cake’, he would be generally be using language in such a way as to tend to cause Mary to infer that he, Tom, thought that Jane did not know they were coming and had not baked a cake. (. . . ) If he does not himself believe both these things, he is guilty of a breach of a linguistic rule, a rule to the effect that we should say ‘If X had been the case, Y would have been the case’ only when we believe that X and Y are not the case, except when it is sufficiently clear from the context that we do not believe this for us to use the unfulfilled conditional without misleading anybody.

Harrison explains that the exemption here covers such contexts as are in the background of (13) and (14) from 7.11, as well as noting the possibility of putting “still” in the consequent to cancel any suggestion that the speaker believes it to be false. Those who have rejected (Cent.) have, as Harrison puts it (p. 374), “confused saying that a speaker should not, unless the context permits it, use an unfulfilled conditional statement if he supposes both the antecedent and consequent to be true, with the view that the unfulfilled conditional statement is false, if both the antecedent and the consequent are true”. Recalling that while Lewis rejects Stalnaker’s uniqueness assumption and therewith the Law of Conditional Excluded Middle, we pause to note (following Sobel [1970], p. 443) that endorsing (Centering), and therewith the rule (Cent.), a weaker version of Conditional Excluded Middle may be recovered, namely: A (A → C) ∨ (A → ¬C).

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For, assuming A is true at a world, either C is or ¬C is, so that we have either A ∧ C or else A ∧ ¬C true there, from which the disjuncts on the right follow respectively, by (Cent.). We can proceed further in this vein, since rather than assuming that A is true at the world in question, we can consider both the possibility that it is, and that it is not, so that we have the following sequentschema valid under (Centering): (A → C) ∨ (¬A → C) ∨ (A → ¬C) ∨ (¬A → ¬C) affirming, for arbitrary A and C, a strong kind of failure for A and C to be ‘counterfactually independent’. Exercise 7.17.2 Suppose we strengthen the condition of (Centering) by adding a stipulation as to what happens if A is not true at a world: {x} if M |=x A + RA (x) = (Centering) ∅ if M |=x A Describe the logic of → which results when (Centering)+ is imposed as a condition on variably strict models. Remark 7.17.3 For the everyday notion of similarity, either relativized to a given respect of similarity, or ‘overall’ similarity (in which the various respects are somehow weighted and combined), there is the possibility that an object may fail to have a uniquely most similar object to it satisfying some condition not just because there are several maximally similar candidates vying for this status, but because there may instead be, for each candidate, a more similar candidate, without limit. Lewis in fact favours a semantics for counterfactuals which makes allowance for this phenomenon when the objects in question are worlds, and the condition in question is that of verifying the antecedent of such a conditional. Thus, in one formulation, to avoid this (as he calls it) Limit Assumption, Lewis regards A → C as true at w just in case some A-verifying world at which C is true is more similar to w than any A-verifying world at which C is false. Strictly speaking, we should disjoin with this truth condition: or there is no A-verifying world sufficiently similar to w for such comparisons to be made. This latter clause is to allow for counterfactuals whose antecedents are, from w’s perspective, impossible. A more elegant formulation, not requiring the Limit Assumption, and also neutral w.r.t. the question (to which we return towards the end of the present subsection) of whether worlds must be comparable in respect of their similarity to a given world, may be found in Lewis [1981], p. 230. Omitting a qualification corresponding to the italicized phrase above, this reads: A → C is true at w iff for any A-verifying world x, there is some A-verifying world y at least as similar to w as x such that for any A-verifying world z which is at least as similar to w as y is, C is true at z. Even with the omission, this condition is too complicated for us to work with here. Having now finally corrected our earlier oversimplified account of Lewis’s semantics, we shall proceed as before with the Limit Assumption (and hence the semantics to which we are accustomed, using variably strict models) in force.

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As Lewis remarks, this makes no difference to the sentential logic (in Fmla or Set-Fmla) of counterfactuals. For readers wishing to follow up some of the literature on the Limit Assumption, we list the following references: Chapter 1 of Pollock [1976], Herzberger [1979], Stalnaker [1981], Lewis [1981], Warmbr¯od [1982]. We turn now from (Cent.) to some further principles, this time (like the Law of Conditional Excluded Middle) involving connectives not available in 7.15, and more specifically, involving disjunction. The first is a subjunctive conditional analogue of the principle traditionally called (Constructive Dilemma) A → C, B → C (A ∨ B) → C which can be thought of as a horizontalized form (for Set-Fmla) of (∨E), and is provable in any Set-Fmla logic closed under that rule as well as (→I) and (→E); accordingly, we call our schema ‘Subjunctive’ Dilemma: (Subj. Dilemma) A → C, B → C (A ∨ B) → C. Part (ii) of the following Exercise is only for those who have read 7.16. Exercise 7.17.4 (i) Using the techniques of 7.15, but, for the completeness proof, making sure to take deductively closed prime sets of formulas (since the language now contains ∧, ∨ and, as we are now writing ⇒, → ), prove the following analogue of 7.15.8 (p. 995): A sequent is provable using the structural rules, (∨I), (∨E), (∧I), (∧E), Prefixing and (Subj. Dilemma) iff it holds in every variably strict model satisfying the condition: RA∨B ⊆ (RA ∪ RB ). (ii ) Add to the structural rules and the rules (⇒) from 7.16 the following ‘Suppositional’ version of (∨E): Γ[A] C

Δ[B] C

Γ, Δ[A ∨ B] C and give a proof of (Subj. Dilemma)—using either the “⇒” or the “→ ” notation, as preferred—from the resulting basis. The condition on R in (i) here is not very edifying, until one imposes the (maximal) similarity interpretation on the relations RA , which makes this condition mandatory—and therefore validates the schema (Subj. Dilemma)—since any world qualifying as a closest A ∨ B-verifying world must be either a closest Averifying world or else a closest B-verifying world. (Subj. Dilemma) itself is an acceptable principle for the same reason, then, that it follows from the hypothesis that all the tallest Armenian children in the school (‘A’) like chocolate (‘C’) and all the tallest Bulgarian children in the school (‘B’) like chocolate, that all the tallest from amongst the school’s children who are either Armenian or Bulgarian like chocolate. Suppose conversely, to stay with this analogy for a moment, that all the tallest Armenian-or-Bulgarian children like chocolate. Does it follow from this

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that the tallest Armenian children like chocolate and the tallest Bulgarian children like chocolate? No: for perhaps all the tallest Armenian-or-Bulgarian children are in fact Bulgarian children, and though all of these like chocolate, the tallest Armenian children (all shorter, ex hypothesi, than these Bulgarians) do not like chocolate. Likewise, from the fact that all the closest(-to-a-givenw) (A ∨ B)-verifying worlds verify C, it does not follow that all the closest A-verifying worlds and also all the closest B-verifying worlds, verify C. Thus (Converse Subj. Dilemma) (A ∨ B) → C (A → C) ∧ (B → C) is not a valid schema on the similarity interpretation of variable accessibility. With the “∧” on the right replaced by “∨”, notice, we do have a valid schema (to which we shall return at the end of this subsection). However, leaving the “∨” as it is, numerous writers have claimed that the informal notion of validity for subjunctive conditional inferences—never mind which sequents hold in all variably strict models, subjected to such additional constraints as the similarity interpretation leads to—is such that not only (Subj. Dilemma) but also (Converse Subj. Dilemma) represent valid inferences: (A ∨ B) → C ought to come out logically equivalent to (A → C) ∧ (B → C). If so, that would suggest serious trouble for the similarity interpretation. The evidence for this claim is arrived at by contemplating such pairs as (1) and (2) below: (1)

If Paul had travelled by train or by plane, he would have arrived on time.

(2)

If Paul had travelled by train he would have arrived on time, and if he had travelled by plane he would have arrived on time.

The idea is that (1) and (2) are just two ways, a short-winded and a long-winded way, of saying the same thing. In view of the above remarks, the controversial implication—that which appears to be an instance of the schema (Converse Subj. Dilemma)—is that from (1) to (2). We have here the problem of disjunctive antecedents, discussed at some length à propos of disjunction in 6.14 (‘Conjunctive-seeming Occurrences of “Or” ’: p. 799). Why is this a ‘problem’ ? Of course, it is a problem for the similarity semantics for subjunctive conditionals, but one might just conclude: so much the worse for that (style of) semantics. This would be a very superficial reaction, however. There is already a problem for the consistency of the intuitions which issue in such verdicts as that (1) and (2) are equivalent. For example, suppose you have spent a year taking some drug which you were authoritatively told would make it all but impossible to conceive a male child; you believe: (3)

If we had had a child, it would have been a girl.

But you do not believe: (4)

If we had had a boy, it would have been a girl.

Yet having a child just is having either a boy or a girl, so (4) amounts to (5)

If we had had a boy or a girl, it would have been a girl.

And by (Converse Subj. Dilemma), (5) implies (4). So there is something wrong already at the level of intuitive judgments as to the validity of counterfactual inferences, before any formal semantic apparatus is brought to bear.

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One might object that it is not permissible to replace “a child” in (4), with “a boy or a girl”, perhaps on the grounds that – when the antecedent in the latter case is made fully explicit as a disjunctive sentence – such a replacement is justified by assuming congruentiality in antecedent position, and the only reason so far given (in 7.15) for this is that it is forced by the Lewis–Stalnaker semantics, from which the present considerations were advertised as being independent. But in fact we do not need to argue in this way, from (3) to (5): there is surely a natural interpretation of (5), with which we could begin, according to which it accurately reflects your beliefs. (Compare the example, from McKay and van Inwagen [1977], “If Spain had fought on the side of the Axis powers or on the side of the allied powers in World War II, it would have been on the side of the Axis powers”.) The (alleged) connection between congruentiality in antecedent position and (Converse Subj. Dilemma)—the latter principle usually going by the name ‘Simplification of Disjunctive Antecedents’ in the literature—has encouraged some to seek a non-congruential logic of subjunctive conditionals. (Nute [1978], [1980].) We saw in the derivation (viii)–(x) in 6.14, that with (Converse Subj. Dilemma), congruentiality leads to a derivation of (Str. Ant.), so that such a reaction represents an alternative route for those who reject Strengthening from the route implicit in Lewis [1973]: instead of retaining (Ant. Cong.) and rejecting (Converse Subj. Dilemma), one adopts the reverse pattern of retention and rejection. The considerations from the end of the preceding paragraph, however, suggest that this is a mistake. Unless, of course, one were to claim that subjunctive conditionals were ambiguous, having one reading on which they obeyed (Converse Subj. Dilemma) but not (Ant. Cong.), and another one on which they obeyed (Ant. Cong.) but not (Converse Subj. Dilemma). (All of this on the assumption—to which we return at the and of the present subsection—that (Str. Ant.) is to be rejected.) But there is an obvious problem for such a suggestion, namely: what becomes of the alleged ambiguity in the case of subjunctive conditionals which do not have disjunctive antecedents? If an ambiguity thesis is to be offered for counterfactuals with disjunctive antecedents, it had better claim an ambiguity resulting from the interaction of ∨ with → , rather than from → alone (a structural rather than a lexical ambiguity, as it was put in 6.14). To this end, consider the possibility (from Åqvist [1973]) of introducing a singulary connective * with *A having the rough informal reading of: “other things being equal, A” or “ceteris paribus, A”. More precisely, we want to have *A come out true at those worlds which belong to RA (w), since we are thinking of this as the set of worlds which differ from w as little as is called for to verify A. This means that we have to employ a doubly indexed or ‘two-dimensional’ semantics for our language. See 3.31.9 (p. 489), and, for the present application, the discussion in §2.8 of Lewis [1973] (already mentioned there). Following the latter discussion, we recall that the world playing the role of w in the above explanation is called the reference world, and the world at which we are interested in evaluating formulas for truth by reference to that world, the world of evaluation, and that we use the notation “|=w x A” to indicate the truth of A when w is taken as the reference world and x the world of evaluation. Thus, where M = (W, R, V ) is a variably strict model, we put: M |=w x *A iff x ∈ RA (w)

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and, in order to decompose → further, a necessity operator with a clause in the truth-definition that combines universal quantification with promoting the lower (‘evaluation world’) index into the upper (‘reference world’) index position: x M |=w x A iff for all y ∈ W, M |=y A.

(The present exposition follows the adaptation of Åqvist’s work in Humberstone [1978] – also summarised in §5 of Humberstone [2004a]; this operator has a normal modal logic, quite interesting in its own right as well as for various other applications. See 3.31.9, p. 489, and the Digression on p. 1260, and references there cited.) The clauses for the boolean connectives and the propositional variables are as in the one-dimensional case. (Thus, in particular, though this is not the only way of proceeding—cf. Segerberg [1973]—we take pi to be true at (lower index) x with reference to (upper index) w just in case x ∈ V (pi ): the reference world is irrelevant.) Putting these ingredients together, let us consider a formula of the form (*A → C). We have x M |=w x (*A → C) iff for all y ∈ W, M |=y *A → C . . . iff for all y ∈ W such that M |=xy *A, M |=xy C . . . iff for all y ∈ W such that y ∈ RA (x), M |=xy C.

Note that the truth-condition we end up with does not depend on the original upper index w, and that, assuming C does not itself contain *, the upper index “x” in “M |=xy C” is also immaterial. Suppressing these, we have that our formula (*A → C) is true (as evaluated) at x iff C is true at every y ∈ RA (x); but these are precisely the truth-conditions of A → C (or A ⇒ C, in the notation of 7.15). The assumption that C does not contain * means that C cannot have a similarly eliminated occurrence of “→ ” within it, but we shall not here concern ourselves with the general case, since we wish to bring this apparatus to bear on the issue of disjunctive antecedents. To translate a disjunctively antecedented counterfactual (A ∨ B) → C into the language sketched above we have: (6) (*(A ∨ B) → C) which represents it as a certain strict conditional whose antecedent is not disjunctive (i.e., does not have ∨ as main connective), though there is of course a strict conditional in the vicinity which does have a disjunctive antecedent, namely (7) ((*A ∨ *B) → C). Note that (7) is, without the benefit of any further assumptions (in the form of conditions on variably strict models) equivalent to (8) (*A → C) ∧ (*B → C). It is perhaps not surprising, then, that counterfactuals in English with disjunctive antecedents should have two readings, for neither of which is there any need to question congruentiality, only one of which supports a (natural language) version of the inference (Converse Subj. Dilemma). An interesting diagnostic test

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for those of the latter type (here formalized as (7)) has been suggested by van Inwagen, as reported in Hilpinen [1981b]: they are expressible in the form (9)

If A or if B, then C.

Thus, for instance, in the sex-selecting drug example above, you might say (to flesh out (4 ) somewhat): (10)

If we had had a boy or we had had a girl, it would have been a girl.

But you would not say: (11)

If we had had a boy or if we had had a girl, it would have been a girl.

because it is not true that if you had had a boy, it would have been a girl. Digression. The informal rendering of *A as “other things being equal, A” was mentioned above. Burks and Copi [1950] anticipate this idea, but take a false step as they say (p. 222) ‘Abbreviate “other conditions remain equal” by “E”. . . ’, this “E” then being treated as a formula itself figuring as a conjunct in a conjunction, the antecedent of a conditional, etc. The mistake is in reading too much into the fact that “Other conditions remain equal” is indeed an English sentence, and though hardly a fit vehicle for assertion outright – a fact well accounted for by the two-dimensional semantic treatment – perfectly suitable as a subordinate clause. (For the background to Burks and Copi [1950], and other related treatments of the same puzzle, see §5.2 of Moktefi [2008].) Geach ([1991], p. 267f.) puts the point well, in a discussion of what he calls the ceteris paribus reading of “if” (that is, of conditionals for which (Str. Ant.) is not acceptable): From: If p, then other things being equal, q we cannot infer: If both r and p, then, other things being equal, q. For, one might say, the ‘other things’ that are supposed ‘equal’ on the supposition that p will not be the same as the ‘other things’ that have to be supposed ‘equal’ on the supposition that (both r and p).

The issue is essentially the same as that presented by the following example. Suppose that a and b are two doctors, and in fact the only two, who treated a patient c. Then both the following are true: a and another doctor treated c. b treated c. But the conclusion which would appear to follow from them if we disregard the context-dependent interpretation of ‘(an)other’: a, b, and another doctor treated c. is false. (Note that Geach’s argument would have little ad hominem force against Burks and Copi, however, since, as noted in 7.15, Burks is already committed to a logic of subjunctive conditionals in which antecedents are subject to strengthening.) End of Digression.

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Hilpinen’s own reaction (mentioned in 6.14) to such data has been to explore the possibility of a formal treatment somewhat closer to English than that derived from Åqvist, and to distinguish two classes of sentences/formulas, called statements and assumptions. The statements are (in our formal language) propositional variables and boolean combinations thereof, while if A is a statement, the singulary operator If yields the assumption If A. (A cruder version of this approach view, according to which If is treated just as a singulary sentence connective, may be found in Sylvan [1992a].) Some boolean compounding of assumptions is also allowed. In particular, where α and β are assumptions, so are α ∧ β and α ∨ β. (Hilpinen does not allow ¬α, or α → β as assumptions). Finally, when α is an assumption, and B is a statement, α then B is a statement. So we have here a formal language which departs considerably from what (1.11) generally occupies our attention under that label, designed to simulate more closely certain features of conditional constructions in natural language. Instead of building up the conditional A → B (or A ⇒ B) in one fell swoop from statements A and B, Hilpinen’s formation rules do so in two stages: first, from the statement A, we form the assumption If A, and next, from this assumption and the statement B, we construct the conditional statement If A, then B. (In the Åqvist language, we in effect build up to subjunctive conditionals in three stages: first applying * to the antecedent, next, linking the result to the consequent with →, and finally applying to the result.) But since some measure of boolean compounding is allowed to intervene between these two stages, we can also form analogues of (11): (12)

(If A ∨ If B) then C

and distinguish them from the analogues of (10): (13) If (A ∨ B) then C. Here (12) and (13) correspond to (7) and (6) respectively, once the semantics is spelt out, to which task we now turn. An advantage of Hilpinen’s language over Åqvist’s is that we can exploit the distinction between statements and assumptions to confine the two-dimensionality to the latter formulas. Given a variably strict model M = (W, R, V ), we can provide the usual one-dimensional truth-definition for propositional variables and boolean connectives; thus, writing !A! for the set of elements of W at which the statement A is true in M, we have the usual effect, listed here just for ∧ and ∨ (14) !A ∧ B! = !A! ∩ !B!

!A ∨ B! = !A! ∪ !B!

while for assumptions α we write !α!w for the two-dimensional notion, the proposition expressed by α from the perspective of reference world w ∈ W : this is what in the double-indexing notation would be written as {x ∈ W | M |=w x α}. Thus for the statement-to-assumption operator If, we put: (15) !If A!w = RA (w) Finally, since then takes an assumption and a statement to a statement, it only requires a one-dimensional clause:


1021

!α then B! = {w : !α!w ⊆ !B!}

This gives the same results as we obtained using Åqvist’s * operator, but with arguably greater elegance, since it does not make for (statement-)formulas which do not participate in the representation of subjunctive conditionals (or anything else), such as **A, (*A →*B), etc. On the other hand, there is one thing we have omitted from the above account, which, when made explicit, signals a point in the other direction as far as such comparisons of elegance are concerned. What we did not mention was that since, as well as statements, assumptions may be compounded by ∧ and ∨, (14) above needs to be accompanied by a parallel stipulation for these formulas. We must say (17)

!α ∧ β!w = !α!w ∩ !β!w !α∨ β!w = !α!w ∪ !β!w

for all assumptions α, β, and all w ∈ W . No such duplication was needed for the Åqvist language. Exercise 7.17.5 (i) As remarked earlier, we derived (Str. Ant.) from (Converse Subj. Dilemma), with the aid of (Ant. Cong.) in 6.14; in fact we worked there in Fmla rather than Set-Fmla, but the alterations required are trivial. Show that, conversely, with the aid of (Ant. Cong.), and any tautologous sequents in the boolean connectives, (Converse Subj. Dilemma) is derivable from (Str. Ant.). (ii ) If the structural ambiguity discerned using either Åqvist’s or Hilpinen’s apparatus for counterfactuals with disjunctive antecedents is a genuine one, what becomes of counterfactuals with conjunctive antecedents? (For example, put ∧ for ∨ in (12), (13)) Are these ambiguous in English in a way which parallels the case of ∨? If not, why not? (Note: this matter is discussed in Hilpinen’s papers.) (iii) On Åqvist’s way of representing subjunctive conditionals, they emerge as a certain kind of strict conditional. But (Str. Ant.) is guaranteed for strict conditionals. Does this mean that subjunctive conditionals on Åqvist’s treatment admit (Str. Ant.)? (Explain why not.) Part (i) of this exercise shows how close the relationship is between strengthening the antecedent and simplifying counterfactuals with disjunctive antecedents in accordance with (Converse Subj. Dilemma). In fact, if we consider the case of either conjunct separately in the conjunctive rhs of that schema, say for definiteness the first conjunct, what we have: (A ∨ B) → C A → C, is something that deserves to be called strengthening the antecedent, A being (in general) inferentially stronger than A ∨ B, though we did not actually call it (Str. Ant.), since—to minimize confusion with the published literature—this label was reserved for the strengthening of a conjunct to a conjunction. Our official term for strengthening in this more general sense in 7.15 was (Suffixing)

AB B → C A → C

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reproduced here with “→ ” in place of “⇒”. This concludes our discussion of (Converse Subj. Dilemma), though for the convenience of readers wishing to pursue the matter more fully, we gather together here some references to the literature on the problem of disjunctive antecedents—including some works already cited above: Loewer [1976], McKay and van Inwagen [1977], Ellis, Jackson and Pargetter [1977], Lewis [1977], Humberstone [1978], Warmbr¯od [1981], Hilpinen [1981a], [1982]; the earliest discussions appeared in reviews of Lewis [1973], to which references may be found in note 4 of McKay and van Inwagen [1977]. A more recent assault on the problem may be found in Alonso-Ovalle [2004] and later papers by the same author (which an internet search will disclose). It was mentioned in 7.15 that two types of counterexample have been alleged to threaten the claim that subjunctive conditionals are congruential in antecedent position. One of these is the case of disjunctive antecedents, in which there was an apparent danger that a plausible principle (Converse Subj. Dilemma) would lead via congruentiality to an implausible principle (Str. Ant.); our reaction (following Hilpinen) to this line of reasoning was that congruentiality was not really implicated, and that counterfactuals with disjunctive antecedents seemed open to two interpretations, distinguishing between which rendered any restrictions on congruentiality unnecessary. It is to the second kind of potential counterexample to antecedent congruentiality that we now turn. In fact, the following cases are not literally potential counterexamples to congruentiality since the replacement of one antecedent by another which yields non-equivalent conditionals is not itself a replacement licensed by provable equivalence in the (sentential) logic of counterfactuals. Nevertheless, these cases do involve a failure of the substitutivity of natural language sentences which are logically equivalent in the informal sense of having the same truth-conditions. The cases in question were raised in the earliest papers ‘in the modern era’ to have addressed the subject of counterfactuals, namely Chisholm [1946] and Chapter 1 of Goodman [1965] (originally delivered as a lecture in 1946, and first published the following year). We present them as Examples 7.17.6(i) Each (says Goodman, p. 15) of the following counterfactuals would normally be accepted: (a) If New York City were in Georgia, then New York City would be in the South. (b) If Georgia included New York City, then Georgia would not be entirely in the South. (ii ) In contexts differing as to which of our beliefs we are supposing to be contrary-to-fact (says Chisholm, p. 303), but in both of which we take it as known that Apollo is immortal and all men are mortal, we would be prepared to accept either of the following (depending on the context): (c)

If Apollo were a man, he would be mortal.

(d)

If Apollo were a man, then at least one man would be immortal.

At first sight, the contrast under (ii) may not seem to the point here, since we have the same antecedent, rather than different but equivalent antecedents. The point of including (ii) will become clear in due course.

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What Goodman says about (a) and (b) of (i) is that the difference in phrasing, despite their antecedents being ‘logically indistinguishable’ directs us to different supplementary suppositions, which, more fully spelt out in the former case would be expressed by “If New York City were in Georgia, and the boundaries of Georgia remained unchanged,. . . ”, but which, for the latter case, amount to “If Georgia included New York City, and the boundaries of New York City remained unchanged,. . . ”. In these elaborations of the originals, notice, we could interchange the original equivalent antecedents, since the work done by the difference between them has now been done by making that difference explicit with the second conjunct. The talk of what we are invited to suppose “unchanged” fits in well with the similarity semantics for counterfactuals: these are respects of similarity which weigh more heavily than others in the determination of overall similarity. Compare the discussion in Lewis [1973], p. 67, of a pair of counterfactuals due to Quine: (e)

If Caesar had been in command in Korea he would have used the atom bomb.

(f )

If Caesar had been in command in Korea he would have used catapults.

Lewis writes that we may call on context (. . . ) to resolve part of the vagueness of comparative similarity in a way favorable to the truth of one counterfactual or the other. In one context, we may attach great importance to similarities and differences in respect of Caesar’s character and in respect of regularities concerning the knowledge of weapons common to commanders in Korea. In another context we may attach less importance to these similarities and differences, and more importance to similarities and differences in respect of Caesar’s own knowledge of weapons.

Thus the first type of context favours counting (e) rather than (f ) true, while for the second, it is the other way around. The difference in wording between Goodman’s (a) and (b) can thus be seen as a way of indicating one rather than another context and directing the hearer/reader to a suitable resolution of the corresponding vagueness in this case. The material added as additional conjuncts “and the boundaries of Georgia [New York City] remained unchanged” makes this (more) explicit, but it is in general a hopeless task to try and avoid context-dependence by linguistic rephrasing – a point well made (for the case of conditionals) in Bennett [1974] and Barwise [1986]. (This is the ‘fleshing out strategy’ of the latter paper, described there as wrong-headed and unworkable. A somewhat more specialized version of the above diagnosis of Example 7.17.6(i) is given at p. 43 of Lewis [1973], invoking Lewis’s theory of counterparts – not discussed here.) Like Quine’s (e) and (f ), Chisholm’s examples (7.17.6(ii)) are subjunctive conditionals with the same antecedent. If that pair, (c) and (d), along with Quine’s pair, differ from (a) and (b) only in that no overt linguistic clue is present to encourage resolution of the vagueness of overall similarity one way rather than another, then we should conclude that there was no informal analogue of the problem of non-congruentiality in antecedent position: for this was supposed to be the problem of change of truth-value in counterfactuals upon substitution of

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one for another of two antecedents which could not themselves differ in truthvalue. The ‘problem’ – we see – is already there in the absence of any such replacement, and so is not a substitutivity problem after all. In 7.15, we noted that the similarity construal of the relations RA forced us to acknowledge the rule (rewritten here with “→ ” for “⇒”) (Ant. Cong.)

AB

BA

B → C A → C

as preserving the property of holding in a (variably strict) model. For the holding of the premiss-sequents in such a model guarantees that !A! = !B!, so for any world, the closest worlds in !B! are precisely the closest worlds in !A!: if the former all lie inside !C!, then so do the latter. We have seen two kinds of considerations which might be held to tell against this rule, one involving counterfactuals (in English) with disjunctive antecedents, and another involving subtle changes of wording (Goodman’s Georgia example etc.) which block the substitutivity of logically equivalent antecedents. But in both cases, we found that a compelling case for the non-congruentiality of subjunctive conditionals had not after all been made out. Proceeding on the basis that (Ant. Cong.) is quite acceptable, we now note that a stronger principle—from which that rule can be derived (given the uncontroversial Prefixing and A → A: this derivation is left to the reader)—can also be justified on the basis of the maximal similarity construal of RA , namely the zero-premiss(-sequent) rule: (Ant. Equiv.)

A → B, B → A, A → C B → C

For suppose that A → B and B → A are both true at w ∈ W , in some model (W, R, V ). We can put this by saying that A and B are counterfactually (or ‘subjunctively’) equivalent at w. Such an equivalence means that RA (w) ⊆ !B! and also RB (w) ⊆ !A!, which are invited to think of as saying that the most similar (to w) A-verifying worlds verify B, and vice versa. Think instead for a moment about geographical proximity amongst cities, and suppose, from the perspective of a particular place, that (1) all the closest cities (to the place in question) with art galleries have zoos, and also that (2) all the closest cities with zoos have art galleries. Then it follows that the closest cities with art galleries are precisely the same cities as the closest cities with zoos. For suppose otherwise, and that there is, say, some closest city (i.e., some city either closer than all others, or tied for maximal closeness with some others) with an art gallery which is not a closest city with a zoo. The city in question certainly has a zoo, by (1). So if it is to fail to be a ‘closest city with a zoo’, that must be because some other city which is even closer has a zoo. But this other city then has an art gallery, by (2), meaning that the first city was not after all the closest city with an art gallery. As with cities, art galleries, and zoos, so with worlds, A, and B: if RA (w) ⊆ !B! and RB (w) ⊆ !A!, then we must have RA (w) = RB (w), as long as R assigns to each formula the function mapping w to the set of worlds verifying that formula than which no worlds outside the set are ‘closer’: and inspection of the reasoning of the previous paragraph reveals this to be the case as long as ‘closer’ is taken to satisfy various reasonable conditions (which will be spelt

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out as we proceed, below, to rework the present semantics in terms of a ternary relation, S, of comparative similarity between worlds). The first and second listed formulas on the left of any instance of the (Ant. Equiv.) schema A → B, B → A, A → C B → C thus guarantee that for any world w at which they are true, RA (w) = RB (w). So if the third formula on the left is also true, meaning that RA (w) ⊆ !C!, the formula on the right must also be true, since in that case RB (w) ⊆ !C!. Counterfactually equivalent antecedents, in other words, are intersubstitutable. It may occur to the reader that since all we needed to infer that RB (w) ⊆ !C! from RA (w) ⊆ !C! and RA (w) = RB (w) was the ⊇-direction of this last equality, the hypothesis that A and B are counterfactually equivalent (at w) was redundantly strong. And indeed, although none of the three left-hand formulas schematically represented in (Ant. Equiv.) can be omitted, some ‘weakening’ is possible, namely to: A → B, B → A, A → C B → C. The left-hand formulas now involve us, for checking whether a sequent of this form holds at a point w in (W, R, V ), in supposing, respectively, that (i) RA (w)∩ !B! = ∅, (ii) RB (w) ⊆ !A!, and (iii) RA (w) ⊆ !C!, and from (i) and (ii), on the maximal similarity construal of the R-relations, imply the unidirectional inclusion (iv) RB (w) ⊆ RA (w). The world-parameter makes this implication look more complicated than it is, but since that parameter is fixed throughout as w, we can ignore it and explain the implication (as in the earlier examples of the Armenian and Bulgarians, and of the art galleries and zoos) by parallel reasoning suppressing reference to it. Instead of “closest to w”, we will have simply “tallest”, and think of “A” and “B” as representing the classes of Americans and basketball players, respectively. Thus we have: (i) (ii )

Some of the tallest Americans are basketball players. All of the tallest basketball players are Americans.

and wish to infer: (iv )

All of the tallest basketball players are amongst the tallest Americans.

So suppose we have a tallest basketball player b. We must show that b is also a tallest American, to establish (iv) . Well, by (ii) , we know that b is an American. So if b is not amongst the tallest Americans, that must be because there is an American a who is taller than b. In fact, we may as well take a to be one of the tallest Americans who are basketball players, since, b not being amongst the tallest Americans, these tallest basketball-playing Americans—and (i) tells us that this class is not empty—are all taller than b. Now we have a basketball player a who is taller than b, contradicting the choice of b as a tallest basketball player. Therefore b must be amongst the tallest Americans after all, establishing (iv) . Returning to (i) and (ii), then, we see that these have (iv) as a consequence, which together with (iii), gives RB (w) ⊆ !C!, so the right-hand formula of our sequent is true at w in the model in question. Remark 7.17.7 The above talk of ‘weakening’ A → C to A → C is not meant to suggest that the sequent A → C A → C holds in every

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variably strict model, even with additional constraints suggested by the maximal similarity construal in force. But this is almost correct: if the would-counterfactual is true at a point and the might-counterfactual is not, then A → C and A → ¬C are both true at that point, and so (by the rule Prefixing) A → ⊥ is true there: RA accordingly assigns to the point in question the set ∅, which is the closest we can come with the present apparatus to saying that A is ‘impossible’ (at the given point). An extensive discussion of modal notions emerging out of the semantics of counterfactuals may be found in Lewis [1973], under the references in the index to “modalities, outer” and “modalities, inner”. Setting aside arguments by analogy from proximity of cities, tallness of basketball players, and the like, let us address more explicitly what we have been calling the maximal similarity construal of the relations assigned to formulas by the R component of a variably strict model (W, R, V ). Let us suppose available a ternary relation S ⊆ W × W × W , thinking of Swxy as saying that x is at least as similar to w as y is. Note that we always compare a pair of worlds— here x and y—in terms of their similarity to some given world—here w. (We are never, that is, involved in having to consider whether x is as similar to w as y is to some possible fourth world z.) To emphasize the fact that x and y are being compared in their similarity to w, we will write “Sw xy” from now on, rather than “Swxy”. For a fixed w, this gives a binary relation between x and y. A minimal assumption, for the intended reading of “S” is that the binary relation in question is always a preorder (a reflexive transitive relation). But what we have in mind in speaking of the maximal similarity interpretation of the variably strict semantics is that the relations RA should arise from !A! and S by accordance with the definition given below when the binary relations Sw satisfy a further condition of ‘comparability’ or ‘linearity’, namely that for any w, we have, for all x, y ∈ W , either Sw xy or Sw yx. That is, that Sw should be a total preorder. (Note that this last condition implies reflexivity, by setting x = y; however, it is useful to isolate the preorder idea first, since not all accept the stronger condition, as we shall note below.) The promised definition of RA in terms of S is obtained by putting X= !A! in the following general definition of RX (w)—the set of worlds then in X ⊆ W which are closest to w in a model conceived of as having the form (W, S, V ): [Def. R]

RX (w) = {x ∈ X | ∀y ∈ X Sw xy}.

That is, we have: RA (w) = {x ∈ !A! : ∀y ∈ !A! Sw xy} or, in more explicitly relational form: wR A x iff both x ∈ !A! and for all y ∈ W if y ∈ !A! then S w xy. Whereas the minimal conditional logic for the variably strict semantics of 7.15 (evaluating conditionals by means of [⇒], though we are now writing ⇒ as → ) involved only Prefixing as a principle governing conditionals—as 7.15.7 informs us—when we use the same procedure for evaluating conditionals, but taking R as defined in terms of some ternary relation S via [Def. R], we automatically

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validate also the ‘identity’ schema A → A, which previously required the special condition (id )

If xR A y then M |=y A.

This follows directly from the form of [Def. R], even without imposing any special conditions on S, such as those mentioned above. The same goes, as the reader may verify, for (Subj. Dilemma) A → C, B → C (A ∨ B) → C as well as for the rule (Ant. Cong.)

AB

BA

B → C A → C

With just these principles, we can go a long way towards the basic logic for subjunctive conditionals described in Burgess [1981b], there proved to be sound and complete w.r.t. the class of all models based on a ternary relation S (as above) whose induced binary relations Sw are reflexive and transitive; a slight complication here is caused by Burgess’s use of a more complicated clause than ours in the definition of truth, namely, that inspired by Lewis in a desire to avoid the Limit Assumption (see the end of 7.17.3 above), which we will now impose on our models in the following form, understood as requiring for all X ⊆ W : (Strong Limit Assumption) If X = ∅ then RX (w) = ∅, for all w ∈ W . In particular, then, where X = !A!, we have: If !A! = ∅ then R A (w) = ∅, for all w ∈ W . In terms of the ternary relation S, this says that if there are some A-verifying points in the model, then for any w ∈ W there are A-verifying points x such that for all A-verifying y ∈ W , Sw xy. Below, when we consider relaxing the connectedness (or ‘comparability’) condition that for any given w ∈ W , Sw xy or Sw yx, we shall have occasion to give a weak version of the Limit Assumption which does not imply connectedness; the Strong Limit Assumption clearly does imply connectedness because we can choose {x, y} as X, and then since [Def. R] above implies that RX (w) ⊆ X, R{x,y} must contain either x or y or both, implying respectively that Sw xy or Sw yx or both. (The need for caution in formulating various principles to retain neutrality w.r.t. the choice of total and arbitrary preorders is stressed in Lewis [1981].) Burgess’s syntactic description of the logic (in Fmla) whose semantic characterization is as the logic determined by the class of all preordered models uses five axioms alongside (as rules of proof), Uniform Substitution, Detachment (Modus Ponens) and a congruentiality rule, together with any axioms sufficient for all tautologies. We rewrite them here, numbered (A0)–(A4), as sequent-schemata in Set-Fmla. (A5) is a streamlining of something Burgess supplements his preorder axioms with to obtain a complete logic for the total preorder case; in the present notation, Burgess’s supplementary principle (called (B5) on p. 83 of Burgess [1981b]) emerges as the schema:

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((A ∨ B) ∨ C) → ¬A ((A ∨ B) → ¬A) ∨ ((A ∨ C) → ¬A). This follows from our (A5) by (Ant. Cong.) after putting “¬A” for “C”, “A ∨ B” for “A”, and “A ∨ C” for “C”. Note that (A5) was mentioned in passing in our discussion of (Converse Subj. Dilemma) above, from which (invalid) principle it differs in having a disjunction instead of a conjunction on the right. (A0) A → A (A1)

A → B, A → C A → (B ∧ C)

(A2)

A → (B ∧ C) A → B

(A3)

A → B, A → C (A ∧ B) → C

(A4)

A → C, B → C (A ∨ B) → C

(A5)

(A ∨ B) → C (A → C) ∨ (B → C).

Of these (A0) is the identity schema already much mentioned, while (A1) and (A2) are consequences of ∧-classicality by the rule Prefixing, and (A4) is of course (Subj. Dilemma), leaving only (A3) and (A5) to be accounted for. Although we are not concerned here with proofs of completeness, the soundnesspertinent fact that instances of (A3) must hold in any model subject to the conditions indicated above is worth demonstrating, since this and (A5), which we ask the reader to review in Exercise 7.17.8(iv) below, are the only schemata on the list for which appeal to those conditions is required. To this end, we should first check that models (W, S, V ) satisfying the preorder conditions and the Limit Assumption must also satisfy the condition (*) below, for all formulas A and B, and all w ∈ W : (*) RA (w) ⊆ !B! implies R A∧B (w) ⊆ RA (w) By way of proof, suppose (1) RA (w) ⊆ !B! and (2) x0 ∈ RA∧B (w), with a view to showing that these imply x0 ∈ RA (w). By [Def. R], (2) is equivalent to the conjunction of (2a): x0 ∈ !A ∧ B!, with (2b): for all y ∈ !A ∧ B!, Sw x0 y. Since (2a) implies that !A! = ∅, we have, by the Strong Limit Assumption, that there is some x1 ∈ W such that (3a)x1 ∈ !A! and (3b) for all y ∈ !A!, Sw x1 y. The conjunction of (3a) with (3b) amounts to the claim that x1 ∈ RA (w), so by (1), x1 ∈ !B!. Hence, by (3a), x1 ∈ !A ∧ B!, and so by (2b), Sw x0 x1 . Therefore, by the transitivity of the relation Sw and (3b), we have that for all y ∈ !A!, Sw x0 y, which is to say, since also x0 ∈ !A! (by (2a)), that x0 ∈ RA (w), as was to be shown. The task of bringing (*) to bear on the validity of (A4) is left for part (i) of the exercise below; here we pause to note that the result of reversing the inclusion in the consequent of (*) also follows from the transitivity of the relations Sw (together with [Def. R]). Call this condition (**): (**) RA (w) ⊆ !B! implies RA (w) ⊆ RA∧B (w). Again, this is left for the reader to prove as part of Exercise 7.17.8 below. In our discussion of what we called (Ant. Equiv.), we showed that instances of that schema were guaranteed to hold in any model satisfying the condition

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(***) RA (w) ⊆ !B! and R B (w) ⊆ !A! together imply R A (w) = RB (w) Now (***) follows from (*) and (**), since those two conditions yield as a consequence: RA (w) ⊆ !B! implies R A (w) = RA∧B (w) and also, putting “B” in place of “A”: RB (w) ⊆ !A! implies RB (w) = RB∧A (w). So: RA (w) ⊆ !B! and R B (w) ⊆ !A! imply R A (w) = RA∧B (w) = RB∧A (w) = RB (w). Exercise 7.17.8 (i) Show that all instances of (A3) hold in any model satisfying the condition (*) above. (ii ) Show that (**) is satisfied by all models (W, S, V ) in which the relations Sw (for w ∈ W ) are transitive. (iii) Show that in all models (W, S, V ) in which the relations Sw are transitive and the Strong Limit Assumption is satisfied, so also is the condition which we showed above sufficed to validate the ‘weakened’ (Ant. Equiv.) schema A → B, B → A, A → C B → C, namely, the condition that RA (w) ∩ !B! = ∅ and R B (w) ⊆ !A! imply R B (w) ⊆ RA (w). (iv ) Verify that every instance of (A5) holds in a model meeting the conditions listed in (iii). The strongest conditions we have imposed thus far (transitivity and the Strong Limit Assumption) on our S-based models may still fall short of qualifying them as incarnations of what we have been calling the maximal similarity construal of the relations RA earlier taken as the results of applying the functions R which figured as primitive in our variably strict models. In particular, part and parcel of this construal, it is plausibly urged that we should secure satisfaction of the condition (mp) which has played such a conspicuous role in our discussions since the start of 7.15, no less than of the converse principle (id ), noted above as a free gift in view of [Def. R]: (mp) If w ∈ !A! then wR A w. More controversially, there is the strengthening of (mp) to (Centering) to be considered: (Centering)

If w ∈ !A! then R A (w) = {w}.

Recall that these conditions guarantee the holding of A, A → C C (in the case of (mp)) and of that schema together with (Cent.): A ∧ C A → C (in the case of (Centering)).

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Adapting some terminology of Burgess [1981b], we baptize the following two conditions on (the frames of) our S-based models (W, S, V ): (Centrality 1 )

Sw wy for all w, y ∈ W .

(Centrality 2 )

If S w xw then x = w for all w, x ∈ W .

The first of these principles is very appealing given that we are reading “Sw xy” as “x is at least as similar to w as y is”, since a world should count as being at least as similar to itself as any world is; the second is slightly less incontrovertible, since it says that any numerical difference counts for some dissimilarity. When [Def. R] is employed, (Centrality1 ) gives us the condition (mp), since if w ∈ !A!, all that remains for w to belong to RA (w) is that Sw wy for all y ∈ !A!, and (Centrality1 ) says that we have this for all y ∈ W , never mind just y ∈ !A!. And (Centrality2 ) gives us that part of (Centering) which does not follow from (mp), namely: If w ∈ !A! then R A (w) ⊆ {w} since, by [Def. R], RA (w) contains precisely those points x ∈ !A! such that for all y ∈ !A!, Sw xy, and assuming w ∈ !A!, this implies that Sw xw, which the condition (Centrality2 ) tells us can only happen for x = w. So on the assumption in question, nothing other than w can belong to RA (w). We turn now to relaxing the condition of connectedness, to allow for the existence of w, x, y with neither Sw xy nor Sw yx : x and y are simply not comparable in respect of their similarity to w. As has already been remarked, this condition was not explicitly imposed on our models but entered via the Strong Limit Assumption, which will therefore have to be weakened. But there is also another change that is called for if incomparabilities are to be allowed, and that is to the definition of truth for subjunctive conditionals. We have been operating with the clause called [⇒] in 7.15, which for this subsection might better be called: [→ ]

M |=w A → B iff R A (w) ⊆ !B!

where, for our recent discussion, M is of the form (W, S, V ) and R is defined in terms of S à la [Def. R]. We should not just keep this clause intact if we are allowing that we may have x, y incomparable w.r.t. Sw because although there may be plenty of A-verifying worlds, if some of these are thus incomparable, RA (w) will be empty and the right-hand side of [→ ] will be vacuously satisfied. Instead of asking for all the closest-to-w A-verifying worlds to verify B, we should be asking of all the A-verifying worlds than which none are closer to w that they verify B. Let us write this, following the form of [→ ], as [→ ]+ M |=w A → B iff R + A (w) ⊆ !B! + (w) to contain precisely the points in !A! than which no Since we want RA points in !A! are more similar to w, the points x to be excluded are precisely those for which there exists y ∈ !A! such that Sw yx and not Sw xy. These x are points for which there is an A-verifying y which is at least as similar to w as x is, though x is not at least as similar to w as y is: thus they are more similar to

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w than x is. We sum this up, for the general case in which X ⊆ W need not be of the form !A!, as [Def. R+ ] wR + X x iff x ∈ X and for all y ∈ X, if S w yx then S w xy It is with this definition of R+ that [→ ]+ above is to understood, and it is + RX (w), so understood, rather than the original RX (w), to whose non-emptiness our revision of the Strong Limit Assumption speaks: (Weak Limit Assumption) If X = ∅ then R + X (w) = ∅, for all w ∈ W, X ⊆ W . For the case in which X = !A!, then, our new assumption demands that if there are A-verifying worlds in the model, then at least some of these must qualify as ‘closest’ to w in the sense, not of being closer than or at least as close as all A-verifying worlds (the original RA (w)), but rather, in the sense of not having any A-verifying worlds closer to w than they are. + It may be helpful to note that if we let Sw be the relation holding between x and y when Sw yx implies Sw xy, then [Def. R+ ] amounts to: + + (w) = {x ∈ X | ∀y ∈ X Sw xy} RX

and so differs from the original [Def. R] only in replacing S by S + . By way + ; of explaining all these ‘+’ superscripts, note that we always have Sw ⊆ Sw this inclusion can be reversed, erasing the distinction between all the present ‘plussed’ notions and their plain namesakes, precisely when for all x, y ∈ W , either Sw xy or Sw yx, as we invite the reader to verify. Now that we have lost the Strong Limit Assumption, and therewith the connectedness of the relations Sw in our models, we need to put back explicitly the condition which previously followed from this, namely, that these relations are reflexive. So what we are working with from now on are models (W, S, V ) with S ⊆ W × W × W , Sw being reflexive and transitive for each w ∈ W , with the Weak Limit Assumption in force, and truth being defined with the aid of [→ ]+ above. If we re-examine the above principles (A0)–(A5) against this new background: (A0) A → A (A1)

A → B, A → C A → (B ∧ C)

(A2)

A → (B ∧ C) A → B

(A3)

A → B, A → C (A ∧ B) → C

(A4)

A → C, B → C (A ∨ B) → C

(A5)

(A ∨ B) → C (A → C) ∨ (B → C),

we find again that (A0) is again delivered automatically by the form of [→ ]+ , and that the Prefixing-products (A1), and (A2), as well as the (Subj. Dilemma) principle (A4), hold in all the models now under consideration. (We noted that these schemata did not depend for their validity on any particular conditions on S; thus they continue to hold when, as in the present semantics, this is replaced by the relation S + .) This is no longer so for (A5), unsurprisingly since this was noted earlier to have as a consequence the principle in terms of

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which Burgess [1981b] gives a completeness proof w.r.t. the earlier total preorder models (though he avoids the use of the Strong Limit Assumption by using a different truth-definition). A model (W, S, V ) in which the instance (p ∨ q) → r (p → r) ∨ (q → r) of (A5) fails is as follows: W = {w, x1 , x2 , x3 , x4 }, Sw is the reflexive transitive closure of { w, x1 , x1 , x2 , w, x3 , x3 , x4 }, so that, in particular Sw holds in neither direction between either of x1 , x2 and either of x3 , x4 , and V (p) = {x1 , x4 }, V (q) = {x2 , x3 }, V (r) = {x1 , x3 }. The left-hand formula of the above + sequent is true at w in this model, since Rp∨q (w) = {x1 , x3 } ⊆ !r!; but neither disjunct of the right-hand formula is true at w. We do not have Rp+ (w) ⊆ !r!, since Rp+ (w) = {x1 , x4 } and x4 ∈ / !r!, and we do not have Rq+ (w) ⊆ !r!, since + / !r!. (This countermodel may be found in Mayer Rq (w) = {x2 , x3 } and x2 ∈ [1981], where essentially the present semantics is employed.) Of course, it is one thing to invalidate a schema by means of a model in which comparability fails, and another to give a plausible example in English in which we are prepared to assent to a subjunctive conditional of the form (A∨B) → C, but not to the corresponding disjunction (A → C) ∨ (B → C). (Pollock [1976], p. 43f., and [1981], p. 254, attempts to do this for a related schema.) In fact the only casualty of the change to the present semantics is (A5), but the continuing validity of (A3) is not obvious, since the argument that every sequent of the form A → B, A → C (A ∧ B) → C held in all of the ‘connected’ models considered earlier made use of the Strong Limit Assumption. However, armed with nothing but the Weak Limit Assumption and the conditions of reflexivity and transitivity for the relations Sw (and of course using now the [→ ]+ clause in the definition of truth), we can show that every such sequent holds, using a somewhat more involved argument. (The proof which follows is due to Thomas Bull.) For the same reasons as before (7.17.8(i)), it will suffice to show that all of the present models satisfy the result of putting R+ for R in the condition we earlier called (*): (*+ )

+ + RA (w) ⊆ !B! implies R + A∧B (w) ⊆ RA (w).

+ + (w) ⊆ !B!, and (2) x0 ∈ RA∧B (w), with a view to showing So suppose (1) RA + that x0 ∈ RA (w). Put Z= {z ∈ !A!: Sw zx 0 }. By the reflexivity of Sw , x0 ∈ Z; thus, as Z = ∅, by the Weak Limit Assumption, there exists x1 such that (3) x1 ∈ Z, which is to say: Sw x1 x0 , and (4) for all y ∈ Z, if Sw yx 1 then Sw x1 y. Note that for all y, Sw yx 1 implies Sw yx 0 , by (3) and the transitivity of Sw ; so for all y ∈ !A!, Sw yx 1 implies y ∈ Z. Then by (4), Sw yx 1 implies that Sw yx 1 implies if Sw yx 1 then Sw x1 y, and so, ‘contracting’: Sw yx 1 implies Sw x1 y, i.e., + (5) for all y ∈ !A!, Sw x1 y. But also, since x1 ∈ Z and Z ⊆ !A!, we have + x1 ∈ !A!, which together with (5) implies that x1 ∈ RA (w). By (1), then, + x1 ∈ !A ∧ B!. Now, (2) tells us that x0 ∈ Rw (A ∧ B), so, since x1 ∈ !A ∧ B!, Sw x1 x0 implies Sw x0 x1 ; thus, appealing to (3) again, we conclude (6): Sw x0 x1 . Therefore, by the transitivity of Sw , for all y, and so in particular for all y ∈ !A!, Sw yx0 implies Sw yx1 , which in turn implies Sw x1 y, by (5), which implies Sw x0 y, by transitivity and (6); so for all y ∈ !A!, Sw yx0 implies Sw x0 y, i.e., + + we have for all y ∈ !A!, Sw x0 y. But since x0 ∈ !A! (because x0 ∈ RA∧B (w)), + this means that x0 ∈ Rw (A), as was to be shown.

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To complete the parallel for ‘partial’ preorders of the earlier ‘total’ semantics, we need to address the conditions (mp) and (Centering) which whose effect was to validate the schemas A, A → C C

and

A ∧ C A → C.

With the current [→ ]+ clause in place, those conditions need to have RA + replaced by RA to have this effect: (mp)+

If w ∈ !A! then wR + A w.

(Centering)+

If w ∈ !A! then R A (w) = {w}.

We leave it as an Exercise to verify that the earlier conditions on S: (Centrality 1 ) (Centrality 2 )

Sw wyfor all w, y ∈ W . If S w xw then x = w for all w, x ∈ W .

suffice for this purpose: Exercise 7.17.9 (i) Show that if a model (W, S, V ) is assumed to satisfy (Centrality1 ), and [Def. R+ ] is used, then the condition (mp)+ is satisfied. (ii ) Show that if the conditions of (i) are strengthened to require that (Centrality2 ) is also satisfied, then (Centering)+ is satisfied. This concludes our study of some of the main schemata that have been held or denied to have a place in the formalization of subjunctive conditionals. A conspicuous feature of all principles mentioned is that none of their schematic formulations involve occurrences of → in the scope of → , though of course many instances of these schemata will exhibit this kind of embedding, since the schematic letters (A, B, C,. . . ) range over arbitrary formulas, including those themselves built up by means of → . In view of this feature, it is sometimes suggested (e.g. van Benthem [1984c]) that we can usefully pursue the logic of conditionals in a language all of whose formulas are conditionals with purely boolean antecedents and consequents. This is, however, a very severe restriction: in the first place, because we lose the ability to relate non-conditional formulas and conditional formulas as in the Modus Ponens schema A, A → C C; in the second place because embedding → in the scope of boolean connectives is also lost – as in A → (B ∧ C) (A → B) ∧ (A → C) (A ∨ B) → C (A → C) ∨ (B → C) ( = (A5) above) and (A → B) ∨ (A → ¬B) (‘Conditional Excluded Middle’). Thirdly, it should be emphasized than even from the principles formulable schematically without iteration of conditionals within the scope of conditionals, there follow principles not falling under this description which are, further, not simply special cases of principles which do fall under the description. A simple example is given by the ‘Contraction’ schema (cf. 7.25 below): (Contrac)

A → (A → C) A → C

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which follows by Prefixing from the Modus Ponens schema just mentioned and the ‘Identity’ principle A → A: (1)

A, A → C C

Modus Ponens

(2) (3)

A → A, A → (A → C) A → C A → A

(1) Prefixing Identity Schema

(4)

A → (A → C) A → C

(2), (3) (T)

Anscombe [1991], p. 127, purports to give a counterexample to the indicative conditional analogue of Contrac, not discussed here. We shall have more to say about Contraction as a structural rule and as a principle in implicational—as opposed to conditional—logics in 7.25 and the related notes to §7.2. (We have in mind nothing more than the very rough and ready implicational/conditional contrast with which this chapter began.) In view of the above adverse citation of van Benthem [1984c], it should in fairness be noted that §3.2.3 of the same author’s [1984a] gives a good discussion of one of the several topics the present survey omits altogether: ‘correspondence theory’ for subjunctive conditionals with the similarity semantics, which is to say: the analogue for the language with → for modal definability theory in the case of the language with singulary (interpreted over Kripke models, as in §2.2 above). In that discussion, van Benthem imposes no restrictions on embedding, and gives definability results for such formulas as the ‘importation’ sequent: p → (q → r) (p ∧ q) → r.

7.18

‘Counterfactual Fallacies’ and Subjunctive/Indicative Revisited

The first half of our title above is that of a famous section (§1.8) of Lewis [1973], in which three patterns of inference are considered – Strengthening the Antecedent, Transitivity, and Contraposition – whose validity is lost on the passage, described in 7.15, from a strict implicational to a variably strict implicational treatment of subjunctive conditionals. In that discussion Lewis (following Stalnaker [1968]) seeks to show that this loss is no sacrifice, by illustrating with concrete examples – actual would-be arguments in English, that is – how the inference-patterns concerned are indeed (in general) fallacious. See also the discussion in the Appendix to Sobel [1970]. One of the most famous of these examples, aimed at showing the invalidity of Transitivity, is Stalnaker’s argument involving J. Edgar Hoover, given below (after 7.18.2). For the reader’s convenience, we list the contested principles here: (Str. Ant.)

B → C (A ∧ B) → C

(Transitivity) (Contrap.)

A → B, B → C A → C

A → C ¬C → ¬A.

As remarked in 7.12, instead of “contraposition”, which elsewhere we have taken to be a rule governing only ¬ (and allowing, e.g., the passage from Γ, A B to Γ, ¬B ¬A: this was the ‘selective contraposition’ of, e.g., 2.32.1(ii) at p. 304), we should really say conditional contraposition for the above, but we shall leave out the “conditional” since it is clear that it is conditionals that

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are under discussion here. Not necessarily subjunctive conditionals, though: whether indicative or subjunctive constructions are involved, inference pattern exhibited in (1) is generally known as contraposition (‘for conditionals’): (1) If A then B. Therefore: If not-B then not-A. Similarly, we would like to be able to use the rest of the above terminology introduced with reference to principles governing → in a way that is neutral as to whether it is indicative or subjunctive conditionals that are involved; thus, where the if then. . . construction in (2) is of either of these two types, we speak of the inference from the former to the latter as strengthening the antecedent. (And likewise in the case of transitivity, or Hypothetical Syllogism, for which we shall not bother to write out the ‘neutral’ form here: see the Digression on p. 992). (2) If A then C. Therefore: If A and B then C. Having granted ourselves this neutral terminology, we can ask whether, in particular, the inference patterns of strengthening the antecedent, contraposition and transitivity, are (i) fallacious for both indicative and subjunctive conditionals, (ii) fallacious for neither indicative nor subjunctive conditionals, or (iii) fallacious for one but not the other kind of conditionals. And by “fallacious”, to simplify the discussion, is meant: not guaranteed to preserve truth (at a world). (This leaves positions according to which indicative conditionals don’t have truth-values but still participate in inferences which deserve to classified in respect of validity out of account.) Position (i) is that adopted in Stalnaker [1968], [1975], and we shall return to some aspects of his exposition below. Position (ii) will be taken by anyone maintaining a strict implication account of both kinds of conditionals, even if the strict implication is different in the two cases (because the relevant accessibility relation is differently conceived, say). Position (iii) is generally held in the specific form: the inferences in question are valid for indicative conditionals but not for subjunctive conditionals. (Lewis [1973], Jackson [1987]). Of course there is also the consideration for Jackson’s account of something like invalidity for the valid inference forms which are invalid in the subjunctive case – the counterfactual fallacies above – namely their failure to preserve, not truth but: assertibility, and Jackson [1981] makes an ingenious attempt to give a parallel account, in terms of similarity. Not similarity of worlds, but of probability functions. For a given speaker, whose degrees of belief are represented by the probability function P : the assertibility of an indicative conditional with A as antecedent depends on the assertibility of its consequent in the closest (or: most similar to P ) probability function P which gives probability 1 to its antecedent, just as the truth of a subjunctive conditional (according to the Lewis semantics with Stalnaker’s assumption from 7.17 in force), again with antecedent A, in world w, depends on the truth of its consequent in the closest (to w) world in which its consequent is true. For this parallel to work, given Jackson’s robustness implicature, we need only concede that P in the indicative case is the probability function obtained from P by conditionalizing on A, i.e., for all B, P (B) = P (B/A). (Here we have written “P ” for brevity in place of the “Pr ” of earlier discussions, most recently in 7.13.) Before returning to Stalnaker, some complexities in Position (ii), distinguished above, deserve mention. In particular, those who would think of the

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logic (understood in terms of validity rather than preservation of assertibility) of indicative and subjunctive conditionals validity of inferences as coinciding need not treat conditionals (of both sorts) as strict implications, and nor need they take exactly the same attitude to all three of the principles listed above as counterfactual fallacies. Lowe [1983] illustrates both of these points at once, presenting as the stronger of two logical treatments of conditionals to which he is sympathetic, KT with the following definition of the conditional, written here as “⇒”: A ⇒ B =Def (A → B) ∧ (A ∨ B). Exercise 7.18.1 (i) Would it have made any difference, given the same choice of modal logic (i.e., KT) if instead of the definiens being as above, the following re-bracketed form had been given? ((A → B) ∧ A) ∨ B. (ii) With Lowe’s definition of ⇒, and taking KT as the inferential consequence relation (‘local truth-preservation’) associated with KT, calculate which of the following claims are correct: q ⇒ r KT (p ∧ q) ⇒ r p ⇒ q, q ⇒ r KT p ⇒ r p ⇒ q KT ¬q ⇒ ¬p. One may be inclined to reject Lowe’s suggestion out of hand since, in virtue of the first conjunct of the proposed definition, the truth of A ⇒ B requires that A should strictly imply B – contradicting the commonplace observation that we accept as true many conditionals (whether indicative or subjunctive, a distinction bearing no weight for Lowe) whose consequents are not implied by their antecedents as a matter of logical necessity. But this objection would be too hasty, overlooking the point made in the second paragraph of 7.15 that the use of the “” notation has nothing in particular to do with logical necessity, giving rise to an ambiguity in the phrase “strict implication” according as to whether any such specific association is intended. Lowe [1983] would still deserve to describe itself (as the title does) as a simplification of the logic of conditionals even if it did not represent an attempt to reduce conditionality to matters of logical necessity (and boolean notions), since a treatment in terms of normal monomodal logic would count as a simplification in by anyone’s reckoning. Exercise 7.18.1(ii) of course asks after the fate of the three ‘counterfactual fallacies’ with which we began, and the answers are not identical for the three cases. Here we ignore contraposition, so as to leave some part of the exercise untreated, and consider the first two: Strengthening the Antecedent and Transitivity. Lowe’s treatment rejects the former as invalid but endorses the latter as valid. The former verdict we highlight as a numbered Example before taking up the latter. Example 7.18.2 As the discussion of counterfactual fallacies in Lewis [1973] made famous, Transitivity together with the acceptance of the logical truth of the corresponding (in his case → ) conditionals with a conjunction as antecedent and one of its conjuncts as consequent yields Strengthening the Antecedent. In view of the verdicts on Str. Ant. and

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Transitivity given by 7.18.1(ii), then, we know in advance that we do not have KT (A ∧ B) ⇒ A. Unpacking the defined “⇒”, this means that ((A ∧ B) → A) ∧ ((A ∧ B) ∨ A) is not always KT-provable, which is naturally due to the second conjunct. A first reaction to someone claiming not to reject a statement of the form If A and B, then A might be to conclude that the individual is unhappy about (∧E) for “and”, a position touched on in 5.15. But since A ∧ B KT A for all A and B, this does not seem to be Lowe’s position. Instead, he is objecting to the transition from this to the claim that the conditional with A ∧ B as antecedent and A as consequent has itself the status of a logical truth, thereby rejecting Conditional Proof as an introduction rule for his preferred conditional (which we continue to write as ⇒: Γ, A B ΓA⇒B Indeed, as we see from the present instance, Lowe rejects even the special case of this rule in which Γ = ∅. For a defence of this unusual position, which, the previous remark notwithstanding, is in some ways reminiscent of the considerations we saw in 5.15, see Lowe [1985]. As for the endorsement of Transitivity, addressed in the Appendix to Lowe [1983] and also in Lowe [1990], amongst other places, Lowe is keen to fault the counterexample Stalnaker [1968] used (and Lewis [1973] endorsed the use of) to show its fallaciousness: If J. Edgar Hoover were today a communist, then he would be a traitor. If J. Edgar Hoover had been born a Russian, then he would today be a communist. Therefore: If J. Edgar Hoover had been born a Russian, then he would be a traitor.

Misgivings about the verdict of invalidity for this argument – usually taking the form of suggesting thee is some contextual equivocation going on, perhaps also with the observation that one is less likely to assent to both premisses if they are presented in the reverse order – have been very common; here, aside from Lowe, we mention only Slater [1992]. (Likewise, many have taken issue with Lowe on this issue, of whom we mention only Ramachandran [1992] here; leafing through volumes of the journal Analysis in surrounding years will disclose many more.) Similarly, in connection with Strengthening the Antecedent, which does come out invalid on Lowe’s account, there is a strong feeling that after conceding that if it had been that A, it would have been that C, the negative reaction to the conclusion “If it had been that A and B, it would have been that C” in the case in which one accepts “If it had been that A and B, it would not have been the case that C” has a tendency to cause second thoughts about the original concession, now that the possibility that B has been raised, that if it had been that A it would have been that C. One might re-consider this, saying: Oh, I wasn’t thinking about its being the case that B when I agreed to that. This is

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remarkably similar to the situation with the claim to know something despite not having ruled a particular piece of contrary evidence that had not occurred to one. When the possibility is raised, one is less likely to continue to claim to have had the knowledge despite not being able to rule the possibility out, and more likely to concede that the claim needs to be retracted or modified now that this new possibility has been made salient. Lewis himself addressed the latter issue with a contextualist theory of knowledge, allowing pragmatic factors to influence what it took for a knowledge-attribution to be true (Lewis [1996]), and yet when considering (subjunctive) conditionals we are apparently allowed to re-set the corresponding parameters mid-argument, as dictated by what is explicit in the antecedent of the conditional under consideration as opposed to the conversational context that played such role in the epistemic case. What justifies such different reactions to the two cases is certainly something of a mystery (for more on which, see Brogaard and Salerno [2008]). We now return as promised to pick up Stalnaker’s relatively early musings on conditionals. Stalnaker [1975] puts the problem of indicative conditionals in very clear terms, and offers his own solution to it. Sticking with his examples, the problem arises because the argument from (4) to (5) “may seem tedious, but it is surely compelling”; ignore (3) for the moment – we will get to it presently and the discussion will be easier to follow if this numerical ordering is employed (which picks up from (1) and(2) after 7.18.1 above): (3)

The butler did it;

(4)

Either the butler or the gardener did it; Therefore:

(5)

If the butler didn’t do it, the gardener did.

The problem arises because if we accept that (5) follows from (4), we will be hard pressed to deny that the truth conditions of indicative conditionals are precisely those of the corresponding material conditionals. (Hard pressed because (4) even less controversially follows from (5): for suppose (4) is false. That means each disjunct of (4) will be false, which makes (5) a conditional with a true antecedent and a false consequent, surely rendering (5) false.) In any case, now bringing (3) into the picture, we are also unlikely to object to the claim that (4) itself follows from (3). But we don’t seem happy – as is evidenced by talk of the ‘paradoxes of material implication’ – to find the inference from (5) itself to (4) compelling. How so, if (3) follows from (5) and (4) from (3)? This is the problem of indicative conditionals. (There is no corresponding problem of subjunctive conditionals because in their case neither of the corresponding sub-inferences seems at all compelling.) We recall that Grice’s response to the problem is to accept both the inference from (3) to (4) and that from (4) to (5) as truth-preserving, though to claim that in conditions under which it would be appropriate to assert (3) it would be equally inappropriate to assert (4) or (5), because of a considerations of conversational implicature (violating the maxim of quantity: roughly assert the stronger of two claims you are in a position to make which both bear on the hearer’s interest, rather than the weaker, unless this has a cost in length ). Similarly Jackson, though this time appealing to the conventional rather than conversational implicature, agrees that both steps are truth-preserving, but that assertibility, in particular, in the passage from (3) to (5), is not preserved. We

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also recall, from our summary in 7.11, that while, since it is conventional rather than conversational implicature that is at issue, the meanings of an indicative conditional and the corresponding material conditional are different, it is a mistake to try and package any such difference as a difference in truth-conditions by offering a ‘possible worlds account’ of the truth-conditions of indicative conditionals. We will return to (possible replies to) his main argument for this position – Jackson’s “actually” argument already detailed in 7.11 – after expounding Stalnaker’s own solution to the problem described above, which employs just such a possible worlds account. Stalnaker suggests that the pragmatics of conversation dictate that one should only assert A or B in a situation in which each of A and not-B and B and not-A represents an open possibility – one not excluded by what is taken to have been established, or else just taken for granted, in the conversational context. He gives the following Gricean reason for why this should be so (Stalnaker [1975], p. 278): The point is that each disjunct must be making some contribution to determining what is said. If the context did not satisfy this condition, then the assertion of the disjunction would be equivalent to the assertion of one of the disjuncts alone. So the disjunctive assertion would be pointless, hence misleading, and therefore inappropriate.

One may well wonder why the appropriateness condition isn’t instead simply that each of not-B and not-A should represent an open possibility, and Stalnaker devotes a footnote appended to the passage just quoted qualifying his claim and discussing various the explicitly non-exclusive disjunction construction ‘A or B or both’, but fortunately for the application to the problem of indicative conditionals as posed above, this simpler condition—that each of not-B and not-A should be an open possibility— suffices for that application, and is easily given the Gricean justification to which he alludes: if not-B were not an open possibility then instead of saying A or B, one should have said just A, and similarly in the other case. As well as the above point about disjunctive assertions, we need a special thesis about indicative conditionals. Although Stalnaker gives the same general semantic clause for indicative and subjunctive conditionals, in either case such a conditional counting as true in w if the most similar antecedent-verifying world to w verified the consequent, he adds the special constraint for indicative conditionals that this closest world should be within the ‘context set’ at the stage of any conversation at which such a conditional is asserted, where the context set is just the set of worlds representing open possibilities as of that stage. (In fact the discussion in Stalnaker [1975] has a slightly different emphasis: it is the subjunctive that is regarded as the marked construction, allowing special permission to stray from the bounds of the context set.) Here for simplicity we go along with ‘Stalnaker’s Assumption’ as to the existence of a unique closest antecedent-verifying world, mentioned in the second paragraph of 7.17; the reader may speculate as to what form the special constraint on asserted indicative conditionals should take if this assumption is relaxed. (Of course, Lewis’s relaxation of this assumption—see again 7.17—was tailored for what he saw as specifically the subjunctive case.) A consequence of the special constraint, since the antecedent has to be true in the closest antecedent-verifying world, is that

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the antecedent is an open possibility: one obtaining in at least some worlds in the context set. These points about asserted disjunctions and indicative conditionals are assembled by Stalnaker to yield an account of what happens with the inference from (4) to (5). An assertion of (4)—repeated for convenience—is only acceptable if the context set contains worlds consistent with the negations of each of its disjuncts (i.e., their negations both represent open possibilities): (4)

Either the butler or the gardener did it.

Now when it comes to the conclusion (5), (5) If the butler didn’t do it, the gardener did. we look at the closest world to the world in which (4) is asserted and find that since there are worlds in the context set at which the antecedent is true, this antecedent being itself the negation of the first disjunct of (4), one of these must be selected as the closest antecedent-verifying worlds on pain of violating the special “Don’t stray from the context set” constraint on indicative conditionals. But since (4) is accepted, all worlds in the context set verify it, and this includes whichever the selected world is, which means that, since the first disjunct of (4) is false there, the second disjunct is true. The second disjunct of (4) just is the consequent of (5). So (5) is a conditional whose consequent is true at the closest world—when closestness is constrained by the context set requirement— verifying its antecedent. On Stalnaker’s account, the inference from (4) to (5) is invalid in the sense that it is not in general preserve truth (at a world), but it is counts as a reasonable inference—so perhaps the description of this as simply ‘fallacious’ would be misleading—where reasonable inference is a pragmatic notion defined by Stalnaker, with illustrative reference to the inference of (5) from (4), as follows: “every context in which a premiss of that form could appropriately by asserted or explicitly supposed, and in which it is accepted, is a context which entails the proposition expressed by the corresponding conclusion” (i.e., the context is a subset of the latter proposition, considered as the set of worlds at which it is true). It is not claimed that (4) entails (5), however, so we need not be committed to the claim that the first disjunct of (4) ( = our earlier (3)) entails (5), as a result of the undisputed fact that (3) entails (4). Exercise 7.18.3 Why would the following be an incorrect summary of Stalnaker’s account of the relations between (3), (4), and (5)? “While from (4) to (5) there is a reasonable inference but no entailment, from (3) to (4) there is an entailment but no reasonable inference.”

Stalnaker’s account of the difference between subjunctive and indicative conditionals in terms of the special pragmatic constraint seems promising as a way of showing how the latter conditionals seem to be especially linked to disjuncts by means of the reasonable inference relation holding between (envisaged particular utterances of) (4) and (5). But in 7.11 we encountered an apparently insuperable obstacle – Jackson’s “actually” argument – in the path of any such account. We did not there, however, consider the possibility of a reply to this argument, and that is what we should do here.

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We will consider first the reply in Dudman [1994b] to the argument, and then a more promising – though still not fully worked out – line of reply, due independently to Brian Weatherson, Daniel Nolan, and David Chalmers (not that their accounts, Weatherson [2001], Nolan [2003], Chalmers [1998], are in unison on everything). We recall that Jackson’s “actually” argument cites the anomalousness of certain indicative conditionals with “actually” in them, where the corresponding subjunctive conditionals are straightforwardly intelligible. Dudman’s initial response is that because the word “actually” can after all appear in the allegedly inhospitable contexts: Jackson’s submission does not rely on facts about usage. Rather, he must be ignoring the profligacy with which some English speakers fling ‘actually’ around, and reserving the adverb to a considered use of his own. (Dudman [1994b], p. 24)

Now in the early studies of the modal logic of “actually”, the profligate usage Dudman alludes to here was already noted and set aside. For instance, Crossley and Humberstone [1977] opened with the words: In this paper we shall be looking at the logical uses of the expressions “actually” and “in fact” – between which we shall not be concerned to distinguish – where, by ‘logical uses’ we intend merely to exclude the speaker’s merely rhetorical use of such phrases in dispelling or forestalling a misapprehension on the part of the hearer, as when one says “Actually it was March, not April, when we bought the house. . . ”, and so on.

It would be interesting, perhaps, to have some account of how the logical use of “actually” is related to its rhetorical-cum-emphatic use, but the need for some such distinction has never been questioned. All of Jackson’s discussion is to be taken as having the logical use in play, exactly as with Crossley and Humberstone [1977]; Dudman’s talk of a ‘considered use of his own’ is out of place. Of course, more work is required than the mere labelling, as here, of a ‘rhetorical-cum-emphatic’ use, and it would involve some subtlety – consider the well-known (British) example of the reply (or part thereof) to a question as to whether one lives in Brighton: “Hove, actually”. In any case, as was remarked in 7.11, what is essentially Jackson’s “actually” argument for the semantic significance of the indicative/subjunctive distinction amongst conditionals can be stated without any use of the word “actually”, and indeed the first published ancestor of the argument – Postal [1974] – is exactly such an “actually”-free version. The following examples are amongst those presented by Postal (pp. 391–393): (6a) If Bob had been taller than he was, he would have made the team. (6b) If Bob was taller than he was, he made the team. (7a) If Bob hadn’t kissed the girl he kissed, he wouldn’t have sighed. (7b) If Bob didn’t kiss the girl he kissed, he didn’t sigh. Postal’s point is that the subjunctive conditionals (6a) and (7a) are ambiguous, each having an interpretation on which the antecedent expresses an impossibility and a more sensible interpretation, whereas the indicative conditionals (6b) and

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(7b) are unambiguous, having only the ‘impossible antecedent’ readings. Postal goes on to propose the obvious account of the ambiguities in the ambiguous cases in terms of possible worlds semantics. Note that, while what we are calling the sensible readings are just those which are forced by the appropriate insertion of “actually” in the antecedent, we do not need to get involved in squabbles over the exactly possibilities of distribution of the latter word in order to state Postal’s version of the argument. In one case there is an ambiguity, and in the other case there isn’t. On top of that, the most natural way of resolving the ambiguity is to see the subjunctive conditionals as introducing into the discussion goings-on at non-actual worlds. Examples 7.18.4The famous contrast between indicative and subjunctive conditionals from Adams [1970] is that between – to continue the line numbering from our discussion above – (8a) and (8b): (8a) If Oswald did not kill Kennedy, then someone else did. (8b) If Oswald had not killed Kennedy, then someone else would have. Anyone thinking that in November 1963, (Lee Harvey) Oswald shot (John F.) Kennedy but not as part of a team with backup co-conspirators ready to act if he had failed will assent to (8a) but not to (8b), apparently establishing a significant difference of meaning between a subjunctive conditional and the corresponding indicative. But, asks Lowe [1979], who says that (8a) is the relevant ‘corresponding’ indicative? Why this indicative, as opposed, in particular (Lowe suggests), to (9): (9) If Oswald has not killed Kennedy, then someone else will have. In defence of this suggestion, Lowe [1979], p.140, writes (with numbering adjusted in this quotation to match that used here): I offer the following argument in support of my belief that (9) is logically equivalent to (8b). This argument appeals ultimately to linguistic intuition, but that is inevitable in view of the nature of the contention it is intended to uphold. Typically (9) would be asserted by someone believing that Kennedy was destined to be murdered at some prior date and suspicious, perhaps, of Oswald’s intentions, but lacking direct evidence that the assassination had occurred. Now it seems likely that such a speaker, on receiving the information that Oswald had in fact committed the crime, would instinctively amend his original assertion of (9) by asserting (8b) instead. But such an amendment would not be intended by the speaker to convey any change in his previously expressed opinion: on the contrary, it would be intended as a reaffirmation of that opinion, albeit within an altered framework of assumptions. This indicated that (9) and (8b) are logically equivalent and differ only in what they indicated as the speaker’s assumptions concerning the truth value of their common antecedent.

In passing we may note that the formulation “lacking direct evidence that the assassination had occurred” suggests Lowe is reading the will in the consequent of (9) as a true future will, as opposed to an epistemic future (“will turn out to have. . . ”), but the main point to be made here

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about this assimilation of subjunctives to indicatives is rather that it fails the test set by Jackson’s “actually” argument; compare (10a) and (10b): (10a) person (10b) person

If Oswald had not killed Kennedy, then someone other than the who actually killed Kennedy would have killed him. ? If Oswald has not killed Kennedy, then someone other than the who actually killed Kennedy will have killed him.

To avoid the specifically past-tense killed in (10b), you may replace “than the person who actually killed Kennedy” with “Kennedy’s actual killer”, understood to mean the person who actually killed, is now killing or will kill Kennedy”. The difference in acceptability between (10a) and (10b) appears to reinforce Jackson’s point that indicative conditionals do not have us consider, as subjunctive conditionals do, what is the case at worlds other than the actual world. We turn briefly to the more serious Weatherson–Nolan–Chalmers line of reply to Jackson’s “actually” argument. Our discussion is based specifically on the version in Weatherson [2001], which uses two-dimensional modal logic as explained in 3.31.9 (p. 489). We write → and ⇒ for the subjunctive and indicative conditional, respectively. Weatherson’s idea is that, concentrating on the ‘diagonal’ case in which the two indices are the same, and, for expository simplicity only, imposing Stalnaker’s Assumption, we evaluate M |=w w A → B by considering the closest world x to w for which it is the case that M |=w x A, w and we then say, for this particular x, that M |=w A → B just in case M |= w x B. On the other hand, for ⇒, with the above background simplifications again in place, we find the closest x to w at which—note the change in the upper index— we have M |=xx A, and say, with this particular x in mind that M |=w w A ⇒ B just in case we have M |=xx B. In addition to the structural difference in the two clauses just given, in respect of whether or not we shift the upper index (‘world of reference’), Weatherson intends that the similarity relations alluded to in the talk of closest worlds meeting a condition might be different in the two cases, in much the same way that Stalnaker [1975] did, with a greater emphasis on epistemic matters for the indicative case. But for present purposes what matters is the fact that the antecedent and consequent in the indicative case are evaluated relative to a diagonal world-pair: the upper and lower indices coincide, explaining why these antecedents and consequents do cannot, by contrast with the subjunctive case, intelligibly have anything of the general form “things are different in this or that respect from the way they actually are”, since the role of such occurrences of “actually” is to invoke a comparison between the upper and lower indices, and in the present case this is not a comparison that could reveal a difference. (See 3.31.9, p. 489.) So this account is directly tailored to make available a reply to Jackson’s “actually” argument and allow for a relatively unified account of subjunctive and indicative conditionals. Whether or not one likes this line of reply – and no comment will be made on that score here – Weatherson [2001] remains a useful source for the parallel between indicative conditionals, epistemic possibility and the a priori on the one hand, and subjunctive conditionals, metaphysical possibility and necessity on the other. (Further illumination on this parallel will be found in Williamson [2006a] and [2009]; Weatherson’s own thinking about

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conditionals has since taken a somewhat different direction – see Weatherson [2009] – turning on recent developments in the philosophy of language which we cannot go into here.)

7.19

‘Ordinary Logic’: An Experiment by W. S. Cooper

Cooper [1968] presents an account of ∧, ∨, →, and ¬ which is intended to provide a closer approximation to the logical behaviour of ordinary conjunction, disjunction, conditionality and negation than that provided by CL . The main focus of his discussion is on finding an account of the conditional which interacts with the remaining connectives in a more realistic way than material implication does (though certain other formal renderings of the conditional are also considered as failed candidates for such an account); it is for this reason that we discuss the account here. A list of arguments is presented which are valid when formulated in CL (rendering conditionals with the aid of →) but which are intuitively invalid, as well as a list of arguments which, though intuitively valid, end up invalid when so formulated. (We will see some examples of both kinds of argument in due course; “argument” here is used in what 1.23 distinguishes as the premisses-&-conclusion sense of the word rather than the course-of-reasoning sense.) Since he wants a better formal approximation to our ordinary informal judgments of validity, the logic proposed by Cooper is described as a system of ‘Ordinary Logic’—a phrase we shall abbreviate to ‘Ord’ when convenient. (We can’t use the abbreviation “OL”, since that’s already in use for orthologic.) There is a syntactic characterization and a semantic characterization, the latter being in terms of a three-element matrix, comprising the tables below for ∧, ∨, and →; the table for ¬ is that associated with the three-valued logic of Łukasiewicz described in 2.11 (Fig. 2.11a: p. 198), repeated here for convenience as part of Figure 7.19a. As the asterisks indicate, the designated values are 1 and 2. (In fact, Cooper writes “T”, “G” and “F” for our 1, 2, and 3, the “G” abbreviating “gap”. We are following our customary practice of reserving “T” and “F” for the two values in the bivalent case.) ∧ *1 *2 3

1 1 1 3

2 1 2 3

3 3 3 3

∨ 1 2 3

1 1 1 1

2 1 2 3

3 1 3 3

→ 1 2 3

1 1 1 2

2 2 2 2

3 3 3 2

¬ 1 2 3

3 2 1

Figure 7.19a: Cooper’s Matrix

The rationale behind the tables is revealed in that for →: it is the idea that unless the antecedent of a conditional is true, the conditional itself is ‘neither true nor false’. The latter phrase is dangerous if taken too seriously, as Dummett pointed out – see the discussion following 2.11.11 (p. 208); that same discussion cited Dummett’s toying with the idea of treating conditionals after the fashion of the → table above. The rather unusual-looking tables for ∧ and ∨ are designed to have repercussions for invalidity and validity of sequents (of Set-Fmla), when taken together with the → table, which are in harmony with what Cooper takes to be our ordinary everyday judgements of argument validity – especially those which clash with the verdicts delivered by CL . We shall see several

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examples of this clash presently. We save for the end of our discussion some remarks on what it is about the matrix of 7.19a that makes its ∧ and ∨ tables so unusual. Although the semantic characterization of what we shall call Ord proceeds via the matrix depicted in Figure 7.19a, Ord is not defined as the consequence relation determined by that matrix. Recall (from 2.11) that this would mean Γ Ord A just in case every matrix evaluation assigning a designated value to all the formulas in Γ assigns a designated value to A. Let us call an evaluation h (in the above matrix) atom-classical if h(pi ) ∈ {1, 3} for each propositional variable pi . (We are thinking of 1 and 3 as the ‘classical’ values.) Then Cooper’s definition (though the terminology is not his) of Ord is that Γ Ord A just in case every atom-classical evaluation assigning a designated value to all the formulas in Γ assigns a designated value to A. For finite Γ we can equally well think of this as Cooper’s definition of validity – Ord-validity, we may call it – for the sequent Γ A. This restriction to the atom-classical evaluations means that we do not (strictly speaking) have a many-valued logic here, and raises the possibility that Ord is not substitution-invariant – or, to put it differently, that the set of Ord-valid sequents is not closed under Uniform Substitution. We shall see below (7.19.3) that this possibility is indeed realized. Note that because of the atom-classicality restriction, a truth-table test for the Ord-validity of a sequent in which there appear k distinct propositional variables requires, as in the familiar two-valued case, only 2k lines, and not the 3k lines one would expect in a three-valued setting. Exercise 7.19.1 Show that a sequent involving only →-free formulas is Ordvalid if and only if it is tautologous (‘CL-valid’). (Hint: notice that if A is such a formula and h is atom-classical, then h(A) ∈ {1, 3}, and then look at the corner cells of the ∧ and ∨ tables, and the top and bottom rows of the ¬-table.) The result of 7.19.1 depends not only on the exclusion of →, but on the restriction in the definition of Ord-validity to atom-classical evaluations: if we could set h(p) = 2, this would allow us (by putting h(q) = 3) to invalidate p ∧ ¬p q, for example. For familiarization with Ord , we include the following Exercise, whose results are noted by Cooper: Exercise 7.19.2 Where = Ord , Verify that for any formulas A, B, A1 , . . . , An : (i) A ∨ B ¬(¬A ∧ ¬B) and A ∧ B ¬(¬A ∨ ¬B) (ii ) ¬(A → B) A → ¬B (iii) A B iff A → B; and more generally A1 , . . . , An B iff (A1 ∧ . . . ∧ An ) → B. While part (iii) of the above exercise cites a familiar property of IL (and CL ), and the De Morgan Laws in part (i) are familiar from CL , part (ii) presents us with something that does not even hold for CL . In particular, the -direction fails for CL . The -direction was in effect discussed for subjunctive conditionals, under the name “Law of Conditional Excluded Middle” in 7.17. As it happens, all of Cooper’s sample arguments involve indicative rather than

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subjunctive conditionals, and we can set the latter to one side for the remainder of our discussion. We return to the controversial -direction of (ii) shortly, after a couple of further points about 7.19.2, and a settling of the issue raised above about Uniform Substitution. The first point to note is that the results cited hold for arbitrary formulas A, B, A1 , . . . , An , and not just for the case in which these formulas are atomic (i.e., for the present language, the case in which they are propositional variables). This means that the matrix-checking procedure has to involve the assignment of 2, and not just 1 and 3, to the schematic letters concerned. The second point to note is that when such a truth-table check is performed, in the cases in which an equivalence (a -claim) is established, the check reveals not just that neither side is designated while the other is not, but that the formulas on both sides always assume exactly the same value. They are thus not just equivalent but synonymous (fully interreplaceable) according to Ord . (Not all equivalent formulas are synonymous according to Ord , as we shall see in 7.19.6.) To the parts of Exercise 7.19.2 we could have added a request to verify that Ord is ∧-classical; the fact that it is entangled in part (iii). When we turn to ∨-classicality, things are different. We shall consider only the ‘∨-introductive’ aspect of ∨-classicality: the question, that is, of whether disjunctions follow from their disjuncts according to Ord . As a special case of 7.19.1, we indeed have p Ord p ∨ q and q Ord p ∨ q. But a glance at the table for ∨ in Fig. 7.19a reveals that 2 ∨ 3 = 3 and that 3 ∨ 2 = 3 (to use the symbol “∨” now for the matrix operation), so that a disjunction with a designated disjunct can be undesignated. As a result, we do not have A Ord A ∨ B for all A, B (or the analogous claim with B on the left). For example, p → q Ord (p → q) ∨ r, since with h(p) = h(q) = h(r) = 3, we have h(p → q) = 2 while h((p → q) ∨ r) = 3. The example establishes: Observation 7.19.3 Ord is not substitution-invariant. Although the class of Ord-valid sequents is not closed under the uniform substitution of arbitrary formulas for propositional variables, we should note that it is closed under variable-for-variable substitution: see the Appendix to §1.2 (p. 180) for the interest of this fact. Though 7.19.3 might be thought not to augur well for the claimed ‘ordinariness’ of Ord , even if not otherwise objectionable, the phenomenon we used to illustrate it – a failure of (∨I) when one of the disjuncts is conditional in form – forms part of his case for Ord as superior to CL . (The general objectionability of non-invariant logics is held against Ord in Cresswell [1969]; problems over Uniform Substitution for a closely related proposal of Belnap’s, discussed below, are noted in Dunn [1970].) As was mentioned in our opening paragraph, Cooper lists a range of arguments which he claims are incorrectly evaluated by the lights of CL . Especially interesting are those involving if in the scope of or, a point alluded to in the discussion of disjunctions apparently exhibiting the behaviour of conjunctions in the Appendix to §6.1, such as (1) ( = (2.19) in Cooper [1968], = (4) in that Appendix): (1)

If the water supply is off then she cannot cook dinner, or if the gas supply is off then she cannot cook dinner. Therefore, if either the water supply case is off or the gas supply is off, then she cannot cook dinner.

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While the sequent (2) below, giving the form of argument (1), is CL-invalid, it is Ord-valid; thus, Cooper maintains, Ord scores well where CL scores poorly, since our ordinary everyday evaluation of (1) is favourable: (2)

(p → r) ∨ (q → r) (p ∨ q) → r.

(See further 7.19.11 below.) Now, whether or not one agrees with Cooper’s confidence about our everyday verdict on (1), it is clear that there is no such favourable verdict, and nor do the tables of Fig. 7.19a validate it, on (3): (3)

p → r (p ∨ q) → r

It is therefore mandatory, given the Ord-validity of (2), that the (∨I) inference recorded in (4) be rejected—otherwise (3) will follow by transitivity (by (T), that is): (4)

p → r (p → r) ∨ (q → r).

The “conjunctive-seeming or ” description suggests that the formula on the left of (2) is an inappropriate representation of the premiss of (1), a more appropriate representation being one in which the aberrant or is corrected to an “∧”: (p → r) ∧ (q → r). We quoted Cooper in the Appendix to §6.1 – p. 812 – as objecting to the need for such ‘correction’: “If the speaker meant ‘and’, why didn’t he say ‘and’ ?” (Cooper [1968], p. 298). The point here not that the conjunctive formulation is illegitimate, but that the disjunctive formulation is legitimate. According to Ord , in fact, the three formulas (5), (6), and (7), are synonymous: (5) (6) (7)

(p ∨ q) → r (p → r) ∨ (q → r) (p → r) ∧ (q → r)

It is a remarkable testament to Cooper’s ingenuity that while preserving as much as they do of the expected logical behaviour of ∧, ∨, ¬ and → as they do, the Ord tables manage to deliver the equivalence of (6) and (7): it is not as though in general Ord blurs the distinction between conjunction and disjunction. By contrast with some of the other accounts of the conjunctive interpretation of disjoined conditionals mentioned in the Appendix to §6.1, Cooper’s account does treat this common interpretation as an explicable processing/performance error, so much as justify it (by providing a theory on which the inferences between (6) and (7) are indeed valid). (It should be noted that Cooper himself moved away from the three-valued treatment, towards a more probabilistic account of the semantics of conditionals in Cooper [1978]; the same movement is evident in the work of another early three-valued theorist: compare Jeffrey [1963] with Jeffrey [1991]. Section III of the latter paper makes some connections with the threevalued approach, citing a suggestion of Bruno de Finetti.) The formal interest of Cooper [1968] remains considerable, however, as the present discussion is intended to show; typical references in the subsequent literature to that paper are almost always to the list of arguments on whose assessment of validity CL is urged to go wrong, rather than to the novel technical proposals—unusual ∧ and ∨ tables, unusual definition of validity in terms of a matrix with restrictions on the class of evaluations considered, etc.) Let us return to the backward direction of the equivalence cited in 7.19.2(ii), listed here as (8); given the Ord -synonymy of any formula and its double negation, we could equally well put this as (9):

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(8) A → ¬B Ord ¬(A → B) (9) A → B Ord ¬(A → ¬B). The ‘contra-classical’ principles (8) and (9) are characteristic of what have been called connexivist logics of implication, or logics of connexive implication – see the references in the notes to this section—and which usually give rise to a tension with (∧E), on the grounds that we could take A as p ∧ ¬p and B as p, in which case the left hand side of (9) would surely be a consequence of the empty set while the right-hand side would not. What happens for the case of Ord , however, is that the fact that the ‘self-negating’ value 2 is designated allows us to have a formula and its negation both in the logic, and in this case we have Ord (p ∧¬p) → p and Ord ¬((p ∧¬p) → p). This phenomenon – not just paraconsistency but outright inconsistency (at least what is called ‘¬-inconsistency’: the provability, for some A, of A as well as ¬A) – gives us another illustration of the failure of Uniform Substitution to put alongside the (∨I) example used to justify 7.19.3 above: Example 7.19.4 A, ¬A Ord B for atomic A, arbitrary B, though not for arbitrary A, B. (For we may put A = (p ∧ ¬p) → p and B = q.) A conspicuous consequence of (9), putting “A” for “B”, and exploiting the fact that Ord A → A (for all A: a fact which, notice, would not be a fact if only the value 1 were designated), is (10): (10) Ord ¬(A → ¬A) (In discussions of connexive logics,(9) – or the corresponding implicational formula, with “→” replacing the turnstile – and (10) are referred to as the principles of Boethius and of Aristotle, respectively.) What Cooper ([1968], p. 304) says in support of (10) is this: A sentence of the form “If A then not-A” is not a contradiction under the classical formalization of ‘if–then’, yet to the man in the street it is as self-contradictory a sentence-form as it is possible to utter.

(See further the notes to the present section, under ‘Connexive Implication’: p. 1056.) This is seen to be a point in favour of Ord because the negation of A → ¬A is a valid formula by the Ord-tables. But something of an anomaly arises. Although, for example, the negation of p → ¬p is Ord-valid (i.e., the sequent p → ¬p is Ord-valid), if we consider instead the conjunction of p → ¬p with q, we obtain something (recalling that Ord is ∧-classical) which has p → ¬p, something of a form than which nothing more self-contradictory can be conceived (by the ‘man in the street’) as a consequence. Yet is the negation of this conjunction Ord-valid? No. We can see that (11) Ord ¬((p → ¬p) ∧ q) by considering an evaluation h with h(p) = 3, h(q) = 1: such an h assigns the value 3 to the formula mentioned in (11). Rather than labour the point further against Cooper, let us pass to an observation of more general theoretical interest: Observation 7.19.5 ¬ is not antitone according to Ord .

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Proof. Although A ∧ B Ord A, for any A, B, we do not always have ¬A ¬(A ∧ B). For example, as above, we can put A = p → ¬p, B = q. The failure of simple contraposition reported in 7.19.5 is only to be expected given the Ord-invalidity of the general (∨I) schema A A ∨ B, in view of the De Morgan equivalences 7.19.2(i). We can bring such failures to bear on the question of congruentiality (§3.3). Observation 7.19.6 Ord is not a congruential consequence relation; specifically, ¬ is not congruential according to Ord . Proof. For A = p → p and B = p ∨ ¬p, we have A B without ¬A ¬B, where is Ord . Since substitution-invariance and congruentiality are two properties often expected of a ‘well-behaved’ consequence relation, it is of some interest to notice that Ord possesses neither of them (7.19.3, 7.19.6). Exercise 7.19.7 Are the remaining primitive connectives (i.e., (∧, ∨, →) of the language of Ord congruential according to Ord ? We have seen some examples of argument-forms to whose informal validity Ord is regarded by Cooper and doing greater justice than CL, but as was mentioned above, it is for examples in the opposite direction—cases of arguments whose standard formalization in CL renders them valid while they are intuitively invalid—that Cooper [1968] is most famous. (Another well known source, of roughly the same vintage, for such examples is Stevenson [1970].) Before closing this subsection with some remarks on the tables in in Fig. 7.19a, we should include some of these examples. Examples 7.19.8(i) If John is in Paris, then he is France. If he is in Istanbul, then he is in Turkey. Therefore, if John is in Paris he is in Turkey, or, if he is Istanbul he is in France. ( = 2.1 in Cooper [1968].) (ii ) If the temperature drops, it will snow. Therefore, it will snow, or, if the temperature drops it will rain. ( = 2.3 in Cooper [1968].) (iii) It is not the case that if the peace treaty is signed, war will be avoided. Therefore, the peace treaty will be signed. ( = 2.4 in Cooper [1968].) (iv ) If both the main switch and the auxiliary switch are on, then the motor is on. Therefore if the main switch is on the motor is on, or, if the auxiliary switch is on the motor is on. ( = 2.5 in Cooper [1968]; rediscovered in Blum [1986].) The example under (i) here is quite interesting in view of our recurrent appeals to the cross-over property (0.11), providing the occasion for another such appeal. We may we represent its form (using mnemonic lettering: the “F” and “T” are intended to suggest France and Turkey, rather than False and True) thus: (12)

P → F, I → T (P → T) ∨ (I → T).

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The implicational arrows here are telling a story of the same pattern as the relation-indicating arrows in Fig. 0.11a (p. 1 above), our pictorial representation of the cross-over condition: if two arrows are such-and-such then one or other of the ‘arrow-crossing’s obtains. To make this more precise, suppose that we have a language with → and ∨ amongst its connectives and a consequence relation on this language. For any valuation v (for the language) we define a relation Rv (A,B) between formulas A and B to hold just in case v(A → B) = T. (Now and henceforth “T” and “F” revert to their use as names of truth-values. “ Rv ” denotes the relation Rv→ defined in 3.34.5.) Schematizing (12) and assuming that is ∨-classical (13)

A → B, C → D (A → D) ∨ (C → B)

we get something which holds for all A, B, C, D if and only if is determined by some class V of ∨-boolean valuations such that for each v ∈ V the relation Rv has the cross-over property. Part (i) of the following Exercise is a variation on 0.14.8. Exercise 7.19.9 (i) Show that for a binary relational connection (R, S, S) in which R is a pre-order, R is strongly connected (a total pre-order) if and only if (R, S, S) has the cross-over property. (ii ) Using part (i), conclude that if the basis for the proof system INat in 2.32 is supplemented by the zero-premiss rule: A → B, C → D (A → D) ∨ (C → B), we obtain a presentation of the intermediate logic LC. We turn now to the unusual structure of the Ord-matrix (Fig. 7.19a), and, in particular, to its ∧ and ∨ tables. Notice that the value 2 serves as an identity element for each of those operations: x∧2=2∧x=x=2∨x=x∨2 for x = 1, 2, 3. This is enough to tell us that the reduct of the algebra of the Ord-matrix to ∧ and ∨ is not a lattice, since, as we saw in 0.21.3(ii), p. 20, no element of a non-trivial lattice could function simultaneously as an identity element for the meet operation and for the join operation. As the semilattice conditions are clearly met in both cases, the problem is over absorption: we want 3 ∧ (3 ∨ 2) = 3, but we have 3 ∧ (3 ∨ 2) = 3 ∧ 2 = 2, for example. (Technically what we are dealing with here is what is called a bisemilattice. See Humberstone [2003b] for other applications and for references to the literature.) Remark 7.19.10 Whereas 0.21.3(ii) addresses the triviality of all lattices with an element simultaneously functioning as an identity element for the meet and join operations, 0.21.3(iii) does the same for the case in which some lattice element simultaneously functions as a zero element for each of the two operations. Writing 2 for such an element, and copying the boolean values (1 for T, 3 for F) gives a three-valued matrix (once we decide on some designated elements) associated with D. Bochvar and also with Kleene, in the latter case under the description weak Kleene matrix (the Kleene matrices of 2.11 – and more particularly K1 – being described as strong); the table for ¬ is just the familiar one

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from Figure 2.11a (p. 198). See Rescher [1969], p. 29f., for diagrams and discussion. The third value, 2 (Rescher writes “I”), is there aptly described as infectious: as soon as a propositional variable in a formula is assigned that value, the whole formula ends up with that value on the evaluation in question. (Thus in Fmla, if only 1 is designated, there are no valid sequents; if 1 and 2 are designated then the valid sequents are just those which are CL-tautologous; the reader is invited to consider what happens in Set-Fmla and Set-Set under these two designation policies.) Because of the ‘infection’ feature noted in the above Remark, the matrices described there have been called on in the service of the unlikely-sounding enterprise of the ‘logic of nonsense’; the idea is that taking the value 2 corresponds to being meaningless, so that compounds with any constituent suffering from this affliction themselves suffer from it. (See Åqvist [1962]; and for a more wide-ranging discussion, Goddard and Routley [1973], Chapter 5.) The interest of this point for our discussion is that one can approach the very different Ord-tables (for ∧, ∨ and ¬) with something like the meaningful/meaningless distinction in mind. (Readers with no interest or experience in (meta-)ethics may care to skip the remainder of this paragraph.) The theory of emotivism is often summarized as holding that moral judgments are without cognitive significance (neither true nor false), having only ‘emotive meaning’. For at least one famous exposition of the view – that to be found in Chapter 6 of Ayer [1946] – this would be an oversimplification, however. Although Ayer takes this line, inter alia, with such statements of general moral principle, his example being “Stealing money is wrong”, for the case of “You acted wrongly in stealing that money”, we have the following discussion (p. 142): Thus if I say to someone, ‘You acted wrongly in stealing that money,’ I am not stating anything more than if I had simply said, ‘You stole that money’. In adding that this action is wrong, I am not making any further statement about it. I am simply evincing my moral disapproval of it. It is as if I had said, ‘You stole that money,’ in a peculiar tone of horror, or written it with the addition of some special exclamation marks. The tone, or the exclamation marks, adds nothing to the literal meaning of the sentence. It merely shows that the expression of it is attended by certain feelings in the speaker.

From the second sentence in this passage, we can gather than Ayer is happy to treat “You acted wrongly in stealing the money” as amounting to the conjunction: (14)

You stole the money and that action was wrong.

whose first conjunct is either true or false depending on the facts of the case and whose second conjunct is without literal (‘cognitive’) significance. Ayer is suggesting that (14) as a whole makes the same claim as—or has the same cognitive significance as—its first disjunct; thus if the first conjunct is true, (14) is true, and if the first conjunct is false, (14) is false. These are the deliverances of the ∧ table in Fig. 7.19a. The ∨ table could be similarly motivated: when it comes to assessing a disjunction, we simply ‘erase’ any disjunct with the value 2 (at the same time erase the “∨”) and look at the value of what we are left

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“IF”

with, since that is where the cognitive significance resides. Of course this talk of erasure is just a way of making graphic the idea of the value 2 as an identity element for the operations concerned. (We ignore some niceties here, such as the ‘presuppositional’ nature of “You acted wrongly in stealing the money” – in respect of the addressee’s having stolen the money on question – as contrasted with (14), and the fact that (14) is perhaps not really conjunctive in form because of the anaphoric role of “that action”.) We turn now to the Ord table for →. This turns out to be none other than the table we obtain from Belnap’s account of conditional assertion, touched on in 7.11, where we reported the proposal of Belnap [1970], [1973], that a conditional A → B (though Belnap uses the notation “A/B” for the present construction) might be understood relative to a world w as asserting the proposition expressed by B if A is true at w, and as asserting nothing otherwise. As was emphasized in van Fraassen [1975], this represents a relatively early venture in two-dimensional semantics, since the truth-value of a sentence is determined by a pair of possible worlds: we need to specify a world (w above) relative to ask which, if any, proposition ( = set of worlds) is expressed, and then we can ask of an arbitrary second world whether or not it belongs to the proposition so determined. But, as emphasized in Dunn [1975b], in the absence of other modal apparatus in the object language, we don’t need this two-dimensionality and can consider the same world playing both roles, which amounts to saying (Dunn [1975b], p. 386) that A → B is true in w iff w is not false in w and B is true in w, while A → B is false in w iff A is not false in w and B is false in w. Since the w is carried along as an idle parameter here, we can summarize the upshot in a three-entry table, and the table we get is the →-table of Figure 7.19a, with “1” for (what has just been called) true, “3” for false, and “2” for what’s left over. A good critical discussion of the informal significance of this proposal may be found in Blamey [1986], esp. p. 41. (Or see the reprinted version, Blamey [2002].) Blamey is especially interested in a variant, which he calls transplication, and whose table is given as 7.19b; though again we should note that Blamey uses the notation “A/B” for this. This results (under the same construal of 1, 2, 3) from changing the above world-based description to: A → B is true in w iff w is true in w and B is true in w, while A → B is false in w iff A is true in w and B is false in w. (In other words, earlier references to non-truth are replaced by references to falsity.) The remarkable applicability of this unusual connective to the topic of presupposition is not something we can go into here, and the reader is referred to Blamey [1986], [2002]. → 1 2 3

1 1 2 2

2 2 2 2

3 3 2 2

Figure 7.19b

The following is something of a curiosity. If we define the binary operation ⇒ on {1, 2, 3} by letting x ⇒ y be ¬x ∨ y, where ¬ and ∨ are the operations depicted in Figure 7.19a (p. 1044), we obtain Figure 7.19c. If we consider the language with connectives ⇒ and ¬ interpreted by the

7.1. CONDITIONALS

1053 ⇒ *1 *2 3

1 1 1 1

2 3 2 1

3 3 3 1

Figure 7.19c

similarly notated matrix operations, then we turn out, restricting attention from Set-Fmla to Fmla, to have described the implication/negation fragment of the relevant-like logic RM, discussed in 2.33. Though differently notated – since we are writing “⇒” here to avoid confusion with earlier uses of “→” ’ – this point may be found on p. 148 of Anderson and Belnap [1975]; the negation and implication matrix in question and the proof that it is ‘characteristic’ for the fragment in question are due to work in 1952 by B. Sobociński and 1972 by Z. Parks, respectively. (See Anderson and Belnap for bibliographical details, as well as the Digression on p. 474 above.) It is not clear what, if anything, to make of this apparently surprising connection between two rather differently motivated enterprises. There is in fact a further independent use of the table in Figure 7.19c in connection with implicational proof systems in a 1963 paper (in German) by Jaśkowski, a partial summary of which is provided in Bunder [1993]. Additional references may be found in the notes to §7.2 (p. 1127), under the heading ‘The 2-Property and the 1,2-Property’. We close with an exercise for amusement’s sake which arises à propos of (1) and (2) in the earlier part of our discussion. Exercise 7.19.11 Since the sequent labelled (2) above, namely: (p → r) ∨ (q → r) (p ∨ q) → r is CL-invalid (not tautologous, that is), it is not provable in the natural deduction system Nat described in 1.23. At which line(s) are the rules of that system misapplied in the following derivation, which, if it were correct, would be a proof of that sequent: 1 2 3 4 3, 4 6 7 6, 7 2, 3, 7 3, 7 1

(1) (p → r) ∨ (q → r) (2) p ∨ q (3) p → r (4) p (5) r (6) q (7) q → r (8) r (9) r (10) (p ∨ q) → r (11) (p ∨ q) → r

Assumption Assumption Assumption Assumption 3, 4 →E Assumption Assumption 6, 7 →E 2, 4–5, 6–7 ∨E 2–9 →I 1, 3–10, 7–10 ∨E

Notes and References for §7.1 Reading for 7.11. The division of conditionals, for purposes of logical investigation, into the two classes of indicative and subjunctive (alias counterfactual)

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is defended in David Lewis [1973], Gibbard [1980], and Jackson [1987]. Jackson provides an extended discussion of the attempts to give possible worlds semantics for indicative conditionals, not only as in Stalnaker [1980] but in other writers (not listed here). Aside from attacking the details of such accounts, Jackson’s argument (cited in 7.11) concerning the distribution of “actually” serves as a general pointer to the inappropriateness of this move, regardless of the details. A further example of work falling prey to this general consideration is Kratzer [1986]. For Vic Dudman’s views on conditionals, all items from 1984 onward listed under his name in the bibliography are relevant; he was kind enough to provide written comments on some of this material: the reader is warned that the gist of several such comments was that his views had not been accurately represented in the summaries to be found here. Some aspects of these views were given an especially sympathetic treatment in Bennett [1988] and Pendlebury [1989]. A propos of examples with “will” in the antecedent like (20) from the Digression preceding 7.11.2, we note that there is also the construction illustrated by “If you will help me, I will help you”, in which the will is volitional rather than future-marking. (Compare “It’s no good—right now she just won’t eat her food”, and see Chapters 7 and 8 of Coates [1983] for a thorough discussion of will and would in English; discussion of a specifically conditional case appears in Radford [1984].) The phenomena involved seem highly language-sensitive and as yet ill understood; informative discussions include Close [1979], Comrie [1982], Declerck [1984], Haegeman and Wekker [1984], and Nieuwint [1986]. More generally, for treatment of conditional constructions in other languages, with comparative remarks on the situation in English, see Akatsuka [1985], [1986], [1997a,b], Haiman [1978], [1983], [1986], Harris [1986], Comrie [1986], §3 of Funk [1985], Dancygier [1985], Cuvalay [1996], Seiler [1997] and Wierzbicka [1997]. The idea that subjunctive and indicative conditionals differ in respect of their amenability to semantic description in terms of possible worlds is found in Gibbard [1981] and Jackson [1987]. For more on this distinction, see, apart from the works of Dudman, Kremer [1987], Jackson [1990], [1991b]; more generally, taxonomic issues are the focus of George [1966], Leech [1971], §§159–168, Barker [1973], Palmer [1974] (esp. §5.5), Yamanashi [1975], Chapter 2 of Pollock [1976], Chapter 8 of Davies [1979], Comrie [1986], §7.3 of Gärdenfors [1988], Dancygier and Mioduszewska [1984], Funk [1985], Chapter 5 of Sweetser [1990], Fillmore [1990], Wakker [1996], Cuvalay [1996], Athanasiadou and Dirven [1996], [1997b], Dancygier [1998] (throughout). Apart from the question of classifying conditionals, there is also the matter of classifying theories of conditionals. The topics overlap somewhat, since a theory of conditionals may be tailored to one particular type (e.g., indicatives) and come to grief as a general account. However, as more specifically attempting classifications of the latter sort, we may list Chapter 3 of Mackie [1973], and Barker [1979]. Conditional Assertion. On conditional assertion accounts, we mention (many of them already cited in 7.11) Quine [1950], pp. 127–165 (chapter ‘On Conditionals’) of von Wright [1957], Jeffrey [1963], Belnap [1970], [1973], Holdcroft [1971], Dunn [1975b], Manor [1974], James [1986b], (see also James [1986a] for background) Cohen [1986], Barker [1993], [1995]. (The author of these last two

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defences of a conditional assertion treatment of indicative conditionals seems to recant in Barker [1997].) An area notorious for its pitfalls is that of conditionals and conditional probability: exactly what Ramsey’s remarks on conditionals amount to (Adams’s Thesis and all that); especially useful warnings against mistakes here are provided by Edgington [1996] and Stalnaker [1992]. More on all these topics may be found in the papers in Eells and Skyrms [1994], an Adams Festschrift devoted to conditional probability and Adams’s Thesis. Focus Constructions. The class of focus particles, whose members even and only we encountered in 7.12, is studied at monograph length in Taglicht [1984] and König [1991a], Erteschik-Shir [1997]; snapshot versions of the first two of these authors’ approaches may be found in Taglicht [1993] and König [1993]; see also de Mey [1996], and the several articles which appeared in a special issue, devoted to focus, of the Journal of Semantics (1991), Vol. 8, as well as the more recent Pulman [1997]. Data on concessive constructions (as with even if and although) in various languages may be found scattered throughout the monograph Rudolph [1996]; for fuller treatments of even and even if, aside from the historically important Horn [1969] and Fraser [1969], include the following, many of them mentioned in 7.12 but collected here for ease of reference: Haiman [1974], König [1986], [1991b], [1992], Dancy [1984], Hazen and Slote [1979], Gärdenfors [1979], (J. A.) Barker [1979], [1980], Pizzi [1981], Bennett [1982], §§3.1 and 5.3 of Jackson [1987], Kjellmer [1989], Kay [1990], Lycan [1991], [2001], Beckmans [1993], (S. J.) Barker [1991], [1994], Francescotti [1995], and Section 1 (esp. 1.3 and 1.4) of Gauker [2005]. For only and only if, some significant contributions have been Horn [1969], Geis [1973], McCawley [1974], [1993], Sharvy [1979], Appiah [1993], Barker [1993], von Fintel [1997]; several further references may be found in the bibliography of the last paper. Horn’s more recent views are in Horn [1996b]. A more localized discussion – making no reference to other work on if and only, that is – may be found in Halpin [1979], Richards [1980], Bunder [1981b]. An experimental study describing reasoning with “only” can be found in Johnson-Laird and Byrne [1989]. Chapter 11 of Kowalski [1979] has some interesting remarks on the role of (if and) only if in logic programming. The following sources provide a discussion of unless, the third of our special conditional-like elements discussed in 7.12: Whitaker [1970], Geis [1973], Fillenbaum [1976] and [1986] (§2.2), Chandler [1982], Brée [1985], Dancygier [1985], Wright and Hull [1986], Berry [1994], Traugott [1997]. Dedicated Texts. Textbook presentations of several logics of conditionals using the ideas of Lewis, Stalnaker and Chellas, are provided by Chapter 5 of Priest [2008] and Chapter 4 of Burgess [2009]. There are more monographs (to say nothing of the numerous anthologies) devoted to the general topic of the conditional than to any other connective, including at least the following: Lewis [1973], Jackson [1987], Sanford [1989], Woods [1997], Dancygier [1998] and Bennett [2003]; for criticisms of the treatment of issues in the vicinity of the Ramsey Test (Adams’s hypothesis vs. Stalnaker’s hypothesis) in this last work, see McDermott [2004]. Modus Tollens. References on Modus Tollens for indicative and subjunctive conditionals include the following: Sadock [1977], Carden [1977], Bosley [1979],

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Adams [1988], Sinnott-Armstrong, Moor and Fogelin [1990], Ohlsson and Robin [1994] and pp. 139–142 of Gauker [1994]; see also Gauker [2005] for objections to Modus Tollens when the consequent of the conditional concerned (the ‘major premiss’) is itself a conditional. Suppositional Accounts. In 7.13 and 7.15 some connections between conditionals and supposition were noted. A good many publications make such connections or supply accounts of supposition that lend themselves to development in that direction. A representative sample: Mackie [1973] (Chapter 3, esp. §9), Rips and Marcus [1977], Skyrms [1987], Chapter 10 of Morriss [1987], Hinckfuss [1990], Kearns [1997], §3.4.2 of Barker [2004]. Confusion about “Strengthening the Antecedent”. Some authors perversely refer to Strengthening the Antecedent as Weakening the Antecedent; these include Nute [1980], van Fraassen [1980]. (Perhaps there is some confusion from the fact that by adding an extra conjunct to the antecedent of a material or strict conditional one typically weakens the conditional. This does not mean that the antecedent has been weakened, though.) Connexive Implication. The topic of connexive logic came up in 7.19. The term is Storrs McCall’s. The pioneering papers are Angell [1962], McCall [1966], [1967b]. For discussion, see §29.8 (written by McCall) of Anderson and Belnap [1975], Mortensen [1984], §2.4 of Routley, Meyer et al. [1982]. Between (10) and (11) in that subsection, we had occasion to quote Cooper [1968] as writing that a “sentence of the form ‘If A then not-A’ is not a contradiction under the classical formalization of ‘if–then’, yet to the man in the street it is as self-contradictory a sentence-form as it is possible to utter.” An amusing confirmation of this claim comes from the fact that Cooper [1975] – whose author, we hasten to add, is a different ‘Cooper’ – asserts (p. 242) that “If p, then not p” is inconsistent – in the sense of uniformly taking the value F – when evaluated in accordance with the standard truth-table account of if and not. The impression thus persists even after the man in the street has been introduces to those truth-tables and is trying to conduct the discussion in terms of them. (On the same page of Cooper [1975], the same claim is made about “If (p or q) and not p, then not q”, which is even further from the truth in that three of the four lines of its truth-table come out as T!) Exercise 7.19.11 was devised by J. A. Burgess, as an example of how Lemmon’s verbal description of how the rule (∨E) is to apply can be misunderstood by students; references to other examples in this vein are cited in the notes to §6.2 (p. 843). General Acknowledgments. For early feedback on the material in this section I am grateful to Vic Dudman (which is not to say that Dudman accepts even all of those parts of our discussion which purport to be summaries of his views); some technical corrections were supplied by Thomas Bull. I am especially grateful to Brian Weatherson for extensive and very helpful comments.

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§7.2 INTUITIONISTIC, RELEVANT, AND CONTRACTIONLESS IMPLICATION 7.21

Intuitionistic and Classical Implication

Since the →-fragments of IL, PL, and ML coincide, we could equally well speak of ‘positive’ or ‘minimal’ implication, though we will in fact be using the label intuitionistic implication throughout. Peirce’s Law is the most famous example of a purely implicational principle which holds for CL but not for these other logics; certainly, then, “classical implication” – to which we return at the end of this subsection – would not be an alternative label for the same thing. Peirce’s Law, which we will go into in more detail in that discussion (as well as in 8.34), is conspicuously a principle requiring for its formulation occurrences of two distinct propositional variables, or if formulated as a schema, two distinct schematic letters (metalinguistic variables over formulas). While IL and CL differ in their one-variable fragments (over, for example, p ∨ ¬p and ¬¬p p), if we consider one-variable purely implicational sequents, the difference vanishes. (Conspicuously also, because of the implication within the antecedent of an implication, Peirce’s Law involves a formula which is not, as we put it in 7.14, ‘clear’; 7.14.5 showed that IL and CL agree on clear formulas, regardless of how many variables they contain.) We state this for Set-Fmla to sidestep the question of what form IL should take in Set-Set (cf. 6.43.8 at p. 899 and the discussion preceding it); the result then follows for the more restricted framework Fmla. Observation 7.21.1 If only a single variable, and no connective other than →, occurs in the formulas Γ ∪ {A} then Γ IL A if and only if Γ CL A. Proof. Since IL ⊆ CL , the direction needing to be proved is the ‘if’ direction. Suppose, without loss of generality, that the only variable in the formulas in Γ ∪ {A} is p. By induction on the complexity of such a formula C (constructed from p by → alone), we see that it is IL-equivalent either to p or to p → p. Let C† be p or else p → p, depending on which of these C is equivalent to. Thus if Γ IL A, then Γ† IL A† (where Γ† = {C† | C ∈ Γ}). Since IL p → p, we know that A† = p and Γ† = {p → p}; so Γ† CL A† , and hence, since the C/C† equivalences hold also in CL, Γ CL A. The above proof can be thought of pictorially by reference to Figure 7.21a, which depicts the Lindenbaum matrix of the one-variable implicational fragment of IL. → *p→p p

p→p p→p p→p

p p p→p

Figure 7.21a

The algebra of Figure 7.21a is the subalgebra of that depicted in Figure 2.13b (p. 226) generated by the element p (or more accurately [p]), and we have added an indication (*) of designated values to turn it into a matrix isomorphic to the

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standard two-element matrix for →, so that the proof of 7.21.1 can be followed with that isomorphism in mind. (The mapping (·)† figuring in the proof of 7.21.1 is an evaluation into this matrix.) Figure 2.13b itself showed what happened in the case of two generators, p and q, on which the implicational fragments of IL and CL do not agree, as we remarked above. The agreement with CL on one-variable implicational formulas can be pushed further down than IL, as we shall see in not only in 7.21.2(iii) of the exercise that follows, but also in 7.25.1 below. Part (ii) of this is made possible by a ‘local finiteness’ result in Diego [1966]; see 2.13.4 above. Exercise 7.21.2 (i) Circle those of the 36 entries in Figure 2.13b (p. 226) which would be incorrect if it were the implicational two-variable fragment of IL rather than CL that was being represented. (ii ) How many pairwise non-equivalent implicational formulas in the two variables p, q, are there in IL? Draw a table like Figures 2.13b, 7.21a, for the equivalence classes. (This involves a considerable amount of work, checking for formulas A, B, on a provisionally complete list of representatives, whether the formula A → B has to be added or is already equivalent to something on the list. Hint: the answer to the “how many?” question should lie between 10 and 20. (Those whose curiosity exceeds their energy can find them depicted on p. 183 of Skolem [1952], with the →-table on p. 184.) (iii) Of the two relevant logics described in 2.33, R and RM, one has the same 1-variable implicational fragment as IL and CL. Which? (Hint: check the implicational claims implicit in Figure 7.21a; for the entry under (row) p → p and (column) p, this means checking that the corresponding formula (p → p) → p is indeed equivalent in the logic in question to that entered in the table (here p).) Casting aside restrictions as to the number of variables involved, we turn to the proof system for IL most readily extracted from the system INat of 2.32; it comprises the two structural rules (R) and (M) and the rules (→I) and (→E): (→I)

Γ, A B

(→E)

ΓA→B

ΓA→B

ΔA

Γ, Δ B

We omit mention of the structural rule (T) in this presentation because its derivability from the above rules was already noted in 1.24. The rule (M) is not similarly avoidable, since the remaining rules yield only a weaker relevant logic (the system RMNat in 2.33.) By changing the introduction rule for → from (→I) to what in 4.12 we called, after Prawitz, (→I)P ra , we can do without (M) as a primitive rule (→I)P ra

ΓB Γ {A} A → B

Observation 7.21.3 Given (R), (→ E) and (→ I), the rules (M) and (→ I)P ra are interderivable


1059

Proof. For the derivability of (M) suppose that Γ B is provable, with a view to obtaining Γ, A B. From the given sequent we have Γ {A} A → B by (→I)P ra , and this together with A A gives Γ, A B by (→E). For the derivability of (→I)P ra suppose that Γ B is provable, with a view to obtaining Γ {A} A → B. If A ∈ Γ, we can write Γ as Γ0 ∪ {A} where A ∈ / Γ0 ; then / Γ, then Γ0 A → B follows by (→I), which suffices, since Γ0 = Γ {A}. If A ∈ since Γ B is provable, so, by (M), is Γ, A B, from which point we can proceed as in the previous case (with Γ0 as Γ). Some parts of the following practice exercise will be appealed to later. Exercise 7.21.4 Prove the following sequents in the above proof system: (i) (p → q) → r, p → r, r → q r (ii ) (((p → q) → p) → p) → q q (iii) (((p → q) → p) → p) → (p → q) ((p → q) → p) → p. Remark 7.21.5 Note that 7.21.4(ii ), which we can call Wajsberg’s Law (for reasons explained in the notes to this section, p. 1121) is a substitution instance of the more general (((p → q) → p) → p) → r r, which is CL-provable because the antecedent of the left-hand formula is (a ‘formula’ form of) Peirce’s Law, which is itself CL-provable. In IL, however, the more general sequent is not provable, as is evident from the fact that if it were, by the substitution-invariance of IL , we could substitute that form of Peirce’s Law for “r” and hence have A → A A IL-provable, where A is the form of Peirce’s Law in question, making A (or A) itself IL-provable. In the proof of 7.21.11 we give a rule form of Wajsberg’s Law of the same character as (Peirce) above; we return to Wajsberg’s Law briefly at the end of this subsection. For more arguments of the above “substitute for r” type, and further reference to Wajsberg’s Law, see the discussion in 9.24.9–9.24.12. The completeness theorem for INat (2.32.8, p. 311) was proved with the aid of a canonical model whose elements were deductively closed consistent prime sets of formulas. In the present purely implicational setting, we do not need to restrict attention to consistent sets or prime sets—restrictions called for by the presence of ¬ and ∨, respectively, in the language of INat. Only the fact that the sets concerned are deductively closed is needed for the to deal with inductive case of → in the proof of 2.32.7 (p. 311). Exercise 7.21.6 By modifying the proof of 2.32.6–2.32.8 as just suggested, show that a purely implicational sequent is valid on every frame (every Kripke frame for IL, that is) if and only if it is provable in the proof system described above. We turn our attention to extending the above proof system to one for the implicational fragment of CL. After that, we close with some remarks about intermediate implicational logics. The most famous principle adding to the

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(→I), (→E) basis for implicational IL is Peirce’s Law, which we consider here in the rule form: (Peirce)

Γ, A → B A ΓA

On the other hand, in 1.18, we defined →-classicality for consequence relations in terms of the three conditions: (i) A, A → B B; (ii ) Γ, A → B C ⇒ Γ, B C; (iii) Γ, A C & Γ, A → B C ⇒ Γ C. It is clear that taking as the consequence relation associated with the above proof system, condition (i) is satisfied (thanks to (→E)), as is (ii), since for any A and B we have B A → B (by (R), (M) and (→I)—or alternatively by (R) and (→I)P ra ). The specifically classical force of the definition is packed into condition (iii), which can be thought of as saying that a conditional, A → B, and its antecedent, A, are subcontraries, in the sense that if is an ∨-classical consequence relation then satisfies (iii) just in case A ∨ (A → B) for all A and B. Digression. Because of the ‘weak claim vs. strong claim’ distinction in §1.2, we cannot put this by saying that any valuation consistent with must verify either A or A → B – though we can do so by requiring that be determined by some class of valuations each of which verifies either A or A → B, for all A, B. Note also that the current condition is stronger than the compositional subcontrariety condition (from 3.18) for →, which amounts to saying that each valuation in the class verifies A, B, or A → B), though we saw that given a contraction equivalence between A → (A → B) and A → B, there is nothing to choose between the two conditions. End of Digression. Let us turn this condition on consequence relations into a sequent-to-sequent rule in Set-Fmla in the obvious way: (Subc)→

Γ, A C

Γ, A → B C ΓC

(Such an ∨-free formulation of subcontrariety appears for example in 6.31.15(ii) p. 850.) The key point to note is that (Peirce) and this subcontrariety rule are equivalent ways of extending the proof system for implicational IL: Observation 7.21.7 The rules (Peirce) and (Subc)→ are interderivable given (R), (M), (→ I), (→ E). Proof. To see the derivability of (Peirce) from (Subc)→ , suppose that Γ, A → B A is provable. (R) and (M) give us the other premiss sequent, Γ, A A, needed for an application of (Subc)→ , taking A as the C schematically indicated in our representation of (Subc)→ , the conclusion of which application is Γ A, as required to show that (Peirce) is derivable. For the converse direction, we show first that any sequent of the form


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(A → B) → C, A → C C is provable given (Peirce), (→I), (→E) and the structural rules. We know from 7.21.4(i) that, putting A, B, C for p, q, r there, that without the aid of (Peirce), all sequents of the form (A → B) → C, A → C, C → B C are provable; from this by (Peirce), taking C → B as the A → B of our schematic statement of the latter rule and {(A → B) → C, A → C} as Γ, we conclude that (A → B) → C, A → C C is provable. Now given premisses for an application of (Subc)→ , Γ, A → B C and Γ, A C, by (→I) we have Γ (A → B) → C and Γ A → C provable, so by two appeals to (T) (with (M)) and the sequent whose provability we have already demonstrated, we obtain the desired (Subc)→ conclusion: Γ C. The interderivability—given the rest of implicational IL—of (Peirce) with (Subc)→ recorded in 7.21.7 is well known. The earliest published reference (known to the author) to what amounts to the same thing may be found at the base of p. 145 in Łukasiewicz and Tarski [1930]. These authors work in the framework Fmla, rather than Set-Fmla, and comment on the interderivability between the formula ((p → q) → p) → p which is a formula version of Peirce’s Law, and ((p → q) → r) → ((p → r) → r), in the context of a pure implicational axiom system with Uniform Substitution among its rules. The way to ‘code’ a disjunction A ∨ B in this setting is as (A → r) → ((B → r) → r) choosing r as a propositional variable not occurring in A or B. (This is the “deductive disjunction”, A B, of A with B: see 1.11.3: p. 50) Thus in the present case, the second of the two formulas cited earlier arises from taking A as p → q and B as p. So this is their (originally Tarski’s, to be more precise) way of expressing the fact that, classically, an implication and its antecedent are subcontraries. Observation 7.21.7 and the above proof of it call for some comment. In the first place, it should be said that in 8.34 (see p. 1275) we shall again derive with the aid of (Peirce) a subcontrariety principle for implication, in the form A ∨ (A → B), there called the (LE) schema. (In fact we derive a representative instance of the schema, p ∨ (p → q).) Evidently, that discussion assumes the presence of disjunction behaving ∨-classically (as it does in IL); the proof there offered is shorter than that given above and trades on specific properties of ∨ to which we cannot, restricting ourselves as here to the →-fragment of IL, help ourselves. Further, the appeal to (Peirce) made in the proof in the discussion in 8.34 is actually to a special case of that rule: the case in which Γ is ∅. So this raises a question we can consider even in the present purely implicational setting: can the full effect of (Peirce) be obtained even if we impose this Γ = ∅ restriction? Secondly, there is the question of how much of what was assumed

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about → is actually required for the derivation of (Subc)→ from (Peirce). In deriving (Peirce) from (Subc)→ in the proof of 7.21.7 we exploited only the structural rules: no use was made of (→I) or (→E). Might there be a similarly austere derivation in the converse direction? The first question is easily answered. We suppose we have only the restricted form of (Peirce) in which Γ = ∅. Let us call it (Peirce0 ). (Peirce 0 )

A→BA A

We want to derive the rule (Peirce) from this, together with the structural and →-introduction/elimination rules. To avoid clashes amongst schematic letters, let us take the rule to be derived as licensing the transition from Γ, C → D C to Γ C. (We reletter the “A → B” in our statement of (Peirce) to “C → D”, that is.) Putting C for p, D for q in 7.21.4(iii) gives us (((C → D) → C) → C) → (C → D) ((C → D) → C) → C, which serves as a premiss for (Peirce0 ), where A = ((C → D) → C) → C and B = C → D, whose conclusion is the formula form of Peirce’s Law, from which (Peirce) follows easily (by reasoning as in the latter part of the proof of 7.21.7). Let us turn to the second question. Is it necessary to appeal to (→I) and (→E) in deriving (Subc)→ from (Peirce)? Here we should set aside such appeals as are only there because we decided to drop (T) as a primitive rule. Suppose, to incorporate all the structural rules, we posed the question in terms of consequence relations. The first part of the proof of 7.21.7 shows that any consequence relation satisfying (Subc)→ , in the sense that for all Γ, A, B, C, if Γ, A C and Γ, A → B C then Γ C, must also satisfy (Peirce), in an analogous sense. What we are asking is whether the converse of this claim also holds. Or do we instead have to insert, for the converse, the additional hypothesis that is →-intuitionistic? To indicate one aspect – that arising from considerations of valuational semantics – of the interest of the question just posed, we introduce a refinement of our talk in 1.25 (and 3.17, 6.31, 6.33) of global and local preservation characteristics, defined in particular in terms of the notions of the local and the global range of a class R of rules – Loc(R) and Glo(R) – from 1.26. (See in particular the discussion before 1.26.11, p. 137.) Recall that when R = {ρ}, we write “Loc(ρ)” rather than “Loc({ρ})”, and similarly in the case of Glo. When Loc(R) ⊆ Loc(ρ), we will say that ρ is a local consequence of R. The set of local consequences of R will be denoted by LCn(R). (The closure operation here is not strictly a consequence operation, since the underlying objects are rules rather than formulas, but we continue with the terminological abuse for conformity with Humberstone [1996b].) All this is tacitly understood as relative to a given language (and a rule is to be taken as the set of its applications in a given language rather than in accordance with the more sophisticated proposal of 4.33); LCn is not a consequence operation on that language, though it is a closure operation in the sense of 0.13. Similarly, we have a global analogue, GCn: GCn(R) = {ρ | Glo(R) ⊆ Glo(ρ)}. In other words, the rules which are global semantic consequences of R are those rules which preserve V -validity for every collection V of valuations for which the


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rules in R preserve V -validity. Notice that if all the rules in R are zero-premiss rules, then GCn(R) = LCn(R). The relationship between the two ‘range’ concepts in terms of which the rule consequence operations LCn and GCn are defined, and our earlier distinction between preservation characteristics is as follows: •

ρ has the local preservation characteristic w.r.t. V ⇔ V ⊆ Loc(ρ)

•

ρ has the global preservation characteristic w.r.t. V ⇔ V ∈ Glo(ρ).

But let us return to the story of (Peirce) and (Subc)→ ; we write “Loc(Peirce)” rather than “Loc((Peirce))” – which is itself an abbreviation for “Loc({(Peirce)})”. Similarly for (Subc)→ . Observation 7.21.8 Loc(Peirce) = Loc(Subc)→ . Proof. ⊆: Suppose v ∈ / Loc(Subc)→ ; we will show that v ∈ / Loc(Peirce). Since v∈ / Loc(Subc)→ , there are Γ, A, B, C with Γ, A → B C and Γ, A C holding on v but Γ C not. So, to use that convenient abbreviation, v(Γ) = T, while v(C) = F. Thus v(A → B) = v(A) = F. But in this case the application A → B A; A of (Peirce) is a transition from a sequent holding on v to one not holding on v. Therefore v ∈ / Loc(Peirce). ⊇: Suppose that v ∈ / Loc(Peirce). So for some Γ, A, B, we have Γ, A → B A holding on v, while Γ A fails on v. Thus v(Γ) = T, v(A) = F, and so also v(A → B) = F. Then the following application of (Subc)→ : Γ, A A; Γ, A → BA; ΓA takes us from premiss sequents holding on v to a conclusion sequent failing on v, showing that v ∈ / Loc(Subc)→ . Note that the application selected of (Subc)→ for the second part of the above proof is just that used in deriving (Peirce) from (Subc)→ in the proof of 7.21.7. Corollary 7.21.9 (i) (Peirce) ∈ LCn(Subc)→ ; (ii) (Subc)→ ∈ LCn(Peirce). Now one might expect that, just as we could derive (Peirce) from (Subc)→ without using any other (non-structural) rules, reflecting the semantic fact here recorded as 7.21.9(i), so we should be able to effect a similarly unaided derivation in the converse direction, reflecting 7.21.9(ii ). As a first step in that direction, we notice that whereas the derivation given of (Subc)→ from (Peirce) in 7.21.7 appealed to (→I) as well as (→E), we can sacrifice the appeal to (→I) if we are prepared to invoke (Peirce) twice in the derivation: Example 7.21.10 A derivation of the rule (Subc)→ from (Peirce), (→E), and the structural rules: (1) Γ, A → B C Given by hypothesis (2) Γ, C, C → A A From (R), (M) and →E (3) Γ, A → B, C → A A 1, 2 (T) (4) Γ, C → A A 3 (Peirce) (5) Γ, A C Given by hypothesis (6) Γ, C → A C 4, 5 (M), (T) (7) Γ C 6 (Peirce)

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Lines (1) and (5) are premisses for an application of (Subc)→ , whose conclusion is obtained at line (7) with the use only of (Peirce), (→E) and the structural rules. Since there is no reference to (→E) in 7.21.9, however—part (ii ) does not say, for instance, only that (Subc)→ ∈ LCn({(Peirce), (→E)})—we may wonder if there is not perhaps some way of using just (Peirce) and the structural rules to derive (Subc)→ . There is no such way. To see this clearly, let us rid our minds of associations with (→I) and (→E) by rewriting the schematic descriptions of our two rules, replacing references to → by references to an arbitrary binary connective #: (Peirce)#

Γ, A # B A ΓA

(Subc)#

Γ, A C

Γ, A # B C ΓC

Observation 7.21.11 While every consequence relation (with binary # as one of the connectives of its language) satisfying (Subc)# satisfies (Peirce)# , the converse is not so. Proof. The first assertion we have already seen in the first part of the proof of 7.21.7. For second assertion, consider IL with A # B = ((B → A) → B) → B. (Peirce# ) is satisfied because IL satisfies Wajsberg’s Law – see 7.21.5 – in the form of the rule: Γ, ((B → A) → B) → B A ΓA while IL does not satisfy (Subc))# ; for example, IL q ∨ (q # p), even though the disjunction here exhibited is an IL-consequence of each of its disjuncts and so would be IL-provable if (Subc)# were satisfied. (In this application of the rule, Γ = ∅.) Note that in writing out the # in → notation, the disjunction used in this proof appears as q ∨ ((((p → q) → p) → p)) → q) → q. There is no need, in fact, to leave the →-fragment to give a counterexample, since instead of using q ∨ (q # p) we could have used (q → r) → (((q#p) → r) → r). (Cf. the remarks following the proof of 7.21.7.) The local consequence operation LCn on rules described earlier does not, then, serve as a reliable indicator of which rules are derivable, with the aid of the structural rules, from a given set of rules. We have just seen that although, as 7.21.9(ii) tells us, (Subc)→ is a local consequence of (Peirce), or, in our neutral formulation, (Subc)# is a local consequence of (Peirce)# , the structural rules do not allow us to derive (Subc)→ from (Peirce). The reader has perhaps anticipated, given our earlier introduction not only of the local consequence operation but also of its global counterpart, where the correct semantic guide to such derivability lies. The (simple) proof of the correct formulation is not included here; it may be found in Humberstone [1996b], where the following appears as Proposition 4.


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Observation 7.21.12 For any rule ρ and set of rules R (for some given language), ρ is derivable from R ∪ {(R), (M), (T)} if and only if ρ ∈ GCn(R). Remark 7.21.13 The above result is also correct for the framework Set-Set, and the analogous claim with “GCn” replaced by “LCn” is also false for that framework. Digression. We pause to provide an analysis in the style of Garson [1990] of the two rules we have been considering for an arbitrary binary connective #. That is, we describe the global ranges of (Subc)# and (Peirce)# , to make it clear how 7.21.12 bears on 7.21.11. But we leave it to the reader to verify that the description is accurate. Here “v u” means “u v”, i.e., for all formulas A of the language concerned, u(A) = T ⇒ v(A) = T. (Subc)# preserves V-validity for a class V of valuations iff for all u ∈ V, for all A, B, C ∈ L, if u(C) = F then there exists v ∈ V with v u, v(C) = F and either v (A) = T or v (A # B) = T. (Peirce)# preserves V-validity for a class iff for all u ∈ V, for all A, B ∈ L, if u(A) = F then there exists v ∈ V with v u, v(A) = F and v (A # B) = T. The proof offered for 7.21.11 amounts to showing that the condition given for (Peirce)# is satisfied, while that for (Subc)# is not, when V is taken as the class of all valuations consistent with IL , and (A # B) is construed as ((B → A) → B) → B. End of Digression. Having now explored two syntactic routes – those signposted by (Peirce) and by (Subc)→ – by which we may strengthen the condition of being an →intuitionistic consequence relation to the condition of →-classicality, we pause to note one important repercussion of arriving at the latter destination. Given a set Γ of formulas in the language of some consequence relation , let us say that Γ maximally avoids the formula C (relative to ) just in case (i) Γ C and (ii ) for all A ∈ / Γ, we have Γ, A C. (Think of (i) as ‘avoiding C’ and (ii ) as maximality in this respect.) We are especially interested in deductively closed sets (-theories) which maximally avoid some formula, and these will be called maximal avoiders. That is, a maximal avoider is a deductively closed set Γ such that for some formula C with C ∈ / Γ, we have: for all A, if A ∈ / Γ then Γ, A C. (Here we exploit the equivalence of “B ∈ Γ” and “Γ B” for deductively closed Γ. Maximal avoiders are called relatively maximal theories in Wójcicki [1988].)) Although we did not use the term, it was maximal avoiders that were delivered by the appeal to Lindenbaum’s Lemma in 2.32.6 on p. 311 (see discussion preceding that Lemma) in order to secure the primeness of deductively closed consistent sets of formulas; we return to this point in 7.21.16 on p. 1066. Here Γ’s being consistent (relative to ) just means that for some B (in the language of ), Γ B. Recall that Γ is called maximal consistent when, although Γ is consistent, every proper superset of Γ is inconsistent. Theorem 7.21.14 Relative to an →-classical consequence relation, any maximal avoider Γ is maximal consistent. Proof. Suppose Γ maximally avoids C, relative to . We want to show that if A∈ / Γ, then Γ ∪ {A} is inconsistent. Thus we have to show that for all B, we have Γ, A B. So, select an arbitrary A and B, with A ∈ / Γ. We need to show

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that Γ, A B. Since Γ maximally avoids C, either A → B ∈ Γ or A ∈ Γ, as , being →-classical, satisfies (Subc)→ , which tells us that if Γ, A → B C and also Γ, A C, then Γ C. (So we must have either Γ, A → B C, in which case A → B ∈ Γ, or Γ, A C, in which case A ∈ Γ.) Now we are given that A∈ / Γ. Therefore A → B ∈ Γ, and so Γ, A B (since is →-classical: though, by contrast with the earlier appeal →-classicality, this time it would be enough that is →-intuitionistic).

Exercise 7.21.15 Show that if Γ is a maximal avoider relative to some →classical consequence relation, then Γ maximally avoids every A such that A ∈ / Γ. Remark 7.21.16 In the proof sketched before 2.32.6, according to which whenever we have Γ IL C, a consistent, deductively closed, prime superset / Γ+ , the idea was to extend Γi to Γi+1 Γ+ of Γ can be found with C ∈ (starting with Γ0 = Γ, and ending by taking Γ+ as the union of all the Γi , i ∈ ω) by adding the formula Bi+1 if this can be done in without the result yielding C as a consequence. This, as there noted, secures the primeness of Γ+ . (If we just wanted Γ+ ⊇ Γ consistent and deductively closed, with C ∈ / Γ+ , we could simply put Γ+ = Cn IL (Γ).) An alternative and more economical way of securing this is to add Bi+1 only if we have to, either because it follows Γi or because some disjunct of a disjunction in Γi needs to be added. A more precise redescription of this strategy involves alternating steps of closing-under-consequences and ‘primification’ as follows. Let Pr ( Δ, C ), for any set Δ (of formulas) such that Δ IL C, be a superset of Δ such that (i) for each D ∨ E ∈ Δ, either D ∈ Pr( Δ, C ) or E ∈ Pr( Δ, C ), and (ii ) Pr( Δ, C ) IL C. The rule (∨E) guarantees the existence of Pr ( Δ, C ) whenever Δ IL C. Now consider Δ0 , . . . , Δn , . . . for a given Γ, C, such that Γ IL C: Δ0 = Γ Δ2i+1 = Cn(Δ2i ) Δ2i+2 = Pr ( Δ2i+1 , C ). Taking the union of all the Δi clearly gives a deductively closed prime superset of Γ, not containing C, and does so more economically than the earlier Γ+ , in the sense that it is (typically) properly included in the latter set. We conclude by looking at the relation between Peirce’s Law and the principle of subcontrariety for conditionals and their antecedents in the framework Fmla and the axiomatic approach. Let us think of them as schemata; we drop the “→” subscript on “(Subc)” here; some dots have been used to reduce parentheses: (Peirce)

(A → B → A) → A

(Subc)

(A → B → C) → (A → C → C).

We shall verify that these two principles are interderivable as candidate new axiom-schemata over BCI logic. It will be a question of reconstructing the


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proof of 7.21.7 in this setting. Getting (Peirce) from (Subc) is a simple matter of putting A for C, which give rise to an inner formula A → A to be permuted to the front and detached by Modus Ponens, leaving (Peirce). As one would expect from the proofs of 7.21.7 and 7.21.11, the converse derivation is a bit more elaborate, so to save space, we offer the following by way of BCI practice: Exercise 7.21.17 Prove the following formula in BCI logic; dots have been used to avoid excessive bracketing: (p → q → r) → ((p → s) → (s → q → r)). Thus putting A for p, B for q and C both for r and for s, in the above BCI theorem, we have the following schema all of whose instances are BCI -provable: (A → B → C) → ((A → C) → (C → B → C)). (Peirce) now allows us to replace the final (C → B) → C) here by just C, delivering (Subc). The result of adding either (Peirce) or (Subc) is the same, then, and this logic – BCIP logic in the (rather casual) style of nomenclature of 2.13 – is the implicational fragment of CL: Exercise 7.21.18 Prove K and W (weakening and contraction principles) from the axioms B, C, I and (Peirce), using Modus Ponens (and Uniform Substitution if these are taken in axiomatic rather than axiom-schematic form). The surprising potency of Peirce’s Law in this regard was first observed by Łukasiewicz in 1948 (see Borkowski [1970], pp.306–310), but it continues to turn heads (Meyer [1990]. For further references, see note 10 of Humberstone [2006b].) Specifically, what Łukasiewicz showed was that any axiomatization using Modus Ponens and uniform substitution from which B (the commuted form of B, taken here as a formula rather than a schema), Peirce’s Law, and any classical tautology of the form pi → (A → B) were provable was an axiomatization in which all implicational formulas of CL were provable. BCI + (Peirce) is easily seen to satisfy these conditions. (For the last, consider ‘Assertion’: p → (p → q → q).) We close by noting that there is a ‘Wajsbergian’ form of subcontrariety principle available in intuitionistic logic, standing to (Subc) in the same relation that Wajsberg’s Law (7.21.5) stands to Peirce’s. Let us recall the latter pair: (Peirce)

(A → B → A) → A

(Wajsberg)

[((A → B → A) → A) → B] → B.

Similarly, whereas the classical principle (Subc) is not IL-provable, intuitionistic logic does provide the variant (Subc)Wajs : (Subc) (Subc)Wajs

(A → B → C) → (A → C → C) [((A → B → C) → (A → C → C)) → B] → B.

The interested reader can confirm using the Kripke semantics or a favoured proof system that this last schema is indeed forthcoming in IL. Quite what to make of it – for example discerning what more general principle is at work here – is another matter.

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Ternary and Binary Connectives Involving Intuitionistic Implication

In this subsection, we discuss the ternary connective • (explained below) and ¨ (‘pseudo-disjunction’, from 4.22.10: p. 555), which can the binary connective ∨ be defined in terms of → in IL (or PL) by the definitions (i) and (ii) respectively, the latter being a special case of the former: (i) •(A, B, C) = (A → B) → C

¨ B = (A → B) → B ( = •(A, B, B)) (ii) A ∨

as well as another special case of (i): the binary connective we shall write as “π”, where (iii) A π B = (A → B) → A (= •(A, B, A)). Also to be touched on is the ternary connective z from 4.14, with z(A, B, C) = (A → B) ∨ C, and, after 7.22.15, a binary connective ⊃ defined by: A ⊃ B = ((B → A) → A) → B, related to an embedding due to C. A. Meredith, of the implicational fragment of CL into that of IL, as well as to the notion of head-implication developed in our 9.25. The notation in (i) is chosen arbitrarily. The two dots in the umlaut notation used in (ii) are intended to suggest the double occurrence of “B” there; ¨ they adorn a “∨” because in CL that is how the definition given would have ∨ behave. As already intimated, we have met this ‘pseudo-disjunction’ connective on several occasions already: see 4.22.10 and the other discussions there cited. Similarly, in CL A π B would be equivalent to A, the new direction (A π B A) being a form of Peirce’s Law, whereas in IL it is in general weaker. The “π” notation is intended to be evoke the initial letter of “Peirce”, even though we are now working with it in a logic without Peirce’s Law. Note that in general A ¨ B, neither of these implicaimplies (in the sense of IL ) A π B, which implies A ∨ ¨ is commutative. (Of course π is tions being reversible, and that neither π nor ∨ ¨ would strengthen not even commutative according to CL ; ‘commutativizing’ ∨ ¨ are idempotent, (full) IL to CL: see further 4.22.11(i), p. 556.) Both π and ∨ and—again in the context of the full language of IL, with ⊥ included—A π ⊥ ¨ ⊥ are each IL-equivalent to ¬¬A. and A∨ Digression. Dummett [1959b], p. 101, writes our A → B as A ⊃ B and uses “A → B” for what in the present notation would be B π A, but he gives the long-winded definiens (A ⊃ B) ∧ ((B ⊃ A) ⊃ A) instead of the simpler form (B ⊃ A) ⊃ B, thus misleadingly suggesting that π is not definable in the implicational fragment. The IL-equivalence just cited is well known, figuring for example in Lemma 2.3 of Wojtylak [1989] – though what is mentioned there is actually the following corollary of that equivalence (reverting to our own notation for this formulation): B → A IL B if and only if B → A IL A and A IL B. The longer form does have the advantage, however, of displaying p π q – again in ¨ q. The our own notation – as IL-equivalent to the conjunction of q → p with p ∨ fact that (A → B) and ((B → A) → A) are collectively IL-equivalent to (B → A) → B was mentioned above on p. 649 as an example of syntactically local conjunction in the implicational fragment of IL (not every pair of →-formulas being thus collectively equivalent to a single formula). End of Digression.


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Many further derived connectives could be considered as candidate primitives in a similar vein, such as for example the a variation on the above z with (A ↔ B) ∨ C in place of (A → B) ∨ C. A particularly interesting special case in which C = B, i.e., the case of the binary connective ⇒ (say) with A ⇒ B behaving like (A ↔ B) ∨ B. As with ⊃ mentioned above à propos of Meredith (not à propos of Dummett), classically ⇒ amounts to → while intuitionistically, we get something non-equivalent, a discussion of which may be found in §4 of Humberstone [2000b]. A similar idea of “disjoining B” can be considered for the case in which what we start with is an implicational connective. Against the background of IL there would be nothing to be gained by considering (to use the same notation again) A ⊃ B defined as (A → B) ∨ B, since ⊃ and → would be equivalent (i.e., form equivalent compounds from the same components). But in a relevant-logical setting, which is to say, in the absence of the K scheme (from BCI, BCK, etc.) a ⊃-compound is weaker than the corresponding →compound and may exhibit interesting and desirable logical behaviour. This is shown in the case of the logic RM in Avron [1986]. (For further discussion, see Blok and Raftery [2004].) A general discussion of issues falling under the heading of connectives “involving” intuitionistic implication would include not only those definable in terms of → in (full) IL, but also those whose logical behaviour is described by rules making essential use of the latter connective, even when they are not so definable. One example would be the ‘anticipation’ connective described in the discussion leading up to 4.38.10 (p. 626). We will have enough on our plate with the de¨ , and π. Of course, the finable case, however, and return to the cases of •, ∨ definitions given above for these connectives make sense for stronger or weaker logics with → in their languages, though we concentrate on the IL case. (An ¨ in the setting especially of BCK logic may be found in algebraic discussion of ∨ Torrens [1988]; cf. also our discussion – of what was there labelled Q – between 2.13.16 and 2.13.17: p. 237. ...Recall also the purely implicational Church disjunction derived connective ∨ from 2.13.13(ii), p. 234.) And our main interest is in taking them as primitive rather than as defined, but behaving as the above definitions dictate. In particular we do not assume that → is present in the language, and want to look at the sequents in Set-Fmla that would be provable if → were present and the above definitions were in force (though of course only insofar as → is absorbed into one of the derived connectives mentioned). Equivalently, for the cases of (i)–(iii), we imagine separately imposing the following three conditions on the relation |= between Kripke models M (assumed = (W, R, V ), the condition (Persistence) in force as in 2.32), points x ∈ W , and formulas: [•] M |=x •(A, B, C) iff for all y ∈ R(x), if ∀z ∈ R(y)[M |=z A ⇒ M |=z B], then we have M |=y C; ¨ B iff for all y ∈ R(x), if ∀z ∈ R(y)[M |=z A ⇒ M |=z B], ¨ ] M |=x A ∨ [∨ then we have M |=y B; [π]

M |=x A π B iff for all y ∈ R(x), if ∀z ∈ R(y)[ M |=z A ⇒ M |=z B], then we have M |=y A.

As usual, a sequent Γ A holds at point in a model iff it is not the case that all formulas in Γ are true at that point and A false there, a sequent holds in a

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model if it holds at every point thereof, and is valid on a frame if it holds in every model on that frame. When Γ = ∅, we also say that formula A is itself valid on the frame. We begin by considering the case of the ternary connective •. (Proof follows – or rather, proofs follow – in the discussion.) Observation 7.22.1 If A is any formula built up using no connectives other than •, and (W, R) is any frame, then: A is not valid on (W, R). We give two proofs of 7.22.1, since this can be done briefly and each argument is of some interest. Firstly, if A were a formula valid on some frame, then the result of replacing all subformulas •(B, C, D) of A by (B → C) → D would be valid on that frame, and this can only happen if the result of making all such replacements is a (classical) tautology. But no formula constructed from propositional variables by means only of this mode of composition can be tautologous, since the corresponding truth-function (b → c) → d = ¬(¬b ∨ c) ∨ d = (b ∧ ¬c) ∨ d always gives the value F when b = c = d = F. Thus a formula of the kind under consideration is false on any boolean valuation on which every constituent propositional variable is false. The second argument does not explicitly involve the detour through CL. Suppose A is a formula built up using only • and A is valid on some frame. Let A be the result of (uniformly) substituting p for every propositional variable in A. Since Uniform Substitution clearly preserves validity on a frame, A must be valid on the frame in question. But this cannot happen, since an induction on the complexity (here: number of occurrences of •) of A reveals that A has the same truth-value as p at any point in a model. (The argument is that any formula built by • from p is equivalent to p. If the formula is of complexity 0, it just is p; and if it is of complexity n > 0, it is of the form •(B, C, D) for B, C, D, of lower complexity and hence by the inductive hypothesis, each equivalent to p. But in that case, as one sees most easily by rewriting •(B, C, D) as (B → C) → D, since (p → p) → p is equivalent to p, the compound is itself again equivalent to p. By “equivalent” here is meant intuitionistically equivalent. However, we recall from 7.21.7 – and this was the reason for saying that CL was not explicitly involved – that IL and CL do not differ in respect of consequences amongst the implicational formulas in any one variable.) A simple consequence of 7.22.1 is worth recording, in terms of the notion of definability introduced in 3.15: Corollary 7.22.2 Intuitionistic implication is not definable in the Set-Fmla logic (with • as sole primitive connective) determined by the class of all frames. Proof. If → were definable, we should have p → p as a formula valid on all frames; but 7.22.1 shows there are no such formulas in the •-language. In fact, 7.22.1 shows that no formula has the property of being valid even on some frame; but this certainly implies that no formula has the property of being valid on every frame. Thus the consequence relation on the pure •-language is atheorematic (as defined in 3.23.10). We turn to the task of providing a complete proof system for the sequents valid on all frames. We assume for this purpose that • is the sole connective, which allows us to assert that the proof system based on the rules which follow


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is complete w.r.t. the class of all frames. For brevity •(A, B, C) will be written as •(ABC) or sometimes just as •ABC. (1) •CAB, A B

(2)

(3) C •(AB•(CAB))

(4)

Γ, A B Γ, •CDA •CDB Γ, A B Γ, •ABC C

The proof system we have in mind takes the above four rules alongside the structural rules (R), (M), and (T). We denote the associated consequence relation by . It is clear that all instances of (1) and (3) are valid on every frame and that (2) and (4) preserve validity on any given frame, so our proof system is sound w.r.t. the class of all frames. Note that rule (2) amounts to saying, in the terminology of 3.32, that • is monotone with side formulas in the third position. For completeness, we use a canonical model (W, R, V ) in which R and V are defined as in 2.32 and W comprises the set of all non-empty deductively closed sets (-theories, that is). Since ∅ is deductively closed, this is a genuine restriction: this particular -theory does not appear in W . We return to this point below (in the proof of 7.22.5). Theorem 7.22.3 For all formulas D, we have in the canonical model M = (W, R, V ) defined above: for every x ∈ W, D ∈ x ⇔ M |=x D. Proof. The proof is by induction on the complexity of D. The basis case, in which D is pi , is settled by the definition of V , as usual. For the inductive step, suppose D is •ABC. By the inductive hypothesis, it suffices to show that •ABC ∈ x if and only if for all y ⊇ x such that for all z ⊇ y, A ∈ z ⇒ B ∈ z, we have C ∈ y. (Here y, z, range over W .) We take the ‘only if’ direction first. Suppose, then that •ABC ∈ x and that we are given y with A ∈ z ⇒ B ∈ z for all z ⊇ y. This is equivalent to saying that y, A B. Thus by appeal to rule (4), y, •ABC C. Since •ABC ∈ x and x ⊆ y, •ABC ∈ y, so y C; thus C ∈ y, as was to be shown. For the ‘if’ direction, we argue contrapositively. Suppose •ABC ∈ / x. We must find y ⊇ x with A ∈ z ⇒ B ∈ z for all z ⊇ y, and yet C ∈ / y. As before, this means we want y ⊇ x such that y, A B while y C. Take some formula E ∈ x (recalling that x = ∅) and let y be the deductive closure of x ∪ {•EAB}. We claim that y, A B and that y C, as required. For the first part of this claim, we use rule (1), which (putting E for the schematic letter C in the formulation of the rule) gives x, •EAB, A B; thus y, A B. For the second part of the claim, suppose—for a contradiction—that y C; that is: x, •EAB C. By rule (2), then, x, •(AB•(EAB)) •ABC. Rule (3) gives: E •(AB•(EAB)). Therefore (by (M) and (T)): x, E •ABC. But E ∈ x, so x •ABC; since ex hypothesi, •ABC ∈ / x, this refutes the supposition that y C and completes the argument.

Remark 7.22.4 The appeals to rules (2) and (3) at the end of this argument could clearly be replaced by an appeal to a single rule, namely:

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Γ, •EAB C Γ, E •ABC Since this is also a validity-preserving rule, it alongside rules (1) and (4), provides a sound and complete rule-system (see the following Corollary). Another variation on the proof system will be mentioned below (7.22.6). Corollary 7.22.5 The proof system with rules (1)–(4) is sound and complete w.r.t. the class of all frames. Proof. Soundness is clear. For completeness, suppose that Γ A is not provable, with a view to showing that this sequent is not valid (on all frames). There are two cases to consider. Firstly, suppose Γ = ∅. Then the sequent is not valid, by 7.22.1. Next, suppose Γ = ∅. Then the deductive closure of Γ, call it x, is an element of the canonical model at which, by 7.22.3, all formulas in Γ are true, but A is not. The sequent is therefore not valid on the frame of this model. The proof of 7.22.3 can be thought of as showing how to simulate intuitionistic “→” in a setting in which it is not definable (7.22.2); although there is no compound in A and B which behaves throughout the canonical model as A → B behaves, for each element of the model, there is a compound behaving appropriately. That is, while (5) is false, (6) is true: (5)

∀A,B ∃C ∀u ∈ W [C ∈ u ⇔ ∀v ⊇ u(A ∈ v ⇒ B ∈ v)]

(6)

∀u ∈ W ∀A,B ∃C[C ∈ u ⇔ ∀v ⊇ u(A ∈ v ⇒ B ∈ v)]

For the C promised by (6) in the case of a given u, A and B, we select E ∈ u and put C = •(EAB); but the choice of E, and therefore of C, varies from one u ∈ W to another, since as we know (7.22.1), there is no formula belonging to every u ∈ W . We can emphasize this aspect of the situation by replacing the rules (1)–(3) above with rules more directly reflecting the idea of •-compounds as local conditionals: Exercise 7.22.6 Show that the proof system with rule (4) above, alongside the following two rules E, •EAB, A B

Γ, E, •EAB C Γ, E •ABC

is sound and complete w.r.t. the class of all frames. (Note that the rules respectively serve as surrogates for →-elimination and introduction rules for the ‘local’ → of the preceding discussion.) ¨ and π, for which the situation Let us turn to the derived binary connectives ∨ is somewhat more problematic, in that, as we shall observe, the straightforward kind of canonical model completeness proof we are accustomed to appears not to be available. (A complication on the usual procedure will see us through, ¨ B as (A → B) → B and A π B as however.) Recall that we interpret A ∨ (A → B) → A, though we do not assume that → is in the language; that is, the definition of truth is cast in such terms as would give the present binary


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¨] connectives the same interpretation as → if the latter were present. (See [∨ and [π] in the second paragraph of this subsection for the appropriate clauses in the definition of truth.) “Sound and complete” above meant sound and complete ¨. w.r.t. the class of all frames. We may begin with some practice in ∨ Exercise 7.22.7 Verify that the following rules all preserve the property of holding in an arbitrary model: (i) (iii)

Γ, A B ¨BB Γ, A ∨ ¨ B AA∨

(ii)

¨BB Γ, A ∨ Γ, A B

¨ B. (iv) B A ∨

Note that Rule (ii) is redundant, given (iii): given a premiss for (ii): ¨ B B, we get the conclusion of (ii) by (M) and (T) using (iii), Γ, A ∨ ¨ B as cut formula. Note also that for (iii) and (iv), showing with A ∨ that the rule preserves holding in an arbitrary model means showing that any sequent of the form in question holds in an arbitrary model, since we are dealing with zero-premiss sequent-to-sequent rules here. To show the completeness of some such collection of rules as those in 7.22.7, one would like to take the canonical model built using all deductively closed sets (all -theories, where is the consequence relation associated the proof system thus presented), the canonical R and V defined as in 2.32, and show – ¨ ] – that for all A, B: corresponding to [∨ ¨ B ∈ x iff for all y such that xRy, A∨ if A ∈ z implies B ∈ z for every z such that yRz, then B ∈ y. The left-to-right direction is straightforward, using the rules of 7.22.7: ¨ B ∈ x and y ⊇ z Suppose, with a view to showing that B ∈ y, that A∨ with A ∈ z ⇒ B ∈ z for every z ⊇ y. (All variables range over W , the set of deductively closed sets of formulas.) This means that y, A B, where is the (finitary) consequence relation associated with the proof system based on the structural rules and those given above, in which case, by rule (i): ¨ B B, y, A ∨ ¨ B ∈ x and x ⊆ y, we have A ∨ ¨ B ∈ y and therefore B ∈ y. so since A ∨ The converse direction – that whenever for all y ∈ R(x), if A ∈ z implies ¨ B ∈ x – is problemthat B ∈ z for every z ∈ R(y), then B ∈ y, we have A ∨ atic. (Changing “deductively closed” to “non-empty, deductively closed” in the specification, to match the treatment of • above, would not help us out of the difficulties to be described.) To see why, consider how an argument aimed at establishing the contrapositive would go, in terms of any rules (defining ) we might want to help ourselves ¨B ∈ to (without jeopardising soundness). Assume A ∨ / x. We must find y ⊇ x satisfying the condition: / y. A ∈ z ⇒ B ∈ z for every z ⊇ y but with B ∈

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Now, the inset condition is equivalent to: y, A B. So what we have to show ¨ B∈ is that whenever A ∨ / x, there exists y ⊇ x such that (i) y, A B while (ii) y B. In the presence of the connective →, we know how to find such a y, namely by taking y as the deductive closure of x ∪ {A → B}, but we are not here assuming that → is available (with →-intuitionistic). Suppose, for a revealing though extreme case, that the only connective in the language of ¨ itself. For there to be, thwarting the line of argument we have been is ∨ considering, no y ⊇ x satisfying (i) and (ii) above, is for it to be the case that for every set Δ of formulas: x, Δ, A B ⇒ x, Δ B. In other words, given the set x, no additional assumptions are available which would help A to yield the conclusion B unless they yielded that conclusion without the need for A. (The earlier y we wanted to satisfy (i) and (ii) is of course {x} ∪ Δ for suitable Δ.) Now this is just the situation we find ourselves ¨ r. This sequent is clearly invalid, and in with, for example, the sequent p q ∨ therefore unprovable in any sound proof system (such that using rules 7.22.7(i)– ¨ r∈ (iv) above). Where x is the deductive closure of {p}, then, we have q ∨ / x. But is there a set Δ such that x, Δ, q r while x, Δ, r: in other words, such that p, Δ, q r while p, Δ, r? Unfortunately, this answer to this is No, as we now show. Let Δ∨ be the set of formulas C∨ resulting from replacing every occurrence ¨ in C ∈ Δ by an occurrence of ∨. If we are to have p, Δ, q r for the of ∨ associated with a sound proof system, then clearly p, Δ, q IL r, so p, Δ, q CL r and so p, Δ∨ , q CL r (since (A → B) → B CL A ∨ B). But IL and CL do not differ where the only connective concerned is ∨; at the end of 6.46 we called this shared fragment ∨ , and showed (6.46.9) it to be left-prime. Thus, since p ∨ r and q ∨ r, it follows from the hypothesis that p, Δ∨ , q ∨ r that for some A∨ ∈ Δ∨ we have A∨ ∨ r. We can see that for any formula A∨ if A∨ ∨ r then A r, where is the consequence relation (on the language ¨ ) determined by the class of all frames. For if A∨ ∨ r whose sole connective is ∨ then A is a disjunction (of disjunctions of . . . disjunctions) whose (disjuncts’ disjuncts’. . . ) disjuncts are occurrences of A (cf. 6.42.10) and since the onevariable implicational fragments of IL and CL coincide (7.21.1), it follows that ¨ C of A as (B → C) → C.) A r. (Rewrite subformulas B ∨ C of A∨ and B ∨ Thus, since A ∈ Δ, we cannot have what we need for any canonical model completeness argument to go through: p, Δ, q r without p, Δ r. Thus to prove the completeness of a syntactic characterization of , some alternative conception of what a ‘canonical’ model might be is called for, to the task of providing which we now turn. Let us begin by recalling the clause: ¨ ] M |=x A ∨ ¨ B iff for all y ∈ R(x), if ∀z ∈ R(y) M |=z A ⇒ M |=z B, [∨ then we have M |=y B. as well as some of the rules – namely the three non-redundant ones from 7.22.7, though differently numbered here: ¨ 1) (∨

¨ B AA∨

¨ 2) B A ∨ ¨ B (∨

¨ 3) (∨

Γ, A B ¨BB Γ, A ∨


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For a consequence relation satisfying the above rules, the canonical model is (W, R, V ) where W is the set of pairs Γ, for which Γ is a -theory; if x = Γ, , we may write Γ, as Γx , x . Thus the novelty here – reminiscent of some parts of Hazen [1996] – is that the points in the canonical model carry a consequence relation with them, so that the logic (as it were) changes as we pass from point to point. This is a somewhat picturesque way of putting things, however, since we do not require that the consequence relations x should be substitution-invariant, or even invariant under variable-for-variable substitutions – cf. §1.2, Appendix (p. 180). However, we emphasize that in demanding ¨ 3) what we are requiring is that whenever Γ, A B, we that should satisfy (∨ ¨ have Γ, A ∨ B B. (This is not ensured by saying that ⊇ 0 , where 0 is the least consequence relation satisfying the rules.) We define Rxy to hold iff both Γx ⊆ Γy and x ⊆ y , and set V (pi ) = { Γ, | pi ∈ Γ}. Since this definition settles the basis case of the proof that for all formulas C, (W, R, V ) |=x C iff ¨ B, which, given the inductive C ∈ Γx , we pass to the inductive case of C = A ∨ hypothesis, amounts to: ¨ B ∈ Γx iff for all y ∈ R(x), if for all z ∈ R(y), A ∈ Γz ⇒ B ∈ Γz , then A∨ B ∈ Γy . ¨ B ∈ Γx and we have y ∈ R(x) satisfying: ‘Only if’ direction: Suppose A ∨ (7)

for all z ∈ R(y), A ∈ Γz ⇒ B ∈ Γz .

We must show on these hypotheses that B ∈ Γy . (7) is equivalent to the claim ¨ 3), Γy , A ∨ ¨ B y B, so since Rxy and A ∨ ¨ B ∈ Γx , we that Γy , A y B, so by (∨ ¨ B ∈ Γy and thus Γy y B, so B ∈ Γy , since Γy is required to be a have A ∨ y -theory. ¨B∈ ‘If’ direction, argued contrapositively: Suppose A ∨ / Γx . We need y ∈ R(x) such that for all z ∈ R(y), A ∈ Γz ⇒ B ∈ Γz , while B ∈ / Γy . Let be the ¨ 1), (∨ ¨ 2) and (∨ ¨ 3), and the least consequence relation extending x , satisfying (∨ further condition that A B. We claim that the above demands on y are met when we take y = Γx , . The first point to check is that y ∈ W , i.e., that Γx is a -theory, which requires checking that we do not have A ∈ Γx while B ∈ / Γx . We know that this does not happen, however, since A ∈ / Γx : for otherwise we ¨ 1)), A ∨ ¨B ∈ should have (by (∨ / Γx , contrary to our initial supposition. Thus it remains to check only that (i) y ∈ R(x), that (ii) for all z ∈ R(y), A ∈ Γz implies B ∈ Γz , and (iii) that B ∈ / Γy . (i) and (ii) are immediate from the way ¨B∈ y was specified, while (iii) follows from our initial supposition that A ∨ / Γx , ¨ 2). Accordingly, we have: via (∨ ¨ 1)–(∨ ¨ 3) alongside the usual strucTheorem 7.22.8 The proof system with (∨ tural rules is sound and complete with respect to the class of all frames, when ¨ ]. truth at a point in a model is understood in accordance with [∨ Proof. Only the completeness half of the claim remains to be shown, and we derive this from the above discussion as follows. That discussion shows that if a sequent Γ C is not provable in the present proof system with, then this sequent fails in the canonical model (W, R, V ) described above, specifically at the point Γ+ , 0 in which 0 is the consequence relation associated with the ¨ 1)–(∨ ¨ 3)) and Γ+ is the proof system (the least consequence relation satisfying (∨

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smallest 0 -theory ⊇ Γ.

When we turn our attention to π, we find that though the precise problem encountered above does not arise in so acute a form, there are again difficulties in establishing that if a compound of the type in question (here A π B, which is to behave like (A → B) → A) does not belong to a point in the canonical model of the traditional kind, then a certain condition which amounts, by the inductive hypothesis, to the formula’s being true at that point, is not satisfied. By contrast ¨ , there are conspicuous examples of assumptions cooperating with the case of ∨ to yield conclusions. For example, we have a ‘converse Modus Ponens’ principle A, B π A B. In fact, we can even weaken A with the aid of B and still have the same rhs: A π B, B π A B. Indeed, given the idempotence of π, noted at the start of this subsection, this schema is a special case – taking C as A – of a transitivity principle: A π B, B π C A π C. (For the description of such a principle in the terminology of transitivity in see: 3.34.4 – p. 502 – and preceding discussion.) The idempotence of π follows from the rules below (see 7.22.9(ii)). The fact that B π A serves as a bridge from A to B does not appear to help with proving, in a standard canonical model argument, the ‘if’ direction of A π B ∈ x iff for all y ∈ R(x), if A ∈ z ⇒ B ∈ z for every z ∈ R(y), then A∈y ¨ ) no difficulties – here using the rule whose ‘only if’ direction presents (as with ∨ (π2) – and rather than explore them at length, we invite the reader (7.22.9(v)) ¨ case here. (However, it would be of interest to to mimic our treatment of the ∨ know whether a standard canonical model argument, without recourse to the Γ, elements, is available.) (π1) A A π B

(π2)

Γ, A B Γ, A π B A

Exercise 7.22.9 (i) Verify that the above rules preserve the property of holding in an arbitrary model. (ii) Show that p p π p, where is the consequence relation associated with the proof system presented by the rules (π1), (π2). (iii) Where this time is the consequence relation determined by the class of all frames, show that π is monotone with side formulas in the first position, and antitone with side formulas in the second position. (These concepts were introduced in 3.32.) What are the corresponding ¨? positional ‘tonicity’ properties of ∨ (iv) With as in (iii), show that for all A, B, C: (A π B) π (B π C) A π B.


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(v) Prove that the proof system with the structural rules and (π1), (π2), is not only sound – as follows from part (i) above – but also complete, with respect to the class of all frames, by adapting the non-standard canonical model argument given for 7.22.8. (Because of this result, the consequence relations introduced in (ii) and (iii) above coincide.) Remark 7.22.10 The example in (iv) of the above Exercise is taken from Kabziński and Wroński [1975], Lemma 1(viii). The authors are discussing (from an algebraic perspective) the ↔-fragment of IL; our A π B is definable in that fragment as (A ↔ B) ↔ B. Note that ↔ is not definable in terms of π, since the of 7.22.9(ii), (iii), is atheorematic in the sense introduced after 7.22.2—as follows from 7.22.1—while for ↔ we have, e.g., IL p ↔ p. (Note also that it is B, not A, which puts in a double appearance in the ↔ formulation.) ¨ and π In summary, we did not succeed with our two binary derivatives ∨ in providing a standard canonical model argument for the completeness of our rules, by contrast with the case of the ternary •, and had instead to resort to some novelties – in particular to allowing different consequence relations to figure in the argument. Such novelties notwithstanding, the solution presented above seems more conservative than a response which would abandon (or liberalize) the familiar conception of rules and sequents, adopting, say the device of “assumption-rules” à la Schroeder-Heister: see the references in 4.14 to this response, where it is mentioned as a possible reaction in the face of an example from Zucker and Tragesser [1978] of a ternary connective (we write as) z with introduction rules conferring upon z(A, B, C) the meaning (A → B) ∨ C. The problem there was to provide suitable elimination rules. A Schroeder-Heister ¨ has indeed been offered in ‘assumption-rules’ style of treatment for the case of ∨ Hazen [1996] (in which Fitch [1952] is cited as an inspiration), which we regard as interesting but as giving up prematurely on the problem of finding a suitable (orthodox) Set-Fmla presentation. At any rate, such a presentation is not hard to find for the case of z, whatever the difficulties may be about appropriate elimination rules in a specifically natural deduction approach (or indeed the sequent calculus approach – discussed à propos of z in 2.32.14, p. 322). The point of interest with our ternary • above was that although we could not define → in terms of it, we could get enough of a ‘local simulation’ of → for a canonical model completeness proof to go through. In the case of z matters are more straightforward, since we can define → with its aid; first, we recall the intended semantics (where M = (W, R, V ) is again a Kripke model of the type used for IL): [z]

M |=x z(A, B, C) iff either for all y ∈ R(x)[M |=y A ⇒ M |=y B], or else M |=x C.

Then A → B can be defined as z(A, B, B). Thus can be defined as z(p, p, p) (or indeed z(p, p, q)); we could then define A ∨ B as z(, A, B). We could then treat these defined connectives in the same way that their primitive namesakes were treated in INat in 2.32, and replicate the relevant parts of the completeness argument there for the resulting system extended by a ‘defining back’ of z: the principle z(A, B, C) (A → B) ∨ C and its converse (in primitive notation:

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z(A, B, C) z(z(p, p, p), z(A, B, B), C) and its converse). We will pursue a somewhat less artificial course, though still exploiting the definabilities noted. We leave the reader to check the following: Observation 7.22.11 The proof system with the following rules is sound w.r.t. the class of all frames, when truth is defined using [z]: (i)

Γ, A B Γ z(A, B, C)

(iii) A, z(A, B, B) B

(ii) C z(A, B, C) (iv)

Γ, C E

Γ, z(A, B, D) E

Γ, z(A, B, C) E

Call a set of formulas z-prime if whenever, for formulas A, B, C, it contains z(A, B, C) it either contains C or else contains z(A, B, D) for all formulas D. This is an analogue of the property of primeness (2.32) for the present setting, since it simulates for the z-language the idea that if z(A,B,C), it either contains C (the second disjunct of (A → B) ∨ C) or else behaves as though it contained A → B (the first disjunct). Lemma 7.22.12 Where is the consequence relation associated with the proof system whose rules are (R), (M), (T), and the rules of 7.22.11, if Γ C then there is a z-prime -closed set Γ+ ⊇ Γ such that Γ+ C. Proof. As indicated for the case of 2.32.6 (p. 311), the z-primeness part of the argument being made by appeal to rule 7.22.11(iv). We take the canonical model for the logic described in 7.22.11 (with the structural rules) to be (W, R, V ) in which W consists of all deductively closed z-prime sets of formulas and R and V are defined in the usual manner. In view of the semantic clause [z], to demonstrate that truth and membership coincide for arbitrary formulas relative to points of the canonical model, we will want to show (for all formulas A, B, C, all points x): z(A, B, C) ∈ x iff either for all y ∈ W such that y ⊇ x, A ∈ y ⇒ B ∈ y, or else C ∈ x. The ‘if’ direction is established with the aid of the rules 7.22.11(i), (ii). For the converse, suppose z(A, B, C) ∈ x and C ∈ / x, with a view to concluding that for all y ⊇ x, if A ∈ y then B ∈ y. Since x is z-prime and z(A,B,C) ∈ x while C∈ / x, z(A,B,D) ∈ x for all formulas D, and hence in particular for D = B. Now if y ⊇ x, then accordingly z(A, B, B) ∈ y also, and so by appeal to 7.22.11(iii), if A ∈ y, B ∈ y (since y is deductively closed). We conclude: Theorem 7.22.13 The proof system described in 7.22.12 is sound and complete w.r.t. the class of all frames. Exercise 7.22.14 In the argument preceding 7.22.13, we appealed to the rule 7.22.11(iv ) in effect only for the special case of D = B. Replacing that rule by this special case, together with the remaining rules, can the general form of the rule be derived? (Either explain why not or give a derivation.)


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We turned, after our earlier successful semantic analysis of the ternary compound •(A, B, C), to an unsuccessful examination of the binary compounds ¨ B and A π B which result from identification of variables (C with B and C A∨ with A, respectively). Identifying A and B in •(A, B, C) gives nothing of any interest, the result being equivalent to C. (Thus the complete logic is obtained by stating this equivalence.) In the case of the ternary z(A, B, C) putting B = A just gives a valid formula (so the complete logic is obtained by means of the schema z(A, B, C)). Identifying B with C would give, as our discussion has indicated, another notation for A → B. The remaining possibility, setting C = A, does indeed give something novel, whose semantic treatment is problematic ¨ and π were: in something of the way that of ∨ Exercise 7.22.15 Explore the problem of finding a set of rules which give a proof system sound and complete w.r.t. the class of all frames, for the language with a binary connective z interpreted in Kripke models by the clause: [z ] M |=x A z B iff either for all y ∈ R(x) with M |=y A we have M |=y B, or else M |=x A. As was mentioned at the start of our discussion, the definitions here used for the binary and ternary connectives we have been considering could instead be understood against the background of some logic for → (and ∨) other than IL. Many aspects of the discussion would be changed by choosing a weaker logic to play this role, such as (positive) R. For example, the equivalence exploited in our discussion of z, of z(A, B, B) = (A → B) ∨ B with A → B, would no longer hold in the setting of relevant logics such as R, since A → B does not (in general) follow from each disjunct of the cited disjunction: specifically, not ¨ -Introduction” inferences (i) from A to from the second. Similarly, of the two “ ∨ ¨ B and (ii) from B to A ∨ ¨ B, only (i) would survive transplantation to this A∨ habitat. Returning to the setting of intuitionistic (or positive) logic, the ternary compounds we have considered fail conspicuously to be what we called clear formulas in 7.14, which means also (7.14.8, p. 987) that they are not topoboolean compounds of their components. No novelties would be raised by investigating, for example, the ternary # for which #(A, B, C) is understood as A → (B → C), or as A → (B ∨ C). But whether, and if so exactly how, the difficulties arising ¨ , π, z ) of • and z are connected with this feature for the binary derivatives (∨ of them, is a question we are not in a position to answer. For the final connective we shall consider, the question of the semantic completeness of a pure proof system will not concern us. We denote it by “⊃”, where A ⊃ B is defined to be the formula ((B → A) → A) → B. And by contrast with the preceding discussion, let us change the framework from Set-Fmla to Fmla, for the sake of historical accuracy. It was in Fmla that C. A. Meredith – see the notes (p. 1122) to this section – made the observation that the behaviour of classical material implication (i.e., → as it behaves in CL) is simulated within pure-implicational IL by ⊃ as just defined. (We give this result as 7.22.29 below, p. 1086.) It should be noticed that while implication is often regarded as ‘weaker’ in CL than in IL, the formula p ⊃ q has as a consequence (according to IL ) p → q, rather than being a consequence of the latter formula.

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Aside from its role in the exercise of translational embedding à la Meredith, the derived connective ⊃ is of considerable interest for the study of IL in its own right; the IL-provability of A ⊃ B corresponds to what in 9.25 we shall call A’s ‘head-implying’ B (in IL), by which is meant roughly that A implies B in the manner in which q implies p → q. (See 9.25 for details, and 9.25.11, p. 1324, for the correspondence just described – though there we put it in terms of B’s being a consequence of (B → A) → A, rather than of the provability of A ⊃ B.) A ⊃ B also proves of interest, defined in terms of → as above, in the context of non-classical logics other than IL, though we shall not pursue that theme; Meyer and Slaney [1989], p. 261, for example, consider its introduction in the relevant(-like) logic RM, under the name of ‘intensional implication’. Digression. The symbol “⊃” is often used in place of “→” when an implication more closely resembling (in one respect or another) implication in CL is being distinguished from an implication being notated with the aid of →. Heyting [1956], p. 116, introduces A ⊃ B to abbreviate ¬(A ∧ ¬B), a formula typically weaker rather, than as our A ⊃ B is, stronger than A → B (according to IL ; a similar remark is made at the base of p. 220 of Meredith and Prior [1968] about a proposed simulation of classical → within IL appearing in Łukasiewicz [1952]). The similarity to the behaviour of such ⊃-compounds to that of the corresponding →-compounds in CL (provided we restrict attention to the framework Fmla) is evident from the fact that IL and CL agree completely on {∧, ¬}-formulas (in the framework in question): see 8.21.5 on p. 1216. (Temporary use of “⊃” was made in the discussion immediately following 2.31.3, p. 301, in order to evoke such ‘material implication’ associations even though no other use was there given to “→”.) End of Digression. In the notation used earlier in this subsection, A ⊃ B is (for any given A, ¨ A) → B, whose CL-equivalence with A → B is evident from B) the formula (B ∨ ¨ A with B ∨ A and the equivalence, even within IL, the CL-equivalence of B ∨ of (B ∨ A) → B and A → B. Exercise 7.22.16 (i) What would happen if, instead, we defined A ⊃ B = ¨ B) → B? (I.e., does this give a formula IL-equivalent to A ⊃ B, (A ∨ as defined above, or to A → B, or to something else again?) (ii ) For simplicity, we invoke ∧ in this exercise, though the point could be made slightly less concisely in the →-fragment. Show that for all formulas A, B, we have A ⊃ B IL (A → B) ∧ (((B → A) → B) → B). Note that the second conjunct on the right is an instance of Peirce’s Law, (taken as a schema), giving a somewhat different perspective on the collapse of ⊃ into → in CL (where this conjunct is always redundant because CL-provable). The fact that ⊃ is stronger than → in IL – in terms of the discussion at the end of 3.24, we are speaking here of deductive or inferential strength – may tempt us to consider, by analogy with the relationship between material and strict implication in, say, S4, the possibility of adding a singulary congruential connective Ω, say, for which we have A ⊃ B equivalent to Ω(A → B). But any such addition would yield a non-conservative extension of IL (in fact collapsing


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it into CL), since we can have A → B IL C → D even though we do not have – what would follow be prefixing the supposedly congruential Ω to each side – A ⊃ B IL C ⊃ D. Example 7.22.17 For a case in which we have A → B IL C → D but not A ⊃ B IL C ⊃ D, take A = p, B = D = q, C = (p → q) → q. The equivalence of A → B with C → D is then the Law of Triple Consequents (2.13.20(ii), p. 239), while A ⊃ B is ((p → q) → q) → p and C ⊃ D reduces to the IL-nonequivalent p → q (to A → B itself, then). Remark 7.22.18 The fact that ⊃ is (in IL) stronger than → should does not mean that the corresponding biconditional, ≡, say, where A ≡ B is defined as (A ⊃ B) ∧ (B ⊃ A), is stronger than ↔. (Here, as in 7.22.16, we step outside the implicational fragment in the interests of concision.) It is not hard to show that in fact we have A ↔ B IL A ≡ B. Let us now, following Meredith, define a translation (·)∗ which replaces → by ⊃: p∗i = pi

(A → B)∗ = A∗ ⊃ B∗ .

Note that although denoted with the assistance of “⊃”, a formula A ⊃ B is itself an →-formula, so * maps →-formulas to →-formulas. (Compare the translation mentioned in 6.42.3, p. 873, from modal to non-modal formulas.) In terms of the primitives, (A → B)∗ is of course ((B∗ → A∗ ) → A∗ ) → B∗ . Exercise 7.22.19 (i) We noted above that A ⊃ B IL A → B. Is it generally true that C∗ IL C? (Provide a proof or a counterexample.) (ii ) Show that there is no function f (on the set of natural numbers) such that for all formulas A, Comp(A∗ ) = f (Comp(A)), Comp(A) being the complexity ( = number of primitive connectives in the construction of) the formula A. For further practice with the behaviour of ⊃ and →, let us consider the Fmla form of Peirce’s Law (continuing our numbering from before Theorem 7.22.8): (8)

((p → q) → p) → p

and ask what the formula (8)* looks like in primitive notation; using our abbreviative convention, (8)* is the formula (9)

((p ⊃ q) ⊃ p) ⊃ p

Rather than unpacking (9) completely in terms of →, which gives a very unwieldy formula, we will satisfy ourselves with finding a formula IL-equivalent to (9). First, we spell out the import of the main occurrence of “⊃”, replacing it in accordance with the definition in terms of →; to reduce parentheses we write p ⊃ q ⊃ p in place of (p ⊃ q) ⊃ p (and similarly elsewhere; likewise p ⊃ q ⊃ p would abbreviate: p ⊃ (q ⊃ p)). We have not generally availed ourselves of such ‘dot conventions’, but the degree of embedding makes their usage here all but mandatory to reduce a clutter of bracketings. (10)

[(p → (p ⊃ q ⊃ p)) → (p ⊃ q ⊃ p)] → p.

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Now the antecedent of (10) itself has an antecedent which when the occurrence of “⊃” which is in the scope of → but not in the scope of another occurrence of “⊃” is spelt out in terms of → looks like this: (11)

p → [((p → p ⊃ q) → (p ⊃ q)) → p].

Since this formula is of the form p → (A → p), it is IL-provable and we can dispense with any further unravelling of it, since (10) will be IL-equivalent to the result of omitting this antecedent of its antecedent, the result of which omission is to simplify (10) to (12): (12)

(p ⊃ q ⊃ p) → p.

Unpacking the main “⊃” in the antecedent reveals (12) as: (13)

[((p → p ⊃ q) → (p ⊃ q)) → p)] → p

and hence, dealing with the remaining occurrences of ⊃: (14)

{[[(p → ((q → p → p) → q)] → ((q → p → p) → q))] → p} → p.

The subformula [(p → ((q → p → p) → q)] can be simplified to p → q, since it is not hard to see that for any formulas A and B p → (A → p → B) IL p → B. (In our example, A and B are respectively q → p and q.) The resultant simplification of (14) is (15)

{[(p → q) → ((q → p → p) → q))] → p} → p.

Finally, to render (15) more conveniently, let us permute antecedents in the subformula between ‘[’ and ‘]’: (16)

{[(q → p → p) → (p → q → q)] → p} → p.

Notice now that the formula between “[” and “]” in (16) is a re-lettered form of the quasi-commutativity principle whose addition (under the name Q) to BCK logic was considered briefly in 2.13. (See the discussion after 2.13.16, p. 237.) That formula is not itself IL-provable, but we shall show that (16) as a whole is. To do so, it suffices to prove the sequent with the antecedent of (16) on the left and the consequent of (16) on the right, in the implicational subsystem of our natural deduction system INat (from 2.32). The effect of the structural rule (M) cannot be recovered from the interplay between (→I) and (→E), since, as we saw in 2.33, these rules (alongside (R)) give the weaker implicational system RMNat, a natural deduction system for the weaker-than-IL (semi-)relevant logic RM. We can conveniently recapture the effect of (M) by using a rule derivable from the primitive →-rules, and which we shall call ‘→-Weakening’: (→-Weakening)

Γ (A → B) → C ΓB→C

Notice that an application of this rule provides a proof of p ⊃ q p → q. The labelling of this rule is intended to recall Left Weakening (from 2.33, p. 354) alias Thinning, or just (M) in Set-Fmla. The horizontal form of this principle is the BCK theorem-schema


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((A → B) → C) → (B → C) which is the result of Suffixing (i.e., invoking B – or B and C – and Modus Ponens) a “C” to the consequent and antecedent of the K schema B → (A → B). Thus, since B is stronger than (or at least as strong as) A → B, we weaken an implication when we replace B in its antecedent with A → B. The question of how to treat (M) itself in a Lemmon-style natural deduction system such as INat suitable for the case in which → is the only connective in play is postponed until after we present the proof. (See 7.22.22.) Example 7.22.20 A proof in implicational INat of the sequent [(q → p → p) → (p → q → q)] → p p: 1 1 1 4 1, 4 6 1, 4, 6 1, 4 1 1

(1) [(q → p → p) → (p → q → q)] → p (2) (p → q → q) → p (3) q → p (4) q → p → p (5) p (6) p → q (7) q (8) p → q → q (9) (q → p → p) → (p → q → q) (10) p

Assumption 1 →-Weakening 2 →-Weakening Assumption 3, 4 →E Assumption 5, 6 →E 6–7 →I 4–8 →I 9, 1 →E

Corollary 7.22.21 The * translation ((8)∗ = (9)) of Peirce’s Law is provable in IL. Proof. The discussion of (9)–(16) above shows that the IL-provability of (9) reduces to the IL-provability of (16), and we have settled that question affirmatively with 7.22.20. Instead of the above syntactic proof, we could have relied on the reader to check that (9) is valid on every Kripke frame (for IL); but such a check is actually not as simple as it might sound, and one can easily come away without noticing the contradiction assuming (9) to be false at a point in a model on such a frame eventually leads to. We might expect to use 7.22.21 as a lemma in the proof of Meredith’s result, mentioned above, and given below as 7.22.29, though we shall not in fact do so for reasons explained below. First, a few words are in order on the status of (M) in the purely implicational subsystem of INat. Remark 7.22.22 As a sequent-to-sequent rule, (M), say in the form ΓB Γ, A B is perfectly satisfactory, but not suitable for use in a Lemmon-style presentation of proofs (such as 7.22.20) in which formulas to appear on the left of the “” in the sequent proved are referred to in the far left-hand column by the line numbers at which they are entered as assumptions. They must therefore always appear as assumptions at some stage, and

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we should take the following form of (M) as the suitable version for such proofs: ΓB

(M2 )

AA

Γ, A B

Thus a proof of p, q p can be presented with Lemmon-style dependency annotation in the following way: 1 2 1, 2

(1) p (2) q (3) p

Assumption Assumption 1, 2 (M2 )

Such a proof could be extended by (→I) to a proof of p q → p at its fourth line, and so for any A, B, one can prove B A → B; the reader may care to use this fact to show the derivability of the above rule of →-Weakening (or indeed also to derive (M2 ) from →-Weakening and (→I), (→E)). We return to the task of showing the adequacy of Meredith’s ⊃-for-→ translation ( )*, sticking initially with Fmla for this purpose. It would be pleasant to be able to give a purely ‘logical’ proof of this, for instance along the following lines. We take for granted, as we did in 2.13, that CL can be presented as SKP logic (i.e., with, as axioms, the formulas labelled as S, K, and P in that discussion—P being what here we have numbered as (8)—and Modus Ponens and Uniform Substitution as rules). (There is of course no connection between the KP part of the label SKP and Kreisel–Putnam intermediate logic KP discussed in 6.42.) Given this, we show by an induction on length of proof (of C from the above axioms) that whenever CL C, we have IL C*. The basis for such an induction concerns the axioms of the chosen axiomatization, and we have already checked the case of P (7.22.21). K presents no difficulties. But when we come to S, matters get a little out of hand, since although the complexity of S is only 6, written out in primitive notation (i.e., with → rather than ⊃), S* is an unmanageably long formula with 51 occurrences of →, and no evident simplifications suggest themselves to make its IL-provability evident. Matters are improved only slightly by passing to an axiomatization of implicational IL with, say, B, C, K and W as axioms. Though W presents no difficulties (that is, while it is easy to verify that IL W *), B—or rather B*—is almost as intractable as S. Reluctantly, then, we opt for a different strategy which will make the present proof not entirely self-contained, since we shall have to appeal to a result from the literature on BCK -algebras without providing a proof of that result itself, which would require a considerable excursus on algebraic matters of no other use to us. (Meredith’s own proof can be found in §2 of Meredith and Prior [1968]. It involves an analysis of the two-variable →-formulas in IL together somewhat unfamiliar axiomatizations of the implicational fragments of CL and IL. Another proof may be found in Prucnal [1974]; see esp. p. 63.) The source on which we draw, and to which the reader is referred for a proof, is Guzmán [1994], in which the following is proved as Theorem 3.3. To minimize terminological differences from Guzmán, we refer to Hilbert algebras (as in 2.13) as positive implicative BCK -algebras, and to Hilbert algebras satisfying


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(x → y) → x = x (the algebraic equation corresponding to P or Peirce’s Law) as implicative algebras. (We have in any case altered Guzmán’s notation to the dual form used in 2.13. See the notes on §2.3 relating to that subsection.) Theorem 7.22.23 Let (A, →, 1) be a positive implicative BCK-algebra and define a binary operation ⇒ on A by putting (for a, b ∈ A) a ⇒ b = ((b → a) → ((a → b) → b) → b. Then (A, ⇒, 1) is an implicative BCK-algebra. Now it may seem that we have changed the subject completely since ⇒ is given by a quite unfamiliar term, and that what we need instead is an analogous result for what we might reasonably notate as (A, ⊃, 1) where for a, b ∈ A, we define a ⊃ b to be ((b → a) → a) → b. (Here we are following the same policy as in the case of “→”, giving “⊃” a double usage and relying on the context to make it clear whether it is the connective or the algebraic operation that is in question.) But this analogous result will be seen to follow if we can show that ⇒ and ⊃ coincide; that is, that every positive implicative BCK-algebra satisfies the equation: (17)

((y → x) → ((x → y) → y)) → y = ((y → x) → x) → y.

And this is something we can easily settle by noting the IL-provability of the corresponding implications (lhs of (17) → rhs of (17), and conversely). We give the proof of one of these, in the style of 7.22.20, leaving the other as an exercise. Actually we prove a sequent one application of (→I) away from the implication itself: Example 7.22.24 A proof in implicational INat of the sequent ((q → p) → p) → q ((q → p) → (p → q → q)) → q: 1 2 3 2, 3 1 1, 2, 3 1, 2, 3 1, 2 1, 2 1

(1) ((q → p) → p) → q (2) (q → p) → (p → q → q) (3) q → p (4) (p → q) → q (5) p → q (6) q (7) p (8) (q → p) → p (9) q (10) ((q → p) → (p → q → q)) → q

Assumption Assumption Assumption 2, 3 →E 1 →-Weakening 4, 5 →E 3, 6 →E 3–7 →I 1, 8 →E 2–9 →I

Exercise 7.22.25 Give a proof in implicational INat of the sequent: ((q → p) → (p → q → q)) → q ((q → p) → p) → q. Note: This can be done, like 7.22.24, in ten lines (and in this case, →-Weakening is not needed). Corollary 7.22.26 The following formula and its converse ((q → p → p) → q) → [((q → p) → (p → q → q)) → q] are both IL-provable.

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Corollary 7.22.27 Every positive implicative BCK-algebra satisfies (17). Proof. From 7.22.26 and 2.13.8, recalling that positive implicative BCK -algebras are just Hilbert algebras, and that IL-provability (of formulas) coincides with SK -provability, we get that every such algebra satisfies each of ((y → x) → ((x → y) → y)) → y ((y → x) → x) → y ((y → x) → x) → y ((y → x) → ((x → y) → y)) → y from which the conclusion follows by the antisymmetry condition (AS) (from 2.13) on BCK -algebras. We are now in a position to recast Guzmán’s theorem (7.22.23) in a simpler form, a form suited to our present purposes: Theorem 7.22.28 Let (A, →, 1) be a positive implicative BCK-algebra and define a binary operation ⊃ on A by putting (for a, b ∈ A) a ⊃ b = ((b → a) → a) → b. Then (A, ⊃, 1) is an implicative BCK-algebra. Proof. Immediate from 7.22.23 and the fact that by 7.22.27 a ⇒ b = a ⊃ b. We can now return to the theme of Meredith’s embedding: Theorem 7.22.29 For any purely implicational formula C, we have CL C if and only if IL C*. Proof. ‘If’: If IL C* then CL C*, since IL ⊆ CL. According to CL , C and C* are equivalent, as A → B and A ⊃ B are CL-synonymous for all A, B. Thus, since CL C*, we have CL C. ‘Only if’: Suppose IL C*, with a view to showing that CL C. Since IL C*, the BCK -algebraic equation tC∗ = 1 fails to hold in the Lindenbaum algebra for implicational IL in Fmla (alias SK logic), as we saw in the proof of 2.13.18 (p. 238). This is a positive implicative algebra so the algebra obtained as in 7.22.28 from it with ⊃ as a fundamental operation is an implicative algebra, as therefore is also the subalgebra of the latter algebra generated by the synonymy classes [pi ] of the propositional variables, in which accordingly the equation tC = 1 fails to hold. (Note that the element 1 from the original Lindenbaum algebra survives intact since [p] ⊃ [p] = [p] → [p].) But for every formula A provable in the implicational fragment of CL, tA = 1 holds in all implicative BCK -algebras, by an argument parallelling that given for 2.13.8; therefore CL C. Meredith’s result can be extended to Set-Fmla: Theorem 7.22.30 For any purely implicational formulas A1 , . . . , An , C: A1 , . . . , An CL C if and only if A∗1 , . . . , A∗n IL C∗ .


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Proof. ‘If’: The same argument as for the ‘if’ half of 7.22.29 works here. ‘Only if’: By induction on n. For n = 0, 7.22.29 gives the result, so let us suppose that the claim is correct for a given n and show that it holds for n + 1. That is, we have A1 , . . . , An ,An+1 CL C and want to show A∗1 , . . . , A∗n , A∗n+1 IL C* . Since A1 , . . . , An ,An+1 CL C, we have A1 , . . . , An CL An+1 → C, so by the inductive hypothesis, we have A∗1 , . . . , A∗n IL (An+1 → C)∗ , i.e., A∗1 , . . . , A∗n IL A∗n+1 ⊃ C∗ , so since A∗n+1 ⊃ C∗ IL A∗n+1 → C∗ , we have A∗1 , . . . , A∗n IL A∗n+1 → C∗ and thus A∗1 , . . . , A∗n , A∗n+1 IL C∗ , as was to be shown.

Exercise 7.22.31 (i) Write out in primitive notation the sequent (p → q → r)∗ , (p → r)∗ r∗ which, by 7.22.30, is provable (since without the “*”s the sequent is tautologous), and provide a proof of the resulting sequent in the proof system used for Examples 7.22.20, 7.22.24, above. (ii ) Show that if B IL A then A ⊃ B IL A → B. We can use part (ii ) of this exercise to bring out a problem with extending 7.22.29 (let alone 7.22.30) to the language which has not only → but also ¬ as a primitive connective. (We do not consider the possibility of incorporating also ∧ and ∨ here, except in a final exercise below.) Since CL ¬¬p → p, the analogue of 7.22.23 for {¬, →}-formulas would lead us to expect that IL (¬¬p → p)*, i.e. IL ((¬¬p → p) → p) → ¬¬p, or, using our abbreviative notation: (18)

IL ¬¬p ⊃ p.

But since p IL ¬¬p, 7.22.25(ii ) would deliver from (18) the false conclusion: (19)

¬¬p → p.

We conclude that Meredith’s translation no longer ‘works’ – in the sense that 7.22.29 no longer holds – when negation is added to the language. (See Meredith and Prior [1968], p. 214, for further remarks in this vein.) One reaction to the above difficulty over ¬ might be to suggest that we expose a further invisible occurrence of “→” which becomes visible—and thus convertible into “⊃”—when we regard ¬A as the formula A → ⊥. (More on this approach to negation may be found in §8.3) In this notation, the principle of double negation elimination as an implicational formula becomes (20)

CL (p → ⊥ → ⊥) → p.

Replacing → by ⊃ throughout and then re-writing the main “⊃” of the result, we have (21)

[(p → (p ⊃ ⊥ ⊃ ⊥)) → (p ⊃ ⊥ ⊃ ⊥)] → p.

Now the resulting emergence of the subformula p ⊃ ⊥ here can be dealt with easily, since in primitive notation this is the formula (⊥ → p → p) → ⊥, which

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in view of the IL-provability of its antecedent’s antecedent, make the formula IL-equivalent to p → ⊥. This is another case falling under 7.22.25(ii). Making this substitution in (21), we get (22)

[(p → (p → ⊥ ⊃ ⊥)) → (p → ⊥ ⊃ ⊥)] → p.

Note now the two occurrences of p → ⊥ ⊃ ⊥ as a subformula. In them, we may again appeal to 7.22.25(ii), this time with p → ⊥ as A and ⊥ as B, to replace the “⊃” by “→”, so that (20) is IL-equivalent to (22), or, when written with the aid of “¬”, (23): (23)

[(p → (p → ⊥ → ⊥)) → (p → ⊥ → ⊥)] → p.

(24)

[(p → ¬¬p) → ¬¬p] → p

Again, in view of the IL-provability of the antecedent of the antecedent (of (22) or (23)), we get the erroneous (18) above as a consequence, and must conclude that even if ¬ is replaced by ⊥ as a primitive in extending implicational IL, 7.22.29 is not available for the extension. Exercise 7.22.32 (i) Does 7.22.30 extend to cover the case in which ∨ is present? (Extend the definition of (·)* so that (B ∨ C)* is understood to be B* ∨ C*.) (ii ) Does 7.22.30 extend to cover the presence of ∧? (With an understanding of (·)* analogous to that in (i).)

7.23

Relevant Implication in Set-Fmla: Proofs of Results in 2.33

The review in 2.33 of the implicational natural deduction systems RNat and RMNat left us with some IOU’s on which we shall make good here. Recall that these Set-Fmla systems have, alongside (R), two primitive rules governing implication. For the stronger system, RMNat, these rules were (→I) and (→E) as in Nat (1.23), while for the weaker system, RNat, the latter rule is accompanied by a stricter version of the former rule, called (→I)d , requiring the discharge of assumptions. The details of how these systems related to the Urquhart semilattice semantics were skimmed over, and will be filled in here. As 2.33.2 (p. 338), we stated and proved: Lemma 7.23.1 For any model, the rule (→ E) preserves the property of holding in that model. We do not here repeat the definitions of model, holding in a model, the condition (Q0 ) and the Π (iterated product) notation from 2.33. As Lemma 2.33.3 (p. 339), we stated but did not prove: Lemma 7.23.2 (i) For any model, the rule (→ I)d preserves the property of holding in that model. (ii) For any model satisfying the condition (Q0 ), the rule (→ I) preserves the property of holding in that model.


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Proof. (i) Suppose that (1) some premiss-sequent Γ, A B of an application of (→I)d holds in a model, but (2) the conclusion-sequent Γ A → B does not. Let C1 , . . . , Cn be an enumeration without repetitions of the formulas in Γ. Then (2) means that there exist x1 , . . . , xn respectively verifying C1 , . . . , Cn but with a product not verifying A → B. In that case, for some element y, y |= A but

n

xi ·y |= B.

i=1

This means that we can take y as an element xn+1 to get a contradiction with (1), taking C1 , . . . , Cn ,Cn+1 as our enumeration of the formulas in Γ ∪ {A}, with Cn+1 as A. (ii ) The part of the above argument at which the discharge restriction on (→I) was exploited was the last step, since if A ∈ Γ then C1 , . . . , Cn ,Cn+1 will not be an enumeration without repetitions of the formulas in Γ ∪ {A}. Specifically, Cn+1 , alias A, will also be Cj for some j n. For definiteness, we can take it in this case that j = n, so that A = Cn , as well as = Cn+1 . Thus our repetitionfree enumeration of Γ ∪ {A} will be C1 , . . . , Cn and for a contradiction with (1) we need n elements, rather than n + 1, respectively verifying these formulas but with a product not verifying B. We borrow the element y for which the proof of part (i) led us to say that y |= A, and appeal to (Q), already seen to be guaranteed by the condition (Q0 ), along with the fact that xn |= A (where xn is the nth element, recalling that Cn = A), to conclude that xn y |= A. This gives a sequence u1 , . . . , un of points, with ui = xi for i < n and un = xn y, respectively verifying C1 , . . . , Cn , but whose product does not verify A → B, contradicting (1). Thus any application of (→I) which is not an application of (→I)d preserves holding-in-a-model as long as (Q0 ) is satisfied, establishing part (ii ) of the Lemma. These two lemmas are needed for the ‘soundness’ half of 7.23.5 below; for the ‘completeness’ half, we need two more, the first of which is from Urquhart [1972]. Lemma 7.23.3 For any finite set Γ of formulas, and any formula A, there is a / Γ. formula A which is synonymous in RNat and RMNat with A, such that A ∈ Proof. Putting A1 = A, define Ai+1 = (Ai → Ai ) → Ai . Each Ai is synonymous with (interchangeable with, as regards provability of sequents) Ai+1 , so, given that Γ is finite, applying the process often enough must lead to a formula Ak not occurring in Γ.

Lemma 7.23.4 The following rule is a derived rule of RMNat: ΘA

ΔA

Θ, Δ A Proof. We have already seen that every sequent of the form A → (A → A) is provable in RMNat, so from Θ A by (→E) we obtain Θ A → A, whence

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from Δ A by (→E) again, we obtain the conclusion-sequent Θ, Δ A.

“IF”

The above rule is a Set-Fmla version of a structural rule (due to M. Ohnishi and K. Matsumoto) presented at 2.33.34, p. 369, as an Mset-Mset rule; for further references see the discussion after that. This brings us to the main result ( = Theorem 2.33.4, p. 339).

Theorem 7.23.5 (i) A sequent is provable in RNat iff it holds in every model. (ii) A sequent is provable in RMNat iff it holds in every model satisfying the condition (Q0 ). Proof. (i) ‘Only if’ (Soundness): Clearly sequents instantiating the schema (R) hold in every model. By Lemmas 7.23.1 and 7.23.2(i), this property is preserved by the rules and (→E) and (→I)d . ‘If’ (Completeness): Consider the model (S, ·, 1, V ) in which S is the collection of all finite sets of formulas, · is set union, 1 is ∅, and Γ ∈ V (pi ) iff the sequent Γ pi is provable in RNat. Note that this is indeed a model because its frame (S, ·, 1) is a semilattice. In general, in this model we have Γ |= A iff the sequent Γ A is provable in RNat. For A = pi this is the effect of our definition of V . Now for the induction step, supposing this is so for B and C, we may conclude it is so for B → C by the following reasoning. If Γ |= B → C, then for all finite sets Δ such that Δ |= B, we have Γ ∪ Δ |= C, and so in particular for Δ = {B }, where B is some formula synonymous with B but not belonging to Γ (7.23.3), since B |= B (using the inductive hypothesis and the provability of B B), we have Γ ∪ {B } |= C, which means (ind. hyp. again) that Γ, B C is provable. Then by (→I)d , the discharge condition being met by the choice of B , we have that Γ B → C is provable, whence by the synonymy of B and B , that Γ B → C is provable. Conversely, suppose that Γ B → C is provable; to show that Γ |= B → C, we must show that for any Δ such that Δ |= B, we have Γ ∪ Δ |= C. By ind. hyp. again, this means showing that the provability of Δ B implies that of Γ, Δ C. But since Γ B → C, this follows by (→E). Thus for all Γ, A, we have Γ A provable in RNat iff (in the model (S, ·, 1, V )), Γ |= A. Now suppose that a sequent C1 , . . . , Cn A is not provable in RNat. Then the set {C1 , . . . , Cn } is an S-element which is the product of elements {C1 },. . . ,{Cn } respectively verifying C1 , . . . , Cn , at which A is not verified. So C1 , . . . , Cn A does not hold in (S, ·, 1, V ). (ii ) ‘Only if’: As in (i), but appealing to Lemma 7.23.2(ii) for the case of (→I). ‘If’: Exactly as in (i), though the detour from B to B is not needed here, and we must check that the model defined in terms of RMNat (in place of RNat, in terms of which V was specified) satisfies the condition (Q0 ). This means checking that if Θ |= pi and Δ |= pi , then Θ ∪ Δ |= pi . But since this amounts to showing that if Θ pi and Δ pi are provable in RMNat, so is Θ, Δ pi , the result follows from Lemma 7.23.4.


7.24

1091

Relevant Implication in Fmla: Various Topics

For convenience, we repeat the axioms (axiom-schemata, rather) for the implicational fragment of R here, with the traditional relevant-logical labelling (for what might otherwise be called I, W , B and C): A→A

Id(entity)

(A → (A → B)) → (A → B)

Contrac(tion)

(B → C) → ((A → B) → (A → C))

Pref(ixing)

(A → (B → C)) → (B → (A → C))

Perm(utation)

The sole rule of proof is Modus Ponens. We also recall two other schemata, namely: (A → B) → ((B → C) → (A → C))

Suff(ixing)

A → (A → A)

Mingle

The first of these is an alternative ‘permuted’ form of Pref – B in the combinatorderived nomenclature of BCK (etc.) logic, where Pref itself is B – and the second is the characteristic schema of the stronger system RM. Here we will be interested initially instead in some relevant logics weaker than R. Historically, the two most important, under this heading, have been the systems E, of entailment, and T of ‘ticket’ entailment – both mentioned briefly in (e.g.) 2.33 above. The former offers Anderson and Belnap’s preferred account of the informal idea of the entailment of one proposition by another, and they thought of R as eliminating ‘fallacies of relevance’ from the suggestion that material (or intuitionistic) implication captured that informal idea, with the move from R to the weaker system E as further eliminating ‘fallacies of modality’. For the implicational fragment of this system, one weakens the schema Perm to allow (the formula schematically represented by) B to be permuted to the front only if B is itself an implicational formula. Without this restriction, we can prove (in R, for example) the formula p → ((p → p) → p), which looks as if it flouts modal distinctions in saying that from the truth of p it follows that p follows from something (namely p → p) which is itself necessarily true: this seems wrong if we are thinking of p as perhaps representing something contingent. By contrast, there would be no objection to allowing p → q to follow from something necessary, since we are now (in E) thinking of implicational formulas as, if true, then necessarily so. (A more formal statement of the distinction between the implicational fragments of R and E may be found at 9.22.1: p. 1303.) An alternative response explored in Anderson and Belnap [1975] to the problem of registering modal distinctions in a relevant logic is that suggested by Lewis’s own strategy: consider a modal extension (with a plausibly behaving operator) of R, and think of A’s entailing B as represented by the formula (A → B). Entailment, on this proposal, would be necessary (or ‘strict’) relevant implication rather than (as for Lewis) necessary material implication. The move to E from R was based on a weakening designed to do justice to the special modal status of entailment-claims; the still weaker system T (of “ticket entailment”) attempts to explicate the even more refined idea that implicational formulas (representing

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such claims) are ‘inference tickets’ rather than points of departure. Roughly this means that they are to be used as major premisses rather than minor premisses in applications of (→E), but for more detail and motivation, §6 and §8.3.2 of Anderson and Belnap [1975] should be consulted. An axiomatic basis for T drops Perm altogether (rather than restricting it, as in E) from that given for R above, and adds to Pref the additional schema—now no longer derivable— Suff. (In the combinator-derived labelling the implicational fragment of T is BB IW logic.) We will not give much attention to these ‘neighbours of R’ here, however. As has been mentioned, Urquhart [1972] treats the Fmla system R (with asides on a Seq-Fmla sequent calculus, more prominent in Urquhart [1971]), rather than the Set-Fmla natural deduction system we have concentrated on. The provable formulas of (implicational) R are shown to coincide with the formulas valid on every semilattice, in the sense of being true at that semilattice’s 1 element. (As it happens, Urquhart writes “0” for our “1”, and “∪” for our “·”.) He also considers weakening the conditions on the structures (S, ·, 1), to the point of considering arbitrary semigroups with unit, instead of only the semilattices, and finds the formulas valid on all such structures to be those provable from Id and Pref by the rules Modus Ponens and the ‘rule’ form of Assertion (from A to (A → B) → B): call this ‘Rule Assertion’). Urquhart calls the proof system here described ‘I’. The semigroups in question are not, however, the most general structures to which his truth-definition applies, since we do not need to require associativity (responsible for validating Pref), and, though without 1 as a left identity element no formulas will be valid, we need not also require it to be a right identity (responsible for Rule Assertion’s preserving validity). Thus we are led to consider structures (S, ·, 1) in which S is a set closed under the operation · with 1 ∈ S satisfying 1x = x, all x ∈ S. For the remainder of this subsection, we will use the term ‘structures’ for these rather than calling them groupoids (with a left identity), to emphasize that they are being deployed in the service of model-theoretic rather than algebraic semantics. It should in fairness be mentioned that Urquhart’s interest in I was in its being a fundamental implicational system from a proof-theoretic perspective, having a Gentzen-style sequent calculus formulation in Seq-Fmla with the usual rules for implication (from 1.27: (→ Left) and (→ Right), understood as adapted to Seq-Fmla) but without the structural rules for permuting or contracting formulas, and of course without thinning or ‘weakening’—alias (M). With respect to classes of structures in the broad sense just isolated, we can consider a question analogous to that of modal definability raised in 2.22: for which classes of structures do there exist collections of formulas with the property that a structure validates all formulas in the collection iff it lies within the class? The class of commutative structures—or, for short, the property of commutativity—is easily seen to be ‘defined’ in this sense, by the Assertion formula p → ((p → q) → q), which we noted in 2.33 could replace Perm in our axiomatization of R. (Actually, we gave the axiomatization with schemata, but the point carries across to a presentation via individual axioms and a rule of Uniform Substitution.) As for Perm itself, we work through the reasoning for a suitable correspondence: Observation 7.24.1 The formula (p → (q → r)) → (q → (p → r)) is valid on a structure iff the structure satisfies: (xz)y = (xy)z, for all elements x, y, z.


1093

Proof. The ‘if’ direction is a pleasant exercise, already involved in the claim that R is sound w.r.t. the class of all semilattices. For the ‘only if’ direction, suppose that we have elements x, y, z ∈ S with (xz )y = (xy)z; we must show how to define V on (S, ·, 1) to have (S, ·, 1, V ), 1 |= (p → (q → r)) → (q → (p → r)). For this it suffices that x |= p → (q → r) while x |= q → (p → r), and the reader is left to verify that this is indeed the result of setting V (p) = {z}, V (q) = {y}, V (r) = S {(xy)z}. It is a useful exercise to find analogues of 7.24.1 for the other axioms of (implicational) R. Indeed one can consider this question with respect to (a representative instance of) the Mingle schema also: that is, forgetting about Urquhart’s way, via the condition (Q0 ) on Valuations, of making p → (p → p) unfalsifiable at the element 1 of any structure, one can ask after the weakest condition on the structures themselves that would have this effect. (Capital ‘V’ on ‘Valuation’ here, as usual, to avoid confusion with valuations.) We pause to record the answer: Observation 7.24.2 The formula p → (p → p) is valid on a structure if and only if the structure satisfies: for all elements x, y, either xy = x or xy = y. Proof. ‘If’: To have 1 |= p → (p → p) in some model, we must have an element x with x |= p and x |= p → p. For the latter, we need an element y with y |= p and xy |= p. But this is impossible if xy must be either x or y. ‘Only if’: Suppose for elements x, y in a structure we have xy = x, xy = y. Put V (p) = {x, y}. Then x |= p, but x |= p → p, as y |= p while xy |= p (because xy ∈ / {x, y}). So 1 |= p → (p → p). Notice that the condition on structures mentioned in 7.24.2 actually implies xx = x (taking y = x), and that this idempotence condition, along with associativity, implies that (for all x, y) (xy)y = xy, a condition which itself secures the validity of all instances of Contrac. Yet from Mingle and Perm (which defines associativity), p → (p → p) cannot be deduced by Modus Ponens. This means that we have on our hands here an incomplete logic on the present semantics, somewhat in the style of the modal logics mentioned in 2.22 which are not determined by any class of frames; the condition of idempotence in Urquhart’s semilattice semantics, would no doubt benefit from further scrutiny in the light of such results, though we shall not give it that attention here. For more in the same vein as 7.24.1–2, see Došen [1989a] (§3.6, §3.10); further discussion of the history of the idempotence condition may be found in Dunn [1986], and, of its precarious status, Humberstone [1988b] (Appendix A), Došen [1989a], p. 59 (second paragraph). We turn to an issue related to the non-monothetic nature of R, pointed out for the implicational fragment (BCI logic) in 2.13 above (also 7.25 below). We can, in R and other relevant logics, have pairs of non-synonymous provable formulas, thus keeping track selectively of inferential dependencies even amongst the formulas provable outright, a kind of selectivity which in (for example) intuitionistic and classical logic is only available amongst unprovable formulas.

CHAPTER 7.

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“IF”

A related sensitivity will emerge in the discussion of suppression, after a brief revision exercise. Exercise 7.24.3 Is R determined by a class of unital matrices? (Justify your answer. For “unital”, see 2.13.) We need the {∧, →}-fragment of R, which is axiomatized by taking the schemata and rules listed for full R from 2.33 in which only those connectives are displayed, to illustrate what Routley et al. [1982] call the fallacy of suppression. Consider the candidate rule of proof: Suppression

A

(A ∧ B) → C B→C

If the second premiss had been A → (B → C) instead, this would be an application of Modus Ponens. (We can rewrite this as a single implication with C as consequent, with the aid of fusion—as in 5.16—as (A ◦ B) → C; so a provable component of a fusion compound, though not of an ordinary (‘extensional’ or additive) conjunction, can legitimately be ‘suppressed’.) To see that as it stands, Suppression is not an admissible rule for R, consider: Example 7.24.4 Take A = C = q → q, B = p. Both premisses are R-provable, but the conclusion, p → (q → q) is not. In this case, the conclusion commits a blatant fallacy of relevance, as judged by Belnap’s variablesharing criterion. The following example, in which there is no variable-sharing problem for the conclusion, is adapted from Dunn [1986]. Example 7.24.5 Take take A = p → p, B = q → r, C = (p ∧ q) → (p ∧ r). The premisses are provable in R but the conclusion, (q → r) → ((p ∧ q) → (p ∧ r)) is not: although provable, p → p cannot be ‘suppressed’ as a conjunct of the antecedent, since without making use of the fact that p implies p alongside the supposition that q implies r, we cannot infer that p ∧ q implies p ∧ r. According to the perspective emphasized in Routley et al. [1982], esp. §2.10, the real interest in passing from irrelevant to relevant logics lies not in the avoidance of fallacies of relevance (in anything like the variable-sharing sense) but in the avoidance of suppression (which in that work covers somewhat more than has been indicated here), of which fallacies of relevance were merely symptoms. (On the other hand, the first use of the term ‘suppression’ in this context may well be that of Smiley [1959], which is considerably less sympathetic to the idea that there is anything fallacious going on.) The fact that R-provability is not preserved by Suppression is a reflection of the fact that the R-provable formulas are not true at each element of a semilattice model in Urquhart’s semantics (or indeed at every point in a model of the ternary relational semantics of Routley and Meyer—see 8.13). Recall that truth is required only at the element 1 in each model (S, ·, 1, V ); even a formula such as p → p can be false at x = 1 in such a model, since we can have y ∈ V (p), xy ∈ / V (p).


1095

Exercise 7.24.6 We extend the truth-definition from 2.33 (for → formulas) to encompass ∧ by stipulating that where M = (S, ·, 1, V ), for all x ∈ S: M, x |= B∧C iff M, x |= B and M, x |= C. Call a formula A ubiquitous if for all semilattice models (S, ·, 1, V ), we have (S, ·, 1, V ), x |= A for all x ∈ S. Show that no formula constructed from propositional variables by means of → and ∧ is ubiquitous. To guarantee that A could be ‘suppressed’ in the sense that passage from formula (A ∧ B) → C to B → C preserved validity, it is not enough to know that A is itself a valid formula, since we need to know that wherever B is true (in a semilattice model), so is A ∧ B. Thus for the transition to preserve validity for arbitrary B and C, what we need is that A is a ubiquitous formula, not just that A is a valid formula. Digression. Avron [1992] has some critical remarks on the issue of suppression. Citing a formulation of Anderson and Belnap’s according to which B should be provably implied by A in a relevant logic when “B follows from A alone”, he says this about how we are to understand the word “alone” here: If we understand it literally, then the only thing that follows from A alone is A and nothing else is A itself. With this interpretation only instances of A → A will be valid entailments. Obviously, therefore, the phrase “A alone” should be understood as “nothing but A, except logically valid sentences and rules”. In a lot of places in their work the relevantists seem to accept this. (. . . ) formal evidence that they officially do is provided by the fact that ((A → A) → B) → B is a theorem of E and R. There is no way that I know to deduce B from (A → A) → B without also using A → A and relying on the fact that the last sentence is logically valid! Thus, an instance of (11) above [ = the “intuitive assertion”: If A entails B and A entails C then A entails B ∧ C] is: If A entails A and A entails B then A entails A ∧ B. Nevertheless the corresponding entailment (A → B) → (A → (A ∧ B)) is not a theorem of R (. . . ) To conclude, there seems to be no conception of what “deducible from A alone” means and to what extent logically valid sentences may be used in deductions from assumptions. (Avron [1992], p. 248.)

In the first case cited by Avron here, of the schema ((A → A) → B) → B we have a proof using Modus Ponens with A → A and (A → A) → ((A → A) → B) → B which is itself a permuted form of ((A → A) → B) → (A → A) → B. (There is a restriction on Perm – or C to give it its combinator-derived label – for E, to the effect that the formula permuted to the front must be an implicational formula, motivated by the earlier remarks about E and R and formulated in 9.22.1 on p. 1303; in this case the permuted formula is A → A so this restriction is satisfied.) Modus Ponens is not Suppression, though, and it is the latter rule which would be required to pass from

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“IF”

((A → A) ∧ (A → B)) → (A → (A ∧ B)) and A → A to the conclusion Avron is demanding, namely (A → B) → (A → (A ∧ B)). Obviously we must be more selective: not just any old way of using a provable formula (a “valid entailment”) is to count as a satisfactory use. Nor can we reformulate the deduction Avron wants so that Modus Ponens does the work rather than Suppression, since this would require the provability for arbitrary A and B of (A → A) → ((A → B) → (A → (A ∧ B))) which is of course not forthcoming in R (let alone in E). Perhaps we can diagnose the flaw in Avron’s diagnosis of confusion on the part of Anderson and Belnap like this. His informal principle If A entails B and A entails C then A entails B ∧ C may be taken with “entails” meaning “provably (relevantly) implies” throughout, in which case it makes the correct observation that whenever we have R A → B and also R A → C, then we have R A → (B ∧ C). This holds of course also for the special case in which B is A. Alternatively, we may understand the ‘entails’ as being an informal allusion to a use of the → connective itself in the object language, and indeed Avron ([1992], p. 248) says after articulating the informal principle that it is “officially translated by the relevantists into the following entailment, which they therefore accept as valid: ((A → B) ∧ (A → C)) → (A → (B ∧ C))”. Note that while the informal principle used the word “entails” three times, there are four arrows in this formula. The main arrow has come in to represent the metalinguistic if then construction. Avron’s argument assumes that (3) should follow from (1) and (2): (1) If A entails B and A entails C then A entails B ∧ C (2) A entails B (3) If A entails C then A entails B ∧ C This is a kind of Suppression for the metalinguistic if -then showing it is simply not safe to think of the fourth (and principal) → in the above schema as translating this construction. Another way of putting the point is to say that this if –then, for Avron’s argument, must be taken to participate in the exportation inference: If S 1 and S 2 , then S 3 . Therefore: If S 1 then if S 2 then S 3 . which as we know (see 5.16), the → of R (when and is represented by ∧ as opposed to ◦) does not satisfy. In terms of negation and disjunction, we can define a connective of ‘material implication’ in the language of R, represented by ⊃, say, with A ⊃ B = ¬A ∨ B, for which the portation inferences are fine, and in terms of which the eventual conclusion of Avron’s reasoning would be that all formulas of the form (A → B) ⊃ (A → (A ∧ B))


1097

should be provable in R – which indeed they are (cf. 8.13.14(ii )). Though the above criticism of Anderson and Belnap seems misguided, Avron’s discussion does usefully direct our attention to the somewhat neglected status of consequence relations and notions of derivability in relevant logic, on which topic see further Batens and van Bendegem [1985], Dunn [1986], and Brady [1994]. End of Digression. We return to the subject of ubiquitous formulas. Although such formulas are, as 7.24.6 reveals, themselves far from ubiquitous, we have already encountered (in 2.33) a formula that must have this status in the Urquhart semantics if there is to be any hope of having a version of R containing the formula that is sound and complete in terms of that semantics. (Indeed, in what follows we restate with some further fleshing out, several points already made in 2.33.) The formula in question is the Church constant T , (“Big T ”), added to a fragment – such as the implication-conjunction fragment of R treated semantically in 7.24.6 – by means of the axiom-schema: A → T . The appropriate semantic account of T is then given precisely by the stipulation that it should be ubiquitous, as one might, for example, by extending the inductive definition of truth for this new case thus: (S, ·, 1, V ), x |= T if and only if x = x. The provable formulas of the {→, ∧, T }-subsystem of R coincide with those which are valid on every semilattice (truth for ∧-formulas and T being understood as we have explained it). We can bring to bear on the discussion of Suppression not only Church’s truth-constant T but also Ackermann’s (“little t”) constant t. Whereas, as may be verified easily by considering either the syntactic or the semantic treatment suggested above, T functions as a (two-sided) identity element for conjunction in R, in the sense that T ∧ A (or A ∧ T ) is there synonymous with A, t is a left-identity for implication (→): t → A and A are similarly synonymous. On the syntactic side, we can see this from the following schemata for t: (1)

(t → A) → A

(2)

A → (t → A)

We could equivalently replace of (1) and (2) by (3) and (4) respectively: (3) t (4) t → (A → A) (Clearly (2) and (4) are permutations of each other. To get (3) from (1), put t for A and detach the consequent from the Id instance t → t; in the other direction, use Assertion.) A rough guide to the appropriate semilattice semantical treatment is given by saying that in any model, t is to be true at the element 1 and nowhere else. (This is only rough, however, a precise formulation requiring apparatus not at our disposal here. See Humberstone [1988b] for details. The rough guide is fine for the purely implicational fragment augmented by the constants T , t, F and f from 2.33, however.) The connection with Suppression is given by the fact that although this rule is not admissible for R, if we (temporarily) define ⊃ by: A ⊃ B = (A ∧ t) → B, then the following variation on that rule is admissible.

CHAPTER 7.

1098 (5)

A

“IF”

(A ∧ B) → C B⊃C

As was mentioned in 2.33, though we did not use this notation there (for this purpose, anyway), the formula A ⊃ B behaves very much as A → B behaves in intuitionistic (rather that relevant) logic: for detailed formulation and proof, see Meyer [1973] (and also Meyer [1995]), who thinks of A ⊃ B in the present context as representing a kind of enthymematic implication. (This follows an idea of Myhill [1953], taken up in a 1961 paper by Anderson and Belnap, discussed and updated in §§35–6 in Anderson, Belnap and Dunn [1992]; an interesting further application may be found in Slaney and Meyer [1992].)

7.25

Contractionless Logics

We devote this subsection to implicational logics without Contraction (alias Contrac or W ), and in particular to the BCI and BCK logics of 2.13, formulated both—as there—in the framework Fmla and recast in Mset-Fmla. We shall also attend to an extension of BCI logic to be called BCIA logic, which is the implicational fragment of the unusual ‘Abelian Logic’ of Meyer and Slaney [1989] and its sequel, Meyer and Slaney [2002]. (We depart from alphabetically ordering axiom names as with “BCI ” “BCKW ” etc., in the present instance to mark the fact that “A” is not a genuine combinator-derived label, just an abbreviation for Abelian axiom.) This logic is not just contractionless, like BCI logic, but positively contraction-resistant in the sense that adding W to it would result in inconsistency (the provability of all formulas, that is). Details follow in due course; the implicational fragment was, it should be mentioned, first considered by C. A. Meredith in 1957: see notes. There are several motives for studying such contractionless logics, or suitable extensions of them to accommodate additional connectives, of which the two best known are (1) the desire to avoid Curry’s Paradox in a set theory with unrestricted comprehension axiom, and (2) a desire to avoid the features that make intuitionistic and classical predicate logic undecidable. Some pertinent references, along with a description of Curry’s Paradox and some variations thereon, may be found in the end-ofsection notes (p. 1123). It should be noted that dropping contraction (W in the list below) from the axioms given for R in 2.33 has very serious effects for the connectives other than →; no {∧, ∨, ¬}-tautologies are provable in the resulting system, whereas with it, all are. The effects of contraction can be obtained by adding instead the axiom-schema (A ∧ (A → B)) → B in the context of the other {→, ∧}-axioms listed in 2.33 (p. 340). Relatedly, a conspicuous feature of the contractionless logics in Fmla is that even when (as for all cases considered here) the logic is closed under Modus Ponens, for an arbitrarily selected formula A, the set {B : A → B} is not itself guaranteed to be closed under Modus Ponens; such a guarantee is in effect given outright by the contraction-related principle S on the list below (and much appealed to in 2.13). The effect of contraction can also be supplied in the presence of negation by adding the schema we call (¬-Contrac) in Chapter 8 (see 8.13.2(ii), p. 1187, and the discussion preceding 8.32.3, p. 1259), namely:


1099

(A → ¬A) → ¬A. However, we mention these points about other connectives only to get them out of the way at the start, since, as already remarked, we shall be looking at purely implicational logics. We begin, in Fmla, with a famous property of BCK logic (from Tarski [1935]) which concerns its Post-complete extensions, passing to a consideration of BCI logic, noting the lack of the corresponding property in its case, and also taking up a matter which led to trouble for a semantics in terms of BCI -algebras at the end of 2.13, discussing en route the system of BCIA logic mentioned above. We end our treatment in Fmla with a statement (without proof) of the celebrated ‘2-property’ and ‘1,2-property’ enjoyed respectively by BCI logic and BCK logic, turning our attention to a natural deduction approach and associated Urquhart-style semantics along the lines given for the implicational fragment of R (namely, BCIW logic) in 7.23; unlike natural deduction systems presented elsewhere in this book, we work with multisets rather than sets as the entities represented on the left of the “”. Since we shall be making frequent use of the combinator-derived labels for implicational axioms, we remind the reader of the main such axioms (from 1.29 and 2.13); of course, since we are here considering especially contractionless logics, the contraction axiom (W ) is included mainly as a reminder of what we are doing without; recall that BCK (BCI ) logic is axiomatized with the aid of Uniform Substitution, Modus Ponens, and the axioms B, C, and K(B, C and I), or, equivalently, with the corresponding axiom-schemata (distinct propositional variables being replaced by distinct schematic letters) and just the rule of Modus Ponens: K: B: B: C: I: W: S:

p → (q → p) (q → r) → ((p → q) → (p → r)) (p → q) → ((q → r) → (p → r)) (p → (q → r)) → (q → (p → r)) p→p (p → (p → q)) → (p → q) (p → (q → r)) → ((p → q) → (p → r))

When questions of rule derivability come up, or exercises requiring proofs of particular formulas, in BCI logic (BCK logic etc.), the axiomatization in terms of which the question is to be understood are those involving axioms B, C, I, (B, C, K, etc.), with Modus Ponens and Uniform Substitution as rules (or just Modus Ponens, in the treatment with axiom schemata). We begin then, with Tarski’s observation, appearing here as 7.25.3 (with parts of the proof incorporated into 7.25.1–2). Understand by a logic for the sake of the latter, a set of implicational formulas closed under Uniform Substitution and Modus Ponens; such a logic is Post-complete when although not itself containing every formula, every logic properly extending it does contain every formula. Given the requirement of closure under Uniform Substitution, this reference to containing every formula (the currently suitable notion of inconsistency) is equivalent to containing the propositional variable p.

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Exercise 7.25.1 Show that for any implicational formula A in which only a single propositional variable occurs (any number of times) A is CLprovable if and only if A is provable in BCK logic. (Hint: look at the proof of 7.21.1 and notice that W is not required, recalling (from 2.13) that IL’s →-fragment is BCKW logic, alias SK logic.) The synonymy classes, for BCK logic, of formulas constructed just using p are accordingly those depicted in Figure 7.21a. Notice, in particular that p → p and p → (p → p) are equivalent. Although the implication from the latter to the former is an instance of the contraction schema W : (A → (A → B)) → (A → B) we do not need that schema since the instance in question follows from K. (General point: a contractionless logic need not lack all instances of W , just as a relevant logic need not lack all instances of K.) A glance at Figure 7.21a gives: Lemma 7.25.2 For any formula A constructed with the aid of a single propositional variable p and the connective →, either A is BCK-provable or else A → p is BCK-provable. Theorem 7.25.3 The only Post-complete extension of BCK logic (in the same language) is the implicational fragment of CL. Proof. We must show that if A is an →-formula not provable in CL, the smallest logic extending BCK logic and containing A is inconsistent. Such a CLunprovable formula A has a CL-unprovable substitution instance A in which the only variable to occur is p, since we may take A as s∗v as in 1.25.12(i) (p. 132), for v an →-boolean valuation with v(A) = F. Since A is not CLprovable, it is not BCK -provable, and so by 7.25.2 we have that A → p is BCK -provable. But, being closed under Uniform Substitution the envisaged extension of BCK logic containing A, contains A , and therefore, being closed under Modus Ponens, it contains also p, and is thus inconsistent. The above proof, it is worth noting, exploits the obvious enough equivalence, for logics S1 and S2 with S1 ⊆ S2 , between the claims (i) that S2 is the (one and) only Post-complete extension of S1 , and (ii) that S2 is Post-complete and for any A ∈ / S1 , the smallest logic extending S2 and containing A is inconsistent. The argument itself shows, in passing, that the S2 at issue here—implicational CL—is itself Post-complete. Remark 7.25.4 To prove 7.25.3 without dipping back into 7.21, one should show that Lemma 7.25.2 holds by proving, by induction on the complexity of A (constructed from p using →) that either A is BCK-provable or else both A → p and p → A are BCK -provable, as Tarski [1935] does (see Lemma 6, p. 398 of Tarski [1956]). This is a typical example of ‘induction-loading’: we are really interested only in the weaker conclusion – dropping reference to the provability of p → A – but the inductive part of the proof requires the stronger hypothesis (that A → p and p → A are both provable). Curiously enough, exactly this proof was


1101

reproduced fifty years later, in Dale [1985a], without mention of Tarski. (Actually, Dale states the result in weaker terms, to the effect that implicational CL is the only Post-complete extension of implicational IL. Somewhat similar themes occupy Porte [1958], Robinson [1959].) Theorem 7.25.3 would not be correct for BCI logic in place of BCK logic, as we can see most easily by a consideration of the implicational fragment of the ‘Abelian Logic’ of Meyer and Slaney [1989], which can be presented as the extension of BCI logic by the formula A below, and which we accordingly refer to as BCIA logic; it will be the subject of our attention up to (and including) 7.25.23 on p. 1116 below: A

((p → q) → q) → p.

The rules presumed for this axiomatization are, as for BCI (BCK, etc.) logic, Uniform Substitution and Modus Ponens. (A single-axiom presentation with the same rules had been given by C. A. Meredith in 1957. See the notes to this section, p. 1122.) Note, that since A is not itself a (classical) tautology BCIA logic represents a rare instance of an interesting contra-classical logic: a consistent logic which is not included in (the corresponding fragment of) CL. See Humberstone [2000a] for a general discussion of this phenomenon – in terms of which BCIA logic is classified as a superficially contra-classical logic. (“Superficially”, because if we interpreted “→”, not as classical “→”, but as classical “↔”, the result would be a CL-tautology, as for B, C and I. Profound contra-classicality requires the unavailability of any such re-interpretation: see the opening paragraph of 4.21 These points concern only the implicational fragment of Meyer– Slaney’s Abelian logic; with the axioms governing further connectives they give, such as conjunction and disjunction, one can no longer simply reinterpret “ →” as “↔”.) By way of informal motivation, however, it is still possible to put the Abelian axiom A in a favourable light. We can think of it as saying that if q follows from the mere hypothesis that p implies it, this can only be because p is true. The justification just sketched on behalf of A is not that if whatever a proposition implies is true then, so is the proposition itself. That would not be controversial, since one of the things the proposition implies is itself; but it does not justify what we need to justify here. The point can be made with an informal use of propositional quantifiers. What the latter straightforward justification justifies is ∀p(∀q((p → q) → q) → p). But we need to justify (taking A as tacitly universally quantified) is rather ∀p∀q(((p → q) → q) → p) which is what our original formulation attempted to justify; a similar confusion over the scope of propositional quantifiers is mentioned in note 42 of Humberstone [2005f ]. (It must be admitted, however, that whatever the merits of this justification for A might be when taken on its own, it does seem to require taking the “→” as representing implication in the sense of entailment, and – as noted in 7.24 – the “C” of BCI logic, or, what is equivalent in this context, the converse of A itself, is not plausible for such an interpretation of “→”.)

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Meyer and Slaney’s own motivation for studying Abelian Logic was rather different. We recall that in substructural logics (see 2.33) allow for the definition of negation in terms of implication and a constant f with the conspicuous property that for any formula B, (B → f ) → f provably implies B, this being a double negation (elimination) principle under the definition of ¬B as B → f . (Outside of the substructural setting, the distinction between F and f evaporates and we just write ⊥ for them, as in §8.3 below.) Meyer and Slaney wanted to explore a logic in which instead of this being a special property of a certain sentential constant (making f ‘special’ in the technical sense of 9.22 below), it was a property that every formula possessed, which is what A in effect declares, since we may substitute any formulas B and C for the variables p and q. (Meyer and Slaney think of B → C as a kind of C-relative negation of B, and call the Abelian axiom A ‘the axiom of relativity’.) As Meyer summarises matters in Meyer and Slaney [2002]: Meanwhile Slaney had the crazy idea that, if negation were really to be defined inferentially, then fundamental principles like full double negation should already be present in the implicational part of logic.

For more on negation in Abelian logic, whose pure implicational fragment (BCIA logic) is our concern here, see the Digression after 7.25.18 below. The study of BCIA logic (and some extensions with further connectives) is motivated rather differently in Casari [1989] – namely from a propositional logic of comparatives – and differently again in Galli, Lewin and Sagastume [2004] – as a logic registering the ‘balance of evidence’. A discussion of the relations between these various logics is provided by Chapter 3 of Metcalfe [2003]. Most of these references consider the full system of Abelian logic (with at least conjunction and disjunction also present), though it is only the implicational fragment that is of concern here. (Nor do we attend to any particular consequence relation for which this fragment constitute the Fmla logic; a very informative survey of the options on this front is provided by Paoli, Spinks and Veroff [2008].) Instead of adding A, one could use the converse of B , as is done in Kabziński [1993]. Yet another axiomatization may be found in Shalack [2004]. Neither of these authors makes any reference to Meyer and Slaney [1989]. The topic has now found its way into the textbook literature, with Schechter [2005] – see Chapter 26 thereof (as well as various index references under “comparative logic”). BCIA logic is very hostile to extension by some familiar implicational principles: Exercise 7.25.5 (i) Show that BCKA logic and also BCIAW logic are inconsistent (in the sense of proving every formula). (ii ) Are the axioms B, C, I and A independent? (The answer to this question was provided by Branden Fitelson and can be found in Meyer and Slaney [2002], §9.) The logic is, nevertheless, consistent, as we can see by interpreting the → as ↔ and observing that, so interpreted, B, C, I, and A are all classical tautologies (and that the rules preserve this property). This is most easily seen by noting that each propositional variable occurring in these axioms occurs an even number of times, and appealing to 7.31.6 (which says that a purely equivalential formula is a tautology iff each propositional variable occurs in it an even number


1103

of times). This point gives, incidentally, a very simple argument for the independence of K and W (each of which violates the ‘even number of occurrences’ condition) in such axiomatizations as BCIW and BCK, as well as showing that W is not BCIA-provable (which is just as well, in view of 7.25.5(i)). Remark 7.25.6 In showing BCI does not have the →-fragment of CL as its sole Post-complete extension by exhibiting a consistent extension BCIA which is not included in CL, we are not claiming that that extension is itself Post-complete. (The full system of Abelian Logic, with additional connectives governed by suitable axioms, is shown in Meyer and Slaney [1989] to be Post-complete, however.) In fact the equivalential fragment of CL (in Fmla, with Modus Ponens and Uniform Substitution as rules), is Post-complete (see Prior [1962a], p. 307), and, when ↔ is rewritten as written as →, is a proper extension of BCIA logic. The “extension” part of this claim was just noted above. For the “proper”, we remark without proof for the moment (though see 7.25.20 below) that the formula (p → q) → (q → p) is not provable in BCIA logic. Nor is this the only Post-complete extension of BCIA logic, as we shall have occasion to note in 7.25.23(ii) below (p. 1116). While, as just mentioned, an implication does not (in general) provably imply its own converse in BCIA logic, there is an analogue of this property which does hold at the level of rules; our proof relies on a syntactical argument instead. We need some preliminaries to begin with. (Part (i) of the following exercise serves in that role; part (ii) illustrates a general lesson about contractionless logic.) Exercise 7.25.7 (i) Show that every formula of the following form is BCI provable: A → [(B → C) → ((A → B) → C)]. (Hint: begin with the ‘Assertion’ schema B → ((B → C) → C) – see the proof of 2.13.26, p. 244 – and prefix A to antecedent and consequent (in a full proof, appealing to B and Modus Ponens), and then permute antecedents appropriately.) (ii ) Here are two rules bearing a resemblance to the missing contraction axiom W . The first is a ‘rule form’ of that axiom, while the second is a rule form of a permuted version of the axiom (i.e., p → ((p → (p → q)) → q)): A → (A → B)

A

A → B

(A → (A → B)) → B

Show that BCI logic is closed under one of these rules but not the other (and of course, say which is which). Lemma 7.25.8 (p → p) → (q → q) is BCIA-provable. Proof. Put q for A and p for B and C in the schema of 7.25.7(i), giving: q → [(p → p) → ((q → p) → p)]. By appeal to the axiom A, we can simplify the consequent, yielding: q → [(p → p) → q]

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“IF”

Remark 7.25.9 From 7.25.8 by Uniform Substitution, we have, for all formulas A, B, that (A → A) → (B → B) is BCIA-provable. While this does not by itself show that BCIA logic is monothetic (all theorems are synonymous), it does go some way toward showing that. The general result is given as 7.25.14(ii) below. By permuting antecedents we get that B provably implies (A → A) → B, so we have the latter provable whenever B is provable, but there is no corresponding argument to the conclusion that the provability of A implies that of A → (B → B). If we had established this, it would follow that whenever A and B are provable, so is A → B, which suffices for the monotheticity since the provability of an implication and its converse secures the synonymy of the antecedent and consequent in any implicational extension of BCI logic. (Recall, from the preamble to 2.13.1 on p. 221, that in calling a logic monothetic, we mean simply that all of its theorems are synonymous.) We resume this line of thought in 7.25.16 below. Exercise 7.25.10 (i) Show that the converse of the formula B , that is, the formula ((q → r) → (p → r)) → (p → q), is BCIA-provable. (Hint: do some permuting of antecedents in the antecedent of this formula to obtain a substitution instance of A.) (ii ) Show that the following formula is BCI -provable: [((p → q) → q) → p] → (p → p). (iii) Let A* be the formula ((p → q) → r) → ((r → q) → p), and show that BCIA logic = BCIA* logic. (Hint: to prove the formula in BCIA logic, in which B is provable by B and C, begin by substituting p → q for p, r for q, and q for r in B .) Note that the formulas under (ii) and (iii) here are counterexamples to a conjecture that might have occurred to the reader to the effect that whenever an implication is provable in BCIA logic, then either that implication or else its converse is provable in BCI logic. The rule form of the formula A* of 7.25.10(ii), we call (Pivot) since it allows us to pivot the initial and final A and C about the central B: (Pivot)

(A → B) → C (C → A) → B

Part (iii) of the preceding exercise can be strengthened to the statement that the closure of BCI logic under (Pivot) is precisely BCIA logic. Clearly we can derive the rule from the formula form A*; conversely, if we begin with A = p, B = q, C = p → q, then a premiss for (Pivot) is available as an instance of I, and the corresponding conclusion is A. This rule also provides for a quick proof of Lemma 7.25.8. Starting with the BCI-provable ((q → q) → p) → p, an application of (Pivot) yields the formula of 7.25.8 as conclusion.


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Observation 7.25.11 Whenever an implication A → B is BCIA-provable, so is the converse implication B → A. Proof. We make use of the fact that since BCI logic is B CI logic (just differently axiomatized), it suffices to show by induction on the length of the shortest proof of A → B from B , C, I, and A (with rules Modus Ponens and Uniform Substitution), that the existence of such a proof implies the existence of a proof of B → A. Basis: A → B is an axiom. I is its own converse, and the converse of C is a relettered version of itself. The converse of A is BCI -provable (as was shown in the proof of 2.13.26, p. 244), and by 7.25.10(i), the converse of B is BCIAprovable. Inductive Step: The rule of Uniform Substitution clearly preserves the property of having a provable converse, so we have only Modus Ponens to check. So suppose B is deduced from A → B and A by Modus Ponens. Since A and B are both provable, each is an implication, and we can represent this application of Modus Ponens as a passage from (i) (A1 → A2 ) → (B1 → B2 )

and (ii ) A1 → A2

to the conclusion B1 → B2 , our task being to show that B2 → B1 is provable. By the inductive hypothesis, since (i) is provable, we have (iii) and (iv ) provable: (iii) (B1 → B2 ) → (A1 → A2 )

(iv ) A2 → A1 .

These together give (v) (B1 → B2 ) → (A1 → A1 ). By 7.25.8, we have the provability of (vi ) (A1 → A1 ) → (B2 → B2 ) and from (v) and (vi ): (vii) (B1 → B2 ) → (B2 → B2 ) Permuting antecedents, we have (viii) B2 → ((B1 → B2 ) → B2 ) whose consequent simplifies to B1 , giving the desired result, by an appropriate substitution instance of A. What does the above proof establish about the status of the rule of Conversion (or, less flatteringly, “Fool’s Contraposition”) Conversion

A→B B → A

for BCIA logic, considered not just as a set of theorems but as a proof system? The key point to note is that it is the admissibility, rather than the derivability, of the rule that is established. Meyer and Slaney [1989] show that Conversion is not derivable since it is not admissible for the full system of Abelian Logic, with additional connectives and axioms (in which for example p ∧ q provably implies but is not provably implied by p). We should notice incidentally that BCIA

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logic is the smallest extension of BCI logic for which this rule is admissible since A’s converse, though not A itself, is BCI -provable. Using 7.25.11 we can show that, in conspicuous contrast with BCI logic, BCIA logic is itself monothetic. We shall do so after an excursus on BCI logic, on whose non-monothetic character we commented at the end of 2.13. Our example concerned the case of (1) and (2): (1)

(p → p) → (p → p)

(2)

(p → p) → (q → q)

whose consequents are both theorems but which differ as to provability (since (2) embodies a violation of Belnap’s variable-sharing criterion of relevance and hence is not even provable in the stronger BCIW, the implicational fragment of R). Thus the two theorems p → p and q → q are not synonymous. (We can easily confirm the unprovability of (2) by appeal to the Urquhart semantics of 2.33, 7.23–24. According to 7.25.8, in contrast, (2) is BCIA-provable; we return to full monotheticity for BCIA logic below. See also 7.25.9.) On the other hand, (2) is easily seen to be valid according to the algebraic semantics provided by BCI -algebras in the sense that the corresponding equation (“t(2) = 1”, in the notation of 2.13), namely (3)

(x → x) → (y → y) = 1

holds in every such algebra. The explanation for this mismatch between the algebraic semantics and the logic is, as we noted in 2.13, not far to seek: if we think in terms of matrix semantics, the matrices we are using have for their algebras BCI algebras and for the sets of designated elements the one-element sets {1}, where we use ‘1’ indifferently to denote the 1-element of different BCI algebras. But no logic can be sound and complete w.r.t. a set of matrices each with exactly one designated value unless that logic is monothetic, which as we have just recalled, BCI logic certainly is not. (One can instead use a nonunital form of algebraic semantics instead, as was noted in 2.13.27, but here we consider sticking with the present semantics and ‘boosting’ the logic to get a system semantically tractable in this style.) Bunder [1983a] accordingly considers extending BCI logic by some further axioms, namely those instantiating the schemata (4) and (5) (which appear as (b) and (c) on p. 20 of Bunder’s paper, though in a different notation): (4)

(A → A) → (B → B)

(5)

A → (A → (B → B)).

One immediately striking point is that we needn’t add both (4) and (5) since we can get (4) from (5) (alongside other BCI principles): just put A → A for A in (5) and Modus Ponens delivers (4) since the resulting substitution gives a formula with a provable antecedent. Perhaps because of this, when summarising the results of Bunder [1983a] in his [1983c], Bunder refers only to (5) and not also to (4) as the needed supplement for turning BCI logic into something algebraically tractable in the manner of BCK logic. (Actually this is all handled rather more syntactically by Bunder than the above description in terms of validity in an algebra may suggest; his formulations effect a bi-directional translation between the theorems of the sentential logic on the one hand and those of the equational theory of the class of algebras on the other.) Now, the trouble


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with picking (5) from the pair {(4), (5)} is that (5) is not valid in every BCI algebra, whereas (4) is. One would have thought that the motive for ‘boosting’ BCI logic here is to obtain a logic which is sound and complete, unlike BCI logic itself, w.r.t. the class of all BCI algebras. To pursue completeness with the aid of (5) is to sacrifice soundness. The rationale behind (5) is perhaps somewhat opaque, though a suggestion will be made below. Bunder is aware of the fact that it is not (as we would put it) valid in every BCI algebra, since in Bunder [1983c] he explicitly moves from the class of BCI algebras to the proper subclass of such algebras satisfying the additional condition (6) x → (x → 1) = 1 It is worth actually verifying that this is indeed, as claimed, a proper subclass; we can readily check this with the aid of a three-element BCI -algebra Example 7.25.12 Consider the BCI algebra with → table given by Figure 7.25a. → 1 2 3

1 1 3 2

2 2 1 3

3 3 2 1

Figure 7.25a

In this BCI -algebra (6) fails: take x = 2 and we get 2 → (2 → 1) = 2 → 3 = 2 = 1. (This matrix and counterexample are due to Brian Weatherson.) Let us return to the question of why (5) might be though pertinent as a way of boosting BCI logic. What may have led Bunder to entertain this idea is that (5) is sufficient to turn BCI into a monothetic logic. It is clear that adding to the usual axiomatization of BCI logic the following rule (7)

A

B

A → B

is both necessary and sufficient for this purpose (necessary because if A and B are both provable they need to be synonymous, so since A → A is provable, we need to have A → B provable too; sufficient because the provability of A → B and B → A suffices for the synonymy of A and B in any extension of BCI in the same language, and (7) gives this for any A and B which are provable). Now, suppose one did not want to add rules, but only axioms (or axiom-schemata); then one might hope to get the effect of (7) with the implicational schema: (8)

A → (B → (A → B))

since given the two premisses of (7), two appeals to Modus Ponens will deliver the corresponding conclusion. And (5) is just a BCI -equivalent form of (8), with the inner antecedents permuted. That, then, is a possible explanation as to why (5) should be being considered at all in the present context. But it is not really a justification, since (5) is stronger than we want for the purpose – assuming

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that our purpose is to bring BCI logic into line with BCI algebra, since, as we have seen, not every BCI algebra validates (5). (Actually what we literally saw was that not every BCI algebra satisfies the condition (6), but this comes to the same thing.) The logic we should be concentrating is not BCI + (5) but rather the smallest extension of BCI closed under not only Modus Ponens but also the ‘monotheticity’ rule (7). Let us call the latter logic: monothetic BCI logic. (For some further information and references, see §4 of Humberstone [2006b], Kowalski and Butchart [2006], and Ulrich [2007]. Monothetic BCI is referred to as μBCI in some of these papers. Exercise 7.25.13 Show (i) that monothetic BCI logic can also be axiomatized by adding to the usual basis for BCI logic the following rule: B A → (B → A) and (ii ) that monothetic BCI logic is sound and complete w.r.t. the class of BCI algebras. Notice that the rule under 7.25.13(i) can be thought of as a special case of the schematic form of the distinctive BCK axiom K: instead of having every formula of the form A → (B → A) provable, we have this only on the proviso that B itself is provable. We return now to the point, remarked on without proof above, that BCIA logic is, unlike BCI logic, already monothetic and in no need of boosting to a special ‘monothetic BCIA’ logic. Exercise 7.25.14 (i) Show that the following rule is a derivable rule of BCI logic: From: B to: (B → A) → A (This is ‘Rule Assertion’ from 7.24.) (ii ) Using (i) and 7.25.11 show that BCIA logic is closed under the rule (7) and is accordingly monothetic. (iii) Understanding by a BCIA-algebra a BCI -algebra satisfying the equation corresponding to A (i.e., ((x → y) → y) → x = 1, or, what comes to the same thing for BCI -algebras, satisfying (x → y) → y = x), show that BCIA logic is sound and complete w.r.t. to the class of BCIAalgebras. (iv ) Show that the class of BCIA-algebras (as understood in (iii)) is equationally definable. (See the proof of 2.13.11, p. 234, and the discussion following 2.13.16 for similar proofs in the case of BCKW - (alias SK -) algebras, BCKP -algebras and BCKQ-algebras.) The characterization given of BCIA-algebras in part (iii) of the above exercise is only one of many equivalent characterizations, usefully collected together as Theorem 2 in Daoji [1987], and partially reproduced in the following: Theorem 7.25.15 Suppose A = (A, →, 1) is a BCI-algebra. Then the following conditions are equivalent: (1) A is a BCIA-algebra (2) A satisfies the quasi-identity x → y = 1 ⇒ y → x = 1


1109

(3) A satisfies the quasi-identity x → 1 = 1 ⇒ x = 1. (Note. There is a double use of “A” here for the universe of the algebra and as the “A” of “BCIA”, but no danger of confusion should arise.) Proof. (1) ⇒ (2): Suppose A is a BCIA-algebra in which for elements a, b, we have a → b = 1. We must show that b → a = 1. The result clearly follows if we can show a = b, which we do as follows: (i) (ii) (iii) (iv)

a→b=1 (a → b) → b = 1 → b (a → b) → b = b a=b

(Given) multiplying both sides of (i) by b From (ii) since 1 → b = b Simplifying the lhs of (iii) by the A-identity.

The reference in the justification given at line (iv) is of course to the identity tA = 1 corresponding to the Abelian axiom A. (See 7.25.14(iii).) (2) ⇒ (3): Take 1 as y. (3) ⇒ (1): Suppose that A satisfies z → 1 = 1 ⇒ z = 1. (We reletter “x” to “z” here to avoid a clash with what follows.) Note that the equation (((x → y) → y) → x) → (x → x) = 1 holds in every BCI -algebra, this being the equation corresponding to the formula under 7.25.10(ii); we can re-write it as (((x → y) → y) → x) → 1 = 1. So putting (((x → y) → y) → x) for z, in the supposedly satisfied quasi-identity z → 1 = 1 ⇒ z = 1: (((x → y) → y) → x) = 1

which means that A is a BCIA-algebra.

Remark 7.25.16 The (1) ⇒ (2) part of the above proof is conspicuously simpler than our proof of the corresponding logical property: that BCIA logic was closed under the rule of Conversion (7.25.11). We could not simulate the algebraic argument in the logical setting at which that proof occurred, however, because of the role played by the distinguished element 1: at that point, we had not shown that all BCIA-provable formulas were synonymous, and indeed used the former result to derive the latter (in the proof suggested for 7.25.14(ii)). An alternative proof of monotheticity, again using 7.25.11, might run as follows: Suppose that A and B are both BCIA-provable. Then by 7.25.9, (p → p) → A and (p → p) → B are both provable, so by 7.25.11, A → (p → p) is provable, and A → B follows by transitivity (or more precisely, by appeal to B and Modus Ponens, twice). Condition (2) of 7.25.15 is, as we have just noted, an algebraic analogue of the Conversion property of BCIA logic. To extract the logical content of condition (3), we may rewrite it as If x → y = 1 and y = 1 then x = 1 which is suggestive of a rule no less unfamiliar than Conversion, a kind of ‘Reverse Modus Ponens’ (called Ponens Modus in Meyer and Slaney [1989]): A→B

B A

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The admissibility of this rule for BCIA logic is an immediate consequence of the admissibility of Conversion, together with (regular!) Modus Ponens. However, since the conditions listed under 7.25.15 are given as equivalent conditions on BCI -algebras, it is of interest to display in its logical setting the converse of the point just made: not only is Reverse Modus Ponens BCI -derivable from Conversion, but Conversion is derivable from Reverse Modus Ponens, given the rest of BCI logic: (i) (ii ) (iii) (iv ) (v)

A→B (A → B) → ((B → A) → (B → B)) (B → A) → (B → B) B→B B→A

ex hypothesi provable subst. instance of B (i), (ii ), Modus Ponens subst. instance of I (iii), (iv ) Reverse M.P.

Thus not only is BCIA logic the smallest logic extending BCI logic and closed under Conversion, it can equally well be described as the smallest extension of BCI logic closed under Reverse Modus Ponens. We can formulate admissibility of Reverse Modus Ponens for a logic in the following terms: No theorem is provably implied by a non-theorem. Thus, in satisfying this condition BCIA logic contrasts rather strikingly with BCK logic, in which every theorem is provably implied by every non-theorem (as well as by every theorem, of course). Digression. The property of being a logic in which no theorem is provably implied by a non-theorem has as a special case the property of having no theorem provably implied by a propositional variable, which is reminiscent of the ‘Ackermann property’ for systems such as E (see Anderson, Belnap and Dunn [1992], p. 156): no →-formula is provably implied by a propositional variable. The latter property is one which the relevant logic R, as well, indeed, as BCI logic itself (R’s implicational fragment being BCIW ), famously lacks. The standard example here is the BCI -provable p → ((p → q) → q) ( = “Assertion” in the traditional relevant-logical terminology). Since the consequent here is not itself a theorem, this is no counterexample to the claim that no theorem is provably implied by a non-theorem (indeed, by a propositional variable). In R (BCIW ), we can make a substitution of p → q for q, getting p → ((p → (p → q)) → (p → q)) which does stands as such a counterexample, but of course this would not serve in the case of BCI logic, since the consequent is the BCI -unprovable formula W . (For another such case, see 7.25.26 below.) Indeed it is clear that we never have p → A and A also provable in BCI logic, since the latter is consistently extended by BCIA logic, in which both would accordingly be provable, and thus also (7.25.11) A → p and so, by Modus Ponens, p itself (contradicting the consistency of BCIA logic). This does not show that for no unprovable formula B are both B → A and also A BCI -provable, and indeed we can offer the following counterexample to any such conjecture: take B → A as (((p → q) → q) → p) → (p → p). See the discussion of Reverse Modus Ponens above for the rationale behind this example. End of Digression. Theorem 2 of Daoji [1987] was only selectively reproduced above as 7.25.15. In fact Daoji lists several conditions which are equivalent, as conditions on BCI -algebras and which therefore give alternative ways of defining the class of


1111

BCIA-algebras. Here we will be concerned only to show, of two of them, that they are conditions satisfied by all BCIA-algebras. Since the axiom A and its converse are both BCIA-provable, all such algebras (as noted in 7.25.14(iii)) satisfy (9) (9)

(x → y) → y = x

which, in the terminology of 3.14.16 (p. 415), says that → has itself as a secondargument reciprocal, and as we noted there, this implies that we have a right cancellation principle (just post-multiply both sides of the antecedent by z): (10)

x→z=y→z ⇒ x=y

This is the first of the two additional conditions mentioned. The second, perhaps somewhat more surprising, is that we also have left cancellation: (11)

x→y=x→z ⇒ y=z

We can reduce the case of (11) to that of (10) by noting that all BCIA-algebras satisfy the equation (12)

(y → x) → 1 = x → y

To see that (12) is a BCIA-identity, note that (13) is—by consideration of the BCIA-provability of the formula A* of 7.25.10(iii): (13)

(x → y) → z = (z → y) → x

Putting y for x, x for y, and 1 for z in (13), we get (14)

(y → x) → 1 = (1 → x) → y

and simplifying 1 → x to x (by 2.13.26, p. 244), we get (12). Using (12) we can turn right-cancellation into left-cancellation: Suppose x → y = x → z; applying (12) to both sides, we get: (y → x) → 1 = (z → x) → 1. So by (10): y → x = z → x and by (10) again: y = z. As far as the implicational logic is concerned, we have BCIA logic closed under two rules corresponding to (10) and (11): (Left Canc.)

(A → C) → (B → C) A → B

(Right Canc.)

(A → B) → (A → C) B → C

Since in (10) the antecedent is x → z = y → z, with “=” and not just “” (the partial order introduced into BCK - and BCI -algebras in the discussion of 2.13), there should be two premiss-sequents for (Left Canc.): that given and also one for the converse implication. But here we can rely on the fact that BCIA logic is closed under Conversion (7.25.11) to simplify the formulation to the one-way implicational premiss above, with a like simplification having been made in the case of (Right Canc.). Similarly, as far as questions of admissibility are concerned, we could equally well have replaced the implications in the conclusions of these cancellation rules by their converses. (This is not to say that such changes make no difference to the derivability of the rules concerned. As it stands (Left Canc.) is clearly a derivable rule of BCIA (in the axiomatization

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suggested by that labelling), since the conclusion follows from the premiss by appeal to Modus Ponens and the converse of B , whose BCIA-provability was noted in 7.25.10(i).) The most striking aspect of the above algebraic development from the point of view of sentential logic is perhaps (12). If we let t be any provable formula of BCIA logic (all such formulas being, we recall from 7.25.14(ii), BCIAsynonymous), then the analogue of (12) tells us that (A → B) → t and B → A are provably equivalent for any A and B. (There is a close affinity between this abbreviative use of “t” and the use of “t” as a special constant introduced in 2.33.) For any formula C, we further abbreviate the formula C → t to κC. κ is a (derived) singulary connective which turns an implication into something equivalent to its converse, and for which A and κκA (for all A) are equivalent; if A is pi , and so not itself an implicational formula, since we have pi equivalent to t → pi , κpi is equivalent to pi → t. Remark 7.25.17 Since κ as just defined is a congruential connective which turns an implication A → B into (something synonymous with) its converse B → A, what we called (1.15) the Converse-Proposition Fallacy is not a fallacy after all in the context of BCIA-algebras, such as, in particular, the Lindenbaum algebra of BCIA logic. (This is noted in Humberstone [2002a]. In Humberstone [2005d], κ is used as a contrary-forming rather than a converse-forming operator, alongside σ an operator forming subcontraries. This usage was followed above in Exercise 6.31.15 (p. 850). And in Casari [1989], referred to in the Digression below, κ is an abbreviation for ¬(p → p).) Exercise 7.25.18 Suppose we add a new 1-ary connective (again written) κ to the language of BCI logic, governed by a congruentiality rule A→B

B→A

κA → κB and the distinctive axioms: (i)

κ(p → q) → (q → p)

(ii )

(q → p) → κ(p → q).

Show that the extension of BCI logic by these principles is not conservative, and in particular that A, the distinctive BCIA principle, is provable in the extended logic. (Suggestion. Begin by applying the congruentiality rule to the BCI equivalent formulas (p → q) → (p → q) and p → ((p → q) → q).) Digression. A version of BCIA logic with negation was axiomatized in §2 of Paoli [2001] (distilling this axiomatization from Casari [1989], esp. §2.6, Thm. 1) by adding axioms governing negation and giving it the usual De Morgan negation properties (double negation equivalences and all forms of contraposition) as well as some unusual properties. In more detail, we have axiom schemes for monothetic BCI : B , C, I ∗ (explained below), with the rule Modus Ponens, and, for ¬, double negation and contraposition axioms ((ii) and the converse of (iii) from 8.12.10, p. 1185, are what Paoli uses), as well as the special axioms:

7.2. INTUITIONISTIC, RELEVANT, CONTRACTIONLESS. . . ¬(A → A)

and

1113

¬(A → A) → (A → A)

∗

with I being the schema: (A → A) → (B → B). Note that the contraclassical surprises on the part of → are here concealed in these special negation principles (in particular the first inset above). Thus this axiomatization does not have the Separation Property, since the → axioms do not suffice for a proof of the Abelian formula A. (Compare the axiomatization in Shalack [2004], in which ¬A is defined as A → (A → A), which Meyer and Slaney [1989] called the canonical negation of A, as opposed to the B-relative negation of A for arbitrary B, namely A → B. Since in the current logic all ‘self-implications’ are synonymous, we can write this negation of A as A → t.) It requires some effort to derive the conversion effects (i) and (ii) from 7.25.18 on this basis, taking κ as ¬. Here we use schematic letters rather than propositional variables. To obtain (A → B) → ¬(B → A) we proceed as follows, the proof being informally sketched. In particular we absorb double negation equivalences in appeals to contraposition, themselves applying to subformulas of the whole formula, and the line annotations mean that the preceding line is transformed into the given line as indicated: (1) (A → B) → ((B → A) → (A → A)) B (2) (A → B) → (¬(A → A) → ¬(B → A)) contraposing the consequent (3) ¬(A → A) → ((A → B) → ¬(B → A)) permuting antecedents We complete the proof and obtain (A → B) → ¬(B → A) by applying Modus Ponens to (3) and the first special axiom governing negation. The converse is trickier, and apparently calls for the full force of I ∗ , which together with the second special axiom inset above gives line (1): (1) (2) (3) (4) (5) (6)

¬(B → B) → (A → A) A → (¬(B → B) → A) A → (¬A → (B → B)) B → (¬A → (A → B)) B → (¬(A → B) → A) ¬(A → B) → (B → A)

as explained permuting antecedents contraposing several permutations contraposing permutation

Let us note in the same spirit that it is possible to proceed directly to a deduction of A from the BCI -provable principle of Assertion in the manner indicated by the following steps, in which we exploit the De Morgan negation ¬’s newly observed ‘conversifying’ effect on implicational formulas. We refer to the latter – i.e. to the equivalence of a negated conditional with its converse, as the converse effect in the following annotations. (One could instead follow the route suggested by Exercise 7.25.18, since ¬ is congruential here.) (1) (2) (3) (4) (5) (6)

B → ((B → A) → A) B → (¬A → ¬(B → A)) B → (¬A → (A → B)) ¬A → (B → (A → B)) ¬(B → (A → B)) → A ((A → B) → B) → A

Assertion contraposing converse effect permuting contraposing converse effect

Finally, we can isolate the converse effect as a pure implicational principle by translating implicational contraposition into negation-free terms, writing the

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negation of an implicational formula as its converse. (For the double negation equivalences there is nothing to write, since the converse of the converse of A → B is just A → B itself.) This gives the BCIA-derivable schema: ((A → B) → (C → D)) → ((D → C) → (B → A)). And from this we may in turn (using BCI resources) derive A, for example going via the principle, stated here as a formula rather than in schematic terms, called Pivot in Humberstone [2002a], and already introduced in 7.25.10(iii) above: (Pivot)

((p → q) → r) → ((r → q) → p).

To obtain this from the principle inset above, begin with the C (alias Perm) formula (p → (r → q)) → (r → (p → q)) and take p as A, r → q as B, r as C and p → q as D. And to obtain A from (Pivot), substitute p → q for r and apply Modus Ponens. End of Digression. Our earlier talk of cancellation properties suggests some connection with groups (0.21); there is in any case the term “Abelian Logic” (for a logic with BCIA as its →-fragment) whose suitability has yet to be made manifest. To turn our BCIA-algebras into groups, indeed, Abelian groups, we shall need a binary operation, which in general won’t be → itself, an identity element, which will just be the BCIA-element 1, and inverses. (15) For a, b ∈ A, where A = (A, →,1), put: a · b = (a → 1) → b

a−1 = a → 1.

For a BCIA-algebra A, we denote by gp(A) the algebra (A, ·,−1 , 1) obtained from A by the definitions in (15). Note that a−1 is just the analogue, for BCIA-algebras, of the formula κA of BCIA logic, as understood in the discussion preceding 7.25.17. Putting 1 for y in (13), we see that the operation · is commutative, so to check that 1 is an identity element (for this operation), it suffices to verify that a · 1 = a, for all a ∈ A. (I.e., we don’t have to verify separately that 1 · a = a – not that this presents any difficulty.) But a · 1 = (a → 1) → 1 = a by (9). Again, because of commutativity, to check that a−1 is an inverse of a, we need only check one of: a · a−1 = 1, a−1 · a = 1; here the former is more readily verified, since it amounts to: (a → 1) → (a → 1) = 1, and so is given by the I-equation. We leave the reader to check that the operation · is associative, which completes the demonstration that for a BCI -algebra A, the algebra gp(A) is an Abelian group. Moving in the other direction, suppose A = (A, ·,−1 , 1) is an Abelian group and we define the binary operation → on A by (16): (16) For a, b ∈ A, let a → b = (a · 1) · b. Then we denote by bc(A) the algebra with universe A, binary operation →, and distinguished element 1. This time we leave to the reader the whole task of verifying that if A is an Abelian group, bc(A) is a BCIA-algebra. For this purpose, it useful to recall that since every group satisfies (17), our Abelian groups satisfy not only (17) but also (18): (17)

(x · y)−1 = y −1 · x−1

7.2. INTUITIONISTIC, RELEVANT, CONTRACTIONLESS. . . (18)

1115

(x · y)−1 = x−1 · y −1

The salient points, summarizing both directions at once, are given by the following (Kalman [1976], p. 195, and Daoji [1987], Theorem 3): Theorem 7.25.19 If B is a BCIA-algebra then gp(B) is an Abelian group; if A is an Abelian group, bc(A) is a BCIA-algebra; further, for any BCIA-algebra B and any Abelian group A, we have bc(gp(B)) = B and gp(bc(A)) = A. Thus we have a simple one-to-one correspondence between BCIA-algebras and Abelian groups. We can use 7.25.19 to check the BCIA-provability of a formula A by considering, instead of the BCIA-algebraic equation tA = 1 (recalling the notation of 2.13), the corresponding group-theoretic equation. For example, in the case of the contraction axiom W , which has the BCIA-algebraic analogue (19): (19)

(x → (x → y)) → (x → y) = 1

we have the group-theoretic equation (20)

(x−1 · (x−1 · y))−1 · (x−1 · y) = 1.

Simplifying the lhs to x · (x−1 · y)−1 · x−1 · y (in which parentheses have been omitted when the associativity of · means no ambiguity is created) and thence to x · x · y−1 · x−1 · y we find that cancelling the occurrence of each variable v with its (not necessarily adjacent) ‘inverse’ v −1 , we are have an occurrence of “x” left over at the end, showing that (20) does not hold in every Abelian group, and hence that (19) does not hold in every BCIA-algebra, showing that W is not BCIA-provable. Of course, we already knew this on the basis of a much cruder syntactic examination, since not every variable in W occurs an even number of times. (See the discussion preceding 7.25.6, p. 1103.) But we can use the same method in the case of other formulas in which the ‘even number of occurrences’ test is passed: Exercise 7.25.20 Use the method described above to show that the formula (p → q) → (q → p) is not provable in BCIA logic. (This point was mentioned in 7.25.6 above.) Remark 7.25.21 The group-theoretic reduction method described above provides a close connection between BCIA logic and the classification of subformulas of a formula as antecedent and consequent parts, as may be found in Anderson and Belnap [1975], pp. 34, 93, 253–4. The occurrences of a propositional variable pi which are consequent parts of a formula C appear as occurrences of xi in the simplified form of the group-theoretic version of the equation tC = 1, while those which are antecedent parts appear as occurrences of x−1 in that form. It follows that i a purely implicational formula C is BCIA-provable if and only if each variable has the same number of occurrences as an antecedent part of C as it has as a consequent part of C. (This remark is included for those familiar with the ‘antecedent part’ vs. ‘consequent part’ distinction, for the precise definition of which other readers are referred to Anderson and Belnap’s discussion. It provides, incidentally, a very simple decision procedure for BCIA logic.)

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Exercise 7.25.22 Let A = (A, →, 1) be an algebra of type 2, 0 . Show that the following are equivalent: (i) A is an associative BCI -algebra (i.e., A is a BCI -algebra in which the operation → is associative) (ii ) A is a commutative BCIA-algebra (that is, a BCI -algebra satisfying x → y = y → x; cf. 7.25.20) (iii) A is a BCIA-algebra satisfying x = x → 1. Kalicki and Scott [1955] exhaustively list the equationally complete varieties of groups, these being precisely the varieties Kp for p prime, with A ∈ Kp just in case A = (A, ·, −1 , 1) is an Abelian group in which ap = 1 for all a ∈ A, where ap is the element a · . . . · a (p times). 7.25.19 allows us to convert this into a classification of the Post-complete extensions of BCIA logic. For p = 2, we have groups satisfying x2 = 1, which can be rewritten as x = x−1 ; the corresponding BCIA-algebraic equation (by (15) above) is that figuring in 7.25.22(ii ). (More information on this class of BCI -algebras may be found in Hu and Iséki [1980] and [1984], and Hoo and Ramanamurty [1987].) Remarks 7.25.23(i) The two-element boolean group (see 6.12.1) is functionally free in the class K2 as just defined. (For ‘functionally free’, see the discussion following 0.24.5.) Thus the algebras isolated under various equivalent descriptions in 7.25.22 provide an algebraic semantics for the pure equivalential fragment of CL (except that one would normally use the notation “↔” rather than “→” here) in the same way as BCIA-algebras do for BCIA logic. (ii ) We can use the Kalicki–Scott result mentioned above to obtain other Post-complete extensions of BCIA logic, e.g., taking p = 3, the logic determined by the class of BCIA-algebras satisfying (x → 1) → x = x → 1 (Here we have taken the BCIA-algebraic translation of x3 = 1 after reformulating this as x2 = x−1 .) That concludes our discussion of BCIA logic, and we return to the more orthodox BCI and BCK systems, concluding our discussion of contractionless logic in Fmla with something the latter logics have in common (and indeed share with BCIA logic). We state this striking result without proof. (For discussion and proofs, see the references provided in the notes, under ‘Meredithiana’, p. 1122.) Theorem 7.25.24 (i) Any formula provable in BCI logic is a substitution instance of a BCI-provable formula in which every propositional variable that occurs, occurs exactly twice. (ii) Any formula provable in BCK logic is a substitution instance of a BCKprovable formula in which no propositional occurs more than twice. The property ascribed to BCI logic by (i) here may be called the 2-Property, and that ascribed to BCK logic by (ii ), the 1,2-Property (since every occurring variable must have either one or two occurrences in some formula of which


1117

our BCK -theorem is a substitution instance). More widespread is the use of these labels to apply to the formulas concerned: a formula has the 2-property (1,2-property) if every propositional variable occurring in it occurs exactly (at most) twice. A more accurate summary of Jaśkowski’s results is then to say that in BCI logic every theorem is the result of a variable-for-variable substitution in a theorem with the 2-property, likewise for BCK logic but with the 1,2-property instead. (See the notes for references: p. 1127.) In the BCI case, the 2-property theorems coincide with the formulas valid in the threeelement Sugihara–Sobiciński matrix: see the Digression following 2.33.5. Thus an arbitrary formula is a BCI theorem just in case it can be obtained by a variable-for-variable substitution from a formula with the 2-property which is valid in that matrix. In the case of BCK logic, Jaśkowski showed that the implicational fragments of BCK logic and CL have the same provable formulas with the 1,2-property – as therefore does IL (observes Hindley [1993]). Thus implicational formulas such as contraction (W ) and Peirce’s Law, on which these logics disagree, are entirely representative in featuring a propositional variable occurring more than twice. To illustrate the 2-Property for BCI logic consider the ‘Law of Triple Consequents’ (see 2.13.20(ii), p. 239) in one direction, namely: (21)

(((p → q) → q) → q) → (p → q)

As is clear from a BCI -proof of (21) which proceeds by applying the derived rule which we have called RSuff (a rule version of B) to the converse of A: (i) p → ((p → q) → q) (ii) (((p → q) → q) → q) → (p → q)

converse of A (alias Assertion) from (i) by suffixing q

we could instead have ‘suffixed’ anything else instead of q, so (21) is a substitution instance of the BCI -theorem (22)

(((p → q) → q) → r) → (p → r)

in which each variable occurs twice. If we consider instead the converse of (21), namely (23)

(p → q) → (((p → q) → q) → q)

we can cite (24) as a BCI -theorem in which each variable occurs twice: (24)

(p → q) → (((p → q) → r) → r)

Notice that (24) is not the converse of (22). We have chosen to illustrate the 2-Property with these two cases, in fact, because they serve as a warning against incorrectly drawing a certain conclusion from 7.25.24(i). First, note that this result can itself be re-expressed by saying: if A provably implies B in BCI logic, then there are formulas A0 and B0 which are substitution instances of A and B and in which every variable (which occurs) occurs exactly twice, such that A0 provably implies B0 in BCI logic. Now the tempting but overhasty conclusion to draw would be this: if A and B are provably equivalent in BCI logic then there are formulas A0 and B0 which are substitution instances of A and B and in which each variable occurs exactly twice, such that A0 and B0 are provably BCI -equivalent. (Recall that the provable equivalence of formulas A and B in an implicational logic means the provability of A → B and B → A.) A little reflection on the case of (21) and (23) will suffice to show that this ‘equivalence’

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version of the 2-Property is not correct (for BCI logic). For more information, consult the end-of-section notes (p. 1127). This marks the end of our discussion of contractionless logics in Fmla. We turn to the task of providing natural deduction systems for (pure implicational) BCI and BCK logics. As mentioned at the start of this subsection, we shall work in Mset-Fmla rather than Set-Fmla, just as we worked in MsetFmla0 and Mset-Mset for sequent calculi for intuitionistic and classical linear logic in 2.33 (before 2.33.10). As there, for the following rules (which we label with an “ms” subscript as a reminder of this fact) the capital Greek letters range over finite multisets of formulas. For introducing and eliminating → we have: (→I)ms

Γ, A B

(→E)ms

ΓA→B

Γ A→B

Δ A

Γ, Δ B

There are also multiset versions of the basic structural rules: (R)ms

AA

(T)ms

(M)ms ΓA

ΓB Γ, A B

Δ, A B

Γ, Δ B

which are simply the rules Identity, Left Weakening and Cut, in their ILL form, from 2.33, as well as of structural rules which lack analogues in SetFmla, the ILL forms of Left Contraction and Left Expansion from 2.33 (from the Digression on p. 354 onwards). (Contraction)ms

Γ, A, A B Γ, A B

(Expansion)ms

Γ, A B Γ, A, A B

A natural deduction system for BCI logic is obtained by taking the rules (R)ms , (→I)ms and (→E)ms . (The rule (T)ms is derivable from these.) The correspondence with BCI logic in Fmla is simpler than that for the Set-Fmla system (RNat) given in 2.33 and further studied in 7.23, since we may take a sequent A1 , . . . , An B to correspond directly to the formula: A1 → (A2 → . . . → (An−1 → (An → B)). . . ). The same correspondence works in the case of BCK logic, a natural deduction system for which is obtained by adding the rule (M)ms to the basis described above for BCI logic. These claims fall out as corollaries to the soundness and completeness theorems given below (7.25.27–28). To obtain analogous natural deduction systems for pure implicational R and the implicational logic RM0, add Contraction for the former and Contraction along with Expansion for the latter. (We gave a Set-Fmla natural deduction system for the implicationconjunction-fusion fragment of RM in 5.16, illustrating it in action with 5.16.6, as well as providing such systems in that framework for the implicational fragment of R and for RM0 in 2.33 – the latter on p. 331; the rule (→ I)ms should be compared with the rule (→I)d to see how a similar effect is achieved in the different logical frameworks. Recall from 2.33 – e.g. the discussion after 2.33.1


1119

– that RM0 is strictly weaker than the implicational fragment of the full logic RM) In fact, before looking at the case of BCI logic, we pause to illustrate these Mset-Fmla proof systems with a proof in that just described for implicational R (alias BCIW logic). The example concerns a formula—we end up discharging all assumptions— whose R-provability is noted in Anderson and Belnap [1975] (§8.13, p. 96). Example 7.25.25 A proof of (p → (p → p)) → (p → (p → (p → p))) in the Mset-Fmla natural deduction system for BCIW logic. Here we write “W ” to indicate an application of the rule (Contraction)ms , and as usual when giving a Lemmon-style formula-to-formula presentation of the proof, start off with justifications of the form “Assumption” (rather than “(R)ms ”) – here abbreviated to “Ass’n” – and we delete the “ms” subscript from the labels for the rules (to avoid clutter). 1 2 3 4 1, 2 1, 2, 3 1, 1, 2, 3 1, 2, 3 1, 2, 3, 4 1, 2, 3 1, 2 1

(1) p → (p → p) (2) p (3) p (4) p (5) p → p (6) p (7) p → p (8) p → p (9) p (10) p → p (11) p → (p → p) (12) p → (p → (p → p)) (13) (p → (p → p)) → (p → (p → (p → p)))

Ass’n Ass’n Ass’n Ass’n 1, 2 →E 3, 5 →E 1, 6 →E 7W 4, 8 →E 4–9 →I 3–10 →I 2–11 →I 1–12 →I

Remark 7.25.26 The formula just proved is quite interesting as a variation on the theme of the Expansion formula (p → p) → (p → (p → p)), since the latter is not R-provable. (This is an implicational formulation of the structural rule of Expansion.) By permuting back the first “p” of the consequent we get another example – the last one having been mentioned in the Digression shortly following 7.25.16 above – of a formula, namely p → [(p → (p → p)) → (p → (p → p)], showing that a propositional variable can provably imply a theorem in implicational R. (The reader wanting further practice with the multiset natural deduction approach might care to construct a proof like 7.25.25 for this new formula.) Indeed the example shows that a propositional variable – and hence, since we have closure under Uniform Substitution – an arbitrary formula, provably R-implies a formula of the form A → A. Section 4 of Humberstone [2006b] asks whether something weaker can be shown for BCI logic: does every theorem of BCI logic provably imply a self-implication (i.e., a formula of the form just cited)? We

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certainly don’t have every formula provably implying a self-implication in BCI logic, since this logic has no theorems of the form p → (A → A). (Exercise: Why not? Hint: think about ↔ in CL.) This leaves open the earlier question about whether every theorem BCI -implies a selfimplication. That question was settled (negatively) in Kowalski [2008], by means of an insightful proof-theoretic analysis. Since our main business for the present subsection is contractionless logics, let us return without further ado to the BCI case. The structures that will serve as the frames on which to construct models here are commutative monoids, for which we use the same notation as for the semilattices of the preceding two subsections. A sequent Γ B of Mset-Fmla , . . . , An ], for holds in such a model M = (S, ·, 1, V ) just in case, where Γ = [A1 n all x1 , . . . , xn ∈ S with M, xi |= Ai (i = 1, . . . , n), we have M, i=1 xi |= B. (The “[. . . ]” notation here is from the notes – under the heading ‘multisets’ – to §2.3, p. 373.) Here we use the same iterated product notation as in 2.33, which denotes the element 1 when n = 0, and presume the same definition of truth at a point (again written on the left of the “|=”, rather than in the more customary subscripted position) as in that section. In fact the only difference from our procedure in 2.33 for Set-Fmla is that here, in writing Γ as [A1 , . . . , An ] we are explicitly refraining from assuming that if i = j, then Ai = Aj . (That is, we allow a formula to have more than one occurrence in Γ.) Validity on (S, ·, 1) is, as usual, a matter of holding, for every Valuation V , in the model (S, ·, 1, V ). Theorem 7.25.27 A sequent is provable in the above natural deduction system for BCI logic if and only if it is valid on every commutative monoid. Proof. ‘Only if’: A soundness proof by induction on the length of the shortest proof of an arbitrary sequent is left to the reader. ‘If’: An adaptation of the canonical model method of 7.23.5 (p. 1090) serves us admirably. Here the relevant model (S, ·, 1, V ) has for S the collection of all finite multisets of formulas, with · as multiset union, 1 as the empty multiset, and Γ ∈ V (pi ) iff the sequent Γ pi is provable in the above natural deduction system for BCI. Note that this is indeed a model because the frame (S, ·, 1) is a commutative monoid (though because · is not idempotent, it is not a semilattice with identity element). As in the proof of 7.23.5, for this model we show Γ |= A iff the sequent Γ A is provable by induction on the complexity of A, the basis case being secured by the definition of V . For the induction step, supposing this is so for B and C, we may infer that it is so for B → C thus: If Γ |= B → C, then for all finite multisets Δ such that Δ |= B, the multiset union of Γ with Δ verifies C, and so in particular for Δ = [B] the inductive hypothesis gives us the provability of Γ, B C, whence by (→I)ms , we get Γ B → C. For the converse, suppose that Γ B → C is provable; to show that Γ |= B → C, we must show that for any Δ for which Δ |= B, we have Θ |= C, where Θ is the multiset union of Γ with Δ. By the inductive hypothesis, this amounts to showing that the provability of Δ B implies that of Γ, Δ C. As Γ B → C, this follows by (→ E)ms . Thus for all Γ, A, we have Γ A provable iff (in the model (S, ·, 1, V )), Γ |= A. Now suppose that a sequent Γ A is not provable, where Γ is [C1 , . . . , Cn ] Then Γ is itself the product (by ·, alias multiset union) of elements [C1 ],. . . ,[Cn ] of S respectively verifying C1 , . . . , Cn , but whose product


1121

does not verify the formula A. So C1 , . . . , Cn A does not hold in (S, ·, 1, V ). Recall that the Mset-Fmla natural deduction system for BCK logic has the same basis as that for BCI logic except that we add the structural rule (M)ms . Recall also, from 2.33, the persistence-like condition (P0 ) on models (S, ·, 1, V ), understood as a requirement in the case of each propositional variable pi : (P0 )

For all x, y ∈ S, if y ∈ V (pi ) then x · y ∈ V (pi )

which was there noted to yield—for semilattice models, though the idempotence condition plays no role in this, so we can appeal to it here also—the general version, understood to hold for arbitrary formulas A: (P)

For all x, y ∈ S, if (S, ·, 1, V ), y |= A then (S, ·, 1, V ), x · y |= A.

Adapting the proof given for 7.25.27 will yield a solution to the following exercise. Exercise 7.25.28 Show that a sequent is provable in the natural deduction system for BCK logic if and only if it is valid on every commutative monoid, when validity on (S, ·, 1) is understood as holding in every model (S, ·, 1, V ) satisfying (P0 ). If we went further in this direction, and added in (Contraction), we should have ended up with the Kripke semantics for IL (since BCKW, alias SK from 2.13, is IL’s →-fragment) by variations on the theme of Urquhart’s semantics for relevant logic. (The interested reader may care to consider how the appropriate binary accessibility relation is to be defined in terms of ·.) We have not attempted to indicate how the present semantical treatments of BCI and BCK logic are related to the semantics in terms of BCI - and BCK -algebras; information on this score may be gleaned from Došen [1989a], Meyer and Ono [1994], and other sources on contractionless logics cited in the notes below (under the heading ‘Doing Without Contraction’, p. 1124). See also Urquhart’s §47, esp. p. 149, in Anderson, Belnap and Dunn [1992].

Notes and References for §7.2 Wajsberg’s Law. Wajsberg’s name (see 7.21.5) is mentioned in connection with the formula ((((p → q) → p) → p) → q) → q in Curry [1963], p. 243. Since the appearance of that book, Wajsberg’s works have been translated into English, and one can see the formula displayed as α under Example 4 on p. 151 of the collection, Surma [1977], of these works. (Our sequent 7.21.4(ii) has the main → in this formula replaced by . For more on Wajsberg’s logical work, see Surma [1987].) Some readers may know the formula from Ono [1990], p. 104, where it appears, by courtesy of G. E. Mints, as a counterexample to a certain proof-theoretic conjecture concerning IL which we need not go into. Figure 7.21a may be found in Setlur [1970a]. I am grateful to Allen Hazen for Remark 7.21.16.

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Meredithiana. The source for what we call Meredith’s translation, figuring in 7.22.29, is note 7 of Prior [1956], in which the pronoun “he” is used, though the reference is not exactly clear since there is no footnote flag numbered 7 in the text. The general tenor of the discussion makes it fairly clear, however, that the intended reference is to (C. A.) Meredith, and the matter is clinched by Prior’s compilation and annotation of Meredith’s sometimes cryptic notes in the form of Meredith and Prior [1968], whose §2 gives the result. (Meredith writes Gpq for our p ⊃ q: a piece of ‘Polish notation’, as explained in the notes to §2.1, under “Primitive Connectives” (p. 269). The same notation is used in McCall and Meyer [1966], in which this derived connective plays an important ancillary role.) Meredith’s result seems to have been rediscovered independently by Prucnal: see Prucnal [1974]. §3 of Meredith and Prior [1968] presents the first axiomatization of what we call BCIA logic (in 7.25), taken from a letter of Meredith’s in 1957, with a single axiom Meredith writes (p. 221), in an adapted Polish notation, as: LLpqLLLrqpr which in our notation would be: ((p → q) → (((r → q) → p) → r) (The rules are: Modus Ponens for L and Uniform Substitution.) The use of “L” is intended to recall ‘left division’, meaning that operation mapping elements a and b of a group to the element a−1 · b. (See (15) from 7.25, and surrounding discussion; further references: Kalman [1960], [1971], [1976], and McCune [1993]. Further historical information on Abelian Logic can be found in §9 of Meyer and Slaney [2002]. The authors cited here, Meredith included, also discuss logical aspects of ‘right division’ (a, b → a · b−1 ), as well as the situation with arbitrary groups in place of Abelian groups.) There is also an ‘equational axiomatization’ (analogous to the algebraic semantics but within the object language) with axiom: LLLLpqrpq = r which corresponds to what in our notation would be the condition (((x → y) → z) → x) → y = z on BCI -algebras – or, more informatively, on algebras of the similarity type of BCI -algebras, since the claim is that satisfying this equation for an arbitrary such algebra makes it a BCIA-algebra (or an Abelian group with left-division as the fundamental binary operation). (The above equation goes back, as Meredith and Kalman mention, to something eventually appearing as Forder [1968].) Note that the equational form inset above with Ls is misprinted (with the second “p” missing) on p. 221 of Meredith and Prior [1968], and that this omission is not corrected in the 1969 corrigendum note (see our bibliography). There is also a brief discussion of this material, with some additional references, in §8.5 of Kalman [2001]. It should be noted that the prefix “BCIA” is used in a completely different way in Raftery and van Alten [2000], for a class of algebras providing an algebraic semantics for the smallest monothetic extension of BCI logic, axiomatizable by adding the axiom (p → p) → (q → q) to B, C and I. The fact that BCI logic is not itself monothetic was mentioned in §2.1, along with the repercussions of this fact for algebraic semantics: see the discussion in following 2.13.25.


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Curry’s Paradox. As was mentioned in the introduction to 7.25, one of the reasons for interest in contractionless logics is a desire to avoid Curry’s Paradox, which arises in a set theory with ‘naive comprehension’ – a principle to the effect that for any formula A(x) in which x appears free, there is a set {x | A(x)} with the property that for any term t, the closed formulas A(t) and t ∈ {x | A(x)} are equivalent, which, since we are concentrating on purely implicational logics, we take to amount to the pair of schemas (1) and (2): (1) t ∈ {x | A(x)} → A(t)

(2)

A(t) → t ∈ {x | A(x)}.

Here “∈”, a two-place predicate symbol, is (we may suppose) the sole item nonlogical vocabulary of the theory, and A(t) is the result of replacing the free occurrences of “x” in A(x) by the term t. We do not disallow the possibility that there are other items of logical vocabulary, and, suppose, in particular, that there is a binary connective #, primitive or defined, which according to the (as yet unspecified) logic of the theory satisfies the following two conditions, stated as schemata in Fmla: (#1)

(A → (A # B)) → (A # B)

(#2)

(A # B) → (A → B).

Now, for the propositional variable q (and any arbitrary but fixed formula would do equally well for this) and use “Q” to abbreviate the term {x | x ∈ x → q}. As instances of (1) and (2), then, we have the following: (3) Q ∈ Q → (Q ∈ Q # q)

(4)

(Q ∈ Q # q) → Q ∈ Q.

By (#1), from (3), assuming → to obey Modus Ponens, we may derive Q ∈ Q#q and thus from (4) we have Q ∈ Q; but by appeal to (#2) and Modus Ponens (for →) again, we have Q ∈ Q → q, so by a final appeal to Modus Ponens, we infer q. But what was done for q here can be done for any formula, so if we want our set theory not to be non-trivial in the sense of not having every formula of its language provable, and we want to keep the comprehension principles (1) and (2), we must make sure that there is no connective # in the language behaving as (#1) and (#2) describe. (This is the rationale behind paraconsistent logics: see our discussion following 8.13.10.) Now, if the underlying logic proves W , then we can take # as → itself and (#1) will be satisfied, being in this case just (the schematic form of) W itself, and if—something less likely to be regarded as negotiable—the logic proves I, (#2) will be provable, being just I itself for this choice of →. Thus implicational R ( = BCIW ), for example, cannot serve as the underlying logic for a non-trivial version of naive set theory. Historically, we should note, Curry’s Paradox is the specific version of the above result for the choice, just indicated, of # as →, and was put forward to show that the problems presented by Russell’s Paradox – which amounts to the special case in which # is → and C is ⊥ – presented for naive set theory are not to be avoided by tinkering with the logic of negation (or ⊥). We have given the more general form since it is possible for some compositionally derived (or independent primitive) # to satisfy (#1) and (#2) even though W does not hold (for →). (See Restall [1993a] for an extended treatment of such possibilities; Restall’s conditions are in fact slightly different from our (#1) and (#2), but

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the tenor of the discussion is as indicated. He also touched on Abelian logic, on whose implicational fragment we are about to comment in a different way. Restall’s derivation of a Curry-like paradox for Abelian logic involves connectives other than → in an essential way.) See further ‘Curry’s Paradox (Resumed)’ below, and also the division of the notes to §8.1 headed ‘Defining a Connective in a Theory: A Further Example’ (p. 1211). Doing Without Contraction. What we had in mind as contractionless logics in 7.25 were logics for which the structural rule of Contraction, in (say) Mset-Fmla was not admissible, and the corresponding implicational logics in Fmla. Important historically in this area are Grišin [1982] and earlier works there referred to, as well as Dardžaniá [1977]. Semantical methods for such logics appeared in Došen [1989a], Ono [1985], Ono and Komori [1985], Komori [1986a] (with Komori [1986b] also being of interest), and, within the relevant logic tradition, in Slaney [1984], Giambrone, Meyer and Urquhart [1987], and Brady [1990], [1991]. There is a good survey in Došen [1993]. (See also Bull [1996], Bunder [1996], Lambek [1993a], Ono [1998a], [1998b], Restall [1993a], [1993b].) Such logics have caught the attention of those interested in computational matters because their predicate-logical extensions fare better than say, classical or intuitionistic logic in respect of decidability; work in this genre includes Ketonen and Weyhrauch [1984], Mey [1990], Kiriyama and Ono [1991], as well as the linear logic tradition (to which references were already given in the notes to 2.33). Here, using his own notation (rather than replacing “” with “”) we quote from Girard himself, in characteristically colourful voice: Contraction is the fingernail of infinity in propositional calculus: it says that what you have, [you] will always keep, no matter how you use it. The rule corresponds to the replacement of Γ Δ by Γ Δ , where Γ and Δ come from Γ and Δ by identifying several occurrences of the same formula (on the same side of the ). To convince oneself the rule is about infinity (and in fact that without it there is no infinite at all in logic), take the formula I: ∀x∃y x < y (together with others saying that < is a strict order). This axiom has only infinite models, and we show this by exhibiting 1, 2, 3, 4,. . . distinct elements; but if we want 27 different elements, we are actually using I 26 times, and without a principle saying that 26 I can be contracted into one, we would never make it! In other terms infinity does not mean many, but always. Another infinitary feature of the rule is that it is the only [one] responsible for undecidability: Gentzen’s subformula property yields a decision method for predicate calculus, provided we can bound the length of the sequents involved in a cut-free proof, and this is obviously the case in the absence of contraction.

This passage comes from p. 8f. of Girard [1995], the “[, ]” enclosing my guesses as to what might have been intended. Now there is also something quite different to be found under the heading of ‘logics without contraction’ or ‘contraction-free’ logics, in which – again often with computational considerations (such as ease of proof-search) in mind – sequent calculus proof systems are devised in which the structural rule of contraction is not derivable (neither a primitive rule nor a derived rule, that is),


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though it is admissible. Dragalin [1988], pp. 11–13 provides an example, using ideas going back to Kleene [1952] (and in particular, aspects of Kleene’s G3 proof system, touched on in our discussion in 1.28: see the end of 2.32; there is a review of the terrain in Dyckhoff [1992], though Dyckhoff’s own contribution was rather different, as noted in 2.32.12, p. 321). The situation here is much as with traditional Cut Elimination methodology. An application of the cut rule (i.e. (T)) erases a formula from the proof of a sequent, so we cannot be sure, starting from the sequent to be proved, of working backwards to find the proof. (Pym and Ritter [2004], especially Chapters 4 and 6, for a dedicated discussion of this theme.) An application of the contraction rule likewise erases a formulaoccurrence, placing a similar obstacle in the way of reconstructing the proof given only its terminus. Since the aim here is to constrain the class of proofs but not that of the provable sequents (where typically it is those of classical or intuitionistic predicate logic), the contraction rule will still be admissible, so we are not in the area of (what we have been calling) contractionless logics. (More selectively, one may try to show that for sequents satisfying a particular condition, a proof without contraction is always possible; an example of work in this style – Kashima [1997] – will be mentioned below.) Even with a structural contraction rule primitive, in a framework such as Seq-Fmla or Seq-Seq, the increasing options for a proof-search do not actually result in undecidability for the propositional case (though they do when rules for the quantifiers are added), for reasons explained – originally in Gentzen [1934], but engagingly set out with a broader range of substructural logics in mind – in §4.1 of Ono [1998a]. (Recall that contraction is only one of several structural rules that can be dropped from a sequent calculus in the original Gentzen style: see 2.33.) Curry’s Paradox (Resumed). In the case of the BCIA logic we spent much of 7.25 pursuing, a trivialization is possible in the style of Curry without even using both halves of the comprehension schema (directions (1) and (2), used above as (3) and (4)) and in which not even the contraction-like (#1) is appealed to. Again we use q to show what can be done for any formula in its place and now take Q* as {x | q → x ∈ x}. Then by (2) we have (5): (5) (q → Q* ∈ Q*) → Q* ∈ Q* whence we immediately obtain q by Modus Ponens and the following instance of the A schema: (6) ((q → Q* ∈ Q*) → Q* ∈ Q*) → q. Discussions of Curry’s paradox may be found in Anderson [1975], Myhill [1975], neither of which is sympathetic to the view that contraction (sometimes called ‘absorption’ in the references to follow – a bad idea since there is no connection with the absorption laws in lattice theory) is to blame, and – where such a view is to the fore – Moh [1954], Prior [1955b], and Meyer, Routley and Dunn [1979]. There is also a semantical version of Curry’s Paradox, related to it in the way the Liar Paradox is related to Russell’s Paradox, uncovered in Geach [1955] (and subsequently discussed in Goldstein [1986], Hazen [1990b]). For some more recent activity in the area of generalized Curry-paradoxicality, see Restall [1993a], Rogerson and Butchart [2002], Rogerson and Restall [2004], and other references listed in Humberstone [2006b].

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A related but different anomaly going under the name of Curry’s Paradox underlies Curry’s proof of the inconsistency of his ‘illative combinatory logic’, which may be conveniently found at p. 575 of Barendregt [1984]. We will not go into the combinatory logic side of the issue here, but instead present the main idea as a non-existence argument (à la §4.2) concerning a connective # satisfying, when added to, for instance, the implicational fragment of IL ( = the implicational fragment of PL), the following pair of rules, somewhat reminiscent of those in 4.21.7 (p. 544), but see more exactly 4.23.4 for a version of the current point: (i)

Γ #A Γ #A → A

(ii )

Γ #A → A Γ #A

I.e., the associated consequence relation is to satisfy: #A #A → A (for all A). To bring out the role of Contraction, we use the Mset-Fmla natural deduction layout of 7.25.25, in the following proof of the sequent p (which shows that every sequent Γ B is provable) and, as there, W indicates an appeal to the framework-appropriate version of that (structural) rule: 1 1 1, 1 1

(1) (2) (3) (4) (5) (6) (7)

#p #p → p p p #p → p #p p

Assumption 1, (i) 1, 2 →E 3W 1–4 →I 5, (ii ) 5, 6 →E

(The rules referred to by means of ‘→I’, ‘→E’ here are the rules (→I)ms and (→E)ms from our discussion in 7.25.) One might reasonably conclude that there is simply no such connective as # behaving in accordance with the rules (i) and (ii ) – or their ‘purification’ in the form of the following two-way rule: Γ #A Γ, #A A The trouble for Curry was that the combinatory logic he worked with in fact guaranteed for every A that there was something behaving as these rules require (what we write as) #A to behave – see the discussion in Barendregt [1984], and also Chapter 17 of Hindley and Seldin [1986]. (Or consult the new version, Hindley and Seldin [2008], of the latter work.) The rules (i) and (ii) above amount to the requirement that #A and #A → A should be equivalent, and more generally one can ask after the effects of postulating for any formula B = B(p, q) a special purpose operator we write as FB satisfying the ‘fixed point equivalence’, for all formulas C FB C B(FB C, C). Here B(D, E) is just the result of substituting formulas D and E for p and q respectively in B (alias B(p, q)). Curry’s Paradox is the special case in which B is p → q. A general approach to the subject which proceeds along these

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lines may be found in Humberstone [2006b]. (In calling FB a special purpose operator, we do not intend any commitment to the claim that for any given B this is a 1-ary connective. Different treatments are possible here, on one of which it is a context for a single formula rather than a singulary connective. The “F” is mnemonic for Fixed-point operator.) The 2-Property and the 1,2-Property. We stated the 2-Property for BCI logic and the 1,2-Property for BCK logic (7.25.22) without proof. Proofs and discussions of these and related results may be found in the following sources: Belnap [1976], Kalman [1982], Bunder [1993], Troelstra [1992], Hindley [1993]. The original source is an article published in German in 1963 by Jaśkowski. (Hindley kindly provided me with his own translation into English of this remarkable paper.) Our exposition did not concentrate on the bearing of this on IL and CL, but Jaśkowski noted the following consequence of the 1,2-Property in the BCK case: any formula in (implicational) CL which was not BCK provable had some propositional variable occurring three or more times. By a close analysis of the corresponding sequent calculus formulations of these logics, Kashima [1997] extracts somewhat more information: if an implicational formula is CL-provable but not BCK -provable then (at least) one of its variables occurs ‘positively’ at least once and ‘negatively’ at least twice (e.g., the formula form of Peirce’s Law), and if an implicational formula is IL-provable but not BCK -provable then (at least) one of its variables occurs ‘positively’ at least once and ‘negatively’ at least twice (e.g., the implicational principle Contrac, alias W ). The notions of positive and negative occurrence here are the same as those of consequent and antecedent occurrences alluded to (though not defined) in 7.25.21. Kashima obtains these results as a corollary to a theorem showing that applications of the Contraction rule on the left or the right (Left Contraction or Right Contraction, from 2.33) can be avoided in sequent calculus proofs when these are proofs terminating in sequents for which the corresponding formulas meet appropriate conditions on positive and negative variable occurrences. In terms of the above distinction between two ways of doing without contraction— having the rule admissible though not derivable and (what we call contractionless logic) having the rule not even derivable—Kashima’s work is a variation on the former theme: it’s not that every provable sequent has a contraction-free proof, but that every provable sequent meeting the conditions specified has such a proof. Many corrections have been made to the material on contractionless logics in 7.25 as a result of suggestions by Sam Butchart and Su Rogerson, to both of whom I am also grateful for their going through the various exercises.

§7.3 BICONDITIONALS AND EQUIVALENCE 7.31

Equivalence in CL and IL

This section concentrates on the biconditional connective “↔”, though no particular attention will be paid to the semantics of the phrase “if and only if” (see the end of section notes, which begin on p. 1161). In the first place, this phrase is more at home in technical (logico-mathematical) settings than in ordinary language, and in that setting is apt to be interpreted in the light of some favoured formal account anyway – such as that provided by CL’s “↔”. Secondly, as we

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were at pains to point out in 7.12 (see in particular 7.12.4(ii) on p. 961, and 7.12.5) this phrase is clearly semantically composite and is best understood in terms of the best account available for the components and, if and only. Rather, we shall look at some aspects of ↔ as it behaves in CL and in IL, as well as at some related ‘equivalential’ connectives, and at the theoretical interest this connective possesses for logical and mathematical discussions. Characteristic of its role in such contexts is its place in certain Fmla incarnations of concepts which are played by the “” notation (for a consequence relation or gcr), such as in providing notions of extensionality and congruentiality for the framework Fmla, and, relatedly—in that a heavy reliance is placed on ↔-compounds to yield the interchangeability of their components—in the theory of definition (of non-logical vocabulary in, e.g., first-order theories). The latter topic (the “criterion of eliminability”) was mentioned, though not discussed extensively, in 5.32. For the topics of extensionality and congruentiality, and specifically for a ↔-based treatment thereof, see 3.21, 3.31; let us add that the notions of a set of equivalence formulas and of an equivalential consequence relation received some discussion after 1.19.9 and in the text surrounding 2.16.5–2.16.7. (But note that in the present discussion, in such phrases as “the equivalential fragment of CL”, “equivalential” simply means pertaining to ↔, rather than signifying the property of being equivalential in the sense just recalled. This double use of the term was noted before 1.29.11.) Amongst the several places ↔ itself has already figured prominently are 1.18 (p. 82), 3.11.2 (p. 378), 3.13.1 p. 386), 3.15.6 (p. 422), and in not just 5.32 (which began on p. 720) but every subsection of §5.3 (especially in reference to the notions of ↔-likeness and ↔-representability), as well as in 7.25.6 on p. 1103 and 7.25.23(i), from p. 1116. See also 9.23 (p. 1307) below, on ↔ in the relevant logic R. Rather than attending further to questions of the soundness and completeness of various proof systems for ↔ in IL or CL or other logics (for which see those sources), we will begin in the more esoteric arena of binary relational connections, and work our way through a number of points of logical interest from there – after a several preliminary paragraphs on three topics not covered in further detail. The equivalential fragment of (especially) classical propositional logic has been a favourite haunt of those in search of axiomatizations with the smallest number of axioms, or, when this number is down to one, the shortest possible axiom. Despite its ‘recreational mathematics’ feel, this kind of search is not without theoretical interest. Łukasiewicz [1939] discussed such questions amongst others, making use of earlier work by Wajsberg and Leśniewski. (Some of the latter’s observations will occupy us below: 7.31.7.) See Surma [1973d] for further information on the early history of these axiomatization issues, and Wos, Ulrich and Fitelson [2002] for information and references as to developments since the advent of automatic theorem proving. A second area of interest, not pursued for more than a few paragraphs in this preamble, concerns the possibilities of defining connectives (see 3.15–3.16) with the aid of ↔, taken as primitive (along with other connectives), on which p. 21 of Kabziński [1982] – or p. 17 of Kabziński [1980] – is particularly informative. Kabziński compares CL, IL and Łω (the infinite-valued logic of Łukasiewicz) as to whether the definitions listed are available. For example, in CL we could define A ∨ B as (A → B) ↔ B, and likewise for Łω , though not for IL, where the required equivalence (or synonymy, to be more accurate for the general case) is not forthcoming. The same pattern of availability obtains in the case of another


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candidate definition of A ∨ B, namely as ((A ∧ B) ↔ A) ↔ B. Turning to the definability of ∧, Kabziński gives two classically available options: •

A ∧ B as (A → B) ↔ A

and •

A ∧ B as ((A ∨ B) ↔ A) ↔ B.

Both of the required equivalences hold also for IL, and neither for Łω : the reverse of the pattern seen in the case of defining ∨. (The first equivalence underlies Curry’s Paradox: see the notes to §7.2, p. 1123.) It is worth adding to the options considered by Kabziński a further possibility (from Hendry [1981], in which it is noted that {∨, ↔, ⊥} is a set of definitionally independent connectives collectively interdefinable with either of the usual sets of primitives – either {∧, ∨, →, ⊥} or {∧, ∨, →, ¬}), also available in IL: •

A ∧ B as (A ∨ B) ↔ (A ↔ B).

(The set {∨, ↔, ⊥} just mentioned is also an independent basis in the case of CL, for which an axiomatization in these primitives can be found in Leblanc [1961].) One might have the reaction that there is nothing to choose between this and the previous option, on the grounds that ↔ is associative: but in IL ↔ is not associative, as we shall be observing below (7.31.5(ii) and 7.31.13(i)), so more attention is required to the particular case to confirm this last equivalence. We leave this to the interested reader – who will observe that the disjunction of A and B anticipates the biconditional formed from them in IL, in the special sense of ‘anticipates’ introduced at the end of 4.38 – and likewise with the status of the latest candidate definition in Łω . One should also consider a re-bracketing of the second candidate definition of ∨ mentioned above, namely defining •

A ∨ B as (A ∧ B) ↔ (A ↔ B).

Since the ← half of this candidate definiens is IL-provable, we are essentially dealing with the converse, which a little manipulation reveals to be IL-equivalent to the implicational compound known...as the Church disjunction of A with B, abbreviated in 2.13.13(ii), p. 234, to A ∨ B, which is classically though not intuitionistically equivalent to A ∨ B. In fact the same goes for another possibility, namely that of defining •

A ∨ B as (A ↔ B) → A

[or, equally well, as: (A ↔ B) → B]

in which the inner “↔” is somewhat incidental (which is perhaps why this does not appear on Kabziński’s list), in that we are considering here just a variation on the Russell–Łukasiewicz theme of A ∨ B as (B → A) → A or (A → B) → B. ¨ ” notation most recently seen in 7.22, – (These are the formulas – in the “ ∨ ¨ A and A ∨ ¨ B. A principle declaring the commutativity of ∨ ¨ appeared as Q B∨ in the discussion after 2.13.16 (p. 237); recall that Q is one of the implicational principles satisfied in Łω though not in the weaker BCK logic.)

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Kabziński’s discussion is recommended for further guidance on such ↔-using definitions, including in Dummett’s intermediate logic LC and Łukasiewicz’s three-valued logic Ł3 . Other logics might also reasonably considered in this connection. In the substructural case – which includes that of the Łukasiewicz logics – one should bear in mind the more thoroughly multiplicative equivalential connective, ↔m , for which, if it is not taken as primitive, we think of A ↔m B as defined to be: (A → B) ◦ (B → A), where ◦ is for fusion (multiplicative conjunction). A well known fact about the relevant logic R is that this formula (for a given A and B) is synonymous with the usual ‘additive’ biconditional A ↔ B (i.e., with ∧ for ◦); we might even want to notate the latter differently – say, as ↔a – to register the distinction even-handedly. Interestingly enough both the provability of the implication from A ↔ B to A ↔m B and provability of the converse implication require an appeal to contraction, so in linear logic neither implication goes through. (See 2.33 for these logics and the issue of contraction.) Digression. An interesting reaction one sometimes encounters is that ↔ is an unnatural primitive connective, because it lacks the property of being, in each of its two positions, either monotone or antitone. (See 3.32.2(ii) – p. 490 – as well as 3.32.2(i), for another such connective, which has nonetheless often been taken as primitive.) It would be interesting to see an argument for taking only “tonicity typed” connectives as primitives. With a definition of ↔ in terms → and ∧ we have primitives of types antitone, monotone and monotone, monotone . Certain approaches to logic, such as that of some versions of the ‘calculus of structures’ – see Guglielmi and Strassburger [2001] – may have difficulty treating such ‘atonic’ connectives, a point that has been made in discussion by Greg Restall. End of Digression. Finally, picking up on the recent intuitionistic theme, Kabziński and Wroński [1975] – mentioned above at 7.22.10 (p. 1077) – undertook an algebraization of the equivalential fragment of IL, calling the corresponding algebras simply equivalential algebras. The equational basis given by the authors for this class of algebras (which is therefore a variety) is as follows, in which the fundamental operation ↔ is omitted to simplify the notation: (xx)y ≈ y

((xy)z)z ≈ (xz)(yz)

((xy)((xz)z))((xz)z) ≈ xy.

(Here we use the ≈ notation from 0.24.7 on p. 32; Kabziński and Wroński and most others working in this field would also reduce parentheses by a convention of association to the left.) As one may gather from this last identity, equational deductions in the theory of equivalential algebras tend to involve terms of considerable length, making the subject something of an acquired taste. It has attracted considerable attention nonetheless, and some references will be found in the end-of-section notes (which begin on p. 1161. Kabziński and Wroński’s own development of the theory makes use of a notion of a filter (in an equivalential algebra) as a set of elements containing b whenever it contains a and ab, and containing (ab)b whenever it contains a. (The filters are then shown to be in a one-to-one correspondence with the congruences on an equivalential algebra. In fact the first condition in Kabziński and Wroński’s presentation is given with “ba” rather than “ab”, but the two formulations are equivalent and


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we use the other form – which can be found, e.g., in Słomczyńska [1996] – for the sake of the remark which follows.) These two conditions correspond to the rules Modus Ponens (for ↔) and the rule: A B ↔ (B ↔ A) These were the rules used in Tax [1973] along with a single axiom (not repeated here) to axiomatize the equivalential fragment of IL. Tax conjectured that no finite set of axioms, with Modus Ponens and Uniform Substitution – or equivalently, no finite set of axiom-schemata with Modus Ponens – could ever comprise a complete basis for the Fmla logic of intuitionistic ↔. The conjecture was finally proved in Wojtylak and Wroński [2001]. While the question of an axiomatization of the pure equivalential fragment of IL as a Fmla logic raises such questions, the provision of a natural deduction system or a sequent calculus for the fragment in question presents no difficulties. (See 1.23.6(ii) for the former.) It is CL that poses more of a challenge (as in the case of →) for the provision of suitable natural deduction rules in Set-Fmla on this front. Leblanc [1966], p. 169 – and in various other publications – presents a pure ↔ elimination rule in an effort to obtain a natural deduction system with the separation property, further simplified in Bernays [1962] (reviewing one of those other publications), but still not rendered simple in Dummett’s ‘pure and simple’ sense (i.e., having only one occurrence of the connective in the schematic formulation of the rule). With these preliminary glimpses of some of the surrounding terrain behind us, we turn now to the promised development of some themes in the logic of ↔, beginning with a consideration from the area of relational connections. The notion of an equivalential combination of objects on the right or left of a binary relational connection was introduced in 0.14.9. In the connection between formulas and valuations (with relation is-true-on), if we restrict attention to ↔-boolean valuations on the right, then we have equivalential combinations on the left, since A ↔ B is such a combination of formulas A, B. Keeping this same restriction in force, we notice that the connection provides equivalential combinations on the right also; the notation “ ˜” here is somewhat unorthodox, standing for a binary operation rather than a binary relation (specifically, an equivalence relation – and usually written larger for that purpose, as “∼”; of course we ask the reader to set aside also any associations with a similar notation for the negation connective, employed more than once in Chapter 8): Observation 7.31.1 If u, v ∈ BV ↔ , then u˜ v ∈ BV ↔ , where u˜ v is defined to be the unique valuation w such that for all formulas A, w(A) = T if and only if u(A) = v(A). Proof. Suppose that u, v ∈ BV ↔ ; we want u˜v ∈ BV ↔ , i.e., for all A, B: u˜v(A ↔ B) = T iff [u˜v(A) = T ⇔ u˜v(B) = T]. The left-hand side was defined to mean: u(A ↔ B) = v(A ↔ B), which since u and v are ↔-boolean, means (1) u(A) = u(B) iff v(A) = v(B). The right-hand side amounts to (2): u(A) = v(A) iff u(B) = v(B). We rewrite (1) and (2) to make their equivalence more obvious: (1)

[u(A) = T ⇔ u(B) = T] ⇔ [v(A) = T ⇔ v(B) = T]

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[u(A) = T ⇔ v(A) = T] ⇔ [u(B) = T ⇔ v(B) = T].

Here “⇔” serves as our metalinguistic material biconditional, and (1) and (2) are easily checked for equivalence by a truth-table test on corresponding CL formulas: (1 )

(p ↔ q) ↔ (r ↔ s)

(2 )

(p ↔ r) ↔ (q ↔ s).

An appeal to the following result – essentially due to S. Surma (see 7.31.3(iii)) – was made already in the proof of Observation 1.18.10 above (p. 87), and alluded to in the discussion leading up to 1.29.11 (p. 170). Observation 7.31.2 Let be the consequence relation Log(BV ↔ ); then for all Γ, A, B, we have: If Γ, A B then either Γ B or Γ, B A. Proof. Suppose for a contradiction that Γ, A B, but Γ B and Γ, B A. Since Γ B there exists u ∈ BV ↔ with u(Γ) = T, u(B) = F; further, as we are given that Γ, A B, we know also that u(A) = F. Since Γ, B A, there exists v ∈ BV ↔ with v(Γ) = v(B) = T, while v(A) = F. In that case, by 7.31.1, u˜v ∈ BV ↔ , and the fact that u˜v(Γ) = T, u˜v(A) = T, u˜v(B) = F, we have a contradiction with the supposition that Γ, A B. While Log(BV ↔ ) in the formulation of 7.31.2 is the consequence relation (1.12), rather than the gcr (1.16), determined by BV ↔ : clearly the result holds either way, though its lopsided look is tailored to the consequence relation case. Something more symmetrical is available in the gcr case, as part (i) of the following indicates: Exercise 7.31.3 (i) Show that, where is the gcr determined by BV ↔ , if Γ0 , Γ1 Δ0 , Δ1 , then either Γ0 , Δ0 Γ1 , Δ1 or Γ0 , Δ1 Γ1 , Δ0 . (ii ) Show directly, by suitable choices of Γ0 , Γ1 , Δ0 , and Δ1 , that 7.31.2 follows from 7.31.3(i). (iii) Show that, where is the consequence relation determined by BV ↔ , Γ, A B implies: either Γ B or Γ A ↔ B. Hints and Notes. For (i) just use the ˜-combination as in the proof of 7.31.2; for (ii), and more in this vein, consult Humberstone [1996a]. (We shall see more of ˜ later on in this subsection, at 7.31.24.) For (iii), use 7.31.2. Essentially, (iii) is presented as a sort of Deduction Theorem for classical equivalential logic in §11 of Surma [1973d]; a slightly different formulation of the same result appeared in that capacity as 1.29.11 above. Except that in one case we write → and in the other ↔, part (iii) here is identical in form to the BCIW Deduction Theorem given at 1.29.10(i), p. 168: something of a surprise, given that W is out of place for ↔ (in CL or any other plausible logic). Note that because of their disjunctive consequents, the condition on consequence relations in 7.31.2 and that on gcr’s in 7.31.3(i) do not correspond to sequent-to-sequent rules—their satisfaction is not equivalent to the admissibility of such a rule—in Set-Fmla or Set-Set respectively. Alternatively put, these conditions fail to be metalinguistic Horn sentences ‘of the second type’ as


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it was put in in 1.13; in Humberstone [1996a] they are described as generalized rule-like conditions. (For a contrast in this respect, see 8.11.6.) 7.31.2 and 7.31.3 present aspects of the pure ↔-fragment of CL which are not very well known, and as we have said, we can think of 7.31.3(i) as the general Set-Set fact which specializes to 7.31.2 in the context of Set-Fmla. A much more widely known fact about this fragment is one originally unearthed by Leśniewski for Fmla, and we shall prove it as 7.31.7 below: the purely biconditional tautologies are exactly those formulas (with ↔ as sole connective) in which each propositional variable occurs an even number of times. But first, we should pause to notice something about the proof of 7.31.1 above. The key point was that stated at the end of the proof in terms of the equivalence of (1 ) with (2 ). We could put the point less linguistically by speaking directly of the truth-functions (here ↔b , in the notation of 3.14), but more generally still, we note that the a binary operation · defined on a set A is said to satisfy the medial law just in case for all a, b, c, d ∈ A, we have: (Medial Law)

(a · b) · (c · d) = (a · c) · (b · d).

Then the equivalence of (1 ), (2 ) in the proof of 7.31.1 amounts to saying that the function ↔b (on the set {T, F}) satisfies the medial law. (See the proof of the slightly more general 7.31.29 below, setting each of #b , ◦b there equal to ↔b .) Exercise 7.31.4 (i) Verify that ∧b and ∨b also satisfy the medial law, and conclude that the classes of ∧-boolean valuations and ∨-boolean valuations are closed under conjunctive and disjunctive combination respectively. (We made much of conjunctive combinations in Chapter 1; see for example 1.14.5.) (ii ) Define corresponding notions of combination of valuations for implication and exclusive disjunction. Are the classes of #-boolean valuations (for # = →, , resp.) closed under these operations (respectively)? (See ˆ below, if the correspondence in question is not clear.) [Def. #] (iii) Give an appropriate generalization of the medial law for n-ary operations (n 1) so that we can conclude from a truth-function’s satisfying the law that the class of valuations in which a given connective is associated with that truth-function is guaranteed to be closed under the corresponding n-ary operation on valuations. (Solution: Set m = n and ◦ = # in the formulation of the general ‘ambi-medial’ law in the Digression following 7.31.32 below.) The notion of ‘corresponding operation’ used in these questions is more fully discussed in Humberstone [1996a] under the rubric “operation Galois-dual (to the given connective)”, where greater detail may be found; see also the discussion below, following 7.31.28 (p. 1146). We pause here to offer a general explanation of the terminology. If # is a boolean connective, that means we have defined a notion of #-boolean valuation (as at p. 65) by specifying a truth-function #b for which a valuation counts as boolean just in case (supposing # to be nary) for all formulas A1 , . . . , An , v(#(A1 , . . . , An ) = #b (v(A1 ), . . . , v(An )). The ˆ operation on valuations which is Galois dual to #, which we shall denote by #, is then defined to map (arbitrary) valuations v1 , . . . , vn to the unique valuation u satisfying

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ˆ u(A) = #b (v1 (A),. . . ,vn (A)) for all formulas A. [Def. #] ˆ and ∨ ˆ are simply conjunctive and disjunctive combination of Plainly, have ∧ valuations, usually notated by us as and , while the nullary operations ˆ from and ⊥ are just vT and vF , the all-verifying and alldelivered by [Def. #] falsifying valuations. Our latest addition, the ‘matching’ operation ˜ appears in the present notation as ↔. ˆ In the following chapter, we shall meet the operation dual to ¬, mapping a valuation v to the ‘complementary’ valuation v, and make use of it to establish properties of the negation fragment of CL similar to those established for the biconditional fragment above. (See p. 1172: 8.11.5, 8.11.6.) Exercise 7.31.5 (i) Show that 7.31.2 would not be correct if is taken as the ↔-fragment of IL . (Suggestion: consider the fact that p (p ↔ q) ↔ q for this choice of .) (ii ) Implicit in the suggestion given for (i) is the fact that (p ↔ q) ↔ q IL p, even though, as one easily verifies, p IL p ↔ (q ↔ q). How does this show that ↔ is not associative according to IL ? (We continue this theme in 7.31.13(i), p. 1136.) We turn to Leśniewski’s observation about the (CL-)tautologies in ↔, here given as 7.31.7. Observation 7.31.6 Suppose v is a ↔-boolean valuation and that A is constructed using ↔ as its sole connective; if there are n distinct occurrences of propositional variables going into A’s construction, we denote these by q1 , . . . , qn . (Note that one and the same variable, e.g. p, may occur as qi and qj even though i = j.) Then v(A) = F if and only if {i | v(qi ) = F} contains an odd number of elements. Proof. By induction on the complexity of A.

(Compare 6.12.2 from p. 783, concerning .) This result could equally well be formulated so as to conclude with the words: Then v(A) = F if and only if {i | v(qi ) = T} contains an even number of elements. Notice that we could have taken A to have the form: q1 ↔ q2 ↔ . . . ↔ qn , suppressing parentheses on the grounds that ↔b is associative. (Indeed we could consider A to have this form with arbitrary formulas B1 , . . . , Bn in place of q1 , . . . , qn , and no restriction on the connectives out of which these Bi are constructed, as long as the v in 7.31.6, thus re-written, is ↔-boolean.) In the case of 6.12.2, the formulation is slightly different because we were not assuming that was the only connective involved (just as in the preceding parenthetical comment). However, we could have given instead the exact analogue of 7.31.6 for : replace all references to “↔” by references to “ ” in the latter result, and conclude with “Then v(A) = T if and only if {i | v(qi ) = T} contains an odd number of elements.” Note also that the ‘temporary’ enumeration, above, of the variable-occurrences in A as q1 , . . . , qn is independent of our once-and-for all enumeration of all the variables in the language as p1 , p2 ,. . . ; in the latter case, we have pi = pj iff i = j.)


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Corollary 7.31.7 A formula built up using ↔ as the sole connective is a tautology if and only if every propositional variable in the formula occurs an even number of times. Proof. ‘If’: Suppose A is an ↔-formula in which every variable occurs an even number of times. By 7.31.6 every ↔-boolean valuation verifies A, since otherwise such a valuation would have to falsify some variable with an odd number of occurrences A: but there are no such variables. ‘Only if’: Suppose pi occurs an odd number of times in A. Consider a ↔boolean valuation v such that v(pi ) = F, while v(pj ) = T for all pj = pi . 7.31.6 implies that v(A) = F, so A is not a tautology. 7.31.7 provides a simple syntactic criterion for deciding membership in the purely equivalential fragment of CL in Fmla – faster than a truth-table test, for example – but it does not directly generalize to Set-Fmla or Set-Set. In the special case of what we might describe as the framework Fmla0 -Fmla (at most one formula on the left, exactly one on the right), 7.31.2 is of considerable assistance, as noted in (ii ) of the following: Exercise 7.31.8 (i) Using 7.31.2 show that, where is the restriction of CL to ↔-formulas, A B if and only if either A ↔ B or B. (Note this is a special case of 7.31.3(iii), included here only for the sake of what follows.) (ii ) Conclude from (i) and 7.31.7 that A B if and only if either every variable occurring in A ↔ B occurs an even number of times in that formula, or else every variable occurring in B occurs therein an even number of times. (iii) Conclude from (ii ) that if A B then any variable occurring an odd number of times in B occurs an odd number of times in A. (iv ) Does the converse of (iii) hold? Give a proof or a counterexample, as appropriate. (v ) Do all purely equivalential classical tautologies – i.e., those constructed from propositional variables using only the connective ↔ – have the 2-property (in the sense of the discussion following 7.25.24, on p. 1116)? Exercise 7.31.9 Give a counterexample to the claim that for all equivalential formulas A, IL A if and only if each variable occurring in A does so an even number of times. (Hint: look back over 7.31.5.) Since we have been concerned here with formulas involving iterated occurrences of “↔”, it is perhaps appropriate to point out that there is a common abbreviative usage of such constructions, especially in informal settings – for which we accordingly write “⇔” rather than “↔” – in which the iteration is not intended to be taken literally. On this usage, one writes A ⇔ B ⇔ C ⇔ D,

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for example, to abbreviate (A ⇔ B) & (B ⇔ C) & (C ⇔ D). Compare the use of “a = b = c = d” to mean “a = b & b = c & c = d”. Because of the danger of confusion, we avoid this type of abbreviation. Of course, one encounters a similar situation with implication, with “A ⇒ B ⇒ C” to mean “(A ⇒ B) & (B ⇒ C)” (cf. “a b c”), though here there is perhaps less danger of confusion since the connective concerned is not associative, so the unbracketed form is less likely to be encountered as an iterated implication – unless special scope conventions (‘association to the left’ and the like) are in force. A simple extension of 7.31.7 is available (for CL in Fmla) if ¬ as well as ↔ is present. The result in question, here requested as 7.31.10(ii ), is sometimes called the Leśniewski–Mihailescu Theorem. Exercise 7.31.10 (i) Show that if A is any formula built using only ↔ and ¬ then there is a formula B built using only ↔ and at most one application of ¬, such that A CL B, and that if B does involve ¬, it can be taken to be of the form ¬C (for ¬-free C, of course). (ii ) Show that if A is any formula built using only ↔ and ¬, then A is a tautology if and only if each propositional variable occurring in A occurs an even number of times, and ¬ itself also occurs an even number of times in A. (Count 0 as an even number for such purposes.) Remark 7.31.11 Part (ii ) of the above exercise can usefully be considered in a context in which rather than ↔ and ¬, we have ↔ and ⊥ as primitives. Subformulas ¬B of the arbitrary A there mentioned appear as B ↔ ⊥ in the new setting, and ⊥ behaves just like any other atomic formula – like any propositional variable, that is – in the sense that 7.31.7 continues to hold if occurrences of ⊥ are treated just like occurrences of any of the pi . Exercise 7.31.12 Modify the proof of 7.31.7 so as to establish the claim made at the end of 7.31.11 directly. (Note that we require all valuations now to be ⊥-boolean, so we must not make any stipulation with the effect that v(⊥) = T.) The equivalence of ¬A with A ↔ ⊥ mentioned in 7.31.11 holds good not only for CL but also in IL, and in each of these logics, negation and the biconditional exhibit similarities which bring out the dissimilarities between the logics. (The {↔, ¬}-fragment of IL itself is discussed in Kabziński, Porębska and Wroński [1981]; see also p. 73 of Czelakowski and Pigozzi [2004].) In particular, the equivalences ¬¬A A and (A ↔ B) ↔ B A which both hold for = CL both fail (in general) for = IL . The former failure (in the -direction) is of course well known; but the latter is equally clear since we may take ⊥ as B to reduce this case to the former. Exercise 7.31.13 (i) Conclude from the point just made that ↔ is not associative according to IL , and that adding an associativity rule to INat would yield a proof system for CL. (Hint: consider the special case of the associative equivalence (A ↔ B) ↔ C A ↔ (B ↔ C) in which C = B; cf. 7.31.5(ii). Note that the conclusion asked for here means that the smallest ⊇ IL according to which ↔ is associative is CL .)


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(ii ) Similarly, show that any congruential binary connective # satisfying, according to ⊇ IL , (A ↔ B) # B A, would make a nonconservative extension of IL . (Hint: put ⊥ for B and appeal to 4.21.1(i) from p. 540.) The hint given for 7.31.13 shows that just as we cannot ‘unpick’, in any logic conservative over IL, an application of ¬ by means of any (congruential) connective—not just the connective ¬ itself—and recover A from ¬A (our summary of 4.21.1(i)), so we cannot unwrap A from the context A ↔ B by means of any binary connective with B as one of its arguments—not just by another application of ↔ itself—and recover A. (Transferring the terminology of 3.14.16, p. 415, from the case of truth-functions to operations on the Tarski–Lindenbaum algebra of IL-synonymy classes, we may put this by saying that the operation corresponding ↔ has no first-argument reciprocal. We could equally well say “no second-argument reciprocal”, since IL and CL agree on the commutativity of this operation. The point about negation in this setting would be that in the case of IL there is no left inverse for ¬.) The proof suggested by that hint is, however, less informative than may be desired. For it leaves open the possibility that the pure equivalential fragment of IL (with, in particular, no ⊥ or ¬ in sight) might be extended conservatively by a # behaving as required by 7.31.13(ii ). We can rule out this possibility by appealing not to the Law of Triple Negation (2.32.1(iii), p. 304) for intuitionistic ¬, which is what 4.21.1(i), p. 540, rested on, but to the analogous principle (‘Law of Triple Equivalents’) for ↔ itself: Exercise 7.31.14 (i) Show (e.g., by means of Kripke models for IL) that for all A, B, we have ((A ↔ B) ↔ B) ↔ B IL A ↔ B. (ii ) Conclude from (i) and the fact that (A ↔ B) ↔ B is not in general IL-equivalent to A, that the restriction of IL to ↔-formulas is non-conservatively extended by the addition of any # satisfying (A ↔ B) # B A. The equivalence in CL of ¬¬A with A, and of (A ↔ B) ↔ B with A, is reflected in the possibility of changing the set of generators of a freely generated boolean algebra and having the same algebra freely generated by the new set. Corresponding to the first equivalence, we can replace a generator by its complement without making a difference to the algebra generated (since the replaced element is recoverable as the complement of the replacing element). The failure of this equivalence (or any alternative à la 4.21.1(i) in IL means that this kind of ‘regeneration’ is not available for freely generated Heyting algebras. Corresponding to the second equivalence is the fact that if a and b are amongst the set of elements freely generating a boolean algebra, then a can be replaced by (what for simplicity we may denote by) a ↔ b without altering the algebra generated, since we recover a as (a ↔ b) ↔ b, while there is again no corresponding possibility in the case of Heyting algebras, as 7.31.14(ii ) shows. Remark 7.31.15 The above examples suggest the following conjecture concerning Heyting algebras: If G1 and G2 are (minimal) sets of elements freely generating the same such algebra, then G1 = G2 . It would be of interest to see a proof or a refutation of this conjecture. (Section 5 of Humberstone [2000a] is relevant.)

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The second CL-equivalence above, involving ↔, lies behind many philosophical arguments falling under the general heading of language-dependence objections, in which a proposed explication of some concept is criticized because the concept in question should apply in the same way to expressively equivalent (interpreted) languages but the proposal under consideration instead makes its application vary with some inessential feature such as the choice of what vocabulary is taken as primitive (which predicates are atomic, for example). Nelson Goodman’s famous “grue” example (e.g., Chapter 3 of Goodman [1965]) can be seen as an objection of this kind, aimed against a partial analysis of the notion of a reasonable inductive inference, as can David Miller’s ‘ArizonanMinnesotan’ example (Miller [1974], [1976]), where the target is an inappropriately language-dependent analysis of the concept of verisimilitude (closeness to the truth). (There have certainly been attempts to reply to Miller’s objection — for example Oddie [1986]. Two items dealing with this matter which are already in our bibliography for other reasons are Urbach [1983], Goldstick and O’Neill [1988].) Miller’s observation with Arizonan-Minnesotan example is rediscovered in Makinson [2009]: see Example 5.1 there; this paper also considers language dependence issues related to theories of belief revision. Numerous such issues have arisen in the history of discussions of logical atomism (Black [1964], esp. p. 64, being perhaps the first; see also Miller [1977], and, for a retrospective discussion of all these debates, Miller [2006b]). Our own brush with this issue occurred in 5.23, in the discussion of logical subtraction, the proposed binary connective (written “−”) governed in particular by the following (set out as a rule display, even though we are using “” rather than “”): Corrected Subtraction Principle: Subject to the proviso that (i ) C is consistent, and (ii ) B and C share no propositional variables. A B ∧ C A − C B We noted a particular consequence of this principle in the discussion following 5.23.6 (p. 687), and raised the possibility of a language-dependence objection in 5.23.8. The consequence was the equivalence of (p ∧ q) − q with p. The above principle allows this subtraction of q ( = C) from p∧q ( = B∧C, alias A) because the variable-disjointness condition—part (ii ) of the proviso—satisfied, B and C here simply being two distinct variables. We can now make the languagedependence objection explicit. Since we are thinking of the language under consideration as an interpreted language, let us call the atomic formulas sentence letters rather than propositional variables. For simplicity, let us suppose that p and q are the only sentence letters in the language to which they belong, and now introduce a second language, also based on two sentence letters, this time r and s. Since in both languages these are thought of as having a particular interpretation (or meaning), there is no harm in making the further supposition that these interpretations are such that r in the second language means what p in the first language means, while s in the second language means what p ↔ q in the first language means. (Imagine a combined language for which any meaningrespecting valuation v, as well as being ↔-boolean, must satisfy the conditions v(r) = v(p) and v(s) = v(p ↔ q).) To say this is to translate the second {r, s}based language into the first {p, q}-based language. But we can also give a


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translation the other way round, translating p as r and q as r ↔ s. So the two languages are intertranslatable. However, if we had been working in the second language rather than the first, the translation of the above appeal to the Corrected Subtraction Principle would have been: r ∧ (r ↔ s) r ∧ (r ↔ s) (r ∧ (r ↔ s)) − (r ↔ s) r and this application would violate the variable-disjointness condition: although the formulas r and r ↔ s, as it was put in §5.2, completely logically independent (appropriately enough since they are the translations of two distinct sentenceletters), they are both constructed with the aid of r; this feature is eliminable (in the second language). But why should the (logical) subtractability of one thing from another depend on the accidental choice of one rather than another of two expressively equivalent languages? The only possible reply would have to be that in some significant respect, two such languages need not after all be on a par. For an elaboration of this reply, see Humberstone [2000d]. Digression. Another kind of occasionally embarrassing language-dependence arises when a point made in the context of one choice of primitive connectives fails to transfer when this is replaced by another expressively equivalent selection of primitives. Examples falling under this heading may be found in Hiż [1958], Makinson [1973b], Milberger [1978], Humberstone [1998a], [2004c]. Hiż’s example was discussed further in Frank [1976], but see also Shapiro [1977]. Two (rather different) reactions to Makinson [1973b] example may be found in §4.6 of Segerberg [1980] and §3 of Humberstone [1993a]. End of Digression. Exercise 7.31.16 Our discussion in §5.2 emphasized the ‘complete logical independence’ of p and p ↔ q (as against p and p → q, say, whose conjunction is also equivalent to p ∧ q). Formulate the corresponding notion of independence for boolean algebras and show that distinct free generators are always independent in the sense defined. Recall that #1 has #2 as a first-argument reciprocal according to when for all formulas A and B, we have (A #1 B) #2 B A. Notice that with as CL , we can take #1 = #2 =↔ (as we have been emphasizing) and satisfy this condition; we could also take #1 = #2 = (exclusive disjunction: 6.12, mentioned above at 7.31.4(ii)). Yet another option would be to take each of #1 , #2 , as the ‘first projection’ connective # with A#B A for all A, B. (In this last case, we do not get an example of ‘regenerating’ a boolean algebra, however. Replacing the element a by the element proj12 (a, b) is just ‘replacing’ a by a: no change of generators here.) Is it a coincidence that in all these examples, #1 = #2 ? It is more informative to give a general argument than to consider all possible choices (16×16 possibilities for the pair #1 , #2 ) of binary connectives truth-functional over the class of boolean valuations. (Contrast the analogous situation with 1-ary truth-functions, where one easily checks by inspection of the cases that if f and g are such functions with f (g(x)) = x for all x ∈ {T, F}, then f = g.) To this end, it is simpler to consider the binary truth-functions themselves. We shall use infix notation, with · and as arbitrary binary truthfunctions; in fact, there is no need to stress any particular interpretation of them, so let us work in the context of an arbitrary two-element set A. In reasoning

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about such a set, for a ∈ A, we denote by a ¯ the element in A other than a; the condition we are interested in for · and (each mapping A × A into A) is this: (Recip)

For all x, y ∈ A, (x · y) y = x.

See 3.14.16 (p. 415) for a more discerning terminology. Lemma 7.31.17 If A is a two-element set closed under binary operations · and satisfying (Recip), then for x, y ∈ A: (i) x · y = x ¯ · y and (ii) x y = x ¯ y. Proof. If the identity denied in (i) held, we could -multiply both sides by y on the right, which (Recip) then tells us would imply x = x ¯, which is not the case. For (ii ), suppose that x y = x ¯ y. Let a be arbitrary. Then one of ¯ · y, by (i). Without loss x, x ¯ is a · y and the other is a ¯ · y, since a · y = a of generality, we may suppose that x = a · y and x ¯=a ¯ · y. (If this is not so for the originally selected a, work with a ¯ instead.) Since x y = x ¯ y, then, (a · y) y = (¯ a · y) y, giving (by (Recip)) the contradiction: a = a ¯.

Observation 7.31.18 Suppose A is a two-element set closed under binary operations · and satisfying (Recip); then · = . Proof. We have to show, on the supposition given, that for all x, y ∈ A, x · y = x y. The elements x and y are either the same or different, and in the latter case, since A only contains two elements, there is only one possible way of being different. Thus it suffices to show for arbitrary a ∈ A, (i) that a · a = a a, and (ii) that a ¯ · a=a ¯ a. To show one, we consider two possible cases, according as a · a does or does not equal a. Case 1: a · a = a. Now (Recip) gives (a · a) a = a, so by the hypothesis of this case, replacing (a · a) by a, we get a a = a; which means, again by the case hypothesis, that a · a = a a. ¯. As in Case 1, (Recip) gives (a · a) a = a, Case 2: a · a = a, i.e. a · a = a so by the hypothesis of the present case a ¯ a = a. By Lemma 7.31.17(ii ), a ¯ a = a a, so a a = a. Then a a = a ¯ = a · a, as was to be shown. We have now shown (i) above, that we always have a · a = a a, and must establish (ii ), that a ¯·a = a ¯ a. Since a · a = a a, we have a · a = a a; but a · a = a · a, by ¯ a, by 7.31.17(ii ), so (ii ) follows. 7.31.17(i), and a a = a Thus, it is no coincidence that the earlier examples of reciprocation amongst boolean connectives featured the same connective as #1 and as #2 in the schema (A #1 B) #2 B A. By 7.31.18, specialized to the case of A = {T, F}, when this schema holds for = CL , #1 and #2 must at least be CL -synonymous. (That is, these connectives make synonymous compounds from the same components.) Exercise 7.31.19 (i) Spell out the details of the argument just gestured toward. (ii ) Although we mentioned the first projection, material equivalence and exclusive disjunction as binary truth-functions satisfying (Recip),


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when taken in each case as · and as , these are not the only ones. Make an exhaustive list. We have seen, with 7.31.18, that the our explicitly mentioned examples ↔b (to use the notation of 3.14 for the truth-function associated with ↔ on ↔boolean valuations) and proj12 are not exceptional in being ‘self-reciprocating’, since as that Observation reveals, any binary operation with a (first-argument) reciprocal is its own (first-argument) reciprocal. (Clearly, we could go through similar reasoning in the case of second-argument reciprocity.) Now one conspicuous feature of ↔b is that it is what we called in the discussion on p. 413 before 3.14.14 superdependence on each of its two arguments, while proj12 does not even depend on its second argument. We shall see (7.31.20) that this is another respect in these two (Recip)-satisfying examples are not exceptional: a binary truth-function superdepending on its first argument, as ↔b and proj12 do, must either (like ↔b ) superdepend on its second argument, or else (like proj12 ) not depend on its second argument at all. Some recapitulation is appropriate first. Using the ‘overlining’ notation of our recent discussion, we can easily say what it is for a (two-valued) truth-function f of n arguments to superdepend on its ith argument (1 i n); it is for the following to be the case for all choices of x1 , . . . , xn from {T, F}: f (x1 , . . . , xn ) = f (x1 , . . . , xi−1 , xi , xi+1 , . . . , xn ). (The contrast was with mere dependence on the ith argument, which we have when the above inequation holds for some choice of x1 , . . . , xn . See 3.14 for further discussion.) 7.31.17 says that the functions · and involved in (Recip) – which 7.31.18 showed to be identical – superdepend on their first argument, so the hypothesis of the following observation is fulfilled in their case. Observation 7.31.20 Suppose f is a function of two arguments defined on a two-element set. Then If f superdepends on its first argument, either f superdepends on its second argument or else f does not depend on its second argument at all. Proof. Suppose that f , with arguments and values in a two-set A, superdepends on its first argument but not on its second. We shall show that f does not depend on its second argument. Since f does not superdepend on its second argument, there exist a, b ∈ A with f (a, b) = f (a, ¯b). Since f superdepends on its first argument, f (¯ a, b) = f (a, b) and f (a , ¯b) = f (a, ¯b). Since there are only two elements in A, f (¯ a, b), f (a, b) (alias f (a, ¯b)), and f (¯ a, ¯b) cannot all be distinct, so from the given inequations, we have f (¯ a, b) = f (¯ a, ¯b). As we already ¯ know that f (a, b) = f (a, b), we conclude that f does not depend on its second argument. The reader may care to ponder what becomes of such points for a function of three or more arguments. Since we shall need it anyway for what follows, we pause to introduce a generalized version of the condition (Recip) above. Let us say that an n-place function g is an ith argument reciprocal of the n-place function f when for all x1 , . . . , xn in the domain of f , we have (Recip)ni

g(f (x1 , . . . , xn ), x1 , . . . , xi−1 , xi+1 , . . . , xn ) = xi .

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The earlier condition (Recip) is, in this terminology, the (Recip)21 ; and (Recip)11 says that g is a left inverse of f . (See the discussion preceding 0.21.1, p. 19.) The following Exercise will not be used later, and is included only for interest. It concerns the generalizability of 7.31.18, for which purpose the order of arguments in (Recip)ni above needs some alteration. Exercise 7.31.21 Clearly if for a given f there is a g satisfying the condition (Recip)ni , then there is a function g satisfying the condition g (x1 , . . . , xi−1 , f (x1 , . . . , xn ), xi+1 , . . . , xn ) = xi . Suppose f and g are functions defined on a two-element set and they satisfy this condition. Does it follow that f = g ? (7.31.18 gives an affirmative answer in the special case that n = 2, i = 1; we used ‘infix’ notation there and wrote f as ·, g as .) Suppose that for some truth-function f there is a g for which the condition (Recip)ni is satisfied, and the n-ary connective # is associated with f on each valuation in some class V , in our usual sense, i.e., that for all v ∈ V , all A1 , . . . , An , v(#(A1 , . . . , An ) = f (v(A1 ), . . . , v(An )). (E.g., # is a fundamental or derived boolean connective and V contains only boolean valuations.) The value of Ai on v ∈ V is uniquely determined by the values (on v) of the compound #(A1 , . . . , An ), together with the values of the other components (A1 , . . . , Ai−1 , Ai+1 , . . . , An ). For the function g in (Recip)ni delivers Ai ’s truth-value from these truth-values. It is useful to have a notation in which to think about such matters concisely. For this subsection, we use “” in this capacity. More explicitly, where V is a class of valuations, let us say B1 , . . . , Bm V C if and only if there exists a function g: {T, F}m −→ {T, F} such that for all v ∈ V : g(v(B1 ), . . . , v(Bm )) = v(C). (Note that the current use of “” is not related to that introduced in 1.29.6(ii) on p. 162, or to other uses of the notation at various other places in our discussion.) We call the relation V here defined the consequence relation superveniencedetermined by V . (This terminology is from Humberstone [1993b], in which the consequence relation V determined by V in the usual sense is for contrast called the consequence relation inference-determined by V . Some remarks motivating the “supervenience” terminology appear below. See also the notes to this section, which begin on p. 1161.) Now, except for the fact that—for simplicity—we are restricting ourselves to finitely many formulas on the left-hand side of the “”, the conditions on consequence relations are indeed satisfied by the relation V which can be thought of as a relation between finite sets of formulas and individual formulas, even though strictly, as defined, is a relation between finite sequences of formulas and individual formulas. (The finiteness restriction will be removed presently, in 7.31.25.) Note in particular that the order of the Bi does not matter, since if there is a function g as required by the above definition for B1 , . . . , Bm to have C as a consequence (by V ), in that v(C) = g(v(B1 ), . . . , v(Bm )) for v ∈ V , then there is also a function g with v(C) = g (v(Bp(1) ), . . . , v(Bp(m) )), where p is any permutation of 1, . . . , m. A more general version of this consideration


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shows that the number of occurrences of elements of a formula on the right is also immaterial. The structural condition (R) – for all formulas A: A V A – is satisfied because we can take g as the identity function. (M) and (T) are justified, respectively, by considerations of ‘dummying in’ argument positions on which the promised function does not depend, and by a generalization of functional composition. Examples 7.31.22(i) For # an n-ary boolean connective with V ⊆ BV # , and any A1 , . . . , An , we have: A1 , . . . , An V #(A1 , . . . , An ). Thus any formula all of whose connectives are boolean is a consequence (according to V ) of the set of propositional variables from which it has been constructed. (ii ) With # and V as in (i), if the truth-function #b has an ith argument reciprocal—equivalently, if #b superdepends on its ith argument—then we can interchange Ai and #(A1 , . . . , An ) in the V -statement in (i). Thus, for instance, for all A, B, we have ¬A V A and A ↔ B, B V A. This last is our new way of saying what was earlier expressed by saying that A ‘can be recovered’ from A ↔ B and B. More accurately, what it says is that, when attention is restricted to the truth-value assignments provided by V , A’s truth-value is uniquely determined by the truthvalues of A ↔ B and B. The examples under (i) here have nothing especially to do with the assignment of a particular truth-function to a connective – an assignment indicated by the word boolean – since the point is simply that if # is truth-functional w.r.t. V , in the sense of 3.11, then we must have A1 , . . . , An V #(A1 , . . . , An ) for all A1 , . . . , An . There is an obvious converse: if this V -relationship holds for all A1 , . . . , An , then # is truth-functional w.r.t. V , since the truth-function required is provided by the function g in our original definition of V . Notice also – a point to be appealed to presently (7.31.28) – that if f is the truth-function associated over V with # and A1 , . . . , Ai−1 , Ai+1 , . . . , An V #(A1 , . . . , An ) then f does not depend on its ith argument (and so is not essentially n-ary). It may seem strange to be thinking, as in 7.31.22(ii), of ¬p as a consequence of p and, as in 7.31.22(i), of p as a consequence of ¬p, because one usually attends, with a given class of valuations (such as BV ) in mind to a particular consequence relation, the consequence relation determined by that class (i.e., Log(BV ), alias CL in this case) in the sense of 1.12. (In the terminology mentioned above, one most naturally thinks of the inference-determined consequence relation rather than supervenience-determined consequence relation.) For parity with the “V ” notation we will use the notation “V ” for the consequence relation determined by V ; so CL , the relation of tautological consequence, is BV . But informally speaking, a class of valuations can be used to ‘determine’ a consequence relation in several ways, determination in terms of truth-preservation being only of them. As we have already noted – setting aside the issue of the consequences of infinite sets of formulas – the “there exists

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a function. . . ” definition above, also gives rise to a consequence relation. The equivalence of p and ¬p according to this consequence relation does not mean that the truth-value of either of these is the truth-value of the other, but that the truth-value of either fixes the truth-value of the other. (Epistemically, think of A BV B as saying, for boolean formulas A, B, any rational individual knowing that A would know that B; the corresponding gloss on A BV B would be that any rational individual knowing whether A would know whether B. Analogously for a multiplicity of formulas on the left. If one thinks of the valuations in V as corresponding to possible worlds in a Kripke model—so that for each point w in the model, vw ∈ V verifies exactly the formulas true at that point—then the relation V is a kind of subject-matter supervenience relation, with A V B meaning that there is no difference between worlds in respect of B without a corresponding difference in respect of A. This is one of several relations that have been called global supervenience. See the notes to this section, starting at p. 1161.) As it happens, there is a preservation-style characterization available for V , with what is preserved being agreement in truth-value rather than the property of being true, understanding valuations u, v, to ‘agree’ on A when u(A) = v(A): Exercise 7.31.23 (i) Show that B1 , . . . , Bm V C iff for all valuations u, v ∈ V , u(B1 ) = v(B1 ) & . . . & u(Bm ) = v(Bm ) ⇒ u(C) = v(C). (ii) Here and in (iii) we pick up the “knowing whether” point parenthetically introduced above. Suppose that V ⊆ BV¬ . Show that for any formulas A, B: A V iff (A V B or A V ¬B) and (¬A V B or ¬A V ¬B). (One could write “¬A V ¬B” without the negation signs as “B V A”, of course, but it seems better not to disrupt the pattern.) (iii) What characterization analogous to that in (ii) be appropriate for the case of A, B C? In fact, by reviving our Galois-dual to material equivalence, the ‘matching’ operation ˜, from the start of this subsection, we can convert this into a preservation-of-truth characterization, since agreement of u, v, on a formula amounts to the truth of that formula on u˜v: Observation 7.31.24 B1 , . . . , Bm V C if and only if for all valuations u, v ∈ V, u˜ v(B1 ) = T & . . . & u˜ v(Bm ) = T ⇒ u˜ v(C) = T. Proof. Immediate from 7.31.23(i) and the definition of ˜.

This characterization of V makes the (finitary versions of) the consequence relation conditions (R), (M) and (T) – argued above on the basis of the original “there exists a function g” definition – very evident, since here we are dealing with truth-preservation (over all valuations of the form u˜v, for u, v ∈ V ). It is also, as is the earlier 7.31.23(i) characterization, much more suggestive than the original ‘functional’ characterization, of a corresponding gcr: Γ V Δ just in case for all u, v ∈ V , if u and v ‘match’ on each formula in Γ, then they match on some formula on Δ. We shall not investigate this gcr here, however.


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Remark 7.31.25 We shall, however, take advantage of another aspect of the characterizations of V : they allow us to make immediate sense of infinitely many formulas on the left. From now on, we understand V accordingly, as a fully-fledged consequence relation holding between arbitrary Γ and C when any two valuations in V which agree on each formula in Γ in turn, agree on C. Its utility notwithstanding, we need to note that the valuations u˜v invoked in 7.31.24, where u, v ∈ V , are typically themselves not elements of V . We did notice an exception in the case of V ⊆ BV ↔ in 7.31.1, however, which we can avail ourselves of now. Recall that V is the consequence relation determined, in the Log(V ) sense, by V , and that the relations V are always consequence relations (understood as in 7.31.25), so the following identification makes sense. Theorem 7.31.26 Where V = BV↔ , we have V = V . Proof. For the ⊆-direction, use the fact that the all-verifying valuation vT is ↔boolean and acts as an identity element for the operation ˜. For the ⊇-direction, use 7.31.1. Thus consequence in the sense of preservation-of-truth (over a class V of valuations: the consequence relation inference determined by V ) and consequence in the sense of preservation-of-agreement (amongst valuations in V : the consequence relation supervenience determined by V ) coincide when V is BV ↔ . It is natural to ask whether ↔ is the only (essentially) binary boolean connective for which such a coincidence obtains. The answer is yes, as we shall see in 7.31.31. But first, a few words about the interest of this question. One might consider whether the consequence relation supervenience determined by BV ↔ is specific to ↔ or whether another boolean # is such that, roughly speaking, this coincides with the consequence relation supervenience determined by BV # . “Roughly speaking” because what we really want to consider is whether ↔ behaves in the same way according to the former consequence relation as # does according to the latter. (This will be made more precise still in due course.) The obvious candidate is . In both cases, when only formulas constructed with just ↔ or with just are concerned, for either of which we use “#” here, we have not only the general truth-functionality schema A, B BV A # B but also the two ‘flipped around’ forms A, A # B BV B and B, A#B BV A, since each choice of #b superdepends on each argument position. Moreover, as we remarked in 6.12.1, the set {T, F} can be turned into a group, taking b as the (fundamental) binary operation, with F as the identity element and each element as its own inverse (a boolean group, then). Now this set can also be turned into a group by taking instead ↔ as the binary operation, with T as the identity element (and again, each element as its own inverse). As logical matrices, with T as the designated element, these two two-element matrices are rather different, but the underlying algebras—the two groups just described—are isomorphic (by a mapping interchanging T and F). (There is, up to isomorphism, only one two-element group, and we have just described it twice over. We also encountered this group in the discussion following 5.34.6. Strictly speaking, since we are considering only a language with one connective, a binary connective, it is the semigroup reducts of these groups that

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are the underlying algebras for the two-element matrices just alluded to.) Now the ‘absolute’ difference between T and F, though it matters much for the relations V , doesn’t matter at all for the corresponding V , since here all we are concerned about is agreement and difference between the effects of valuations in V . A match between two Fs is just as good as a match between two Ts, for example. We shall confirm the speculation prompted by these considerations using a variation on 6.12.2 (p. 783) and 7.31.6, tailored to that end. Digression. The isomorphism between ({T, F}, ↔b ) and ({T, F}, ) just enb countered suggests an interesting alternative to the usual way of counting the number of binary truth-functions; instead of arriving at the number 16, we should reach a lower tally if we counted the number of two-element groupoids which are distinct to within isomorphism. (It doesn’t matter, therefore, whether the universe is {T, F} or any other two-element set.) The reader may care to compute this new total. The answer may be found by consulting p. 767 of Shafaat [1974]. End of Digression. We are concerned with two relations of the form V , where V is in the one case BV ↔ , and the language concerned is the pure ↔-language, and in the other BV , where the language concerned is the pure -language. The result of replacing every occurrence of ↔ in a formula of the former language is a formula of the latter language, which we shall denote by A . Lemma 7.31.27 For any formula A of the ↔-language and any boolean valuation v, if the complexity of A is even, then v(A) = v(A ), and if the complexity of A is odd, then v(A) = v(A ). Proof. By induction on the complexity of A.

Exercise 7.31.28 Show how it follows from 7.31.27 that for any boolean valuations u and v, and any formula A of the ↔-language u˜v(A) = u˜v(A ). Theorem 7.31.29 For all formulas A1 , . . . , An , B of the ↔-language, where ↔ and are the consequence relation supervenience-determined by BV↔ and BV respectively, we have: A1 , . . . , An ↔ B if and only if A , . . . , A B . Proof. An immediate corollary of 7.31.28.

We could equally well have stated this result working with a single language with a sole binary connective # and classes of valuations interpreting differently, V , over which # is associated with ↔b , and W , over which it is associated with b , in which case the formulation would be: For any formulas A1 , . . . , An , B of this language: A1 , . . . , An V B if and only if A1 , . . . , An W B.


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(Of course, we could equally well have allowed infinitely many formulas on the left, both here and in 7.31.29.) The upshot of this discussion, then, is that the BV behaviour of ↔ does not single that connective out uniquely from amongst the boolean connectives, and that is why we are interested instead in seeing if the relationship between the BV and the BV behaviour of ↔, as described in 7.31.26 above, might instead be something peculiar to ↔. We resume that line of enquiry here. The proof of 7.31.2 (p. 1132), showing that Log(BV ↔ ) satisfies a certain ‘Flip-Around’ condition on consequence relations , namely (1): (1) For all Γ, A, B: if Γ, A B then either Γ B or Γ, B A. used only the fact (7.31.1, appealed to again in the proof of 7.31.26 above) that BV ↔ was closed under ˜. Thus what is really shown is that for any class V of valuations closed under ˜, the consequence relation determined by V satisfies (1). There is a weaker condition than (1) which it is not hard to see is satisfied by any consequence relation of the form V , regardless of whether V is closed under ˜, namely the special case of (1) in which Γ = ∅. This says that whenever B follows from A, unless this reference to A was redundant, then we can flip this consequence statement around and conclude that A follows from B. (2) For all A, B: if A B then either B or B A. Exercise 7.31.30 Show that if is V for any V , then satisfies condition (2) above. (If in difficulty, consult Humberstone [1993b], p. 329.) To relate the semantic properties we have here noted to be sufficient for the syntactic conditions (1) and (2), it is helpful, where V is a class of valuations, to write V ˜V for {u˜v | u, v ∈ V }. Then to say, as in 7.31.30, that is V , is to say that is V V . Turning from (2) to (1): to say that V is closed under ˜ ˜ is to say that V ˜V ⊆ V . Now this inclusion can be reversed, because the converse inclusion is automatic if V = ∅, and if V = ∅ then where v ∈ V , the assumption that V ˜V ⊆ V gives v ˜v = vT ∈ V , so we can represent v as v ˜vT , showing v ∈ V ˜V . Thus the semantic condition which was seen to suffice for (1) was that is determined (in the sense of 1.12) by some class V with V = V ˜V whereas that seen to suffice to (2) was that is determined by some V of the form U ˜U . The latter is the weaker condition since we do not require that U = V . We can now close our discussion of whether 7.31.26 singles out ↔ uniquely. Strictly this should be put in terms of singling out ↔b uniquely, and this answer to the question incorporates that refinement. Our proof makes reference to Figures 7.31a, b. p T T F F

# x1 x2 x3 x4

Figure 7.31a

q T F T F

p T T F F

# T F F x4

Figure 7.31b

q T F T F

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Theorem 7.31.31 Suppose that #b is an essentially binary truth-function and BV # = BV # . Then #b is ↔b . Proof. Make the suppositions of the Theorem. (We omit the subscripts on the turnstiles.) They will allow us to piece together #b and identify it as ↔b . We fill in the gaps (xi ) in the truth-table description of Figure 7.31a. As #b is essentially binary, we do not have p p # q or q p # q, so since = , the entries x1 or x2 in Figure 7.31a must be F and also x1 or x3 must be F. But (7.31.22) since p, q p # q, (and thus p, q p # q) we have x1 = T. This settles the first three lines, giving Figure 7.31b. As for the final line, if x4 were F, we should have p # q p, so we now show that this is not the case, which gives us #b = ↔b . p # q p would mean, since = , that p # q p. Then by 7.31.27, we would have either p or else p p # q. Clearly we do not have p (since for any # there are #-boolean valuations verifying and also valuations falsifying a propositional variable). But nor do we have p p # q since (as already noted) this would mean that # was not essentially binary. Thus x4 = T. Just how interesting is this, however? After all, the matching operation ˜ is Galois-dual to boolean ↔, so an especially tight connection is perhaps only to be expected. This raises the question of whether or not, for binary connectives #1 , #2 , the class of #1 -boolean valuations is ever closed under the operation ˆ 2 Galois-dual to #2 when (#1 )b = (#2 )b . To reduce subscripting, let us write # #1 and #2 as # and ◦ respectively. The key condition we need to consider is a variegated form of the medial law cited just before 7.31.4 above: (Ambi-medial Law)

(a # b) ◦ (c # d) = (a ◦ c) # (b ◦ d), all a, b, c, d ∈ A.

We will say that # and ◦ (binary operations on A) satisfy the ambi-medial law when the above holds. There is no need to say “# and ◦ in that order . . . ” because interchanging b and c, and reversing the two sides of the equation gives the result of interchanging instead “#” and “◦”: thus if # and ◦ (in that order) satisfy the ambi-medial law, then so do ◦ and # (in that order). Compare the ‘self-dual’ nature of the distributive law for lattices (0.13.3(ii)). In fact, we note in passing that if # is idempotent then this implies that # distributes over ◦, since putting b = a we get (a # a) ◦ (c # d) = (a ◦ c) # (a ◦ d), and hence a ◦ (c # d) = (a ◦ c) # (a ◦ d). Observation 7.31.32 If for binary boolean connectives #, ◦, the truth-functions #b and ◦b satisfy the ambi-medial law, then the class of #-boolean valuations is closed under ˆ ◦. ˆ Proof. This is simply a matter of following out the definitions (esp. [Def #] above). We need to check that uˆ◦v ∈ BV # , for u, v ∈ BV# , i.e., to check that for all formulas A, B, we have uˆ◦v(A # B) = uˆ◦v(A) #b uˆ◦v(B), on the assumption that (since u and v are #-boolean), for all A, B, we have u(A # B) = u(A)#b u(B), v(A # B) = v(A) #b v(B). The lhs of the equation we wish to verify, uˆ ◦v(A # B) = uˆ◦v(A) #b uˆ◦v(B), amounts to u(A # B) ◦b v(A # B) and hence to (u(A) #b u(B)) ◦b (v(A) #b v(B)). The rhs amounts to:


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(u(A) ◦b v(A)) #b (u(B) ◦b v(B)). But by the ambi-medial law: (u(A) #b u(B)) ◦b (v(A) #b v(B)) = (u(A) ◦b v(A)) #b (u(B) ◦b v(B)).

Digression. Though we are concentrating here on binary truth-functions and binary operations on valuations, it is worth remarking that the ambi-medial law can be given a more general formulation for which a straightforward adaptation of the proof of 7.31.32 yields the correspondingly generalized result. Suppose # is m-ary and ◦ is n-ary; then the general ambi-medial law for # and ◦ is the statement that for all aij (1 i m, 1 j n) ◦(#(a11 , . . . , am1 ), . . . , #(a1n , . . . , amn )) = #(◦(a11 , . . . , a1n ), . . . , ◦(am1 , . . . , amn )). What was given as the ambi-medial law above (and what we shall continue to refer to under that name) is the special case in which m = n = 2. End of Digression. The question remains whether the ambi-medial law is ever satisfied for distinct #b , ◦b . This can be settled affirmatively with the example #b = proj12 , ◦b = proj22 , though in this case 7.31.32 becomes trivial since every set is closed under the application of the projection functions. Another case would be that of #b = proj12 and ◦b = ∧b , so that here at least one of the functions is essentially binary. This consideration motivates part (iii) of the following exercise. Exercise 7.31.33 (i) Show that if, for binary boolean connectives #, ◦, the truth-functions #b and ◦b do not satisfy the ambi-medial law, then the class of #-boolean valuations is not closed under ˆ◦. (ii ) Conclude from (i) that if the class of #-boolean valuations is closed ˆ under ˆ ◦ then the class of ◦-boolean valuations is closed under #. (iii) Investigate the possibility of finding a pair of distinct essentially binary truth-functions satisfying the ambi-medial law. (See notes to this section.) (iv ) Is the class of -boolean valuations closed under ˜? (Hint: Consider the example in the proof of 1.14.6, p. 69.) On a slightly different topic: it was noted above that for # ∈ {↔, }, we have all of the following: A, B BV A # B; A, A # B BV B; and B, A # B BV A. It is clear even independently of 7.31.26 which asserts this phenomenon to obtain generally, that the analogous BV -statements are also correct for # = ↔, though not (because the first fails) for # = : (3)

A, B BV A # B;

A, A # B BV B;

B, A # B BV A.

This illustrates a property of sets of formulas which lies at the opposite extreme from the property of independence as this is understood à propos of sets of axioms (e.g. for a -theory). The latter is that property Γ possesses just in case: (4) For all C ∈ Γ: Γ {C} C.

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For arbitrary A, B, on the other hand, taking Γ as {A, B, A # B}, what (3) illustrates is: (5) For all C ∈ Γ: Γ {C} C. When Γ satisfies (5), its elements are, adapting terminology from McKee [1985], generalized equivalent according to . This is a generalization of the usual notion of logical equivalence in the sense that when Γ consists of two formulas A, B, (5) amounts to the condition that A B. However, as McKee illustrates, the more general notion frequently arises in practice and so it is useful to have a succinct terminology for it. We deployed the notion (though not the terminology), for example, in 5.31.11 on p. 719. (In fact, McKee is more interested in a connective expressing generalized material equivalence than in this relation of generalized logical equivalence; the intended interpretation is that a compound formed by this connective is true on a valuation unless there is a way of selecting one of the components which is false while all the rest are true. In accordance with our policy of not treating multigrade connectives, we do not discuss this further here.) Now ↔ is not the only boolean connective for which (3) holds; we could also put # = ∧. However ↔ is the weakest connective for which (3) holds. In the context of BV (alias CL ) this is clear from a consideration of determinants and truth-tables (see 4.14.3–5 (p. 530 onwards); the three parts of (3) are the determinant-induced conditions on corresponding to the first three lines of the truth-table for ↔, which are the same as those for ∧). But the point also holds for IL , as we invite the reader to check in (i) of the following exercise. Exercise 7.31.34 (i) Suppose that for all formulas A and B of the language of IL there is some formula A # B satisfying (3) above, with “BV ” replaced by “IL ”, and such that for any formula C such that A, B IL C and C, A IL B and C, B IL A, we have C A # B. Show that A # B IL A ↔ B. (ii ) Let # be a ternary boolean connective with the property that for any formulas A, B, and C, the four formulas A, B, C and #(A, B, C) are generalized equivalent according to CL . How many choices of #b are there for which this is the case? What are the weakest and strongest among them (in the sense of 4.14.3–5)?

7.32

A Connective for Propositional Identity

There have been several attempts at describing the logic of a binary sentence connective with a role analogous to that played in first order logic with identity by the identity predicate (‘=’). While the latter connects two singular terms to form a sentence true (relative to an interpretation) just in case those terms have (relative to that interpretation) the same denotation, so we are to think of the envisaged connective, which we shall write as “≡” (this symbol’s association with material equivalence in some quarters notwithstanding) as yielding a truth when what is expressed by the sentences flanking it is the same proposition. The idea of such an analogy was mentioned in 4.36, with the following Set-Fmla rules (alongside the structural rules (R), (M), (T)) as dictated by the analogy: (≡I) A ≡ A

(≡E)

Γ A≡B

Δ C(A)

Γ, Δ C(B)


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These rules were seen (4.36.5, p. 604) to characterize ≡ uniquely, which was the reason for our earlier interest in them. (Recall from the discussion surrounding 3.16.1, p. 424, that the ‘context’ notation C(B) is intended to stand for any formula differing from the formula C(A) in having one or more occurrences of A replaced by B.) We noted the combined effect of (≡I) and (≡E) in 4.36: they were noted to yield S4 when the “≡” was interpreted as strict equivalence, in the presence the additional assumption that we were also dealing with a normal modal logic. (We will look at the effect of (≡I) on its own, which is of some interest in illuminating a technical concept of ‘reducibility’ from McKinsey and Tarski [1948], in 7.33.) The latter really is an additional assumption: Observation 7.32.1 No sequent of the form A ≡ B is provable using (alongside the structural rules) only the rules (≡ I) and (≡ E) unless A and B are the same formula. Proof. By induction on the length of proofs, one can verify that all sequents provable in the system described hold on every stringent valuation, where a stringent valuation is defined to be a valuation v satisfying the condition: v(A ≡ B) = T if and only if A and B are the same formula. Thus no sequent of the form A ≡ B is provable in which A and B are distinct formulas, since such sequents do not hold on all (indeed, on any) stringent valuations. It follows from this that the (≡I)–(≡E) proof system does not yield the strict equivalential fragment of S4, since the S4-provable sequent (p ≡ p) ≡ (q ≡ q), for example, is unprovable on the basis of those rules, by 7.32.1. More generally: Exercise 7.32.2 Show that, by contrast with the case of S4, no two formulas are synonymous in the present system. (Cf. 7.33.3, p. 1159.) Exercise 7.32.3 Show that all sequents of the following forms are provable in using (≡I), (≡E) and the structural rules: (i) A ≡ B, C(A) C(B) (ii ) A ≡ B B ≡ A (iii) A ≡ B, B ≡ C A ≡ C (iv ) A ≡ B C(A) ≡ C(B). Hint (for (iv )): think of C(A) ≡ C(A) as itself a context into which the second exhibited occurrence of A is replaced by B. 7.32.3(i) is just a horizontalized form of the rule (≡E); the “C(·)” notation is to be understood here, and in (iv ), as in the above formulation of the latter rule. Since we assume the presence of the familiar structural rules, we could equally well have used 7.32.3(i) in place of (≡E). The Set-Fmla logic of ≡ we have been exploring is a close cousin of a logic in Fmla developed by Roman Suszko and several co-workers, under the name SCI (‘Sentential Calculus with Identity’). References may be found in the end-of-section notes (p. 1161); Suszko did not think of this venture in terms of propositional identity so much as in terms of the identity of whatever sentences refer to (on some non-Fregean theory of what that might be: the word situation

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is used in some of the literature – see Wójcicki [1984], [1986], Suszko [1994]). Since these logics are presented in Fmla, additional connectives (assumed to behave classically), including in particular → (behaving classically), are taken to be available, and the axiom-schemata used include one for each primitive connective asserting a congruence property (adapting the term congruence from its algebraic habitat – introduced above for 0.23.1, p. 27), called (#) below. In the first place, we need the analogues of the conditions on equivalence relations, taken as axiom-schemata: (1)

A≡A

(2)

(A ≡ B) → (B ≡ A)

(3)

(A ≡ B) → ((B ≡ C) → (A ≡ C)).

These are of course Fmla versions of (≡I), 7.32.3(ii ), (iii), respectively. The only rule of proof is Modus Ponens (for →), but there is also a Modus-Ponenslike principle, taken as an axiom schema, for ≡ itself: (4)

(A ≡ B) → (A → B)

whose resemblance to Modus Ponens (or more accurately, to a horizontalized version of (→E), stated for ≡, is clearer in its purified Set-Fmla incarnation: (mp)

A ≡ B, A B.

Note that (mp) is the special case of 7.32.3(i) in which C(·) is taken as the null context. For the congruence axioms proper, in the case of n-ary #, the required axiom-schemata take the following form, in which we have also used ∧ to avoid a cumbersome formulation. Note that amongst the n = 2 cases for # is included the case of # as ≡ itself: (#)

(A1 ≡ B1 ∧ . . . ∧ An ≡ Bn ) → (#(A1 , . . . , An ) ≡ #(B1 , . . . , Bn )).

If we give this as a Set-Fmla sequent-schema: A1 ≡ B1 , . . . , An ≡ Bn #(A1 , . . . , An ) ≡ #(B1 , . . . , Bn ), we can see that it is provable from our own basis by means of successive replacements using (≡E). We illustrate this for the case of n = 2. Example 7.32.4 A proof of A1 ≡ B1 , A2 ≡ B2 (A1 # A2 ) ≡ (B1 # B2 ); for reasons of space we supply the justifications of these lines below the proof rather than as side annotations: (1) A1 ≡ B1 (A1 # A2 ) ≡ (B1 # A2 ) (2) A2 ≡ B2 , (A1 # A2 ) ≡ (B1 # A2 ) (A1 # A2 ) ≡ (B1 # B2 ) (3) A1 ≡ B1 , A2 ≡ B2 (A1 # A2 ) ≡ (B1 # B2 ) Justifications – line (1): 7.32.3(iv ); line (2): ≡E; line 3: from 1, 2, by (M), (T).


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We have shown how to derive all of the principles of Suszko’s SCI from (≡I), (≡E), the structural rules, and #-classicality for such boolean connectives (principally ∧, →) as those principles involve. A derivation of (Fmla versions of) our (≡I) and (≡E) from Suszko’s basis is more straightforward – essentially just a matter of showing that all instances of a Fmla version of 7.32.3(i), (A ≡ B) → (C(A) → C(B)) are forthcoming on that basis (by induction on the complexity of C(p)) – and we shall accordingly take the liberty of calling the consequence relation associated with our own proof system SCI . There is a certain amount of vagueness here, since we have not specified the language of that consequence relation; for simplicity let us resolve the vagueness by concentrating for the moment on the case in which the only connective of the language is ≡ itself. Thus, for instance, the only cases of the schema mentioned in 7.32.4 that arise are those in which # is ≡. We can make explicit provision for the presence of additional connectives as the need arises. The recent talk of simulating the properties of a congruence relation in principles governing our ‘congruence connective’ ≡ should not be confused with the question of congruentiality in the sense of §3.3. The connective ≡ is most definitely not congruential (according to SCI ) in that sense (though {p ≡ q} does count as a family of equivalence formulas in the special sense – see the end of 1.19 – required for a logic to be equivalential): Example 7.32.5 We can adapt the example given in the discussion following 7.32.1 to show that SCI is not congruential. By (≡I) and (M) we have p ≡ p SCI q ≡ q, but whereas SCI (p ≡ p) ≡ (p ≡ p), 7.32.1 tells us that SCI (p ≡ p) ≡ (q ≡ q). For another example, 7.32.3(ii ) gives p ≡ q SCI q ≡ p, while SCI (p ≡ q) ≡ (q ≡ p), again by 7.32.1, even though SCI (p ≡ q) ≡ (p ≡ q). A semantic description of SCI in terms of matrices has been given by Suszko, and we present it here, somewhat reformulated. The algebras of the matrices will be groupoids, (A, =); ˙ that is, = ˙ is a binary operation – N.B., a binary operation and not a binary relation – under which A (= ∅) is closed. If other connectives were present in the language, these algebras would need to have additional fundamental operations corresponding to them. A matrix (A, =, ˙ D) based on such an algebra we will call a Suszko matrix if it satisfies the following condition, for all a, b ∈ A: a= ˙ b ∈ D D if and only if a = b. Note that even if | D | = 1, this condition does not uniquely determine = ˙ (as it does not fix which element, outside of D, a = ˙ b is to be when a = b). Theorem 7.32.6 SCI is determined by the class of all Suszko matrices.

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Proof. For soundness, it suffices to note that all instances of (≡I) are valid in any Suszko matrix and that (≡E) preserves this property. For completeness, suppose Γ SCI C. For any formula A, let [A] = {B | Γ SCI A ≡ B}, and define [A1 ]=[A ˙ 2 ] to be [A1 ≡ A2 ]. If we let A be the set of all these [A] (A any formula), then the algebra here defined is expanded to a matrix by putting D = {[B] | Γ SCI B} and we can verify that this is a Suszko matrix since [A] = ˙ [B] ∈ D ⇔ [A ≡ B] ∈ D ⇔ Γ A ≡ B ⇔ [A] = [B]. Now consider the evaluation h with h(A) = [A]. We have h(A) ∈ D for all A ∈ Γ, while h(C) ∈ / D, establishing completeness.

˙ [A2 ] to be [A1 ≡ A2 ]; but Exercise 7.32.7 In the above proof, we define [A1 ] = this would be an incorrect definition if it could happen that [A1 ] = [B1 ] and [A2 ] = [B2 ] while [A1 ≡ A2 ] = [B1 ≡ B2 ]. Show that this possibility cannot in fact arise. Our own proof system in terms of (≡I) and (≡E) is a Set-Fmla version of a logic for propositional identity developed in Fmla as an extension of CL in that framework by Cresswell ([1965], [1967]). Cresswell concentrates on a further extension of the system with what in our preferred framework would be the following rule:

(Cresswell )

AB

BA

A ≡ B

Let us call the resulting system SCI + and the associated consequence relation SCI + . Whereas, as we saw above (7.32.5) SCI is not congruential, this new consequence relation is; in fact: Observation 7.32.8 SCI + is the smallest congruential extension of SCI . Proof. To see that SCI + is congruential, suppose that A SCI + B; by the rule (Cresswell), we have SCI + A ≡ B. The synonymy of A with B according to SCI + then follows by (≡E). To show that SCI + is the least congruential consequence relation extending SCI , we show that any such consequence relation is closed under (Cresswell). Suppose that for such a congruential we have A B. By congruentiality, we must have A ≡ A A ≡ B. By (≡I) and (T), A ≡ B. The following result may be found in Cresswell [1965] or Wiredu [1979]; see the discussion after 4.36.5 above (p. 604). Theorem 7.32.9 SCI + is the (inferential) consequence relation associated with the strict equivalential fragment of S4.


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Proof. Calling the latter consequence relation S4 , it is clear that SCI + ⊆ S4 . For the converse inclusion it suffices to show that any SCI + -unprovable sequent is invalid on some transitive reflexive Kripke frame. We can use a canonical model argument for this purpose, where the canonical model (W, R, V ) is defined by taking W as the set of all SCI + -deductively closed sets of formulas, and (for x, y ∈ W ) iff for A, B, we have A ≡ B ∈ x only when A ∈ x iff B ∈ x. For more on SCI and modality, see Suszko [1971a], Malinowski [1985], and Chapter 4 of Ishii [2000]. (This last reference actually discusses a weaker logic for ≡ than SCI.) Exercise 7.32.10 Consider for simplicity the language with only two connectives, the binary connectives ∧ and ≡, and a proof system with the standard structural rules as well as the introduction and elimination rules for these connectives given in 1.23 and the present subsection, respectively. We can think of the associated consequence relation, which we denote just by here, as the SCI –conjunction consequence relation. Either prove or refute, in turn, the claims that for this relation: (i) (p ∧ p) ≡ p; (ii) (p ∧ q) ≡ (q ∧ p); (iii) p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r. (Ponder before proceeding. Solution follows.) A tougher version of the above exercise – though easier in respect of giving away the answer to the question about the correctness of the -statements mentioned there – would ask for a single three-element matrix invalidating all three formulas listed under (i)–(iii), while at the same time validating all sequents in the SCI –conjunction consequence relation . Remark 7.32.11 Any way of filling the dashes in the following tables by the undesignated values 2 and 3 yields a matrix as asked for by the tougher version of 7.32.10 just mentioned. Thus since there are two such ways of replacing each dash and there are eleven dashes, the following partial matrix gives 211 answers to the tougher question (which is not to say that all of the resulting 211 matrices determine distinct consequence relations): ∧ *1 2 3

1 1 − −

2 − − 3

3 − 2 2

≡ 1 2 3

1 1 − −

2 − 1 −

3 − − 1

All admissible ways of completing the first partial table still give a conjunction the designated value just in case each of the conjuncts receive that value, validating all required ∧ principles, while all admissible ways of completing the first table make the whole thing a Suszko matrix. It remains only to check that we have invalidated the three formulas in 7.32.10. Since 3 ∧ 3 = 2, the idempotence formula (i) is invalid. And the fact that (3 ∧ 3) ∧ 3 = 3 ∧ (3 ∧ 3) (as the lhs evaluates to 2 and the rhs to 3) knocks out the commutativity and associativity formulas (ii) and (iii).

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Some ventures in the literature connect with these themes. Hugly and Sayward [1981] explore the possibility of extending the Anderson–Belnap system of tautological entailment (see the discussion after 2.33.6 above – p. 340 above) by a propositional identity connective which they write as ‘=’ but which we shall re-notate as “≡”, the latter governed in the envisaged extension by axioms instantiating the schemas: A ≡ A;

A ≡ ¬¬A; A ≡ (A ∧ A); (A ∧ B) ≡ (B ∧ A).

The connective ∨ does not appear explicitly as it is being taken as defined (De Morgan style) in terms of ¬ and ∧ (which Hugly and Sayward write as ∼ and &). In addition there are rules for the symmetry, transitivity, and replacement properties (securing congruence) for the relation holding between A and B when A ≡ B is provable. The system of Hugly and Sayward is not really a sentential logic in that just as the “→” of Anderson–Belnap’s tautological entailment system cannot appear in the scope of another connective, so the “≡” (or “=”) of Hugly and Sayward. They are just describing two binary relations between boolean formulas, and since a consequence relation is not given, there is no way of expressing the requirement that A ≡ B should have A → B (and its converse) as a consequence, short of the weaker metatheoretical statement to the effect that whenever the combined system proves the former for a given A and B, it proves the latter (which statement is certainly correct). The authors remark (p. 278, adapting notation here) that “if (A ∧ B) ≡ (A ∧ C) then B ≡ C,” which again, given the resources they offer, should presumably be interpreted to mean that if the former is provable, so is the latter. This is reminiscent of the ‘Unacceptable Cancellation Condition’ of 5.23.1: see the case, discussed just before that, of the equivalence of q ∧ (p ∧ q) with q ∧ p (take A as q, B as p ∧ q and C as p; we have reversed some conjuncts from the discussion in 5.23 to fit the principle quoted from Hugly and Sayward). The equivalence cited here holds for Anderson and Belnap’s tautological entailment; Hugly and Sayward are of course discussing (see the title of their [1981]) precisely a logic in which (propositional) “identity diverges from mutual entailment”, the latter being what we have been calling equivalence. And they are at pains to point out that examples such as the would-be propositional identity above would only be forthcoming if in addition to the idempotence and commutativity axioms (i.e., the last two in the inset list above) we added an associativity axiom, conspicuously missing from their list precisely because of its untoward effects for the cancellation principle. Digression. In fact Hugly and Sayward [1981] give the cancellation condition only as an example of a more general principle (clashing with the associativity axiom) namely closure under the rule (p. 278) which passes from C(A/B) ≡ C to A ≡ B, where the notation “C(A/B)” has been introduced (p. 271) to stand for a formula which “differs from C only in that one occurrence of A in C has been replaced by B”. Note that this is a more stringent interpretation for such replacement notations than is usually offered. On the usual account “C(A/B)” denotes formula in which resulting from C by replacing one or more occurrences of A by B (or, on, another convention: vice versa). The latter amounts to fixing the formula C(q) as a context the uniform substitution of A for q in which gives C(A) and of B for q gives C(B); then on this second account C is C(A) and C(A/B) is C(B). In a different notation, using the Łukasiewicz ‘variable


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functor’ notation “δ”, used extensively in the theory of propositional identity of Prior [1971], this is the content of a striking propositional identity principle (again we change “=” to “≡”, and also “⊃” to “→”, but retain the propositional quantification): ∀p∀q((δp ≡ δq) → (p ≡ q)) which Hugly and Sayward [1976], p. 183, describe as being “arguably a logical truth”. (Here δ corresponds to the C(·) of the metalinguistic formulation.) See Freeman [1977] for some further discussion. End of Digression. A principle for propositional identity even stronger than the cancellation principle toyed with by Hugly and Sayward, written here in the stronger form not available in their 1981 paper (with some re-lettering from the version above, for the sake of a comparison with the stronger principle) is the following: (A ∧ C) ≡ (A ∧ D) C ≡ D. The stronger principle is an ‘ordered pairs’ condition we called [OP∧] at the start of 5.34, whose ⇐ direction, translated from its quasi-identity form becomes the following: (A ∧ C) ≡ (B ∧ D) (A ≡ B) ∧ (C ≡ D), and more generally, for n-ary #: #(A1 , . . . , An ) ≡ #(B1 , . . . , Bn ) (A1 ≡ B1 ) ∧ . . . ∧ (An ≡ An ). Such principles find their way into a Russell-inspired theory of propositional identity in Church [1984], on p. 514 of which they are given (as items 9 and 10) for the case of # as ∨ (with n = 2: Church lists the implication to the first conjunct and to the second separately). Note that the version for ∧ as # already conflicts with the commutativity axiom of Hugly and Sayward [1981], since from (p ∧ q) ≡ (q ∧ p) we can now infer p ≡ q. We have seen now objections from various quarters to associativity and commutativity (for ∧, with similar considerations in the case of ∨) as principles of propositional identity – (ii) and (iii) of 7.32.10. In fact, even idempotence (7.32.10(i)) has come under fire, as in this quotation from p. 192f. of Prior [1963], in which we translate the Polish notation (see the notes to §2.1, p. 269) afterwards: Negatively, it also seems to be a law that the proposition that not not p is not the same proposition as the proposition that p (i.e., NINNpp), and quite generally that no proposition can be a logical complication of itself (e.g. NIKppp).

Translating from Polish notation to that in use in our discussion, the principles Prior is endorsing here in the first and second parenthetical passages, respectively, are ¬(¬¬p ≡ p) and ¬((p ∧ p) ≡ p), negations of representative instances of the second and third schemata, inset above, endorsed in Hugly and Sayward [1981] – though with the left-right order round ≡ reversed. (Unlike Hugly and Sayward, Prior insists that ≡ – or I, in his notation – is a connective and not a relation symbol.) Quite what being a “logical complication of” consists in is rather unclear, but if it implies having greater complexity than, according to

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some measure of complexity, then it certainly doesn’t sound as though anything could stand in such a relation to itself – and, yes, we probably do need to use the language of relations here – but to an adherent of the principle A ≡ (A ∧ A) the only place where there is greater complexity as we compare “p” and “p ∧ p” is in the notation used in the latter case, rather than in the proposition this notation is used to express. (More discussion of Prior’s views on propositional identity can be found in Routley and Routley [1975], esp. p. 206.) In any case, as we already saw in 7.32.11, Suszko’s SCI (with ∧) does not in fact satisfy (i)–(iii) of 7.32.10, and so provides the flexibility to be extended or otherwise in various directions answering to the several principles we have been considering. Our treatment finishes here. Amongst the most interesting applications of SCI it leaves untouched is to the accommodation of what at first seem to be alternatives to CL (such as Łukasiewicz’s Ł3 ) as instead extensions of SCI, itself an extension of CL. The references in the notes, which begin on p. 1161, contain some discussion – and a more careful description – of this topic; especially useful is the brief survey article Malinowski [1985]. Contemporary abstract algebraic logic (introduced briefly in 2.16) has its own version of Suszko’s distinction between Fregean and non-Fregean logics. Chapter 6 of Czelakowski [2001] gives an account of some of this work and of the Suszko background; see further Czelakowski and Pigozzi [2004]. As mentioned in Remark 3.23.5 above (p. 456), these authors’ Fregean consequence relations (or Fregean deductive systems, as they prefer to say) are simply the weakly extensional consequence relations of Chapter 3.

7.33

Reducibility without Tabularity: An Austere Propositional Identity Logic

At the start of 7.32, we were concerned with the collective effect of (≡I) and (≡E), and as remarked there, we discuss here the former rule in isolation, since this provides a convenient opportunity to elaborate on an interesting example given by McKinsey and Tarski [1948] in their discussion of algebraic semantics for intuitionistic and modal logic. (The background needed for this discussion is 2.11.) The framework for that discussion was Fmla, and we shall follow suit; this is possible because (≡I) makes sense as a schema in this framework, there being no formulas to the left of the “”. At p. 6 of McKinsey and Tarski [1948], the authors introduce the concept of an n-reducible logic (in Fmla) meaning by this a logic in which any unprovable formula has an unprovable substitution instance in n or fewer propositional variables; they go on to define a logic as reducible if it is n-reducible for some n. (A more general definition, suited to arbitrary logical frameworks rather than just Fmla would have a logic count as n-reducible when any unprovable sequent has an n-variable unprovable substitution instance, with reducibility defined in terms of n-reducibility as before. Except for 7.33.8 below, our discussion is based on Fmla.) Exercise 7.33.1 Show that CL, in the language with connectives ∧, ∨, →, ¬, is 1-reducible. (Hint: See the discussion immediately following the proof of 1.25.7, p. 130.) A propos of this Exercise, note that if we have either of the nullary connectives ⊥, , primitive as well as those listed, then we can turn the claim into one of


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0-reducibility for CL. (See Theorem 1.23 of Chagrov and Zakharyaschev [1997], as well as Chapter 13 for some more recent results in the area we are about to describe the first work in.) McKinsey and Tarski [1948] prove that neither IL nor S4 is reducible; since the non-tabularity of both logics was well known at the time (thanks to Gödel [1932] and Dugundji [1940] respectively), they take an interest in showing that non-reducibility is a stronger property than non-tabularity; i.e., that every tabular logic is reducible, though not conversely. We follow McKinsey and Tarski’s discussion fairly closely here, filling out some details they allude to as obvious. Observation 7.33.2 Every tabular logic is reducible. Proof. We establish the result by showing that if S is a tabular logic (in Fmla) determined by some matrix with n elements, then S is n-reducible. Suppose that A ∈ / S. Then the supposedly available determining matrix invalidates A by providing an evaluation mapping the propositional variables of A to the matrix elements, on which evaluation A receives an undesignated value. If A contains more than n distinct propositional variables, then since there are only n matrix values, the evaluation in question must assign the same value to distinct variables. Whenever this happens, pick one of those variables and substitute it for the others taking that value. This gives a substitution instance of A containing no more than n variables, which that same evaluation assigns an undesignated value, a substitution instance which does not belong to S, since it is invalid in the determining matrix. The logic we choose, following McKinsey and Tarski, to show that the converse of 7.32.2 is false, has as theorems exactly those formulas which are of the form A ≡ A for some formula A. For simplicity we assume that ≡ is the only connective in the language of this logic. In other words, we explore the effects in Fmla of (≡I) and no other rules. We call this rather austere little logic I, which can be thought of as abbreviating “≡I”, but which can also be given another etymology which is more closely suggested by the way McKinsey and Tarski formulate the example, in which instead of “≡”, they write “→”. Recall the combinator-derived nomenclature for logics such as BCK and BCI (see 2.13); these are implicational logics which can be presented in Fmla by axiom schemata or axioms as suggested by the letters in the label, together with the rule Modus Ponens. In particular, I is a label for the schema A → A, or alternatively the axiom p → p; in the presentations using (finitely many) axioms rather than schemata, the rule of Uniform Substitution is added to Modus Ponens to complete the basis. For the moment, calling the logic axiomatized by the schema just mentioned I, we note that is closed under Uniform Substitution and that even if the rule Modus Ponens is not used in the basis, this rule is admissible (though not derivable). For suppose that A → B and A are both provable. Then since A → B is provable, B must be the formula A itself, in which case the Modus Ponens conclusion B is of course provable. We now resume the discussion in the ≡-language, with I understood as the set of all formulas of the form A ≡ A. Lemma 7.33.3 No two formulas in the language of I are synonymous according to I.

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Proof. Suppose A and B are distinct formulas. Then we have I A ≡ A but I A ≡ B, so A and B are not synonymous according to I. A particular sequence of formulas in a single propositional variable will be of service to us below, in the proof of 7.33.7. Let A1 be the formula (p ≡ p) ≡ p, and let Ai+1 be the formula (p ≡ p) ≡ Ai . Before proceeding, we pause to define a substitution which will be used in that proof. Exercise 7.33.4 Where A1 , . . . , An , . . . is the sequence of formulas just defined, let s be the substitution uniquely determined by: s(pi ) = Ai . Thus, for example (recalling that p and q are p1 and p2 respectively), s(p) is (p ≡ p) ≡ p, s(q) is (p ≡ p) ≡ ((p ≡ p) ≡ p), and s(p ≡ q) is ((p ≡ p) ≡ p) ≡ ((p ≡ p) ≡ ((p ≡ p) ≡ p)). Show that if A and B are distinct formulas, then s(A) = s(B). Although we are mainly interested in A1 , . . . , An , . . . because of the property just mentioned, which will allow us to show that I is reducible (7.33.7), we can press this sequence into service immediately, as part of the demonstration that I is not tabular; the concept of local finiteness was introduced for 2.13.5 (p. 228): Observation 7.33.5 I is not locally finite. Proof. No two of the infinitely many formulas A1 , . . . , An , . . . defined above are synonymous according to I, by 7.33.3; but all are constructed from a single propositional variable.

Corollary 7.33.6 I is not tabular. Proof. Immediate from 7.33.5, since tabular logics are locally finite (by 2.13.5). To complete the presentation of the example of I as a logic which is reducible without being tabular, we show that I is reducible. In giving the example, McKinsey and Tarski [1948] remark that I is 2-reducible; we shall see that a stronger claim can be made, to the effect that I is 1-reducible. Observation 7.33.7 I is 1-reducible. Proof. Suppose that I C. We must show that there is an I-unprovable onevariable substitution instance of C. If C is constructed from a single propositional variable, we are done (since C itself will serve as the relevant substitution instance). So suppose that there is more than one variable in C. Then C must have the form A ≡ B, and since I C, A and B are different formulas. Let s be as defined in 7.33.4. We claim that the formula s(A ≡ B), which is constructed from the variable p alone is I-unprovable. This formula is s(A) ≡ s(B), so by the result of 7.33.4, since A and B are distinct, the formula is not provable in I, establishing the claim.


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The traditional way of speaking of tabular logics is of those with a finite characteristic matrix (as was mentioned in the notes to §2.1, p. 272). A corresponding terminology for reducible logics would be those with a finitely generated characteristic matrix, though we do not go into why such a description is apposite – for which the interested reader is referred to McKinsey and Tarski [1948]. (See also Porte [1965], p. 79.) We close with an exercise suggested by the description. Exercise 7.33.8 Show that for any logic S (in Fmla) S is tabular if and only if S is both locally finite and reducible.

Notes and References for §7.3 The terminology of the ‘medial law’ in 7.31 is fairly standard (though groupoids satisfying it are also called entropic); it may be found, along with much else in the way of properties of binary operations, in Dénes and Keedwell [1972], pp. 58–60. The same subsection’s ‘ambi-medial law’ is an innovation, though the property (of pairs of binary operations) involved is occasionally encountered either unnamed or under a different name – for example, at p. 347 (line 2 from base) of Vaughan [1968]. With regard to Exercise 7.31.33(iii), the author does not know a nice general argument, and is grateful to (his son) Bryn Humberstone for writing and running a computer program checking all 256 pairs of binary connectives, revealing 90 of these pairs to satisfy the ambi-medial law, and none of these 90 pairs to consist of two distinct truth-functions both of which were essentially binary. (Thus the only pairs ◦, # of essentially binary truthfunctions to satisfy the ambi-medial law are the cases in which ◦ = #, and the medial law is satisfied: these are the cases of ∧b , ∨b , b and ↔b .) A propos of the notion of global supervenience mentioned in 7.31, see Kim [1993], Chapters 4, 5, 7, 9 for clarification and applications; see also Humberstone [1998d]. As mentioned in that subsection, the relations V and V are called the consequence relation inference-determined by V and the consequence relation supervenience-determined by V in Humberstone [1993b], which, along with Humberstone [1992] and [1996a] may be consulted for further points of interest connected with the matching operation ˜. (Rautenberg [1993], p. 71, also mentions a supervenience-style consequence relation for arbitrary matrices, or the underlying algebras, which can be regarded as a generalization of the notion with which we have been working. See the Digression after 2.15.9.) The material on Suszko’s SCI referred to in 7.32 may be found in Suszko [1971a], [1971b], [1973], [1974], [1975], [1977], [1994] Bloom and Suszko [1972], Michaels and Suszko [1976], Słupecki [1971], Wasilewska [1976], Malinowski [1985], [1997], Wójcicki [1984], [1986], Ichii [2000]. For the case of the pure ≡ logic of SCI, taken as consequence relation, Kabziński suggests [1993] a version of BCIA logic (see 7.25) presented under the name BCII, a strange label already remarked on in the end of section notes on the (BCK etc. Material in) 2.13, which begin at p. 274. The connections with Suszko’s ideas are explained on pp. 21 and 23 of Kabziński [1980]. The description here of the relation to the BCIA logic of 7.25 is vague because that was a logic in Fmla, whereas Kabziński is proposing a particular consequence relation for which the consequences of the empty set comprise the theorems of that logic; this is of course true of many distinct con-

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sequence relations. (Three of them are studied systematically in Paoli, Spinks and Veroff [2008], for the full language of Abelian logic.) Though it considers a relational (rather than a connectival) treatment of something at least in the vicinity of propositional identity, Anderson [2001] contains a discussion of issues related to the commutativity, idempotence, and associativity principles touched on in 7.32 for ∧ ((A ∧ B) ≡ (B ∧ A), etc.). As usual, many topics connected with the biconditional or equivalence connective(s) have not been touched on in this section. The topic of equivalential algebras (called intuitionistic equivalential algebras in Kabziński [1980]), though barely touched on in our discussion, has spawned a considerable literature, from which here we simply repeat (for convenience) the references cited in 7.32: Kabziński and Wroński [1975] and Słomczyńska [1996]. In addition, various other special-purpose equivalence-like connectives, not touched on at all in 7.32, have been considered. A good example is provided by Aczel and Feferman [1980]; see further Bunder [1982] for a discussion of the logical properties of those authors’ novel equivalence connective. At the start of §7.3 it was said that “we shall not be taking an interest in biconditionality in the sense of investigating the logical properties of the phrase ‘if and only if’.” The occasionally heard suggestion that S1 only if S2 should be treated as If not-S1 then not S2 , may prompt interest – as remarked in 7.12.4(ii) on p. 961 – in a biconditional A ≡ B defined as, or taken as primitive but with such a candidate definition in mind (A → B) ∧ (¬A → ¬B). For example, this is not IL-equivalent to the usual definiens for A ↔ B. The investigation of its logical behaviour against various backgrounds, such as that of IL, is another of the topics not covered in our discussion.

Chapter 8

“Not” §8.1 INTRODUCTION TO NEGATION 8.11

Contraries and Subcontraries

For a direct parallel with the preceding three chapters, we should open with an extended discussion of the points at which the treatment of negation in formal logic has been held inadequately to represent the behaviour of not (or it is not the case that) in English or corresponding locutions in other natural languages. We shall in fact pursue a different course. The reason is not that there is less to say, but that the discussion would inevitably take us too far outside the domain of sentential logic for its inclusion here to be fitting. For foremost amongst the respects in which the treatment of negation in, say, classical logic, has been held to oversimplify the facts of natural language negation are certain details concerning quantificational constructions, definite descriptions, non-referring names, predicate vs. sentential negation, and other such matters. An extensive discussion of these and much else besides may be found in the encyclopedic treatise on negation, Horn [1989]. Many issues arise over the question of whether there is more than one kind of negation, these kinds distinguishing themselves over whether or not the presuppositions of the sentence negated are inherited by the negated sentence. Since we do not wish to get embroiled in questions about presupposition, we avoid the whole area here. (For those interested, some discussion may be found in Seuren [1988], Horn [1990]; see also Chapter 4 of Carston [2002].) Associated with these complications is what Geach [1956] has referred to as “the virtual absence of propositional negation from ‘ordinary language’” (p. 75), meaning here and in the following quotation (from the same page) by ‘propositional’ what we should mean by ‘sentential’. Fifty years later – e.g., Barker [2004], p. 89f. – one continues to find this sentiment voiced; here we continue with the quotation from Geach: Now though propositional negation has been made familiar by modern logic, it a rather sophisticated notion. In ordinary language it is rather rare to negate a statement by prefixing a sign of negation that governs the whole statement; negation is almost always applied primarily to some part of a statement, thought this often has the effect of negating the statement as a whole.

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As well as the behaviour of negation in natural language, numerous other topics no less worthy of treatment will be omitted. We say nothing of the allegations made from time to time linking (especially classical ) negation with various social and political problems: examples may be found in Korzybski [1958], Hayakawa [1969], Jay [1981], and the works by Val Plumwood cited in the critical discussion of them provided in §7.5 of Goldstsein, Brennan, Deutsch and Lau [2005]. Many connectives have fallacies named after them. For → we have “misconditionalization” (Stove [1972]) as well as the conditional fallacy (Shope [1978]); for ∧ and ∨ we have the conjunction fallacy (Tversky and Kahneman [1983], Tentori, Bonini and Osherson [2004], Costello [2007], Crupi, Fitelson and Tentori [2008]) which, in its original form, consists in assigning a higher probability to a conjunction than to one of its conjuncts. (See, however, Schwartz [1996], esp. Chapter 3.) Also pertaining to probabilistic reasoning (cf. 5.15.1, p. 660), we have a proposed disjunction fallacy (see Piatelli-Palmarini [1994], p. 93). But negation would is perhaps the only connective to have had a delusional state named after it; the following passage is from p. 125 of Enoch and Trethowan [1979], in which various psychiatric syndromes are described. They mention that Cotard’s syndrome, from which the discussion of which this quotation comes, is sometimes known by the French phrase délire de negation. The central essential symptom of Cotard’s syndrome is a delusion of negation. This can vary in severity along a continuum from a belief by the patient that he is losing his powers of intellect and feeling, to the severest form when he believes that he no longer exists, leading his to deny both his own existence and that of the cosmos.

This is, interestingly, not the only context in which negativity is associated specifically with negative existence claims. We return to this point in our discussion of whether there are inherently negative propositions – as opposed to merely negative sentences – in 9.13 below (not that even the classification of natural language sentences as negative or otherwise is trivial). More seriously, we do not touch on such questions regarding negation, falsity, and bivalence, as were raised for example in Dummett [1959a], [1963]. (See the discussion of Dummett in 2.11 for a cursory introduction.) Later work of considerable logical interest relates to the treatment of negation (“negation as failure”) in logic programming and database theory, a discussion of these matters would take us too far afield to be appropriate here. The interested reader is referred to Lloyd [1987], Shepherdson [1988], [1989], Fine [1989], Gabbay [1991]. Finally, we shall not treat numerous negation-like connectives having weak inferential powers when these do not come up naturally for consideration in the course of discussing more full-blooded species of negation. Set aside under this description are various notions of seminegation and semicomplementation, which may be found in Rasiowa [1974], p. 192f., Wójcicki [1988], §2.7 (the same notion), and Humberstone and Lock [1986] (a different notion), as well as the split negation pairs of C. Hartonas (see Restall [2000], p. 62 for exposition and references). We begin instead with by considering the conditions of ¬-classicality on gcr’s and ¬-booleanness on valuations, introduced in §1.1. Some aspects of negation as it fares in the setting of consequence relations will be introduced in 8.11.2 (part (iii) onwards), and will occupy us further in later sections.

8.1. INTRODUCTION TO NEGATION

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A gcr, , it will be recalled, is ¬-classical iff the following two conditions are satisfied for all formulas A in the language of : (1) A, ¬A and (2) A, ¬A. Corresponding to these conditions are conditions on valuations v, namely that for all A: (1) At most one of v (A), v(¬A) is equal to T. (2) At least one of v (A), v(¬A) is equal to T. The valuations satisfying (1) and (2) are precisely the ¬-boolean valuations defined in 1.13. The correspondence in question is given by: Every v ∈ Val () satisfies (1 ) iff satisfies (1) and similarly in the case of (2) and (2) . Some traditional terminology may be used to describe these conditions. We can say that formulas A and B are contraries according to when A, B . Or, thinking now more semantically, that these formulas are contraries w.r.t. (some class of valuations) V when for no v ∈ V do we have v(A) = v(B) = T. This idea of contrariety, or of being mutually exclusive, is what is conveyed by (1), (1) , for the relation between a formula and its negation. Similarly, we may say that formulas A and B are subcontraries according to when A, B; and that they are subcontraries w.r.t. V when for no v ∈ V do we have v(A) = v(B) = F. The idea of subcontrariety, or of being jointly exhaustive, is, then, what (2), (2) convey. If A is a contrary of B (i.e., A and B are contraries) and also a subcontrary of B, then the traditional terminology would have it that A is a contradictory of B (relativized to or to V , as desired). Thus, the conditions of ¬-classicality for a gcr, and ¬-booleanness for a valuation, amount to requiring that ¬A be a contradictory of A. These notions of contrariety and subcontrariety occupied us at the end of 6.31 – from 6.31.10 (p. 848) on; see especially 6.31.15 (p. 850). Indeed, already in 3.18 we saw that IL had a hard time – in a sense made precise there – accommodating a 1ary connective # which made #A simultaneously a contrary and a subcontrary of A (though there, in the interests of a more general discussion, with an nary connective and n + 1 formulas to be considered as mutually exclusive or jointly exhaustive, we spoke of compositional contrariety and compositional subcontrariety determinants). Remark 8.11.1 In the context of traditional (Aristotelian) logic, the terminology described here is normally taken to have a rather restricted application, to such statements as can be the premisses and conclusions of ‘syllogistic’ inferences (“All F s are Gs”, etc.). Warning: in addition, a frequent convention in textbooks is to use contrary not in the above sense but as meaning – as we would put it – “contrary but not subcontrary”, and similarly subcontrary as meaning “subcontrary but not contrary”. This was mentioned, with the first of these two examples given, in 3.34.9, p. 506 (with a slightly different gloss from that about to be provided). Thus it would be said that contraries are statements which could not both be true but could both be false, and subcontraries are statements which could both be true but could not both be false. On this convention, the relationships of contrariety, subcontrariety, and contradiction, are mutually exclusive. (The present usage follows, for


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example, that of Smiley [1996]; on the opening page of McCall [1967a] it is described as the most common usage of the terms. In 3.34 (at p. 503) we saw that this had been the usage in Lemmon [1965a]. For further discussion of this issue, see Humberstone [2003c], and, for a rather different perspective, tailored to square-of-opposition concerns, Humberstone [2005f ]. This reference to the Square of Opposition is not only to the syllogistic square, but to its modal analogues. See also Humberstone [2005d].) Exercise 8.11.2 (i) Show that (1) and (2) above, taken as zero-premiss rules of Set-Set (i.e., replacing “” with “”) uniquely characterize ¬, in the sense of §4.3. (ii ) Show that this is not so for either (1) or (2) taken separately. (iii) This and the remaining parts of this exercise look at issues in SetFmla; to avoid explicit mention of the structural rules (R), (M) and (T), we formulate them in terms of consequence relations , but still employ the rule-figure notation. Picking up the theme of 7.21.11, consider the following conditions on consequence relations with a 1-ary connective # in their languages: (Peirce)#

Γ, #A A

(Subc)#

ΓA

Γ, A C

Γ, #A C ΓC

Show that Any consequence relation (with # in its language) satisfying (Subc)# satisfies (Peirce)# , and that the converse is not so. (Suggestion for the second part of this question: consider M L , as in 8.32, with #A as ((⊥ → A) → ⊥) → ⊥ and appeal to Wajsberg’s Law.) (iv) Show that for any consequence relation satisfying (Peirce)# , we have (for all formulas A in the language of : ##A A. (v) Show that any consequence relation satisfying (Peirce)# and the condition (EFQ)# = A, #A B, also satisfies the following conditional condition: (RAA)#

Γ, A C

Γ, A #C

Γ #A

(Suggestion: Using part (iv), transform the premisses for an application of (RAA)# to (1) Γ, ##A C and Γ, ##A #C and appeal to (3) C, #C #A as an instance of (EFQ)# ; appeal to the cut condition (T) on consequence relations twice to infer from (1), (2) and (3) that Γ, ##A #A and then finally appeal to (Peirce)# .) (vi) As a culmination of (iii)–(v), show that if a consequence relation satisfies (Peirce)# and (EFQ)# , then it satisfies (Subc)# . (Suggestion: take two premisses for the latter rule, (1) Γ, A C and (2) Γ, #A C. From (2), not only C but also #C is a -consequence of Γ ∪ {#A, #C}, so by (RAA)# ), Γ, #C ##A, and by (iv) we can remove these two #s. Complete the argument by appeal to (T) with (1), using A as the cut formula, and then invoke (Peirce)# .) (vii) Suppose is the least consequence relation (so Γ A iff A ∈ Γ) on a language one of whose 1-ary connectives is #. Show that for no


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formula A are A and #A contraries according to , while for every A, A and #A are subcontraries according to . (A related theme occupied us in 6.31.11.) The ¬-classicality conditions on consequence relations in 1.13 were a subcontrariety condition, (Subc)# for # = ¬ and a contrariety condition, (EFQ)# for # = ¬. This is named after the Ex Falso Quodlibet from the discussion following 1.23.3; its status as contrariety principle for consequence relations was in effect noted immediately before 6.31.13. (It results from applying the method of ‘arbitrary consequences’ to the corresponding gcr formulation, just as (Subc)# is obtained by the method of ‘common consequences’ from the corresponding gcr condition, as in 3.13.) Part (vi) of 8.11.2 shows that the former condition can be weakened to (Peirce)¬ (i.e., (Peirce)# with # as ¬), the term “weakened” being appropriate here by 8.11.2(iii). (8.11.2(vi) should be compared with 7.21.7, p. 1060, in this respect.) With or without this weakening, the rules corresponding to conditions (Peirce)¬ and (Subc)¬ (i.e., put “” for “” in the above formulations) count as pure and simple – albeit oblique – rules in Dummett’s sense (see 4.14), so we see that such discussions as the following, from Michael [1999], p. 368, oversimplify somewhat: Similarly double negation elimination cannot be proven with pure and simple rules for negation. In the sequent framework this corresponds to the fact that when we restrict our sequents to a single sentence on the right hand side, the corresponding sequent cannot be proved if the rules are pure and simple.

Our sequent-to-sequent rules may not be sequent calculus rules, but the first sentence of this passage seems wrongly to suggest that ¬¬pp cannot be proved using them and the standard structural rules. However, if oblique rules are disallowed and primitive connectives are chosen appropriately – here ⊥ fares better than ¬ – then the sentiment just cited from Michael can indeed be vindicated, as is shown in Humberstone and Makinson [forthcoming]. The condition (RAA)# amounts, for the case of # = ¬, to the admissibility of the rule called Purified (RAA) in 4.14 (and at the start of 8.13 below). The label “(Peirce)¬ ” is intended to recall its implicational prototype (Peirce), or more explicitly (Peirce)# (with # as →) from 7.21, since it would be a special case of the latter rule if ¬A were understood as A → ⊥, a theme to which we take up in 8.12 and more fully in §8.3. For the moment we return to the Set-Set example that led to parts (i) and (ii) of 8.11.2. A reflection of 8.11.2(ii) for the case of interpreted languages is that a statement has nothing that can be called its contrary or its subcontrary, whereas – reflecting 8.11.2(i) – there is such a thing as the contradictory of a given statement. Geach [1969] urges this as a consideration against efforts, as in McCall [1967a], to investigate the logic of contrariety by adding (to a logic according to which ¬ already behaves ¬-classically) a new singulary contrary-forming operator on formulas. However, there is of course no formal obstacle to the study of a gcr (¬-classical or otherwise) satisfying the analogue of the earlier (1) for some new connective in place of ¬ (as remarked in note 6 of Humberstone [1986a]). Its consistent valuations are thereby subjected to the corresponding analogue of (1) . A more interesting logical treatment would no doubt impose a requirement

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of congruentiality (3.31) on the new connective, so that it represented an operation on the propositions (in the sense of -equivalence classes) expressed by formulas, and described the behaviour of an arbitrary contrary to a proposition. An even stronger move would be to add a requirement of extensionality for the envisaged (or, at least, for the new connective according to ). This would give Log(VN ∪ VF ), understanding this notation as in 3.22.3 (p. 450), 3.22.6–7, and means treating the envisaged connective as a hybrid (in the sense of 3.24, p. 461) of negation and the (1-ary) constant-false connectives. Similar points hold for (2), and subcontrariety. Remarks 8.11.3(i) The above description of the fact that a statement has no unique contrary as a reflection of 8.11.2(i) is an expository oversimplification. One might hold, w.r.t. any given notion of necessity (e.g., metaphysical necessity) that a statement has a unique necessitation, even though principles appropriate to the logic of that notion (in the most familiar logical frameworks) fail to characterize it uniquely (4.36). (ii ) Geach [1969] is certainly right to criticize McCall’s cavalier talk of “the” contrary of a proposition, in the informal introduction to McCall [1967a]. (The contrary of A is represented by RA.) McCall seems to recognise the prospect of uniqueness difficulties in this passage from p. 122: It is, however, the idea of contraries existing in pairs that interests us here. This idea implies that if q is the contrary of p, q must be the sole contrary of p. In that case p would be the sole contrary of q, in other words the contrary of the contrary of p. In symbols, EpRRp.

(Translated out of Polish notation, this last formula is p ↔ RRp.) This is certainly confused. There is no difference (except in emphasis) between saying “the contrary of p” and “the sole contrary of p” and from either description’s applying to q, nothing follows about the value of this would-be “contrary of” function for q as argument. Digression. McCall [1967a] goes on from the somewhat unpropitious motivating remarks just quoted to consider a suitable modal logic in which “R” would amount to (what we are accustomed to writing as) “¬”, and is led by his conditions on how a contrary-forming operator should behave into some novel territory, which we shall not survey here. We can give some idea of the need for novelties, however, by noting that the conditions that RA should have ¬A as a consequence and that A and RRA should be equivalent, mean that the corresponding modal principles A → A (the T schema) and A ↔ A (since written with the negations explicit the left-hand side of the latter is ¬¬A) will need both to be present, and the only normal modal logic answering to these two demands is the system KT! in which A and A are equivalent for all A, and hence R is after all no stronger than ¬. §2 of Zeman [1968] also undertakes the investigation of a singulary operator he calls ‘strong negation’ and again writes as “R”, and likewise looks at matters from a modal ‘R as impossibility’ point of view, considering what becomes of this notion in KT, S4, and some other normal modal logics. (A different use of the phrase ‘strong negation’ will appear in 8.23.) More recently, some have considered various versions of negation not


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so much via languages in which is present, or with a view to rendering it definable, as just directly interpreted in terms of the accessibility relations of Kripke models: the negation of A as true at a point just in case A is false at all accessible points. By analogy with the terminology of strict implication, it is natural to speak of strict negation in connection with such work – of Došen [1986a] and Lock [1988] are examples. Perhaps the first occurrence of this terminology is in Hacking [1963]: see p. 68. (The introduction of the terminology on Hacking’s p. 52 has a misprint: “ ∼ ∼” should read “ ∼ ”.) The phrase is used in an unrelated sense in some work in many-valued logic: see Definition 5.2.1 of Gottwald [2001], which also introduces a sense of ‘strong negation’ different from that mentioned above or that featuring in 8.23 below. Various much discussed principles governing negation (the law of excluded middle, double negation equivalence, non-contradiction, etc.) can then be considered in terms of the conditions on the accessibility relation they correspond to. End of Digression. The projects sketched above for the logic of ‘contrary negation’ and ‘subcontrary negation’, while perfectly feasible, seem less interesting as alternatives to contradictory negation in lacking the latter’s unique characterization; certainly for those who see uniqueness of this kind as a necessary condition for a connective to count as a bona fide logical constant, and want this description to apply to anything deserving to be called negation, we should look elsewhere. A recurrent theme in this chapter will be the success or failure of rules governing something going by the name of negation to characterize that connective uniquely. For example, we shall see that the rules of an accepted proof system MNat for Minimal Logic (8.32) do not characterize ¬ uniquely (8.32.7). Returning to the above contrary/subcontrary theme, and beginning with the former, let us note that we can boost up the idea of a contrary-forming negation given by (1) into something uniquely characterized if we think of the negation of a formula as its deductively weakest contrary (cf. 4.14.1, 2 – p. 526 – and surrounding discussion). We rewrite (1) in rule form as (1a) and give this ‘weakest’ property by (1b): (1a)

A, ¬A

(1b)

A, B B ¬A

Exercise 8.11.4 (i) Show that (1a) and (1b) uniquely characterize ¬. (ii ) Suppose we start with a language containing →, and the rules (→I), (→E), alongside the structural rules – the implicational fragment of IL (in Set-Fmla) in other words, and introduce a new singulary connective R governed by a rule like (1a) with R in place of ¬. We are saying that RA and A are always contraries, that is. Now introduce ¬ by the definition of ¬A as A → RA, and show that as thus defined, ¬ satisfies (1a) itself, as well as the rule (1c) of the following paragraph. (This means that we have ended up with intuitionistic negation – a ‘weakest contrary’ forming operation – by a different route. This idea is from §1 of Zeman [1968], §2 of which was touched on in the Digression above. For this weakest contrary characterisation, see the discussion after (1c) below.) We should observe that the rules (1a) and (1b) are sequent-to-sequent rules of the logical framework Set-Set, and also of the framework Set-Fmla0 . Con-


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sidering them in the latter context first, we note that there is a failure to meet the condition (arguably a desideratum) of generality in respect of side formulas from 4.14. The formula-variable “B” should be replaced, to obtain this generality, by a set-variable: (1c)

Γ, A

Γ ¬A Now (1c) is just the sequent calculus rule (¬ Right), as applied in Set-Fmla0 , and (1a) is derivable from (¬ Left) in that framework: so what we have now ended up with is a pair of rules for intuitionistic negation. (See the end of 2.32; 8.11.4 should now be compared with 0.21.5(ii), p. 22: algebraically viewed, our ‘weakest contrary’ negation is pseudocomplementation.) Later (8.22), we shall consider a dual conception of negation, not as forming the weakest contrary, but as forming the strongest subcontrary, of what is negated. To judge by the late appearance on the scene of dual intuitionistic negation, the latter is a notion somewhat less natural psychologically than the former. Even in classical logic, in which the two notions coincide, the ‘weakest contrary’ characterization is the one that comes first to mind, as we see in the following quotation from Brandom [1994], a work which takes no special interest in non-classical logic. This sentence appears on p. 115; Brandom uses “incompatible” as a (relational) noun meaning what we mean by contrary: “The formal negation of a claim is constructed as its minimal incompatible, the claim that is entailed by each one of the claims incompatible with the claim of which it is the negation”. Digression. In 4.14 we saw some ideas from Peacocke [1987] in which the idea of an inference’s being found primitively compelling inference was used to characterize of various connectives. We did not touch on another notion used to similar effect in certain other cases, pre-eminently that of negation. This is the idea of a pair of statements being found “primitively incompatible”, in the sense that a reaction of incompatibility is not felt to call for justification or explanation in terms of anything more fundamental. Peacocke [1987] sketches a reason for thinking that if we conceive the negation of a statement as the weakest statement primitively incompatible with the given statement, which we shall here simply amount to the statement’s weakest contrary, then this conception forces us to acknowledge that a statement follows from its double negation. The following quotation of the argument is from a later publication clarifying it (Peacocke [1993], p.177). Peacocke is notating the weakest contrary of a statement A by OA, and asks us to suppose we had a case in which the transition from OOA to A did not preserve truth: In this case we would have that OOA is true and A is not true. Since OOA is incompatible with OA, OA must not be true in the given case. But if A and OA are both not true, OA is not the weakest proposition incompatible with A, for being the weakest proposition incompatible with A entails that OA is true in any case in which A is not. Since by hypothesis OA is the weakest such proposition, there can be no case in which the analogue of classical negation elimination fails.

Since, as we have seen, the idea of—as we have been understanding it—a weakest contrary of (“proposition incompatible with”) a given statement is captured


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by the rules governing ¬ in IL (in INat, say, to be specific) and from these rules one cannot prove for arbitrary A, the sequent ¬¬A A, Peacocke’s understanding of the “weakest” candidate satisfying a certain condition must be different from our own. In Peacocke [1987] the idea was cashed out in terms of truth-functions, as we saw with the partial truth-tables in 4.14.4 and 4.14.5 (pp. 531, 532). This is clearly a different matter from the inferential strength notion of weakestness operative in our discussion. That idea is not especially visible in the version of the argument quoted above, however, raising the question of what might be meant by “weakest” here. One difficulty is over the “not” italicized in the following re-quotation of a crucial passage: “being the weakest proposition incompatible with A entails that OA is true in any case in which A is not”. We can make precise sense of this by imagining what properties the logic of the metalanguage in which it is cast is to taken as conferring on not, as well as on the conditional buried under the relative clause construction, which presumably means: if A is not true, then OA is true. Two prominent possibilities for construing the metalinguistic discussion are provided by CL and IL. In either case, we construe the latter conditional as amounting to ¬A → OA, with different logical properties in the two cases. Taking CL as our guide, the latter is equivalent to A ∨ OA, an assumption that an adherent of IL is unlikely to find congenial if OA is any kind of contrary-forming operator. If, attempting to match this apparently neutral talk of weakestness with a logical neutrality by taking the weaker logic IL as our guide, a problem arises – as noted in Wright [1993] – at an earlier stage, namely in the first line of the inset passage above, where the invalidity of a one-premiss is claimed to entail the possibility of a case in which the premise is true and the conclusion not true, since on our current understanding of this not, (¬¬E) is straightforwardly a counterexample to the claim. There is, according to the intuitionist, no valid argument from ¬¬A to A, but equally not possibility of a case in which ¬¬A is true and A is not true, where the latter is taken to mean that ¬A is true. End of Digression. Turning now to Set-Set, we note that (1c) is in this framework still not fully general in respect of side formulas; we need (1d)

Γ, A Δ Γ ¬A, Δ

This is the sequent calculus rule (¬ Right) for Set-Set, given in 1.27. Given (R), (M) and (T), this rule is interderivable with the original (2), so we are back where we started, with classical negation. We can make a similar investigation with subcontrariety, boosting (2), here rewritten as (2a), with a uniqueness-securing rule to the effect that the negation of A is the strongest subcontrary of A: (2a)

A, ¬A

(2b)

A, B ¬A B

Again, we can consider fully general versions of (2b) in various frameworks. In Set-Set, we get the rule (¬ Left) from 1.27, and are back with classical negation. Obviously Set-Fmla0 would be an unpromising place to look, being inhospitable to the premiss-sequents of (2b). But a mirror image of this framework, Fmla0 -Set, in which any finite set of formulas may appear on the right of , but only a set with no formulas or one formula on the left. This gives the logic

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of dual intuitionistic negation, to be presented in 8.22. From the perspective of this logic, classical logic appears as making a the gratuitous further assumption that the strongest subcontrary of a statement is also a contrary of that statement, just as, if we start with IL, classical logic takes the further gratuitous step of supposing that the weakest contrary of a statement must be also be a subcontrary of that statement. The talk of gratuity here may be understood informally in terms of lack of justification for the assumption in question, or more formally in terms of providing negation with inferential powers which are ‘stronger than needed’ (4.32) to characterize this connective uniquely. For the remainder of this subsection, we consider some aspects of the behaviour of classical and intuitionistic negation in a language without other connectives, beginning with the classical case. Let ¬ be the gcr determined by the class of all ¬-boolean valuations, on the language with ¬ as sole connective. (We could allow additional connectives without this making a difference to anything, since the only restriction we impose on valuations is that they be ¬-boolean.) We shall see (8.11.6) that the relation ¬ is symmetric. The is-true-on relational connection between the language and the class of boolean valuations has negative objects on the left (see discussion preceding 0.14.3), since for a formula A, the formula ¬A is the corresponding negative object (or, more accurately, is a corresponding negative object: the formula distinct formula ¬¬¬A would do just as well, since it is true on exactly the ¬-boolean valuations that A is not true on). This connection also has negative objects (‘complementary valuations’) on the right; for v ∈ BV ¬ , we denote by v¯ the valuation assigning T to those formulas assigned F by v (and vice versa, we may add redundantly). This is the negative object corresponding to v. (And here, we can say ‘the’ negative object: v¯ is, for a given v, the unique valuation verifying exactly the formulas not verified by v.) Exercise 8.11.5 Verify that if v is ¬-boolean, then so is v¯, as just defined. We are now ready for the promised result: Observation 8.11.6 If Γ ¬ Δ, then Δ ¬ Γ. Proof. Suppose Δ ¬ Γ. Since ¬ = Log(BV ¬ ), there exists v ∈ BV ¬ with v(Δ) = T, v(Γ) = F. Thus, by the way v¯ is defined, v¯(Γ) = T and v¯(Δ) = F. But (8.11.5) v¯ ∈ BV ¬ . Therefore Γ ¬ Δ. In the terminology introduced just before 1.19.3 (p. 93), this means that ¬ is its own dual. For the following Corollary, we use the ‘unselective contraposition’ property of classical negation; ¬Γ is of course {¬C | C ∈ Γ}. Corollary 8.11.7 If Γ ¬ Δ, then ¬Γ ¬ ¬Δ. Note that whereas analogous conditions obtained for Log(BV ↔ ) in 7.31.2 and 7.31.3 (p. 1132) counted only as ‘generalized’ rule-like conditions because of their disjunctive consequents, 8.11.6 and 8.11.7 record the admissibility of straightforward sequent-to-sequent rules. It is worth seeing where the argument for 8.11.6 breaks down if we try to venture outside the negation fragment of CL. Such a result clearly fails if we


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consider, for example, the {∧, ¬}-fragment, since p ∧ q CL p though not conversely. The problem with running through the above argument in this setting is that although v¯ is ¬-boolean whenever v is, the assumption that v is {∧, ¬}-boolean does not secure the conclusion that so also is v¯: specifically, v¯ will fail to be ∧-boolean. We worked through the complementary valuations v¯ to obtain 8.11.6 and 8.11.7 because of the inherent interest of operations Galois-dual to boolean sentence connectives—see the discussion following 7.31.4—but of course a more purely syntactic argument would be available once we have a proof system known to be sound and complete w.r.t. the class of ¬-boolean valuations. The simplest (in Set-Set) for such purposes would be that based on (1) and (2) – or the corresponding sequent-schemata (with “” replacing “”)– from the beginning of the present subsection. Coupled with the structural rules, one easily sees, by induction on the length of proof, that 8.11.6 and 8.11.7 hold, since ¬ is the gcr associated with this proof system. The latter technique is probably the simplest available for establishing something along the lines of 8.11.7 for the pure negation fragment of IL. Because Set-Set has less intrinsic significance for IL than it does for CL (see 6.43.8: p. 899), we consider the Set-Fmla version. Accordingly, let the consequence relation IL¬ be the restriction of IL to formulas constructed from propositional variables with the aid of ¬. A simple adaptation of 2.32.6–2.32.8 shows that this is the consequence relation associated with the proof system with, alongside the structural rules, (EFQ) and purified (RAA); the latter rule was introduced in 4.14. An easy argument by induction on the length of proofs then gives: Observation 8.11.8 If Γ IL¬ A, then ¬Γ IL¬ ¬A. A proof along the lines suggested is probably much easier than one in terms of valuations. The latter would involve starting with a collection V of valuations satisfying the condition (*): (*) For all u ∈ V, all formulas A: u(¬A) = T iff for all v ∈ V, if u v, then v (A) = F. in which “u v” (alternatively: “v u”) means (as usual) that every formula true on u is true on v. Such a V is obtainable by taking the characteristic functions of a suitable canonical Kripke model for IL or the negation fragment thereof. One would then explore the possibility of adding in additional valuations v ∼ for v ∈ V (playing a role analogous to that played in the boolean case by to v¯), defined by: (∼) For all u ∈ V, all formulas A: u∼ (A) = T iff for all v ∈ V , if u v, then v(A) = F. If V satisfies (*), then (∼) dictates that u∼ (A) = u(¬A), for every formula A. We should have to have our eventual extension of V closed under the operation (∼), which can be achieved by taking it as V + = V ∪ {u∼ | u ∈ V } ∪ {u∼ ∼ | u ∈ V }, since a third application of this operation brings nothing new (u∼ ∼ ∼ = u∼ ). One could then show without difficulty that if V satisfies (*), so does V + , but to be able to use this to conclude that 8.11.7 holds, one would also need to show


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that (∼) preserves consistency with IL¬ , and it is not clear how to do this without using 8.11.7 itself.

8.12

Negative Miscellany

In this section we shall discuss the following three topics: the tendency to give an inappropriately privileged status to negation; the use of what are called signed formulas; and three different ways of treating a Falsum (or Absurdum) symbol (we can’t quite say “connective” here, since that does not apply in all treatments, though it certainly does for our extended presentation of this topic in §8.3). Under the heading of inappropriate privilege what we have in mind is the tendency for people to forget that a negation connective – whether there is only one (then typically represented by ¬) in the language or whether there are (as in 8.22, 8.23) several deserving that description – is just that: a connective. Dummett [1977] and Hart [1982] are close to overlooking this point when they use negation to deal with falsity in encoding as Set-Fmla principles what we call (3.11) the various determinants of a truth-function. In our earlier discussion, we gave a procedure for obtaining the condition on a gcr induced by such a determinant. This can immediately be rewritten as a zero-premiss rule (sequentschema, that is) in Set-Set, and the authors mentioned, concerned to move to Set-Fmla, deal with multiplicity on the right by negating all but one of the formulas there and placing the negated formulas on the left. In Hart’s discussion (Hart [1982], p. 131) some rules for negation are included so that the result of doing this does not depend on which formula is allowed to remain on the right, rules he describes by saying that while “(n)othing in our bare notation forces us to, but if we insist that ‘¬’ be interpreted by a truth-function”, then the truthfunction N (= ¬b ) is the only candidate. (Actually, Hart uses “ ∼” rather than “¬”.) Certainly, nothing in the notation forces this – and the Strong Claim, as we called it in 1.14 (see also 3.13.13, p. 394), does not hold for ¬. The discussion should remain in Set-Set, or move to the conditions on a consequence relation (essentially Set-Fmla rules) induced by a determinant in the sense of 3.13, rather than giving a special status to negation, so that the latter pokes its nose into the formulation of determinant-capturing principles for other connectives, “impurifying” those principles. Passing now to Dummett’s discussion, we read the following from p. 16 (of Dummett [1977]): In general, the truth-tables for the connectives are correct intuitionistically in the following sense. Construe each entry as a rule of inference with two premisses: one premiss is A if A is assigned the value True, and ¬A if A is assigned the value false, and the other premiss is, likewise, either B or ¬B; the conclusion is either the complex statement or its negation according as it receives the value True or False. Then all such inferences are intuitionistically correct.

Dummett gives as examples, in the case of ∧ rules such as the two premiss rule of Fmla which takes us from A and B to A ∧ B, corresponding to the “first” determinant T, T, T of the truth-function ∧b . In the spirit of our general discussion and Dummett’s intentions (since he is clearly thinking of the A and


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B here as dependent on arbitrary assumptions, on whose union the conclusion depends) we should represent this either by the two-premiss rule (∧I), or else by its zero-premiss ‘horizontal’ formulation: A, B A ∧ B. Let us opt for the latter representation, so that we can put Dummett’s point most simply by saying that each of the four schemata the above recipe leads to is IL-provable: A, B A ∧ B; A, ¬B ¬(A ∧ B); ¬A, B ¬(A ∧ B); ¬A, ¬B ¬(A ∧ B). Similarly, as Dummett further illustrates, the recipe yields the following in the case of →: A, B A → B; A, ¬B ¬(A → B); ¬A, B (A → B); ¬A, ¬B A → B. And finally (still Dummett [1977], p. 16) he concludes with the remark: “Of course, the classical assumption that the various assignments exhaust all possibilities is not intuitionistically correct.” (The simplest way to note that sequents instantiating these schemata are all intuitionistically provable is to rewrite them as representative instances, with distinct propositional variables in place of distinct schematic letters, and appeal to Observation 8.21.1 – p. 1215 – especially as reformulated in the Digression which follows it in our exposition. We are dealing with the two-variable version of the example treated there as A(p).) Now, our earlier complaint about the impurity of formulation resulting from allowing ¬ to intrude into what are supposed to be principles about (in the above examples) ∧ and → applies here also. But it is interesting, further, to see that in the inset quotation from Dummett given above, we have this talk of “two premisses”: why “two”? Not all connectives are binary connectives! What do we get if we apply the corresponding recipe in the case of ¬ itself, a particularly salient non-binary connective in this discussion? For the determinant T, F of ¬b we get A ¬¬A, while for F, T , what we get is ¬A ¬A. By focussing on the latter, we can see that there is something seriously wrong with the idea that these sequent-schemata are telling us anything about truth-functionality. It is a substantive constraint on a valuation v that we should have, for every formula A, v(¬A) = T whenever v(A) = F. But it is no constraint on a consequence relation that for every formula A, ¬A should be a consequence of ¬A. (We are not claiming here that there is any technical error—or false claim—in Dummett’s discussion. For further variations on the “reading off a logic from a set of truthtables” idea, where ¬ is given a similarly special role, see Tennant [1989]. See Sylvan [1992b] for further discussion of Tennant’s paper.) We pass from the topic of inappropriate privileges for negation to what may be seen as a response to that problem. The idea is to start with a language in the usual sense (1.11) and then introduce for every formula A of that language two new things which will be called signed formulas, denoted (say) by [+]A and [−]A. Signed formulas are not, then, a kind of formulas, but obtained from formulas by what we call positive and negative signing, respectively. Some authors write the signs as “t” and “f”, or “T” and “F”, but since that makes signed formulas look like semantic statements in the metalanguage we do not follow suit. Instead, following Bendall [1978] and Smiley [1996], we regard the appending of the signs [+] and [−] as indicating assent to and dissent from what follows. (Alternatively, they can be thought of as explicit indicators of illocutionary force, the force being that of assertion in the former case and of denial or ‘rejection’ in the latter.) We can construct a natural deduction system in which what are proved are not sequents of Set-Fmla but what we may call


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signed sequents, meaning that on the left of the separator () appears a finite set of signed formulas, and on the right a single signed formula. Define a function c from signed formulas to the set {1, 0} to be a correctness function just in case for no signed formula α do we have c(α) = c(¯ α), where α ¯ is the result of changing the sign on α (from [+] to [−] or from [−] to [+]); we reserve “α”, “β”, “γ”, to range over signed formulas and “Γs ”, “Δs ”, etc., over sets of signed formulas. A signed sequent Γs β holds on a correctness function c if and only if it is not the case that c(α) = 1 for all α ∈ Γs and c(β) = 0. The set of signed sequents determined by a class C of correctness functions is the set of sequents holding on each c ∈ C. Any such set of sequents is evidently closed under the following rules: (R)s (T)s (±)

α α

(M)s

Γs α

Γs β Γs , α β

Γs , α β Γs β

Γs , α β

Γs , α β¯

Γs α ¯

and in fact any set of signed sequents closed under these rules is determined by some class of correctness functions. Actually, this assertion can be simplified: Exercise 8.12.1 Show that the rule (T)s is derivable from the other three rules above. Given a valuation v for a language L, we have the function c, easily seen to be a correctness function, on the set of signed formulas [+]A, [−]A, for A ∈ L, induced by v in the sense that c([+]A) = 1 iff v(A) = T

c([−]A) = 1 iff v(A) = F

and we can uniquely recover v from a c behaving as these two conditions dictate (for all A ∈ L), so terminology can be easily transferred back and forth between classes of valuations and classes of correctness functions. In particular, we can call c a boolean (or more selectively, #-boolean, for various connectives #) correctness function just in case the unique v which induces c is boolean (#-boolean). Say for definiteness, we fix L as the language with primitive connectives ¬, ∧, ∨, →. A proof system for the set of signed sequents which are valid on every boolean correctness function will be a version of CL in this ‘signed sequent’ framework. Many such proof systems are possible. (See Bendall [1978] for two, Smiley [1996] for a third, and Humberstone [2000c] for a comparative discussion, filling out the present brief survey.) One can be obtained by supplementing the structural rules (R)s , (M)s , (±) above by some zero-premiss rules for the connectives. Example 8.12.2 For ∧ it is enough to take the familiar looking principles: [+]A ∧ B [+]A; [+]A ∧ B [+]B; [+]A, [+]B [+]A ∧ B, with the following for ¬:


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[+]¬A [−]A [−]A [+]¬A. (Note that we do not write, for example “[+](A ∧ B)”, as though we had to prevent ‘[+]A ∧ B’ from being read as “([+]A) ∧ B”. The latter is not a signed formula at all. The signs (or force indicators) cannot occur in the scope of connectives (or in the scope of other signs). These last two principles can be thought of as telling us that asserting notA is tantamount to denying A. Such an equivalence may well make us think that denial need not be acknowledged in its own right, since we always have a reformulation available in terms of assertion; vigorous rebuttals of this view (which was defended by Frege) may be found in Price [1994] and Smiley [1996]. Exercise 8.12.3 (i) Show that, given the structural rules already mentioned, the first two rules listed for ∧ in 8.12.2 are interderivable with the following pair: [−]A [−]A ∧ B; [−]B [−]A ∧ B. (ii ) Show that again given those structural rules, the second of the rules listed for ¬ in 8.12.2 is interderivable with: [−]¬A [+]A. (iii) Show that the following rule preserves for an arbitrary ∨-boolean correctness function c, the property of holding on c: Γs [+]A ∨ B

Γs , [+]A γ

Γs , [+]B γ

Γs γ (iv ) The above rule is just a version of the usual (∨E) rule suited for signed sequents. (The “Γs ” appearing in the second and third premisssequents could of course be replaced, respectively, by “Δs ” and “Θs ”, if desired, and in the conclusion by listing all three set variables.) Show that the rule in (iii) is interderivable, given the structural rules, with the zero-premiss rule: [+]A ∨ B, [−]A [+]B. (Suggestion: First derive from the given structural rules a rule allowing passage from Γs , α β and Γs , α ¯ β to Γs β, and a zero-premiss rule α, α ¯ β. These two structural rules will assist in the derivation of the inset rule above from the rule in (iii) and the converse derivation.) Part (iv) of the above exercise together with the fact that the structural rules preserving holding on an arbitrarily selected correctness function means that we have a conspicuous contrast with 1.26.3 where we saw that the class of ∨-boolean valuations was not sequent-definable: Observation 8.12.4 The class of ∨-boolean correctness functions is signedsequent-definable, in the sense that there is a class of signed sequents holding on precisely the ∨-boolean correctness functions. Proof. Where L is the language under consideration, about which the only assumption that we make is that ∨ is amongst its connectives, take Σ = {[+]A [+]A ∨ B | A, B ∈ L} ∪ {[+]B [+]A ∨ B | A, B ∈ L} ∪ {[+]A ∨ B, [−]A [+]B | A, B ∈ L}. Then c is ∨-boolean if and only if every signed sequent in Σ holds on c.


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Exercise 8.12.5 Show that, again by contrast with ordinary Set-Fmla, in the signed sequent framework the class of all ¬-boolean correctness functions is sequent-definable. The reprieve indicated by 8.12.4–5 from the failure for Set-Fmla of the Strong Claim (1.14) in the cases of disjunction and negation illustrates why we have no need of a signed sequent framework analogous to Set-Set, with more than one signed formula on the right. It is not hard to see that we can get the same effect as this would provide by leaving one signed formula behind and moving the rest across to the other side of the “” while changing their signs. In fact, that is what is going on with the principle in 8.12.3(iv); take a Set-Set sequent such as p ∨ q p, q, and positively sign everything: [+]p ∨ q [+]p, [+]q. To avoid multiplicity on the right, pick either of the two signed formulas there and move it over while changing signs, getting either [+]p ∨ q, [−]p [+]q or (interderivably, given the structural rules) [+]p ∨ q, [−]q [+]p. Quite independently of any illumination the use of signed formulas may throw on negation via the above equivalence between [+]¬A and [−]A, Smiley [1996] is impressed by the theoretical benefits we have just been discussing – the ability of the signed sequent framework to force the valuations (inducing correctness functions) on which sequents hold to be #-boolean for various previously intractable cases of # (such as ∨ and ¬). Bendall [1978] cites a different, though related, aspect of the signed sequent framework (related because essentially this is again a matter of being able to simulate Set-Set) and that is the availability of proof systems with the separation property (4.21), even for CL, where there are difficulties for separable proof systems with pure and simple rules. It is characteristically the need to bring ¬ into the proofs of tautologous ¬-free sequents that spoils the picture (as in Nat), and it turns out that all such intrusions of negation can be replaced by (or, less charitably, encoded as) negative signings. (For further discussion, see Bendall [1978], Humberstone [2000c].) We return briefly to an aspect of the use of signed formulas below, in our discussion of different ways of interpreting a Falsum symbol. Before embarking on that discussion, however, we close the present discussion with an interesting observation by Bendall on the idea that negation should in some way be interpreted as a sign of dissent (negative judgment) or of denial (rejection). The idea comes from Bendall [1979] and represents a cruder version of what we have been discussing up to this point, though it is a point well worth noting. Our discussion has been of the formal side of a particular proposal whose philosophical underpinnings are roughly as follows: we have a primitive understanding of what it is to reject a proposition which is on all fours with our understanding of what it is to assert a proposition. Then we introduce into our language a 1-ary connective ¬ which will allow us to convert (a sentence expressing) one proposition into (one expressing) another, in such a way that asserting the latter will be taken to amount to rejecting the former. (This is the [+]¬A, [−]A equivalence from 8.12.2 again.) That means that we can now embed the newly available propositional content (¬A) within the scope of other connectives, and so on, something we were formerly unable to do. Of course there are numerous questions about this philosophical story, such as, in particular, what features of rejection make it an available possibility to


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introduce negation as described. We refer the interested reader to the discussion in Price [1990] and [1994]. The point of going through the above background is merely to draw attention to the fact that although negation is introduced via a sign of rejective force or ‘negative judgment’ (namely ‘[−]’), the story at no point construes negation (¬) itself as such a sign. (The phrase ‘rejective negation’, which is associated with approaches inspired by this idea, has an unrelated use in Martin [1989], Pollard and Martin [1996]. Nor should either of these uses be confused with rejection à la Łukasiewicz: see the top paragraph of p. 377 in Humberstone [2000c].) There is a very obvious reason for not doing so, namely, that negation can occur embedded within its own scope as well as within the scope of other connectives, and we have been assuming – without wanting to go into the merits and demerits of this assumption here – that force indicators are inherently unembeddable. (The last paper cited also entertains, not very sympathetically, force-linked binary propositional attitudes/speech-acts standing to conjunction and disjunction in the way that rejection stands to negation, under the names ambi-assertion and alterjection; a more sympathetic treatment of this idea appears in note 24, p. 57, of Richard [2008].) With that assumption in place, however, we may consider a subset of the set of all formulas of the language L introduced above, which we will call L0 ; the formulas in L0 are precisely those formulas of L in which there is no occurrence of ¬ in the scope of any occurrence of any other occurrence of a connective (whether of ¬ itself or of ∧, ∨, or →). The formulas in L0 are thus of two kinds: (i) those containing no occurrence of ¬, and (ii) those containing exactly one occurrence of ¬, which are of the form ¬B where B is of kind (i). Note that L0 is not a language in the technical sense of 1.11, but that cannot prevent us from noticing various things about L0 , including in the first place that we have so defined membership in L0 that occurrences of ¬ could be construed as occurrences of a force indicator without violating the embargo on embeddings. The formulas A of kind (i) just distinguished above could be understood as signed formulas [+]A, while those A of kind (ii), of the form ¬B (with B ¬free) could be understood as signed formulas [−]B. That is, having restricted its possibilities of occurrence so drastically, what we started off thinking of as a normal sentence connective can now be re-construed as a force indicator. Against this background, Bendall [1979] makes the following: Observation 8.12.6 For every formula C of L there is a formula C0 of L0 which is CL-equivalent to C. Proof. By induction on the complexity of C.

Remarks 8.12.7(i) Note that while there is a familiar process of converting formulas into what is sometimes called negation normal form, meaning that all occurrences of negation immediately attach to atomic formulas, and can be driven there by CL-equivalences such as De Morgan’s and Double Negation laws, Bendall’s procedure in effect operates these transformations in reverse, pulling all occurrences of negation out to the front of the formula (and then cancelling them in pairs, leaving either none or one left standing).

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CHAPTER 8. “NOT” (ii ) The language L had to be chosen quite carefully for the inductive part of the proof of 8.12.6 to go through. If we had allowed only the connectives ¬, ∧, and ∨ (say), without admitting →, then the proof would not go through and the result would not hold: there would be no L0 equivalent for ¬p ∨ q or for ¬p ∧ q, for instance.

The second of the above remarks shows that selecting any old functionally complete set of connectives will not guarantee an analogue to 8.12.6 (and the same holds for the claim that every formula has a negation normal form classically equivalent to it). If we think about this in the context, instead, of IL rather than CL, we should therefore not immediately be put off finding a ‘Bendall normal form’ (L0 -style equivalent) for every formula. We simply allow, as we do for CL in having so many connectives, some connectives additional to the familiar primitives (which are those of L, in fact, with the possible addition of ↔ or replacement of ¬ by ⊥), which makes for a certain redundancy from the point of view of expressive completeness but which allows the result to be obtained. For example, to deal with the stacking up of iterated initial negations we shall need to have available—as a new primitive—a connective, O, say, which behaves exactly as though it had been defined by: OA = ¬¬A. (Cf. Došen [1984]. This will deal with any number of pairs of “¬”s in one application, because of the Law of Triple Negation: 2.32.1, p. 304.) While we should not be put off immediately, however, we should be put off eventually. For consider the formula ¬p ∨ ¬q (or more generally, formulas of the form ¬A ∨ ¬B). We want to find, in our language-with-additional-primitives for IL, a formula containing either no occurrences of ¬ or at most one initial occurrence of ¬. Consider the first possibility first. We have some formula A(p, q), say, of the enriched language, constructed out of p and q and free of ¬, for which A(p, q) and ¬p∨¬q are equivalent (according to a version of IL couched in this language). But that means we can simply obtain the effect of negation directly, however deeply embedded, by putting A(B, B) for ¬B. (Note the contrast with the classical case in Bendall’s treatment: negation is certainly not definable in terms of the remaining connectives used, namely ∧, ∨, and →.) So let us turn to the second possibility. We have again a ¬-free formula A(p, q), and this time it is ¬A(p, q), rather than A(p, q) itself, which is equivalent to ¬p ∨ ¬q. We again invoke the Law of Triple Negation to notice that such an equivalence would imply (since ¬¬¬A(p, q) IL ¬A(p, q)) that ¬¬(¬p ∨ ¬q) IL ¬p ∨ ¬q, which is not the case. Thus even providing additional primitives to facilitate an extraction-of-¬ procedure in the style of Bendall [1979], IL does not behave as CL does. Exercise 8.12.8 (i) Why could the above argument involving ¬p ∨¬q not have been run with ¬p ∧¬q instead? (ii ) What is the weakest intermediate logic in which the implication, just noted to fail for IL, from ¬¬(¬p ∨¬q) to ¬p ∨ ¬q, goes through? (I.e., characterize this logic using the descriptive resources of 2.32 above.) In addressing this question, it will help to simplify the first formula by making use of Glivenko’s Theorem. The contrast between the resulting formula and the second formula above will occupy us throughout 8.24. For our final topic in this miscellany – except for some exercises at the end – we consider the idea of treating negation using a special constant for falsity,


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contradiction or absurdity, which we refer to as a Falsum symbol. (This goes somewhat against the preference for the abstract conception of languages described in 1.11, but seems the most convenient description. It does, however, oversimplify the situation in that it useful to continue to use the Falsum terminology in such contexts as that of Minimal Logic, for which the above semantic glosses would be inappropriate; see the following paragraph and §8.3.) In order to subsume the various incarnations of this idea to be compared here, we need to keep the formulation somewhat vague; in vague terms, the idea is that the derivability of this constant from a formula A is to suffice for us to infer ¬A. The comparison to be made is in respect of how well-integrated the envisaged Falsum symbol is to be amongst the formulas of the language under consideration. At the most fully integrated end of the spectrum, we have the option of treating the Falsum symbol as a formula in its own right – a sentential constant or, alternatively put, a zero-place connective. You can’t get much more fully integrated amongst the formulas of a language than by actually being one of those formulas. When discussing this option, as we shall be doing much more fully in §8.3, we write the constant in question as ⊥. If implication is present in the language, the idea sketched above is usually in this case executed by defining ¬A as A → ⊥, and more may (IL, CL) or may not (ML) be said about the logical behaviour of ⊥ other than it is just one of the formulas of the language (an atomic formula alongside the propositional variables). If from A together with other assumptions Γ we manage to derive ⊥, in a natural deduction proof for definiteness, then by an (→I) rule of the type featuring in Nat and INat, we obtain A → ⊥, which is what ¬A was defined to be on the current proposal. Indeed we may allow several constants ⊥1 , ⊥2 , . . . and either define ¬A as the disjunction of the A → ⊥i or take ¬ as primitive but behaving as though it had been so defined. Another possibility is to take ¬A as the implication with the disjunction of the ⊥i as consequent, rather than the disjunction of the implications just considered. We go into more detail on such possibilities in 8.33, here only noting their existence, and the fact that constants ⊥i playing this role have been variously described in the literature as: anti-axioms (Johansson [1953]), counteraxioms (Curry [1963], even ‘taboos’ (Ono [1966]). When, as in the version we are concerned with here, there is just one, and we write it as ⊥, the idea of treating the negation of A as A → ⊥ is not unlike the construction “If A then I’m a monkey’s uncle”, except that the crucial tacit Modus Tollens appealed to here – with “I’m not a monkey’s uncle” as the silent premiss – is not reconstructible if we are really using ⊥ to define ¬: the silent premiss simply becomes “If I’m a monkey’s uncle, then I’m a monkey’s uncle”. Similarly in the context of formal arithmetic, where ¬A is apt to be defined as A → 0 = 1, we seem to have lost the ability to express as a substantive truth the fact that 0 is not equal to 1. This point is made on p. 217 of Hossack [1990]. The whole idea, just entertained and briefly mentioned earlier in 3.16.3(iii) (on p. 427), of defining logical vocabulary in a particular (non-logical) theory is one not touched on in our discussion of defining connectives (3.16) though it is certainly worthy of attention. An even more dramatic example – cf. 3.15.4, 3.15.5(iii) (see p. 419), for the contrast with propositional IL (which extends to IL with quantifiers over individuals) – is the definability of disjunction within the same theory just referred to as formal arithmetic, and more precisely identified as Heyting Arithmetic, which comprises the intuitionistic consequences of the Peano axioms (suitably formulated, since we must remember that what

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are classically trivial reformulations give rise to intuitionistically non-equivalent versions). See Dummett [1977], p. 35 and – for a fuller discussion – Troelstra and van Dalen [1988], p. 127. For another example, see the end of section notes (‘Defining a connective in a theory’, p. 1211). However, Hossack’s point could perhaps equally well be made in terms of the purely logical definition of ¬A as A → ⊥. At any rate, that is enough for the moment about the fully integrated treatment of a Falsum symbol as one formula amongst others. There are two more options to be considered as to how well-integrated amongst the formulas our Falsum symbol should be, and both may be introduced by considering the treatment of such a symbol in various works by Neil Tennant, though we shall single out Tennant [1978] as representative, using the notation “Λ” from Tennant [1987a]. When at p. 29 of the former work he gives (p. 29) the usual inductive definition of the satisfaction of a formula by an assignment of objects to variables (since Tennant is dealing with predicate logic) relative to an interpretation (or model) and has inductive cases of ¬, ∧, ∨, and → (in our notation) but none for Λ. The latter only makes its appearance in a later chapter dealing with (natural deduction) proofs, its first mention being in connection with the rule (“¬-elimination”) which allows passage from a formula and its negation (depending on various assumptions) to Λ as conclusion (depending on those same assumptions). Similarly, there is an introduction rule which allows the inference of ¬A from some assumptions when from those assumptions together with A, Λ has been derived. This is like the procedure for obtaining ¬A on the “fully integrated Falsum” option described above, except that on the fully integrated option, having obtained ⊥ from A and Γ we derive A → ⊥ by (→I) from Γ, and regard this conditional as being the formula ¬A. The intermediate step here with → is what is conspicuously missing from Tennant’s treatment, and indeed there is no such formula as A → Λ: the special Falsum symbol never appears embedded within another formula. And that means – at least if languages are defined along anything like the lines of 1.11 – that Λ is not itself a formula. No wonder, then, that there is no mention of it in Tennant’s definitions or proofs which proceed by induction on the complexity of formulas. In Dummett’s ‘pure and simple’ terminology (see 4.14 above), Tennant [1979] criticizes the rules for the Sheffer stroke offered in Webb [1970] as not being simple, replacing them with rules simultaneously involving that connective and Λ. Thus if the latter were also taken as a connective – and a nullary connective is what it would have to be if treated as a formula at all – the proposed replacement would fall even further from the ideal, not even being pure. (Tennant [1979] actually cites Corcoran [1969], rather than Dummett’s work, for the pure-and-simple desideratum. Tennant’s paper, which aims to give a constructive treatment of the Sheffer stroke, seems to have overlooked the need for an Ex Falso style rule governing Λ itself. Further references on this subject may be found in the notes and references to §4.3, which begin on p. 626.) Well, if it’s not a formula, what is Λ? The following answer is not Tennant’s, as we shall see below, but provides us with a useful intermediate position to consider, in which Λ is not as fully integrated amongst the formulas as ⊥ was, since it’s not actually one of them, but is more fully integrated than it is on Tennant’s own position. The ‘intermediate’ answer requires that we recall the signed formulas—which were also not actually formulas—of earlier in this subsection. We considered only one-place force indicators or ‘signs’, [+] and [−], that attached to a single formula and made a signed formula. But we can readily


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imagine n-ary force indicators which make a signed formula from n formulas, when n 2, (see §5 of Humberstone [2000c]), and indeed why not when n = 0? If we declared Λ to be a zero-place force indicator, this would make Λ already a signed formula in its own right, and hence able to appear in sequents (of some suitable signed sequent framework, as in the above discussion of Bendall [1978] and Smiley [1996]), and yet not able to appear embedded within other (signed) formulas. The point about appearing in sequents means that the transition described above in which the earlier appeal to (→I) is by-passed can be represented as sanctioned by a rule along the lines of Γ, A Λ Γ ¬A or, since we should probably have all the signing explicit, perhaps better: Γs , [+]A Λ Γs [+]¬A This, then, is the middle position in our range of options available for the treatment of a Falsum symbol. It is not so well integrated amongst the formulas as to be embeddable within the scope of sentence connectives, as on the first option, but at least it can appear on the left or right of the sequent separator as a (signed) formula can, which we take as an index of a certain degree of integration amongst the formulas. (There is an obvious semantic treatment that would be appropriate in the style of our earlier discussion. Since it is not a formula, a valuation assigns Λ no truth-value, but since it is a signed formula, a correctness function c must assign it a correctness value – 1 or 0 – and the intended semantics is obtained by insisting that c(Λ) = 0.) The most extreme position, on which our Falsum symbol is least well integrated amongst the formulas, will be one according to which it cannot even, as it can on the middle option just reviewed, appear in sequents. A proposal along these lines is advanced with varying degrees of explicitness in the work of Neil Tennant (for example Tennant [1978], [1987a], [1987b], Chapter 23) is that one avails oneself of the convenience of rules along the lines of (¬I), (¬E) below, but without regarding ⊥ as a sentence connective at all. Since as a connective ⊥ is 0-ary, not regarding it as a connective amounts to not acknowledging ⊥ as a formula at all, which makes the above sequent-to-sequent formulations problematic. Tennant uses a formula-to-formula representation of his version of the rule (¬E), and more generally adopts the rule figures of Prawitz [1965] to deal with assumption-discharging rules (such as (¬I)); representing (¬E) at p. 40 of Tennant [1978] thus (though with different schematic letters, and, as already remarked, a different notation from “Λ” – though we retain the latter even in direct quotation from Tennant):

(¬E)

A

¬A Λ

(¬I)

[A] · · · Λ ¬A

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Tennant’s gloss on (¬E) is that from the premisses A and ¬A “we may immediately infer ‘Contradiction!’, for which we use the symbol Λ”. The exclamation mark is very suggestive, though it is not clear that the word infer exactly suits his purposes here: we are precisely not involved, on this novel conception, in drawing a conclusion from premisses, so much as in indicating a feature of the proof so far, namely, that it has yielded contradictory results. (What Tennant in the passage just quoted says we may infer is given by an exclamation rather than a declarative sentence – a sign that inference was not really what was at issue; the use of the term constant in the passage quoted below likewise runs somewhat counter to the intended proposal.) In Tennant [1987a], the actual source of the “Λ” notation we have been using, this is again not listed as being amongst the logical operators, and we have the following remark (p. 673): Note also that Λ, the absurdity constant, appears only on its own as a line in a proof; it never occurs as a subformula (nor, therefore, as a formula).

The reference to proofs here is to natural deduction proofs; Tennant supplies a corresponding sequent calculus from whose proofs Λ is entirely missing. (The proof systems Tennant is here discussing are systems for his own ‘Intuitionistic Relevant Logic’, though that is immaterial to our present interest in his remarks, which echo the policy of Tennant [1978] in which there was no such restricted focus.) The sequent which ‘corresponds’ to the stage in a natural deduction proof at which Λ has been derived from assumptions A1 , . . . , An is not the sequent A1 , . . . , An Λ (though Tennant actually uses “:” rather than “” as a sequent separator), but rather the Set-Fmla0 sequent A1 , . . . , An ∅ (which we usually write as “A1 , . . . , An ”). We could employ the same understanding in connection with the sequent-to-sequent representation of natural deduction proofs themselves. (Recall the independence of logical framework from ‘approach to logic’ stressed in 1.21 and elsewhere: this would be the natural deduction approach in the somewhat less familiar framework Set-Fmla0 .) Such a representation of (¬I), (¬E), above would amount to deleting the “⊥” from our schematic formulation of the rules, and regarding its appearance in the formula-to-formula representation of proofs as something having the status of a proof-annotation, in the same style, for example, as the crosses appearing below closed paths through a tree in the proof system of Jeffrey [1967]. It is never suggested that these are occurrences of some symbol belonging to the language for which this is a proof system. The remark quoted above from Tennant to the effect that his absurdity symbol “appears only on its own as a line in a proof” considerably understates the novelty of his approach, since imposing this condition on an ordinary formula would allow that formula, for example, to be assumed in the course of a natural deduction proof – as is the case with the conventional ⊥ constant, for instance. The passage quoted went on to rule this out with the parenthetical comment that the absurdity symbol does not qualify as a formula, of course. (At p. 359 of Tennant [2004], the suggestion is made that ⊥ – as he there writes, rather than Λ – is a scope-bearing element in a proof, since it indicates of some particular set of assumptions that they are collectively inconsistent.) This completes our survey of three positions one could take with regard to the Falsum symbol, though it may well occur to the reader that there is another option we have left out of consideration – a sort of variant on Tennant’s radical


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position. Instead of disallowing this symbol to appear anywhere in a sequent, we allow it to appear on the right only (and unembedded there). That would make the lines on which a Tennant-style “Λ” appears (depending on assumptions Γ) correspond in the expected manner to a sequent with “Λ” on the right (and Γ on the left), but of course something else rather unexpected would have happened at the same time: we should have entered the domain of heterogeneous logic, in which the language from which the formulas on the left of a sequent and that from which those on the right are drawn no longer coincide. This is more of a novelty than we care to embrace here, referring the interested reader referred to Humberstone [1988a] for an exploration of such possibilities. In 8.31 we return to the most conservative position, on which the Falsum (⊥) is a fully integrated atomic formula – which is not to say that the alternatives canvassed above do not merit further attention. We conclude the present subsection with a couple of interesting exercises on aspects of negation. Exercise 8.12.9 (i) (Adapted from Avron [1991b], p. 279.) Say that a 1-ary connective # in the language of a gcr is a right negation according to if for all sets of formulas Γ, Δ and all formulas A, we have Γ, A Δ if and only if Γ #A, Δ and that # is a left negation according to when for all Γ, Δ, A: Γ A, Δ if and only if Γ, #A Δ. Show that # is a left negation according to a gcr just in case # is a right negation according to . (This corrects an oversight in Kuhn [1977], p. 24 and p. 63, on each of which pages both conditions are imposed together, as though independent.) (ii) Say that a 1-ary connective # in the language of a gcr is a weak right negation according to if for all sets of formulas Γ, and all formulas A, we have Γ, A if and only if Γ #A and that # is a weak left negation according to when Γ A if and only if Γ, #A . If # is a weak left negation according to a gcr does it follow that # is a weak right negation according to ? What about conversely? Justify your answers. (Suggestion: For (i) here, see Exercise 1.28.1, p. 154.) Exercise 8.12.10 (Adapted from Došen [1986a].) Consider adding ¬ to the purely implicational language of BCI logic, and the following candidate axioms (given here as schemata) to supplement the usual axiomatization of that logic: (i) (A → B) → (¬B → ¬A) (ii) A → ¬¬A (iii) (A → ¬B) → (B → ¬A). Show that adding both (i) and (ii) to BCI logic is equivalent to adding just (iii). (The resulting logic is what we might consider ‘BCI logic with minimal negation’. Adding just (i) gives instead BCI logic with subminimal negation. The latter phrase is Allen Hazen’s; see the discussion after 8.33.5 below.


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8.13

Negation in Quantum Logic and Relevant Logic

We begin with a reminder from Chapter 0. Bounded lattices expanded by a 1ary complementation-like operation ¬ for which the following conditions (from 0.21.4) are satisfied, for all lattice elements a, b: ¬0 = 1, ¬1 = 0, ¬¬a = a, ¬(a ∧ b) = ¬a ∨ ¬b, ¬(a ∨ b) = ¬a ∧ ¬b are called ortholattices if the complementation-like operation is indeed complementation in the full sense that we also have a ∨ ¬a = 1 and a ∧ ¬a = 0. Bounded distributive lattices expanded by a ¬ satisfying the inset equational conditions—but not necessarily the last two full complementation complementation conditions—are called quasi-boolean algebras, as was mentioned in 0.21. The two varieties of algebras have been associated with quantum logic and (the {∧, ∨, ¬}-fragment of) relevant logic respectively, and we shall be looking in this subsection at the behaviour of negation – the logical analogue of the complementation-like operation above – in these two logical settings. We shall not be considering the algebraic semantics provided by these varieties in any detail, preferring to concentrate on the more illuminating model-theoretic semantic characterizations that have been provided, but starting here gives a feel for how close the algebraic aspects of the two situations are. (A thorough discussion of the relationship between the algebraic and the model-theoretic semantics, as well as of the connections between the semantics—of both types—for orthologic and for relevant logic may be found in Dunn [1993], [1996]. Further discussion of Dunn’s approach, and other aspects of negation in relevant logic, is provided by Chapter 5 of Mares [2004].) We should add also that we will not hesitate to consider, at least in passing, the pride and joy of relevant logic, →, absent from the fragment mentioned above. The discussion of negation in relevant logic begins after 8.13.10 (p. 1192) below. We should mention also that orthologic is really a sort of ‘baby’ quantum logic, with the role played algebraically by ortholattices (alias orthocomplemented lattices) being played in a more serious context—but somewhat problematically for the model-theoretic semantics (due to R. Goldblatt) we shall be describing here—by the class of orthomodular lattices, as was mentioned in 2.31 (cf. also 8.13.10, p. 1192). When we dealt with quantum logic, or more accurately, orthologic, on earlier occasions, in 2.31 and 6.47 we did not attend to negation, concentrating instead on conjunction and disjunction. The latter was governed by a specially restricted rule (∨E)res in our Lemmon-style natural deduction system QNat, and (in 6.47) treated semantically by a special clause we called [∨]Gldb after Goldblatt. This appellation notwithstanding, Goldblatt [1974a] does not actually have a primitive disjunction connective, treating it instead as defined à la De Morgan, in terms of the only primitive connectives which are ∧ and ¬. (Our [∨]Gldb simply recorded the effect of this definition.) Accordingly, let us forget about disjunction and consider what needs to be said about conjunction and (especially) negation. The Distribution Law (1), rewritten in terms of ¬ and ∧ becomes the unfamiliar-looking conjunction-negation schema (2): (1) A ∧ (B ∨ C) (A ∧ B) ∨ (A ∧ C) (2) A ∧¬(¬B ∧¬C) ¬(¬(A ∧ B) ∧¬(A ∧ C)) and for the purposes of blocking, as required, any proof of (2), no amount of restricting the ∨-Elimination rule will be of any assistance. Instead, we shall


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have to block the rules governing ¬. The rules (∧I) and (∧E) from QNat and Nat can remain intact. As a glance at the algebraic equations above suggests, we shall not be wanting to interfere with the double negation equivalence, so the rules (¬¬I) and (¬¬E) from Nat are not to be tampered with, leaving (RAA) from Nat as the point on which to concentrate our attention. (In 1.23.5, p. 121, we saw that with the latter rule in its original formulation, the rule (¬¬I) is redundant; but since we want it to be derivable and are about to consider tampering with (RAA), we lay it down explicitly, to be on the safe side.) Actually, it will be more convenient to consider ringing the changes on the ‘purified’ version of (RAA) introduced in 4.14, and repeated here for convenience: Purified (RAA)

Γ, A B Δ, A ¬B Γ, Δ ¬A

In Mset-Fmla, as one might be for various substructural logics (see 2.33, 7.25), it is often important to distinguish this rule, in which the formula to be negated (A) appears in each premiss-sequent, from a version of the rule in which this formula appears only in one of the premisses. Two versions, in fact, since we have a choice as to which premiss it is to appear in: (RAA)1

Γ, A B Δ ¬B Γ, Δ ¬A

(RAA)2

Γ B,

Δ, A ¬B

Γ, Δ ¬A

We emphasize that we are working here, as usual, in Set-Set, and with the structural rules (R), (M) and (T) in force. Exercise 8.13.1 Show that in the presence of the structural rules just alluded to and the rule (¬¬I), (RAA)1 and (RAA)2 are interderivable. We can obviously derive (RAA)1 and (RAA)2 from purified (RAA), since given premiss-sequents for either of the former rules, we simply ‘thin in’ the required occurrence of A (i.e., appeal to (M)) and then apply the latter rule to obtain the desired conclusion. Exercise 8.13.2 (i) Show that given not only all the rules mentioned in 8.13.1 as well as (¬¬E), (∧I), and (∧E), the purified (RAA) rule is not derivable. (Hint: Argue by induction on the length of proofs that from the rules cited, no sequent Γ C is provable in which Γ = ∅, whereas if purified (RAA) were derivable, the sequent ¬(p ∧ ¬p) would be provable.) (ii ) Consider the rule (¬-Contrac): From Γ, A ¬A to Γ ¬A. (The reason for the name will be explained below, just before 8.32.3, p. 1259.) Show that, given the structural rules the following two sets of rules are interderivable: {purified (RAA)}, {(RAA)1 , (¬-Contrac)}. (We could equally well say “(RAA)2 ” in the latter case.) Since we do not want the full force of (RAA), purified or otherwise, to be available – that would allow for a proof of (2) above – 8.13.2(ii) tells us that we must not adopt as rules governing negation, both (RAA)1 and (¬-Contrac).

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To see what sort of restricted versions of these rules might be appropriate, we turn to the semantics for orthologic given in Goldblatt [1974a], except that we use the notation and terminology with which this semantics was discussed in 6.47 (though there we considered only ¬-free formulas and took ∨ as a primitive connective). We have models (W, R, V ) in which W is a non-empty set on which R is an irreflexive symmetric relation (called orthogonality in Goldblatt, and represented by “⊥”, which would be confusing here since we use this as a nullary connective), and V is a function assigning subsets of W to propositional variables, subject to the following special condition, in which for brevity we have used the quantifier notation ∀ rather than writing out “for all”: ∀pi ∀x ∈ W : [∀y ∈ W ((V (pi ) ⊆ R(y) ⇒ x ∈ R(y))] ⇒ x ∈ V (pi ). As in 6.47, we reformulate the condition by defining X R x to hold (where X ⊆ W and x ∈ W ) just in case for all y ∈ W for which X ⊆ R(y), we have x ∈ R(y), and then stating the above condition as the requirement that V (pi ) be R -closed, in the sense that whenever X ⊆ V (pi ) and X R x, then x ∈ V (pi ). (Goldblatt [1974a], p. 25, calls such sets ⊥-closed, and suggests that the requirement that the sets V (pi ) be thus closed represents a “restriction on what sets of outcomes may be identified with propositions or ‘events” ’, when the orthogonality relation is thought of as holding between distinct alternative outcomes of the performance of the same physical operation.) Truth for conjunctions at a point in a model of kind just introduced – henceforth, an orthomodel – is defined inductively, with the clauses for propositional variables and conjunctions of formulas being those familiar from the Kripke semantics for modal logic. Disjunction was treated in 6.47, and here we focus on the novelties of negation, with the following clause, for all M = (W, R, V ), all x ∈ W , all formulas A: [¬]Gldb

M |=x ¬A if and only if for all y ∈ W, if M |=y A then xRy.

The most obvious thing to notice about this clause is that, especially by comparison with the treatment of or ¬ in the Kripke semantics for modal or intuitionistic logic, is that the accessibility condition occurs in the consequent rather than the antecedent of the right-hand side of [¬]Gldb . Though striking, this difference is superficial, because we could have formulated everything with the complement of R and then obtained the same effect by saying that is re¯ quired for the truth of ¬A at x is that A should fail to be true at all R-related (“non-orthogonal”) points y. (See further 8.13.8 below and the Digression following it.) This contraposed form of the right-hand side of [¬]Gldb makes it evident that we are dealing here with a manifestation of the phenomenon called strict negation in the Digression following 8.11.3. As well as bringing out the similarities between ¬ as treated by [¬]Gldb and the behaviour of impossibility in the Kripke semantics for modal logic, the reformulation is also highly reminiscent of the treatment of negation in the Kripke semantics for intuitionistic logic, even though the background frame conditions are very different—here irreflexivity and symmetry for the accessibility relation (or reflexivity and symmetry, if we want to think in terms of the complement), there reflexivity and transitivity (± antisymmetry)—as well as the special conditions on models—there Persistence and here the R -closure condition. The latter condition, just like Persistence,


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spreads from the propositional variables for which it was postulated, to compounds of arbitrary complexity. Another similarity with the IL case is that a formula and its negation are never both true at a point in a model. We combine these findings as (i) and (ii) in: Lemma 8.13.3 Letting !A! for some orthomodel M (understood from the context) be the set of points in the model at which the formula A is true, we have for all formulas A: (i) !A! is R -closed. (ii) !A! ∩ !¬A! = ∅. (iii) !A! = !¬¬A!. Proof. We omit the details, which are a matter of following through the definitions. For (i) argue by induction on the complexity of A; as with IL (2.32.3, p. 308) the case of ∧ requires the induction hypothesis while that for ¬ does not. (ii ) exploits the irreflexivity of R, while the ⊆ direction of (iii) uses the symmetry of R, and its ⊇ direction uses (i) – indeed this direction, which amounts to the validity (in the sense defined below) of (¬¬E) sequents ¬¬A A, is the whole rationale behind the R -closure condition which (i) says generalizes to all formulas A once stipulated for the propositional variables. We intend 8.13.3 as a lemma towards a soundness proof for an as yet unspecified proof system, with a view to tailoring the specification to secure just such a result. The valid sequents are to the those Γ C such that for any orthomodel M = (W, R, V ) and any x ∈ W , if M |=x A for each A ∈ Γ, then M |=x C. Remark 8.13.4 We could have defined in an obvious way the intermediate concepts of a sequent’s holding in an orthomodel and being valid on an orthoframe—the (W, R) part of such a model—notions which we shall take the reader to grasp on the basis of similar definitions elsewhere (e.g. in modal logic) when we need them (briefly) below. Likewise for the concept of a logic’s being determined by a class of orthoframes. Validity tout court is, in terms of these concepts, validity on all orthoframes and 8.13.6 below says that the proof system we are working towards is determined by the class of all orthoframes. Let us go back the problem of what rules to lay down for negation, with a view to keeping out proofs of invalid sequents (and preferably allowing in proofs of all the valid ones). Clearly, in view of 8.13.3, the rules (¬¬I) and (¬¬E) are all right. What about (RAA)1 and (¬-Contrac)? If all the left-hand formulas Γ ∪ Δ of the sequent conclusion of an application of (RAA)1 are true at a point x in an orthomodel (W, R, V ) but the formula, ¬A, on the right is not true at x, does this conflict with the hypothesis that the two premiss sequents for that application, Γ, A B and Δ ¬B are valid? Since all elements of Δ are true at x, the validity of the latter sequent guarantees (i) that (W, R, V ) |=x ¬B. Since (W, R, V ) |=x ¬A, there exists a y ∈ W such that (ii) (W , R, V ) |=y A and not xRy. To get a contradiction from (i) and (ii) by using the hypothesis that Γ, A B is a valid sequent, we would need to know


1190

that the formulas in Γ were all true at y, since we could then conclude that B was true at y, which would contradict (i) and (ii). (By (i) all points at which B is true bear the orthogonality relation R to x, so y would have to, contradicting the second part of (ii).) Unfortunately, all we know about the formulas in Γ is that all of them are true at x, which does not bear at all on their truth at y. Of course in the special case in which Γ is empty, the problem disappears and we do have the desired contradiction from supposing (RAA)1 preserves validity. The situation is entirely analogous in the simpler case of the rule (¬-Contrac). Accordingly, in both cases, we adopt the course of restricting the rules to the Γ = ∅ case: (RAA)res

AB

Δ ¬B

Δ ¬A

(¬-Contrac)res

A ¬A ¬A

In view of 8.13.1, we could equally have based our restricted version of (RAA) on (RAA)2 rather than (RAA)1 . It may be wondered why, since we started with attempts to rein in (RAA), we have let (¬-Contrac) and the above restriction of the rule into the picture at all. It is not hard to see that on the above semantics, the sequent ¬(p ∧ ¬p) is valid. This sequent has nothing on the left. But even with all the other rules – rules other than (RAA)1 , that is – listed in 8.13.2(i), we noted that the above sequent (like any “∅ on the left” sequent) was unprovable. We have now restricted (RAA)1 even further, to (RAA)res , so certainly the sequent remains unprovable. It can easily be proved with the aid of (¬-Contrac)res , however. Exercise 8.13.5 Show that given the structural rules (RAA)res and the rule of Simple Contraposition (From A B to ¬B ¬A) are interderivable. (The admissibility of the latter rule amounts to saying, in the terminology of 3.32, that ¬ antitone according to the associated consequence relation.) The rule of Simple Contraposition mentioned here is taken as primitive in Goldblatt’s proof system, which is cast in Fmla-Fmla rather than Set-Fmla. We have preferred to work in the latter framework for continuity with the earlier discussions of QNat, as in 2.31. (This difference also accounts for our use of (¬Contrac)res . Note that taking disjunction as primitive alongside conjunction, we used the Fmla-Fmla system Lat in 2.14 to illustrate some points about algebraic semantics of what we called the -based kind.) However, the completeness proof in Goldblatt [1974a] can easily be adapted to give the ‘only if’ half of the following result. One uses a canonical model argument in which the points in the model are deductively closed consistent sets of formulas and the relation R is defined to hold between two such sets just in case there is some formula which is an element of one and whose negation is an element of the other. Of course it must be verified that the special R -closure condition on models is satisfied. Lemma 8.13.3 will help with any unobvious parts of the ‘if’ direction. Theorem 8.13.6 A sequent is valid if and only if it is provable using, alongside the structural rules, the rules (∧I), (∧E), (¬¬I), (¬¬E), (RAA)res and (¬Contrac)res .


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Exercise 8.13.7 Prove sequent (i) below in the proof system described in 8.13.6, and sequents (ii)–(v) in the proof system (for the language with ∨ as an additional primitive connective) we get by adding to the rules of 8.13.6 the rules (∨I) and (∨E)res of 2.31 (p. 299) and 6.47: (i) ¬(p ∧ ¬p) (iii) ¬p ∨ ¬q ¬(p ∧ q) (v) ¬p ∧ ¬q ¬(p ∨ q).

(ii ) ¬(p ∧ q) ¬p ∨ ¬q (iv ) ¬(p ∨ q) ¬p ∧ ¬q

We have already had occasion to notice that the contraposed form of [¬]Gldb directs our attention to the complement of the orthogonality relation of an orthoframe, which will be a symmetric and reflexive relation. Since we know that the inferential consequence relation (on the language of modal logic) determined by the class of all symmetric and reflexive frames is KTB , the following result – details of which are in §4 of Goldblatt [1974a] – is not unexpected. Exercise 6.42.3, referred to here, appeared on p. 873: Observation 8.13.8 As in 6.42.3, we describe a translation of the language (for simplicity, without ∨) of the present subsection into the language of monomodal logic, which faithfully embeds the Set-Fmla logic of 8.13.6 into KTB (conceived of as a logic in Set-Fmla). The translation is the unique map τ such that: τ (pi ) = pi (i.e., = ¬¬pi ); τ (A ∧ B) = τ (A) ∧ τ (B); τ (¬A) = ¬τ (A). Then, where is the consequence relation associated with the proof system of 8.13.6, we have, for all formulas Γ ∪ {A} of the language of : Γ A if and only if τ (Γ) KTB τ (A). (The word “faithfully” in our description of how this embedding works refers to the fact that we have the ‘if ’ direction and not just the ‘only if ’ direction.) Digression. The condition on τ pertaining to ¬-formulas brings out the ‘strict negation’ character of negation in orthologic. One might have thought of the following variation on the representation described in the Digression on p. 1260. We explain the semantics of as there (or in the two-dimensional material in 7.17, p. 1018) and then represent ¬A as (A → R), where the boldface R is a sentential constant for the orthogonality relation. This would be a treatment of negation somewhat in the style of the Falsum-based accounts which will occupy us in §8.3 (R as ⊥). There are, however grave difficulties in the face of conservatively extending these non-distributive logics by the addition of an implicational connective (the → in “(A → R)”), as was remarked in 2.31. End of Digression. Exercise 8.13.9 (i) Show that the consequence relation associated with the proof system of 8.13.6 is congruential. (ii ) Show that the Lindenbaum algebra this consequence relation, (say), yields is an ortholattice (with partial ordering [A] [B] iff A B, as in 2.14), when a suitable choice – please specify – of the meet, join, top and bottom elements is made.


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(iii) Show that ¬ is not uniquely characterized (in the sense of §4.3) by the rules listed under 8.13.6. (Cf. 0.21.5(iv), p. 22.) Remark 8.13.10 In 2.31 we called the consequence relation mentioned under 8.13.6(ii) OL and at the end of that section briefly mentioned its ‘orthomodular’ extension OM L , for which we have: A, A ⊃ B OM L B where A ⊃ B is the formula ¬A ∨ (A ∧ B) (“⊃” being the so-called Sasaki hook, as in the discussion following 2.31.3, p. 301). Goldblatt [1974a] obtains a completeness result for this logic w.r.t a class of general orthoframes – the analogues in the present context of the general Kripke frames mentioned in the Digression on p. 290 – and leaves open the question as to whether there is a determining class of orthoframes (simpliciter ). Goldblatt [1993], Chapter 3, has this question still open, but does provide information on the class of all frames for the logic (in particular to the effect that it this class is not first-order definable). We turn now to another area of logic in which negation is not uniquely characterized by the rules governing it (cf. 8.13.9(iii)), namely relevant logic. (This comment will be clarified in the Digression below.) Here there is arguably more than one candidate to qualify as a negation connective, and the preceding description was offered with so-called De Morgan negation in mind: the negation symbolized by “¬” in our earlier discussions of relevant logic (especially 2.33). Later (pp. 1203–1205) we will encounter another negation connective, usually called Boolean negation, which we will be representing by “ ∼”, and we repeat our warning that in much of the literature these two notations are used the other way around, or else Boolean negation is represented by ¬ and De Morgan negation by overbarring. We choose “¬” to represent De Morgan negation because we have always used this notation for what – even if only for historical reasons – has been regarded as the negation connective of principal interest in a given area. (Thus for example, intuitionistic negation was represented by “¬”, rather than dual intuitionistic negation – see 8.22 – or strong negation – see 8.23 – even though these other connectives have had some role to play in discussions of IL. Note that we refrain from attempting to give a formal definition of the idea of a negation connective both because this would not be true to the ‘family resemblance’ nature of the informal notion – cf. Humberstone and Lock [1986] – and because such definitions are not conducive to open-minded investigation. Cf. the comments on essentialism at p. 427 above.) Digression.The above comment about negation not being uniquely characterized in relevant logic stands in need of clarification because already the ILL (and a fortiori CLL) sequent calculus rules of 2.33 uniquely characterize ¬. This point was made there, before we had the concept of unique characterization available, via asking in 2.33.28(ii) whether the linear logical rules for ¬ (and other connectives) were regular (‘identity-inductive’), to which the answer is easily seen to be affirmative. As noted in 4.31.1, this is tantamount to securing unique characterization of ¬ by rules in question. The possibility of having an empty succedent is essential here. (The framework for CLL, recall, was Mset-Mset while that for ILL was Mset-Fmla0 .) So a more careful formulation would be


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that no rules for ¬ in Mset-Fmla (or Set-Fmla) uniquely characterize that connective in R (or for that matter, classical linear logic), a proof of which can be given model-theoretically by using two separate *-functions – see below – in the models; this is analogous to the proof of 8.13.9(iii) suggested by the hint there given, or the proof for Minimal Logic at 8.32.7 below. Similarly, ¬ is not uniquely characterized in a Fmla incarnation of classical linear logic or R, in terms of the remaining connectives, exempting f from the latter. Of course with f there is the explicit definition ¬A = A → f , but then we can make the present point for f itself, which unlike F , resists unique characterization in the absence of empty-succedented sequents. (For F we have F B for all B, so in the combined system pertinent to checking unique characterization, with F as duplicate candidate, we can choose F for B; by symmetry we get both sequents F F .) But are things really any different from the case of intuitionistic logic itself? At the start of 4.32 we gave the obvious uniqueness proof for ¬ as governed by the IGen rules (¬ Left), (¬ Right), in which the application of (¬ Left) gives rise to an empty succedent, so this proof works only in Set-Fmla0 and not in Set-Fmla. Yet a reformulation is possible here by taking as primitive ⊥, conveniently blurring the f /F distinction in participating as consequent in the implicational definition of ¬ (like f ) while being uniquely characterized without recourse to empty succedents (like F ). Alternatively we can use only ¬ itself and replace (¬ Left) with initial sequents of the Ex Falso form – A, ¬A B – and (¬ Right) with (¬-Contrac) from 8.13.2(ii), p. 1187. These rules give us the subformula property, the second of them being reminiscent of the Kleene G3 style rules of 2.32.13 (p. 321). The reference to the non-uniqueness of ¬ in Minimal Logic is specifically to the Set-Fmla natural deduction system MNat, this being the subject of 8.32.7. An alternative approach sometimes encountered to ML is in a variation on the Set-FMLA0 sequent calculus we get from IGen by disallowing ‘Right Weakening’, or (M)-on-the-left (so we cannot pass from the (¬ Left) conclusion A, ¬A ∅ to the EFQ sequent A, ¬A B. The uniqueness proof (1)–(3) for ¬ in IGen goes through in this system, but notice that a sequent can no longer be semantically treated by saying that it is holds at a point in a model just in case whenever all its left-hand formulas are true at that point at least one of its (zero or one) right hand-formulas is true there. The ‘zero’ subcase has to be treated separately, with Γ ∅ meaning that the truth of all the formulas in Γ implies that the point in question belongs to the distinguished subset Q (of the models described just before 8.32.5 below). Why all the fuss about f , F , and ⊥ – and no fuss about t, T , and ? Well, the situations should be treated analogously, mutatis mutandis, though the mutations involved tend to escape one’s notice because of the weight of tradition, which tends to make less fuss about empty right-hand sides than empty left-hand sides. See §4 of Humberstone [2005d] for further remarks on this. See also the index entries for Set1 -Fmla (under “logical frameworks”) – though Mset1 -Fmla would be pertinent for some of the above discussion, and in any case neither of these is dual to the frameworks considered there, as a full sub specie aeternitatis perspective would dictate. (That is, any problem about Set-Fmla or Set-Fmla0 should be compared with the corresponding problem, if there seems to be one, with Fmla-Set or Fmla0 -Set, for example.) End of Digression.

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There are approaches—usually called the Australian Plan and the American Plan—to the semantics of De Morgan negation which are equally easy to grasp when we are considering the question of which {∧, ∨, ¬}-formulas follow from which according to the tradition of relevant logic reviewed in 2.33 and revisited at various intervals since (5.16, 6.13, 6.45, 7.23, 7.24), though perhaps not equally simple when (iterated) → formulas are taken into account (where the American Plan looks a little complicated). The question of the logical relations between {∧, ∨, ¬}-formulas is often called the question of first-degree entailments, since the protagonists have usually concentrated on Fmla and thought of this as the question as to which formulas of the form A → B should be provable when A and B are formulas in the {∧, ∨, ¬}-fragment. We will use the same terminology (though elsewhere we have already used the alternative terminology of tautological entailments), but think of this as the question of which sequentsA B, for A and B in the fragment, should be provable. We will begin by describing one particular motivation for work on relevant logic in the tradition alluded to: paraconsistency – already alluded to briefly in the notes to §7.2, p. 1123 (and in the second paragraph of 1.19.2, p. 92). Paraconsistent logic, as a general project, covers work done in logic from a certain motive: that of providing an account of logic serviceable for theories which, while inconsistent in the sense of containing some formula and also its negation – henceforth ¬-inconsistent – are not inconsistent in the sense of containing every formula. (The latter notion of inconsistency is often in this context called triviality, as noted in 1.19.2.) The reasons for being so motivated have been various, as have also been the suggestions on how best to implement the idea. Among these reasons, there are the radical and the moderate. In the latter category, we may take an interest taken in ¬-inconsistent but non-trivial theories on the theoretical ground that a such a theory – naive set theory, for instance – may be worth studying in its own right: not as collapsing into any other such theory (in the same language) as it would if seen through the lights of a consequence relation according to which A, ¬A B for all A, B. There is also the more practical ground that we do not want databases, expert systems, and other information processing systems to yield affirmative answers to every query simply because inconsistent information happens to have been entered. (Belnap [1977], or §81 of Anderson, Belnap and Dunn [1992].) The more radical reason for taking an interest in paraconsistent logic is the idea that some ¬inconsistent but non-trivial theories are not only (as on the moderate approach) worth studying, but are actually true. If we call a statement false when its negation is true, this involves the philosophical position baptized by one its more prominent defenders (Priest [2006a], first published in 1987) dialetheism: the view that some statements are both true and false (‘false’ here meaning: such that its negation is true – for some privileged negation connective). From this perspective, it is not just that some ¬-inconsistent non-trivial theories are interesting objects of investigation, but that (considered as a theory in an interpreted language) some such theories are true. We will not enter into the philosophical debate this provocative proposal has understandably triggered. (See Weir [2004] for a critical discussion which nonetheless takes the proposal seriously.) As remarked, various suggestions have been made on the question of how to develop a paraconsistent logic. One obvious suggestion, in view of the unprovability in R of such ex falso principles as (A ∧ ¬A) → B, is that paraconsistency be implemented by the selection of a relevant logic. However, certain hallmarks


1195

of relevance, such as the general unprovability of A → (B → A) and its permuted form, B → (A → A), have little connection with the paraconsistent motivation, while on the other hand, certain features of relevant logics like R and E actually go against that motivation. (For an axiomatization of E, the originally favoured Anderson and Belnap ‘logic of entailment’, add the conjunction, disjunction and negation axioms given for R in 2.33 – p. 340 – to the implicational axioms given in 9.22.1, p. 1303.) This is notably the case with the principle Contrac (or W in the combinator-derived terminology), as was remarked in the notes to §7.2, à propos of our discussion (7.25) of contractionless logics (see p. 1124). There are also many approaches to paraconsistency which have nothing to do with relevant logic, well exemplified in the collection Priest et al. [1989], and ably compared in the editors’ own contributions to that volume (see especially Priest and Routley [1989]). See also Priest [2006a] and Priest, Beall, and Armour-Garb [2004]. We shall not be considering these other approaches. Restricting attention to the first-degree entailments, to set aside such complications (occasioned by formulas with → in the scope of → or indeed of other connectives), we have already declared our preference for considering the sequents A B rather than the formulas A → B, and noted in 2.33.6(iii), p. 340, that such a formula is provable in the relevant logic R just in case the corresponding sequent is provable in the Kleene-inspired system Kle introduced in 2.11.9, p. 207, and observed to have a four-element characteristic matrix in 3.17.5 (p. 430). As far as first-degree entailments go, the American Plan and the Australian Plan are different ways of repackaging this consideration in more palatable terms. The American Plan (which originated in Dunn [1976]) has a suggestive labelling for the matrix elements, in terms of sets of ordinary truthvalues (T and F), while the Australian Plan (which originated in Routley and Routley [1972]) looks initially somewhat different, and draws on a special operation, the *-function, in the frames of a model-theoretic semantics, in order to interpret negation. (With various historical references to hand, Dunn [1999], note 19, suggests that the American and Australian Plans might more accurately be called the Indian Plan and the Polish Plan, respectively.) Beginning with the Australian Plan, we need models of the form (S, *, V ), in which S is a non-empty set on which * is a function satisfying a** = a, for all a ∈ S. (A function satisfying this condition is called an ‘involution of period 2’, or more commonly just an involution, for short. Note that we write a* for *(a).) That deals with the frame (S, *) of the model; V , as usual, is just a function assigning to each propositional variable a subset of S. Intuitively, we are to think of the elements of S as some kind of generalization of possible worlds – the Routleys’ favoured term for its elements being set-ups – and we now give the inductive definition of the truth of an arbitrary formula (in connectives ∧, ∨ , ¬) relative to one of these elements: (S, *, V ) |=x pi iff x ∈ V (pi ); (S, *, V ) |=x B ∧ C iff (S, *, V ) |=x B and (S, *, V ) |=x C; (S, *, V ) |=x B ∨ C iff (S, *, V ) |=x B or (S, *, V ) |=x C; (S, *, V ) |=x ¬B iff (S, *, V ) |=x∗ B. The novelty lies, of course, in the last clause, where the *-function makes its appearance, allowing for a formula and its negation (or neither of them) to be true at a point x, as long as x = x*. (See below.)


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A Set-Set sequent holds in a model (S, *, V ) just in case any element x ∈ S at which all the formulas on the left are true is an element at which at least one formula on the right is true, and a sequent is valid on a frame (S, *)validity!on a Routley–Routley frame if it holds in every model on that frame. (Note that by the way holding in a model is here defined, the commas are interpreted extensionally or additively rather than – as in the natural deduction systems and sequent calculi of 2.33 – intensionally or multiplicatively.) Because of the concern with first degree entailments, we are especially interested in the validity of sequents which happen to belong to the more restrictive framework FmlaFmla, as with all but the first listed in the following: Exercise 8.13.11 Which of the following sequents are valid on every frame? (The first five are taken from 8.13.7: in the interests of becoming familiar with the semantics, work through the exercise rather than relying on the solution given in the following paragraph.) (i) ¬(p ∧ ¬p) (iii) ¬p ∨¬q ¬(p ∧ q) (v) ¬p ∧ ¬q ¬(p ∨ q) (vii) p ∧ ¬p q.

(ii ) ¬(p ∧ q) ¬p ∨ ¬q (iv ) ¬(p ∨ q) ¬p ∧ ¬q (vi ) p ∨ q, ¬p q

Working through the above examples will provide a good feel for how the *-function deals with negation, managing to deliver as valid (on any frame) De Morgan’s Laws, as well as—not on our list—the double negation equivalences, while at the same time denying validity, as one would expect from a relevant logic, the Ex Falso sequent (vii) with invalidation of its more innocent looking ‘partner in crime’, the disjunctive syllogism sequent (vi). On the other hand, the loss of (i)—especially as the same invalidity verdict would arise if we had chosen p ∨ ¬p as the formula concerned—may make it look as though the casualties are unexpectedly high. This can be fixed, however, and we should notice for the moment that the semantics is designed for the first-degree entailments, and (i) does not represent such an entailment. If instead we insert a formula on the left, we should check whether any intuitive verdict conflicts with the official verdict of the semantics. (In the excluded middle case, for instance, we have p p ∨ ¬p valid and ¬p p ∨¬p valid but q p ∨ ¬p invalid. For we need only specify V (q) = S and V (p) = {x*}, where x ∈ S and x = x*, and we have the left-hand formula true at x without the right-hand formula true there.) Exercise 8.13.12 Show that for any model M = (S, *, V ) and any x ∈ S, the model Mx = {x, x*}, *, V ) in which * and V in the latter case are the restrictions of their namesakes in M to the set {x, x*}, satisfies the following: Mx |=y A iff M |=y A for y ∈ {x, x*}, for all A. (Hint: this is a simple variation on the theme of 2.22.4 (p. 285), made simple because, by the involution condition (x = x**), the submodel Mx generated by x contains at most two elements.) Although in the Routley–Meyer semantics, an extension of the semantics we have been considering designed to cope with the full language of R and in particular with the presence of (arbitrarily embedded) →, we cannot limit the size of the models, 8.13.12 tells us that in the present context, no more than two-element models need to be considered. (For a modal perspective on such models, see the discussion after 8.23.10, p. 1241 below.)


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Observation 8.13.13 A sequent is valid on every frame just in case it is valid on the two-element frame (with distinct elements a, a*). Proof. Suppose Γ Δ is invalid on some frame with more than two elements. There is then some model M and some x in (the universe of) M with all formulas in Γ true at x (in M) and all formulas in Δ false at x (in M). By 8.13.12, this distribution of truth-values persist when we consider the model Mx , whose frame is either isomorphic to the two-element frame with elements a, a*, or else (because x = x*) to the one-element frame. In the former case, we are done, and in the latter, we can construct a model, with Valuation V , on the two-element frame which behaves like Mx by putting V (pi ) = {a, a*} when V (pi ) = {x} and V (pi ) = ∅ otherwise. (Here V is the Valuation of Mx .) The four ‘propositions’ provided by the two-element frame are its subsets {a, a*}, {a}, {a*}, and ∅. If, relative to a particular model, !A! and !B! are the propositions expressed by formulas A and B (i.e., the subsets comprising precisely those elements at which these formulas are true in the model), then the truth-definition above tells us that !A ∧ B! = !A! ∩ !B!;

!A ∨ B! = !A! ∪ !B!;

!¬A! = ¬!A!.

For the third equation, ¬ on the right-hand side is to be interpreted thus: ¬Y = {x | x∗ ∈ / Y }. Whenever we have an involution * on a set, the operation ¬ thus defined on its power set, together with union and intersection as join and meet, produce a quasi-boolean algebra with ¬ as the quasi-complementation operation. (The affirmative cases under 8.13.11 are a semantic reflection of this.) Indeed, Białynicki-Birula and Rasiowa [1958] showed that all quasi-boolean algebras can be represented as being of this form for a suitable collection of sets – though the fact that this representation theorem amounts to a completeness theorem for a simple relevant logic was not realized until considerably later: see Dunn [1986] for details. (By contrast, Goldblatt put his semantic completeness proof for orthologic and his representation theorem for ortholattices, distilling the algebraic content of the proof, onto the market more or less simultaneously: see Goldblatt [1974a] and Goldblatt [1975], respectively.) In our special case where the four propositions from the two-element Routley–Routley model are the elements of the algebra, we can see that this is the algebra of the matrix (‘Smiley’s Matrix’) described in 3.17.5 (p. 430), once we re-name the elements: {a} will be 1, {a, a*} will be 2, ∅ is 3 and {a*}, 4. (Smiley originally supplied a table for → as part of his matrix as well, which we are ignoring. See Dale [1980] for some discussion. Meyer, Giambrone and Brady [1984] discuss the expanded matrix with not only implication but also, alongside De Morgan negation, an operation for what will introduced below as Boolean negation. For philosophical discussion, see Fox [1990], esp. p. 78, and for technical discussions, Pynko [1995], Font [1997].) Recall that the designated elements are 1 and 2, so here we are very much “taking a’s point of view” in the sense that this pattern of designation corresponds to preserving truth-at-a (whether truth at a alone or truth at a along with a*). Figure 8.13a gives the Hasse diagram we obtain with joins and meets correspond to disjunction and conjunction. The information in Figure 8.13a could equally well be presented in tabular form, of course, and the meet and join operations are precisely those of the


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2(={a,

?1(={a}) ??? ?? ?? ? ? a*}) ?? 3(=∅) ?? ?? 4(={a*})

Figure 8.13a

direct product of the two-element chain with itself. The join table is in fact to be found on the left of Figure 2.12a (p. 213). On the right of that figure is the complement table for the four element product matrix which is by contrast very different from our current quasi-complement ¬, for which we have ¬2 = 2 and ¬3 = 3. On the remaining elements, 1 and (the lattice zero) 4, ¬ behaves the same way as in the product case, mapping each to the other. So we have here a De Morgan algebra (and not just a quasi-boolean algebra, as these notions were distinguished after Exercise 0.21.6, p. 22). Note in particular that the of the above lattice is not the relation ⊆ on the subsets of {a, a*}. (The same holds for Figure 8.13b below.) To get that correspondence Figure 8.13a would have to be rotated clockwise through 90◦ . It is possible to view the diagram as embodying these two lattices at once, and this perspective has led to the idea of a ‘bilattice’, with a range of applications. However, we do not give the definition – due to M. Ginsberg – here, and refer the interested reader to Arieli and Avron [1996]. To move from the Australian Plan, with its *-functions, to the American Plan, we think of the semantic fate of a formula at the element a from whose perspective the choice of designation (1, 2) above was guided. We record the fact that for a formula A with !A! = x (x ∈ {1, 2, 3, 4}) we have A true at a by writing a “T” by x, and if we have ¬A true at a, we write “F”, collecting such of these values T, F, as are thereby associated with x into a set. Figure 8.13b records the resulting relabelling of the De Morgan algebra in 8.13a. The partial order involved can be described thus: x y if and only if T ∈ x ⇒ T ∈ y and F ∈ y ⇒ F ∈ x (where “⇒” is metalinguistic material implication). The ‘re-labelling’ has been done, as we have been putting it, from a’s point of view—where a is as in Figure 8.13a: in terms of matrix semantics, for which the new occupants of roles 1 and 2 ({T} and {T, F}) are again the designated elements, an evaluation h assigns a subset of {T, F} according to the principle that if |=a A, for some two-element Routley–Routley model as above, then T goes into h(A), while if |=a∗ A (so |=a ¬A), then F goes into h(A). Dunn [1976] has urged the intuitive superiority of the present way of casting the semantics over the Routleys’, with the query “But just what is this ‘star operation’ and why does it stick its nose into the truth conditions for negation?” There is no unnaturalness about the behaviour of the present evaluations – sometimes called ‘ambivaluations’, for obvious reasons – under Dunn’s description: since truth and falsity (T and F) on the present conception are neither mutually


1199

?{T} ??? ?? ?? ? ? {T,F} ?? ?? ??

∅

{F}

Figure 8.13b

exclusive nor jointly exhaustive, we don’t express the customary semantic idea that a conjunction is true iff both conjuncts are true by saying v(A ∧ B) = T iff v(A) = T and v(B) = T, but instead by saying that T is amongst the elements of h(A ∧ B) just in case T is amongst the elements of h(A) and T is also amongst the elements of h(B). Thus a conjunction is (at least) true— never mind about falsity—iff each of its conjuncts is at least true. We need to add something about falsity separately, since falsity should infect a conjunct whichever conjunct it afflicts (and regardless of whether that conjunct is also ‘true’). So we should say that F is amongst the elements of h(A ∧ B) iff F is amongst the elements of h(A) or amongst the elements of h(B) or both. Dunn’s summary – also appearing at p. 37 of Makinson [1973a] – of the analogues of the familiar #-booleanness conditions for ∧, ∨ and ¬ in this style is then: T ∈ h(¬A) iff F ∈ h(A), F ∈ h(¬A) iff T ∈ h(A); T ∈ h(A ∧ B) iff T ∈ h(A) and T ∈ h(B), F ∈ h(A ∧ B) iff F ∈ h(A) or F ∈ h(B); T ∈ h(A ∨ B) iff T ∈ h(A) or T ∈ h(B), F ∈ h(A ∨ B) iff F ∈ h(A) and F ∈ h(B). Note that if we if we restrict attention to assignments with h(A) a singleton subset of {T, F}, we are essentially dealing with traditional bivalent valuations, and the above conditions are just the usual #-booleanness requirements (except that T and F are being called {T} and {F}: in fact in some presentations the top and bottom elements of Figure 8.13b are just called T and F, with those depicted on the left and right by B and N—for both and neither. respectively). The idea of not imposing this restriction comes for Dunn [1976] and Belnap [1977] from what in our introductory remarks on paraconsistency we described as the ‘moderate’ motivation, thinking of T ∈ h(A) as meaning that h records some person or information-processing device having been told that A is true, and F ∈ h(A) that it has been told that A is false. Thus h(A) = {T, F} when it has been told both that A is true and that A is false, and h(A) = ∅, when it has been told neither. The latter—a truth-value gap rather than a truth-value glut, in terminology which has gained a certain currency—is not a contingency needing to be accommodated for paraconsistency, though it is needed for the duality-respecting nature of relevant logic, to invalidate p q ∨¬q. Accordingly

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Priest [2006a] (and elsewhere), with a specifically paraconsistent motivation, just uses the three values {T}, {T, F} and {F}, which simply means working (for ∧, ∨ and ¬) with the Kleene matrix K1,2 of our 2.11. We leave the reader to check that the above conditions do indeed yield Figure 8.13b’s verdicts on meets and joins, after considering one illustrative case: that of why the join of {T, F} with ∅ is {T}. We have to see, that is, why if h(A) = {T, F} and h(B) = ∅, we must have h(A ∨ B) = {T}. Since T ∈ h(A), we have to put T into h(A ∨ B), by the first of the pair of conditions for ∨. The second of that pair says that F only enters h(A ∨ B) if F ∈ h(A) and F ∈ h(B), so since F ∈ / h(B), we do not have F ∈ h(A ∨ B). Therefore h(A ∨ B) = {T}. Of course as well as the meets and joins, there are also the quasi-complements, not explicitly recorded in the diagram, to verify. But this is very straightforward. Again by way of illustration, we look at why {T, F} is its own quasi-complement. Obviously if h(A) = {T, F}, then since T ∈ h(A), the second condition on the list puts F into h(¬A), and since F ∈ h(A), the first condition puts T into h(¬A). So h(¬A) = {T, F}. For more information on how the current four-element matrix is related to some famous three-valued logics, see §7 of Dunn [1999] (§6 of which gives much more detail on the four-element story than has been presented here). Digression. An interesting four-valued logic, known as BN4, emerges when the tables for conjunction, disjunction, and negation implicit in the above discussion are supplemented by one for implication in the following way: T ∈ h(A → B) iff h(A) h(B), with as in Figure 8.13b, and F ∈ h(A → B) iff T ∈ h(A) while F ∈ h(B). Information on this logic may be gathered from Brady [1982], §4 of Slaney [1991] and §2 of Restall [1993a], as well as further references given in the last two of these sources. (Note that there is a misprint in the table given for →, on p. 1056 of Slaney [1991]. The correct table appears on p. 383 of Restall’s paper. The reader may like to figure out what the table should look like on the basis of the above description in terms of the conditions under which T or F belongs to h(A → B).) Slaney’s discussion includes a non-finitely valued version of BN4 called simply BN, which is a close cousin of the system of intuitionistic logic with strong negation described in 8.23 below. End of Digression. The American Plan and the Australian Plan clearly differ more in how the semantical machinery is being described than in the machinery itself. Nevertheless, the former, with its ambivaluations, is generally regarded as more cumbersome in the context of the full semantics for relevant logic (with → in the language) than the latter. (See Sections X and XI of Meyer [1979], Routley [1984] and Restall [1995a] for detailed developments.) Since the only model-theoretic semantics we have described for relevant logic in general and for (some fragments of the axiomatic Fmla system) R in particular, along with some natural deduction proof systems for its implicational fragment in Set-Fmla—the RNat and RMNat of 2.33 and 7.23, 7.24, have been Urquhart’s semilattice semantics, we have not actually seen the natural habitat of the *-function, which is in the Routley–Meyer semantics for these (and other) systems. In fact Urquhart [1972] could not find any way of treating negation in terms of his apparatus which allowed for a soundness and completeness proofs for R (with negation), from which he concluded that the Anderson–Belnap intuitions leading them to isolate this system had been confused. We saw in 6.45 that a similar conclusion for disjunction might also be drawn, but that a minor and rather natural emendation to the Urquhart semantics was available which accommodated disjunction and


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its R-dictated interrelations with other connectives satisfactorily. We will not consider such emendations here, instead briefly describing the Routley–Meyer semantics (as in Routley and Meyer [1973] and many other papers—summarized in Anderson, Belnap and Dunn [1992], Chapter IX—with later simplifications suited to various logics in, for example, Priest and Sylvan [1992], Restall [1993b], Meyer and Mares [1993]). We introduce this via an intermediate notion of a model, already present in Routley and Routley [1972]. Exercise 8.13.14 Just before 8.13.11, we introduced the simple Routley models (S, *, V ) – or Routley–Routley models, if you prefer – and we now consider expansions thereof to structures (S, *, 0, V ) in which 0 ∈ S and, as well as the involution condition a** = a for all a ∈ S, we have a special condition regarding 0, namely that 0* = 0. We trade in the sequents of our earlier discussion for formulas of the language with connectives ∧, ∨, ¬ and →, but only such formulas as have no occurrence of → in the scope of an occurrence of → (‘first-degree formulas’). Such a formula holds in a model if it is true at the 0 element of that model, in the sense of the earlier truth-definition augmented by the clause: (S, *, 0, V ) |=0 A → B iff for all x ∈ S, if (S, *, 0, V ) |=x A then (S, *, 0, V ) |=x B. (i) Show that all tautologies constructed from ∧, ∨ and ¬ hold in all models. (ii ) Which of the six formulas listed below hold in all models? p → ((p ∧ q) ∨ (p ∧¬q)); ¬(p ∧ q) → (¬p ∨¬q); p ∨ (q → p); ¬(((p → q) ∧ p) ∧¬q); ((p ∨ q) ∧¬p) → q; ¬(p → q) ∨ (p → (p ∧ q)). (The last formula appeared as (p → q) ⊃ (p → (p ∧ q)) at the end of the Digression following 7.24.6, p. 1095.) Notice that truth of → formulas was only defined for the 0 element of the model. The restriction to first-degree formulas means that in evaluating a formula at this element, we never need to consider the truth of → formulas at other elements, since ∧ and ∨ do not take us away from the point, x say, we start with, and ¬ does, but only to x*, which when x is 0, is again 0. Routley and Routley [1972] think of 0 as representing the real (or actual) world, amongst a host of more abnormally behaving set-ups; at these other elements a formula and its negation can both be true, and a formula and its negation can both fail to be true, but since 0* = 0, such possibilities are ruled out for 0. In respect of the “A and ¬A both true” half of the picture then, this corresponds to endorsing the moderate and rejecting the radical motivation for paraconsistency. (The set-ups other than 0 would represent how things may be, compatible with this or that theory.) We remain in Fmla for the rest of the discussion. To extend the semantics of 8.13.14 to cover formulas with arbitrarily embedded occurrences of → it is necessary to expand the models with some further interpretive machinery. As reported earlier, adding a binary operation à la Urquhart [1972] does not work out in the presence of disjunction alongside conjunction and implication, so Routley and Meyer [1973] employ instead a ternary relation R with models now taking the form (S, R, *, 0, V ), where certain further conditions have to be satisfied, for the formulation of which two definitions are usually made; first, define for a, b ∈ S:


1202 a b if and only if R0ab. Secondly, define a quaternary relation R2 on S by:

R2 abcd if and only if ∃x ∈ S(Rabx & Rxcd ). The role played by “Rabc” in the present semantics is that played by “a·b = c” in the Urquhart semilattice semantics (as notated by us in 2.33 and elsewhere; we changed Urquhart’s ‘0’ to ‘1’ for conformity with this multiplicative notation, but leave the Routley–Meyer ‘0’, which plays a similar role, as it is). In fact Fine [1974] (or see Fine’s §51 in Anderson, Belnap and Dunn [1992]) gives a semantic account in which is primitive rather than, as above, defined, as well as an Urquhart-style binary operation ·, in terms of the two of which the Routley–Meyer ternary relation can be defined: Rabc ⇔ a · b c. But here we are following Routley and Meyer. (Some other ways of motivating the use of the ternary relation as it figures in the clause for implication above are outlined in Chapter 3 of Mares [2004].) The conditions that must be satisfied by the frames (S, R, *, 0) of the Routley–Meyer models, in addition to the conditions given in 8.13.14, are the following, understood as requiring for all a, b, c, d ∈ S: (P1) (P2) (P3) (P4) (P5)

aa Raaa R2 abcd ⇒ R2 acbd R2 0abc ⇒ Rabc Rabc ⇒ Rac∗ b∗

(In fact, Routley and Meyer [1973] do not themselves impose the condition that 0* = 0, from 8.13.14, on frames, instead calling a frame—or as they say, a “relevant model structure” normal when it satisfies this condition, and show that the same logic is determined whether or not the condition is imposed. For the tautologous {∧, ∨, ¬}-formulas all to be valid on every frame, in particular, we don’t actually require the stipulation that 0 = 0*; it suffices that we always have R00*0, which follows from (P1)–(P5). To see this begin with the special case R0*0*0* of (P2), which yields R0*00 by (P5) and the involution condition a** = a. The desired result then follows by 8.13.16(ii) below.) Models on these frames have a special persistence-like condition to satisfy too: V (pi ) must be not just any subset of S, but a subset so chosen that whenever a ∈ V (pi ) and a b, we have b ∈ V (pi ). The new clause in the definition of truth at a point – this time an arbitrary point, and not just at 0 – deals with → and runs as follows, for M = (S, R, *, 0, V ): M |=x A → B iff for all y, z ∈ S for which Rxyz, if M |=y A then M |=z B. A formula is valid on a frame (S, R, *, 0) if it is true at the element 0 in every model on that frame. We should check that the persistence-like condition stipulated for propositional variables in the definition of model generalizes to arbitrary formulas: Exercise 8.13.15 Show by induction on the complexity of A that for any model M = (S, R, *, 0, V ) with a, b ∈ S, if a b and M |=a A then M |=b A. We will not spell out in full the axioms for R again here, having done so in 2.33, but remind the reader that, for the language with connectives ∧, ∨,


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¬, and → (so we will ignore, inter alia, fusion and the Church and Ackermann constants) the story can be summed up as follows. The purely implicational axioms are those of Identity, Contraction, Permutation and Prefixing, with the rule Modus Ponens, all of which is equivalent to saying that we have the implicational logic BCIW (see 2.13, 7.25). We have also have a rule of Adjunction, some ‘formula’ formulations of the introduction rules for ∧ and ∨, as well as a Distribution axiom governing the interaction of the last two, and for ¬ we will repeat the axioms here; they are given in schematic form: (A → ¬B) → (B → ¬A)

¬¬A → A.

Let us check that an arbitrary instance of the first schema is valid on every frame. If not, then there is a model M = (S, R, *, 0, V ) with the formula in question false at 0, so there are a, b ∈ S with a b and M |=a A → ¬B while M |=b B → ¬A; thus for some c, d ∈ S we have Rbcd and M |=c B while M |=d ¬A, so M |=d* A. By (P5), since Rbcd, we have Rbd *c*. By 8.13.15, since M |=a A → ¬B and a b, we have M |=b A → ¬B, so, as Rbd *c* and M |=d* A, we get M |=c* ¬B. But this contradicts the fact that M |=c B, since c** = c. Exercise 8.13.16 (i) Is the relation defined as above on the elements of S in a model a partial ordering of those elements? (ii ) Show that for any frame (S, R, *, 0), if Rabc then Rbac, for all a, b, c ∈ S. (iii) Check that all instances of the schema Contrac are valid on every frame. (Note: (ii ) will be useful, as well as (P2), and devious appeals to (P3) are also required.) We have done as much as will be done here towards the proof of the soundness part of the soundness and completeness theorem for R relative to the present semantics. The reader may consult Routley and Meyer [1973] for the rest, where it is shown that the formulas valid on every frame are exactly those provable in R. Although the ternary relation in the frames is a striking novelty (for more information on the history of which, consult Chellas [1975] and Dunn [1986]), we have only introduced the Routley–Meyer semantics here for the sake of looking at what happens to De Morgan negation in a richer setting than just the firstdegree formulas (as in 8.13.14). We have already seen an eyebrow raised at the intrusion of the *-function, in quoting Dunn’s remarks about the comparative merits of the American Plan over the Australian Plan. Stronger objections still were voiced in Copeland [1979], basically lamenting the inadequate motivation for treating negation via this function, while offering hardly anything to connect it up with our ordinary linguistic practices. A soundness and completeness result with respect to a semantics lacking any such motivation can hardly be seen as furnishing any justification for a logic. An occasionally acerbic interchange followed, in which several interesting points emerged which we cannot review here; see Routley, Routley, Meyer and Martin [1982], Copeland [1983], Copeland [1986], Meyer and Martin [1986]. One subsequent reaction – Mares [1995] – was to eliminate the *-function without moving to the American Plan’s ambivaluations, by using the idea of ¬A as A → f (where f is the Ackermann constant from in 2.33), and


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then finding a way of explaining the semantic significance of f which did not re-introduce the *’s. The point behind Copeland’s criticism was antecedently familiar amongst workers in relevant logic – even the architects of the Australian Plan, as we see from the following remarks from Meyer [1974b], p. 80: “Granted that negations should be true if what they negate is false, whence comes this ¯ true at third base iff A is false at first base?”. (“ A” ¯ crazy switch that makes A is our “¬A”.) One strand in Copeland’s criticism, alluded to above, was that relevant logicians were supposed to be keeping the meanings of ∧, ∨ and ¬ “classical” while introducing a non-classical novelty in the shape of a relevant implicational (or in the case of E, entailment-expressing) connective: but the appearance of the clause for negation which uses the * function betrays this project. After all, for a conjunction or a disjunction to be true at a ‘set-up’ it is the truthvalues of the components at that set up which matter, and in the Routley–Meyer semantics (as in the simpler Routley–Routley semantics) they do indeed matter in exactly the expected way. Suddenly, when it comes to negation, the truthvalue at some other point of the formula negated is what we are told to attend to. Now this anomaly had not escaped the attention of the pioneers of the Routley– Meyer semantics, and in Meyer and Routley [1973], [1974], they developed what they called “classical” relevant logics, by which was meant relevant logics in which some 1-ary connective, and here represented by ∼, behaved more like ¬ according to CL than De Morgan negation itself does. This new connective (new to relevant logic, at least) came to be called Boolean negation. (We capitalize “Boolean” here, by contrast with our use in talk of boolean valuations, partly in deference to the tradition, and partly for parity with the contrasting “De Morgan negation”.) The idea is basically semantic. Instead of saying, as for De Morgan negation— or relevant negation as it is called in Meyer and Routley [1973]—that the negation of a formula is true at a point a (in a model) if the formula negated is not true at a*, we simply make the much more familiar demand that the formula negated is not true at a itself. (Meyer [1974b] calls De Morgan and Boolean negation inferential and semantic negation, respectively.) However, the persistencelike property of 8.13.15 threatens trouble in the following way: we can never have points a, b with a b unless as well as everything true at a is true at b (in some given model), we also have everything true at b true at a. The reason is that if, when a b, we have a formula C true at b, then ∼C will not be true at b, so by 8.13.15—if it held—we would have ∼C not true at a, and hence C true at a. But in any case how would we prove the inductive step for ∼ in a version of 8.13.15 for this setting? Meyer and Routley solve this problem by revising the notion of a frame, imposing the requirement that a b (i.e., R0ab) only if a = b, and checking that this does not make any difference to the positive fragment of R – in particular, that no new ∼-free formulas become valid when attention is restricted to validity on frames satisfying this special condition. The upshot of this is that adding ∼ with suitable axioms either to positive R or to full R produces a conservative extension. In fact the supplementary principles which are suitable (i.e., which render the extension sound and complete w.r.t. the class of all frames) suggested by Meyer and Routley comprise one axiom(-schema), here called (∼ ∼E) and one rule (for which we supply their label): (∼ ∼E)

∼ ∼A → A


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(Antilogism) From: (A ∧ B) → C to: (A ∧ ∼C) → ∼B Note that since we are in Fmla, the Antilogism rule is meant to be a rule of proof, yielding new theorems from old. (The status of rules in relevant logic is thoroughly discussed in Brady [1994]. On the specific case of Antilogism itself, but with De Morgan negation – the standard negation of relevant logic – in place of Boolean negation, see Anderson and Belnap [1975], p. 263, where the disastrous effects of such a rule are noted.) Exercise 8.13.17 Prove from the axiomatization of R with Boolean negation just provided, formulas (i)–(iv ), and the derivability of the contraposition rule (v): (i) p → ∼ ∼p (ii ) (p ∧ ∼p) → q (iii) p → (q ∨ ∼q) (iv ) Taking for granted the provability in R of all formulas of the form (A ∧ (A → B)) → B, show that we always have, in the present extension of R, (A → ∼A) → ∼A. (Hint: begin by substituting ∼A for B in the schema to be ‘taken for granted’. That schema is actually interdeducible with Contrac given the other implication-conjunction principles standardly used to axiomatize R, and the schema we are asking for a proof of might be called ∼-Contrac since it is a Fmla analogue, with ¬ replaced by ∼, of the Set-Fmla rule ¬-Contrac from 8.13.2(ii), p. 1187, and subsequent discussion.) (v) From: B → C to: ∼C → ∼B (Note: since we are not presuming the presence of the constant T which is an identity element for ∧, in R, an argument which takes the “A” of the above statement of Antilogism to be T is not acceptable.) Of course, 8.13.17(ii) and (iii), which are obviously valid (on every frame) according to the semantics given, do not look like the sorts of things that have a place in a relevant logic. Even setting aside the fact that ∼ is being described as some kind of negation, so that (ii), for instance, must be some kind of Ex Falso principle traditionally regarded as anathema in relevant logic, we have conspicuous violations in (ii) and (iii) of the Belnap variable-sharing criterion of relevance (see 2.33). How should we react to this? Is a choice called for? (“Will the real negation please stand up?” – Dunn [1986], p. 214.) Or can the two negations cohabit peacefully? Is it even obligatory upon the relevantist to concede the intelligibility of Boolean negation? (What indeed is a relevantist? See Dunn and Belnap [1981], Meyer [1978].) It may have seemed bad enough that in discussing the ‘Lewis derivation’ of Ex Falso Quodlibet a distinction had to be made between two kinds of disjunction—the extensional disjunction ∨ for which (∨I) was fine, and disjunctive syllogism (A–or –B, not–A ∴ B) was not, versus intensional disjunction (fission), for which these possibilities were reversed. But now we hear that it is not just the or (in the parenthetical representation of disjunctive syllogism just given) that can be taken in either of two ways, but also the not. And

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in fact, if fission is regarded as a disjunctive notion and can be defined in terms of negation and relevant implication by setting A + B = ¬A → B, perhaps there is even room for a third disjunction-like connective, ⊕, say, where A ⊕ B = ∼ A → B. (By 8.13.18 below, this would be a non-commutative disjunction. On the other hand, so would a definition with definiens ¬A → B in IL: but no-one criticizes intuitionistic logic for ‘proliferating disjunctions’, so this line of complaint may not stand up on examination. We looked at another non¨ B, CL-equivalent to A ∨ B, in 7.22.) Perhaps, commutative IL compound, A ∨ however, we don’t have to welcome all of this menagerie after all. In §4.2 we argued that, pace Belnap [1962], conservative extension of a favoured logic should not be regarded as a sufficient condition for one favouring that logic to concede the intelligibility of the connective(s) added in the extension. A principle ground for this conclusion was that extensions each separately conservative might not be conservative when taken together. (See the discussion preceding 4.23.1, p. 569.) Priest [1990] gives an argument of this form for not being swayed by the Meyer–Routley conservative extension result into conceding the intelligibility of Boolean negation. The situation is a little less straightforward than in §4.2, however, because while one extension of R is the logic we have just been discussing, R with Boolean negation added, the other ‘extension’ is not what would normally be thought of as a logic, but a theory, namely a naive theory of truth in a suitable relevant logic, non-trivial without Boolean negation in the language (governed by the principles above in the logic) would become trivial if Boolean negation (subject to those principles) were added. And matters are further complicated by the fact that the ‘suitable’ relevant logic with which to make this point had better not be the far-from-contractionless R itself, since a semantic analogue of Curry’s Paradox (see notes to §7.2 – on 7.25 – and Geach [1955]) actually trivializes the naive theory of truth when cast as an R-theory. But regardless of the details (for which see Priest [1990]) the general moral of our discussion of the conservative extension criterion in §4.2 applies here: there may be reasons of any number of kinds for rejecting the intelligibility of a connective (in the ‘inferential role’ sense of connective, of course), of which non-conservative extension of a favoured logic is only one. Yielding an extension which is unable to support a non-trivial theory of some notion to whose intelligibility one is otherwise committed would be another. Note that this requires a paraconsistent logic if the theory in question is inconsistent (in the sense of proving some formula and its negation – here taking negation as ¬). In the present case this would be a notion of truth as expressed by a truth-predicate which applies to all sentences of the language, including those in which this predicate appears, and is also subject to a disquotational principle. This is what we had in mind under the name of ‘naive theory of truth’ above. The situation thus turns out to be somewhat different from the case investigated in 4.32 of the idea of intuitionistic and classical negation (¬i and ¬c , as we provisionally denoted them) living side by side. There we found that, because of what amounts from the perspective of the Kripke semantics for IL to the non-persistence of ¬c -formulas, certain assumption-discharging rules whose holding-preservation properties required that the undischarged assumptions all be persistent, had to be restricted. We concluded that adherents of IL were well within their rights to refuse such restrictions, and to adopt an attitude of intolerance to the interloping ¬c . But this is because with the rules unrestricted, we


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get a non-conservative extension, and that is not what – at the level of logics – happens in the present instance, even though at first sight it is again precisely a question of persistence that is involved. On the other hand, Dunn suggests ([1986], p. 214) that it is only because of—what we would call—a restriction to Fmla that the non-conservativeness result is available, and that it disappears on passing to Set-Fmla. Set-Fmla relevant logic in anything but the pure implicational fragment is something we are studiously avoiding because of the conceptual complications, however, so we refrain from discussing that suggestion here. With the epistemic gloss on the Kripke semantics for IL in mind, it may be said that trying to have a clause to the effect that something, ¬c A, say, is true at a point iff A is not, is a doomed attempt at manufacturing a new proposition deemed magically to have been established at a given stage of research by the mere non-establishment of A. With the American plan and its subsets of {T, F} understood on the Told-True, Told-False (and combinations thereof) pattern, Boolean negation of A would be something such that not being told A guaranteed that you were told it – no gaps, and such that once you were told it, that guaranteed that you wouldn’t be told that A – no gluts. It seems again pure magic for anything to play such a role. The philosophical issues arising here would appear to warrant closer attention than we can give them here. Turning, then, to more tractable matters, we may ask—following Meyer and Routley [1973], p. 57—whether, where A(¬) is any formula containing occurrences of ¬ (and for simplicity, not containing ∼) and A(∼) is the result of replacing all those occurrences of ¬ by ∼, we have the following, CR denoting the result of extending R, in the language with both negations, by the above principles governing ∼: If R A(¬) then CR A(∼), for all formulas A(¬). Here we have used the A(#) notation from our discussion in 3.24 of the subconnective relation. The question asks, in that terminology, whether according to CR, De Morgan negation is a subconnective of Boolean negation, keeping all other connectives fixed. The “C” here is for “classical”; Meyer, Routley and others have used several variations on the label “CR” to denote the version of R with Boolean negation, sometimes with and sometimes without De Morgan negation as well. (See note 4 of Meyer and Mares [1993].) Meyer and Routley give a simple counterexample to the above implication, returning a negative answer to the subconnective question: Example 8.13.18 R (p → q) → (¬q → ¬p) but CR (p → q) → (∼q → ∼p). Although we do not in general have CR (B → C) → (∼C → ∼B), whose semantic validation would require the unprecedented permutation condition Rabc ⇒ Racb, we do have the ‘rule (of proof) form’ of this schematic implication, as noted at 8.13.17(v). (Semantically, this is possible because the 0 element of the frames, validity on all of which corresponds to provability, is subjected to special conditions not applying to arbitrary elements.) Digression. Unprecedented though the permutation condition alluded to in 8.13.18 might be—and notice that together with 8.13.16(ii) it has the effect that Rabc ⇔ Rdef where def is any permutation of abc—it still does not collapse

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the → into the → of CL or IL, as the following example, provided by Meyer, reveals. Let (S, R, *, 0) be as follows, S is a two-element set {0, a}, R is the relation { 0, 0, 0 , 0, a, a , a, a, 0 , a, 0, a , a, a, a }, and * is the identity map. Consider any model on this frame obtained by setting V (p) = V (q) = {a}. In such a model we have p → (q → p) false at 0, so → is still not behaving as in CL. In fact, the extension of CR which is called KR by Meyer and which has the added axiom ¬A ↔ ∼A, identifying De Morgan and Boolean negation, does not therefore have p → (q → p) as a theorem. (Treat A ↔ B here as abbreviating (A → B) ∧ (B → A).) The semantic side of KR is taken care of by restricting attention to those frames in which the *-operation is the identity map, the effect of which is that we might just as well drop it from the models, along with one of the two negations from the language, and just use the Boolean clause in the truth-definition, along with the ‘unprecedented’ permutation condition on the relation R. (Usually—e.g. in Urquhart [1984] KR is formulated in a language with just the one negation connective, in view of the above equivalence. See also §65—written by Urquhart—of Anderson, Belnap and Dunn [1992].) Philosophically, this is an interesting response to Dunn’s challenge “Will the real negation please stand up?”, since now there is only one candidate in the room. It also shows that the following passage from Meyer and Routley [1973], p. 57, is in error. In quoting the passage, we adjust notation to match our own. We are in the neighborhood of apparent breakdown, since all principles mentioned thus far valid for the relevant negation ¬ are also valid for ∼, which also admits paradoxes; but if paradoxes are added to R, it breaks down all the way to classical logic. Conclusion: there must be some classical negation principle which is valid for the relevant negation ¬ in R which is not valid for ∼ in CR. That’s true, and further inspection reveals the principle in question to be contraposition.

The contraposition principle in question was the subject of 8.13.18, but the above reasoning (about breaking down “all the way to classical logic”) is flawed, as the case of KR shows, since the CR-unprovable contraposition principle is KR-valid and KR exhibits no such breakdown. There remains the question— not answered here—of whether the positive fragment of KR is the same as that of R, or whether in adding this simultaneously De Morgan and Boolean negation connective, we have effected a non-conservative extension. (Thanks to Bob Meyer for most of the content of this Digression.) End of Digression. The fact that CR (p → q) → (∼q → ∼p) means that there is no way of representing ∼A in any conservative extension of CR as A → X for some constant X (since otherwise the missing contraposition theorem here would be forthcoming via Suffixing); in particular the Church constant F , with its characteristic axiom-schema F → B, cannot be play the part of such an X. (See further 8.13.19(iii).) On the other hand, if we have ∼ to hand, as in CR, then in view of 8.13.17(ii), we can indeed define F as (e.g.) p ∧ ∼p, and all instances of the F schema will be provable. This in turn means, of course, that in general ∼A is not synonymous in CR with A → (p ∧ ∼p). This means that one of the following three possibilities must obtain, where for definiteness, we have taken A as q:


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(1) Formulas B and C can be such that CR B → C and CR C → B, even though B and C are not synonymous according to CR (i.e., not inter-replaceable in theorems in such a way as to preserve provability); (2) CR (q → (p ∧ ∼p)) → ∼q; (3) CR ∼q → (q → (p ∧ ∼p)). We can easily rule out possibility (2). A substitution instance of 8.13.17(ii) gives p ∧ ∼p provably implying ∼q, so CR (q → (p ∧ ∼p)) → (q → ∼q), but then the ∼-Contrac schema 8.13.17(iv), putting q for A, allows us to replace the consequent of the above theorem by ∼q itself, refuting (2). Exercise 8.13.19 (i) Which of the remaining possibilities, (1) and (3), obtains? (ii ) Abbreviating (A → B) ∧ (B → A) to A ↔ B, are all instances of the following extensionality-like schema provable in CR? (A ↔ B) → (∼A ↔ ∼B). (Note: The way one would normally expect to show such a thing is blocked by 8.13.18.) (iii) For the reasons given, Boolean negation cannot be defined in CR with the Church constant F by: ∼ A = A → F , but let us regard the latter as defining a new kind of negation (‘Church negation’, say – though Church [1951a] actually has more than one suggestion to make here), and write it as A. Since only → and F are involved in the definition, it makes sense for R, and indeed for linear logic, as in 2.33. Understanding it in the latter context, which of the following negationlike properties does enjoy (i.e., which of the following sequents is provable in CLL (classical linear logic) with the present understanding of )? p, p q; p → p p; p → p p; p p; p p. (iv) Which of the sequents listed under (iii) would change in status from unprovable to provable if we moved from linear logic to R? (For present purposes, think of R as CLL with the structural rule of Contraction, not worrying about issues of {∧, ∨}-distribution.) (v) Discuss the possibility of providing left and right insertion rules for , taken as primitive, to be added to those for classical or intuitionistic linear logic as given in 2.33 and which when so taken, render provable p p → F and the converse sequent, while extending CLL or ILL conservatively. (Comment: there is no problem in the case of ( Left); compare 2.32.14 – p. 322 – though there we were not in a multisetbased framework.) Do matters change significantly if Left Contraction is added to the rules? The arguably mysterious *-function can be expressed directly by a oneplace connective in the object language. We write this connective as ‘*’. The appropriate clause in the truth-definition (where M = (S, R, *, 0, V )) is: M |=x *A iff M |=x∗ A.

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Then if we had taken ∼ and * as primitives, we could have defined ¬A as *∼A, or equivalently, as ∼*A. On the other hand, since we take the language of CR to have both ∼ and ¬ as primitive, we could define * by: *A = ¬ ∼A (or again by: *A = ∼ ¬A). Exercise 8.13.20 Can we define ∼ in terms of ¬ and * (and if so, how)? For a treatment of relevant logic in which * and ∼ are taken as primitive, see Meyer and Routley [1974]. (The authors write A* in place of our *A, and ¬A in place of ∼A.) More on the interrelations between the two negations and * may be found in Meyer [1979]. See also Dietrich [1994]. Since in this discussion of negation in relevant logic we have had a brief look at the Routley–Meyer semantics, we take this opportunity to close by presenting the example – due independently to J. M. Dunn and R. K. Meyer (see Dunn [1986], p. 198f., or Dunn and Restall [2002], p. 65f.) – of a formula in which negation (Boolean or De Morgan) makes no appearance, but which, though not provable in R (and thus not valid on every frame in the Routley–Meyer semantics), is valid on every semilattice frame à la Urquhart [1972], when the same clauses are used for conjunction and disjunction as in the Routley–Meyer semantics. (6.45 above examines the possibility of making a change to the Urquhart semantics in respect of disjunction in order to avoid this mismatch between validity and provability in R). Example 8.13.21 The formula [(p → (q ∨ r)) ∧ (q → r)] → (p → r) is valid according to Urquhart’s semantics though not provable in R. For, omitting the initial stage of the semantic evaluation, what we soon arrive at asking is whether we can have a point x in an Urquhart model (with semilattice operation ·) at which the antecedent is true but not the consequent. This means that p → (q ∨ r) and also q → r are to be true at x, but there exists a point y with p true at y, and r false at x · y. Thus since p → (q ∨ r) is true at x and p is true at y, q ∨ r must be true at x · y, and since r isn’t true there, q must be. Now since also q → r is true at x, since q is true at x · y, x · (x · y) has r true at it. Since by idempotence and associativity x · (x · y) = x · y, this makes r true at x · y, contradicting our earlier assumption which had r false there. Thus the formula is valid on every semilattice frame. When we compare this with the Routley–Meyer semantics, we begin the same way (i.e. omitting the very first stage of the reasoning) with the assumption that the formula is invalid on (S, R, *, 0)—not that * will enter the reasoning—leading us to a model on this frame with x ∈ S verifying p → (q ∨ r) and q → r but not p → r. Thus there are y, z ∈ S with Rxyz and p true at y, r false at z. As before, the assumptions force q ∨ r to be true at z, and hence q to be true at z. But the truth at x of q → r is now powerless to give us a contradiction, since this would only arise if we had to make r true at z on the grounds that Rxzz (since q → r is true at x and q true at z). Nothing in the semantic apparatus tells us that Rxxz, however. (Recall that we have Rxxx, Ryyy, and Rzzz, by the contraction-validating condition (P2) on frames, but nothing to force Rxyz ⇒ Rxzz.) A somewhat more complicated version of the formula discussed here is cited in note 3 of Routley and Meyer [1972]; see also Urquhart [1972], p. 163, and Anderson, Belnap and Dunn [1992], p. 151.


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Notes and References for §8.1 Reading. For all matters pertaining to natural language negation, the indispensable reference is Horn [1989]. Particularly useful references for the more formal side of things are (in order of publication) Meyer [1973], Vakarelov [1989], Dunn [1993] and [1996], and Vakarelov [2006]. Chapter 6 of Curry [1963] is devoted to negation and is highly regarded, though somewhat impenetrable and with various eccentricities. Vakarelov [1989] works with ∧ and ∨ (in the framework Fmla-Fmla); a purified discussion (in Set-Fmla) may be found in Béziau [1994]. (Béziau [2006], p. 19, line 7 from base, seems to deny that intuitionistic negation forms contraries – by contrast with the clam of 8.11 that the intuitionistic negation of a formula is its weakest contrary – but readers will have to make up their own minds on this score.) Several collections of papers are devoted to the subject of negation, including Wansing [1996a], Gabbay and Wansing [1999]. Aside from the references given in 8.12 on the subject of approaching negation via rejection, the discussions in Rumfitt [1997], [2000], Gibbard [2002], and Dummett [2002] may be of interest, as will be Wansing [2010] which discusses rejection, strong negation, dual intuitionistic negation and intuitionistic negation itself.

Defining a Connective in a Theory: A Further Example. In the discussion between 8.12.8 and 8.12.9 the topic (set aside in Remark 3.16.3(iii), p. 427) came up of defining a connective, not, as described in (3.15 and) 3.16, within a logic, but rather within a non-logical theory: defining ⊥ in Heyting Arithmetic (a version of Peano arithmetic with intuitionistic predicate logic as the underlying logic) as 0 = 1. A further example of considerable interest in its own right is given here: the Hinnion–Libert paradox for naive set theory, as purified of explicitly appearing sentence connectives by Greg Restall. Restall [2008] discussed treatments of set theory, arithmetic, and semantics (a theory of the predicate “is true”) in a logical setting presupposing no particular stock of connectives answering to prescribed rules (for example, a →intuitionistically behaving implication connective). To make up the expressive weakness that results from abandoning any familiar apparatus of connectives, it is necessary to depart from the characterization of theories given in 1.12 – as sets of formulas closed under a consequence operation/relation – essentially by taking them as sets of Set-Set sequents closed under the structural rules definitive of gcr’s. (The account of theories in Restall [2008] is different from this, we note in passing.) Further non-structural rules thus give the content of particular theories, as for the following version of naive set theory, in which the usual conventions of predicate logic are followed (and t and u are any terms, v any variable); Restall’s notation has been used for the rules (and their names – sequent calculus syle naming for the ∈ rules, (Ext∈ ) for a set-theoretic extensionality principle), with “” retained rather than being replaced by the separator “”. The primitive predicates are ∈ and =, and we have terms {v | ϕ} (“the set of all v such that ϕ”) where ϕ is a formula, in which occurrences of v not already bound become bound. (In the discussion of Curry’s Paradox in the notes to §7.2, p. 1123 and beyond, we wrote “A(v)” for ϕ(v), but here we need lower case letters to make room for a ‘cap’/‘hat’ on top of them – “ ϕ ” – in the developments below.)


1212 Γ, ϕ(t) Δ

(∈ Left)

(Ext∈ )

(∈ Right)

Γ, t ∈ {v | ϕ(v)} Δ Γ, v ∈ t v ∈ u, Δ

(= Left1 )

Γ, v ∈ u v ∈ t, Δ

Γ t = u, Δ Γ, φ(t) Δ Γ, t = u, φ(u) Δ

(= Left2 )

Γ ϕ(t), Δ Γ t ∈ {v | ϕ(v)}, Δ v not free in Γ ∪ Δ Γ φ(t), Δ Γ, t = u φ(u), Δ

Restall’s aim here is to avoid getting the development of set theory (or in other examples from Restall [2008], arithmetic and the theory of truth) getting bogged down in the details of this or that preferred implicational or negational (etc.) connective with prescribed logical behaviour. The above rules illustrate how to do this for the case of (naive) set theory. Restall’s discussion shows how to purify completely the discussion of Hinnion and Libert [2003] of any explicit appeal to any such connectives (including also ∧ and ∨), by deriving q – the “q” notation explained presently – using the rules above, structural rules included. The ingenious derivation will not be reproduced here, since it can be found in Restall [2010a], should no form of Restall [2008] itself be available, but its gist can be conveyed informally in the following terms. Let Q be the following (admittedly, somewhat impenetrable) term: x {y | x ∈ x} = {y | q} where q is a place-holder for any sentence of the language of the current theory (i.e. constructed using the primitive vocabulary in play above), which is allowed to contain free variables but with y not being one of them. This is intended as a notational reminder of the use of the (non-italically written) sentence letter “q” (the propositional variable p2 , that is) for continuity with the Curry’s Paradox discussion in the notes to §7.2, where it was similarly paired with Q as the name of a set. (Restall [2008] writes the above Q as “H” in honour of Hinnion.) Now, to make trouble, unpack the statement that Q ∈ Q by substituting Q for x in the inset equality. Since q was chosen arbitrarily, the result is that everything is provable, as with Curry’s Paradox in its original form. Restall concludes that one or other of the non-structural rules given above must be abandoned: the troubles of (this version of) naive set theory are not to blamed on injudicious assumptions about the behaviour of any particular connectives in one’s primitive vocabulary (on a contraction principle for →, for example: see the discussion of Curry’s Paradox in the notes to §7.2: p. 1123). One should not conclude that no connectives are definable, however, or that it is not useful to attribute various pathological features of these theories to the behaviour of such connectives. One might think that since the primitive vocabulary for this part of Restall [2008] included ∈, =, and the abstraction notation {v | . . . v . . .} but no sentence connectives, that the latter can play no role in the derivations of anomalous results such as that just informally described. This would be wrong. Not only is the binary connective ↔ of material equivalence (i.e., ↔ with its familiar classical properties, as in 3.11.2, p. 378) definable from these resources – whence the status of the present example as one of definability of connectives in a theory – but the above Hinnion–Libert


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paradox uses in fact precisely the key features of this connective that yield a certain ‘biconditional’ variant of Curry’s Paradox. (This variant is discussed in Rogerson and Butchart [2002] and Humberstone [2006b].) For the variant in question, using the above notation, we let Q be {x | x ∈ x ↔ q} and then note that taking Q as x we get Q ∈ Q ↔ (Q ∈ Q ↔ q), from which q follows by classical logic (though not by intuitionistic logic, let us recall from the discussion preceding 7.31.13, p. 1136). Putting this in terms of consequence relations the point would be that from Q ∈ Q Q ∈ Q ↔ q we obtain q when is taken as the consequence relation of classical logic. Now all the required properties of ↔ can be obtained from the above rules from Restall [2008] for doing set theory. When ϕ is a formula not containing the variable x free, the notation {x | φ} still makes sense, we are assuming, and rules (∈ Left) and (∈ Right) mean that in this case, for arbitrary terms t and u, the formulas t ∈ {x | ϕ} and u ∈ {x | ϕ} are equivalent (in the sense), each for being equivalent to ϕ itself. Now we introduce the (ad hoc) abbreviation ϕ the term {v | ϕ} formed by choosing v as a variable not occurring free in ϕ; note that we do not require ϕ to be a closed formulas – i.e., variables other than v may be free in ϕ. Then we can define ϕ ↔ ψ as ϕ = ψ and from (Ext∈ ) and (= Left), the result is that ↔ enjoys its usual classical behaviour. That is, for all ϕ, ψ, we have the following conditions satisfied (as in 3.11.2, p. 378, though given in a different order here): (1) ϕ ↔ ψ, ϕ ψ; (3) ϕ, ψ ϕ ↔ ψ;

(2) ϕ ↔ ψ, ψ ϕ; (4) ϕ ↔ ψ, ϕ, ψ.

(Note that even positing the existence of compounds ϕ ↔ ψ with properties (1) and (4) alone is non-conservative over intuitionistic logic, when the commas on the right of are construed disjunctively: the conditions of 3.18.1 – due to S. Pollard – are satisfied in this case.) For example, we get (1) thus: From t∈ϕ and thus (by the ∈ t∈ϕ t∈ϕ by (= Left2 ) we infer ϕ = ψ, t ∈ ψ, rules (and cut), ϕ = ψ, ϕ ψ, i.e., (1). The derivation for (2) is similar, using (= Left1 ). We get (3) as follows By the ∈-rules we get ψ v ∈ ψ and so by weakening: ϕ, ψ, v ∈ ϕ v ∈ ψ, from which (Ext∈ ) yields with a similar derivation for ϕ, ψ, v ∈ ϕ v ∈ ψ, ϕ, ψ ϕ = ψ, i.e., (3). Dual moves give (4). The formulation with “i.e.” is intended to stress that there is no suggestion that we add a new connective “↔” to the object language. The suggestion is rather that compounds we can reasonably designate (in the metalanguage) by ϕ ↔ ψ are already available and for arbitrary components ϕ and ψ in the shape of the formulas ϕ = ψ, that this enables us to see clearly that the Hinnion–Libert paradox just is the biconditional variant of Curry’s paradox. (Here we employ the metalinguistic as opposed to the object-linguistic conception of definition, as contrasted in the second and third paragraphs of 3.16.) In the formulation of this above, we had Q = {x | x ∈ x ↔ q}. In the current primitive notation “x ∈ x ↔ q” is

“x ∈ x = q”. Choosing y as a variable free in neither of the formulas here, this


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amounts to {y | x ∈ x} ={y| q}. Plugging this into the above definition of Q, what we get is that Q = x {y | x ∈ x} = {y | q} , which is how it appeared in our earlier summary of Restall [2008]’s presentation of Hinnion–Libert. Thus, what is crucial to the pattern of reasoning in a language without primitive connectives is the definability of a connective whose inferential behaviour makes that pattern available. (Accordingly ↔ here plays a role like that of in modal provability logic, as in Smoryński [1985], Boolos [1993]: it delivers us from an inability to see the wood for the trees – to extract the modal ‘fixed point’ pattern from a mass of detailed Gödel numbering, quantifiers, provability predicates. . . ) We have already seen (parenthetically) some reasons – nonconservativity considerations from 3.18 – independent of the Curry’s Paradox issue, for an advocate of IL to be unhappy about the use of the multiple succedents in the set-theoretic principles in play here. They are not as (propositional-)logically innocent as they seem, because of the interaction between formulas and terms provided by the abstraction terms. But – we may close by observing – even if all principles were restricted to having at most one formula on the right of the , trouble could still be made for the intuitionist if we supplemented or replaced a singleton-succedent version of (Ext∈ ) with suitable principles for inserting “⊆”, taken as a new primitive relation symbol, into sequents were supplied. (The ∈-rules and =-rules are retained.) In particular, consider this pair, the second of which we understand as subject to the proviso that v not free in Γ: (⊆ Left)

Γv∈t

Γ, v ∈ u ϕ

Γ, t ⊆ u ϕ

(⊆ Right)

Γ, v ∈ t v ∈ u Γt⊆u

Now we could define an intuitionistically behaving → connective, thinking of ϕ → ψ as ϕ ⊆ ψ and, if we wanted to, we could present the original Curry’s Paradox, with Q = {x| x ∈ x → q}, in Hinnion–Libert style taking Q as x {y | x ∈ x} ⊆ {y | q} . But since the main point here has been to illustrate the idea of defining a connective using the non-logical resources of a specific theory, and this has already been done with the (classical) ↔ example, further details are omitted.

§8.2 NEGATION IN INTUITIONISTIC LOGIC 8.21

Glivenko’s Theorem and Its Corollaries

The title of the present section notwithstanding, our discussion of the behaviour of ¬ in IL is scattered over a considerable area, beginning with 4.32; in fact the notion of (what we shall call) a ¬-intuitionistic consequence relation appears in 8.31. In the present subsection we shall be concerned principally with intuitionistic double negation (“¬¬”), and in the following two subsections, with negation connectives other than ¬ in IL. Results analogous to Glivenko’s Theorem have been found in connection with various substructural logics, but we do treat them here; comprehensive discussion and references can be found in Chapter 8 of Galatos, Jipsen, Kowalski and Ono [2007]. Glivenko’s Theorem, relating IL to CL by the erasure of formula-initial “¬¬”s, was mentioned but not proved in 2.32; as remarked after 2.32.2 (p. 306) this is only one of a cluster of results that are apt to go by the name “Glivenko’s

8.2. NEGATION IN INTUITIONISTIC LOGIC

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Theorem”. We fill that lacuna here, making use of some of the results of our earlier discussion. Also needed will be the following Kolmogorov-style result (see the notes to §8.3, p. 1284): Observation 8.21.1 Let Γ be a set of formulas and A be a formula, and Δ be the set of all formulas pi ∨ ¬pi for pi a propositional variable occurring in any of the formulas in Γ ∪ {A}; then Γ CL A ⇒ Γ, Δ IL A. Proof. With Γ, A, and Δ as described, suppose Γ, Δ IL A, with a view to showing that Γ CL A. By 2.32.8 (p. 311), there is a model M = (W, R, V ) with x ∈ W – and we may without loss of generality take M to be generated by x – with all formulas in Γ ∪Δ true at x, and A false at x. Since the formulas in Δ are true at x, for each pi occurring in any formula of Γ ∪ {A}, we have either V (pi ) = W or V (pi ) = ∅. From this, we claim, it follows that every formula in Γ ∪ {A} is either true throughout or false throughout M, and that the valuation v M assigning T to a formula in the former case and F in the latter is a boolean valuation. (Actually, since we have not defined v M for all formulas, and in particular, amongst the propositional variables, only such as occur in Γ ∪ {A}, we have only partially specified a valuation; however, for definiteness, we may suppose that for all pi not so occurring, V (pi ) = W , so that v M is defined for all formulas.) Now we have v M (C) = T for all C ∈ Γ while v M (A) = F, establishing that Γ CL A. Digression. For various applications—e.g., to fragments of IL without disjunction, it is convenient to use a different formulation of 8.21.1. Let q1 , . . . , qn be all the propositional variables occurring in Γ ∪ {A} and understand by ±q1 , . . . , ±qn any listing of these variables and their negations, in the sense that each ±qi is either qi or ¬qi (for i = 1, . . . , n). Then by the reasoning of the above proof – of, if preferred, as an immediate corollary to 8.21.1 (since each ±qi has the ‘excluded middle’ disjunction qi ∨ ¬qi as an IL-consequence) – we have the following, for any such selected ‘signing’ ±q1 , . . . , ±qn of the qi : Γ CL A ⇒ ±q1 , . . . , ±qn , Γ IL A. To illustrate the interest of this in the simplest possible case, we may infer that whenever A is a formula constructed out of no variables other than p, to emphasize which we write it as A(p), p CL A(p), we also have p IL A(p). Here we are taking Γ as {p} and n = 1, with ±q1 = q1 = p, and the formulation inset above gives us that since p CL A(p), we have p, p CL A(p), i.e., p IL A(p). End of Digression. We are now in a position to prove Glivenko’s Theorem. (“¬¬Γ” denotes the set of all formulas ¬¬C, for C ∈ Γ.) Theorem 8.21.2 For any formula A and any set of formulas Γ: Γ CL A implies ¬¬Γ IL ¬¬A. Proof. Suppose Γ CL A; then by 8.21.1, with Δ as in the statement of that Observation: Γ, Δ IL A. So by 2.32.1(iv) (p. 304), ¬¬Γ, ¬¬Δ IL ¬¬A and since ¬¬Δ consists of formulas of the form ¬¬(B ∨¬B), all of which are intu itionistically provable, we appeal to (T) to conclude that ¬¬Γ IL ¬¬A.

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Corollary 8.21.3 Γ CL ¬A iff Γ IL ¬A. Proof. The unobvious (‘only if’) direction follows from the Theorem, which, given that Γ CL ¬A, implies that ¬¬Γ IL ¬¬¬A, from which it follows that ¬¬Γ IL ¬A by the ‘Law of Triple Negation’ (2.32.1(iii): p. 304), whence Γ IL ¬A by ‘double negation introduction’ (the rule ¬¬I from 1.23). The above proof answers Exercise 2.32.2(ii), p. 306. Remark 8.21.4 Calling a formula refutable if its negation is provable, we see from 8.21.3, taking Γ = ∅, that the same formulas are refutable according to IL as are according to CL ; given the role of refutability in any of the usual definitions of what it is for a set of formulas to be inconsistent w.r.t. a consequence relation, it follows that a set of formulas is classically inconsistent iff it is intuitionistically inconsistent. (The definition of a refutable formula as one whose negation is provable is an informal explanation, since there is no such thing in general as “the negation” of a formula. Of course, here what we have in mind for A as its negation in the case of IL as are and CL is the formula ¬A. In a more luxuriant setting, say with intuitionistic, dual intuitionistic and strong negation all present, a more careful formulation would need to be given. Cf. discussion following 8.13.10 at p. 1192 above.) Corollary 8.21.5 If A is a formula built up using only the connectives ¬ and ∧, then CL A if and only if IL A. Proof. We address the unobvious (“only if”) half of the claim that for A constructed using ¬ and ∧, A is classically provable if and only if intuitionistically provable. Any A so constructed can be written as a conjunction B1 ∧ . . . ∧ Bn , in which perhaps n = 1, with each Bi not itself a conjunction. Clearly, to have CL A, no Bi can be a propositional variable; therefore, since A is built using only conjunction and negation, each Bi is of the form ¬C for some formula C. Thus by 8.21.3, 4, we have IL Bi in each case, and hence IL A. Coro. 8.21.5, as proved here, appeared in Gödel [1933b]. There is a certain laxity in the proof, since it uses multiple conjunction, and for us ∧ is binary. We could take the formula B1 ∧ . . . ∧ Bn to be bracketed in some specific way (say, with parentheses grouping to the left) and allow the usual abuse of language in referring to the Bi as ‘conjuncts’ of this formula – a selected equivalent (both in IL and CL) of the formula A – and still insist that they not themselves have ∧ as main connective. In 2.32, we warned against regarding IL as an extension rather than sublogic of CL in the light of this result. (See the discussion before 2.32.2: p. 306.) Such would-be philosophical applications of 8.21.5 aside, we can use this result— or, more accurately, 8.21.4—to throw some light on the lattice of intermediate logics in Fmla, such logics being thought of as collections of formulas (in the language of IL) containing every formula A such that IL A and closed under Modus Ponens and Uniform Substitution. We will need to make a preliminary observation, with whose content the reader is probably familiar, concerning classical logic.


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Observation 8.21.6 For any non-tautologous formula A (of the language of Nat) there is a substitution s such that ¬(s(A)) is tautologous. The proof of 8.21.6 is left as an exercise; its content is sometimes expressed by saying that any non-tautologous formula has a contradictory substitution instance. (Compare the notion of Lindenbaum completeness of logics as defined in Anderson and Belnap [1975], p. 121: if no substitution instance of A is provable, then the negation of A is provable. We could say “A is refutable” for this last part, but either way, note the dependence on a particular connective selected to play the role of negation here, and the need to be more explicit should several candidates present themselves in that capacity. One would prefer a purer characterization.) Now, the point of interest we have in mind concerning intermediate logics can be conveyed by noting another terminology (especially popular with Russian-speaking logicians), in which such logics are discussed, namely as ‘superintuitionistic’ or ‘superconstructive’. Let us use the former term, defining it in accordance with its etymology: a superintuitionistic logic is a logic extending intuitionistic logic. Then the point to note is that the consistent superintuitionistic logics are precisely the intermediate logics.(Note that we are restricting attention to sentential logic here.) “Consistent” here means that not every sequent (sc. formula, since we are in Fmla) is provable in the logic. It is clear that any intermediate logic is superintuitionistic; the converse is less obvious. Might there not be, that is, a consistent logic – call it X – which extends IL but is not a subsystem of CL? Then (recall we take X as a logic in Fmla) for some formula A, A ∈ X but A is not a tautology. Thus by 8.21.6, A has a substitution instance s(A) with ¬s(A) a tautology. By 8.21.4, then, IL ¬s(A), so since IL is included in X, ¬s(A) ∈ X. But wait: s(A) ∈ X also, by Uniform Substitution, since A ∈ X, and so, since X extends IL and IL s(A) → (¬s(A) → B)) for any formula B, by Modus Ponens twice, we get B ∈ X: so X is inconsistent after all. Thus, setting aside the inconsistent logic, the superintuitionistic logics are just the intermediate logics. (Note that we are speaking of sentential logics here; an analogous claim could not be made in the case of predicate logic. For one thing, the fact that every unprovable formula has a refutable substitution instance does not hold in that setting; for another, 8.21.4 does not hold for intuitionistic and classical predicate logic: see the discussion after 2.32.1 on p. 304.) In spite of the heavy work done by negation in the above argument, a similar property can be established for the ¬-free (and ⊥-free) fragment of IL (Positive Logic) and indeed for the purely implicational fragment; the interested reader is referred to Dale [1985a] – cf. 7.25.4 above (p. 1100). For part (i) of the following Exercise, which Glivenko’s Theorem will be of assistance (for the only if direction; for the if direction, think Peirce). For part (ii), invoke 8.21.5. Exercise 8.21.7 (i) Show that a sequent Γ A is classically provable if and only if for some formula B, the sequent Γ, A → B A is intuitionistically provable. (Additional hint: for the ‘only if’ direction, choose B as ¬A.) (ii ) Show that the {∧, ¬}-fragment of IL, identified as the consequence relation IL restricted to formulas in the fragment concerned, is structurally incomplete (as this phrase was defined before 1.29.6 on p. 162). (Hint: recall that ¬¬p IL p.)

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Since we have been concerned lately with the {∧, ¬}-fragment of IL, which (Gödel’s) 8.21.5 tells us coincides with the same fragment of CL when attention is restricted to the framework Fmla, it is worth seeing what happens when we compare the fate of this fragment in the Set-Fmla versions of these two logics. Whereas Set-Fmla IL (alias IL ) has no finite characteristic matrix, as follows from another famous observation of Gödel’s (establishing this point for the Fmla logic), for the {∧, ¬}-fragment matters stand very differently; although not 2-valued, something Exercise 8.21.8 below asks should be shown, this logic turns out to be 3-valued: see Theorem 8.21.12 below. The Set-Fmla incarnation of IL does not of course coincide with its incarnation in CL, as we emphasised (with the different status of ¬¬p p vis-à-vis these logics) in our introduction to Glivenko’s Theorem in 2.32, in opposition to the suggestion that intuitionistic logic should be regarded as a strengthening rather than a weakening of classical logic. (See the discussion following 2.32.1, p. 304.) Exercise 8.21.8 Show that there is no 2-element matrix at all, in which validity for Set-Fmla sequents coincides with provability in the {∧, ¬}fragment of IL. Before looking at the (matrix) semantics of the intuitionistic logic of conjunction and negation in Set-Fmla, let us begin by rehearsing the classical side of the picture. We need a proof system in each case and choose the natural deduction systems Nat and INat for this purpose, restricted to the rules involving ∧ or ¬ (including (RAA), which in the ‘unpurified’ form considered for these systems when they were introduced in 1.23 and 2.32 respectively, involved both ∧ or ¬). Sequent calculi or proof systems from other approaches to logic, as reviewed in 1.28, would have served equally well. In the case of CL, we recall that the completeness (w.r.t. the class of ∧boolean valuations) of the subsystem of Nat with its only operational rules being (∧I) and (∧E) can be established (1.12.3: p. 58) by the ‘instant’ technique, as we called it, of defining, for unprovable Γ C, a valuation v by v(A) = T iff Γ A is provable, then this v manages, thanks to the rules mentioned, to be an ∧-boolean valuation on which Γ C does not hold. Bringing negation into the picture, we would want also that such a v is ¬-boolean, so it must not verify both B and ¬B for any formula, and it must verify at least one of these two in each case. The former is no problem for v as just defined since, as the negation-conjunction subsystem of Nat renders provable the Ex Falso schema B, ¬B C (see 1.23.4, on p. 119), at least one of Γ B and Γ ¬B must be unprovable. But the latter is more demanding, requiring – the other half of the condition of ¬-classicality (1.13, 62) for the associated consequence relation – that if Γ, B C and Γ, ¬B C are both provable then so also would be (contrary to hypothesis in the present instance) the sequent Γ C. The derivability of this last sequent from using the rules of Nat was Exercise 1.24.2(i) on p. 126; to get our desired invalidating v using this, the we cannot use the ‘instant’ route, but must extend the Γ of the presumed unprovable Γ C to a Γ+ ⊇ Γ, which ‘maximally avoids’ C in the sense that for any A ∈ / Γ+ there is a finite subset Θ + of Γ for which Θ, A C is provable. The characteristic function of Γ+ is then both ∧-boolean and ¬-boolean. In the case of IL with ∨, a similar maximalization strategy was needed to secure the primeness at a corresponding point (2.32.6: p. 311) in the completeness proof for that logic (in Set-Fmla), though if we were doing a completeness


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proof of the present fragment w.r.t. the Kripke semantics, that is not necessary: in the terminology of 2.32, one simply takes the elements of the canonical model to be deductively closed consistent sets of formulas. However, for the completeness result we have in mind here, namely completeness w.r.t. the three-element matrix in Example 2.11.3 (p. 202) – despite the connection there noted to a particular Kripke frame – something along the lines of maximalization, at least to the extent that a Lindenbaum construction is involved does appear to be called for. (See 8.21.13.) We can take for granted the equivalence of the provability of a Set-Fmla sequent in a suitable proof system for IL in the connectives ∧, ¬, with its validity (‘on all frames’) in the Kripke semantics, since as already remarked, the completeness proof given earlier (2.32.8, p. 311) survives – sometimes in a simplified form – the dropping of any of the connectives there taken as primitive. Suitable proof systems include the {∧, ¬} subsystem of INat, with operational rules (∧I), (∧E), (RAA), (EFQ), and also the corresponding subsystem of IGen with operational rules (∧ Left), (∧ Right), (¬ Left) and (¬ Right). Although the equivalence of these two proof systems (over the provability of Set-Fmla sequents, since we recall that IGen is actually couched in the more generous framework Set-Fmla0 ) is most easily seen when all three of the structural rules (R), (M) and (T) are available, what we actually need for some arguments here is the redundancy of (T), though as in 1.27, by analogy with which the system without (T) would be called IGen − , we take this point – essentially the Cut Elimination theorem for Gentzen’s LJ – on trust here, rather going into the somewhat laborious proof. (See Notes and References for §1.2, under ‘Cut Elimination’: p. 191.) Lemma 8.21.9 Where is the restriction of IL to formulas in the {∧, ¬}fragment, for any set of (such) formulas Γ, if Γ pi then either Γ C for all C, or else there exists B ∈ Γ with B pi . Proof. For convenience here we think of as the consequence relation associated with the {∧, ¬}-subsystem of IGen. It suffices, accordingly, to prove the Lemma for Γ assumed finite, and we may argue by induction on the length of a shortest cut-free proof of Γ pi in the indicated proof system. In the cases of a proof of length 1, by (R), we certainly have B ∈ Γ with B pi , since in this case Γ = {pi }. The inductive cases provided by the rules (∧ Right), (¬ Right), do not arise for Γ pi , since the right hand formula is neither a conjunction nor a negated formula). Similarly with (¬ Left), since the rhs is not empty. This leaves (∧ Left) and (M) and. In the case of (∧ Left) Γ pi is then Γ0 , D∧E pi , derived either from Γ0 , D pi or Γ0 , E pi : subcases which are sufficiently similar that it suffices to attend to the first. The inductive hypothesis tells us that since Γ0 , D pi has a shorter proof than the sequent currently under consideration, either (i) Γ0 , D C for all C, or else (ii) there exists B ∈ Γ0 ∪ {D} with B pi . In case (i), by (∧ Left), we have Γ0 , D ∧ E C for all C, i.e., Γ C for all C. That leaves case (ii) to check, which falls into two subcases, depending as whether the promised B with B pi is an element of Γ0 or is the formula D. If B ∈ Γ0 , then since Γ = Γ0 ∪ {D ∧ E}, we have B ∈ Γ with B pi as required. If B = D, then since B pi we have D ∧ E pi by (∧ Left), so again there is B ∈ Γ with B pi . Finally, suppose the current sequent was obtained by (M), which again gives two cases: thinning on the left and thinning on the right. In

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the former case the premiss-sequent for the application of (M) was Γ0 pi , say, with Γ pi being Γ0 , D pi , in which case either Γ0 C for all C giving the result by (M) again – thinning in D on the left, or else there is B ∈ Γ0 with B pi , and thus we have B ∈ Γ ( = Γ0 ∪ {D}) with B pi , as required. In the latter case, the premiss-sequent was Γ ∅, and by thinning on the right we immediately get Γ C for all C. In what follows, instead of saying that Γ IL C for all C, as in the formulation (and proof) of Lemma 8.21.9, where this locution appeared though with “IL ” replaced by “” since only formulas of the current fragment were concerned, we will for brevity simply say that Γ is IL-inconsistent. (This is equivalent to the notion of IL-inconsistency introduced for Lemma 2.32.6, p. 311, in our survey discussion.) Exercise 8.21.10 (i) Show that Lemma 8.21.9 fails if is understood as the {∧, ∨}-fragment of CL rather than of IL . (ii) Show that if, for some set of formulas Γ and formula D, both Γ ∪ {D} and Γ ∪ {¬D} are IL-inconsistent, then so is Γ. Lemma 8.21.11 With as in 8.21.9 and variables ranging over (sets of ) formulas in the fragment concerned, we have: If Γ, ¬A C and Γ, ¬¬A C then Γ C. Proof. It suffices to show the result in the case of C = pi and in the case of C = ¬D for some D in order to obtain the general result claimed, since any C is of the form C1 ∧ . . . ∧ Cn with each Cj of one of these two types. So we could argue that if Γ, ¬A C and Γ, ¬¬A C, then Γ, ¬A Cj and Γ, ¬¬A Cj for each j = 1, . . ., n, in which case assuming the result for the Cj themselves Γ Cj for j = 1, . . . , n, so Γ C1 ∧ . . . ∧ Cn , i.e., Γ C. Let us deal with the case of C = ¬D first. We are to suppose that Γ, ¬A ¬D and Γ, ¬¬A ¬D. Since ⊆ CL , we conclude that Γ CL ¬D by 8.21.3. Finally, suppose that C is pi , and that we have (1) Γ, ¬A pi and (2) Γ, ¬¬A pi . We want to show that Γ pi . By (1) and Lemma 8.21.9 either there is B ∈ Γ ∪ {¬A} with B pi , or else Γ ∪ {¬A} is IL-inconsistent. Similarly, by (2) and 8.21.9 either there is B ∈ Γ ∪ {¬¬A} with B pi , or else Γ ∪ {¬¬A} is IL-inconsistent If the possibilities represented by what follows the “or else” in these two cases both obtain then Γ is IL-inconsistent by 8.21.10(ii) (putting D = ¬A), in which case we have the desired conclusion that Γ pi . So we may assume that at least one of Γ ∪ {¬A}, Γ ∪ {¬¬A}, is IL-consistent. Taking the first case first, since there is B ∈ Γ ∪ {¬A} for which B pi , we note that B cannot be ¬A, as 2.32.9(iii) – p. 315 – would imply that {¬A} was IL-inconsistent, contradicting the IL-consistency of Γ ∪ {¬A}, so the promised B is an element of Γ, in which case Γ pi . The same argument works for the second case (Γ ∪ {¬¬A} consistent). We proceed to show that the sequents provable in this system are precisely those valid in the reduct to ¬ and ∧ of the three-element matrix mentioned in Example 2.11.3 (p. 202), which we shall here call the three-element Gödel matrix. That means: the Kleene–Łukasiewicz ∧-table and ¬ interpreted as mapping 1 to 3, 2 to 3, and 3 to 1. (The designated value is 1.) As mentioned there, this


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is just another way of representing the (point-generated) two-element Kripke frame, the three values corresponding to the three subsets of its universe closed under the accessibility relation. This is the most useful way of establishing the soundness of the present proof system w.r.t. the matrix in question, in fact, and the proof below addresses only the completeness half of the claim. The present (soundness and) completeness result appeared with a different proof in Rautenberg [1987] (Corollary 3, p. 7). Theorem 8.21.12 The {¬, ∧}-fragment of IL in Set-Fmla is sound and complete w.r.t. the corresponding reduct of the three-element Gödel matrix. Proof. As indicated above, we take only the completeness half of this claim as in need of proof. Given IL-unprovable Γ C in the {¬, ∧}-fragment, we begin by extending Γ to a set Γ+ of formulas in this fragment with the property that Γ+ IL C and for every formula A, either ¬A ∈ Γ+ or else ¬¬A ∈ Γ+ . Taking A1 ,. . . ,An ,. . . as an enumeration of all the formulas in the fragment we can do this by a Lindenbaum construction in which Γ0 = Γ and Γi+1 = Γi ∪ {¬Ai } if Γi , ¬Ai IL C and Γi+1 = Γi ∪ {¬¬Ai } otherwise. Taking Γ+ as the union of all these Γi we clearly get ¬A ∈ Γ+ or else ¬¬A ∈ Γ+ for each A, so it remains only to check that Γ+ IL C. Since IL is finitary and Γ+ is the union of the Γi , is suffices to show that for each Γi , Γi IL C, which can be done by induction on i. For i = 0 this is secured by the initial hypothesis that Γ C is unprovable, and, for the induction, supposing that Γi IL C gives Γi+1 IL C since otherwise this means that Γi , ¬Ai+1 IL C and also Γi , ¬¬Ai+1 IL C, contradicting the hypothesis that Γi IL C, in view of Lemma 8.21.11. With this Γ+ ⊇ Γ to hand, we may define a mapping h from the set of formulas into the set {1, 2, 3} by setting h(A) = 1 if Γ+ IL A is IL-provable, h(A) = 3 if Γ+ IL ¬A, and h(A) = 2 otherwise. This does define a function h, the only possible failure of functionality coming from the case in which our instructions lead to putting h(A) = 1 because Γ+ IL A and also h(A) = 3 because Γ+ IL ¬A: but this would imply that Γ+ IL C, contrary to hypothesis (since A, ¬A IL C). If we can show that h is a matrix evaluation into the Gödel matrix, we are done, since from the unprovability of Γ C, our specification of h yields the result that h(B) = 1 for all B ∈ Γ while h(C) = 1. This procedure much as indicated for an analogous argument sketched in 2.11.9 (p. 207): each of the nine cases presented by the ∧-table must be checked, as well as each of the three cases arising for the ¬-table. We will go through only the latter three here. Suppose h(A) = 1. Since the table requires that for h to be a matrix evaluation, h(¬A) = 3, this is what we must check. The supposition that h(A) = 1 means that Γ+ IL A, and what we require for h(¬A) = 3 is that Γ+ IL ¬¬A, so we have the desired result from the fact that A IL ¬¬A. Second case: Suppose h(A) = 3. This means that Γ+ IL ¬A is provable, and what we must be able to conclude is that h(¬A) = 1, which we already have, by the way h was specified. Third case: h(A) = 2, meaning that neither Γ+ IL A nor Γ+ IL ¬A. We need to show that h(¬A) = 3, i.e. that Γ+ IL ¬¬A. Here we use the special property of Γ+ , that either ¬A ∈ Γ+ or ¬¬A ∈ Γ+ . Since not Γ+ IL ¬A, the first possibility does not obtain, so ¬¬A ∈ Γ+ and hence Γ+ IL ¬¬A, as desired.


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Remark 8.21.13 If we had used the specification of h in the above proof, but just applied it to Γ+ as the deductive closure of Γ there would be no way of establishing the implication at the end of the proof, to the effect that h(A) = 2 ⇒ h(¬A) = 3. As an alternative, we might, keeping fixed the conditions under which h(A) = 1, h(A) = 3 as in the above proof, try putting h(A) = 2 iff (i) Γ+ IL ¬¬A and (ii) Γ+ IL A. This then secures that h(A) = 2 ⇒ h(¬A) = 3 by virtue of (i), part (ii) having been added to secure that the conditions under which a formula assumes each of the three values remain mutually exclusive. But the problem arises of showing that they are jointly exhaustive (and hence that h is a total function). Supposing that none of the conditions is satisfied for a given A means that we have Γ+ IL A, and Γ+ IL ¬A and also – the new case – either Γ+ IL ¬¬A or Γ+ IL A, so in particular Γ+ IL ¬¬A (since we already have Γ+ IL A). To rule this out we must accordingly be able to show that Γ+ IL A or Γ+ IL ¬A or Γ+ IL ¬¬A, which is equivalent (since Γ+ IL A ⇒ Γ+ IL ¬¬A) showing that either Γ+ IL ¬A or Γ+ IL ¬¬A, so we are back where we started with a need for something along the lines of the above Lindenbaum construction exploiting Lemma 8.21.11. Of course 8.21.12 would fail if we allowed “∨” in the language because the ‘weak law of excluded middle’ or KC axiom (see 2.32) ¬p ∨ ¬¬p is valid on the two-point Kripke frame underlying the present three-element matrix, though not IL-provable. (The argument given for the negation conjunction fragment would break down at Lemma 8.21.11, with a counterexample in taking this disjunction as C, p as A, and Γ = ∅.) The addition of → would also jeopardise the result since we could simulate the preceding disjunction example by using ‘deductive disjunction’ (1.11.3 on p. 50). In any case, we already have an assurance, a priori relative to any such details, that these fragments of IL are non-tabular: see Observation 6.42.23 (p. 890).

8.22

Dual Intuitionistic Negation

We closed 8.11 with the idea of investigating an extension of Positive or Intuitionistic Logic with a connective which forms the strongest subcontrary of the formula to which it is applied. Let us write ¬d (“d” for “dual”) for this connective, and consider what rules might appropriately supplement the Set-Fmla system INat to achieve the desired effect. Note that in this case the associated consequence relation is already guaranteed to be ∨-classical, enabling us to record the subcontrariety of A and B by the provability of their disjunction. Thus, rule (i) below makes the dual intuitionistic negation of A, as we shall call ¬d A, a subcontrary of A, while rule (ii ) makes it the strongest such subcontrary, since it says that given any subcontrary, B, of A, B follows from ¬d A: (i)

A ∨ ¬d A

(ii)

A∨B ¬d A B

We have now obtained, in ¬d A a formula qualifying as a strongest subcontrary of A, as in 8.11 we observed ¬A to constitute (in IL) a weakest subcontrary of A. Again, we can change “a” to “the” (up to synonymy):


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Exercise 8.22.1 (i) Show that the rules (i) and (ii ) above uniquely characterize ¬d (in terms of ∨). (Hint: show that for any proposed duplicate ¬d we have ¬d A ¬d A and the converse sequent provable, and appeal to part (ii) below.) (ii ) Show that ¬d is congruential according to the consequence relation associated with the proof system INat + (i), (ii ). (Hint: show that the rule of ‘Simple Contraposition’, from A B to ¬d B ¬d A is derivable from this basis.) (iii) Show that for any formula A, the sequent ¬A ¬d A is provable. (Hint: use (∨E) starting from (i), employing (RAA) en route.) (iv ) Show that rule (i) can be replaced by an upside-down form of rule (ii ). Considered as natural deduction rules, (i) and (ii ) above are somewhat unusual. The rule (i) has no premiss-sequents, which may raise an eyebrow— though compare the structural rule (R) in this regard. In any case, we could replace it with (i)

Γ, A C

Δ, ¬d A C

Γ, Δ C

to avoid this feature, at the same time as purifying the rule of the involvement with disjunction. Rule (ii ), though again impure, is a perfectly good rule in various approaches to logic in Set-Fmla, but hardly what one expects in the natural deduction approach, in view of the appearance, on the left-hand side of the conclusion-sequent, of ¬d A. We can fix this feature by passing to (the equivalent): (ii )

A∨B

Δ ¬d A

ΔB

Note that we cannot replace this rule by a pair of rules with left premisses A, on the one hand, and B, on the other, since the present logic obviously lacks the Disjunction Property (6.41), in view of (i). (Thus according to Gabbay’s criteria, reviewed in 4.38.1, p. 615, dual intuitionistic negation does not qualify as a ‘new intuitionistic connective’.) There is also a conspicuous failure exhibited by (ii ) of generality in respect of side formulas. Remark 8.22.2 A variation on (ii ) which is general in respect of side formulas would be: (ii )

ΓA∨B

Δ ¬d A

Γ, Δ B

This rule, a sort of Disjunctive Syllogism for ¬d , is by no means equivalent to (ii ) and (ii ) , and would render provable all sequents of the form A, ¬d A B. (Put Γ = {A}, Δ = {¬d A}; cf. 6.13.1, p. 789.) In effect, then, ¬d A would end up as a contrary of A, and hence as implying ¬A, the weakest contrary of A in IL (8.11): but in view of 8.22.1(iii), this would make ¬A and ¬d A equivalent, for all A.

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We have not, of course, shown that ¬A and ¬d A are not equivalent according to our logic of (intuitionistic and) dual intuitionistic negation, though any such result would deprive the present study of interest since we are keen to note here the possible cohabitation of non-equivalent connectives each uniquely characterized by rules making them, in the one case, a weakest contrary-forming operator (¬), and, in the other, a strongest subcontrary-forming operator (¬d ). (Compare the failure of the analogous possibility of ‘non-equivalent cohabitation for intuitionistic and classical negation’ reviewed in 4.32, 4.34.) This will, however, easily be demonstrable (8.22.4) with the aid of a semantics for our proof system, and the Kripke-style semantics we give, which is due to C. Rauszer (see notes, p. 1250), is of considerable interest in its own right. To get a feel for this interest, one should ponder the proportionality Tense Logic is to (Mono)Modal Logic as ??? is to Intuitionistic Logic. The point is to find what should fill the “???” position here. Recall from 2.22 that to obtain a tense logic from a normal monomodal logic, we introduce a new modal operator which quantifies over frame-elements which stand in the converse of the frame’s accessibility relation. Enrichments of the language of IL, as interpreted by Kripke models, can be contemplated along similar lines. For a model M = (W, R, V ) as introduced for IL in 2.32, we had M |=x ¬A iff for all y ∈ W such that xRy, M |=y A. An obvious suggestion for a converse operator might be to replace “xRy” here by “yRx ”. However, this makes for non-persistent compounds and so jeopardizes those aspects of the proof theory of IL (e.g., (→I)) which require persistence. (Cf. 4.34.2, p. 593.) What we shall say for ¬d is the result of making this replacement while at the same time trading in universal quantification for existential quantification: M |=x ¬d A iff for some y ∈ W such that yRx, M |=y A. The transitivity of R then guarantees that formulas of the form ¬d A are persistent in the sense of being true at any point R-related to a point at which they are true. Let us baptize the proof system with which we have been working, presented as INat with (i) and (ii ), DINat (“D” for dual : no connection with DLat from 2.14, where “D” abbreviated distributive). The remark just made about persistence means that we need only check that all instances of (i) hold in every model, and that (ii ) preserves the property of holding in a model, understanding truth as defined with the above clause for ¬d , in order to establish a soundness result: Theorem 8.22.3 All provable sequents of DINat are valid on every frame. Corollary 8.22.4 The sequent ¬d p ¬p is not provable in DINat. Proof. Consider the two-element model on the frame of Figure 2.32a described à propos of that figure (p. 309). ¬d p is true at the point w2 in that model, whereas ¬p is not, so the sequent does not hold in the model, and so is invalid on the frame of the model. Thus, by 8.22.3, this sequent is not provable in DINat.


1225

Corollary 8.22.5 DINat is a conservative extension of INat Proof. If σ is a ¬d -free sequent provable in DINat then, by 8.22.3, σ is valid on every frame in the sense of the present subsection. In that case, σ is valid on every frame in the sense of 2.32, since the present semantics for all connectives other than ¬d is as there; accordingly σ is provable in INat by 2.32.8. We state the converse of 8.22.3, a completeness theorem for DINat, but leave the proof as an exercise in adapting the proof given for a Set-Set system below (8.22.7). Theorem 8.22.6 All sequents valid on every frame are provable in DINat. The sense in which the connective ¬d is dual to ¬ can be brought out via the treatment of these two in the algebraic semantics for DINat, as well as by a consideration of appropriate rules in a Set-Set formulation of the logic. The algebraic perspective originally motivated Rauszer and we will say something about it first, before passing to the promised Set-Set logic. We should recall from the end of 0.21 that a pseudocomplement of an element a in some bounded distributive lattice is an element ¬a whose lattice meet with a is 0, and which is greater than or equal to (in terms of the associated partial ordering) every element having this property. In this chapter, we have preferred to express this in more ‘logical’ terms by saying that the intuitionistic negation of a formula is a contrary of that formula which is deductively weakest among such contraries. An algebraic semantics in the style of §2.1 for the →-free fragment of IL can be obtained by considering the class of all pseudocomplemented lattices (alias pseudo-boolean algebras), i.e., structures expanding bounded distributive lattices with an operation ¬ assigning to each element its (unique: 0.21.5(ii), p. 22) pseudocomplement; this operation then interprets intuitionistic negation in the same way that the join and meet operations interpret disjunction and conjunction. With this background in mind, we can consider the notion of a dual pseudocomplement ¬d a of an element a of a bounded distributive lattice, which is ‘dual’ in precisely the usual lattice-theoretic sense (§0.1). That is, we require that a ∨ ¬d a = 1 and that ¬d a is less than or equal to () every element with this property (i.e., the property of having its join with a being 1). Again, we have preferred a logical over an algebraic description, and described the dual intuitionistic negation of a formula as its deductively strongest subcontrary. The appropriate algebraic structures for a study of the →-free fragment of DINat would be lattices which are both pseudocomplemented and (in the obvious sense) dually pseudocomplemented; considerations of symmetry alone would direct our attention to this class of algebras, even without the logical interest provided by a desire to see ‘equal time’ for contrariety and subcontrariety as fundamental aspects of negation (8.11). Digression. In fact, of course, we have → to contend with also. As mentioned in 2.32, certain expansions of pseudo-boolean algebras called relatively pseudocomplemented lattices, or Heyting algebras (defined for before 0.21.6: p. 22), are considered for an algebraic treatment here. The above-mentioned symmetry motivation for attending to ¬d thus leads Rauszer to dualize the notion of a relative pseudocomplement (the operation interpreting intuitionistic implication),


1226

so that her logics incorporate also a dual intuitionistic implication, →d as we notated it the discussion between 4.21.8 and 4.21.9 (p. 547), where a suitable (model-theoretic) semantic treatment is described. Incidentally, these logics she calls “H-B logics”, the initials standing for Heyting, with whom she associates ¬, and Brouwer, with whom she associates ¬d . The association is mediated by an early use of the term Brouwerian algebra to describe algebras dual to Heyting algebras, rather than being intended to call attention to different notions of negation in play in the writings of these authors. End of Digression. Another way of bringing out the aptness of talk of ‘dual’ intuitionistic negation comes from considering a Set-Set version of DINat in which certain leftright symmetries are brought out; of course given the -based algebraic semantics of 2.14, this is not really a new perspective on the subject from that given before the above Digression—but it puts matters in a slightly different light. For definiteness, we will pick on the sequent calculus approach to logic in Set-Set, given for classical logic in 1.27 and for intuitionistic logic at the end of 2.32. In the latter discussion we noted that instead of pursuing the sequent calculus development of IL in Set-Fmla0 , we could equally well have chosen Set-Set, provided that the rules (¬ Right) and (→ Right) are restricted so that their conclusion sequents have only one formula on the right of the “”. (Of course this change of framework means that the rules are no longer fully general in respect of side formulas.) We repeat the rules for ¬ here, alongside those to be offered for ¬d : (¬ Right)

(¬ Left)

Γ, A Γ ¬A ΓA Γ, ¬A

(¬d Left)

(¬d Right)

A, Δ ¬d A Δ AΔ ¬d A, Δ

We have placed (¬ Right) and (¬d Left) next to each other so as to make the fact that these rules are each others’ mirror images (the choice different set-variables, Γ on for the left-hand side and Δ for the right, being merely conventional), and similarly with (¬ Left) and (¬d Right). As remarked in 8.11, the rules governing ¬d here can be seen as rules in the framework Fmla0 -Set, just as those governing ¬ can be seen as belonging to the framework Set-Fmla0 . However, we take the present framework to be Set-Set, since we wish to consider both pairs of rules in the same proof system. Some authors seem to have assumed that, perhaps because both ¬ and ¬d have “something to do with negation”, they represent competing conceptions of negation at most one of which deserves a place in the language, so they write “¬A” and consider replacing the usual intuitionistic treatment of such formulas by a dual intuitionistic treatment. Examples are Goodman [1981], Dummett [1976b], and Miller [2000c], esp. §3. (Perhaps there is a tacit appeal to the desideratum of generality in respect of side formulas involved here. But on the face of it, the opportunities for viable cohabitation here seem conspicuously better than with the case, discussed in 4.32, of what we there called ¬i and its cannibalistic cousin, ¬c .) By contrast, incidentally, the connective →d behaves, according to the logic semantically characterized in our discussion in 4.21, referred to above, hardly at all like an implication connective. This reflects the fact that of the corresponding


1227

truth-functions, ¬b and →b , only the former is self-dual. (Recall that the Kripke semantics for IL interprets ¬ and → using what in 4.38.6, on p. 620, are called the topoboolean conditions induced by the corresponding truth-functions; for self-duality, see 3.14.8 on p. 409. We touch on →d again briefly below, at and just before 8.22.8.) The rule (¬d Right) does the work of our earlier Set-Fmla rule labelled (i). To derive the latter, take Δ = {A} and apply (¬d Right) and then (∨ Right). Similarly, (¬d Left) is the present analogue of Rule (ii ); note that the change of framework has allowed us to dispense with the intrusion of “∨” in the formulation of this rule. We take the above four rules, together with the structural rules and the rules for ∧, ∨, and → from the Set-Set formulation of IGen (2.32) to define our sequent calculus for dual intuitionistic logic in Set-Set. We call this proof system DIGen. A sequent Γ Δ holds (as usual) in a model as long as no point in the model verifies all of Γ while falsifying all of Δ; it is valid on a frame if it holds on every model on that frame. Theorem 8.22.7 A sequent is provable in DIGen iff it is valid on every frame. Proof. The soundness (‘only if’) proof is left to the reader; it requires only a check that (¬d Left) and (¬d Right) preserve the property of holding in a model. For the completeness proof, we use a canonical model built up from maximal pairs Γ, Δ w.r.t. the gcr associated with DIGen. (I.e., such pairs as the Γ+ , Δ+ of 1.16.4, p. 75; see the end of 2.23, for an analogous construction in modal logic.) The accessibility relation is given by Γ, Δ R Γ , Δ iff Γ ⊆ Γ . Note that this is equivalent to saying that Δ ⊆ Δ. V (pi ) = { Γ, Δ ∈ W | pi ∈ Γ}. We need to show that for this model M we have: M |= Γ, Δ A iff A ∈ Γ. The only step of the induction to which we attend here is the novel case of A = ¬d B. First, suppose M |=Γ,Δ ¬d A. Then (by the inductive hypothesis) there exists / Γ . Therefore ¬d B ∈ Γ since B, ¬d B, by Γ , Δ ∈ W with Γ ⊆ Γ and B ∈ (¬d Right), and so ¬d B ∈ Γ, since Γ ⊆ Γ. / Γ . Conversely, suppose ¬d B ∈ Γ. We want Γ , Δ ∈ W with Γ ⊆ Γ and B ∈ Let Δ0 = {B} ∪ Δ. The pair ∅, Δ0 is -consistent, since otherwise Δ, B and so, by (¬d Left) ¬d B Δ, which would contradict the -consistency of Γ, Δ since ¬d B ∈ Γ. We can take as our desired Γ , Δ any maximal pair coordinatewise extending ∅, Δ0 , since B ∈ Δ0 and so B lies outside any such Γ , and Γ ⊆ Γ since Δ ⊆ Δ for any such Δ . To wrap up the argument, we note that if we have an unprovable sequent Γ Δ, then by 1.16.4 (since the associated gcr is finitary) we can extend Γ, Δ to a maximal pair Γ+ , Δ+ which is therefore an element of the canonical model at which the given sequent does not hold, as Γ ⊆ Γ+ and Γ+ ∩ Δ = ∅. In 8.11.4(ii), p. 1169, we reviewed the idea, from Zeman [1968], of starting with an operator R for forming contraries, and obtaining the weakest contrary (or intuitionistic negation) ¬A of A by defining this as A → RA. The question naturally suggests itself as to whether something similar is possible for the strongest subcontrary. Accordingly, let us introduce a new 1-ary connective S governed by the Set-Set condition (to avoid extraneous connectives):


1228 A, SA

telling us that SA is a subcontrary of A. Blindly following the example of Zeman’s R, we might consider A → SA as a candidate definiens for ¬d A, with → governed by the usual intuitionistic rules. Since those rules give SA A → SA, we do get A, ¬d A from the above schema, so ¬d forms a subcontrary. But now the further hypothesis that A, B is provable should lead us to a proof of B ¬d A, i.e., B A → SA; plainly this is not forthcoming on the basis of our definition. A second stab, since now we wanted the strongest formula meeting a condition rather than, as with R, the weakest, would reverse the implication. We venture SA → A as the proposed definiens for ¬d A. Again, however, no route is available to get from A, SA and A, B to the desired conclusion: A → SA. The reader may check that if → is replaced in this latest attempt by the →d of the Digression on p. 1225 above (and discussion leading up to 4.21.9) then all is well. For a fuller discussion of related matters see Humberstone [2005d], where, as already remarked under 7.25.17 (p. 1112), “κ” and “σ” are used as arbitrary contrary and subcontrary forming 1-ary connectives – paradigm cases of operators we do not expect to be uniquely characterized by the totality of rules they satisfy. The first source in which such operators appear is perhaps Goddard [1960], esp. p. 105. Exercise 8.22.8 Suggest some suitable rules (i.e., preferably giving a complete and certainly giving a sound proof system for →d as characterized semantically in the discussion leading up to 4.21.9 on p. 547) from which, given A, SA as a further principle, A →d SA B is derivable from A, B. We return to ¬d on its own merits for our final practice exercises, first examining some double negation phenomena, and then some De Morgan principles. Exercise 8.22.9 Say, for each of the following six sequents, whether or not it is valid on every frame (in the sense of this subsection), and give a proof of one sequent (for which the answer to that question is “yes”) in the natural deduction system DINat, and a proof of another in the sequent calculus DIGen. (i) ¬d ¬d p p (ii) p ¬d ¬d p (iii) ¬d ¬p p (v) p ¬d ¬p (vi ) p ¬¬d p. (iv) ¬¬d p p Exercise 8.22.10 Again, divide these into those that are valid (on every frame) and those that are not, and prove one of the former sequents in either of the proof systems mentioned in 8.22.9 (i)

¬d (p ∨ q) ¬d p ∧¬d q

(iii) ¬d (p ∧ q) ¬d p ∨¬d q

8.23

(ii ) ¬d p ∧¬d q ¬d (p ∨ q) (iv ) ¬d p ∨¬d q ¬d (p ∧ q).

Strong Negation

At the end of the 1940’s, the idea occurred independently to D. Nelson and A. Markov that the familiar negation connective of IL might be replaced or supplemented by one with a more direct constructive interpretation. (See notes.) Rather than interpreting the negation of a statement as recording the fact that a


1229

contradiction can be deduced from the statement, we could take it as reporting the possession of a counterexample. For example, the negation of a conjunction A ∧ B would be equivalent to the disjunction of the negations of A, B, and the negation of a universally quantified formula would be equivalent to the existential quantification of the negation of the formula following the universal quantifier. (Here we concentrate, as usual, on the sentential logic to which these considerations gave rise.) Since the disjunction and existential formula here involved are themselves interpreted constructively, and so as assertible only in the presence of an assertible disjunct or assertible instantiating formula (respectively), the negated conjunction or universal quantification receives a stronger interpretation than the usual intuitionistic negation would allow, and so the current conception is often referred to as “strong negation”. Nelson’s term was “constructible falsity”, and he thought of the negation of a statement A as saying that it is constructibly false that A, to be understood in the same way that the unnegated statement asserts its own constructible (constructive) truth. Negated and unnegated formulas are treated as ‘on a par’, rather than, as in conventional accounts, having the former treated indirectly in terms of what having a proof of the formula negated would lead to (i.e., absurdity). This is reflected in the Kripke semantics described below by the fact that strong negation is not treated (as implication is) by quantification over points R-related to a given point. This semantics derives from work by Thomason, Routley, and Gurevich, referred to in the notes, under ‘Strong Negation’ (p. 1251). It proves a hospitable environment for traditional intuitionistic negation, which we (following some of these later writers) can include as an additional primitive, under the usual notation (¬A); the strong negation of A will be written as “−A”. (This “−” is of course not to be confused with the similarly notated binary connective of §5.2. Nor is the present (and standard) use of the phrase “strong negation” to be confused with various other usages, such as that of Zeman [1968] and Lenzen [1996] on the one hand, or Sandu [1994] on the other. This list of examples of different uses of the same terminology could be extended considerably. See also the first footnote of Hollenberg [1997].) A Set-Fmla proof system for strong negation, in the language with ∧, ∨, ¬, → and the new “−” as primitive connectives, can be obtained by adding as zero-premiss rules to the basis for our intuitionistic natural deduction system INat the following sequent schemata: (0) (1a) (2a) (3a) (4a) (5a)

−A ¬A −(A ∧ B) − A ∨ −B − (A ∨ B) − A ∧ −B − − A A − ¬A A −(A → B) A ∧−B

(1b) (2b) (3b) (4b) (5b)

−A ∨ −B − (A ∧ B) −A ∧ −B − (A∨ B) A − −A A − ¬A A ∧−B − (A → B)

Some of the duality-related equivalences one might miss in passing from classical to intuitionistic logic such as the De Morgan Laws are restored by the introduction of strong negation: with ¬ replacing - only (1b) and (2a, b) are intuitionistically valid, whereas with strong negation we also have (1a), for example. To provide a semantics for the proof system just described, we can embody the ideas in our opening paragraph in a kind of Kripke semantics which is, as


1230

we might put it, locally three-valued. That is, while using the definition of truth (at a point in a model) associated with the Kripke model theory for IL, we think of this is explaining the possession of a formula of the value 1 in what is, as far as conjunction, disjunction, and (strong) negation are concerned, the three-element Kleene matrix, with the value 1 designated, called K1 in 2.11 (see 2.11.2, p. 201); note that it is the present “−” that corresponds to the “¬” of that discussion. (¬ and → will have to be treated somewhat differently, since they have ‘forward-looking’ clauses in the Kripke semantics. In particular since → is not at issue for the moment, the name of Łukasiewicz is just as appropriately invoked as that of Kleene; indeed at the end of this section, we shall see a system of strong negation for which it is Łukasiewicz rather than Kleene that needs to be mentioned, because of the three-valued behaviour of → there.) We will be writing the analogue of A’s possession of the values 1 and 3 at a point x in − a model M as respectively “M |=+ x A” and “M |=x A”; there is no separate need for a notation corresponding to taking the value 2, and no need to attend separately to it in our inductive truth definition. (The formulas taking the value 2 at a point are just those left over as taking neither of the values 1, 3, at that point.) We return to the explicitly three-valued perspective below (in the discussion following 8.23.7). Since we want to treat A and the strong negation −A, of A, in an evenhanded way, we begin our even-handed treatment at the level of propositional variables, and take models to have the form (W , R, V + , V − ) in which (W, R) is a frame exactly as in the models of 2.32 (or 8.21, 8.22, for that matter) but now we think of assigning both truth-sets and falsity-sets; hence the need for V + and V − . (Note: think of this use of “falsity” as heuristic; it departs from our customary use of “false” to apply to whatever is not true. Instead, here it serves merely to encode the truth of the strong negation of a formula. See further the discussion of Dummett towards the end of 2.11.) Each of V + and V − is a function assigning subsets of W to the propositional variables. Since we still think of truth and falsity as mutually exclusive (though a variant called ‘paraconsistent strong negation’, abandoning this constraint, will be mentioned before 8.23.6 below), we impose the following condition on models: (Exclusion)

V + (pi ) ∩ V − (pi ) = ∅, for each propositional variable pi .

The even-handedness desideratum leads to a simultaneous persistence condition in both cases, positive and negative: (Posve Persistence)

For any x, y ∈ W : (x ∈ V + (pi ) & xRy) ⇒ y ∈ V + (pi ).

(Negve Persistence)

For any x, y ∈ W : (x ∈ V − (pi ) & xRy) ⇒ y ∈ V − (pi ).

We extend V + and V − to all formulas by inductively defining two semantic relations, |=+ and |=− , for the truth of a formula at a point in a model, and for the falsity of a formula at a point in a model: For any model M = (W, R, V + , V − ) and any propositional variable pi , any formulas A and B, and any x ∈ W :


1231

+ M |=+ x pi iff x ∈ V (pi )

− M |=− x pi iff x ∈ V (pi )

+ + M |=+ x A ∧ B iff M |=x A and M |=x B

− − M |=− x A ∧ B iff M |=x A or M |=x B

+ + M |=+ x A ∨ B iff M |=x A or M |=x B

− M |=− x A ∨ B iff M |=x A and M |=− B x

M |=+ x ¬A iff for all y ∈ R(x), M |=+ yA

+ M |=− x ¬A iff M |=x A

− M |=+ x −A iff M |=x A

+ M |=− x −A iff M |=x A

M |=+ x A → B iff for all y ∈ R(x) for + which M |=+ y A, we have M |=y B.

+ M |=− x A → B iff M |=x A and B M |=− x

In some of the references cited in our notes, the superscripted plus is omitted and the turnstile is reversed rather than being superscripted by a minus; we avoid this notation since reversing a relational symbol is too suggestive of symbolizing the converse of the relation concerned. In some of what follows we will refer to a semantics of the present sort as involving two truth-relations, meaning by this the two ternary relations (each holding between model, point, and formula) |=+ and |=− . One can imagine more elaborate simultaneous inductive definition of more than two such relations, but we shall not investigate this possibility here. (Nor do we attempt a more precise characterization of what is at issue. The informal idea is certainly to exclude the availability of many different binary relations |=x – one for each model elements x – in the Kripke semantics for modal or intuitionistic logic as constituting such a ‘many truth-relations’ semantics. For other examples of the genuine article, see Appendix C of Humberstone [2003b] and Section 4 of Humberstone [2004a].) We make some formal comments, and then some informal comments, on the above definition. The formal commentary consists in a ‘Persistence and Exclusion’ Lemma, noting that the above conditions on Valuations extend to the truth and falsity sets of arbitrary formulas. We denote by !A!+ the set − of all points x in some model M for which M |=+ the set of x A, and by !A! − + + − − points x such that M |=x A. (Thus !pi ! = V (pi ), !pi ! = V (pi ).) Parts (i) and (ii) of 8.23.1 show that Positive and Negative Persistence generalize to arbitrary formulas, while part (iii) does the same for Exclusion. Lemma 8.23.1 For any model M = (W, R, V + , V − ), and any formula A: (i) !A!+ is closed under R. (ii) !A!− is closed under R. (iii) !A!+ ∩ !A!− = ∅. Proof. By induction on the complexity of A; in the case of (i) and (ii), what should be proved inductively is their conjunction, rather than each separately. By way of informal comment, we begin by clarifying the claim that the above semantics essentially imposes the K1 matrix structure at each point of

1232


the models. This claim applies to the treatment of conjunction, disjunction, and negation, in the last case it being the strong negation − (rather than intuitionistic negation ¬) that corresponds to the ¬ of our discussion in 2.11. The point is that for any point x in one of our models, if we write hx (A) = 1 for M |=+ x A, hx (A) = 3 for M |=− x A, and hx (A) = 2 otherwise, then hx is a K1 -evaluation (provided that A is a formula in the three connectives mentioned). The justification for the subscript ‘1’ here, which indicates the pattern of designation rather than the algebra of the matrix, emerges from the special interest work on the logic of strong negation has taken in the consequence relation baptized SN below, defined by preservation of truth (|=+ ). Secondly, now proceeding to address the two remaining connectives in play, we note that the “|=− ” clauses for ¬ and → here do not direct one to R-related points, by contrast with the corresponding “|=+ ” clauses. One might have expected instead, in the former case, a condition to the effect that some R-related point verifies A, and, for the latter, that some R-related point verifies A while falsifying B. (We use “verify”, “falsify”, here to indicate |=+ , |=− , respectively, as opposed to our practice elsewhere of using falsify to mean fail to verify.) Notice that this would interfere with the persistence properties (8.23.1(i), (ii)), however. (See the notes and references to this section, which begin on p. 1250 for a variation in Veltman [1984] along just these lines.) Moreover, the “|=− ” clause for ¬ is exactly that for −, even though their “|=+ ” clauses differ. This has the consequence that −¬A and − −A are true at the same points. But the latter is true at the same points as A itself. So each of the formulas −¬A and A is a consequence of the other, by the consequence relation SN defined thus: Γ SN A iff for every model M = (W, R, V + , V − ), for all x ∈ W : + if M |=+ x C for each C ∈ Γ, then M |=x A.

We will also say that a sequent Γ A is valid (according to the present semantics) when Γ SN A in this sense. (Obviously, a frame-relative notion of validity could also be defined if desired.) Since the evaluation of formulas not containing strong negation proceeds entirely in terms of |=+ , and the clauses defining this relation are exactly as for |= in the Kripke semantics for IL, SN is a conservative extension of IL . (Cf. the proof of 8.22.5.) Now, in 4.21.1(i) (p. 540), we saw that IL cannot be conservatively extended by the addition of any singulary connective which ‘undoes’ ¬ according to a congruential consequence relation, so the fact, just noted, that −¬A SN A alerts us to the non-congruentiality of SN . Specifically, strong negation is not congruential according to SN , and the proof of 4.21.1(i) displays a counterexample: Example 8.23.2 ¬p SN ¬¬¬p but it is not the case that −¬pSN −¬¬¬p. (3a)–(4b) above, all instances of which are valid on the present semantics, provide us with another example: we have p SN − ¬p, but we cannot ‘strongly negate’ each side, or we could cancel the two strong negations on the right (3a, b) to get −p SN ¬p, which holds only in the left-to-right direction (à la (0) above). (In view of the ‘local K1 ’ character of the {conjunction, disjunction, (strong) negation} fragment of SN , we expect some non-congruentiality. This example was (in a different notation) mentioned in 3.31.5(ii) on p. 487 (originally in 3.23.4, p. 456): p ∧ −p SN q ∧ −q, though it is not the case that


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−(p ∧ −p) SN −(q ∧ −q). One might attempt a reply to the effect that such failures of congruentiality show only that equivalence cannot be expected to secure synonymy when equivalence (as with SN ) is understood as sameness of truth-conditions in a semantic setting in which this (taken as sameness in respect of |=+ -status at arbitrary points) does not itself suffice for sameness of falsity-conditions (understood in terms of |=− -status). But this does not sound very plausible in the case of Example 8.23.2, for instance: there seems nothing to choose between saying ¬p and saying ¬¬¬p, as far as the intuitionist is concerned. In 4.38.1 we suggested that a reasonable condition to impose on ‘new intuitionistic connectives’ is that they be congruential. This suggestion deserves to be refined a little. If we consider the passage from IL to some consequence relation on an expanded language, say (for the sake of illustration) with an additional 1-ary connective O, the demand that O be congruential according to is the demand that A B should always imply OA OB. Let us weaken this to the condition that if A IL B (i.e., A IL B and B IL A), then we always have OA OB. In other words, even if the new logic creates equivalences where the equivalents are not replaceable in the scope of O, it at least respects any such equivalences as are already IL -provable. (The reader will have no difficulty in formulating this weakening of the congruentiality condition for nary O, for arbitrary n. The condition is analogous to what Segerberg [1982] calls ‘classicality’ of a connective, except there it is CL -equivalences that are to be respected.) Whereas the example of p ∧ −p’s equivalence, according to SN , with q ∧ −q was a case of non-congruentiality, this weakened demand is not thereby violated because the equivalence in question is not given by IL itself. On the other hand, what 8.32.2 illustrates is a violation of the weaker condition, and it is this feature of the example that makes strong negation seem like a ‘merely sentential’ as a opposed to a genuinely propositional operator. Even though with SN we have, as noted above, a conservative extension of IL , there is something we might informally describe as ‘non-conservative’ going on: the identity conditions for propositions are not being conserved. We can state this condition of ‘conservation of synonymy’ in the following terms, in which the proposition [A] expressed by A is taken, as in 2.13, to be the class of formulas synonymous with B. More precisely, when 1 ⊆ 2 and the language of (the consequence relation) 1 is properly included in the language of 2 , we denote by [A]i the class of formulas in the language of i which are synonymous according to i with A. Then to say that synonymy is conserved in the passage from 1 to 2 is to say that: [A]1 = [B]1 implies [A]2 = [B]2 , for all A, B in the language of 1 . It is this condition which Example 8.32.2 shows is not met in the case of the passage from IL to SN . Notice, in passing, that the condition makes sense— and might or might not be satisfied—even when 1 is not itself congruential. (Of course our present 1 , namely IL , is in fact congruential.) There will be a bit more to say on the topic of congruentiality after we settle the question of how our semantic characterization of SN is related to the proof system outlined above for IL with strong negation. The answer is that SN is in fact the consequence relation associated with that proof system:


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Theorem 8.23.3 A sequent Γ A is provable on the basis of INat supplemented by (0)–(5b) above if and only if it is valid according to the semantics given (i.e., iff Γ SN A). Proof. ‘Only if’: This (soundness) direction is left to the reader. ‘If’: For completeness, we make a modification of the canonical model argument used for Thm. 2.32.5 (p. 309). Define the canonical strong negation model, M, to be (W, R, V + , V − ) with W the set of all deductively closed prime consistent sets of formulas of the present language (deductively closed w.r.t. the consequence relation associated with the proof system described, that is). R is ⊆. Put V + (pi ) = {x ∈ W | pi ∈ x}, V − (pi ) = {x ∈ W |−pi ∈ x}. A straightforward inductive argument (on the complexity of arbitrary A) gives: − M |=+ x A iff A ∈ x and M |=x A iff −A ∈ x.

for all x ∈ W . We need to use the inductive hypothesis on the “|=− ” side to prove the inductive step for A = −B on the “|=+ ” side; the inductive steps on the “|=− ” side are provided by (1a)–(5b). Positive and negative persistence are guaranteed from the definition of R; Exclusion is secured by (0). Thus M is indeed a model and when Γ A is unprovable, we have a deductively closed superset of Γ which is prime and does not contain A (and so is consistent), at which, considered as an element of W , all formulas in Γ are true (in the “M |=+ ” sense) while C is not. Since we now have two ways of thinking of SN , the original semantic way, and the syntactic route via the proof system mentioned in 8.23.3, there are two ways of setting about answering part (i) of Exercise 8.23.4 (i) Show that for any formula A there is a formula A in which the only occurrences of “−” are those which immediately precede propositional variables, with A SN A . Does this result hold with the demand that A and A be SN -equivalent replaced by the demand that they be SN -synonymous? (ii ) Show that if A SN B and −A SN −B, then A and B are synonymous according to SN . (Suggestion: where C is any formula in which there appears some propositional variable for which A and B are uniformly substituted to give C(A) and C(B) respectively, we want to show that if A SN B and −A SN −B, then C(A) SN C(B); the best way to do this is to argue by induction on the complexity of C that if A SN B and −A SN −B, then both C(A) C(B) and −C(A) −C(B). Although we want specifically the conclusion concerning C(A) and C(B), the conjunctive consequent is useful for the inductive step in which C is taken to be −C0 for some C0 for which the inductive hypothesis is assumed to hold.) The result in 8.23.4(ii) is generally offered as something of a consolation prize for the failure of congruentiality tout court. It makes explicit the idea, mentioned above, that sameness of truth-conditions and of falsity-conditions is


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what is required (and is indeed sufficient) for synonymy. But this serves only to emphasize that “−” is itself non-congruential according to SN , and does nothing to relieve dissatisfaction over the fact that some of this non-congruentiality is indicative of the failure to conserve IL-synonymy complained of above. We can, however, make use of 8.23.4(ii) to examine the status of the equivalences given by (1a) and (1b), here collected as (1), (2a) and (2b), here collected as (2), etc. ( is SN .) (1) −(A ∧ B) −A ∨−B (4) −¬A A

(2) −(A ∨ B) −A ∧−B (5) −(A → B) A ∧−B.

(3) −−A A

(1) and (2) do not give us all we might want under the name of De Morgan’s Laws, if what we are interested in is the synonymy – and not just the equivalence – of their left-hand sides with their right-hand sides. Non-congruentiality is, after all, precisely the failure (in general) of equivalent formulas to be synonymous. Let us examine (1) with this question in mind. By 8.23.4(ii), the left- and right-hand sides are synonymous if not only they but also their strong negations are SN -equivalent. Strong(ly) negating both sides of (1) gives (1 ) − −(A ∧ B) −(−A ∨−B) But, making use of (2), we have (1 ) if we have: (1 ) − −(A ∧ B) −−A ∧ − −B And (1 ) indeed holds, in view of (3) and the fact that ∧ is congruential according to SN . Thus formulas equivalent in virtue of (1) are indeed synonymous. More generally, Observation 8.23.5 (i) Formulas SN -equivalent in virtue of (1), (2) or (3) above are SN -synonymous. (ii) Formulas SN -equivalent in virtue of (4) or (5) above are not in general SN -synonymous. Proof. (i) Arguing as in the case of (1 ), (1 ) above. (ii ) For (4): see the discussion immediately following 8.23.2 above. For (5): similar. Since most of the ‘trouble’ we have been making for SN arises from (4) and (5)—the latter essentially being the general case with (4) arising by putting B = ⊥, one might give some consideration to just laying down (1)–(3) for “−”. In a language in which no other connectives are present, this is just what certain authors have in mind when they speak of strong negation; this is true, for example, of Pearce and Wagner [1990], [1991]. Since there is no ‘ordinary’ negation ¬ for strong negation to be stronger than (and nor do the characteristic non-congruentiality issues arise), a better description here would be ‘De Morgan negation’: cf. 8.13. The latter label is especially apposite because the works just cited do not even have the ¬-free surrogate for (0) above, namely the Ex Falso Quodlibet schema: (0 ) A, −A B, for which reason this variant of the enterprise is often called paraconsistent strong negation. (As already intimated, the condition of Exclusion would be

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dropped in the semantic treatment for this variant, and we would lose the generalized form of this condition given in Lemma 8.23.1(iii).) In virtue of 8.23.5(i), one can say that ∧ is definable in SN , since for any A and B, the formula A ∧ B is not just equivalent to but synonymous with the formula −(−A ∨ −B); ∨ is similarly (‘dually’) definable in terms of strong negation and conjunction. Remark 8.23.6 In fact, even ¬ is definable according to SN , since ¬A is again not just equivalent to but synonymous with A → −A. (For an elegant presentation of the proof system without ¬ as primitive, replace the original (0) by (0 ) .) In this subsection, we have argued that although the logic of strong negation passes several of the tests proposed by D. Gabbay (see 4.38.1, p. 615) for whether one is dealing with a bona fide ‘new intuitionistic connective’, in particular in possessing the Disjunction Property and in extending IL conservatively, the particular form of non-congruentiality exhibited by strong negation according to SN represents an intuitionistically unacceptable kind of failure to ‘conserve synonymy’. The idea that formulas as they appear in the scope of strong negation are not given the interpretation they receive when outside of its scope will be reinforced by an example—intended as something of a parody of the way SN itself is described—in our closing paragraph. But first, we pause to note that two of Gabbay’s own criteria for ‘new intuitionistic connective’ status are not satisfied by strong negation: (i) the criterion that he puts by saying that the new connective should “not be non-classical”, and which in 4.38 (see especially the discussion leading up to 4.38.5, p. 618) we preferred to formulate in terms of not being the intuitionistic analogue of a classical connective; and (ii) the criterion that the rules governing the new connective should uniquely characterize that connective. We can leave (ii) as an exercise and get straight to (i): Exercise 8.23.7 Show that the rules of the proof system given in this subsection for IL with strong negation do not uniquely characterize the latter connective. (Suggestion: Writing − and ∗ for strong negation and another singulary connective satisfying the same rules ((0)–(5) above) consider models (W, R, V + , V − , V ∗ ) satisfying conditions analogous to those introduced for SN , using which, the sequent −p ∗p can be invalidated.) We turn now to the question of whether “−” is the intuitionistic analogue of a classical connective, meaning: whether there is some truth-function f such that the extension of SN by ¬¬A A is determined a class of valuations over which “−” is associated with f (i.e., for v in the class v(−A) = f (v(A)) for all A; if so then “−” is an intuitionistic analogue of #, where f is #b ). We approach the matter indirectly with an observation from Vakarelov [1977], who considers what he calls ‘classical logic with strong negation’. This means the logic in Set-Fmla (actually, for Vakarelov, the logic in Fmla) determined by the class of models (W, R, V + , V − ) in which W consists of a single point, w say. In this case, the earlier forward-looking clauses, giving the |=+ -conditions for ¬ and → formulas, are equivalent to the following ‘classical’ clauses (since R(w) = {w}):

8.2. NEGATION IN INTUITIONISTIC LOGIC + M |=+ w ¬A iff M |=w A

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+ + M |=+ w A → B iff either M |=w A or M |=w B

Now let us record, relative to any such model M on our one-element frame ({w}, { w, w }) the status of a formula A as an ordered pair of truth-values, with a T or an F in the first position according as to whether M |=+ w A, or not, and a T or F in the second position according as to whether M |=− w A or not. By the Exclusion condition on models (and 8.23.1(iii)) the status can never be T, T , leaving only the three possibilities T, F , F, F and T, T which we rebaptize as 1, 2 and 3 for alignment with our labelling of values in 2.11. To work the case of disjunction—by way of example—in this setting, we rephrase the earlier conditions as: the pair-value of A ∨ B has a T in its first position iff at least one of A and B has a T in its first position, and a T in its second position iff both of A and B have a T in their second position. Thus for instance if A has the value T, F and B has the value F, T , this means that A ∨ B has the value T, F , corresponding to the Kleene–Łukasiewicz calculation 1 ∨ 2 = 2. A ‘T’ in either position just means that the corresponding truthrelation (|=+ for the first position, |=− for the second) does indeed hold between the point w and the formula in question. Thus to say that A has the value T, F

is to say that w bears the |=+ relation to A but not the |=− relation to A, while to say that B has the value F, T is to say that w does not bear the first of these two relations to B but does bear the second. It may seem odd to associate a ‘T’ in any position (here the second) with something that is supposed to amount to a kind of falsity (‘constructible falsity’) but the intention is to make clear the general situation with multiple truth-relation semantics: for each such relation we need to indicate a Yes or a No as to whether the relation, and we use “T” and “F” as all-purpose recorders of these respective indications. The resulting pairs (or, in the general case, n-tuples) have accordingly a rather different significance from the pairs in (pairwise) direct products of matrices (Figure 2.12a from p. 213, etc.), as is reflected in our numerical abbreviations; thus here F, F is 2 rather than 4, and there is no T, T , leaving T, F , alias 1, to serve as the sole designated value. The resulting calculations give, as was mentioned at the start of the present subsection, the Kleene–Łukasiewicz tables (from 2.11) for conjunction, disjunction and negation, meaning in this last case that the table given in 2.11 for ¬ applies here for −. The two remaining connectives to deal with are ¬ and →, whose simplified one-element-model |=+ -clauses appear inset above. In the case of ¬, we have a T in the first position of ¬A’s pair-value if and only if A’s pair-value does not have a T in the first position; and, by the original |=− -clause for ¬, a T in the second position under ¬A if and only if there is a T in the first position under A. Thus the full story for ¬ is that its three-valued matrix function maps T, F to F, T , and F, F to T, F , and F, T to T, F ; a tabular representation is given on the left of Figure 8.23a, with the corresponding table for → appearing on the right. (With reference to the interpretation of such tables, the remarks made in connection with Figure 2.11a, back on p. 198, continue to apply.) These may seem somewhat unfamiliar, but the functions depicted are definable using the three-valued Łukasiewicz functions for negation and implication which we shall write – using the same notation for the connectives concerned – as ¬Ł and →Ł . (A more explicit notation would have “Ł3 ” rather than simply “Ł”.) We have, for all elements a, b ∈ {1, 2, 3}:


1238 ¬ 1 2 3

→ 1 2 3

3 1 1

1 1 1 1

2 2 1 1

3 3 1 1

Figure 8.23a

(6) ¬a = a →Ł ¬Ł a

and

(7) a → b = a →Ł (a →Ł b).

(In fact, the table on the right of Fig. 8.23a may be found as a table for implication in Thomas [1967], taking up ideas of V. Vučković and B. Sobociński.) Since we already have ∧ and ∨ and, in particular, “−” (which in this three-valued setting is just ¬Ł itself), to see that, conversely, all the Łukasiewicz functions can be defined, it suffices to show that →Ł is definable (compositionally derivable); but it is not hard to check that for all a, b: (8) a →Ł b = (a → b) ∧ (¬Ł b → ¬Ł a) Thus Łukasiewicz’s three-valued logic and Vakarelov’s ‘classical logic with strong negation’ are just different notational versions of each other—a point made in Vakarelov [1977], on which the above discussion has been based. In fact, as we shall see below (after 8.36), the definitions in virtue of which these two logics are definitionally (or ‘translationally’) equivalent were described already in Nelson [1959] – but with the strong negation extension of IL rather than CL on the one side (and a certain substructural logic on the other, though of course the term logic here needs to be understood in such a way that distinct logics can still be definitionally equivalent; see pp. 69–71 of Wójcicki [1988] for explanation – if needed – of the terminology here). For some remarks on translationally embedding CL into not only Ł3 but into all the Łn , see Tokarz and Wójcicki [1971], esp. Section III summarizing work in Polish by Pogorzelski. Now we can return to the question of whether or not strong negation is the intuitionistic analogue of a classical connective (“is not non-classical”, in Gabbay’s terminology). The only possibility left open by (0)–(5) is that “−” ends up equivalent to “¬”, in the smallest extension of SN according to which ¬¬A and A are equivalent for all A. But we have just seen a consequence relation extending SN in which (i) ¬¬A and A are equivalent for all A, and (ii) we do not have -A equivalent to ¬A for all A. Though we need to check claims (i) and (ii) here, our conclusion from them must be that strong negation is not the intuitionistic analogue of a classical connective. As to (i), the point is that although we do not have ¬¬a = a for all a ∈ {1, 2, 3}, the only differences do not affect designated-value status, so we still have ¬¬A A for all formulas A, where is the consequence relation determined by the three-element matrix above. Turning to (ii) the point is that (on this same understanding of ), although we do we have, for example, −p ¬p (alias ¬Ł p ¬p), we do not have the converse (consider a matrix evaluation h with h(p) = 2). As Vakarelov ([1977], p. 119) remarks, “This shows that the ‘strong negation’ is really still strong” (“still”, meaning: even now we have boosted the properties of ¬ from intuitionistic to classical – and note that is indeed ¬-classical). We turn to an imaginary example – in the sense that no actual proposal has been made with this content – to underscore the problem mentioned above


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with semantical accounts in terms of several (here, two) truth-relations. We will notate them as |=+ and |=◦ . Though we give the semantics in terms of models (W, R, V + , V ◦ ) in which there is no restriction on the size of W , this is only done for the sake of a resemblance to the models discussed above, and in fact only one-element models need to be considered. Similarly, the relation R will not make an appearance. Thus there will be no persistence condition to worry about. Nor do we impose anything analogous to the condition (Exclusion) above. We could impose such a condition, or instead impose the contrasting condition that V ◦ = V + ; but such conditions are immaterial to the point being made. We suppose the language has one binary connective and one singulary connective: ∧ and O respectively. For a model M = (W, R, V + , V ◦ ), we define the two truth-relations thus: for any x ∈ W , we have: + M |=+ x pi iff x ∈ V (pi );

M |=◦x pi iff x ∈ V ◦ (pi );

+ + M |=+ x A ∧ B iff M |=x A and M |=x B;

M |=◦x A ∧ B iff M |=◦x A;

◦ M |=+ x OA iff M |=x A;

M |=◦x OA iff M |=+ x A.

Thus O, like strong negation in the original semantics, toggles us back and forth between the two truth-relations. In the original semantics (for strong negation) and in the current example, ∧ behaves conjunctively w.r.t. the |=+ relation. Whereas in the original semantics ∧ behaved disjunctively w.r.t. the |=− relation, here ∧ behaves like the first projection truth-function w.r.t. the |=◦ relation. We could equally well have it behave implicationally or biconditionally or . . . , for the present point, since that point is precisely to emphasize how unconstrained the treatment by one truth-relation is given that by another (or, in the general case, by the others). The earlier non-congruentiality of SN has an analogue here, reflecting the fact that it is the merest accident that we think of A ∧ B as the same compound of A and B in its appearance in certain embedded contexts (namely, in the present case, when embedded under the truth-relation-switching operator O). For consider the consequence relation defined by in terms of this semantic apparatus in the same way as SN was defined – by truth(-in-the-sense-of-|=+ )preservation at all points in all models. We have p ∧ q p ∧ q of course since there is no ‘switching’ going on; but we do not have O(p ∧ q) O(q ∧ p), since the this would amount to p q, given the behaviour of ∧ w.r.t. the |=◦ relation. So O is non-congruential according to : equivalent formulas are not guaranteed to yield equivalent results when substituted one for another within its scope. Exercise 8.23.8 Give a many-valued version of our imaginary ∧ with O example; that is, recast the frame of a one-element model in the above as a matrix. (Note that there should be four rather than three matrix elements, since no combination of |=+ and |=◦ ‘values’ is ruled out.) We have been poking a bit of fun at the logic of strong negation, especially in respect of congruentiality issues, and in all honestly need to conclude by saying that this has not been quite fair. Thinking of the Fmla logic for a moment, adding suitable implicational formulas (corresponding to sequent schemata (0 – 5b)) to an axiomatization of IL with the additional connective −, the noncongruentiality point amounts to a failure of ↔-congruentiality; this reflects


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a failure of contraposition for −: A → B does not provably imply −B → −A. But Nelson [1959] realized that if we introduced a different implicational connective, ⇒, say, with A ⇒ B defined to be (A → B) ∧ (¬B → ¬A), then ⇔-congruentiality would be secured; further, if we started with ⇒ as primitive, then we could recover the original intuitionistic → by defining A → B as A ⇒ (A ⇒ B). With ⇒ as primitive we can define the consequence relation relating Γ to C just when C can be obtained from theorems of the logic together with formulas in Γ by means of Modus Ponens (as a rule governing ⇒), and this consequence relation is congruential. (See the discussion of Figure 8.23a above for a later three-valued version of this discovery.) A contemporary presentation of such facts (and more) is provided by Spinks and Veroff [2008]; for the main logical points, divorced from the issue of equivalent algebraic semantics (in the sense of 2.16 above), see pp. 401–403. As the authors point out, what we have in the ⇒-based case is a substructural logic, contraction (alias W ) failing for this implicational connective – though a weakened version on the contraction theme survives (“3-to-2” as opposed to “2-to-1” contraction, to use some terminology which will be intelligible without further explanation). Naturally if one wants a smooth sequent calculus for such a logic, Mset-Fmla would be a more suitable framework than Set-Fmla (cf. 7.25). We close with something else that might be thought of as an attempt at getting something like classical negation up and running in the setting of intuitionistic logic. Unlike strong negation, this 1-ary connective, which we shall simply write as “∼” and call Moisil negation, will be made to be congruential in the most straightforward way by means of the third (a contraposition rule) of the following three principles. We may use them to axiomatize what has been called symmetric-intuitionistic logic, but will here be called intuitionistic logic with Moisil negation, by adding them to any standard basis for IL in Fmla: ∼∼ A → A

A → ∼∼ A

A→B ∼B → ∼A

We could equally well present the ideas in Set-Fmla, replacing all these main “→”s with “”s. Moisil [1942] considered the extension of Positive Logic by these principles; for references and an extensive discussion, see Monteiro [1980]. (The above principles appear on p. 60 of Monteiro [1980].) More accessible, as well as being of considerable general interest, are Galli and Sagastume [1999], Ertola, Galli and Sagastume [2007]. Algebraic and model-theoretic semantics are described in these works, but will not be presented here. Here are the principles used earlier in describing strong negation, recast in Fmla, and with (3a) and (3b) omitted, since these would just be the first two of the principles governing ∼ inset above: (0) (1a) (2a) (4a) (5a)

∼ A → ¬A ∼ (A ∧ B) → (∼ A∨ ∼B) ∼ (A ∨ B) → (∼ A ∧ ∼B) ∼ ¬A → A ∼ (A → B) → (A ∧ ∼B)

(1b) (2b) (4b) (5b)

(∼ A∨ ∼B) → ∼ (A ∧ B) (∼ A ∧ ∼B) → ∼ (A ∨ B) A → ∼ ¬A (A ∧ ∼B) → ∼ (A → B)

Exercise 8.23.9 Which of the seven schemata listed here is derivable (i.e., has only provable instances) in intuitionistic logic with Moisil negation?


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One may wonder about the status of contraposition in ‘theorem’ form, as opposed to the postulated ‘rule’ form above. Observation 8.23.10 Formulas of the form (A → B) → (∼ B → ∼ A) are not in general provable in IL with Moisil negation. Proof. Interpret ∼ A as ¬A (“strict negation”) and everything provable in IL with Moisil negation is provable in the normal modal logic KB!. (Nomenclature as explained in 2.21.) But the formula (p → q) → (∼ q →∼ p), where the consequent means “¬q → ¬p”, is not. (Note that we could equally well have said “KBc ”, since KBc = KB!; for more on this logic, see the proof of 9.24.6: p. 1314.) The appearance of in the above proof may seem extraneous to the subject under discussion, but Moisil [1942] (introducing ∼) was called ‘Modal Logic’ – or ‘Logique Modale’, to be precise. (To be sure, Moisil’s grasp of modal notions left much to be desired, as Turquette [1948] points out.) The proof of 8.23.10 shows that we could start with classical logic (with its boolean negation ¬) and add ∼ governed by the principles – axioms and rule – laid down for Moisil negation, and that neither (0) above, nor its converse, would then be provable. These principles are accordingly very weak, far weaker than would suffice for unique characterization (of “∼”). In fact with the example of KB! figuring in the proof of 8.23.10, we have bumped into the De Morgan negation (there written as ¬ rather than ∼) in play in 8.13: see in particular 8.13.13 (p. 1197), and compare this with the (readily established) fact that KB! is determined by the class of two-element frames each of which is accessible to the other but not to itself. The Moisil negation understood as the necessary (i.e., -prefixed) boolean negation of the formula is true at one of these points just in case the formula negated fails to be true at the other. But this is just the effect of the (‘Australian Plan’) *-semantics of 8.13 when the * function maps each of the two elements to the other.

8.24

Variations on a Theme of Sheffer

In 4.38.5(iii), p. 618, several intuitionistic analogues of the classical Sheffer stroke were noted, the first two of which gave compounds of A and B amounting respectively to ¬(A ∧ B) and ¬A ∨ ¬B, intuitionistically understood. (We are not concerned with any kind of analogy with the functional completeness of the classical Sheffer stroke – cf. 3.15.8, p. 423, and the discussion following 4.38.5 – but with intuitionistic analogues in the technical sense introduced for 4.38.5 itself.) It is not clear that either of these (or indeed various other candidates) deserves to be considered as ‘the’ intuitionistic Sheffer stroke, though we did point out that the first of them has a certain special status as satisfying topoboolean condition corresponding to the truth-function associated with |. Still, to avoid confusion, we will not use that notation and will instead write A B to suggest the (“nand ”) idea of negated conjunction, for the compound behaving like ¬(A ∧ B). This leaves us having to come up with an alternative notation for the compound behaving like ¬A ∨ ¬B, and we adopt the ad hoc solution of writing A ∗ B in this case. Avoiding “| ” will also prevent confusion with our discussion in 4.37 of the supposedly new intuitionistic connective from Bowen


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[1971] there written as |. Recall that Bowen proposed some sequent calculus rules for | which had the effect we can record in the notation just introduced thus for the associated consequence relation : A∗BA|B

and

A | B A B.

In what follows, we will look at the logical behaviour of ∗ and in their own right, instead of at this non-uniquely characterized intermediary (which received our attention in 4.37, especially 4.37.7). This means that we operate with the following two clauses in the definition of truth at a point, in a model M = (W, R, V ), presumed to satisfy for the propositional variables the condition (Persistence) from 2.32. The first strengthens the sufficient condition, (S) in 4.37 for Bowen’s | -compounds into a necessary and sufficient condition for ∗-compounds, while the second makes a similar strengthening of the necessary condition (N) given there for Bowen’s | -compounds. [∗]

M |=x A ∗ B iff either for all y ∈ R(x), M |=y A or for all z ∈ R(x), M |=z B.

[]

M |=x A B iff for all y ∈ R(x), M |=y A or M |=y B.

(Of course, we could equally well have formulated the rhs of [] as saying that for no y ∈ R(x) do we have both M |=y A and M |=y B. Here we rely on the fact that the logic of our own discussion is classical.) We begin by looking for syntactic conditions on consequence relations to capture the behaviour in IL of ∗. To make free use of the rule notation for conditional conditions, we work in Set-Fmla, and for simplicity take our sequents to have no connectives other than ∗. We want the associated consequence relation associated with the proof system on offer to be that determined by the class of all frames when [∗] is employed in the truth-definition. We shall want a similar proof system for the case of , and will then go on to raise questions about transferring the syntactic description across to a more traditional sequent calculus approach, thereby providing a corrected version of the discussion of Bowen [1971] on ‘the’ intuitionistic Sheffer stroke (reviewed in 4.37). A key starting point is that intuitionistic negation ¬ is definable using ∗ as currently interpreted, since ¬A can be expressed by A ∗ A. (That is, the latter formula is true at a given point in a model just in case A is true at no accessible point.) This allows us to formulate an (∨E) or (∨ Left) style rule for ∗, given as (∗1) below, and an ‘Ex Falso Quodlibet’ principle, appearing as (∗2), as well as some rules, (∗3), (∗4), which combine the effect of (∨I) or (∨ Right) with (¬-Contrac) from 8.13.2(ii), p. 1187; these references to ¬ and ∨ are made with the equivalent (in full IL) ¬A ∨ ¬B in mind. (∗1)

(∗3)

Γ, A ∗ A C

Γ, B ∗ B C

Γ, A ∗ B C Γ, A A ∗ A Γ A∗B

(∗2)

(∗4)

A, A * A C Γ, B B ∗ B Γ A∗B


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Exercise 8.24.1 Where ∗ is the consequence relation associated with the proof system having the above rules alongside the structural rules (R), (M), and (T): (i) Show, by a suitable derivation in this proof system, that for all formulas A, B, we have A ∗ B ∗ B ∗ A. (ii) Show similarly that for all A, B, C, we have A, B, A ∗ B ∗ C. (A rule to that effect could accordingly replace (∗2), since the latter is the special case in which A = B.) (iii) Show that in the proof system described at the start of this exercise, the rule (∗3) is interderivable, given the remaining primitive rules, with (∗3a) and (∗3b), taken together: (∗3a)

Γ, A A ∗ A Γ A∗A

(∗3b) A ∗ A A ∗ B.

(iv) Show that every sequent provable in this proof system is valid on every frame (when truth in a model is governed by [∗]). To show that our proof system is not only – as the last part of the above Exercise reveals – sound, but also complete, w.r.t. the class of all frames, we use the canonical model method, constructing the model required, (W, R, V ) in the following way. Counting a set of formulas as consistent when it does not contain every formula and deductively closed when it is closed under the associated consequence relation ∗ , we take W to comprise the consistent deductively closed sets of formulas which are also ∗-prime in the sense of the following definition. Γ is ∗-prime just in case whenever A ∗ B ∈ Γ, either A ∗ A ∈ Γ or B ∗ B ∈ Γ. (Since A ∗ A behaves like ¬A, we are thus thinking of A ∗ B as the disjunction of A’s negation with B’s, which gives the connection between ∗-primeness and primeness in the usual sense, as in 2.32.) Lemma 8.24.2 If Γ A is not provable in the current proof system, then there is a consistent deductively closed ∗-prime set of formulas Γ+ ⊇ Γ such that Γ+ ∗ A. Proof. The only novelty for a Lindenbaum argument here (see the discussion leading up to 2.32.6, p. 311) is ∗-primeness, for which (∗1) should be appealed to. Having described the W of our canonical model, we define R and V exactly as in the discussion following 2.32. Lemma 8.24.3 For any formula C, and any element x of the canonical model just described, we have C ∈ x if and only if (W, R, V ) |=x C. Proof. The key step, invoking the inductive hypothesis (of an induction on the complexity of C, for the case in which C is A∗B requires us to show for arbitrary x ∈ W: A ∗ B ∈ x ⇔ for all y ∈ R(x), A ∈ / y or for all z ∈ R(x), A ∈ / z.


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⇒: For a contradiction, suppose that A ∗ B ∈ x(∈ W ) but there exist y, z ∈ R(x) with A ∈ y, B ∈ z. Since x is ∗-prime and A ∗ B ∈ x, we have (i) A ∗ A ∈ x or (ii) B ∗ B ∈ x. In case (i), this puts A ∗ A into y, making y contradicting the consistency of y, in view of (∗2), since A ∈ y; case (ii) is argued similarly. ⇐: We must show that if for all y ∈ R(x), A ∈ / y, then A ∗ B ∈ x, and similarly for the case in which for all z ∈ R(x), B ∈ / z. Taking the first case for illustration, if no y ∈ R(x) contains A, this must be because y ∪ {A} is inconsistent; then for some finite Γ ⊆ y, Γ, A has every formula as a consequence by the relation ∗ , so in particular, Γ, A ∗ A ∗ A, in which case by appeal to (∗3), Γ ∗ A ∗ B and so A ∗ B ∈ x. The other case is similarly argued, but with an appeal to (∗4) replacing this appeal to (∗3).

Theorem 8.24.4 A sequent is provable in the present proof system for ∗ if and only if in case it is valid on every frame. Proof. We assume the ‘only if’ direction taken care of (Exercise 8.24.1(iv)). Now suppose that σ = Γ A is not provable. Since Γ ∗ A, by 8.24.2 there is a point x in the canonical model with Γ ⊆ x while A ∈ / x, and at which further, by 8.24.3, all formulas in Γ are true and A is not, showing that σ is not valid on the frame of that model. Turning our attention to , we offer the following rules, the second of which is that mentioned under Exercise 8.24.1(ii), with ∗ replaced by : (1)

Γ, A, B A B Γ AB

(2)

A, B, A B C

Theorem 8.24.5 A sequent is provable using the above rules for and the structural rules (R), (M), and (T) if and only if in case it is valid on every frame. Proof. The soundness (‘only if’) direction being evident, we concentrate on completeness, which is proved by taking the canonical model (W, R, V ) to comprise the deductively closed consistent sets of formulas where deductive closure is in respect of the consequence relation , understood analogously to ∗ above. (No primeness related considerations here, then.) As usual, the key step in the proof that membership and truth coincide in this model is showing A B ∈ x ⇔ for no y ∈ R(x), A ∈ y and A ∈ y. For the ⇒ direction, we invoke (2), and for the ⇐ direction, (1), the rest of the argument being as in the proof of 8.24.4. We can combine the rules for ∗ and with each other, alongside the structural rules, to obtain a proof system for both connectives together. Let , for the following Exercise, be the consequence relation associated with this proof system.


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Exercise 8.24.6 (i) Show by a syntactic argument that for all formulas A, B, we have A ∗ B A B, where is as defined before this exercise. (ii) Show by either a semantic or a syntactic argument that for as above, we have the following for all formulas A and B: A B ((A ∗ B) ∗ (A ∗ B)) ∗ ((A ∗ B) ∗ (A ∗ B)). The converse of the -statement in (i) here would not be correct, in view of the intuitionistic invalidity of, e.g., the sequent ¬(p ∧ q) ¬p ∨ ¬q. In classical logic, for any two definable connectives which are not equivalent in the sense of forming equivalent compounds from the same components, neither is a subconnective of the other, even when attention is restricted to sequents of Set-Fmla. Recall that #0 is a subconnective of #1 (both connectives of the same arity) according to a logic conceived of as a set of sequents if whenever σ(#0 ) and σ(#1 ) are sequents in each of which the only connective to occur is that exhibited in parentheses, and #0 occurs in the first precisely where #1 occurs in the second, then if σ(#0 ) is in the logic, so is σ(#1 ). That this result holds for CL in Set-Fmla is a reformulation of 3.13.17, p. 397 (a corollary of a result of W. Rautenberg, given as 3.13.16). It does not, however, hold for IL, as we may illustrate with the two connectives with which we have been concerned. Example 8.24.7 Taking ∗ and as defined connectives of the usual language of IL, we find, despite their non-equivalence, that ∗ is a subconnective of , by the following argument. Let F2 be the two-element point-generated Kripke frame (for IL). We write “valid” to mean “valid on every frame”, recalling that this is equivalent to the IL-provability of a sequent, and also make use of a temporary extension of the “F |= A” notation to replace the formula by a sequent and write “F |= σ” to mean that σ is valid on F; this allows for a succinct presentation of the argument as: σ(∗) valid ⇒ F2 |= σ(∗) ⇒ F2 |= σ() ⇒ σ() valid. The first implication just records that a sequent valid on every frame is valid on F2 , the second exploits the fact that ∗ and are equivalent in any model on this frame (or any other convergent – or ‘directed’ – frame), and the justification for the third is given by 8.21.12 (due to Rautenberg), on p. 1221, where validity on F2 is described in matrixtheoretic terminology, and seen to suffice for the validity of any sequent in the {¬, ∧}-fragment of IL – including therefore our sequent σ(). Remark 8.24.8 A propos of the above example, it is worth remarking that is not, conversely, a subconnective of ∗ (in IL). To see this, note ¬(A ∧ B, like any other negated formula, is an IL-consequence of its own double negation, whereas ¬A ∨ ¬B does not enjoy this property. Thus we may take σ() as ((p q) (p q)) ((p q) (p q)) p q to have σ() but not σ(∗) provable. (By the equivalence noted under 8.24.6(ii), the latter amounts to p q p ∗ q.)


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Although ∗ is a subconnective of , the rules we would obtain by replacing by ∗ are not derivable from those given for ∗ (alongside the structural rules); more specifically, the rule “(1)-for-∗”, as we may call it, is not derivable, as can be seen from the fact that this rule does not preserve the property of holding in a model. (“(∗1)”-for- can be seen to be underivable from the rules by a similar argument.) Let us turn to the question of suitable sequent calculus rules for ∗ and , beginning with the latter. We would like suitable left and right insertion rules, they should be pure and simple, taken together they should ideally also be cutinductive and regular. A consideration of these rules will enable us to make a general observation about the ‘intuitionistic restriction’ on Set-Set rules as we transfer a formulation of such rules from that framework for CL over to Set-Fmla0 for IL, as in 2.32. The rule (1) took us from Γ, A, B A B to Γ A B. As is evident from the proof sketched for 8.24.5, the role played by this rule is to ensure that if Γ ∪{A, B} is inconsistent then AB follows from Γ. We couldn’t have expressed this by putting a schematic letter on the right of the premiss sequent, making this Γ, A, B C, since although this would make the resulting rule simple (in the sense of only having one occurrence of in its formulation), it would not have the desired effect: we are trying to say that if everything follows from Γ ∪{A, B} then A B follows from Γ, and we have inadvertently ended up saying instead that if anything follows from. . . (A similar issue arises again in 8.31.) The idea of putting A B itself on the right in our Set-Fmla rules works because the other rule, (2) tells us that a set containing A and B has every formula as a consequence if and (of course) only if that set has A B as a consequence. But moving to Set-Fmla0 we can rely on (M) to leave the right-hand side empty to convey the inconsistency of the set on the left. That gives us a suitable ‘Right’ rule for ; and we have dressed up (2) suitably as a ‘Left’ sequent calculus rule: ( Left)

ΓA

Γ B

( Right)

Γ, Γ , A B

Γ, A, B Γ AB

Exercise 8.24.9 (i) Check that these rules are regular and cut-inductive. (ii) Making use of Theorem 8.24.5, or adapting its proof, show that with the Set-Fmla0 structural rules (R), (M), and (T), the above rules ( Left) and ( Right) render provable precisely such Set-Fmla0 sequents (of the language with as its sole connective) as are valid on every frame. The rule ( Left) is what was offered in Bowen [1971] and recorded in 4.37 above – with written “|” – as (| Left). It was there paired with the following rule (or pair of rules): (| Right)

Γ, A Γ A| B

Γ, B Γ A| B

which is, we observed, the appropriate right insertion rule for what we are now writing as ∗. (This will appear again below, with the ∗ notation, as two rules,


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(∗ Right 1) and (∗ Right 2).) What, then, might a corresponding (∗ Left) rule look like? Here we encounter a difficulty. There seem to be no suitable candidates for (∗ Left). (We have met this situation before, for another connective, in 2.32.14, p. 322.) Writing out the transition from premiss sequents to conclusion that we want to represent using ¬ explicitly, so that it would be a case of (∨ Left) if A ∗ B were written as the disjunction of these negated formulas, we have: Γ, ¬A C

Γ, ¬B C

Γ, A ∗ B C we have the problem of getting rid of the occurrences of ¬. In the case of the earlier rule (∗1), we encoded these negated formulas as A ∗ A and B ∗ B, making for a non-simple rule. If we were looking at a classical version of the rule, we could put the formulas to be negated on the right of the turnstile (unnegated) instead: Γ A, C

Γ B, C

Γ, A ∗ B C though of course for a classical version of this rule we should have “C” replaced by “Δ”. But even the case in which Δ is ∅, so that we have a Set-Fmla0 rule: ΓA

ΓB

Γ, A ∗ B does not have the desired effect, essentially because of the fact that the intuitionistic sequent calculus rule (¬ Left) is not invertible. (See 2.32.19(ii), p. 325.) In particular, we cannot replace Γ¬A – to take the left premiss by way of example – with Γ A, since the hypothesis that latter sequent is provable is stronger than the corresponding hypothesis concerning the former sequent. Remark 8.24.10 On the assumption that there is indeed, as the evidence reviewed here suggests, no suitable pair of sequent calculus rules for ∗, the following conjecture is worth investigating – though we shall not look into it here: The connectives which are amenable to treatment by such rules are those whose semantic treatment can be formulated using a topoboolean condition. As noted in 4.38.6, p. 620, for (what we are now calling) this is the topoboolean condition induced by the nand truth-function; while there is no truth-function inducing a topoboolean condition which is equivalent to the semantical clause for ∗. Despite the lack of a left-hand partner, we can learn a bit from the above (| Right), here split into two rules and re-notated with ∗: (∗ Right 1)

Γ, A Γ A∗B

(∗ Right 2)

Γ, B Γ A∗B

What we can learn about is the care with which the ‘intuitionistic restriction’ on Set-Set rules for CL needs to be formulated, summarised in the slogan: “no overcrowding on the right”. Here are two possible ways of spelling out this


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restriction, either of which is perfectly satisfactory for our CL and IL sequent calculi Gen and IGen from 1.27 and 2.32: The intuitionistic restriction.

:::::::::::::::::::::::::::

Liberal Version: Any proof in the Set-Set system in which no there appears no sequent outside of the framework Set-Fmla0 qualifies as meeting the intuitionistic restriction. Stringent Version: The application of any Set-Set rule with the potential to increase the cardinality of the right-hand side is, for the intuitionistic restriction to apply, restricted to applications in which before the rule applies, the righthand side is empty. What is meant by a rule’s having the “potential to increase the cardinality of the right-hand side” is that the rule – here assumed to be an n-premiss rule – has some application σ1 , . . . , σn , σ in which the cardinality of the right-hand side of σ exceeds the maximum of the cardinalities of the right-hand sides of σ1 , . . . , σn . Amongst the operational rules of Gen with the potential to increase the cardinality of the right-hand side there is only one, namely (¬ Right): Γ, A Δ Γ ¬A, Δ If IGen is formulated instead without ¬ as primitive, taking ¬A to abbreviate A → ⊥, using ⊥ with ⊥ as a further initial sequent and achieving the effect of (¬ Right) by means of (→ Right), then there are no such potentially rightincremental operational rules. But to illustrate the present point, we stick with ¬ primitive and the rule (¬ Right). Example 8.24.11 (An innocuous case.) Suppose we have reached a stage in a proof (in IGen) at which a sequent of the form Γ, A ¬A and wish to apply (¬ Right) to remove the (exhibited) A from the left and transplant it to the right, transformed into ¬A. Thus the conclusion of the proposed application will be Γ ¬A. Can we legitimately do this? According to the liberal version of the intuitionistic restriction, we may indeed do so. Since, making the right-hand side fully explicit after the application of the rule, what we have there is {¬A, ¬A} which is of course the same one-element set as {¬A}, so there has been no departure from Set-Fmla0 . However, the right hand side was not vacant before this application (¬ Right), so according to the stringent version of the restriction, this is not a correct application of the rule. However, the stringent interpretation intuitionistic restriction not block the proof any sequent, as one may check by examining the possible ways in which the sequent displayed inset above will reveal. The most important such way is that ¬A got to be on the right because of an earlier application of (¬ Right) itself, in which case we could have obtained Γ ¬A already from the premiss sequent for that application of the rule.


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In a multiset-based framework (as in 2.33, for instance) the sequent inset in 8.24.11 would have arisen by (¬ Right) in which there were two occurrences of A on the left, in which case what has to be done before the last step suggested above is that a contraction on the left replaces them with one (which is then moved over to the right and negated): but there is no problem with Left Contraction for IL. (Sometimes it is easy to overlook the possibility of a left contraction’s having been applied, however: see 2.32.9(iv), p. 315.) Hence the description of the above example as innocuous. Contraction on the Right, however, is another matter, though to illustrate this we need to depart from the primitive connectives presumed for IGen and enter the territory of the present subsection: Example 8.24.12 (Danger – Hidden Right Contraction! ) We can use the rules ( Left), ( Right) and (∗ Right 1), (∗ Right 2) to illustrate a more dramatic contrast between the liberal and the stringent versions of the intuitionistic restriction, and to see that it is the stringent version which is required. The rules whose formulation is sensitive to this distinction are (∗ Right 1) and (∗ Right 2), so we begin with their appropriate classical (Gen or LK) analogues: (∗ Right 1)

Γ, A Δ

(∗ Right 2)

Γ A ∗ B, Δ

Γ, B Δ Γ A ∗ B, Δ

The stringent version of the intuitionistic restriction is then the requirement that Δ = ∅; this coincides with our earlier formulation of the rules (∗ Right 1, 2). The liberal version – to prevent our straying from Set-Fmla0 – requires only that in the conclusion sequents of these rules we have |{A ∗ B} ∪ Δ| 1. Thus with the liberal version we can apply (∗ Right 2), taking Δ = {A ∗ B}, as follows: pp

qq

p, q, p q q, p q p ∗ q pqp∗q

( Left) (∗ Right 1) (∗ Right 2)

!!

The step marked “!!” would not be legitimate on the restrictive version of (∗ Right 2), and the sequent proved using the liberal version is intuitionistically invalid (as one sees by re-writing it, using the more primitives, as ¬(p ∧ q) ¬p ∨ ¬q. This sequent is of course classically valid and in a classical version of the proof done in Mset-Mset what we would have after the corresponding application of (∗ Right 2) would be p q p ∗ q, p ∗ q, followed by an appeal to Right Contraction ( = (RC) in 2.33) to reach the terminus of the above proof. In the classical case – cf. Riser [1967] – the distinction between “” and “∗” evaporates and we might just as well write “|” for them/it; note that there is no problem about a left insertion rule in this case, and there is a free choice for the right insertion rule: the (∗ Right 1, 2) rules as above would be fine, as would be the rule ( Right) given before 8.24.3, but kitted out with a set variable “Δ” for side formulas on the right of premiss and conclusion sequent. These rules are like the Gentzen and Ketonen (∧ Left) rules


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contrasted in 1.27 (after 1.27.4, p. 141), except that the compound is inserted on the other side of the “” since we are dealing with negated conjunctions. That concludes our business for this subsection, but the last reference to Ketonen suggests a final exercise in linear logic: Exercise 8.24.13 (i) Let us consider additive and multiplicative forms of the Sheffer stroke, in an extension of the system CLL of classical linear logic from 2.33, we write the corresponding compounds of A, B, as A |a B and A |m B respectively, intending that the sequent pairs p |a q ¬(p ∧ q) and p |m q ¬(p ◦ q) should be provable when the CLL rules for the connectives appearing here are supplemented by sequent calculus rules for |a and |m . Formulate such rules, in the style of those given in 2.33 (i.e., the framework should be Mset-Mset and the rules should be pure and simple left and right insertion rules). (ii) As in (i) but this time formulate suitable rules for the dual connective, nor, in additive and multiplicative incarnations, for which we write ↓a and ↓m , and with the aid of which we want the following to be provable: p ↓a q ¬(p ∨ q); p ↓m q ¬(p + q). (iii) What, if anything, goes wrong if parts (i) and (ii) here are attempted, in the latter case for ↓a only (since the target equivalence for ↓m is formulated using the ILL-unavailable +), against the background of ILL rather than CLL?

Notes and References for §8.2 Intuitionistic and Dual Intuitionistic Negation. Various difficulties for the intuitionistic treatment of negation, not taken up in the text, are alleged in Belding [1971], Hossack [1990], Price [1990], Lenzen [1991]. For more information on double intuitionistic negation, which featured prominently in 8.21, see Došen [1984], [1986b]. Rauszer’s work on dual intuitionistic logic (8.22) may be found in her papers Rauszer [1974a], [1974b], [1977a], [1977b], [1977c], [1980]. See also López-Escobar [1985], [1981a], Goré [2000], Brunner and Carnielli [2005], and Shramko [2005], Wansing [2010]. Some further references were supplied in the discussion before 4.21.9: see the Digression on p. 1225. A curious fact about the literature on this topic is that the subject is frequently rediscovered de novo by authors who make no reference to the earlier work of others; examples include Dummett [1976b] (see p. 126), Goodman [1981], and Urbas [1996]. A study of some extensions of dual intuitionistic logic is provided by Szostek [1983]. A generalized version of dual intuitionistic negation appears in Sylvan [1990]. The analogy with tense logic, so that we have not only forward but also backward looking operators (in terms of the Kripke semantics) mentioned in our discussion is mentioned in a related connection on p.143 of Gabbay [1981]. (Gabbay is interested in a connective T interpreted by setting T A true at a point x if there is some y = x such that yRx and A is true at y.) The duality underlying the contrast between intuitionistic negation and dual intuitionistic negation (or Heyting and Brouwerian negation, as Rauszer puts it) cross-cuts another relationship—also deserving the name of duality, in fact— which arises in substructural logics without permutation (of formulas in a sequent, as in Seq-Fmla), corresponding to the two implication connectives we


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get in such a setting – one for which we have the schema A, A →1 B B and the other for which we have the schema A →2 B, A B. Such a (“bilinear”) logic is reviewed in Lambek [1993b], and in view of the two dualities, Lambek remarks (p. 222) that it supports four distinct negation connectives. Following the work in categorial grammar (cf. Lambek [1958]) of which this is an outgrowth Lambek writes A →1 B as A\B and A →2 B as B/A. (Unlike the simple categorial grammar described in 5.11.2, p. 636, and 5.11.3, in an ordersensitive approach such as this, we distinguish the category labels C-over-D, written there as C/D, for functor categories depend as they make an expression of category C when supplied with an expression of category D on their left – the label here being “D\C” – from those which make an expression of category C when supplied with one of category D on their right: “C/D”. The symbols “→” and “←” are sometimes used for this distinction, as in Goré [1998a] and p. 209f. of Zielonka [2009],. The former paper also provides further remarks, from a display-logical perspective, on the “four natural negations within Intuitionistic Bi-Lambek Logic” (p. 479). Our discussion does not touch treat these refinements. Strong Negation. On strong negation (8.23): Nelson [1949], Thomason [1969], Vorob’ev [1972], Routley [1974], Gurevich [1977], Hazen [1980], Almukdad and Nelson [1984], Akama [1988], Tanaka [1991]. There are affinities with the system of Fitch [1952], studied model-theoretically in Routley [1974] and prooftheoretically in Thomason [1967]; for Fitch’s logical and philosophical motivation, see Anderson [1975], Myhill [1975]. Nelson’s semantic account was a variation on the theme of Kleene’s ‘realizability’; see also López-Escobar [1972]. [1981a] Algebraic semantics for strong negation were provided in Rasiowa [1958] and further studied in Białynicki-Birula and Rasiowa [1958], Vakarelov [1977], Sendlewski [1984]. Vakarelov [2006] provides a helpful survey of variations on the theme of negation specifically oriented around Rasiowa’s work. Suggested applications for what the authors are happy to call – though we have already expressed reservations about this in 8.23 – strong negation in logic programming may be found in Pearce and Wagner [1990], [1991]. Though neither the phrase “strong negation” nor “constructible falsity” appears in von Kutschera [1983], this is the subject of the paper. Part II of Odintsov [2008] is devoted to strong negation, and the paraconsistent variant thereof in particular. For further references, see Spinks and Veroff [2008]. Veltman [1884] develops a logic like that provided by SN with strict negation as the only primitive negation connective, but with the clause: + − M |=− x A → B iff M |=x A and M |=x B

replaced by the following (our notation here, not Veltman’s): + − M |=− x A → B iff for some y ∈ R(x), M |=y A and M |=y B.

The effect of this change is that we lose the persistence properties recorded in parts (i) and (ii) of 8.23.1 (p. 1231), which is Veltman’s intention, as he wants to model statements which cease to be assertible with the acquisition of new information, such as “It may be that p”. (Veltman’s operator may is defined by setting may A = −(A → −A). Paired with must defined by must A = −A → A, we have a treatment of epistemic modality in several respects more plausible


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than that provided by and in the conventional approach to these matters – in a normal modal logic such as KT or S4, that is. Further background is supplied by Section 2 of Humberstone [2005f ].)

§8.3 THE FALSUM 8.31

Negation and the Falsum Connective

We foreshadowed the subject matter of the present section at the end of our discussion in 8.12, with a treatment of several options according to which a Falsum symbol was more or less integrated amongst the formulas of the language, and here we go into detail with the proposal according to which the Falsum is as fully integrated as possible, being itself a perfectly ordinary formula (written ‘⊥’: of course, although this is a formula, it is atypical in being a nullary connective). Nothing will be said about the other options canvassed in 8.12. Before introducing ⊥ in this capacity, we must back up a little. Recall from 4.12 that the language of Positive Logic has connectives ∧, ∨, and →, and that a Set-Fmla version of this logic can be presented in the natural deduction approach by choosing just the rules of INat from 2.32, or equivalently just those of Nat from 1.23 (since these systems differ only over rules for ¬), which govern the three connectives mentioned—the proof system PNat, that is. (Reminder: since the ¬-rules of Nat extend this system non-conservatively, PNat is not a proof system for the positive fragment of CL in Set-Fmla, on which see 8.34.) Suppose that we wanted to introduce ¬ into this language with the idea that ¬A was to follow from Γ just in case every formula followed from Γ ∪ {A}. This would be the ‘(simple) absurdity’ approach to negation, according to the terminology of Curry [1963]. We are interested, then, in supplementing the above rules in such a way that the consequence relation associated with the extended proof system satisfies, for all formulas A and all sets of formulas Γ: (1) Γ ¬A ⇔ for every formula B: Γ, A B Now the ⇒-direction of (1) can easily be converted into a sequent-to-sequent rule in Set-Fmla, thus: (2)

Γ ¬A Γ, A B

And this is equivalent to a rule with which we are familiar (though Γ is now, given our conventions, required to be finite – not a serious restriction, in view of 1.21.4(i), p. 109): Exercise 8.31.1 Show that (2) and the rule (EFQ) from 1.23 are interderivable given (R), (M) and (T). When we turn our attention to the ⇐-direction of (1), we might initially expect to capture its force by simply inverting (2): (3)

Γ, A B Γ ¬A

8.3. THE FALSUM

1253

But of course this doesn’t work, amounting instead to the principle that if anything follows from Γ ∪ {A}, then ¬A follows from Γ. (A similar point arose in the discussion after 8.24.8 above.) We wanted, not “if anything”, but: “if everything”. The trouble is that the ⇐-direction of (1) is not a universal strict Horn formula of the metalanguage (a metalinguistic Horn formula ‘of the second type’—1.13), because of the universal quantifier “ for every formula B” in its antecedent. It is true that even the ⇒-direction of (1), as written, has this universal quantification in its consequent: but we here rely (in effect) on the metalinguistic version of the equivalence of p → ∀x(F x) with ∀x(p → F x), an equivalence with no analogue for a universal quantifier in the antecedent. (The equivalence in question holds in IL and so a fortiori in CL; for the latter case, there is something of an analogue in the equivalence of ∀x(Fx) → p with ∃x(Fx → p), which does not hold intuitionistically. However, what we wanted for our rule formulation was an outer universal quantifier.) For those readers who prefer the Prawitz-style description of rules to the sequent-to-sequent formulations we generally adopt, (3) would become: (3 )

[A] · · · B ¬A

The problem is the same: the dots indicate a derivation of any given formula B from A, while we wanted to express the much stronger condition that all formulas can be so derived. We could solve our problem if we had some special formula with the property that if it was derivable from a set Γ, then every formula was. That is, if we had a formula ¢ such that for all Γ: (4) Γ ¢ ⇔ for every formula B, Γ B. Here the ⇒-direction is what we actually wanted, and the ⇐-direction is a triviality: of course if every formula follows from Γ then so does our special formula ¢. And the ⇒-direction of (4) has the right form, just like the ⇒direction of (1), for us to convert it into a rule: (5)

Γ¢ Γ B

Since “¢” is the name of a particular formula, and not a schematic variable ranging over arbitrary formulas, the problem about the unwanted “any”-as-opposedto-“every” interpretation does not arise, and we could use our new formula to fix up the situation with rule (3) above. We replace (3) with (6)

Γ, A ¢ Γ ¬A

Provided that satisfies (4), Γ, A ¢ iff for every B: Γ, A B; so our new premiss-sequent says exactly what we wanted it to say: that every formula follows from Γ ∪ {A}.


1254

As the reader will have observed, the condition (4) is satisfied by a consequence relation just in case ¢ B for every formula B; this is to say that satisfies (4) iff treats ¢ ⊥-classically, to use a locution from 1.19. We might as well use the familiar notation and write the desired ¢ as “⊥” from the start. What we have seen, then, is how a desire to extend P(ositive) L(ogic) in such a way as to satisfy (1) above for the newly added connective of negation, leads naturally to introducing first the zero-place falsum connective ⊥ with suitable (⊥-classicizing) rules, and using this to formulate the rules for ¬. The point has been to forestall the impression one sometimes encounters that recourse to ⊥ is a highly artificial ‘trick’ designed to streamline the logical study of negation; as we have seen, such recourse is rather an all but inevitable response to one way of thinking about what we might want—namely, condition (1)—from a negation connective. (Of course, we might not want this at all; it would be anathema to a traditional relevant logician, whose treatment(s) of negation will occupy us in §8.13.) ‘One way of thinking’ about this project: but not the only way. For, as we shall have occasion to observe in a moment, the same effect can be obtained without ⊥, by simply adding the rules (RAA) and (EFQ) governing ¬. In 2.33 (p. 329), we defined a consequence relation to be →-intuitionistic iff for all Γ, A, B: Γ A → B iff Γ, A B. Clearly if is the consequence relation associated with a proof system in Set-Fmla, then is →-intuitionistic just in case rules (→I) and (→E) are admissible for that proof system. (When people speak of a connective behaving like intuitionistic implication, they have in mind as much the inferential powers it lacks as those it has, so the above definition does not capture their intentions; on it any consequence relation which is →-classical is a fortiori →-intuitionistic.) There is of course no parallel need for terms like “∧-, ∨-, ⊥- intuitionistic”, since the corresponding concepts would coincide with their classical analogues, as applied to consequence relations. However there is such a need in the case of ¬, and we here enter the following definition: is ¬-intuitionistic iff for all Γ, A: Γ ¬A if and only if for every B: Γ, A B. Exercise 8.31.2 Verify that IL is ¬-intuitionistic. Now this definition of ’s being ¬-intuitionistic just amounts to the satisfaction of the condition (1) above. So the effect of rules (2) and (6), in the latter case with “⊥” for “¢”, alongside the elimination rule (⊥E)

Γ⊥ ΓB

is to extend PL to IL, in a language containing not just ∧, ∨, →, and ¬ (our favoured language for 2.32) but also ⊥. (The rule (⊥E) simply ⊥-classicizes the associated consequence relation; recall that IL is ⊥-classical.) Traditionally, it has not been usual to employ such a language, but instead to exploit the equivalences (7) ¬A IL A → ⊥

and

(8) ⊥ IL q ∧¬q

to treat either ⊥ as primitive, with ¬ as a defined connective (exploiting (7)) or vice versa (exploiting (8)). Here we make use also of the fact that IL is congruential (3.31). Reasons against such attempts to economize on the

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primitive connectives were given in 3.16, but of the two suggested definitions it is the first that sits especially well with the way we have been building up to IL from PL. For (EFQ) now comes to be seen as the effect of (→E) and (⊥E), since ⊥ follows from A and A → ⊥ (alias ¬A) by the former rule, and then anything follows from this by the latter rule. And (RAA), the other primitive ¬-governing rule of INat, gets a similar reconstruction: a contradiction derived from Γ and A again enables us to derive ⊥ from these formulas, so A → ⊥ (i.e., ¬A) follows from Γ by, this time, (→I). If, on the other hand, we do employ both ⊥ and ¬ as undefined connectives, it is possible to obtain economies—not in respect of number of primitives, but in respect of length of proofs—if we give rules for ¬ in terms (in part) of ⊥; the resulting rules, while very natural in terms of the above-mentioned definition of ¬A as A → ⊥, are accordingly impure rules. Let us call them (¬I) and (¬E), since they would correspond to the introduction and elimination rules for the → were that definition in force:

(¬I)

Γ, A ⊥ Γ ¬A

(¬E)

ΓA

Δ ¬A

Γ, Δ ⊥

A complete presentation of the {¬, ⊥}-fragment of IL (in this language with both taken as primitive) is obtained by using the structural rules, (⊥E) above, and these new rules (¬I) and (¬E). (For the corresponding fragment of ML, omit (⊥E).) An illustration of the economical effect of this basis follows; we use a Lemmon-style format for the proof in the style of 1.23, and related to the explicitly sequent-to-sequent format as there indicated (in respect of such things as the use of the dash between line numbers for the assumption-discharging rule (¬I)). Example 8.31.3 A comparison of optimal derivations in Nat and the new proof system of the sequent ¬p ∨ ¬q ¬(p ∧ q). Nat proof: 1 2 3 2 2, 3 3 7 2 2, 7 7 1

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

New proof:

¬p ∨ ¬q p∧q ¬p p p ∧ ¬p ¬(p ∧ q) ¬q q q ∧ ¬q ¬(p ∧ q) ¬(p ∧ q)

Assumption Assumption Assumption 2 ∧E 4, 3 ∧I 2–5 RAA Assumption 2 ∧E 8, 7 ∧I 2–8 RAA 1, 3–6, 7–10 ∨E


1256 1 2 3 2 2, 3 6 2 2, 6 1, 2 1

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

¬p ∨ ¬q p∧q ¬p p ⊥ ¬q q ⊥ ⊥ ¬(p ∧ q)

Assumption Assumption Assumption 2 ∧E 4, 3 ¬E Assumption 2 ∧E 7, 6 ¬E 1, 3–5, 6–8 ∨E 2–9 ¬I

A similar shortening will be found on comparison of two proofs along these lines of the sequent ¬p ∧ ¬q ¬(p ∨ q). (Start by assuming the left-hand formula along with p ∨ q.)

Now although the new proof displayed above is only one line shorter than the original, it is shorter in a way that illustrates what one may think of as an inconvenience in the system Nat, brought out by the interaction between (∨E) and the rules for negation. (∨E) requires us to obtain a conclusion from each of the two disjuncts of a disjunction before claiming it follow from the disjunction itself (or whatever that depends on, should it not have the status of an assumption in the proof). At line (5) in the first proof above, we get a contradiction from the first disjunct, and at line (9), we get a contradiction from the second disjunct. But we cannot perform (∨E) before (RAA) because it’s not the same contradiction, and we need to have a single formula derived twice over. (By using assumption-rigging we can get from the one contradiction to the other, but this will only make the proof longer.) Thus the first proof employs (RAA) in each half of the ∨-elimination so that it is the negation of assumption (2) which is derived—the same formula both times—from each disjunct, thereby allowing an application of (∨E). It is because there are two applications of (RAA) when the rule-applications are so ordered that the first proof is longer. The second proof allows the derivation of a contradiction from each disjunct to lead to the derivation of that contradiction from the disjunction because we use the same ‘all purpose’ contradiction, ⊥, both times; thus the (∨E) step is not unnaturally postponed in the way that it is in the first proof of 8.31.3. Remark 8.31.4 Those familiar with the rules for the quantifiers in Lemmon [1965a] will be able to observe a similar though more dramatic shortening of the proof if they consider the Lemmon-style proof of the predicatelogical sequent ¬∃x∀y(Sxy ↔ ¬Syy), and compare this with a proof in which negation is handled with (¬I) and (¬E). (In both cases, follow Lemmon’s practice of treating A ↔ B as an abbreviation for (A → B) ∧ (B → A). The sequent just mentioned gives the form of the standard resolution of the ‘barber paradox’, according to which it is a logical truth that there is no-one who shaves (“S”) all and only those who do not shave themselves.) Here it is the interaction of (∃E) with negation that causes the Lemmon proof to be longer than one might wish.

8.3. THE FALSUM

8.32

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Minimal Logic

The proof system MNat was mentioned in passing in 2.32; rules for this system may be obtained from those for Nat by deleting the rule (¬¬E). To extend this to a presentation of INat, we must add (EFQ); so as it stands, the only primitive rule we have for ‘Minimal Logic’ is (RAA). This is a natural deduction system in Set-Fmla, corresponding to an axiom-system in Fmla for Johansson’s Minimalkalkül, proposed (in Johansson [1936]) as a response to the perceived unnaturalness of one aspect of Heyting’s logic, that embodied in our Set-Fmla presentation of IL in the rule (EFQ). Animadversions on this score are of course familiar from relevant logic. But from this viewpoint, minimal logic does not do so well after all, for, letting the Tarski consequence relation associated with MNat be M L , note Exercise 8.32.1 (i) Show that for any formulas A, B: A ∧ ¬A M L ¬B. (ii ) Using (i), show that any set which is deductively closed w.r.t. M L contains some formula and its negation, iff it contains every negated formula. (iii) Using (i) show that the consequence relation M L is not determined by any single matrix. (Solution to (iii) appears in on p. 1261 below.) 8.32.1(i) undermines the claim of ML (a label we use analogously to “IL” and “CL”) as a protector of the intuitive reaction against (EFQ) which partly motivates relevant logic – the paraconsistent motivation, as we call it: it just isn’t true that from a contradiction, everything follows (2.33). For those who react like this would be no less hostile to the suggestion that from a contradiction, everything of the form not B follows. The usual worry is that A ∧ ¬A may have ‘nothing to do with’ B, and this worry carries over to the case of ¬B: the Belnap criterion of relevance for sentential logic – shared propositional variables in the antecedents and consequents of all provable implications – is after all equally blatantly violated. (One should not reply that A ∧ ¬A and ¬B both involve negation: negation is not something a negated statement is about.) We should also note that traditional (EFQ), say in the form A ∧ ¬A B, would in any case follow from this special ¬B form in the presence of (¬¬E), which is generally accepted in work on relevant logic. The upshot of the above discussion is that considerations of the counterintuitive nature of (EFQ) cannot serve to motivate ML, since ML serves up other principles which cannot reasonably held to differ from (EFQ) on this score (whichever way one is inclined to react to such claims of counterintuitiveness). The interest of ML is better displayed in a somewhat different light, suggested by 8.31. There we saw that all of the properties possessed by ¬ according to IL could be seen as resulting from the equivalence of ¬A with A → ⊥, as long as ⊥ obeyed the (EFQ)-like rule (⊥E). Suppose we dropped this last proviso, and considered just those properties ¬ enjoyed in virtue of the equivalence in an extension of PL (with ⊥ as an additional connective) between ¬A and A → ⊥. For example, the (¬¬I) sequent p ¬¬p would be in and the (¬¬E) sequent ¬¬pp would be out, since we have p P L (p → A) → A, for any formula A, and hence also for ⊥, once this is added to the language, whereas it is not similarly the case that (p → A) → A P L p for arbitrary A. Here we have used “P L ” to indicate the consequence relation of PL – though we could equally make the


1258

point with “IL” since there is no difference here as far as {∧, ∨, →}-sequents are concerned. Again, the (‘Modus Tollens’) sequent p → q, ¬q ¬p is in, while the (EFQ)-like p ∧ ¬p q is out, since (respectively) p → q P L (q → A) → (p → A) for all A, but (e.g., taking A as r) p → (p ∧ (p → r)) P L q. Now it turns out that the list of sequents which are ‘in’ can be obtained by extending the PNat rules by (RAA). This gives the system MNat mentioned above, and is perhaps a better way of thinking of Minimal Logic than as a sort of failed relevant logic: Minimal Logic extends Positive Logic by giving negation just those properties it has in virtue of the equivalence of ¬A with A → ⊥, where no special properties are assumed for ⊥. Of course, if, as in the language of MNat, ⊥ is not present at all, we have to put this somewhat differently, for example, by choosing some propositional variable to play the role of ⊥. But which variable is chosen will have to depend on the sequent concerned, to avoid interference with the variables already present: Observation 8.32.2 A1 , . . . , An M L B if and only if A∗1 , . . . , A∗n IL B∗ for all formulas A1 , . . . , An , B of the language of MNat, where we use (·)∗ to indicate → pi for some that all contexts ¬ occurring in formulas are replaced by propositional variable pi not occurring in A1 , . . . , An , B. 8.32.2 is an easy corollary to the completeness theorem for ML below (8.32.6); alternatively, consult Prawitz and Malmnäs [1968], where it is observed that no ‘uniform’ translation in the style of that found in 6.42.3 (p. 873) is available here (the mapping (·)∗ depending on the given A1 , . . . , An , B). (This is the correction, referred to in 3.15.5(vii) – see p. 421 – of an earlier claim by Prawitz.) It is important to note that negation in ML picks up its properties from intuitionistic (or ‘positive’) implication and not classical (‘material’) implication, which would yield more. For example, since CL p ∨ (p → A), for all A, we would obtain the Law of Excluded Middle for a ¬ thereby introduced. For a purely implicational example, consider Peirce’s Law in the form: for all A, (p → A) → p CL p. This would yield the sequent (a form of what used to be called Consequentia Mirabilis): ¬p → p p. This is not part of ML, or even IL, which its addition would collapse into CL, as we already have ¬¬p ¬p → p in IL, which by (T) would yield ¬¬p p. The latter sequent form of (¬¬E) incidentally shows that the conjecture that Γ CL A iff Γ* CL A* (with (·)* as in 8.32.2) would not be correct, since for example (p → q) → q CL p. (For a logic according to which A does follow in general from (A → B) → B, see Meyer and Slaney [1989], or our discussion of the implicational fragment of this ‘Abelian Logic’ – or BCIA logic – in 7.25; indeed, Meyer and Slaney use this consideration to motivate the logic. It means that the constant f as we might write it here, as in 2.33, rather than ⊥, in terms of which the negation of A is defined by A → f would allow double negation elimination for the defined negation without itself being special, as we put it in 9.22.) A variation, interchanging p and ¬p in the above Consequentia Mirabilis sequent is worth a moment’s attention: p → ¬p ¬p. Unlike the original, this sequent is not only intuitionistically provable, but is even in ML, since p → (p → ⊥) p → ⊥ is a case of the PL ‘contraction schema’ (see 7.25): p → (p → A) p → A.

8.3. THE FALSUM

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We can turn p → ¬p ¬p into a purified rule, governing only ¬ (assumed primitive): (¬-Contrac)

Γ, A ¬A Γ ¬A

This rule participates in an interesting analysis of (RAA) – à propos of which we already encountered it briefly in 8.13.2(ii), p. 1187 – will play a guiding role in 8.33: Exercise 8.32.3 Show that, given the rules of Positive Logic (as presented above) (RAA) is interderivable with the combination of (¬-Contrac) and the rule of Selective Contraposition:

Γ, A B Γ, ¬B ¬A

(The labelling is intended to emphasize the distinctness of this rule from the weaker rule of Simultaneous Contraposition from 2.11.9 (p. 207): we can ‘select’ at will any formula from the left of “” to contrapose with the formula on the right. Another way of describing this—from 3.32—is to say that according to the associated consequence relation, ¬ is antitone with side formulas.) We summarize the relations between PL, ML, IL and CL here, via the proof systems we have considered: Logic CL

Situation regarding ⊥ As for IL.

IL ML

Satisfies (⊥E). If in the language: no special properties.

PL

(Not in language.)

Situation regarding ¬ Satisfies (RAA), (EFQ) and (¬¬E). Satisfies (RAA) and (EFQ). Satisfies (RAA); this gives ¬ precisely the properties it would have if defined by ¬A = A → ⊥ (⊥ as in ML, → as in PL). (Not in language.)

In 8.33 we will also consider certain further systems with ¬ which are weaker than ML. Remark 8.32.4 In saying that ⊥, in a formulation of ML in which this connective appears (as a primitive connective of the language), ⊥ has ‘no special properties’, we mean that Γ M L A only if for every result Γ , A , of uniformly replacing ⊥ in Γ, A, by a formula B, we have Γ M L A . (Note that here we require that ¬ is not also taken as primitive, or there would be counterexamples such as the fact that p, ¬p M L ⊥.) This is a stronger claim than the claim that ⊥ is completely undetermined (3.11) according to M L , as can we see from the following example. Extend the language of modal logic (§2.2) by a zero-place connective r and let be the inferential consequence relation determined by the class of all frames, when the definition of truth is extended by the clause


1260 (W, R, V ) |=x r ⇔ xRx

Clearly we do not have r B for all B, and we do not have r, either; so r is completely undetermined according to . But r does have some ‘special properties’ according to , since for example r p → p, whereas it is not in general the case (i.e., for arbitrary B) that B p → p. (A more wide-ranging discussion of special vs. non-special treatment of connectives, including those of arbitrary arity, appears in 9.22. A constant with the above semantics for r appears, under the name “loop” in §3 of Gargov, Passy and Tinchev [1987].) Digression. The above treatment of “r” (or “loop”) is suggestive of something more general and of considerable interest in its own right: having a sentential constant – we will use a boldface capital R here – to represent the accessibility relation. Since this is a binary relation, we should work with two-dimensional modal logic, and may use the following clause in the definition of truth relative to a pair of points (notated “|=xy ”) in a model M with accessibility relation R: M |=xy R if and only if xRy. The idea is that the lower index represents the point at which we are considering formulas to be true or false in the intuitive sense (so we have M |=xy pi iff y ∈ V (pi )), with the upper index keeping track of the last point to have played that role; thus for we say: M |=xy A iff for all z such that yRz, we have M |=yz A. (We encountered this clause in 7.17, p. 1018, when discussing a two-dimensional treatment of conditionals.) The latter represents absolute necessity in the sense that if we started with a model interpreting by means of accessibility relation R, we can thinking of as necessity relative to R, and instead of writing A, we write (R → A). Some care needs to be taken over Uniform Substitution in the present context, since substituting R for a propositional variable represents a kind of confusion of types: the latter represent sets of worlds, the former, a set of pairs of worlds. Thus anyone favouring the identification of propositions with sets of worlds would describe the nullary connective R as sentential constant which is not a propositional constant. In the terminology of Humberstone [1981b] it would be a ‘di-propositional constant’. Full details may be found in that paper, which is inspired by Smiley [1963a]; some questions left open there are closed in Kuhn [1989]. A similar approach could be taken in the case of the Urquhart semilattice semantics for relevant logic (2.33, 7.23), adding with the clause above and a new 1-ary connective O, say, where to avoid confusion we write “⊃” for ‘material’ implication—forming compounds true w.r.t. a pair of elements just in case either the first component fails to be true w.r.t. the pair, or the second component is true w.r.t. that pair. For “O”, we lay down that M |=xy OA iff M |=xxy A, where the semilattice operation is indicated by juxtaposition. Then the relevant implication A → B emerges as (A ⊃ OB). This point as well as a review of some of the references just given appears in Humberstone [2004a]. End of Digression. The talk of uniform replacement in 8.32.4 is suggestive of Uniform Substitution, though of course ⊥ is not a propositional variable, but a propositional (or sentential) constant – alias zero-place connective – which behaves like such a variable. This difference matters especially in respect of questions of matrix characterization, and it stops M L (unlike P L ) from having a single determining matrix; as we saw in the Digression on Shoesmith and Smiley toward the

8.3. THE FALSUM

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end of 3.11 (see p. 380), it causes a failure of their Cancellation Condition. The point has nothing to do with the presence of ⊥ (for which the failure of that condition was illustrated in the Digression), and arises for ML with only ¬ (and ∧, ∨, →: as in MNat): p ∧¬p M L ¬q, by 8.32.1(i), but we do not have p ∧¬p M L A for arbitrary A, and we do not have M L ¬q; all this even though p ∧¬p and ¬q do not share a propositional variable. (This answers 8.32.1(iii) above.) Our preferred approach to the semantics of ML will not be matrix-based, however, but model-theoretic. This brings out the relationship between ML and IL in a particularly revealing way, as we shall see. Kripke’s semantics for IL is modified in an obvious way to treat ML. Except for various asides (8.32.5, 8) the ⊥-free language of MNat will be used; we aim at a soundness and completeness proof for this proof system, w.r.t. a certain class of frames. The treatment follows Segerberg [1968] in essentials, who takes ⊥ to be present, and assigns to it a truth-set in the models, calling this set Q. But we can use this idea even in the absence of ⊥. Thus, what we shall call Q-frames have the form (W, R, Q) with (W, R) a Kripke frame for IL as in 2.32 and Q ⊆ W satisfies the condition: (QPers) x ∈ Q & xRy ⇒ y ∈ Q, for all x, y ∈ W . (Actually, this condition can be dropped without altering the soundness and completeness result below; see 8.32.5(ii).) A Q-model on (W, R, Q) is a structure (W, R, Q, V ) in which V obeys the condition (Persistence) from 2.32 in assigning subsets of W to the propositional variables. The clauses in the inductive definition of the truth of a formula A at a point x ∈ W in a Q-model (W, R, Q, V ), notated as “(W, R, Q, V ) |=x A”, run exactly as in the Kripke semantics for IL for A = pi , A = B ∧ C, A = B ∨ C, and A = B → C, whereas, predictably, for A = ¬B, we say something different, namely: (W, R, Q, V ) |=x ¬A iff for all y such that x Ry and (W, R, Q, V ) |=y A, y ∈ Q. Remarks 8.32.5(i) With ⊥ in the language one says: (W, R, Q, V ) |=x ⊥ iff x ∈ Q. Since the use of ⊥ reminds one of something that functions as a constant false or impossible statement—which is why it is called the Falsum—elements of Q are sometimes called queer points (whence the “Q”). The above condition (QPer) ensures that the formula ⊥ is persistent (true at all points R-related to points at which it is true), (ii ) For the ⊥-less language of MNat, the condition (QPer) is not needed in order to show that all formulas are persistent. The reader is urged to check this for the inductive case (in a proof analogous to that of 2.32.3, p. 308) of ¬. The notions of truth of a formula, and holding of a sequent, in a model, and of validity of a sequent on a frame are taken over intact from 2.32, except now with reference to Q-models and Q-frames, and with the above semantic treatment of ¬. Validity over the class of all such frames coincides with provability in MNat: Theorem 8.32.6 A sequent Γ C is provable in MNat iff it is valid on every Q-frame.

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Proof. ‘Only if’: check that (RAA) preserves the property of holding in a model. Here it is necessary to observe that if B ∧¬B is true at a point, that point belongs to the set Q of the model. In the absence of (EFQ), this is the only rule involving ¬ and hence the only point at which there is a difference from the case of IL, though it is necessary to observe (even for the case of the rule (→I), not governing ¬), that all formulas are persistent: see 8.32.5(ii). ‘If’: We need a canonical Q-model, as for 2.32.8 (p. 311) we needed a canonical model. Let MM L = (WM L , RM L , QM L , VM L ), where WM L is the collection of all deductively closed (w.r.t. M L ) prime sets of formulas, and QM L comprises those elements of WM L which contain some formula together with its negation. RM L and VM L are defined exactly as in 2.32, except restricted to WM L rather than WIL . It remains only to check that MM L |=x A iff A ∈ x, and the only point at which we should examine this verification is the case in which A is ¬B. Exploiting the inductive hypothesis, we have to show that ¬B ∈ x iff for all y ∈ WM L such that xR M L y and B ∈ y, y ∈ QM L . The ‘only if’ direction is clear, since if ¬B ∈ x and B ∈ y ⊇ x then y contains some formula and its negation / x; we (namely B) and so y ∈ QM L . For the converse direction, suppose ¬B ∈ want to find a deductively closed and prime y ⊇ x with B ∈ y and y ∈ / QM L . But any prime extension of the deductive closure of x ∪ {B} will meet these conditions, since no formula and its negation belong to this closure (keeping y out of QM L ). The argument for this claim runs: if x ∪ {B} M L C and also x ∪ {B} M L ¬C, for some formula C, then for some finite Γ ⊆ x we have Γ, B C ∧ ¬C provable in MNat, whence by (RAA), Γ ¬B would be provable, contradicting the supposition that ¬B ∈ / x. From the coincidence of truth at, with membership in, a point in the canonical Q-model, the completeness of MNat follows as for INat in 2.32. We pause to observe a contrast in respect of unique characterization (§4.3) between the rules of MNat and those for INat: Corollary 8.32.7 The rules of MNat do not uniquely characterize ¬. Proof. Consider a proof system with rules like those of MNat but in a language containing another singulary connective ¬ alongside ¬, and a rule (RAA) like (RAA) but with ¬ in place of ¬. Semantically, we use frames (W, R, Q, Q ), with the clause for ¬ in the definition of truth in a model requiring for the truth of ¬B at x ∈ W that all R-related B-verifying points lie inside Q . As in the soundness proof for MNat, we can see that all provable sequents of this composite system are valid on each such frame; but since the sequent ¬p ¬ p is not, it is unprovable.

Remarks 8.32.8(i) If ⊥ is primitive instead of ¬ (with no rules except those of PNat), then the set QM L in the canonical model for the system should be taken to comprise those x ∈ WM L with ⊥ ∈ x. (ii ) As noted in 8.32.5(i), here the requirement (QPer) is mandatory for the soundness proof, since otherwise the formula ⊥ will not be persistent, which is needed to justify the passage by (→I) from Γ, AB to ΓA → B when ⊥ ∈ Γ. (In the ¬-primitive version, without ⊥, QM L does satisfy

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this condition: but imposing it makes no difference to the class of valid sequents since it was not required for the soundness proof—8.32.5(ii)— and was not exploited elsewhere in the completeness proof.) (iii) Instead of considering ML with ¬ but without ⊥, or with ⊥ but without ¬, we could consider a version with both connectives primitive. Rules allowing passage (in either direction) between Γ, A ⊥ and Γ ¬A should be added to a basis for PL; (¬I) and (¬E), given before 8.31.3, suffice for this purpose. Then not only is ¬ definable (3.15) in terms of ⊥, ¬A and A → ⊥ being synonymous, but ⊥ is definable in terms of ¬: ⊥ and ¬p ∧¬¬p are synonymous. (Note that the “¬¬” cannot be omitted here, since ⊥ M L p.) An alternative definiens for ⊥ is ¬(p → p).) (iv) See the Digression (p. 1192) after 8.13.10 for a sequent calculus treatment of ML which does allow (by contrast with the content of 8.32.7) for unique characterization of ¬. Notice that in the light of the current semantics, Theorem 2.32.8 (p. 311), saying that INat is determined by the class of all frames, can be rephrased as saying that INat is determined by the class of all Q-frames (W , R, Q) in which Q = ∅. This raises the possibility of investigating logics between ML and CL in terms of classes of Q-frames (when such classes are to be found) determining them. Such an investigation undertaken for a great many of these systems (in Fmla, as it happens) in Segerberg [1968], and we will look briefly at this area in 8.34. We close this subsection with a familiarization exercise in ML. Outside of the present section, further discussion of ML may be found in toward the end of 9.24. Exercise 8.32.9 (i) Does the Law of Triple Negation (2.32.1(iii): p. 304) hold for ML? (ii ) Does ML have the Disjunction Property (6.41.1, p. 861)? (iii) Is Glivenko’s Theorem correct for ML? (I.e., Does 8.21.2, from p. 1215, hold with “IL ” replaced by “M L ”?) If in difficulties, see Remark 8.33.11 below (p. 1271), and, further 9.24.10 (p. 1317). For part (i) above, see the discussion following 9.24.8, p. 1315; for part (ii), the discussion following 8.34.12, p. 1281.)

8.33

Variations on a Theme of Johansson

We will present here three variations on the theme of Minimal Logic. The first arises from a natural experimental urge, namely to explore the logical properties conferred on a singulary connective O by the equivalence in an extension of PL of OA with ⊥ → A, where ⊥ itself has no special properties (8.32.4: p. 1259). In other words, we put our unspecified constant in antecedent rather than consequent position. We confine this variation in the first instance to an exercise (8.33.1) returning at the end of our discussion to its solution and to further efforts in this direction. The second variation is closer to the treatment of negation in ML, except that instead of using a nullary connective in the consequent, we make use of a singulary connective, thus allowing the formula whose implication by A counts as A’s negation to depend on the formula A. The third


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variation will generalize the ML treatment of negation in a different way, allowing for various different zero-ary connectives in the consequents of implications, the disjunction of which gives the negation of the formula in their antecedent. For our initial glance at the first variation, then: Exercise 8.33.1 For a language like that of MNat but with singulary “O” in place of “¬”, we provide semantics using the Q-frames and Q-models of 8.32, with the following clause in the definition of truth: (W, R, Q, V ) |=x ¬A iff for all y with xRy & y ∈ Q: (W, R, Q, V ) |=y A. Find some rules to extend the rules of Positive Logic for which you can show (by suitably adapting the proof of 8.32.6) that the resulting proof system provides proofs for precisely the sequents valid on every frame, with the above truth-definition in place. One solution to the above Exercise will be found at 8.33.8 below, at which point we will look at some other ventures in the style of the ‘first variation’. We turn to the second variation. As already mentioned, instead of adding a zero-place connective ⊥ to the language of PL, we add a one-place connective, ⊥1 . A way of thinking of the role of ⊥ in ML is as some unwanted (or ‘taboo’ – see 8.12) formula, so that ¬A can be thought of as saying that A implies this unwanted formula. But why should the unwanted formula be independent of A? We could have, for each formula A, some formula whose being implied by A is taken to amount to A’s negation. Using our new connective, we write this A-specific taboo formula as ⊥1 A. The idea would then be to arrange matters so that ¬A is equivalent to A → ⊥1 A. This procedure would correspond to a formulation of ML with both ¬ and ⊥. For a closer parallel with 8.32, we shall instead take only ¬ as primitive. As there, the role of ⊥1 can be taken over by a suitable clause in the definition of truth. We will use a variation of the Q-models, called Q1 -models, in which a function Q1 assigns to each set of points a set of points; were ⊥1 explicitly present in the language, then the truth-set (in a model) of ⊥1 A would be the result of applying this function to the truth-set of A. In fact, we could instead let the function assign sets of points to formulas rather than to their truth-sets; but this excessively linguistic approach is avoided here. The result is that if it were taken as a primitive connective, ⊥1 would have to be rendered congruential by any complete proof system. We have the same result for ¬, which we do propose to take as primitive. This leads to the following system. MNat 1 is the Set-Fmla proof system in the same language as MNat, which has that system’s rules for ∧, ∨, and →, and for ¬, the rules (¬-Cong.)

AB

BA

¬A ¬B

and (¬-Contrac), given before 8.32.3. As we saw in that Exercise, the latter rule, taken in conjunction with Selective Contraposition, gives the force of (RAA). Taking Γ = ∅ in Selective Contraposition, we get that ¬ is antitone (3.32) according to the consequence relation associated with any proof system closed under that rule, and is therefore a fortiori congruential according to that relation. Since, however, we have taken away Selective Contraposition, and left

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only the (¬-Contrac) ‘half’ of (RAA), we have had to add (¬-Cong.) here as a separate primitive rule. As we shall see, this gives exactly the right proof system for the notion of validity described informally above, and more precisely in what follows. A Q1 -frame is a triple (W, R, Q1 ) with (W, R) as in 8.32 and Q1 : ℘(W ) −→ ℘(W ) satisfying for all x, y ∈ W : (Q1 Pers)

x ∈ Q1 (X) & xRy ⇒ y ∈ Q1 (X) for all X ⊆ W .

A Q1 -model on (W, R, Q1 ) is a structure (W, R, Q1 , V ) with V as in 8.32 (or indeed 2.32). Truth is defined as in 8.32 except that for ¬ we say, where M = (W, R, Q1 , V ): M |=x ¬A iff for all y such that xRy and M |=y A, y ∈ Q1 (!A!). Here !A! is the set of points in M at which A is true. We understand the notions of holding in a model and of validity on a frame in the usual way. Theorem 8.33.2 A sequent Γ C is provable in MNat1 iff it is valid on every Q1 -frame. Proof. Soundness – the ‘only if’ direction of the Theorem – involves only a routine check that the rules preserve the property of holding in a Q1 -model, so let us concentrate on how the completeness proof of 8.32.6 can be modified, for the ‘if’ direction. Call the canonical model (W, R, Q1 , V ). W , R, and V here are as before (except that deductive closure is understood relative to the present logic). For Q1 we say: if X ⊆ W is {x ∈ W | A ∈ x} for some A, then Q1 (X) = {y ∈ W | A, ¬A ∈ y}. This is a correct definition, thanks to (¬-Cong.), which implies that if X is of the given form for A and A , then A, ¬A ∈ y ∈ W iff A , ¬A ∈ y. For X not of the given form, Q1 (X) is arbitrary. It remains only to show that for all x ∈ W , ¬B ∈ x iff ∀y ⊇ x, B ∈ y ⇒ y ∈ Q1 ({z | B ∈ z}). The ‘only if’ direction is given by the way Q1 was specified. For the ‘if’ direction, suppose ¬B ∈ / x. Then any prime, deductively closed superset of x ∪ {B} will do, as long as this does not also contain ¬B (which would put the element in question into Q1 ({z| B ∈ z})). To establish the existence of such sets, we need to know that x ∪ {B} ¬B, where is the consequence relation associated with MNat 1 . But if x ∪ {B} ¬B then x ¬B, by appeal to (¬-Contrac), contradicting the assumption that ¬B ∈ / x. We turn to our third variation on Johansson. Going back to the simpler idea of an equivalence between the negation of A and A’s implying some particular formula (⊥), which we may think of as expressing some ‘unwanted’ proposition, so that ¬A means that A has an untoward consequence, we could consider the use of several such propositions. For illustrative purposes, let us consider the case of two such, and let ⊥1 and ⊥2 be formulas for expressing them. (Thus, the subscripted ‘1’ here is not, as in the previous example, supposed to suggest a singulary connective—both ⊥1 and ⊥2 being 0-ary—but simply as an index to single out the first of a range of formulas ⊥1 , . . . , ⊥n ; for the present we are setting n = 2, though below we shall allow arbitrarily many, even infinitely


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many.) Three ways come to mind for explicating negation in terms of such a range of ‘taboo’ constants. We could try to arrange things so that (1) Γ ¬A iff Γ A → (⊥1 ∨ ⊥2 ) for the consequence relation we are envisaging (and all Γ, A in its language), or perhaps instead, so that (2) Γ ¬A iff Γ (A → ⊥1 )∨ (A → ⊥2 ) or, finally, with the aim of securing (3) Γ ¬A iff Γ A1 → ⊥ or Γ A → ⊥2 . Now suggestion (1), though arguably the most natural way of proceeding, lead to nothing new, since it in effect treats the disjunction ⊥1 ∨⊥2 like the single ⊥ of ML (with ⊥ primitive). While we could equip our frames with two Q-type subsets Q1 , to be the truth-set of ⊥1 , and Q2 , to be the truth-set of ⊥2 , as far as the evaluation of negated formulas is concerned, it is just the single set Q1 ∪ Q2 that ever needs to be considered, so that we might as well have just the one set Q in the frames—at least if our attention is on the logic with ¬ rather than the ⊥i as primitive, as it soon will be. As for suggestion (3), this leads to the conclusion that either ⊥1 → ⊥2 or else ⊥2 → ⊥1 , by reasoning of the kind encountered in 4.22, in which case we can again make do with only one Q-set in our frames. (In more detail: (3) implies that for all A, (i )

A → ⊥1 ¬A and A → ⊥2 ¬A,

and (ii )

¬A A → ⊥1 or ¬A A → ⊥2 ;

so for all A, either A → ⊥2 A → ⊥1 or else A → ⊥1 A → ⊥2 . Putting p for A we get: p → ⊥2 p → ⊥1 or else p → ⊥1 p → ⊥2 . In the former case the claimed result follows on substitution of ⊥2 for p, and in the latter case, on substitution of ⊥1 for p. This appeal to Uniform Substitution is essential to the argument, enabling us, as it does, to pass from “for all A, either A → ⊥2 A → ⊥1 or else A → ⊥1 A → ⊥2 ” to “either for all A: A → ⊥2 A → ⊥1 or else for all A: A → ⊥1 A → ⊥2 ”.) Accordingly, what we shall consider is the intermediate proposal (2). This makes ¬A equivalent to (A → ⊥1 ) ∨ (A → ⊥2 ). Now let us deal with the suggestion in a language which has ¬ behaving as such an equivalence dictates, but without the taboo constants in the language. We obtain this effect by using models (W, R, Q1 , Q2 , V ) and the following clause for ¬ in the definition of truth, relative to a point x ∈ W in such a model (call it M): M |=x ¬A iff either R(x) ∩ !A! ⊆ Q1 or R(x) ∩ !A! ⊆ Q2 Observation 8.33.3 (i) Formulas of the form ¬A are persistent, even though we have not imposed the requirement that Q1 , Q2 , be closed under R. (ii) Of the two rules ¬-Contrac and Selective Contraposition seen in 8.32.3 were collectively equivalent to (RAA), the first does not preserve the property of holding in a model of the type currently under consideration, while the second does. Remark 8.33.4 If we consider a sequent-schema form of Contrac, namely: A → ¬A ¬A

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and then restore the sentential constants we are supposing not present in the current language, we get: A → ((A → ⊥1 ) ∨ (A → ⊥2 )) (A → ⊥1 ) ∨ (A → ⊥2 ) Since, as in ML itself, we treat our Falsum constants as satisfying only such logical principles as hold for arbitrary formulas (8.32.4, p. 1259) in PL, we are left considering the schema: A → ((A → B) ∨ (A → C)) (A → B) ∨ (A → C) which one easily checks is not provable in PL, despite its superficially contraction-like appearance. In fact, if added to INat this schema gives precisely the intermediate logic LC mentioned in 2.32 and elsewhere. (Compare 2.32.10(i), p. 318.) This is a reflection of a general and wellknown fact about IL, namely that for any n-ary connective # definable from the familiar primitives, we have for all formulas A, B1 , . . . , Bn : A → #(B1 , . . . , Bn ) IL A → #(A → B1 , . . . , A → Bn ). This equivalence is most readily appreciated semantically: the common antecedent on both cases, A, restricts our attention, from the point of view of a point x in a Kripke model, to points y in R(x) ∩ !A!, and for these y there can be no divergence in truth-value between Bi and A → Bi . (The same goes for Bi and A ↔ Bi , Bi and as well as for A∧Bi , giving us some variations on the above IL principle.) In particular, in the present case, with # as ∨, we have an equivalence between A → (B ∨ C) (which of course does not have (A → B) ∨ (A → C) as an IL-consequence) and the left-hand side of the -schema inset above. The current choice of frames is obviously a very special case of the idea of allowing arbitrarily many subsets Qi of ‘queer’ points, since we have settled on having exactly two. This means that certain sequents are validated which are not valid in the general case; for example, all those of the form: ¬A ∧ ¬B ∧ ¬C ¬(A ∨ B) ∨ ¬(B ∨ C) ∨ ¬(A ∨ C). The reasoning is as follows. For the conjunction on the left to be true at a point x in a model, we must have R(x) ∩ !A! ⊆ Q, R(x) ∩ !B! ⊆ Q , and R(x) ∩ !C! ⊆ Q , for some choice of Q, Q , Q out of {Q1 , Q2 }. Let us take as a representative possibility the case in which Q = Q = Q2 . Then, since R(x) ∩ !A! ⊆ Q2 and R(x)∩ !B! ⊆ Q2 , (R(x) ∩ !A!) ∪ (R(x) ∩ !B!) = R(x) ∩ (!A! ∪ !B!) ⊆ Q2 , which suffices for the truth at x of ¬(A ∨ B), and hence of the disjunction ¬(A ∨ B) ∨ ¬(B ∨ C) ∨ ¬(A ∨ C). A convenient way of giving the general version of the semantics whose special (n = 2) case we have been considering is in terms of what we shall call Q-frames, these being structures (W, R, Q) with W and R as before, and Q a collection of subsets of W . Q collects such sets as Q1 and Q2 from the foregoing discussion, but allows more (or fewer) such subsets. Models on such frames are constructed as usual, and in the definition of truth we say for negation:


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(W, R, Q, V ) |=x ¬A iff ∃Q ∈ Q such that ∀y ∈ R(x) ∩ !A! y ∈ Q In other words, thinking of the various Q ∈ Q as ‘unwanted’ or ‘rejected’ propositions, ¬A is true at x when A leads us from x only into one of these unwanted regions. Note that if Q is ∅ in some Q-frame then all sequents ¬A B are valid on that frame (since ¬A cannot be true at any point in a model on that frame in view of the above “∃Q ∈ Q ’), and if Q contains exactly one Q ⊆ W then models on the frame are (essentially) just models on a Q-frame, so that the logic determined by the class of such “|Q| = 1” Q-frames is ML. Finally, we observe that if W ∈ Q, then all formulas of the form ¬A are true throughout any model on the frame. Exercise 8.33.5 Check that Observation 8.33.3 remains correct for models on Q-frames. It would be pleasant to include here a completeness result, as suggested by 8.33.5, namely that a proof system delivering precisely the sequents valid on every Q-frame is obtained from one for PL by adding the rule (Selective Contraposition). This result has indeed been proved in Hazen [1992]– he calls the present system the logic of ‘subminimal negation’ – but it involves complications making its inclusion here unsuitable. Accordingly we proceed instead to a slightly generalized version of the Q-frame semantics, in terms of which a simple completeness proof is readily available. This seems worthwhile in view of intrinsic interest of the present proof system. To elaborate for a moment on the main point of interest: since we saw in ML the combined effect of (¬-Contrac) and (Selective Contraposition) and our second variation on ML looked at the former rule in isolation, it seems appropriate to fill out the picture by attending to a semantic account of what happens if we take the latter rule in isolation. (In isolation from other ¬-governing rules, that is; of course we are supposing the usual introduction and elimination rules for ∧, ∨, and →.) The generalization of Q-frames we need for the following argument takes them again to be structures (W, R, Q), but this time Q is not a collection of subsets of W but a function assigning to each x ∈ W some such collection. Here we shall need a persistence-securing condition for negated formulas, namely (QPers)

xRy ⇒ Q(x) ⊆ Q(y), for all x, y ∈ W .

Since the set of ‘queer’ propositions is allowed now to depend on the points x ∈ W , we will call the structures here described point-sensitive Q-frames. The clause for ¬ in the definition of truth in a model on such a frame is: (W, R, Q, V ) |=x ¬A iff ∃Q ∈ Q(x) R(x) ∩ !A! ⊆ Q The original Q-frame semantics can be recovered by ruling out the sensitivity we have allowed for, with the condition Q(x) = Q(y), for all x, y ∈ W . It is easy to see that (Selective Contraposition) preserves the property of holding in any model on one of these point-sensitive Q-frames, which shows that the proof system supplementing PL with that rule is sound w.r.t. the class of all such frames. Letting be the consequence relation associated with this proof system, we give a canonical model argument to show that the system is complete w.r.t. all point-sensitive Q-frames. This canonical model we denote simply by (W, R, Q, V ), with W , R, and V as in the case of ML (except that deductive closure is understood relative to ). For an element x ∈ W, Q(x) will

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consist of sets Qx,B for a formula B and an element x of W , (partially) defined in the following way: If ¬B ∈ x then Q x,B = {z ∈ W | B ∈ z}. If ¬B ∈ / x then Q x,B is not defined. Truth and membership can be shown to coincide: Lemma 8.33.6 For this canonical model M = (W, R, Q, V ), we have, for all formulas A, M |=x A iff A ∈ x. Proof. We consider only the inductive case of A = ¬B. Suppose ¬B ∈ x. We need Q ∈ Q(x) such that for all y ∈ W such that y ⊇ x, if B ∈ y then y ∈ Q. Since ¬B ∈ x, the set Qx,B exists and will serve as the desired Q ∈ Q(x). Suppose ¬B ∈ / x. We must show how, given Q ∈ Q(x), there exists y ⊇ x, B ∈ y but y ∈ / Q. Say the given Q is Qx,C for some formula C. This tells us that ¬C ∈ x, and Qx,C is the set of all W -elements containing C. We obtain the desired extension y of x by taking some prime deductively closed superset of x ∪ {B}. We need to make sure that some such superset lies outside of Qx,C ; but if all lay within Qx,C that would mean x ∪ {B} C, and so, by (Selective Contraposition), x ∪ {¬C} ¬B. But ¬C ∈ x, and we are supposing that ¬B ∈ / x. This contradicts x’s being deductively closed. Following in the usual way from this and our earlier observations on soundness, we have Theorem 8.33.7 A sequent is provable in the proof system extending Positive Logic with (Selective Contraposition) iff it is valid on every point-sensitive Qframe. As already remarked, Hazen [1992] shows that if the qualification ‘point-sensitive’ is deleted from the Theorem, the result still holds – a considerably more satisfying completeness result than 8.33.7 itself. Returning to our first variation: in Exercise 8.33.1, suitable rules governing the 1-ary operator “O” were requested in an extension of PL, interpreted semantically in terms of Q-models M = (W, R, Q, V ) thus: OA is true at x in M iff every y ∈ Q which is R-accessible to x verifies A. We begin with a solution to this problem. Observation 8.33.8 The set of sequents valid on all Q-frames with the above clause for O in definition of truth at a point is given by extending PNat with the rules (i) OOA OA (ii) From: Γ, Δ C to: Γ, OΔ OC (where OΔ = {OD | D ∈ Δ}). Proof. Soundness: clear. Completeness: the only variation in the canonical model argument given for MNat above (8.32.6) arises over “O” and the Q-component of the models. Here we define the Q of the canonical model to be:

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CHAPTER 8. “NOT” {u ∈ W | for all formulas A,OA ∈ u ⇒ A ∈ u}.

It suffices to show, on the assumption (the inductive hypothesis) that subformulas of a formula OB belong to an element of the canonical model iff they are true at that element, that this is so for OB itself. Invoking this assumption in particular for B, all we have to show is that OB ∈ x ⇔ ∀y ⊇ x(y ∈ Q ⇒ B ∈ y). ⇒: If OB ∈ x and y ⊇ x then OB ∈ y, so if y ∈ Q then, by the way Q was defined, we must have B ∈ y. ⇐: Suppose OB ∈ / x. We must find y ⊇ x, y ∈ Q, B ∈ / y. So let y be {A|OA ∈ x}. That this set is deductively closed (relative to the present logic) follows from rule (ii ), taking Γ = ∅. The sequent C OC is provable for all formulas C, using the rule (ii ) with Δ = ∅, Γ = {C}, from which it follows that y ⊇ x: for if C ∈ x, then in view of this sequent we must have OC ∈ x, so C ∈ y by the way y was specified. We also need to know that y ∈ Q. But if OA ∈ y, that means OOA ∈ x, so by rule (i), OA ∈ x, so A ∈ y; hence y ∈ Q. Finally, we note that B ∈ / y, since OB ∈ / x (again, by the way y was specified). Now although for old times’ sake we retained the “Q” in notating the above models, it must be admitted that O has very little to do with negation, so we did not retain the “¬” notation as well. Sticking with this same notation for a moment (for that very reason), we should pay some attention to alternatives to thinking of our O as giving by thinking of OA as A → ⊥ (the original ML idea, where we wrote “O” as “¬”) or as ⊥ → A (which is what the version just considered amounts to, thinking of ⊥ as the formula whose truth-set Q represents in any model). What other options are there for ‘reducing’ the 1ary O to the 0-ary ⊥, again making no special assumptions about the logical behaviour of the latter? We present two simple options for consideration for returning to something which makes O behave much more like negation. Exercise 8.33.9 (i) Find some rules with which to extend PNat in such a way that the resulting is determined by the class of all Q-frames when truth at a point in a Q-model is defined with the aid of the following clause for O: (W, R, Q, V ) |=x OA iff x ∈ Q and (W, R, Q, V ) |=x A. (ii ) As in (i), but for the clause: (W, R, Q, V ) |=x OA iff x ∈ Q or (W, R, Q, V ) |=x A. Parts (i) and (ii ) here correspond of course to possible definitions of OA as ⊥∧ A and ⊥∨ A respectively. (Since in these cases, the accessibility relation is not invoked, nothing hangs on the use of PL as opposed to CL in the role of background logic to be extended by suitable O-rules; in the latter context, part (i) has been used to treat the necessity operator of modal logic – so O is written as : see Porte [1979].) We revert to the “¬” notation to consider a final subvariation in this style, combining the standard ML case with that of 8.33.8 (or 8.33.1). The former involves a reference to Q-membership in the consequent of a conditional, the latter to Q-membership in the antecedent; we can combine them by using instead

8.3. THE FALSUM

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a biconditional: Given Q-model M = (W, R, Q, V ) with x ∈ W , let us stipulate that: M |=x ¬A if and only if for all y ∈ R(x), M |=y A iff y ∈ Q. Notice that in the case of Q-models with Q = ∅ (i.e., the case of Kripke models for intuitionistic logic), this is equivalent to the usual clause for negation (since the condition for the ‘if’ half of the final ‘iff’ is never satisfied). Putting this another way, with ⊥ primitive in the language, it wouldn’t matter whether we defined ¬A as A → ⊥ or as A ↔ ⊥ in the setting of intuitionistic logic. It would make a difference in the ‘minimal’ setting, however, in which no assumptions are made about ⊥ (about Q, to revert to semantic description). But because the closer kinship remains with what one expects on the part of negation, we write “¬” rather than “O” (as in 8.38, 8.39). Let us refer to the ‘biconditional’ version of Minimal Logic, with ¬ understood as above, as ML*. Observation 8.33.10 Neither M L ⊆ M L∗ nor M L∗ ⊆ M L . Proof. As a counterexample to the inclusion M L ⊆ M L∗ , we have the fact that while M L ¬(p ∧¬p), M L∗ ¬(p ∧¬p); as a counterexample to the converse inclusion, we have: M L∗ ¬¬(¬¬p → p) (see below, after 8.33.11) while M L ¬¬(¬¬p → p). (See immediately below.) To check that, as claimed in the above proof, M L ¬¬(¬¬p → p), it suffices to show that M L (((((p → q) → q) → p) → q) → q), since this is what we get from the “implies ⊥” version of notation, and then replace ⊥ by a new propositional variable, as we may since in ML, ⊥ exhibits no distinctive logical behaviour (8.32.2). One can use the Kripke semantics to check that the formula ((((p → q) → q) → p) → q) → q is invalid; more simply, note that, since we are now in the pure implicational fragment, where ML and IL do not differ, if that formula were intuitionistically provable, so would be all sequents of the form (where A, B, replace p, q): (((A → B) → B) → A) → B B. Now put q for A and (p → q) ∨ p for B to obtain an IL-provable right-hand side with an IL-unprovable left-hand side.

Remark 8.33.11 Note that what we have just shown, namely that M L ¬¬(¬¬p → p), reveals that Glivenko’s Theorem does not hold for ML. (This answers 8.32.9(iii); for counterexamples of a somewhat different type, see 9.24.10 (p. 1317). The erroneous claim that ML does satisfy Glivenko’s Theorem is sometimes encountered, e.g., at p. 46 of Tennant [1992].) According to Troelstra and Schwichtenberg [1996], p. 41, for negative formulas C we have CL C iff M L C, which may seem to contradict what we have just found, until it is noted that the authors mean by a ‘negative’ formula, not a negated formula, and not a formula in which the only


1272

connective used is ¬, but rather an ∨-free formula in which all propositional variables (which occur) are negated. So understood, the above claim from Troelstra and Schwichtenberg is an early result of A. N. Kolmogorov; see the notes to this section, which begin on p. 1284. The ML-unprovability of ¬¬(¬¬p → p) was already noted in the original paper on ML, Johansson [1936], who used a matrix argument to show this, later simplified in Nemesszeghy [1976] by using the matrix mentioned in Example 2.11.3 (at p. 202) with the following modification: the ‘output column’ in the table for ¬ should be changed from 3, 3, 1 (reading downwards) to: 2, 1, 1. Adding this formula to ML gives the logic called JP in Segerberg [1968], there described as the least extension of ML for which Glivenko’s Theorem – see 8.21 – holds, and axiomatized by means of the special case, called P , of Peirce’s law (itself called P in Segerberg’s discussion): ((⊥ → p) → ⊥) → ⊥, or, more concisely: ¬¬(⊥ → p). The ⊥ version of our formula ¬¬(¬¬p → p) is, correspondingly, ((((p → ⊥) → ⊥) → p) → ⊥) → ⊥, and, putting a new variable, q, for ⊥ (as in the proof of 8.33.10), one has ((q → p) → q) → q IL ((((p → q) → q) → p) → q) → q, giving the equivalence of the axiomatizations mentioned here of Segerberg’s JP , for more information on which, see Woodruff [1970]. (Of course we could equally well have used the subscript “PL”, or indeed “ML”, rather than “IL” here, since we are concerned only with implicational formulas.) As is often the case, a reformulation of the equivalence ¨ ” (from 4.22.10, at inset above using the pseudo-disjunction notation “ ∨ p. 555, and elsewhere) is quite revealing: ¨ q IL ((p ∨ ¨ q) → p) ∨ ¨ q. (q → p) ∨ The rhs here could be further simplified, and made to resemble the lhs, if we write it with the aid of the Meredith implication ⊃ (from 7.22, especially 7.22.16 and environs) which simulates classical → in the ¨ q. Now each side, of the setting of pure implicational IL ): (q ⊃ p) ∨ equivalence, if taken as a candidate logical principle in its own right, says for something recognizably implicational, that an implication and its antecedent are pseudo-subcontraries – or, more precisely (see 4.22.10), that the implicational formula is a subcontrary of its antecedent. Since they are IL-equivalent, either principle would classicize IL; the left-hand formula is of course a form of Peirce’s Law. This same formula, ¬¬(¬¬p → p), is of further special interest to us: it gives us, when we remove the outermost negation, a formula true at precisely the points in the set Q in any model. To put it another way, we could introduce ⊥ into the language by defining ⊥ as ¬(¬¬p → p), for quite generally we have, for = P L :

8.3. THE FALSUM

1273 (((A ↔ B) ↔ B) → A) ↔ B B.

(To see the direction most easily, note that (A ↔ B) ↔ B is IL-equivalent to (A → B) → A, and invoke Wajsberg’s Law, from 7.21.5.) This can be exploited to provide a sound and complete presentation of M L∗ (in 8.33.12 below, for a solution to which the claim just made will need to be verified). The idea will be to define Q in the canonical model (which is otherwise as before – though of course using deductively closed sets under a different closure operation) to be {u ∈ W | ¬(¬¬p → p) ∈ u}. Exercise 8.33.12 Show that the least consequence relation ⊇ P L in the language with additional connective ¬ satisfying for all formulas A: ¬A A ↔ ¬(¬¬p → p) is determined by the class of all Q-frames, when the ML* definition of truth is employed (in other words, that this is M L∗ ). (Suggestion: the harder part is the soundness direction, showing that all instances of the sequent-schema ¬AA ↔ ¬(¬¬p → p) and its converse are valid on every frame. For the completeness half, use the definition of the canonical Q suggested just before this exercise.) The proof-theoretic treatment of ML* embodied in the above exercise leaves much to be desired. The reader may like to explore the possibility of pure ¬rules, for example, so that (e.g.) the →-free fragment of ML* can be isolated; another respect of awkwardness is raised in the following exercise. Exercise 8.33.13 The form of the truth-definition for ML* makes it clear that the set of sequents valid on every frame is closed under Uniform Substitution, but the syntactically specified consequence relation of 8.33.12 is not obviously substitution-invariant (though this follows from the result of that Exercise, given what was just said about validity). For example, it is not immediately clear how to prove, using ¬A A ↔ ¬(¬¬p → p) and its converse (for arbitrary A) together with the rules of PNat together, such sequents as: (i) ¬q q ↔ ¬(¬¬r → r)

and (ii ) q ↔ ¬(¬¬r → r) ¬q.

Find proofs of (i) and (ii ). Analogues of the last two exercises could have been posed for ML itself. 8.32.8(iii), p. 1262, noted that ⊥ was ML-definable by ¬p ∧¬¬p; so we could have given a proof system for ML distinct from MNat, by adding to PNat the following rules (in the style of 8.33.12): ¬A A → (¬p ∧¬¬p)

A → (¬p ∧¬¬p) ¬A.

Apart from including a query (part (iv)) on this alternative system for ML, what follows is just a version of 8.32.9 (p. 1263) rephrased to enquire about ML* rather than ML. Exercise 8.33.14 (i) Does the Law of Triple Negation hold for ML*? (ii ) Does ML* have the Disjunction Property? (iii) Is Glivenko’s Theorem correct for ML*? (iv ) Derive the rule (RAA) of MNat from the alternative proof system for ML described immediately before this exercise.


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8.34

Extensions of Minimal Logic

The reader will recall from 2.32 that the consistent superintuitionistic (or ‘superconstructive’) logics turned out to be precisely the intermediate logics: those between IL and CL. We are thinking of logics here primarily as either SetFmla proof systems closed under certain rules (those of INat, MNat, or PNat, as appropriate) or the substitution-invariant consequence relations associated therewith. In the present subsection we will look briefly at some, let us say, supraminimal logics: extensions of ML. (We change ‘super-‘ to ‘supra-’ because the former prefix suggests extreme minimality, whereas we are moving in the opposite direction, having already looked ‘downward’ from ML in 8.33.) Does a similar point hold in the present setting: that, the inconsistent case to one side, the supraminimal logics are precisely those which are ‘intermediate’ in the analogous sense of lying between ML and CL? The answer is clearly no, since we could extend the system MNat of 8.31 by adding as a new rule the zeropremiss rule (a zero-premiss rule in both the usual ‘vertical’ sense of having no premiss-sequents, and the ‘horizontal’ sense of having nothing to the left of the “”): ⊥. Alternatively, if ⊥ is absent from the formulation but ¬ is present, we could take this rule in the form of the schema ¬A. Then the consequence relation associated with this proof system is clearly an extension of M L which is not included in CL , and which is consistent (i.e., not every sequent belongs to the consequence relation). Exercise 8.34.1 State and prove a theorem like 8.32.6 for the above proof system (the ¬-based formulation), finding a condition on Q-frames and showing soundness and completeness for the system with respect to the class of frames satisfying the condition. (For assistance, see the comment following 8.34.9 below.) Aside from those supraminimal logics which extend IL, the best known are, unlike that just considered, sublogics of CL, and we shall concentrate on two of these, the logics LD and LE, to use the nomenclature of Curry [1963]. As in the above discussion, our preferred framework will be Set-Fmla. If we added either the Law of Excluded Middle or the rule (¬¬E) to our intuitionistic natural deduction system INat, we should obtain a formulation of CL. These additions have different effects on MNat, however. Double Negation Elimination immediately catapults us all the way from ML to CL, since it delivers us the EFQ formulation A∧¬A B from the sequent A∧¬A¬C whose ML-provability was noted at 8.32.1(i): take C = ¬¬B and then apply (¬¬E)); thus we are already in IL, and we already know that (¬¬E) yields CL in this context. With the Law of Excluded Middle (say in the form A ∨ ¬A) things are different. Adding this to MNat gives a proof system for the logic Curry calls LD, and describes as the logic of ‘complete refutability’. Exercise 8.34.2 (i) Show that Γ A is LD-provable iff it is valid on every Q-frame (W, R, Q) in which for all x, y ∈ W : (xRy & x = y) ⇒ y ∈ Q. (ii ) Conclude from (i) that p ∧¬p q and ¬¬p p are both unprovable in LD. (Describe countermodels.) (iii) Do the rules used to present a proof system for LD above uniquely characterize ¬?

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(iv ) Let DIL be the restriction to ¬-free formulas of the consequence relation associated with the dual intuitionistic proof system DINat of 8.22, with all occurrences of dual intuitionistic negation then rewritten as ¬. Is either of DIL , LD included in the other? (Give a proof or counterexample to each potential inclusion, as appropriate.) If we formulated the Law of Excluded Middle, which gets us from ML to LD, in terms of ⊥ rather than ¬, we should have the schema A ∨ (A → ⊥); adding the latter schema to IL instead would give the same result as adding the more general schema A ∨ (A → B), since ⊥ IL B for any B. But again, with ML rather than IL as the background logic, the effects of the two additions are different. The system obtained by adjoining A ∨ (A → B) to MNat, Curry [1963] calls LE; he describes it (p. 261) as the logic of negation as ‘classical refutability’, crediting this description and some examination of the system to unpublished work of Kripke – though there is a brief discussion in Kripke [1959b]. (See Odintsov [2008], Chapter 3.) Note that the (LE) schema above can be read as saying that a conditional and its antecedent are subcontraries, and that this is the same effect as was obtained (without appeal to disjunction) in that part of the definition of the →-classicality of a consequence relation in 1.18 (we numbered it ‘(iii)’) which demanded that for all sets of formulas Γ, and all formulas A, B, C: If Γ, A C and Γ, A → B C then Γ C These conditions on are also intimately related to one statable using Peirce’s Law, as in the following alternative presentation of LE: add to the rules of MNat the rule form of that principle (last discussed in 7.21): Γ, A → B A

(Peirce)

ΓA

Clearly this rule is derivable in any extension of MNat in which all instances of the schema (A → B) → A A are provable. So let us derive the latter schema from the (LE) rule A ∨ (A → B) above; below is a (schematic) proof in the style of those from Nat in 1.23. (Note that in calling this sequent schema – or zero-premiss rule – the (LE) rule, there is no intention to recall the unrelated structural rule from 2.33.23 of the same name, abbreviating Left Expansion. Likewise there is no connection with the similarly denoted Left Extensionality condition of 3.23.)

2 3 4 2,4 2

(1) (2) (3) (4) (5) (6)

A ∨ (A → B) (A → B) → A A A→B A A

LE schema Assumption Assumption Assumption 2, 4 →E 1, 3–3, 4–5 ∨E

Conversely, we can derive the (LE) schema using (Peirce); it will be clearer if we derive a particular but representative instance: p ∨ (p → q). For this, an ML-provable sequent of the form (p ∨ (p → q)) → B p ∨ (p → q)


1276

would be helpful. Whatever solution is suggested for B, the left-hand side is ML-equivalent (indeed PL-equivalent) to (p → B) ∧ ((p → q) → B). If we take B as p → q, the first conjunct becomes p → (p → q) which accordingly contracts to yield the second disjunct of the right-hand formula. So for this choice of B, the sequent inset above is provable, and we can therefore apply (Peirce), with Γ = ∅, A = p ∨ (p → q), to derive: p ∨ (p → q) In our discussion in 7.21, this Γ = ∅ form of (Peirce) was referred to as (Peirce0 ); we had occasion to note that even in the pure implicational setting of that discussion, the full effect of (Peirce) can be recovered from this special case. (Clearly, we could use the proof given there for 7.21.7, p. 1060, to derive the (LE) schema from (Peirce) by going via (Subc)→ , though this would be less economical than the derivation given above.) Digression. Although we are here concerned with extending ML rather than IL by (Peirce) or the (LE) schema above, notice that since ¬¬A, A → ¬A IL B for any formulas A and B, we have in particular, for any A: ¬¬A, A → ¬A IL A, so that (Peirce) applies, in which the Γ of our schematic description of that rule is for this application {¬¬A} and the B there is ¬A, giving the conclusion ¬¬A A. (We have already observed that the law of excluded middle is a special case of the (LE) schema and that this and the distinctively classical Nat rule from 1.23, (¬¬E), are equivalent against the background of IL; but the more direct derivation using (Peirce) is perhaps of interest. End of Digression. Exercise 8.34.3 (i) Show that Γ A is LE-provable iff it is valid on every Q-frame (W, R, Q) in which for all x, y ∈ W , xRy implies x = y. (ii ) Using (i), show that, while LD ⊆ LE , the converse inclusion does not obtain. (iii) Using (i), show that a sequent is LE-provable iff it is valid on each of the two 1-element Q-frames (in the one of which Q is empty, and in the other of which, Q has the sole frame-element as a member). (Hint: Think about generated subframes.) Remark 8.34.4 From 8.34.3(iii) we get that LE is Halldén-incomplete (6.41), with, for example, the Halldén-unreasonable ¬p ∨ (¬¬q → q). (Since it has the Disjunction Property, ML itself is automatically Halldéncomplete.) Using 8.34.3(iii) we can construe LE in Fmla as a four-valued logic, a point made by Lemmon and Porte (see Prior [1969], Porte [1984]). The two oneelement frames described in the exercise give two two-element matrices (with matrix-elements corresponding to the empty set and the set containing the sole point), each with the familiar boolean table for ∨, ∧, and →, and differing over the ¬ table (or, if present, the ⊥ table). In the matrix for the ‘Q non-empty’ frame, ¬ has the constant-true table (⊥ must be true), while for the ‘Q empty’ case ¬ has the usual boolean table (⊥ must be false). Thus we can form the product table with entries a, b where a is T or F according as the formula in question is true or false at the sole point in a model on the ‘Q non-empty’ frame, and b is T or F according to the truth-value at the point in the ‘Q empty’ frame.

8.3. THE FALSUM

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We abbreviate these pairs in accordance with the conventions of 2.12 (thus 1 = T, T , etc.), giving tables for ∧, ∨, → as in the ‘homogeneous’ product tables of Figure 2.12a, while for ¬ we have a ‘heterogeneous’ product table in the style of Figure 8.34a. ¬ *1 2 3 4

2 1 2 1

Figure 8.34a

In terms of our VFIN-notation for the 1-place truth-functions (3.14) we have here the product of V and N in that order; if we had reversed the order, thinking of the pairs as Q-empty value, Q-non-empty value instead of as Qnon-empty value, Q-empty value , then we would have had a different table (as with 3.24a and 3.24b on pp. 465 and 468). The formulas valid in the product matrix are then just the formulas valid in each of the factors, and thus those valid on each of the two 1-element frames Q-frames, and thus those which are provable in LE (by 8.34.3(iii)). All of this is very interesting as far as it goes, but—as we noted in 3.24 on the subject of treating hybrids as products—it goes no further than Fmla. The logic LE in Set-Fmla (or the consequence relation LE ) is certainly not four-valued; indeed this logic is determined by no matrix of whatever size, since it does not satisfy the Cancellation Condition: we have, exactly as for ML itself (see the discussion following 8.32.4, p. 1259) – p, ¬p LE ¬q without either: p, ¬p LE A, for all A

or:

LE ¬q,

even though there are no propositional variables in common on the left and right (of the claim that p, ¬p LE ¬q). Thus, to bring out the point that ¬ in LE hybridizes the properties of V and N it is better to forget about four-valued truth-tables and put the point in terms of intersection. (This point was foreshadowed at 3.24.5, p. 467.) Let us write CL⊥ for the extension of the {∧, ∨, →}-fragment of CL by the addition of ⊥; that is, CL⊥ is the consequence relation determined by the class of all {∧, ∨, →}-boolean valuations which assign the value T to ⊥. We use the same labelling even if ¬ rather than (or as well as) ⊥, the determining class in this case comprising the {∧, ∨, →}boolean valuations v with v(¬A) = T for all A (and if it is present, with v(⊥) = T). Then we have the following Observation (made in Kripke [1959b], Parsons [1966]), whose proof is left as an exercise; note that because of ambiguity in the labelling as to whether ⊥ or ¬ (or both) are primitive, there are really several connected claims involved, one per disambiguation. (Obviously the same assumption about primitives should be made on each side of the ‘=’.) Observation 8.34.5 LE = CL ∩ CL⊥ .

1278


This concludes our discussion of the supraminimal logics LD and LE. (For discussion of an LD-like logic with relevant rather than minimal logic as the underlying negation-free system, see Mares [1995].) Before proceeding we interpose some information on alternative labellings (a sequel to that provided in 2.32.11 on p. 319): Remark 8.34.6 Fmla versions of LD and LE may be found in Segerberg [1968] under the names JX and JE (or JP) respectively, and in Parsons [1966] under the names MC+ and MCC, respectively; see also Prior [1969]. (Segerberg calls ML “J”, while Parsons calls it MC.) LE also appears without a special name—other than as “the system with axioms 1–11”— on p. 103 of Kanger [1955]. Kanger is mainly interested in giving independent axiomatizations with the separation property for the various logics; see also Robinson [1968] and Meredith [1983]. LE appears in Ono [1967] and several further papers by the same author cited there under the name LN; Ono also considers the extension of Positive Logic by Peirce’s Law. See further Odintsov [2001] and Chapter 4 of Odintsov [2008]. Since neither CL ⊆ CL⊥ nor CL⊥ ⊆ CL , our experience in 6.41 with the lattice of modal logics roughly ‘predicts’ the Halldén-completeness of LE noted in 8.34.4 above, given 8.34.5. (Recall Lemmon’s formulation of 6.41.4: a modal logic is Halldén-incomplete iff it is the intersection of two ⊆-incomparable modal logics.) “Roughly”, because we are not here assuming the CL background there current and because we were there working in Fmla rather than Set-Fmla. Together, these differences actually mean that although LE does not present such an example, it will be possible to find Halldén-complete logics which are the intersections of ⊆-incomparable logics. In fact, as we shall now go on to show, this is the case for ML itself. The historical situation is as follows. As is evident from the failure of the Cancellation Condition, M L is not determined by a class of matrices of cardinality 1, so the question arises as to what the cardinality of the smallest class of matrices determining the logic—its ‘degree of complexity’—exactly is. Hawranek and Zygmunt [1981] showed that the answer here is: 2. This makes M L the intersection of two consequence relations each of degree of complexity 1 (each determined by a single matrix), and Hawranek [1987] identifies these two consequence relations, both syntactically and algebraically. Actually the description is in terms of consequence operations (rather than relations) and runs as follows (in our notation). •

Cn 1 is the consequence operation defined by: for any set Γ of formulas, Cn 1 (Γ) is the least set of formulas which includes Γ ∪ CnM L (⊥) and is closed under Modus Ponens.

•

Cn 2 is defined by: for any set Γ of formulas, Cn 2 (Γ) is the least set of formulas which includes Γ ∪ Cn M L (∅) and is closed under Modus Ponens and the rule: from ⊥ to A (A any formula).

We will give a model-theoretic account of one of these two consequence operations (or rather, the associated consequence relations), and of a slight variant on the other, which we re-baptize with more suggestive names. We shall call the

8.3. THE FALSUM

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logic associated with Cn 1 “ML⊥”, by analogy with the use of “CL⊥” above; the other logic we are interested in we call weak intuitionistic logic and denote by “WIL”. The latter is somewhat stronger than that associated with Hawranek’s Cn 2 , which we shall accordingly call doubly weak intuitionistic logic and denote by WWIL. The consequence relations we are interested in are M L⊥ and W IL , and M L will be seen (8.34.12: p. 1281) to be the intersection of these two. We take pay only fleeting attention to W W IL . We begin with our own quasi-syntactic definitions of ML⊥, WWIL and WIL, whose common language we call L for present purposes: •

M L⊥ is the consequence relation associated with the proof system we obtain from that for MNat by the addition of the zero-premiss rule: ⊥.

•

W W IL is M L ∪˙ {⊥ A|A ∈ L}.

•

W IL is M L ∪˙ {A ∨ ⊥ A | A any formula}.

Note that ∪˙ is the join operation in the lattice of consequence relations on the language in question, which has connectives ∧, ∨, → and ⊥. (We leave the interested reader to give the ¬-based alternative formulations.) Exercise 8.34.7 Show that, as just defined, M L⊥ and W W IL coincide with the consequence relations associated with the consequence operations Cn 1 and Cn 2 introduced above. Observation 8.34.8 W W IL ⊆ W IL . Proof. It suffices to show that for any A, we have ⊥ W IL A. This follows by (T) from the facts that ⊥ W IL A ∨ ⊥ (by the first term in the above ˙ ∪-characterization of W IL ) and that A ∨ ⊥ W IL A (by the second term in ˙ the ∪-characterization). Thus in weak and doubly weak intuitionistic logic alike, ⊥ lives up to its description as the falsum in allowing the ex falso quodlibet principle ⊥ A (for arbitrary A). But for the sake of a smooth model-theoretic description, it is weak rather than doubly weak intuitionistic logic we shall be attending to from now on, though at the end of our discussion we return to the task of showing that these logics are in fact different (that the inclusion in 8.34.8 is a proper inclusion, that is). Nor will we be asking about matrix characterizations of W IL ; information on these matters for W W IL and M L⊥ may be found in Hawranek [1987]. Although W IL ⊇ M L , weak intuitionistic logic is not a supraminimal logic in the sense introduced at the beginning of the present subsection. (These points apply also to W W IL .) In particular. because, as we shall observe, W IL does not have the ‘Deduction Theorem’ property (that Γ, A W IL B always implies Γ W IL A → B: one half of what it takes for W IL to be ‘→-intuitionistic’) it cannot be, as our definition of supraminimality (by analogy with intermediateness) required, the consequence relation associated with a proof system extending MNat (which would therefore include the rule (→I)). It follows from 8.34.10 and 8.34.11(ii) that we get a counterexample by putting Γ = ∅, A = ⊥, B = p.

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For a model-theoretic description of ML⊥ and WIL, we use the same notion of a Q-frame employed in the semantics for ML itself (in 8.32), but with two novel notions of validity. Given a Q-frame (W, R, Q), we say that a sequent is Q-valid on (W, R, Q) if it holds at each x ∈ Q in every Q-model on (W, R, Q), and Q-valid on (W, R, Q) if it holds at each x ∈ W Q in every Q-model on (W, R, Q). Actually, the novelty is really in the second of these two notions, in view of Observation 8.34.9 A sequent is Q-valid on every Q-frame if and only if it is valid on every Q-frame (W, R, Q) with Q = W . Proof. ‘If’: Suppose a sequent σ is not Q-valid on (W , R , Q ); thus for some Valuation V and some x ∈ Q, σ does not hold at x in M = (W , R , Q , V ). Then σ does not hold at x in the submodel (W , R, Q, V ) of M generated by x. Because Q is closed under R = R ∩ W × W (“Q-Pers”) and x ∈ Q, the Q-frame (W, R, Q) satisfies the condition that Q = W , and σ is not valid on (W, R, Q). ‘Only if’: Suppose σ is not valid on (W, R, Q) where Q = W . Since Q = W , validity and Q-validity coincide; so σ is not Q-valid on (W, R, Q). 8.34.9 and 8.34.11(i) below together answer, given a modulation from the ⊥based to the ¬-based language, Exercise 8.34.1. And in reducing the notion of Q-validity on every frame to validity on a class of Q-frames, 8.34.9 reassures us that the familiar rules (such as, in particular, (→I)) will preserve Q-validity on a Q-frame, since they preserve validity on an arbitrary Q-frame (indeed, ¯ preserve holding in any given Q-model). It is Q-validity, the semantic correlate of WIL-provability, which is unusual in this respect; in particular, we have: ¯ Example 8.34.10 ⊥ p is Q-valid on every frame while ⊥ → p is not. ¯ (Thus the class of sequents Q-valid on a frame is not closed under (→I), essentially because, while the Q of a Q-frame (W , R, Q) is required to be closed under R—this being the condition (QPers) of 8.32—its complement W Q is not.) Theorem 8.34.11 For any sequent Γ A: (i) Γ M L⊥ A iff Γ A is Q-valid on every Q-frame ¯ (ii) Γ W IL A iff Γ A is Q-valid on every Q-frame. Proof. We prove just the ‘if’ (completeness) direction of (ii), using the canonical model method. Rather than construct a new canonical model, we simply use the canonical Q-model for ML, described in the proof of 8.32.6 (though now since ⊥ is primitive we put QM L = {u ∈ WM L | ⊥ ∈ u}). Suppose that Γ W IL A. Then Γ W IL A ∨ ⊥ (since A ∨ ⊥ W IL A); thus, certainly Γ M L A ∨ ⊥, so for some x ∈ WM L , we have Γ ⊆ x and A ∨ ⊥ ∈ / x. Therefore A ∈ / x and also ⊥ ∈ / x, the latter giving us that x ∈ / QM L . Since truth and membership coincide, we have MM L |=x C for each C ∈ Γ, while MM L |=x A. Thus the unprovable sequent Γ A fails to hold at x ∈ WM L QM L in MM L , showing it not to be ¯ Q-valid on the Q-frame (WM L , RM L , QM L ).

8.3. THE FALSUM

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Corollary 8.34.12 M L = M L⊥ ∩ W IL . ¯ Proof. Since a sequent is valid on a Q-frame iff it is Q-valid and also Q-valid on that Q-frame, the result follows from 8.32.6 and 8.34.11. Now, to elaborate the point about Hallén-completeness made above: although the intersection of M L⊥ and W IL is M L , and neither of M L⊥ , W IL , is included in the other, M L is Halldén-complete; “although” here, because – as already intimated – expectations created especially by working in Fmla (and common properties of ∨ in that framework) would seem to rule this possibility. If the ⊆-incomparability of M L⊥ and W IL were illustrable exclusively with formulas, so that for some A and B we had both (1) and (2): (1) M L⊥ A while W IL A

(2) W IL B while M L⊥ B

then we should have a Halldén-unreasonable disjunction on our hands in the intersection logic, in the shape of A ∨ B , where B has any propositional variables of B occurring also in A relettered away. Now while (1) is illustrated by choosing A as ⊥, there can be no B for which (2) holds, since we can see at once from the proof of the Disjunction Property for IL (6.41.1, p. 861) that the same argument establishes the Disjunction Property, and hence a fortiori Halldén-completeness—for ML. (This answers 8.32.9(ii), p. 1263.) To illustrate the failure of the inclusion W IL ⊆ M L⊥ we must use examples such as (most obviously) the fact that (3) ⊥ W IL p while ⊥ M L⊥ p, which cannot be converted into a contrast in respect of the formulas which are consequences of the empty set according to the two relations. We cannot, that is, reformulate (3) as (4): (4) W IL ⊥ → p while M L⊥ ⊥ → p, because what precedes the ‘while’ is false: see 8.34.10. (There is a perfectly good Halldén-incomplete supraminimal logic which is the intersection of the two supraminimal logics ML + ⊥ ( = ML⊥) and ML + ⊥ → A, with such disjunctive theorems as ⊥ ∨ (⊥ → p); it just happens not to ML but a proper extension of ML itself. This logic is discussed under the name JE in pp. 39–40 of Segerberg [1968].) We close by returning to the question of whether the inclusion described in 8.34.8 above (W W IL ⊆ W IL ) is proper. A negative answer amounts to the claim that A ∨ ⊥ W W IL A for all A. Note that one cannot argue for such an answer by appeal to the rule (∨E) used in the presentation of MNat. This would tell us C ∨ D W W IL E whenever C M L E and D M L E. In the present instance C, D, and E are respectively A, ⊥ and A, so we have the relevant C M L E but not (in general: e.g., take A as p) D M L E. We shall see that the we above claim is false: we do not have A ∨ ⊥ W W IL A for all A. Recall that W W IL was introduced as M L ∪˙ {⊥ A| A ∈ L}. We shall show that for the much stronger logic = CL ∪˙ {⊥ A | A ∈ L}, where CL is the consequence relation determined by the class of all {∧, ∨, →, ¬}-boolean


1282

valuations, over which class of valuations the sequent p ∨ ⊥ p is not valid. (Note that the “CL” is something of a misnomer, since we assume ⊥ is in the language but CL is not ⊥-classical, since we did not require the valuations in the determining class to be ⊥-boolean.) We argue this by reference to the products of the two-valued tables for ∧, ∨, → and ¬ with themselves; to fill this out to a matrix for the whole language of , we add that h(⊥) = 2 for any evaluation h. (Reminder: “2” means T, F ; the value 1 ( = T, T ) is the sole designated value.) For any sequent Γ A belonging to , we have that Γ A is valid in the four-element matrix just described. For the rule (T), built into ˙ the ∪-notation in defining as CL ∪˙ {⊥ A | A ∈ L}, preserves validity in any matrix and the sequents provided by the first term of the join, CL are valid in the matrix (by the first part of 2.12.6(i)), as are those provided by the second, since h(⊥) is undesignated. But the sequent p ∨ ⊥ p is not valid in the matrix, since we may take h(p) = 3, in which case h(p ∨ ⊥) = 3 ∨ 2 = 1, which is designated while h(p) is not. So even with the help of all of the zero-premiss sequent-to-sequent rules of classical logic in ∧, ∨, ¬, →, we cannot derive the schema A ∨ ⊥ A from the schema ⊥ A; we certainly cannot do so, then, with the more restricted resources of ML. To sum up: Observation 8.34.13 W IL ⊆ W W IL ; in particular p ∨ ⊥ W W IL p even though p ∨ ⊥ W IL p.

8.35

The Falsum: Final Remarks

We have been thinking of negation in minimal and intuitionistic logic as a matter of ‘implying the Falsum (⊥)’, with ⊥ in the former case having no special properties and in the latter as an all-implying constant. If we temporarily entertain the idea of quantifiers binding propositional variables we can relate these two ways of thinking of the Falsum differently: we could think of the intuitionistic ⊥ as ∀q.q, where q is now a bindable propositional variable, and of the minimal ⊥ as q itself. We will implement this strategy for the most part without officially introducing such explicit quantifier-and-variable notation, writing ⊥ rather than a bindable propositional variable (‘q’) and using a oneplace connective to do the binding (rather than “∀q”). Thus ⊥ is to behave as in ML while ⊥ behaves the way ⊥ itself behaves in IL. We can avoid having to write this in explicitly variable binding terms because only (various occurrences of) a single propositional variable need to be bound by a quantifier (so we don’t need to simulate “∀p”, “∀r”, . . . alongside “∀q”). We revert to the “∀q” notation only for occasional expository asides. The reason for using “q” in this capacity, incidentally, is the mnemonic connection with the “Q” of the Q-models of 8.32. In semantic terms, the transition from ML to IL amounts to considering the set Q of such a model as an arbitrary R-closed (‘persistent’) set of points for ML but specifically as ∅ for IL. For the perspective of the present subsection, Q is again a matter for arbitrary (R-closed) choice, but because of the presence of , we can consider within a single formula all possible ways of making that choice. (This semantic gloss will be made precise below.) The logic to be developed will be called IML for “Intuitionistic and Minimal Logic”, since we have, in the formulas ⊥ and ⊥ versions of an intuitionistic and a minimal Falsum formula, respectively. (No doubt, such a description would equally aptly to a logic with two atomic formulas, ⊥i and ⊥m , say, with

8.3. THE FALSUM

1283

the appropriate rules—in the latter case, with no rules; but this is not what we have in mind.) The language of IML has for its primitive connectives the binary connectives ∧, ∨, and → of Positive Logic, as well as the 0-ary ⊥ and 1-ary . We provide a proof system for IML by subjoining to the rules of PNat an elimination rule and an introduction rule for . ⊥ will be mentioned in the former rule, and for the formulation of the latter rule, we need to introduce the following terminology. We say an occurrence of ⊥ as a subformula of A is exposed in A when it does not occur in the scope of any occurrence of in A (i.e., when the occurrence in question lies in no subformula B where B is a subformula of A). In the explicitly quantificational notation, an exposed occurrence of ⊥ thus corresponds to a free occurrence of the bindable propositional variable. (E)

Γ A Γ A(B)

(I)

ΓA Γ A

subject to the proviso below

For (I) the proviso alluded to takes the form: provided no formula in Γ has an exposed occurrence of ⊥; and in the statement of (E), A(B) denotes the result of replacing all exposed occurrences of ⊥ in A by the (arbitrary) formula B. Note that the latter formula may itself contain occurrences of (and ⊥). The consequence relation IM L associated the present proof system is not substitution-invariant, as we illustrate with Example 8.35.1 The sequent p p is clearly provable, by one application of (I) to the premiss sequent p p. But its substitution instance ⊥ ⊥ is not provable. Because of the failure of Uniform Substitution, we are not dealing here with a normal modal logic (as defined in 2.21)—thinking of as , and ignoring the fact that the underlying logic is IL rather than CL—though the ‘rule of normality’ for is derivable: Exercise 8.35.2 (i) Show that the normality rule: A1 , . . . , An B A1 , . . . , An B is a derived rule of the above proof system for IML. (ii ) Conclude that all sequents of the following forms are provable: (A ∧ B) A ∧ B; A ∧ B (A ∧ B); A ∨ B (A ∨ B); (A → B) A → B. The above exercise shows some similarity between IM L and the inferential consequence relation K of 2.23 (with as ); but we can extend this further to an analogy with S4 : Exercise 8.35.3 Show that for all formulas A in the language of IML: A IM L A and A IM L A. Example 8.35.4 A proof of the sequent (p ∨ ⊥) p ∨ ⊥: (1) (2)

(p ∨ ⊥) (p ∨ ⊥) (p ∨ ⊥) p ∨ ⊥

(R) 1 E


1284

Setting out this proof in the style of Lemmon [1965a], as in our 1.23, we should have: 1 1

(1) (2)

(p ∨ ⊥) p ∨ ⊥

Ass. 1 E

The formula schematically represented as B in the statement of (E) is here the formula ⊥. (Let us observe in passing that we could equally well, at the last step, passed to p∨⊥, so, unusually for an elimination rule, the rule (E) can lead to conclusions with more occurrences of the eliminated connective than there are in the premiss.) We recall from 3.15.4 (p. 419) that although there are no interdefinabilities amongst ¬, ∧, ∨ and → in IL as we are accustomed to it (by contrast with CL), the first three connectives are definable in terms of → and the universal (propositional) quantifier available in second-order intuitionistic propositional logic. See 3.15.5(v) on p. 420 for appropriate definitions; each involves only a single universally quantified variable, so they can be simulated here: Exercise 8.35.5 Show that for all A, B: (i) A ∧ B IM L ((A → (B → ⊥)) → ⊥). (ii ) A ∨ B IM L ((A → ⊥) → ((B → ⊥) → ⊥)). Remark 8.35.6 To make the right-hand sides of 8.35.5(i), (ii ) look more familiar, we could agree to write subformulas C → ⊥ as ¬C. Conspicuously, the way ⊥ occurs in the above equivalences allows such a rewriting to deal with all of its occurrences. (We don’t, for instance, have any occurrences of ⊥ in antecedent position.) What we get are, respectively: (i) A ∧ B IM L ¬(A → ¬B) (ii ) A ∨ B IM L (¬A → ¬¬B). In the same vein, notice that although the principle of double negation fails in either of the forms ¬¬p IM L p, ¬¬p IM L p, it holds in the ‘mixed’ form: ¬¬p IM L p. (Think of the left-hand side here as ∀q((p → q) → q), and take q as p to derive the right-hand side.) Exercise 8.35.7 (i) Do we have IM L (¬¬p → p)? (ii) What about IM L ¬¬p → p? Exercise 8.35.8 Devise a semantics for IML, in the sense of providing a notion of model and an inductively defined notion of truth at a point therein, with the property that Γ IM LA if and only if every point in any such model at which all formulas in Γ are true, also has A true at it. (Note: This should really be labelled as a ‘Problem’ rather than an ‘Exercise’, in that the answer – or an answer – is not clear to the author.)

Notes and References for §8.3 As mentioned in the text, our semantic treatment of Minimal Logic (8.32) is based on that of Segerberg [1968]; some of this material appears in Georgacarakos [1982]. ML was first formulated Johansson [1936], though its {¬, →}

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fragment was axiomatized (in Fmla) eleven years earlier by A. N. Kolmogorov. Detailed references, historical information and further discussion, may be found in Plisko [1988a], [1988b], Troelstra [1990], Uspensky [1992]. This last paper also explains (p. 389) the solution to Exercise 26.20 of Church [1956], which asks for a proof that if, restricting attention to {→, ¬}-formulas, every propositional variable in such a formula is replaced by its double negation, the result is CL-provable if and only if it is ML-provable. (Church credits this observation, which we have not included in 8.32, to Kolmogorov, who proved a closely related result, mentioned in 8.33.11 by an argument like the proof of 8.21.1 (p. 1215). For further details, see again Uspensky [1992], at the page already cited.) These issues are related to various translations between Minimal, Intuitionistic and Classical Logic discussed in the Kleene and Prawitz–Malmnäs references given at the end of Exercise 6.42.3, p. 873. For the application of Kripke semantics to logics between Minimal and Classical Logic, Segerberg [1968] is invaluable; see also Woodruff [1970], already mentioned in 8.33.11, as well as Goldblatt [1974b], on matters arising from that paper. Note that the ‘minimality’ of ML relates to its status vis-à-vis CL and IL; it has no pretensions toward minimality in any more absolute sense. (The contrast here is with the ‘minimal logic’ of Church [1951b].) A famous discussion of falsum-based negation occurs may be found in Chapter 6 of Curry [1963]; see also Meyer [1973], Bunder [1984]. Meyer [1977] is devoted to aspects of Curry’s LD, also discussed in Curry [1952] and [1963]. Further discussion of Curry’s sequent calculi for supraminimal logics may be found in Bernays [1953] and Kripke [1959b]. Chapter 4 of Odintsov [2008] is devoted to extensions of ML.

Chapter 9

Universally Representative and Special Connectives §9.1 UNIVERSALLY REPRESENTATIVE CLASSES OF FORMULAS 9.11

Introduction

This chapter introduces and compares two properties a connective may have or lack in a logic: the property of being ‘universally representative’ and the property of being what we shall call ‘special’. It will be sufficient to treat logics as individuated by the set of sequents they contain. The logical frameworks from which our examples are taken are Fmla and Set-Fmla. In the latter case, closure under the structural rules (R), (M) and (T) will be assumed, so that we could equally well treat logics as consequence relations, and say—parallelling the usage introduced at the outset of 3.11—that a connective in the language of such a consequence relation is universally representative (or, is special) according to . When we have the ‘collection of sequents’ interpretation in mind (for Fmla or Set-Fmla) we will usually say ‘universally representative’ in such-and-such a logic. The extension of these concepts to Set-Set (or to gcr’s) is entirely routine; to avoid clutter, no further mention will be made of this fact. We need to recall the notion of synonymy, as applied when logics are conceived in the above way: formulas are synonymous when the replacement, not necessarily uniform, of one by the other in a sequent belonging to the logic yields a sequent also belonging to the logic. Now the basic notion for the present subsection is that of a class Δ of formulas being universally representative in a logic, which we define to be the case when every formula in the language of that logic is synonymous with some formula in Δ. Our Set-Fmla formula examples will all be congruential logics (i.e., their associated consequence relations – à la 1.22.1, p. 113 – are congruential in the sense of 3.31), for which cases we can alternatively describe Δ as universally representative when every formula is equivalent (in the sense) to some formula in Δ. And our Fmla examples all feature logics whose languages contain the connective → and in which membership of both A → B and B → A is both necessary and sufficient for the synonymy of 1287

CHAPTER 9. UNIVERSAL AND SPECIAL

1288

A with B according to the logic, in which case we will again describe A and B as equivalent, and again have the above alternative description of universal representativeness available. (Sometimes there will also be the connective ↔ and the given relation holds between A and B when A ↔ B is in the logic; these are the ↔-congruential logics of 3.31. Note that strictly speaking we should say in all these cases that the sequents A → B,

B → A,

and

A↔B

are in the logic: but as usual we disregard the distinction in Fmla between a formula and the corresponding sequent.) Examples 9.11.1(i ) The class of all formulas in the language of some logic is universally representative in that logic (since synonymy is reflexive). (ii ) For classical logic in Fmla or Set-Fmla, assuming the presence of (e.g.) ∧ and →, the class of formulas containing at least twenty distinct propositional variables is universally representative (since—whether we interpret this counting by types or by tokens—any formula falling short can be fleshed out to an equivalent in which the additional variables occur ‘inessentially’, conjoining the given formula with suitable many formulas pi → pi until the required quota of variables is present). (iii) With the same logic in mind, the class of conjunctive formulas is universally representative, since A is synonymous with A ∧ A. We shall be especially interested in examples of type (iii) here, and introduce the following terminology to facilitate their study. We call an n-ary connective # in the language of some logic universally representative in that logic when the class of formulas of the form #(A1 , . . . , An ) is a universally representative class in that logic. Thus 9.11.1(iii) shows that the connective ∧ is universally representative in CL: every formula has a conjunctive equivalent. Note that we could equally have chosen A ∧ (p → p) as a conjunctive equivalent of A—or for that matter A ∧ (A → A)—given the assumption there in force as to the language; if is present, A ∧ would have done as well. Exercise 9.11.2 (i ) Assuming the presence of all the connectives ∧, ∨, ¬, →, ↔ in the language of CL (in Set-Fmla, for definiteness): which of the connectives listed are universally representative in CL? (ii ) Which of the above connectives is/are universally representative in IL? (iii) Moving to the framework Fmla, answer the question in (ii) for the logic R. We return to the notion of universally representative connectives in 9.12, whose opening paragraph (along with 9.13.3, p. 1299) answers the questions in (i) and (ii) of this Exercise; as for (iii), only the last of the connectives listed (“↔”) presents any difficulty, and we deal with it in 9.24. In the meantime, we note another case of some interest: it may be, for some connective #, that the class of formulas which do not contain # is universally representative (in some logic). In this case, let us call # eliminable (in that logic). If a connective # is definable, in the sense of 3.15, then # is obviously

9.1. UNIVERSALLY REPRESENTATIVE CLASSES OF FORMULAS 1289 eliminable: every formula has a #-free formula synonymous with it, namely that provided by using the definition. As remarked in 3.15, definability is uniform eliminability. We can obtain the converse implication (eliminable ⇒ definable) on the further assumption that our logic is closed under Uniform Substitution. For in that case, the eliminability of n-ary # implies that #(p1 , . . . , pn ) is synonymous with some #-free formula A, substituting Bi for pi (1 i n) in which gives a formula synonymous with #(B1 , . . . , Bn ). This equivalent can then replace #(B1 , . . . , Bn ) in any formula in which the latter occurs, to yield a synonymous formula containing one less occurrence of # than the original. By repeating the procedure, we eventually arrive at a #-free equivalent. Remark 9.11.3 The assumption of closure under Uniform Substitution is essential: in its absence it is easy to have a connective which is eliminable, but where the mode of elimination is not uniform and definability is accordingly lacking. An example is the singulary connective F (for “it is fixedly the case that”) in the Fmla logic called S5AF in Davies [1981], at p. 264 of which there appears a proof its eliminability. Fp is synonymous with the plain p, though it is not in general the case (illustrating the failure of Uniform Substitution) that FB and B are synonymous, for arbitrary B. (The “A” in the label “S5AF” is for “actually”, the idea being that the composite operator “fixedly actually” would express a kind of necessity – or something like it – in virtue of meaning “Whatever world is taken as the actual world, it is true in the actual world that ”. See Humberstone [2004a] for further discussion and references.) A similar example arose in 8.35, with playing the role of F.

9.12

Universally Representative Connectives

Returning to our main topic, the universally representative nature of certain connectives in various logics, we begin with some examples from CL. Here we have, as already noted various choices for a conjunctive representation of A: as A ∧ A, as A ∧ (A → A), A ∧ (p → p), A ∧ and so on (cf. also 9.13.1), the precise range of options depending in part on which connectives are available. Now of these, the first representation enjoys a special property: it shows that ∧ is universally representative ‘all by itself’, without the help of extraneous connectives. We may define an n-ary connective # in the language of some logic, which language also has connectives #1 , . . . , #k , to be universally representative with the help of {#1 , . . . , #k } in that logic, when every formula is synonymous with some formula #(B1 , . . . , Bn ) in which the only connectives that appear, aside from # and any connectives in A itself, are #1 , . . . , #k . Thus (assuming in the language) the choice of A ∧ as a formula CL-synonymous with A reveals ∧ to be universally representative in CL with the help of . When # is universally representative in a logic with the help of ∅, we will say that # is universally representative all by itself in that logic. The choice of A ∧ A, then, as a formula synonymous with A, reveals ∧ to be universally representative all by itself in CL, because this particular conjunctive equivalent of A contains only ∧ and such connectives as occur in A. Likewise, the choice of A ∨ A (rather than, say, A ∨ ⊥) shows that ∨ is universally representative all by itself in CL. Similarly, to complete the inventory of connectives mentioned under 9.11.2 (p. 1288), we note that A and (A → A) → A are CL-synonymous, as are A and

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(A ↔ A) ↔ A, and, finally, A and ¬¬A. Note that all but the last claim hold for IL in place of CL. We return to the question of ¬’s universal representativeness in IL in 9.13.3 (p. 1299), and to the possibility of a connective’s being universally representative but not so all by itself in 9.12.9–10. For certain logics, it can be shown that the ‘help’ just described, on the part of extraneous connectives, is never needed: any universally representative connective is universally representative all by itself. Indeed, we shall show this for CL (9.12.4). To do so, it is simplest to think of CL as the consequence relation CL = Log(BV ), so that we may reason about truth-functions rather than connectives. Theorem 9.12.1 For any non-constant n-ary truth-function f, there exist truthfunctions g1 , . . . , gn , each of them singulary, compositionally derived from f such that for all x ∈ {T, F}, x = f (g1 (x), . . . , gn (x)). Remarks 9.12.2(i) It is clear that the non-constancy condition is not only sufficient, but also necessary for a representation of the promised form x = f (g1 (x), . . . , gn (x)), since if f constantly assumes the value T, this equation cannot hold for x = F, and vice versa. (ii) As will be evident from the proof which follows, the gi whose existence is claimed are all of them of a particularly simple kind, namely, in each case, either the identity truth-function gi (x) = x or else the result of applying f itself to x, taken appropriately many times: gi (x) = f (x, . . . , x) where, on the right, there are n occurrences of x. In the proof, we shall abbreviate such a right-hand side to f (xn ); similarly, we write f (y k , z m ) for f (y1 , . . . , yk , z1 , . . . , zm ) when k+m = n and y1 = y2 = . . . = yk = y and z1 = z2 = . . . = zm = z. (We illustrate the procedure given in the following proof, under 9.12.5.) Proof. (Proof of 9.12.1, that is.) Let f be a non-constant n-place truth-function. Then either (1) f (Tn ) = T and f (Fn ) = T or (2) f (Tn ) = T and f (Fn ) = F or (3) f (Tn ) = F and f (Fn ) = T or (4) f (Tn ) = F and f (Fn ) = F. In case (2), we have (for x ∈ {T, F}), x = f (xn ); in terms of the formulation of the Theorem, each of g1 , . . . , gn is the identity function. In case (3), f (xn ) takes the argument T to the value F and conversely, so in this case, we may take each gi to be given by gi (x) = f (xn ), and we shall have x = f (g1 (x), . . . , gn (x)). Cases (1) and (4) require more work. Suppose we are faced with f falling under case (1), so that: f (Tn ) = T and f (Fn ) = T. Since f is not a constant function, there must be some way of choosing z1 , . . . , zn from {T, F} so as to have f (z1 , . . . , zn ) = F. There is no loss in generality in assuming that the zi = T precede the zj = F in the listing z1 , . . . , zn . (Otherwise we merely consider an appropriately permuted form of f for which this is the case, e.g. defining f (x, y, z) = f (z, y, x) and ‘permuting back’ at the end of the argument.) Suppose that the first k

9.1. UNIVERSALLY REPRESENTATIVE CLASSES OF FORMULAS 1291 truth-values in our listing are T, and that the remaining m(= n − k) are F. Accordingly: (*) f (Tk ,Fm ) = F. Then for x ∈ {T, F} we have x = f ((f (xn ))k , xm ). For, if x = T, then, as we are dealing with case (1), in which f (Tn ) = T, f (xn ) = T, so that the rhs here works out as f (Tk , xm ). But again, since x = T, f (Tk , xm ) is f (Tn ) = T. Next, suppose x = F, then again we have x = f ((f (xn ))k , xm ), since now the rhs works out as f (Tk , xm ), because under case (1) f (Fn ) = T, and with x = F (*) tells us that f (Tk , xm ) = F. The reasoning for f falling under case (4), in which f (Tn ) = F and f (Fn ) = F, is similar, and we are led via the existence of k, m, for which (by the assumption of non-constancy) (**)

f (Tk ,Fm ) = T

to the conclusion that for x ∈ {T, F} we have x = f (xk ,(f (xn ))m ).

The Corollary in which we are interested uses the notion of the truth-function associated with a connective over a class of valuations, for which see 3.11. Corollary 9.12.3 If = Log(V ) for some class V of valuations over which an n-ary connective # in the language of is associated with some non-constant truth-function f# , then for every formula A of that language there exist formulas B1 , . . . , Bn , such that A #(B1 , . . . , Bn ); further, such a sequence B1 , . . . , Bn can be found in which each Bi is either A itself or else is the formula #(A, . . . , A), where A occurs n times. Proof. Apply 9.12.1 taking f as f# , and using the specific terms mentioned under 9.12.2(ii). Note that what we have called f# here would, by an obvious extension (to cover non-primitive #) of the notation introduced in 3.14 (p. 403), be called #b . Corollary 9.12.4 If # is an n-ary connective which is fully determined according to a congruential consequence relation , and we have neither: #(A1 , . . . , An ), for all formulas A1 , . . . , An , nor #(A1 , . . . , An ) C for all A1 , . . . , An , C, then # is universally representative all by itself according to . Proof. If # is fully determined according to , then by 3.13.9(iii) (p. 391), we can find a V with = Log(V ) and apply 9.12.3. The conditions requiring that we have neither #(A1 , . . . , An ) nor #(A1 , . . . , An ) C for all formulas A1 , . . . , An , C, imply that the truth-function associated over V with # is not constant. The condition that is congruential is included to secure that from

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A #(B1 , . . . , Bn ) (as in 9.12.3) it follows that A and #(B1 , . . . , Bn ) are synonymous according to . We will illustrate the procedure suggested by the proof of 9.12.1 for finding the promised x = f (g1 (x), . . . , gn (x)), and then comment on two respects in which 9.12.4 understates the universal representativeness property with which it deals. Example 9.12.5 Suppose f is the ternary truth-function defined by f (x, y, z) = T iff exactly two out of x, y, z are T. Thus this is a non-constant truth-function for which f (T,T,T) = f (F,F,F) = F, and so f falls under case (4) of the division of cases in the proof of 9.12.1. We have to find z1 , z2 , z3 for which f (z1 , z2 , z3 ) = T, and one possibility is z1 = z2 = T, z3 = F. The proof suggests that the first and second arguments of f are to be filled by x, and the third by f (x, x, x) to give the solution to the ‘universal representativeness problem’ for f in the form: x = f (x, x, f (x, x, x)). (Note: the wording “exactly two out of x, y, z are T” doesn’t literally make sense, since there can’t be more than one amongst x, y, z which is equal to T; the more careful formulation would run: “for all x1 , x2 , x3 , f (x1 , x2 , x3 ) = T iff for exactly two i ∈ {1, 2, 3}, xi = T.) For this example, the sequence z1 , z2 , z3 is of the type explicitly treated in the proof, since the T values come before the F values. Suppose instead, our attention had landed on the choice T, F, T, of z1 , z2 , z3 for which f (z1 , z2 , z3 ) = T; in this case, we have an appropriately permuted form of the previous solution: x= f (x, f (x, x, x), x). We turn now to the first of two respects in which Corollary 9.12.4 understates what its proof establishes. This is the matter of uniformity. Imagine a language with three singulary connectives O1 , O2 , and O3 , and a logic on this language in which every for formula A which is either a propositional variable or a formula of the form O1 B, A is synonymous with O3 A, and for every formula A of the form O2 B, A is synonymous with O3 O3 A. Then, since every formula not of the forms listed here for A is already of the form 03 B for some B, the connective O3 is universally representative in this logic. However, it is not uniformly so, since whether we have to prefix one, two, or zero occurrences of O3 to obtain an equivalent O3 -formula depends on the form of the given formula. (One could also consider cases in which transformations other than prefixing are involved.) Now the point about the equivalents furnished by invoking 9.12.1 is that they do not in depend in this way on the form of the given formula A. For instance, if we consider the class of valuations over which the ternary truth-function f mentioned in Example 9.12.5 is associated with a three-place connective #, then according to the consequence relation thereby determined, we have, for all formulas A, A #(A, A, #(A, A, A)). Similarly, for the case of ∧ in CL mentioned in 9.11.1(iii), we have A CL A ∧ A for all A, regardless of what form A takes. This is also an application of 9.12.1, since CL = Log(BV ), and over BV, ∧ is associated with a non-constant truth-function (called ∧b in 3.14) falling under case (2) in the proof thereof (the ‘idempotent case’ we might say). What we have, then, for the connectives of a meeting the conditions of 9.12.4 is not only that they are universally representative all by themselves, but that they are uniformly universally representative all by themselves.

9.1. UNIVERSALLY REPRESENTATIVE CLASSES OF FORMULAS 1293 The availability of a uniform scheme attesting to the universal representativeness of a connective in a logic is an immediate consequence of its universal representativeness simpliciter in that logic together with closure of the logic under Uniform Substitution, which is why it was a feature of all the examples we have given of universal representativeness prior to the artificial example of the preceding paragraph. For given such closure (or substitution-invariance, as we call the corresponding property for consequence relations), we need only ask what form a #-formula synonymous with some propositional variable is to take, when is universally representative, and then use this form, substituting any formula A for that variable, to obtain a uniform scheme. For the O1 ,O2 ,O3 example of the previous paragraph, for instance, if we close the logic there envisaged under Uniform Substitution, we note that, since it was stipulated that a propositional variable is synonymous with the result of prefixing a single occurrence of O3 , in the envisaged extension of that logic, this will be so for every formula. Remark 9.12.6 There is a related point about uniformity in the statement of 9.12.1 for truth-functions themselves. The claim was not just that for a non-constant n-ary truth-function f and truth-value x there are (compositionally) derived singulary truth-functions g1 , . . . , gn , for which x = f (g1 (x), . . . , gn (x)). Such a formulation allows the gi to depend on the choice of x ∈ {T, F}, and would be utterly trivial (given the nonconstancy of f ). Rather, the claim was that g1 , . . . , gn can be chosen on the basis of f alone, and will verify: x = f (g1 (x), . . . , gn (x)) for x = T and for x = F. For further practice with the effects of Uniform Substitution on universal representativeness, we include: Exercise 9.12.7 Suppose for two connectives #1 and #2 in some logic closed under Uniform Substitution, the union of the class of #1 -formulas with the class of #2 -formulas is a universally representative class. Show that in that case one or other of these classes is universally representative (so that either #1 or #2 is a universally representative connective in this logic). There is a second respect in which the full strength of 9.12.1 has not been extracted in Corollaries 9.12.3, 9.12.4. Our discussion of universally representative connectives has taken the connectives at issue to be fundamental (primitive) connectives of the languages of the logics concerned. But we can apply the concept of universal representativeness also to compositionally derived connectives (indeed to algebraically derived connectives, though without much interest, in this case). For example, consider the derived ternary connective # of the language of CL where #(A, B, C) is A ∧ (B → ¬C). This connective meets the conditions of 9.12.4, and Theorem 9.12.1 again applies since the f spoken of there was any non-constant truth-function; the function associated with this choice of # over BV is such a truth-function. Thus this derived connective is universally representative in CL. Note that this claim would more usually be expressed in some such terms as the following: every formula is classically equivalent to some formula of the form A ∧ (B → ¬C). 9.12.3 gives us a stronger result, namely that the desired equivalent can always be obtained by starting

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with the formula, D, say, for which we want such an equivalent, and forming the formula D1 ∧ (D2 → ¬D3 ) in which each Di is either D itself or else is D ∧ (D → ¬D). Exercise 9.12.8 (i) State for i = 1, 2, 3 exactly which form Di should take in the case of the example just given. (ii) Find a logic in which each of two connectives is universally representative but some connective compositionally derived from them is not. (Hint: Examples can be found in CL.) We have seen many examples of connectives universally representative all by themselves in various logics, but have yet to identify a case in which connective is universally representative but not so all by itself. Exercise 9.12.9 Take the smallest normal modal logic containing all formulas of the form A ↔ A. Would it be correct to say that is universally representative but not universally representative all by itself in this logic? (Suggestion: if in difficulty, consult the proof of 9.24.6 (p. 1314) after verifying that the logic described here is none other than KBc .) Without wanting to give away any more of the solution to the above exercise, we give a simple illustration of the possibility under consideration. Example 9.12.10 Consider the class of Kripke frames (U, R) in which each u ∈ U has exactly one t ∈ U for which tRu and exactly one v ∈ U for which uRv. (This is a bi-directional version of Example 2.22.9, p. 288.) We call such t, v, the predecessor and the successor of u, respectively. Interpret a bimodal language whose two -operators we write as Y and T (mnemonically: Yesterday and Tomorrow – note the title of Segerberg [1967]) in models on such frames by deeming Y A true at a point when A is true at its predecessor and T A true at a point when A is true at itself successor. In virtue of the equivalences TY A ↔ A

and

Y T A ↔ A,

and the fact that their provability secures the synonymy of their leftand right-hand sides, each of Y , T , is universally representative, so to show that T (for example, though a parallel argument would go through for Y ) is not universally representative we need only check that q is not equivalent to any formula T A(q) with A(q) constructed from q using only the connective T : but since any such A(q) has the form T k q for some k, this is clear on consideration of a model based on the integers, in which we can always verify q at a point and falsify all such T A(q) there. In fact it is easy to see that there is no Y -free formula A(q), allowing arbitrary boolean connectives in its construction as well as T , which is equivalent to q, by a variant of Lemma 2.22.4 (on generated submodels: p. 285) tailored to the bimodal – indeed, tense-logical – case with two accessibility relations, one for each of the two -operators, T and Y , (U, R, R−1 , V ).

9.1. UNIVERSALLY REPRESENTATIVE CLASSES OF FORMULAS 1295 Note that the inset equivalence in the Example 9.12.10 means that Y and T are mutually inverse – each is an inverse (both a left and right inverse) of the other, in the logic concerned; see the discussion following 4.22.23. Exercise 9.12.11 (i) Modify the above example so as to have T Y A equivalent to A (for all A), but Y T A not (in general) equivalent to A. (Use the more familiar tense-logical notation “F”/“G” (for T ), “P”/“H” (for Y ) if desired.) (ii) Show that in linear logic (say, for definiteness, the system CLL of 2.33), the multiplicative conjunction connective ◦ is universally representative, but not so all by itself.

9.13

Are There Conjunctive (Disjunctive, etc.) Propositions?

The universal representativeness of such connectives as conjunction and disjunction (across a broad spectrum of logics) is a familiar source of potential embarrassment for proponents of theories of propositions, facts, and states-of-affairs, who want to regard these entities as non-linguistic and yet still non-trivially partitionable into ‘conjunctive’, ‘disjunctive’, ‘negative’ (etc.) propositions, facts, or states-of-affairs. We consider a proposed account of genuine conjunctivity (genuine disjunctivity, etc.) to be trivial, then, if has the effect either of ruling every sentence to be conjunctive (resp. disjunctive), or of ruling no sentence to be conjunctive (resp. disjunctive). Let us focus on the illustrative case of whether some statements express distinctively conjunctive propositions, or, as it was just put, whether some sentences (of an interpreted language) are genuinely conjunctive. The existence of conjunctive equivalents for arbitrary statements means that if we regard as conjunctive whatever can be expressed conjunctively, every statement is conjunctive, while (again with a broad range of logics in the background) if we reserve this label for those only expressible in the form of an explicit conjunction, then no statement is conjunctive. In response to such difficulties, Anderson and Belnap ([1975], p.36f.) suggest the following non-trivializing definition of conjunctive proposition: a sentence expresses such a proposition when it is equivalent (according to one’s preferred logic) to some conjunction to neither of whose conjuncts it is again equivalent. In the case of formal languages such as those of concern to us here, we should replace sentence by formula; we may also want to replace the talk of equivalence with talk of synonymy. The upshot of Anderson and Belnap’s proposal is then, as they note, that the genuinely conjunctive formulas, by the lights of a given logic, are those which are meet-reducible (i.e., not meet-irreducible in the sense explained at 0.13.9(ii), p. 11) in the Lindenbaum algebra of that logic – or, strictly, in the lattice reduct of that algebra. There is also the promise that by switching to a relevant logic and in particular to the entailment system E being defended in Anderson and Belnap [1975], the prospects of not having every formula count as genuinely conjunctive are improved – as indeed they are, to some extent (see 9.13.1). With this criterion of conjunctivity in force, we see that the A∧A representation of A, from 9.11.1(iii), fails to show that (arbitrary) A expresses a conjunctive proposition in the above sense, since though this is a conjunctive formula synonymous with A (in E or any of the relevant logics discussed by Anderson and Belnap, no less than in IL or CL), A stands in

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this relation with one—in fact both—of the displayed conjuncts. If the truth constant T mentioned in 2.33 is present, the synonymy of A with A ∧ T likewise fails this part of the test, though here only on one of the conjuncts. (Recall that we use the symbols “T ” and “t” for the Church and Ackermann truth constants each of which is a partial analogue of .) Another example in the same vein – if one thinks of T as a disjunction of all formulas – is given by the absorption laws, which hold in E and render equivalent any formula A and A ∧ (A ∨ B) for any B: but again since A is equivalent to the first conjunct, this does not trivialise the notion of genuinely conjunctive formulas when this is understood in terms of meet-irreducibility. We shall be seeing shortly – 9.13.2 – that this project is indeed doomed to triviality nonetheless. (The absorption equivalences are often used to show that we can – in a phrase of J. M. Dunn’s – ‘dummy in’ any new variable not in A, by making that variable be the B of the above absorption equivalence, or the dual form, so that every formula is provably equivalent, even in perfectly respectable relevant logics, to a formula containing as many extraneous variables as you want. This typically does not arise in their multiplicative fragments, however, which have the relevant equivalence property: provably equivalent formulas are constructed from precisely the same variables. See Humberstone and Meyer [2007] and references therein.) For the synonymy of formulas A and B in the Fmla logic E (for “entailment”), which serves as Anderson and Belnap’s preferred logic for this discussion, and concerning which the information in 9.22.1 (see p. 1303) and 9.22.2(ii) below will suffice, we may use as a necessary and sufficient condition for the synonymy of A with B, the equivalence (in E) of A with B, in the sense of: the provability of the two implications A → B and B → A. Since equivalence in E is much weaker than CL-equivalence, the options for the selection of putative equivalents with any given connective as main connective are reduced. Example 9.13.1 In CL every formula A is equivalent to the conjunctive formula (A ∨ B) ∧ (A ∨ ¬B), for any choice of the formula B. Now, moving to E, if we select B as a propositional variable not occurring in A, A is equivalent to neither of the two conjuncts (in view of 2.33.6(iv) from p. 340: the variablesharing condition on provable implications, which holds for E as well as R). But this representation does not show an arbitrary formula A to be genuinely conjunctive in E, since A is no longer guaranteed to be equivalent (in E) to the formula inset above. (If we delete the “∨ ¬B” from the second conjunction, we do have the equivalence, this being the absorption equivalence considered above, but of course we now have A equivalent to one of the conjuncts.) The suggestion is, then, that relevant logic may help where classical – and, we might add, intuitionistic – logic let us down, in making possible an articulation of the intuitive idea of some but not all sentences (formulas) expressing conjunctive propositions of some but not all being ‘genuinely conjunctive’. Unfortunately, as it stands, the idea does not work out:

9.1. UNIVERSALLY REPRESENTATIVE CLASSES OF FORMULAS 1297 Example 9.13.2 Again, choosing a propositional variable pi foreign to an arbitrarily selected formula A, and for definiteness let us suppose that i = 2 (so this variable is q) consider the formula: (A ∨ q) ∧ (A ∨ (q → A)) This formula and A itself are equivalent in E, as one may see most easily by looking at its ‘distributed’ form: A ∨ (q ∧ (q → A)). But for A to be equivalent to one of the conjuncts is for A to be equivalent to A ∨ q or else to A ∨ (q → A). The first option is ruled out as it would imply the provability of q → A in E which would be a violation of the variablesharing requirement 2.33.6(iv). (The result is given there for R, but as E is a sublogic of R, we may appeal to it here; notice in passing that contraction – or Contrac – is used essentially here, without which (q ∧ (q → A)) does not provably imply A.) The second would similarly imply the provability in E of (q → A) → A. Notice that in any logic closed under Uniform Substitution, Modus Ponens, and containing A → A we may substitute A for q in (q → A) → A, leaving the original two occurrences of A undisturbed, since q does not occur in A, to infer that A is provable by Modus Ponens from the resulting (A → A) → A and the assumed provable A → A. Thus only the theorems of such a logic would have a chance of expressing conjunctive propositions by the lights of Anderson and Belnap’s suggestion. But in the case of the logic they themselves have in mind, namely E, the situation is even more unfortunate since E has no theorems of the form (pi → A) → A with pi foreign to A. (Indeed even the stronger logic R no theorems of the form (q → A) → B when q does not occur in either A or B, whence the present claim follows a fortiori for the special case that B is A.) A dual form of this example shows that arbitrary A is equivalent in E to a disjunction to neither of whose disjuncts it is equivalent, namely: (A ∧ q) ∨ (A ∧ ¬(A → q)), with q chosen as before (or replaced by the first variable not occurring A). A simple variation on Anderson and Belnap’s criterion for genuine conjunctivity is to classify A as genuinely conjunctive (or as expressing a conjunctive proposition) when A is equivalent to some conjunction B ∧ C to neither of whose conjuncts it is equivalent, and in which the conjuncts B and C have no propositional variables in common. (Cf. the role of such variable-disjoint conjunctions in 5.23.) This criterion yields a non-trivial division into conjunctive and non-conjunctive even if the logic in terms of which equivalence is understood is classical logic. For example, the formulas p and p ∧ q are respectively non-conjunctive and conjunctive. The latter is already exhibited as B ∧ C with B, C meeting the conditions required. If such B, C exist for the former, we may assume without loss of generality that p occurs in B and not C, writing therefore: p ↔ (B(p) ∧ C). (p cannot occur in both since we assume B, C, share no propositional variables; it must occur in at least one, to give an equivalent to p.) Thus p → C must

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be provable and so (substituting a tautology for p) C must be also. In CL this gives the conclusion that B(p) ∧ C is equivalent to B(p), which is therefore itself equivalent to p, contrary to the hypothesis that neither B nor C should be equivalent to p in the conjunctive representation B ∧ C of p. However, Millertype examples (Miller [1974], [1976], [2000b]) show that when this criterion is used to determine the genuine conjunctiveness of formulas in an interpreted language, the determination is sensitive to the stock of atomic formulas and is not invariant under translation into intertranslatable languages. For this reason, the present proposal does not deliver a language-independent notion of conjunctive proposition. Another variation on the Anderson–Belnap proposal is to de-trivialize the conjunctive/non-conjunctive distinction it underwrites by fixing a limit on the number of propositional variables available. Instead of identifying a formula as non-conjunctive when it is meet-irreducible in the Tarski–Lindenbaum algebra whose generators are the synonymy classes of all the sentence letters, we consider the n-variable case with generators [p1 ],. . . ,[pn ], where [A] is the equivalence class (or more generally, synonymy class) of the formula A, relative to the logic under consideration. Jankov [1969] calls such formulas conjunctively indecomposable in the (expansion of the) lattice concerned. For example, when n = 2 and the logic is CL, we can see these elements in Figure 2.13a (on p. 225) as the top element and the four elements of the next row down. The remaining 11 elements are meet-irreducible. Jankov proves a few results about such formulas in various (expansions of) lattices, including alongside those corresponding to CL, the cases of ML, IL, and their ∨-free fragments. (Also considered are the lattices with the partial order not defined, as here, to relate [A] to [B] when A provably implies B in the logic in question, but also that defined to hold when B can be derived from A by means of Modus Ponens and the theorems of the logic in question – which amounts to the previous case for the logics covered – but allowing also Uniform Substitution to apply to A in the derivation.) How much interest we should take in such results depends on how worried we are by the fact that the equivalence class of a single formula changes in status from meet-irreducible to meet-reducible as we pass from a given algebra to one with generators [p1 ],. . . ,[pn ] to that generated by [p1 ],. . . ,[pn+1 ]. Of course we presume the formula concerned has all its variables amongst [p1 ],. . . ,[pn ], and perhaps matters can be made to seem less arbitrary if these are chosen to be precisely the formulas in A, or again those occurring essentially in A (i.e., in every formula equivalent to A, according to the logic chosen). Again, dual considerations apply in the case of disjunctiveness as join-reducibility in these various finitely generated lattices – finitely generated but not necessarily finite, since Jankov’s discussion includes logics which are not locally finite. We turn to the case of ‘negative’ propositions. (The case of genuine disjunctivity will be touched on later.) We have already occasion to contrast the equivalence of ¬¬A with A in CL to show that ¬ is universally representative in CL with the unavailability of this equivalence in IL. Thus, if “expressing a negative proposition” is understood as “being synonymous with some negated formula”, then every proposition is negative by the lights of CL. There is, one can’t help noticing, a lot of popular talk of it being ‘impossible to prove a negative (claim)’ and so on; what someone saying this usually has in mind (and we need not address its merits here) would be more accurately put by saying that it is impossible to prove a negative – i.e., negated – existential claim.

9.1. UNIVERSALLY REPRESENTATIVE CLASSES OF FORMULAS 1299 Now, the fact that the particular equivalence in question – A A case – is not available does not of course show that there is no other way of finding a formula ¬B equivalent in IL to an arbitrarily selected formula A; hence, for all we know so far, IL may be in the same boat as CL in trivializing the negative/nonnegative proposition distinction. This is not the case, however: Observation 9.13.3 The connective ¬ is not universally representative in IL. Proof. We saw already in 4.21.1(ii), p. 540, that since negated formulas are all IL-equivalent to their own double negations (2.32.1(iii): the Law of Triple Negation), if p (for example) were IL-equivalent to some formula ¬B, it would be IL-equivalent to ¬¬p, which is not the case. (In 4.21.1(ii) we wrote B as #p to emphasize the non-conservative effect of adding any such # as a right inverse for intuitionistic ¬: a fortiori there is no derived connective already available in IL which would satisfy the equivalence #¬p IL p.) Digression. The reference in the above proof to 4.21.1(ii), in which the existence of a right inverse for intuitionistic negation was discussed, raises some terminological questions. In the fist place, the right (and for 4.21.1(i), left) inverse terminology really applies to the operation corresponding to the connective ¬ in the Tarski–Lindenbaum algebras of the logics concerned (IL and extensions, for the present case). Secondly, we know that having a right inverse amounts to being a surjective mapping, so it is not uncommon to speak of universally representative connectives as surjective (again eliding the difference between a connective and the corresponding Tarski–Lindenbaum operation); this raises no difficulty in the case of 1-ary connectives, but the terminology of surjectivity is less clear in the case of functions of more than one argument, as was pointed out in 3.14.18, where the two-place case is addressed, and the required usage of “surjective” to match “universally representative” would be the second of the two uses there distinguished. This terminology would rule an n-ary operation on a set surjective just in case every element of the set is the value of this operation for some n arguments drawn from the set. An algebra A each of whose fundamental operations is in this sense is surjective, is called full as is any matrix (A, D) based on it, in Applebee and Pahi [1971] and Pahi [1971a]. (In fact the authors use the phrase “full model” rather than “full matrix” – which is used in Pahi [1972b] – for the latter. The phrase “full model” is defined differently in Font and Jansana [1996], p. 31.) We return to Pahi’s ideas in 9.27. End of Digression. The proof of 9.13.3 exploits a feature of negated formulas in intuitionistic logic – that they are equivalent to their own double negations – which is not shared by arbitrary formulas and thereby shows that the negated formulas cannot make up a universally representative class. Having a special feature of this sort is of interest in its own right as well as for its negative bearing on the question of universal representativeness, and we devote §9.2 to this issue, after a few words on the case of disjunction. The universal representativeness of disjunction in CL and indeed in any reasonable logic parallels that of conjunction in rendering problematic any attempt at a non-trivial partition of the formulas of a formal language or the statements of an interpreted language into those which are ‘genuinely disjunctive’ (“express

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disjunctive propositions”) and those which are not. One additional consideration which may deserve more attention than we shall devote to it here arises out of the ‘plus’ semantics for ∨ of 6.45, 6.46. There is a structural similarity between the clause [∨]Plus provided by that semantics and the clause [Interval ∧ ] suggested in 5.12 for temporal conjunction (‘and then’). In the latter case, we required for the truth of A ∧ B w.r.t. an interval x that x was the concatenation of adjacent intervals y and z verifying respectively A and B. Concatenation (symbolized in 5.12 by ‘+’) is a partial operation, being defined only for pairs of intervals the first of which ‘abuts’ the second, in the sense of being earlier than and adjacent to it. One could borrow some of the structure provided here by the temporal ordering from the notion of similarity between possible worlds that is exploited in Lewis [1973] for the semantics of counterfactuals. More pertinent to the present project would be an analogous notion of similarity between possibilities in the sense of 6.44. For the case of possibilities construed as sets of possible worlds (rather than taken as primitive as in 6.44) such a notion may be extracted from Bigelow [1977], [1979]. Supposing it makes sense to talk of the similarity-distance between worlds in an absolute (rather than world-relative) way, we can think of the modal analogue of a temporal interval as a collection of worlds which contains, whenever it contains w1 and w2 , any world ‘between’ w1 and w2 in terms of this metric of similarity. Call such a collection of worlds a connected region of logical space. (A similar notion is that of a ‘convex proposition’, as deployed in Goldstick and O’Neill [1988]; see also Oddie [1987]. Similar again, though cast in the somewhat extraneous terms of borderline cases and applied to disjunctivity of predicates, is the approach of Sanford [1970b]. (See also Sanford [1981].) A fuller treatment of natural properties in terms of convexity nay be found in Gärdenfors [2000], esp. Chapters 3 and 4.) Just as the union of the set of moments preceding midnight on January 1st, 1800 A.D., with the set of moments following midnight on January 1st, 1956, fails to constitute a temporal interval, so we may suppose that taking all the worlds in which Napoleon Bonaparte visited Egypt together with all those worlds in which Neptune has 6 moons fails to give us a connected region of logical space. It is only in an attenuated sense that all the worlds here collected have ‘something in common’, namely that in each of them either Napoleon visits Egypt or Neptune has 6 moons. Disjunctions of the latter sort make their most frequent (if not their only) occurrence in the discussions and exercises of logicians on the subject of disjunction. Remark 9.13.4 The unnaturalness just alluded to also arises, and has been given some attention, in connection with disjunctive predicates. McCawley [1968], p. 150, notes that while it is perfectly natural to say ‘John and Harry are similar in that both are fools’, it would be odd to say ‘John and Harry are similar in that I met their respective sisters on a prime-numbered day of the month’. In this case, there is no explicitly disjunctive predicate. By contrast, in refuting the view that properties are either possessed essentially by everything possessing them or else possessed accidentally by everything possessing them (sometimes called ‘the Principle of Predication’), Plantinga [1974], p. 61, asks us to consider the property of being prime or prim, possessed essentially by the number 7 and accidentally by a certain Miss Prudence Allworthy. Whatever one says about ‘disjunctive properties’ it is hard to see it as

9.2. SPECIAL CLASSES OF FORMULAS

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respect of resemblance between these two individuals that the predicate is true of each of them. (Lewis [1983] is relevant here, and the development of its ideas – which require us to take seriously the primitive notion of a ‘perfectly natural’ property – in Section III of Langton and Lewis [1998].) For discussion of ‘disjunctive concepts’ in cognitive psychology, the reader is referred to Anisfeld [1968] Bar-Hillel and Eifermann [1970], and Hampton [1988b]. (In fact, it should be added, there is no need to look to artificial examples, like the property of being prim or prime, to make trouble for the Principle of Predication. Consider, for instance, the property of being the same height as the Eiffel Tower, possessed essentially by the Eiffel Tower and accidentally by any other structure of the same height. Note: what is meant here is the property of being the same height as the Eiffel Tower, and not the property of being the height that the Eiffel Tower actually is.) Returning to the case of disjunctive propositions, and transposing the considerations preceding 9.13.4 to the possibilities semantics, let us just suppose that the ‘+’ from [∨]Plus stands for a partial operation, summing possibilities y and z to y + z only when the latter possibility is a connected region. (We take the latter notion as primitive, so as not to have to think of possibilities as sets of worlds; appropriate principles governing it would have to be explored for a thorough treatment of the present topic.) Then we can imagine a kind of ‘connected disjunction’, symbolized, say, by “∨ ”, and subject to the clause that A ∨ B is true w.r.t. to a region x just in case for some y and z verifying respectively A and B, x = y+z. (A related idea may well be that of ‘subsumable disjunction’, as in §5 of Levesque [1988].) Since we are not presenting a formal semantic account in any detail here, it is far from clear precisely what the logic of connected disjunction should look like.

Notes and References for §9.1 More information on the universal representativeness of in normal modal logics than is given here may be found in Humberstone and Williamson [1997]. It is worth noting that there is an obvious notion of universal representativeness in the setting of arbitrary algebras, where the notion would apply to an algebraic operation (fundamental or -compositionally derived) when every element of the algebra is the value of that operation for some suitable arguments. See Adams and Clark [1985], whose Theorem 1.1 is an algebraic version of our 9.12.1. Our 9.13.3 is essentially the same as Theorem 12.6 of Koslow [1992], p. 97.

§9.2 SPECIAL CLASSES OF FORMULAS 9.21

Introduction

We introduce the second of the two concepts to be discussed in this chapter with the intuitive idea that sometimes a logic treats a class of formulas in a special way, in the sense of saying something about all formulas of the given class which it does not say about all formulas of the language in which it is cast. We understand ‘special’, then, rather weakly, since we do not require that what is said of all members of the class in question should be said only of

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its members—merely that it should not be something asserted with complete generality for all formulas of the language. (We return to this point in 9.22.8.) The above idea can be made precise in the following way, which we give first for logics in Fmla. A class Δ of formulas of the language—call it L—of such a logic is special in that logic if there is some formula A ∈ L containing a propositional variable—q, say—such that the results A(B) of substituting B for q uniformly in A, belong to the logic for all B ∈ Δ, but not for all B ∈ L. Think of the formula A = A(q) as representing a context (3.16.1, p. 424); then this context attests to the special treatment accorded by the logic to formulas in Δ: whenever this context is filled by a formula in Δ, the result is provable in the logic, though the same cannot be said for arbitrary formulas. To reformulate the above example concerning intuitionistic negation in Fmla, we see that the Fmla logic IL treats formulas of the form ¬D in a special way (“{C | ∃D C = ¬D} is special in IL”) because taking A as q ↔ ¬¬q, the result of substituting for q any formula of the given form is intuitionistically provable, whereas this is not so for arbitrary formulas. Put in terms of contexts, the point is that filling the blanks in ↔ ¬¬ by a negated formula, though not by just any old formula, yields something in IL. Note that if the logic is itself (like IL) closed under Uniform Substitution, then we can give a slightly simpler equivalent to the above definition: for Δ to be special in the logic is for there to be a formula A(q) such that A(q) is not itself in the logic, but every result of substituting a formula in Δ for q is in the logic. We can now reformulate the above definition for application to the framework Set-Fmla (or Set-Set, for that matter). Recall from the opening of 9.11 that we think of a logic in this framework as simply being a collection of sequents. Letting L be the language of such a logic, we say that Δ ⊆ L is special in the logic if there is some sequent σ in at least one of whose constituent formulas there occurs the variable q, and which such that the results σ(B), of substituting B for q uniformly throughout σ, belong to the logic for all B ∈ Δ, but not for all B ∈ L. The IL example of the previous paragraph can now given the following form, thinking of IL as the collection of sequents provable in the natural deduction system INat (or in the sequent calculus IGen). The class of formulas with ¬ as main connective is special in IL in virtue of the fact that substituting any such formula for q in ¬¬q q yields a sequent in IL, whereas it is not true that the substitution of every formula for q in this sequent yields a sequent in IL. Again, we can exploit closure under Uniform Substitution to put this more simply: while the results of substituting any negated formula for q in ¬¬q q yields a member of IL, this sequent itself is not in IL. We return to cases in which a special class Δ consists of all formulas having a certain connective as main connective in 9.22, after some further words on Uniform Substitution. Of a logic not closed under Uniform Substitution, one is inclined to say that it treats the propositional variables in a special way. With the above definition in force, we can almost say that such logics are precisely the logics in which the class of propositional variables is special. The ‘almost’ can be removed if a suggestion from the end of the Appendix to §1.2 (p. 180) is enforced as a condition on logics. Feeling that closure under Uniform Substitution was too stringent a condition – violated by many examples (see the notes to §1.2, p. 191)


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of what have been proposed as logics – the weaker condition of closure under (uniform) substitution of propositional variables for propositional variables was suggested. Exercise 9.21.1 Show that the above requirement of closure under variablefor-variable substitutions is not equivalent to the requirement of closure under invertible substitutions, where a substitution s (of formulas for variables is) invertible just in case s has a left inverse (0.21.1: p. 19) which is also a substitution. Exercise 9.21.2 Show that in a logic closed under variable-for-variable substitutions, the class of propositional variables is special if and only if the logic is not closed under Uniform Substitution.

9.22

Special Connectives

As with universally representative classes of formulas we were primarily interested in the case in which the class whose representativeness was at issue consisted of precisely the #-formulas for this or that connective #, so here the focus of our attention will be on the question of whether the class of all #-formulas is special in a logic, and when it is, we shall similarly describe the connective # as itself special (in the logic). Thus from 9.21, we have: ¬ is special in IL (on either construal of IL there contemplated); a possible misconception arising out of the terminology ‘special connective’ will be dealt with in 9.22.8. In the meantime, we proceed with further examples. Example 9.22.1 The implicational fragment of R was axiomatized in 2.33 with the axiom-schemata Pref, Perm, Id and Contrac (alias B, C, I and W ) together with the rule Modus Ponens. The implicational fragment of E can be axiomatized with the following restricted version of Perm: [A → ((B1 → B2 ) → C)] → [(B1 → B2 ) → (A → C)]. In other words, only implicational formulas can be ‘permuted out’ to the front of a formula. The unrestricted form of this schema is not derivable. (See Anderson and Belnap [1975], esp. Chapter Four, for more information.) Thus the results of substituting any implicational formula for q in A = (p → (q → r)) → (q → (p → r)) is a theorem of E, even though the formula A is not itself a theorem of E. Accordingly, the connective → is special in E. Part (ii) of the following Remarks is included for background information; it is not strictly needed for following the main development of the present section. Remarks 9.22.2(i) In the stronger logic R, the unrestricted form of Perm (C in the “B, C, I,. . . ” terminology) can be used to show that → is universally representative, since (as noted in 7.23.1) any formula A is equivalent to (A → A) → A in R. (ii) The full system E with ∧, ∨, and ¬ can be axiomatized by subjoining to the implicational basis given in 9.22.1 precisely the same axioms and

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CHAPTER 9. UNIVERSAL AND SPECIAL rules as were given in 2.33 for R. The idea behind this system (see Anderson and Belnap [1975]), already mentioned at the beginning of 7.24, is that whereas R formalizes relevant implication, E formalizes entailment, which is supposed to avoid not only ‘fallacies of relevance’— already weeded out by R—but also ‘fallacies of modality’, of which the paradigm case may be taken to be the fact (noted in (i)) that A provably implies (A → A) → A, which, since its antecedent is provable, can be construed as saying (or at least implying) that it is necessarily the case that A: thus to have such a formula follow from A itself is to obliterate the distinction between truth and necessary truth.

The properties ascribed to → in E by 9.22.1 and in R by 9.22.2 can never both be possessed by a connective in a single logic. Let us work through the reasoning for this claim with the case of → at hand. (The claim implies that → is not universally representative in E, and is not special in R.) If → is universally representative in a logic, this means that every formula is synonymous with an →-formula. There could then be no formula A(q) with every substitution instance A(B) in the logic, where B is of the form C → D and yet some formula A(B0 ) out of the logic. After all B0 is ex hypothesi synonymous with some formula C0 → D0 of the form in question so A(C0 → D0 ) is in the logic while A(B0 ) is not: but this contradicts the supposed synonymy of B0 and C0 → D0 . This argument is of course quite general, having nothing to do with the choice of → for illustrative purposes, or with the choice of Fmla as a logical framework. We state the general result as Observation 9.22.3 No connective universally representative in a logic is special in that logic. We can now re-phrase the proof of 9.13.3 (p. 1299), establishing that negation is not universally representative in intuitionistic logic, in the present terms: the result is an immediate consequence of 9.22.3 and the fact that negation is special in IL. (We emphasize that this is a description of the proof already given of 9.13.3, not an alternative proof. Note that throughout, when we speak of negation in IL, we are of course thinking of ¬, and not the dual intuitionistic negation or the strong negation of 8.22, 8.23, resp.) 9.22.3 says that, for a given connective and logic, the possibilities universally representative and special are mutually exclusive; this raises the question of whether they are jointly exhaustive: must a connective either be universally representative or else special in a given logic. We can put this another way. Now that we know that every universally representative connective is non-special, we may wonder if the converse also holds: that every non-special connective is universally representative. The following structural difference between the two concepts suggests a negative answer. The property of being universally representative in a logic is one which a connective possessing which will continue to possess in any stronger logic (in the same language), since #’s having this property is a matter of certain things being provable. In this respect, it is a property like completeness w.r.t. some notion of validity. On the other hand, the property of being special in a logic is a matter of certain things being provable and certain other things being unprovable. Thus a connective may be special in one logic, non-special in a stronger logic, and special again in some still stronger


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logic. For example, for the chain of normal modal logics (in, for definiteness, Fmla) K ⊆ KD ⊆ S4 ⊆ KT!, is special in K, non-special in KD, special again in S4, and non-special in KT!. Once we have established this, we shall see that the properties universally representative and non-special cannot be coextensive, since as already noted, the former property is preserved on passage to extensions. Our first example of a connective neither special nor universally representative in 9.24 will be that of in KD. The other parts of the claim just made about these various logics (on which see §2.2 for background and 2.21 in particular for the normal modal logics entering into the following discussion) can be established here. The discussion is considerably simplified by the fact that normal modal logics are all closed under Uniform Substitution and are ↔-congruential. Observation 9.22.4 The connective is (i) special in K and (ii) special in any normal modal logic extending K4 if that logic is not an extension of KTc , and (iii) not special in KT!. Proof. (i) To see that is special in K, take A(q) in our definition of special as the formula ⊥ → q. This formula is not provable in K, but the result of substituting any -formula for q turns it into a provable formula. (ii) If S is a normal modal logic extending K4, then we have S B → B for any formula B of the form C, so if S is not an extension of KTc , and does not accordingly contain B → B for every formula B, is special in S. (In terms of our definition, A(q) is q → q.) (iii) In KT! A ↔ A for all A, so is universally representative in KT! and hence, by 9.22.3, not special in KT!. Note that 9.22.4(ii) implies that is special in S4 (as mentioned in the above chain) but also in K4, KD4, and S5, since all these logics extend K4 but not KTc . Remark 9.22.5 We can think of the special treatment given to in the logics just mentioned from a semantic point of view as follows: in every model (W, R, V ) for one of those logics, the truth-sets of -formulas are closed under R, while the same cannot be said for the truth-sets of arbitrary formulas. In 9.24, we shall use reasoning about the Kripke semantics for normal modal logics to establish the non-special nature of in KD. Exercise 9.22.6 Show that is special in KVer. We return to modal logic to continue with the example of a case of being neither special nor universally representative, in 9.24, after a discussion of how ↔ is special in the relevant logic R. We end the present subsection with an exercise parallelling 9.12.6 (p. 1293) in indicating the spuriousness of a putative generalization of the notion of being special in a logic (as did 9.12.6 for the notion of universal representativeness).

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Exercise 9.22.7 We have said that a connective # is special in a logic (in Fmla) if there is some formula A(q) in the language of that logic such that every result A(B) of substituting a #-formula B for q in A(q) is in the logic, while this is not so for arbitrary formulas B. Let us say that # is special in the generalized sense in a logic if there exists a family {Ai (q)}i∈I of formulas (each containing q) of the language of that logic such that for any #-formula B, we have Ai (B) in the logic for some i ∈ I, whereas it is not the case that for an arbitrary formula B, Ai (B) is in the logic for some i ∈ I. (Thus, being special tout court is the case in which I contains only one element.) Show that if a logic is closed under Uniform Substitution, any connective # in the language of that logic which is special in the generalized sense in the logic is special in the logic. We turn to the correction of two possible misapprehensions, the first of them roughly to the effect that special connectives do not need to be what one might call ‘idiosyncratically special’: Remark 9.22.8 Although we require a connective which is special in a logic to form compounds which behave according to the logic in ways which not all formulas behave, we do not require that only such compounds behave in that way. For instance, in arguing that ¬ is special in IL because negated formulas are equivalent to their own double negations, it is not suggested that only negated formulas enjoy this property. Any formula of the form A → ¬B is, for example, also equivalent to its double negation. In this case, it is easy to see that A → ¬B is itself IL-equivalent to a negated formula (to the negation of A ∧ B. Indeed in the present instance, from the interaction between the connective whose specialness is at issue and the context (namely ¬¬ ) used to establish that specialness, it is immediately clear that any formula C equivalent to its double negation is itself equivalent to a negated formula (because ¬¬C is after all a negated formula). But our definition of specialness does not require even that only formulas equivalent to #-compounds exhibit the behaviour in virtue of which # counts as special: simply that not all formulas exhibit that behaviour. Exercise 9.22.9 Find a normal modal logic in the Fmla framework according to which has the property mentioned in the last sentence of 9.22.8 (i.e., there is some A(q) such not A(B) is not provable for all formulas, but is provable if and only if B is equivalent to a -formula). The second misapprehension to be addressed can be illustrated by the following line of thought. Conjunctions (in IL, CL, etc.) provably imply their conjuncts, whereas for example disjunctions do not similarly imply their disjuncts. So there is something special about conjunction (in the logics in question): it forms a compound which has its components as consequences, as is not the case for all connectives – e.g., not for disjunction. This of course is not the intended use of the terminology in play here, for which we have already seen that no universally representative connective (such as ∧ in the logics concerned) can count as special. The observations about conjunction do not single out a property that conjunctive formulas distinctive possess, but rather a relation they distinctively


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stand in to their components. There is a natural and interesting phenomenon here, which we discuss under the heading ‘special relations between formulas’ in 9.25.

9.23

The Biconditional in R

In this subsection we shall see that biconditional ↔ is special in R, and then go on to relate this to the fact that the implicational fragment of this logic (BCIW logic, in other words) is not algebraizable in the sense of Blok and Pigozzi [1989], as was noted, following Blok and Pigozzi, in 2.16.4(iii). We begin with an exercise which will be of considerable utility later on. Exercise 9.23.1 Call a formula A for which A → (A → A) is provable in a logic S a Mingler in S. Show that if S is any extension of BCI logic, closed under Modus Ponens (and Uniform Substitution), then the converse of any implicational theorem of S is a Mingler in S. (Thus the class of converses of theorems of S, like the class of theorems itself in any consistent S, is special in the sense of the start of 9.21. We are mainly interested in special connectives, however.) Recall from 2.33 that the main logic of interest in the relevance tradition (setting aside the E that was a major preoccupation in Anderson and Belnap [1975]), namely R, has a proper extension RM which can be axiomatized by adding the Mingle schema to an axiomatization of R, the content of which is that every formula is a Mingler in the sense of 9.23.1. As the latter exercise shows, while plenty of formulas are Minglers in R, not all are (e.g., take A = p). Unlike that exercise, the following requires contraction: Exercise 9.23.2 Prove the following in BCIW logic: (p → q) → ((q → p) → (p → q)). Use the axiomatic presentation of this logic, or the sequent calculus given in 2.33 – as for linear logic but with Left Contraction – or the natural deduction system illustrated in 7.25.25. (In fact the simplest way to tackle both this exercise and 9.32.1 together is to provide a BCI proof of (q → p) → [(p → q) → ((p → q) → (p → q))]. Then we get 9.23.1 by substituting A for q and B for p, given A → B provable, and Modus Ponens yields the Mingle formula for B → A. To derive the formula in the present exercise, contract on p → q in the consequent here and then permute antecedents.) If the sequent calculus or natural deduction suggestions made for the above exercise are taken, then prior to two final applications of (→ Right)/(→I) we have a proof of the sequent: p → q, q → p p → q. If we are thinking of ↔ as defined, as usual, via A ↔ B = (A → B) ∧ (B → A), then to insert ↔ on the left here – moving now to a specifically sequent calculus setting – we need to use (∧ Left) from 2.33, with the inset sequent above as premiss:


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(p → q) ∧ (q → p), q → p p → q and by (∧ Left) again: (p → q) ∧ (q → p), (p → q) ∧ (q → p) p → q. (Recall from early 7.31 that two possible definitions of ↔ in R, that mentioned above and the multiplicative variant with ◦ for the main ∧, are actually equivalent there, thanks to contraction.) Rewriting the last inset sequent in terms of ↔, we have: p ↔ q, p ↔ q p → q. Similarly, starting with a re-lettered version of our original sequent: q → p, p → q q → p, and following the same path, we have a proof of p ↔ q, p ↔ q q → p. These two sequents have the same left-hand side, so the additive rule (∧ Right) can be applied, giving us our conclusion, rewritten with ↔ on the right: p ↔ q, p ↔ q p ↔ q. As two applications of (→ Right) reveal, this says that any formula with ↔ as main connective is a Mingler in R. Since not every formula enjoys that property, we have proved: Observation 9.23.3 The connective ↔ is special (and therefore not universally representative) in R. Compare this with the following. Observation 9.23.4 The connective ↔ is universally representative (and therefore not special) in RM. Proof. The result follows from the RM-provability for all formulas A, of the formula A ↔ ((A → A) ↔ A). The backward direction of this equivalence is provable in R (indeed, in linear logic), in view of the BCI -provability of ((A → A) → A) → A. For the forward direction, we need both A → ((A → A) → A)

and

A → (A → (A → A).

Again the first of these is already available in BCI logic, while the second requires the Mingle axiom A → (A → A). Prefixing any B twice gives: (B → (B → A)) → (B → (B → (A → A))),


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so in particular taking B as A itself, the antecedent becomes an instance of the Mingle schema and we can detach the consequent, the second of the two formulas inset together above. This contrast between R and RM may call to mind the striking difference in respect of algebraizability of their implicational fragments disclosed in Blok and Pigozzi [1989]. As was reported in 2.16.4(ii)–(iii), the latter is algebraizable while the former is not. Since algebraizability is a matter (2.16.3) of the availability of equivalence formulas, with some formulas (or, in the general case, sets of formulas) δ(A), ε(A), for which any formula A is itself equivalent to the equivalence of δ(A) with ε(A), it might seem obviously jeopardised by logics in which the prototypical equivalential connective ↔ is not universally representative, as with R: if every A is equivalent to an equivalence then the equivalential connective (e.g. ↔) would ipso facto be universally representative. But on closer inspection this line of thought is seriously oversimplified, for the following related reasons. First, we have been discussing logics in Fmla, whereas the issue of algebraizability pertains to consequence relations (or Set-Fmla logics with suitable structural rules). Secondly, the wording “any formula A is itself equivalent to the equivalence of δ(A) and ε(A)” makes a dangerous double use of the terminology of equivalence. The pertinent equivalence of δ(A) with ε(A) involves E, in which E(p, q) may well be something along the lines of {p ↔ q} (or {p → q, q → q}), but the sense in which equivalence must in turn be equivalent to A has nothing to do with E (↔ etc.) but instead with the relation , where is the consequence relation whose algebraizability is at issue. (This same contrast was drawn in the Digression after 2.16.4 as one between sequents ΓA and consequence statements Γ A; see also the end of this subsection.) Thirdly, and finally, the contrast in respect of algebraizability between R and RM concerns only their implicational fragments: as soon as conjunction is added, we have algebraizability in both cases, as mentioned in 2.16.4. (In fact, for the record, everything in this section bearing on this fragment in the case of RM could be said of RM0 instead – i.e., the implicational logic from 2.33 which extends BCIW logic with the Mingle axiom.) We must work harder than this if we are to show how the Mingle axiom makes the difference it does for the pure implicational fragment of R (i.e., BCIW logic, but now thought of as the consequence relation specified immediately below) is not algebraizable, at least with equivalence formulas Δ = {p → q, q → p}. (This apparent proviso will be lifted in 9.23.6–7 below.) We have to rule out the existence of (possibly derived) 1-ary connectives δ, ε, for which all of the following hold, where is the consequence relation associated (as in 1.29) with the axiomatic proof system BCIW, with Modus Ponens as sole rule: p δ(p) → ε(p); p ε(p) → δ(p); δ(p) → ε(p), ε(p) → δ(p) p. Recall from 1.29.10 (p. 168) that a (sort of) deduction theorem for this consequence relation logic runs as follows Γ, A B ⇒ (either Γ B or Γ A → B). Thus we have the following possible cases: (a)

δ(p) → ε(p)

or

(a )

p → (δ(p) → ε(p));


1310 and (b) ε(p) → δ(p) and, (1) (2) (3)

or

(b )

p → (ε(p) → δ(p));

from the last of the three inset -conditions above, (1) or (2) or (3): (ε(p) → δ(p)) → p, (δ(p) → ε(p)) → ((ε(p) → δ(p)) → p), (δ(p) → ε(p)) → p.

(The condition (2) here may seem wrong, since it in effect presents p as implied by the multiplicative conjunction (or fusion) of δ(p) → ε(p) with its converse, instead of their additive conjunction, which is what was required for δ(p) ↔ ε(p); but although the multiplicative and additive conjunctions of a pair of formulas are not in general BCIW -equivalent, as we saw in early 7.31 with the discussion of ↔m , and have already had occasion in this subsection to repeat, when the two conjuncts are implicational converses – of which we shall hear more at the end of 9.25 – this distinction evaporates.) Altogether, we have 12 combinations in all, then, some of which can be discarded immediately as the existence of δ, ε, satisfying them would imply p, contradicting the consistency of BCIW logic. The 12 combinations are listed in Figure 9.23a, with those excluded out on the ground just given crossed out. For example, the two combinations with a and 1 are crossed out because these two conditions yield (by Modus Ponens) p. ab1 a b1 ab 1 a b 1

ab2 a b2 ab 2 a b 2

ab3 a b3 ab 3 a b 3

Figure 9.23a: Some Combinations Quickly Ruled Out

All surviving combinations in Figure 9.23a can be excluded by the fact that for our current choice of , p → (p → p), the Mingle axiom not being Rprovable. In the case of a b2, in view of b (and Perm or C ), we have δ(p) → ε(p) provably implying p, so, in view of a , p is provably equivalent to this formula, which, in view of b, is the converse of a theorem, so by 9.23.1, b is a Mingler and so p, being equivalent to b, is itself a Mingler: which is not the case. The combinations a b3, ab 1 and ab 2 yield to similar (somewhat simpler) arguments. (Note that by “are equivalent” here, is meant “are provably equivalent”, which in the current purely implicational setting means that each provably implies the other, which suffices for their synonymy in view of B and B .) To deal with a b 1 we use a substitution instance of a variant of the first inset formula in 9.23.2, with antecedents commuted, namely: (δ(p) → ε(p)) → ((ε(p) → δ(p)) → (ε(p) → δ(p))). By b and 1, ε(p) → δ(p) is equivalent to p, so we have (δ(p) → ε(p)) → (p → p). But a says that the antecedent of this formula is provably implied by p, so again we get the contrary-to-fact conclusion that the Mingle axiom is provable


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for p. The two remaining cases, a b 2 and a b 3 are ruled out by similar reasoning. Thus we have the following, proved as (part of) Theorem 5.9 in Blok and Pigozzi [1989] by a much quicker argument, involving concepts from abstract algebraic logic not presented in our summary in 2.16 above: Theorem 9.23.5 BCIW logic is not algebraizable with equivalence formulas {p → q, q → p}. Corollary 9.23.6 BCIW logic is not algebraizable. Exercise 9.23.7 Show that the above Corollary does indeed follow from Theorem 9.23.5, by showing that algebraizability with any set E(p,q) of equivalence formulas would imply that {p → q, q → p} could play this role. Remarks 9.23.8(i) In fact it is Coro. 9.23.6 that corresponds to part of Blok– Pigozzi’s Theorem 5.9, the rest of that result using the fact that algebraizability of a consequence relation is inherited by any stronger consequence relation on the same language. It includes, for example, BCI logic as unalgebraizable on the grounds that BCIW logic is. (Here BCI logic is to be understood, as in the BCIW case, as the consequence relation associated with the Fmla logic with Modus Ponens.) (ii) Blok and Pigozzi [1989] assert correctly that any fragment of R including both → and ∧ is in their sense (as explained in 2.16) algebraizable, taking δ(p) as p ∧ (p → p) and ε(p) as p → p, and Δ(p, q) as {p → q, q → p}. The consequence relation, call it R they have in mind takes the axioms of R as consequences of ∅ and the rules Modus Ponens and Adjunction; thus we are dealing with the consequence relation associated à la 1.29 with the axiomatic system for the fragment in question. Blok and Pigozzi begin on p. 49 with a perfectly correct derivation showing that p R (p → p) → (p ∧ (p → p)). We now want, given the choice of Δ: p R (p ∧ (p → p)) → (p → p), and they correctly remark that the formula on the right is R-provable, which by itself suffices for the claim, but instead of noting that, they try and use the “p” on the left to do a Modus Ponens and claim that we thereby get p R (p ∧ (p → p)) Δ p, using the infix notation for Δ, and continue “for the inference in the other direction. . . ”, by which time they have forgotten that the defining equation was p ∧ (p → p) ≈ p → p, and continue as though it had been: p ∧ (p → p) ≈ p

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CHAPTER 9. UNIVERSAL AND SPECIAL and head for what the “other direction” would be if this were the equation in question, concluding that: (p ∧ (p → p)) Δ p R p. But this doesn’t even hold for CL , and is in any case beside the point since the wrong defining equation – specifically, the wrong ε(p) – has been inserted into the discussion by this stage. These inadvertent formulations notwithstanding, the claimed result being easily rescued. Returning to correct the fourth last claim inset above, but (since we have ∧ and → available), writing ↔ for Δ and writing out the “” form, we arrive at what is needed for algebraizability: p R (p ∧ (p → p)) ↔ (p → p). The reader will have no difficulty in giving a correct demonstration that this condition is satisfied.

When A R B this can happen in either of two ways. First, it may be that R A → B (which always implies that A R B). In this case we convey this further information by writing A → R B. Alternatively, it may be that this ‘provable implication condition’ is not satisfied, in which case we will superscript a crossed-out arrow in place of the arrow. That is, we write “A → R B” to mean “A R B but not A → R B”. Bringing these more informative versions of simple R -statements to bear on the last inset condition in 9.23.8(ii), what we have is, in the forward direction: p → R (p ∧ (p → p)) ↔ (p → p), whereas for the backward condition, we have instead: (p ∧ (p → p)) ↔ (p → p) → R p. Indeed we know a priori relative to the considerations about the biconditional being special in R that we could not have had both the forward and the backward directions holding in the → R form, since this would have meant that we did indeed have every formula provably equivalent (provably ↔-equivalent, to be absolutely explicit) to a ↔-formula, which would have been inconsistent with the fact that the latter formulas are special. There is no corresponding inconsistency with the weaker claim that every formula is R -equivalent to such a formula. The truth of the latter claim does not, by contrast, guarantee that the formulas involved are synonymous. (For example, we know that R not monothetic, whereas all R-theorems are R -equivalent.) This is just the point already made with our complaint that “the wording ‘any formula A is itself equivalent to the equivalence of δ(A) and εA’ makes a dangerous double use of the terminology of equivalence.”

9.24

Connectives Neither Universally Representative Nor Special

Examination of the proof of 9.22.4(i), showing to be special in K, shows the argument to depend on the unprovability of ¬⊥ in that logic, since this formula is provable in a normal modal logic if and only if ⊥ → C is provable, for all


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formulas C, whereas the ‘special’ nature of in K relied on the fact that this was so only for formulas C subject to the restriction that they be of the form B for some B. The smallest extension of K in which ¬⊥ is, by contrast, provable, is KD, and we shall show that is non-special in this logic. The discussion will not quite be self-contained, since we shall need to supplement the completeness results for KD obtainable by the means of §2.2 by a further such result. From 2.21.8(v) on p. 281, and the discussion of 2.22, we have the result that KD is determined by the class of serial frames, and appealing to 2.22.4 (from p. 285), we get the result that any non-theorem of KD is false at some point in a model generated by that point, this being a model on a serial frame. What we need here, however, is that such a model can be converted into a model M = (W, R, V ) also falsifying the given formula, and whose frame satisfies the ‘unique predecessor’ condition (*), mentioned above (p. 563) between 4.22.20 and 4.22.21: (*) For all x, y, z ∈ W , if xRz & yRz, then x = y This is not a feature of point-generated submodels of the canonical model for KD, and we shall not here give the proof of the following result, for which see the discussion of ‘unravelled frames’ in Sahlqvist [1975]; the unravelling of a point-generated serial frame satisfies (*) and any model on a point-generated frame gives a model verifying the same formulas as the original. From these claims, the result follows. Theorem 9.24.1 KD is determined by the class of all serial frames satisfying (*). What we want to show is that there is no formula A(q) in the language of KD such that A(B) is not provable in KD for all formulas B, but A(B) is provable for all formulas B of the form C. In view of closure under Uniform Substitution, we can replace the reference here to A(B) not being provable for all formulas B by a reference simply to the unprovability of A(q) itself, and instead of speaking of the provability of A(B) for every -formula B, we may equivalently speak of the provability of A(q). Thus to establish that is not special in KD it will be sufficient to show that for any formula A(q), if A(q) ∈ / KD, then A(q) ∈ / KD. Recall that A(q) is simply the result of substituting q for q in A, alias A(q). We begin with a Lemma recording the effect of such substitutions. (Recall that “q” is an abbreviation for “p2 ”.) Lemma 9.24.2 Let (W, R) be a serial frame satisfying (*) above, and M = (W, R, V ) and M = (W, R, V ) be two models on this frame satisfying: V (pi ) = V (pi ) for all i = 2 while V (q) = {z ∈ W | ∃y ∈ W yRz & y ∈ V (q)}. Then for all y ∈ W , M |=y B(q) if and only if M |=y B(q), for any formula B = B(q) and the result, B(q), of substituting q uniformly for q in B. Proof. By induction on the complexity of B. The inductive cases are straightforward, as are all those subcases of the basis case in which B = pi for i = 2. We examine the case of B = p2 = q.

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Suppose M |=y q but M |=y q (q being B(q) for B = q). Then for some z ∈ W such that yRz, M |=z q. This contradicts the stipulation that V (q) contains all R-successors of points in V (q), y being such a point. Conversely, suppose M |=y q but M |=y q. Then for all z ∈ R(y), M |=z q. Fix on one such z (guaranteed to exist since R(y) = ∅, by seriality); z ∈ V (q), so for some y ∈ W , y Rz and y ∈ V (q), by the stipulation on V . Yet, since M |=y q, y ∈ / V (q), so y = y . This contradicts (*), since z can only have one R-predecessor. The use of the substitution of q for q here and in the proof of 9.24.3 should be compared with the more general ‘inner cancellation’ rule of Humberstone and Williamson [1997], p. 47, which involves the substitution of pi for all pi . These are both restricted substitutions in the sense explained in 9.27 below. Theorem 9.24.3 is not special in KD. Proof. In view of the remarks preceding the 9.24.2, it suffices to show that, where A(q) is any formula unprovable in KD, containing q, and A(q) is the result of substituting q for q throughout A(q), A(q) is not provable in KD. So suppose KD A(q). Then by 9.24.1, we have M |=y A(q) for some model whose frame is serial and satisfies (*). Consider the model M related to M as in the hypotheses of Lemma 9.24.2; the Lemma then allows us to conclude that M |=y A(q). But since M is a model on a serial frame, KD A(q). Now, since is special in KD4 (9.22.4(ii)), is not universally representative in KD4, and hence is not universally representative in the weaker logic KD. Thus we have our first example of the phenomenon promised by the title of the present subsection: Example 9.24.4 is neither universally representative nor special in KD. Exercise 9.24.5 (i) Where would an attempted proof along the lines of that of 9.24.3 break down as applied to the system KD4 instead of KD? (ii) Consider the tense logic extending the system Kt described in 2.22 by the axiom schemes (analogous to D) GA → FA and HA → PA. Show that, by contrast with the monomodal case of KD, in this logic G and H are both special. (Hint (for G): consider A(q) as GPq → q. This formula was mentioned in 4.22.25(ii), p. 566.) (iii) Returning, inspired by the example in (ii), to the case of monomodal logic, show that in any normal modal logic extending KB but not KBc , is special. For those with a particular interest in modal logic, we note the following strengthening of 9.24.5(iii): Observation 9.24.6 For any normal modal logic S ⊇ KB, is special in S iff not S ⊇ KBc .


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Proof. The ‘if’ direction is 9.24.5(iii). For the ‘only if’ direction, suppose S ⊇ KB and S ⊇ KBc . The first of these conditions is actually redundant, since KBc ⊇ KB; in fact KBc = KBD!. More useful for present purposes is the equation KBc = K + A ↔ A. (See below.) Thus for an S meeting our conditions, every formula is synonymous with its double necessitation, and is accordingly universally representative (all by itself, in fact), and so not special (9.22.3). As for the fact cited here, that KBc = K + A ↔ A, the reader is invited to verify this. (If in doubt, consult Proposition 2.4 of Humberstone [2009a] and surrounding discussion.) It is worth pausing here to note the very different behaviour of our two properties of universal representativeness and specialness as far as logics and their extensions are concerned. Whereas, at least for substitution-invariant logics, universal representativeness for a connective is passed from a logic to any extension thereof, a connective’s being special in a logic comes and goes as we consider stronger logics, as was already mentioned before 9.22.4: is special in K, not special in KD K, special again in KD4 KD, and then non-special again in KT! KD4 (in this last case because is universally representative in KT!). The second illustration we offer of the possibility of a connective’s being neither special nor universally representative in a logic is that of the zero-place ⊥ in minimal logic. Looking back at 8.32.4 (p. 1259) with the present section’s definition of ‘special’ in mind reveals that ⊥ is not special in ML: there is only one formula whose main connective is ⊥, and that is the formula ⊥ itself. But any ML-provable sequent remains provable on uniformly replacing ⊥ by any formula. Next, note that for a zero-place connective to be universally representative in a logic would be for every formula to be synonymous with that formula in the logic, which would imply that all formulas are synonymous with each other, as is certainly not the case in ML. Hence: Example 9.24.7 ⊥ is neither universally representative nor special in ML. By contrast, of course, ⊥ is special in IL and CL, since every formula is a consequence of ⊥ (but not of q) according to these logics. This fact, most naturally stated in Set-Fmla, also holds for Fmla, since we can use implication (→) to express it. We mention this last point because, interestingly enough, ⊥ is not special in the {↔, ⊥}-fragment of CL in Fmla: see 7.31.11. Exercise 9.24.8 (i) Could ¬ serve as an example of a connective which is neither universally representative nor special in ML (in a formulation thereof taking ¬ as a primitive connective)? (Hint: look back at 9.13.3 on p. 1299, with 8.32.9(i) from p. 1263 in mind. Solution follows below, in any case.) (ii) Recall from early 7.31 that there are at least two candidate biconditional connectives – there called the additive and multiplicative biconditionals (subscript a and m) – available in linear logic, neither stronger than the other in the absence of contraction, and equivalent in its presence (as in R):

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CHAPTER 9. UNIVERSAL AND SPECIAL A ↔a B = (A → B) ∧ (B → A) A ↔m B = (A → B) ◦ (B → A). Using what was learnt about the biconditional in R in 9.23.3, can we add either of these connectives, as they behave according to linear logic (CLL, say) to our list of examples of connectives neither universally representative nor special in a logic? (This should perhaps be labelled “Problem” rather than “Exercise”, in that the author does not have an answer in mind.)

Though it perhaps simplifies matters, there is really no need for the parenthetical restriction (to the case in which ¬ is primitive) in part (i) of the last exercise, since one may ask of a defined connective whether or not it is universally representative or special, as indeed part (ii) thereof does. Returning to part (i), all that matters for such purposes is the equivalence, where = M L : ¬A A → ⊥. The solution to 9.24.8(i) is that since the law of triple negation holds for ¬ in minimal logic though in general a formula is not equivalent to its double negation, the situation is exactly as in IL: ¬ is special and therefore not universally representative. (Minimal negation is therefore not a candidate illustration of something, to use the words of the exercise, neither universally representative nor special.) The law of triple negation, unpacked in accordance with the equivalence inset above, becomes: A → ⊥ ((A → ⊥) → ⊥) → ⊥ Clearly this equivalence holds for all A, with = M L , just in case the sequent p → ⊥ ((p → ⊥) → ⊥) → ⊥ and its converse are ML-provable. This in turn holds just in case for all formulas B, replacing ⊥ by B in these sequents gives us provable sequents of ML (since ⊥ is ‘non-special’ in ML). Selecting without loss of generality to do duty for all such B a new propositional variable, q, we can easily verify that this is so by constructing with the aid of (→I) and (→E), proofs of the sequents (i) and (ii) (i)

p → q ((p → q) → q) → q

(ii )

((p → q) → q) → q p → q.

(Alternatively, use the Kripke semantics for IL; since neither ¬ nor ⊥ is involved, one might just as well say “the Kripke semantics for Positive Logic”. In fact the equivalence provided by (i) and (ii) – the Law of Triple Consequents – is already available in BCI logic, as was mentioned in 2.13.20(ii), p. 239.) This concludes our discussion of 9.24.8(i); we pass to another application of the ‘non-specialness’ of ⊥ in ML, namely to the question – to be precise, question 8.32.9(iii) from p. 1263 – of whether Glivenko’s Theorem holds for ML. The following observation is formulated so as to be neutral as to which of ⊥, ¬ (or both) is take as primitive. Observation 9.24.9 For any formula A not constructed with the aid of either ¬ or ⊥, if M L ¬¬A, then M L A.


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Proof. Pick a propositional variable not occurring in A; for present purposes let us assume that q is such a variable. By the previous discussion, we have M L ¬¬A iff M L (A → ⊥) → ⊥ iff M L (A → q) → q. Hence, substituting A uniformly for q, we get that if M L ¬¬A then M L (A → A) → A; thus, since M L (A → A), we have M L A. ML is accordingly very different from IL in respect of provable formulas of the form ¬¬A. In particular (à propos of Exercise 8.32.9(iii), p. 1263): Remark 9.24.10 As a Corollary to 9.24.9, we can see – though we already saw this at 8.33.11 (p. 1271) – that Glivenko’s Theorem fails for Minimal Logic. In 8.21.2, on p. 1215 (and in 2.32), we took Glivenko’s Theorem to be the claim for IL whenever Γ CL A, we have ¬¬Γ IL ¬¬A. (As usual, ¬¬Γ is simply {¬¬C | C ∈ Γ}. The counterexamples can be redescribed as refuting Glivenko’s Theorem in the sense of 8.21.3, p. 1216) We can exploit 9.24.9 to get counterexamples to the result of replacing “IL” by “ML” here, when Γ = ∅, by choosing a ¬-free (and ⊥-free) A which is classically but not intuitionistically (and therefore not minimally) provable; e.g., (p → r) ∨ p. (By contrast, the example in 8.33.11 made essential use of negation inside the A concerned.) Is the restriction to ¬, ⊥-free formulas essential? The above argument—the proof of 9.24.9, that is—won’t work without it, since the replacement (e.g.) of subformulas ¬B of the formula A for which it is assumed that M L ¬¬A will have to be by B → q (with q assumed, as above, foreign to A). Let A* be the result of making such replacements. Then instead of asserting that M L (A → q) → q, on the basis of the given assumption, we have rather: M L (A* → q) → q. Now we cannot continue the earlier reasoning. We might hope to substitute A for q, as before, but the result of so doing is not M L (A* → A) → A, whatever help that might have been, since the (uniform!) substitution of A for q will disrupt A*, as the latter contains (when A was not ¬-free) q. For the same reason, we cannot even substitute A* for q to obtain M L (A* → A*) → A* (and hence M L A*), since the first of the three occurrences of A* here should be replaced by the formula obtained from A* by substitution of A* for q. It is not hard to see that not only does any argument along the above lines break down, but that the result (M L ¬¬A ⇒ M L A) itself does not hold for arbitrary A: Example 9.24.11 A simple illustration is provided taking A = p ∨¬p. Here M L ¬¬A, since M L ((p ∨ (p → q)) → q) → q. (Replace the antecedent by the ML-equivalent (p → q) ∧ ((p → q) → q) to see this easily.) Clearly, for this case, we do not have M L A. Exercise 9.24.12 Show that for another example of M L ¬¬A without M L A, we may take A = (¬p → p) → p. We note that the corresponding ‘new propositional variable’ versions of the A and ¬¬A of the above Exercise are respectively Peirce’s Law and Wajsberg’s Law. (On the latter, see 7.21.5 and the proof of 7.21.11.)


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Exercise 9.24.13 Adapt the proof of 9.24.9 to show that if there are no occurrences of ¬ or ⊥ in A or in (any formula in) Γ, then Γ M L ¬¬A ⇒ Γ M L A. By a similar argument, we have: Observation 9.24.14 For any ⊥, ¬-free A, B: ¬¬A M L ¬¬B ⇒ A M L B. Proof. Again taking q as a variable foreign to A, B, we have that ¬¬A M L ¬¬B implies (A → q) → q M L (B → q) → q, so substituting B for q we get: (A → B) → B M L (B → B) → B, and hence (A → B) → B M L B; so, since A M L (A → B) → B, we have A M L B.

9.25

Special Relations Between Formulas

In 9.11 we defined a class of formulas to be universally representative in a logic when every formula was, according to that logic, synonymous with one in the class. This is the n = 1 case of the following more general notion. We can call an n-ary relation amongst formulas universally representative whenever any n formulas A1 , . . . , An are respectively synonymous with formulas B1 , . . . , Bn which stand in the relation in question. We also defined a class Δ of formulas of the language of some logic to be special (in or according to that logic) if there is some sequent σ the result, σ(B), of substituting a formula B in which for each occurrence of a certain propositional variable, was always provable in the logic if B ∈ Δ, though not provable for all B. This notion too can be generalized. We can say that an n-ary relation R between formulas of some language is special according to a logic (in that language) if there exists a sequent σ containing propositional variables q1 , . . . , qn such that not every result, σ(B1 , . . . , Bn ) of uniformly substituting formulas B1 , . . . , Bn for q1 , . . . , qn (resp.) is provable in the logic, though whenever B1 , . . . , Bn ∈ R, we do have σ(B1 , . . . , Bn ) provable. The same considerations as led to 9.22.3 give Observation 9.25.1 No relation universally representative in a logic is special in that logic. Proof. Suppose (n-ary) R is universally representative and also special, in some logic. Since R is special, there exists a sequent σ(q1 , . . . , qn ) with σ(B1 , . . . , Bn ) provable whenever B1 , . . . , Bn ∈ R, yet there exist formulas C1 , . . . , Cn for which σ(C1 , . . . , Cn ) is not provable. This contradicts the hypothesis that R is universally representative, since C1 , . . . , Cn must then be respectively synonymous with formulas D1 , . . . , Dn , say, with D1 , . . . , Dn ∈ R, implying that σ(D1 , . . . , Dn ) is provable, and hence, by the synonymy in question, that the sequent σ(C1 , . . . , Cn ) is. For the remainder of this discussion, we shall concentrate mostly on congruential logics which are also closed under Uniform Substitution, in particular CL and (especially) IL, taken in their Set-Fmla incarnations. We shall also


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consider only binary relations. In this simple setting, the claim that a relation R is universally representative (in a logic) amounts to the claim that any two formulas are equivalent (according to the logic: here -related for the appropriate consequence relation ) to formulas standing in the relation R, and the claim that R is special (in the logic) amounts to the claim that there is some unprovable sequent σ(q1 ,q2 ) for which σ(B1 ,B2 ) is provable for any B1 , B2 , standing in the relation R. We now introduce a pair of relations in terms of which to illustrate these concepts. For any logic with → in its language, we define formulas A and B to be explicitly tail-linked if they are implicational formulas with a common antecedent, and to be explicitly head-linked if they are implicational formulas with a common consequent. We are associating the terms tail and head with the antecedent and consequent respectively because of their appearance at the tail and head of the arrow “→” in the formula in question. (The reverse convention is equally intelligible, on the grounds that the ‘head’ should come first. It is followed in some discussions, but not here. If there are several implicational connectives in the language of a logic, a more refined terminology would be called for, indicating which implication is at issue.) Formulas A and B are head-linked in the given logic if they are respectively equivalent (according to the logic) to formulas A and B which are explicitly head-linked; mutatis mutandis for “tail-linked”. That is, according to , A and B are tail-linked when for some formulas A0 , B0 , C, we have A C → A0 and B C → B0 , and head-linked when for some formulas A0 , B0 , C, we have A A0 → C and B B0 → C. Note that we may say the same thing for a logic by saying that being tail-linked (or headlinked) is universal – in the sense that any two formulas stand in the relation – as by saying that the relation of being explicitly tail-linked (resp. head-linked) is universally representative. Examples 9.25.2(i) According to IL and CL the relation of being tail-linked is universal. For, given A and B, we can take C as any provable formula (such as q → q, or , if assumed available) and then we have A C → A0 and B C → B0 taking A0 and B0 as A and B themselves. (ii ) According to CL the relation of being head-linked is universal. For, given A and B, we can take C as any refutable formula (such as q ∧ ¬q, or ⊥, if assumed available) and then we have A A0 → C and B B0 → C when we take A0 and B0 as respectively ¬A and ¬B. While in CL and IL, we see from 9.25.2(i), tail-linkage is universal, only CL puts in an appearance in part (ii) as illustrating the universality of head-linkage. Obviously the argument given in support of the claim there for CL (essentially involving the intuitionistically unacceptable double negation equivalence) does not adapt to the case of IL; but this leaves open the question of whether some less obvious argument might not establish that any two formulas are nonetheless head-linked. We can settle the question (negatively) by invoking 9.25.1, to which end we now show that the relation of being head-linked is special in IL. In fact the data we need for this purpose can be gathered from some of the material in 4.22. We were there concerned, in the discussion leading up to 4.22.7 (p. 554), with the existence of a connective ∗ satisfying, for as IL or some conservative extension thereof:

1320 (1∧)

CHAPTER 9. UNIVERSAL AND SPECIAL (A ∧ B) → C (A → C) ∗ (B → C)

(for all formulas A, B, C) and found, in 4.22.7 that ∗ could be taken, with as ¨ . In terms of → IL itself, as the connective baptized in 4.22.10 (p. 555) as ∨ itself, the desired rhs of (1∧) appears as (i)

((A → C) → (B → C)) → (B → C)

Since the left-hand side of (1∧) is symmetrical in respect of the occurrence of the schematic letters “A” and “B” there, it is also IL-equivalent to the result of interchanging those letters in (i): (ii )

((B → C) → (A → C)) → (A → C)

Thus (i) and (ii ) are equivalent for any formulas A, B, and C. (Of course we could have run this argument without introducing ∧ by replacing the lhs of (1∧) with A → (B → C) and permuting antecedents.) But this is to say that ¨ is not generally commutative in IL (4.22.11(i) on p. 556), in that, although ∨ ¨ q (alias (p → q) → q)) IL-equivalent to q ∨ ¨ p, for example, we do not have p ∨ ¨ we do have, for any formulas A and B, (A → C) ∨ (B → C) IL-equivalent to ¨ (A → C); but that is to say: (B → C) ∨ Observation 9.25.3 The relation of being head-linked is special in IL (and therefore, by 9.25.1, not universal in IL). In more detail, and using the notation from our earlier definition of the ¨ q2 q2 ∨ ¨ q1 , and observe that whenever B1 , concepts concerned, we put σ = q1 ∨ B2 are head-linked they are equivalent to explicitly head-linked formulas A → C and B → C respectively, so in virtue of the equivalence of (i) and (ii ) above, σ(B1 , B2 ) is IL-provable even though σ itself is not. That established 9.25.3. The commutativity of pseudo-disjunction is not the only distinctively classical (and thus ‘specialness’ conferring) feature head-linkage secures in intuitionistic ¨ coincide.) All classical properties from the logic. (Recall that in CL, ∨ and ∨ implicational fragment similarly hold in IL for head-linked formulas, as 9.25.4(i) illustrates; part (ii) shows some of the difference between being head-linked and being explicitly head-linked: Exercise 9.25.4 (i) Show that if A and B are head-linked formulas in the language of IL, then we have Peirce’s Law for them, i.e., IL ((A → B) → A) → A. (ii) The formulas p → s and q → (r → s are not explicitly head-linked though in the presence of conjunction (with IL for → and ∧) we could rewrite the latter as (q∧r) → s and have explicit head-linkage. However, already in the pure implicational fragment, we can write a formula of the form B → s IL-equivalent to the second formula. Find such a formula. (Hint: This is possible even in BCI logic, thanks to a close cousin of the Law of Triple Consequents available there.) The relation of being head-linked is of some interest quite apart from its connection with its being special in IL. We will explore it a little, occasionally drawing on that connection to illustrate or establish various points. One type of question that suggests itself takes the form, for this or that n-ary connective


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#: does it follow from the hypothesis that A1 , . . . , An are head-linked with common head C, in the sense that for each i (1 i n) there is Bi with Ai IL-equivalent to Bi → C, that #(A1 , . . . , An ) is also head-linked with that head to its components, i.e., that this compound is equivalent to B → C for some formula B. If the answer to this question is affirmative, we will say that # preserves head-linkage (in IL). Examples 9.25.5(i) ∧ preserves head-linkage in IL; here n = 2 in the general definition and we write A, B for A1 , A2 . The hypothesis in this case is that A and B are respectively equivalent to A0 → C and B0 → C for some formula C. Then A ∧ B is equivalent to (A0 ∨ B0 ) → C, and implication with the same head. (ii ) → preserves head-linkage in IL. Indeed in this case, we do not need the hypothesis that each of A, B, is IL-equivalent to an implication with C as consequent in order to infer that A → B can be. The formula A is irrelevant here, since if B is equivalent to B0 → C, then A → B can be written as A → (B0 → C), and hence as (A ∧ B0 ) → C. Let us consider the case of disjunction. If we have A and B respectively equivalent to A0 → C and B0 → C, then in CL – or indeed in the intermediate logic LC – we could have argued that A ∨ B was equivalent to (A0 ∧ B0 ) → C; but this equivalence is not available in IL. To settle the question let us look at a special case, in which A and B are p → r and q → r respectively. An affirmative general answer would require for this special case that there is some formula D with D → r IL (p → r) ∨ (q → r) But this is impossible, since the forward direction would imply that either D → r IL p → r or D → r IL q → r (see the discussion immediately following 6.46.1), which together with the backward direction yields the incorrect conclusion that either p → r IL q → r or else q → r IL p → r. This gives: Observation 9.25.6 ∨ does not preserve head-linkage in IL. There is also the 1-ary ¬ to ask our question about; this amounts (see below) to asking whether A and ¬A can always be written as implications with a common consequent C. Observation 9.25.7 ¬ does not preserve head-linkage in IL. Proof. We show that p and ¬p are not IL-equivalent to respectively to A0 → C and A1 → C for any A0 , A1 , C. Suppose that they are; then we should have ¨ ¬p and ¬p ∨ ¨ p IL-equivalent, by the discussion preceding 9.25.3 above. But p∨ ¨ ¨ p. IL p ∨ ¬p while IL ¬p ∨ ¨ ¬p is a reflection of what is called (¬-Contrac.) Note that the fact that IL p ∨ ¨ p to be IL-provable amounts to the failure of (¬¬E) in 8.32; the failure of ¬p ∨ for IL, since in primitive →-notation, this formula is (¬p → p) → p, whose antecedent is IL-equivalent to ¬¬p.

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What the proof of 9.25.7 shows is that p and ¬p are not head-linked, and hence that in general it is not the case that an arbitrary formula and its negation are head-linked (according to IL). How is this supposed to show that ¬ does not preserve head-linkage? If we look back at the definition of the latter concept, specializing to the n = 1 case, we get: 1-ary # preserves head-linkage if and only if for any A1 , if there is a C such that A1 is IL-equivalent to B1 → C, then for some formula B, #A is equivalent to B → C. And this is tantamount to the claim that A and #A are head-linked, provided that we know for every A1 , there is a C such that A1 is IL-equivalent to B1 → C for some B1 : but of course we can choose B1 = , C = A1 . To put this another way: the relation of being head-linked (according to IL) is reflexive. It is evident from the definition that this relation is also symmetric. Thus the question arises as to whether we are dealing with an equivalence relation here; Observation 9.25.9 will settle this negatively. To say that the relation of being head-linked (according to IL) is transitive is to say that for any formulas A, B, C (of the language of IL) if for some A0 , B0 , D, we have A IL A0 → D and B IL B0 → D, and for some B1 , C0 , D we have B IL B1 → D and C IL C0 → D , then for some A1 , C1 , D , we have A IL A1 → D and C IL C1 → D . To show that this is not so, it will be enough to show that for any A and C a B can be found which is head-linked to A (so that a formula D as above exists) and head-linked to C (so that a formula D exists, as above), since we know already that not every pair of formulas is head-linked in IL (9.25.3). Observation 9.25.8 If IL B then A and B are head-linked, for any formula A. Proof. Under the stated hypothesis, A IL B → A and B IL A → A, and the formulas on the right-hand sides of these -statements are explicitly head-linked.

Corollary 9.25.9 The relation of being head-linked in IL is not transitive. Proof. Take A and C not head-linked in IL. (For example, p and ¬p, as in 9.25.7.) Take B for which IL B. By 9.25.8, A is head-linked to B and B is head-linked to C, so the relation cannot be transitive. We leave the interested reader to make any further investigations of the relation of being head-linked (and similar relations) in IL (or in other logics), turning our attention to a closely related phenomenon we shall call ‘head-implication’, mentioned in passing in the discussion following 7.22.15 (p. 1079) above. By virtue of the K axiom of BCK logic (see 2.13), A → (B → A) is provable for any A, B, in any extension of that logic, such as IL. We consider the →fragment of IL (‘BCKW logic’), as well as (implicational) BCK logic itself, and on the Set-Fmla incarnation of the former. In terms of IL , then, the point is that A IL B → A for any A, B. We shall introduce a relation of head-implication according to which this says that A stands in this relation to B → A in virtue of A’s being the consequent of – at the ‘head’ of the arrow in – the latter formula (as in the ‘head-linkage’ terminology introduced before


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9.25.2). But we don’t want the relation to depend on the syntactic form of the formulas involved, depending instead only on the proposition expressed. That is, if A head-implies B according to a consequence relation , then we should have for any A , B respectively synonymous (according to ) with A, B, that A head-implies B . We shall only consider congruential consequence relations, so that the reference to synonymy can be replaced by one to logical equivalence (A A , etc.). This motivates the following definition: A head-implies B according to just in case for some formula B0 in the language of , we have B B0 → A. Note that we assume → to be a connective in the language of . To transfer this notion across to logics in Fmla (such as BCK logic in 2.13 or 7.25), replace the requirement that B B0 → A by the requirement that each of B → (B0 → A)

(B0 → A) → B

should be provable. The terminology of head-implication for this relation would be infelicitous in logics without K (linear and relevant logics as in 2.33, and their implicational fragments BCI and BCIW logics) since A could head-imply B without actually (provably) implying B – for instance if A and B were respectively q and p → q. We begin our exploration of head-implication at the other extreme, in terms of strength, by considering instead CL. Part (i) of this exercise is included for the sake of an interesting contrast. Exercise 9.25.10 (i) Find a formula A whose sole connective is →, or show that none can be found, with the property that A CL p and A CL q. (ii ) Find a formula A whose sole connective is →, or show that none can be found, with the property that p CL A and q CL A, while we do not have B CL A for all B. (Hint. For (i): First show that if there is an A as required by (i) then such an A can be found in which no propositional variables other than p and q occur. This follows by the substitution-invariance of CL . Next, examine all possible candidates to see if any of them have p and also q amongst their CL-consequences. Since there are only – to within CLequivalence – six such candidates (see Figure 2.13b, p. 226), this quickly yields either a suitable A or else. For (ii ), a solution follows.) One way to think about 9.25.9(i) is to notice that without the restriction to →-formulas, we could take A, our desired common consequence of p and q, as p ∨ q. (The restriction that we do not have B CL A for all B is added simply to eliminate the easy answer p → p.) But of course – a point which is certainly turned up by an examination of Figure 2.13b – we can re-express the latter disjunction in CL as a purely implicational formula, say as (p → q) → q, or again (CL-equivalently) as (q → p) → p. Although in IL these two implicational formulas are equivalent neither to each other nor to the disjunction p ∨ q, either of them will serve as an example of a common IL-consequence of p and q. In fact, we can weaken the logic still further, to BCK logic, and still return the same answer to (the analogue of) 9.25.9(i). It would be pleasant to be able to say, for these weaker logics at least, that there is something ‘special’ about the way (p → q) → q follows from q as opposed to the way it follows from p. It is for this reason that we have introduced the terminology of head-implication.

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We would like to be able to say that q head-implies (p → q) → q, though p does not. The first part of this claim is clear from the definition, since, taking q as A and (p → q) → q as B, we have p → q as the B0 required by the definition for A to head-imply B. This holds for all extensions of BCK logic. But as to the negative part of the claim, to the effect that p does not head imply this formula, we must tread more carefully. We have already seen enough to know that it would be false for the case of = CL . In CL we can, after all, for the A as p and B as (again) (p → q) → q, choose B0 = q → p and have A CL B0 → A. Thus p, no less than q, head-implies (p → q) → q according to CL. The point can be generalized: Observation 9.25.11 For any formulas A, B, such that A CL B, A headimplies B according to CL . Proof. If A CL B then clearly we have B CL A ∨ B, and thus, re-expressing this as B CL (B → A) → A we have A head-implying B, with the formula B0 of the definition above being B → A. The equivalence mentioned in this proof, B (B → A) → A, holds only in the forward direction for = IL , and the additional assumption that A IL B does not help with the reverse direction. We can nevertheless put the equivalence to good use in the IL-setting: Observation 9.25.12 For any formulas A, B, in the language of ⊇ IL : A head-implies B according to if and only if (B → A) → A B. Proof. ‘If’: If (B → A) → A B, then B (B → A) → A, on the assumption that ⊇ IL , so we can take B → A as the B0 in the definition of head-implication. ‘Only if’: Suppose A head-implies B according to ⊇ IL . Thus for some B0 , we have B B0 → A. Now, by the Law of Triple Consequents (see 2.13.20(ii), p. 239) B0 → A and (((B0 → A) → A) → A) are equivalent, but the former is equivalent to B and the latter to (B → A) → A, so we have (B → A) → A B.

Remark 9.25.13 Note that for = CL the condition given here as necessary and sufficient for A to head-imply B (according to ), namely, (B → A) → A B, holds iff A B, as we expect from 9.25.10. A Fmla version of this condition, ((B → A) → A) → B gives a necessary and sufficient condition for A to head-imply B according to BCK logic, since none of the proof of 9.25.11 exploits W (Contraction). The formula involved here is what would be written as A ⊃ B in the notation of 7.22


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(see the discussion leading up to 7.22.16 on p. 1080 above); note incidentally that every formula of this form is actually provable in the ‘contraclassical’ system of BCIA logic described after 7.25.4 on p. 1100. A partial ordering corresponding to the logical relation of head-implication is introduced into the discussion of BCK -algebras (and distinguished from the usual -relation from our discussion of such algebras in 2.13) to striking effect in Guzmán [1994], Proposition 3.2. (In fact, Guzmán’s discussion is couched in the dual presentation of BCK -algebras mentioned in the notes to §2.1 above – p. 274 – so the correspondence is not immediately obvious.) We can use 9.25.11 to show, for a contrast with the case of CL reviewed above, according to IL , p does not head-imply (p → q) → q. For if it did, then by 9.25.11, taking A as p and B as (p → q) → q, we should have: (((p → q) → q) → p) → p IL (p → q) → q. To see that this is not the case, substitute ⊥ for p; the lhs then reduces (in IL) to ¬¬q and the rhs to q. Thus we might naturally say that not only is implication a special relation in IL and CL (take σ(p, q) as just p q, and of course the relation holds between A and B just in case σ(A, B) is provable, which is no so for arbitrary A, B. Not only this, but further, head-implication is a special relation in IL but not in CL relative to implication, in the sense that for σ (p, q) = (q → p) → p q and σ as just given, whenever σ(A, B) is CL-provable, σ (A, B) is CL-provable, whereas this does not hold for IL. (Of course, σ is our certificate of head-implication, taken from 9.25.12.) Whereas the condition that B’s being a consequence of (B → A) → A, which 9.25.11 says is not only sufficient but also necessary for A to head-imply B in IL, holds also for the weaker BCK logic, as mentioned in 9.25.12, the necessity of the condition in the following exercise does require Contraction: Exercise 9.25.14 Show that if A head-implies B according to IL , though not for arbitrary A, B, with A IL B, then for all C we have: (C → A) → B (C → B) → B. Because, except for occasional asides, we are not treating propositional quantification, we do no more here than mention the following worthwhile project. One could consider what happens in second-order propositional IL (i.e., with such quantification admitted) the possibility of replacing the metalinguistic existential quantifier binding the variable “B0 ” in our definition of the relation of head-implication (as holding between A and B when for some B0 , we have B logically equivalent to B0 → A) by an existential (propositional) quantifier in the object language. This would allow us to trade in the binary relation for a binary connective, →h say, of head-implication, with A →h B defined to be the formula ∃q(B ↔ (q → A)). But we shall not explore the logical behaviour of this connective here, concluding instead with a brief look at another special relation. Formulas B and C are explicit (implicational) converses if there are formulas D and E for which B is the formula D → E and C is the formula E → D. Formulas B and C are implicit converses – though we generally omit the “implicit”

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– just in case they are respectively equivalent (so this is a logic-relative notion) to formulas B and C which are explicit converses in the sense just defined. Accordingly to say that the relation implicit converse of is special in a logic – for present convenience taken to be a logic in Fmla in which the provability of such an B and C suffices for he synonymy of the D and E concerned (i.e., essentially a ↔-congruential logic) – is to say that there is a formula A(p, q) which is not provable but for which A(B, C) is provable if B and C are provably equivalent, in the logic concerned, to explicit converses. From Humberstone [2002a], q.v. for proofs, we take such results as are summarised here; for part (iv), see also the discussion preceding 7.25.17 (p. 1112): Observation 9.25.15 (i) B and C are implicit converses in IL if and only if IL (B → C) → C and also IL (C → B) → B. (ii) The characterization given for IL in (i) is also correct for the ‘semi-relevant’ logic RM. (iii) The same characterization holds also for CL and for the intermediate logic LC, in both of which cases the condition supplied can be simplified to: “if and only if IL B ∨ C”. (iv) B and C are implicit converses in Abelian BCI logic (the system BCIA of 7.25) if and only if (B → (B → B)) → C and the converse implication are BCIA-provable. Note that in (iv) the reference to the converse of a provable implication also being provable in BCIA logic is redundant, since the converse of any provable implication is provable in that logic (see 7.25.11, p. 1105). The following is clear enough not to require proof: Corollary 9.25.16 In all the logics mentioned in 9.25.15, the relation of being a converse is special. Recall that by a converse is meant an implicit implicational converse. Exercise 9.25.17 Is the binary relation of being a converse of a converse of special in IL? Justify your answer. (A solution follows form Coro. 1.4 in Humberstone [2002a].) What 9.25.5 provides for the relation ‘is an implicit converse of’ is what we might call a simple provability characterization: it gives us a set of formulas Γ(p, q) and tells us that the relation holds between B and C just in case all formulas in Γ(B, C) are provable in the logic in question. (More generally, a setof-sequents formulation with Σ(p, q) could be given.) There are, in particular, no existential quantifiers over formulas in the characterization, as in the original definition of the (implicit) converse-of relation. 9.25.12 similarly provides a simple provability characterization of head-implication for IL. But for our first example of a special relation in IL, namely, head-linkage, no such characterization was provided. Nor does the author know of such a characterization, at least for the pure implicational fragment of IL. If ∧ is present, then we can see that the provability of the Peircean formulas ((B → C) → B) → B

and

((C → B) → C) → C


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will serve in this capacity, since we can reformulate these as ((B → C) → (B ∧ C)) → B

and

((C → B) → (B ∧ C)) → C,

whose converses are also IL-provable. We already know that head-linked formulas satisfy the Peircean principles (9.25.4(i)), and therefore these last two formulas are also provable for them. But so are the converses of these two implications, so B and C are then equivalent to explicitly head-linked formulas, the common head being B ∧ C. But this leaves open the question of whether any such simple syntactic characterization be provided of head-linkage in the pure → fragment of IL. We do learn from the above, however, that also in the conjunction–implication fragment of CL, any two (and so any finite number) of formulas are head-linked, without having to invoke ¬ or ⊥ and use the fact that A and A is equivalent to (A → ⊥) → ⊥, since we can use the conjunction of the formulas in question to serve as the desired head – a kind of local simulation of ⊥.

9.26

Special ‘in a Given Respect’

We return from the subject of special relations between formulas to special classes (“special 1-ary relations”) of formulas, and in particular to the case in which the class of all formulas with a given connective as main connective is a special class according to this or that logic. This was what we agreed to speak of more concisely by describing the connective itself as special in the logic concerned. Recall that this amounts to defining n-ary # as special according to a consequence relation (to concentrate on the case of Set-Fmla, and assuming substitution-invariance as well as the usual structural rules) when there is some sequent σ such that although σ ∈ / , for all formulas B1 , . . . , Bn , we have concerning the result σ(#(B1 , . . . , Bn )), of substituting the formula #(B1 , . . . , Bn ) uniformly for some propositional variable occurring in σ, that σ(#(B1 , . . . , Bn )) ∈ . We noticed, for example, that ¬ was special in IL because we could take σ in the above definition as ¬¬p p: every substitution instance which replaces the variable p by a formula of the form ¬B belongs to IL , though σ itself does not. The choice of σ (and of the propositional variable to be replaced in σ, in case there are several variables in σ) can be thought of naturally as specifying a respect in which ¬ special in IL: here, we are saying that the respect in which formulas of the form ¬B are special in IL is that they, unlike arbitrary formulas, satisfy a double-negation elimination principle. But merely to talk of a connective as special (in a logic) is just to make the existential claim that there is some respect in which it is special. That way of talking carries a risk, then, of directing attention away from the possibility that although special in two logics, the same connective may be special in different respects in the two cases, or that it is special in more respects in the one logic than in the other. Example 9.26.1 ¬ is special in the intermediate logic KC in a respect in which it is not special in IL. For consider σ = p ∨ ¬p; σ(¬B) is KC-provable for all B, though σ itself is not KC-provable. So σ indicates a respect in which ¬ is special in KC, though not a respect in which ¬ is special in IL (where σ(¬B) is not provable for arbitrary B): all this notwithstanding the fact that ¬ is special tout court in both logics (in respect of double negation elimination).

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The above example our main focus for discussing respect-specific ‘specialness’. We will return to it after what may initially seem a change of subject. The topic to which we shift attention is persistence of formulas in Kripke models for intuitionistic and intermediate logics. Recall (2.32.3: p. 308) that all formulas A are persistent in such models in the sense that for M = (W, R, V ), x, y ∈ W , whenever M |=x A and xRy, we also have M |=y A. Let us now call a formula A inversely persistent in M if for all x, y ∈ W , whenever M |=y A and xRy, we also have M |=x A. (If you visualise the models ‘growing downwards’ [growing upwards], then you will think of inverse persistence as upward [downward ] persistence; as in the discussion before 6.44.6 (p. 904), a more neutral description would be simply backward persistence.) Since the above condition amounts to saying that if M |=x A and xRy, then M |=y A, one could also think of this as a kind of ‘negative persistence’ – a phrase avoided here as it was used in a different sense in 8.23.) Which formulas of the language of IL are inversely persistent in all models? Clearly if a formula is true at no point in any model, or is true at every point in any model, then it can never go from not being true to being true in a model and so counts as inversely persistent in all models. These degenerate cases are in fact the only cases that arise: Observation 9.26.2 If A is inversely persistent in all Kripke models, then either IL A or IL ¬A. Proof. Suppose IL A and IL ¬A. Thus (2.32.8, p. 311) there are models M1 = (W1 , R1 , V1 ), M2 = (W2 , R2 , V2 ) with w ∈ W1 , x ∈ W2 , M1 |=w A, M2 |=x ¬A, and hence for some y ∈ W2 such that xRy, M1 |=y A. Take the submodels of M1 and M2 generated by the points w and y respectively and adjoin a new point u to get a larger model of which these are generated submodels, exactly as in the proof of 6.41.1 (p. 861). In the resulting model A fails to be true at u (since w is accessible to u and A is not true at w), so since A is true at y, another point accessible to u, A fails to be inversely persistent. If we make no restriction to the class of models under consideration, then, there is not much to be said about inverse persistence. But if we restrict attention to models on frames of a certain type, non-degenerate cases of inversely persistent formulas emerge. In particular, suppose we restrict attention to piecewise convergent frames; that is, frames in which any two R-successors of a common point themselves have a common R-successor (R being the accessibility relation of the frame). If we suppose in a model on such a frame a formula of the form ¬B goes from not being true to being true as we pass from x to y ∈ R(x), this means that for some u ∈ R(x), B is true at u—since otherwise ¬B would be true at x): but since xRy and xRu, by the (piecewise) convergence condition, u and y in turn have a common successor, z say, at which B must be both true and not true (compare the discussion leading up to 6.42.15). This contradiction establishes the (1) ⇒ (2) direction of: Observation 9.26.3 The following two conditions are equivalent for any frame (W, R): (1) (W, R) is piecewise convergent; (2) All negated formulas are inversely persistent in every model on (W, R).


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Proof. Given the preceding discussion, all that remains to be shown is that (2) implies (1). So suppose that (W, R) is not piecewise convergent, in the hope of finding a model on (W, R) in which some formula ¬B is not inversely persistent. Since (W, R) is not piecewise convergent, there exist x, y, z ∈ W with y, z ∈ R(x) but R(y)∩R(z) = ∅. Put V (p) = R(y). Then since R(y)∩R(z) = ∅, ¬p is true at z in (W, R, V ), though it is not true at x in (W, R, V ): a failure of inverse persistence for a negated formula. Speaking entirely informally, we might say that there is something ‘special’ about negated formulas – something not holding for all formulas – in models on piecewise convergent frames: they are inversely persistent. This is not our official use of the term special, however, since no logic has been singled out, nor any sequent noted to be absent from that logic although every result of substituting a negated formula for some variable in the sequent is present in the logic. Can we convert the semantic peculiarity of negated formulas in respect of piecewise convergent frames into an observation about the specialness of ¬ in a logic? To begin with, let us consider some promising candidate sequentschemata: Exercise 9.26.4 Show that for any formulas A, B, and any formula C inversely persistent in a model M, that (i) A → (B ∨ C) (A → B) ∨ C holds in M (ii ) (C → B) ∨ C holds in M (iii) C ∨¬C holds in M. Next, we need to find one or more logics to consider. Obvious candidates are the intermediate logic KC and some of its extensions (see the discussion after 2.32.8, p. 311). Their canonical models have piecewise convergent frames (6.42.15) and so all negated formulas are inversely persistent in these models (9.26.3). Thus if in (i)–(iii) of 9.26.4 we replace the schematic “C” by “¬D” we get schemata whose instances all hold in such models, and thus, in particular, are provable in KC. Without such a replacement, this cannot be said for KC, so we have now exposed a respect in which ¬ is special in KC. In particular, we have returned, with 9.26.4(iii), to our starting point in this subsection: that the law of excluded middle is provable for negated formulas but not for arbitrary formulas in KC. (Recall that KC is sometimes called the logic of the Weak Law of Excluded Middle.) Clearly the same goes for such formulas in any extension of KC which is properly included in CL (where the special status of ¬ vanishes). What we have learnt in the meantime is that the particular respect here isolated in which ¬ is special according to these logics is a reflection of a respect in which negated formulas behave distinctively in models on piecewise convergent frames: inverse persistence.

9.27

Pahi’s Notion of Restricted Generalization

The Digression after Observation 9.13.3 (see p. 1299) mentioned Pahi’s notion of a full matrix – one in whose underlying algebra every fundamental operation is in a certain sense (recalled there) surjective. This subsection picks up the idea and summarises a few of the things Pahi does with it, as well as connecting

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the discussion of Pahi [1971a] with the themes of the present chapter. (Also relevant: Applebee and Pahi [1971], Pahi [1971b], [1972a], [1972b].) But let us begin somewhere apparently quite different, with a familiar phenomenon in propositional logic, and for simplicity we address it for the case of classical logic. For continuity with Pahi’s discussion, the logical framework Fmla will receive our primary attention here (and take all logics to be closed under Uniform Substitution and Modus Ponens). With the most customary choices of primitive boolean connectives those encountering CL for the first time – for example {∧, ∨, ¬, →}, as remarked in the discussion leading up to Example 1.23.2 (itself appearing on p. 119), or some functionally complete subset thereof – the tautologies have the conspicuous feature that in each of them some propositional variable occurs more than once. (Note that we are only saying that some propositional variable occurs more then once, not that every propositional variable to occur occurs more than once, as happens in the equivalential fragment of CL – 7.31.7 (p. 1135) – and the 2Property for BCI logic – 7.25.24(i) on p. 1116.) The explanation offered below comes at Coro. 9.27.4. In equational logic, a term in which no (individual) variable occurs more than once is called a linear term. By way of example, see Ðapić, Ježek, et al. [2007] (or the phrase “linear equation” in, e.g., Taylor [1979]). It is convenient to use this terminology here and mean by a linear formula, a formula in which no (propositional) variable occurs more than once. Note that there is no connection (other than etymological) between linear formulas in the present sense and linear logic (as in 2.33). The phenomenon just alluded to, then, is the fact that no linear formula is a tautology (for the choices of primitives just alluded to). Let us look at this for the case of purely implicational formulas first; the desired result will appear after that, at Coro. 9.27.4. Observation 9.27.1 For every linear formula A in the language with → as sole primitive connective, there exist boolean valuations u, v, with u(A) = T and v(A) = F. Proof. By induction on the complexity of A, assumed to be a linear formula. The basis case (A = pi ) is clear, so for the inductive step, take A = B → C, where B and C, being subformulas of a linear formula, must also be linear, are guaranteed by the inductive hypothesis each to have verifying and falsifying boolean valuations. If u is a verifying valuation for C, then since u is →boolean, u(C) = T, so we only need to find a boolean valuation v with v(C) = F, i.e., v(A) = T and v(B) = F. So take a boolean valuation, v , say with v (B) = T, and a boolean valuation v for which v (C) = F (both, as promised by the inductive hypothesis). Since A is a linear formula, B and C have no propositional variables in common and we may take the desired v to be any boolean valuation agreeing with v on the propositional variables in B and with v on the propositional variables in C, and we have v(B → C) = F. (This is a case of ‘amalgamating’ (e)valuations, as in 2.11.6.)

Exercise 9.27.2 Now suppose that we have, in addition to →, further primitive connectives ∧, ¬, ∨. Show that, as 9.27.1 says for the pure → formulas, for each linear formula in this language, there exist verifying


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and falsifying boolean valuations. (Suggestion: just add to the list of cases in the inductive step of the proof above.) Evidently this kind of reasoning extends to a great many sets of boolean connectives, though it would clearly fail if, for example ⊥ or were present, since these are already formulas A for which it is not the case that there exist boolean valuations u, v, with u(A) = T, v(A) = F, A in each case being a linear formula (no variables occurring at all, so none occurring other than exactly once). The issue is not their nullary status, since we could make the same point with, e.g. the binary connective ⊥2 with associated truth-function the constant false function of two arguments. (This is a good reason for speaking of linear formulas rather than, as Pahi [1971a] does, of variable-like formulas, which is used for this syntactic property in Pahi’s writings: not all formulas with the property behave like propositional variables in the relevant respect.) The point is rather their constancy. Applying the reasoning behind 9.27.1, 9.27.2, we get: Observation 9.27.3 For all linear formulas A constructed using only connectives # for which the associated truth-function #b (associated over BV with #, that is) is non-constant, there exist u, v ∈ BV with u(A) = T, v(A) = F. Corollary 9.27.4 No A satisfying the conditions in 9.27.3 is a tautology. Like 9.27.3 itself, this corollary would fail in the presence of such connectives as (already a linear tautology, as is p → ) and ⊥ (consider ⊥ → p), as well as the binary ⊥2 mentioned above (with linear tautologies such as ⊥2 (p, q) → r). (For related considerations, see Schütte [1977], Theorem 1.7.) Exercise 9.27.5 Suppose we have a Set-Set sequent A1 , . . . , Am B1 , . . . , Bn in which no propositional variable occurs more than once (in the whole sequent, that is) and in which the formulas Ai , Bj are constructed using ‘non-constant’ connectives (to abbreviate the formulation in 9.27.3). Does it follow that the sequent is non-tautologous? (Explain why, or else give a sequent meeting these conditions which is tautologous.) Observation 9.27.3 ends by promising boolean valuations u, v, with u(A) = T, v(A) = F, which could be reformulated as promising boolean valuations u, v, with u(A) = v(A), which says directly that A is not a constant formula over the class of boolean valuations, taking constancy over a class of valuations to consist in being assigned the same value by all of them. Thus an informal rendering of 9.27.3 would be: linear formulas constructed using non-constant connectives are themselves non-constant. So much for familiarization with the notion of a linear formula and one simple example its utility; we now turn to some of the developments of the papers by Pahi listed at the start of this subsection, and their connections with the main business of the present chapter. In the two-valued case the non-constancy of a function and its surjectivity in the sense recalled in the Digression on p. 1299 coincide, since a function (of however many arguments) which does not uniformly return one of the values T, F, must have both in its range. But in the area of matrix semantics à la 2.11 in which more than two values are involved non-constancy is a weaker condition than surjectivity: consider the negation table in the matrix of 2.11.3 (p. 202), for example, in which both 1 and 3 appear amongst the outputs (non-constancy)


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while 2 does not (failure of surjectivity). As mentioned in the earlier Digression, Pahi calls a matrix full whenever each fundamental operation of its underlying algebra is surjective in the relevant sense – every element being the output for some sequence of inputs, that is. (Fullness is thus really a matter of the algebra rather than the matrix concerned, as is noted in Applebee and Pahi [1971] which partially overlaps Pahi [1971a], though in what follows explicit reference is made only to the latter paper.) Such matrices, Pahi [1971a] observes, have the interesting property that whenever they validate a formula they validate any formula which is a restricted generalization of that formula, where this phrase is understood as follows. Let A(q1 ,. . . ,qn ) be a formula in which only the variables displayed occur, and A(B1 ,. . . ,Bn ) be the result of uniformly substituting Bi for qi (1 i n). This substitution instance of A(q1 ,. . . ,qn ) is called a restricted substitution instance of A(q1 ,. . . ,qn ) if the formulas Bi are all linear formulas and further, when i = j then Bi and Bj have no propositional variables in common. Under the same conditions, A(q1 ,. . . ,qn ) is called a restricted generalization of A(B1 ,. . . ,Bn ). Any matrix validating a formula validates all of its substitution instances, of course, and so a fortiori all its restricted substitution instances, but the point of current interest, mentioned above and appearing below as 9.27.7, is that in the latter case we have an implication in the converse direction as long as the matrix in question is full. First we need a generalized version of 9.27.1; the following is Lemma I of Pahi [1971a]. As indicated already, the feature generalized on is surjectivity rather than non-constancy. (For the notation which follows, recall that “A” stands for a formula and is not connected with the use of “ A” for an algebra or “A” for the universe of the latter.) Lemma 9.27.6 If M = (A, D) is a full matrix with A an algebra of the same similarity type as some language in which A is a linear formula, then for each a ∈ A there is some M-evaluation h with h(A) = a. Proof. By induction on the complexity of A. If A is pi (the basis case) the result is clear, so suppose A is #(B1 , . . . , Bn ) for some primitive n-ary connective, where #M is the corresponding matrix operation. We need to show that there for each possible value a ∈ A there is an M-evaluation h with h(A) = a. So take such an a ∈ A; since M is full, #M is surjective and there are b1 , . . . , bn ∈ A with #M (b1 , . . . , bn ) = a. By the inductive hypothesis, since each Bi is also linear, we have evaluations h1 , . . . , hn with hi (Bi ) = bi (1 i n). And since A is linear, these Bi are variable-disjoint from each other so the evaluations hi can be amalgamated to give the desired h with h(Bi ) = bi for all i, for which accordingly h(A) = #M (b1 , . . . , bn ) = a, as required.

Theorem 9.27.7 Suppose A(q1 ,. . . ,qn ) is a restricted generalization of the for→ mula A(B1 ,. . . ,Bn ), which formulas we now refer to more succinctly as A(qi ) →

→

and A(Bi ) respectively, and that M |= A(Bi ) for some matrix M appropriate to → the language concerned. Then M |= A(qi ). →

Proof. Suppose, arguing contrapositively, that M |= A(qi ). Thus for some M→ / D (where M = (A, D)). For bi = h(qi , Lemma 9.27.6 evaluation h, h(A(qi )) ∈


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gives evaluations h1 , . . . , hn , with hi (Bi ) = bi , for i = 1, . . . , n. As the distinct Bi are variable-disjoint (from the definition of ‘restricted generalization’) these hi can be amalgamated into a single evaluation h* with h*(Bi ) = bi ; but then →

→

→

h*(A(Bi )) = h(A(qi )), an undesignated element, showing that M |= A(Bi ). Pahi [1971a] goes on to draw numerous corollaries, of which the first is that any purely implicational logic (in Fmla), extending BB logic, closed under Modus Ponens and Uniform Substitution and having, for some formula A distinct from, p both A → p and its converse as theorems, is closed under restricted generalization. (Pahi actually cites BB I logic in this connection but the provability of I follows from the conditions given here.) The BB and Modus Ponens conditions guarantee that (since no further connectives are involved) a formula and anything provably implying it as well as provably implied by it are synonymous in the logic and thus that p and the formula A of the last condition(s) are synonymous, thus giving the same element in the Lindenbaum matrix of synonymy classes of formulas (as in 2.13, except that since the logic need not be monothetic, when the synonymy classes of theorems are taken as designated elements, there may be more than one element). Since A is distinct from p, setting aside the possibility that A is some other propositional variable, A is an implicational formula in which we can assume the only variable appearing is p itself, so writing A as A(p) by Uniform Substitution every formula B is synonymous with A(B) and the Lindenbaum matrix just mentioned is full, whence by Theorem 9.27.7, the logic concerned is closed under restricted generalization. Exercise 9.27.8 How does the conclusion just drawn follow even if A is a variable, q, say, distinct from p? And why are we entitled to assume that if p is provably equivalent in the logic in question to some formula A distinct from it that this holds for some A in which p is the only variable to appear? As a special case of the corollary just described, Pahi concludes that the implicational fragment of R (BCIW logic) is closed under restricted generalization, since we may take A as the formula (p → p) → p, and further that the positive ( = {∧, ∨, →}) fragments of all intermediate logics are similarly closed, in view of the synonymy therein of p with the implicational formula (p → p) → p (again), with the conjunctive formula p ∧ p and with the disjunctive formula p ∨ p. This line of thought will be starting to sound familiar, being more than faintly reminiscent of the fact that no connective universally representative in a logic can be special in that logic (9.22.3) or perhaps more generally that no relation among formulas which universally representative relation in a logic can be special in that logic (9.25.1). Naturally, in view of the orientation of the present work, emphasis was given to the ‘local’ connective-by-connective contrast special vs. non-special (according to a given logic), rather than to the ‘global’ (logic-by-logic) contrast of interest to Pahi: being closed under restricted generalization. Still, there is a close formal similarity between saying that a connective is special in a logic S (in Fmla, assumed closed under Uniform Substitution) when for some formula A = A(q): S A(q) while S A(B) for all B with the given connective as main connective,

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and saying that S is not closed under restricted generalization, which, in the n = 1 case of the general definition above, means that again for some formula A(q): S A(q) while S A(B) for some linear formula B. One complication is that in connection with the definition of being special A(q) (or more generally σ(q) was not required to be a formula (more generally a sequent) in which the exhibited variable was the only variable to occur, whereas in defining the restricted generalization relation, A(q1 ,. . . ,qn ) was to be a formula in which only the variables displayed occur, and A(B1 ,. . . ,Bn ), the generalization thereof, had distinct qi replaced by variable-disjoint linear formulas Bi . So in the n = 1 case of the latter, q was the to be only variable occurring in A(q). In the examples actually explicitly mentioned in Pahi [1971a], however, it turns out that only one of q1 , . . . , qn is replaced by a linear formula other than a propositional variable. Proposition 9.27.9 If some connective is special in a (substitution-invariant, Fmla) logic S, then S is not closed under restricted generalization. Proof. Suppose k-ary # is special in S, with A(q1 ) being S-unprovable for some A(q1 ), in which further variables q2 , . . . , qn may also appear, whereas A(B) is provable in S for any #-formula B. Then in particular A(#(r1 , . . . , rk )) is provable in S, where r1 , . . . , rk are k distinct variables chosen so as not to overlap with q2 ,. . . ,qn , so as to make this a restricted substitution instance of the original S-unprovable formula A(q1 ), and showing S not to be closed under restricted generalization.

Exercise 9.27.10 Would the converse of 9.27.9 be correct – i.e., is every logic (in Fmla, closed under Uniform Substitution) which is not closed under restricted generalization have a (possibly derived) special connective? One might suspect that by using more than just the first one amongst the variables q1 , q2 ,. . . ,qn an active parameter, any logic in which some syntactical relation amongst formulas was special would thereby fail to be closed under restricted generalization, but this is not so, as the following illustrates. Example 9.27.11 Let S be the implicational fragment of IL in Fmla (the logic of ‘Positive Implication’). The relation holding between an implicational formula and its antecedent is special in this logic, since while S p → (p → q), we do have S (A → B) → B whenever B is an implicational formula with A as antecedent. (This is then a case of contraction, or W .) But as we have already seen, this S is closed under restricted generalization. (The point here is that putting, e.g., p for p and p → q for q in p → (p → q) does not give a restricted substitution instance, as although the formulas substituted for variables are linear, they are not variable-disjoint.) Given a logic S – for definiteness in Fmla – let the Pahi completion of S be the smallest logic extending S and closed under restricted generalization. Thus


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for example in the language with sole connective →, the Pahi completion of the implicational fragment of E has as its Pahi completion the implicational fragment of R. (See 9.22.1; this example is also cited in Pahi [1971a] and elsewhere.) Exercise 9.27.12 Do all intermediate logics (in Fmla, with primitive connectives {∧, ∨, →, ¬}) have the same Pahi completion. If so, what is it? If not, give two intermediate logics and explain how their Pahi completions differ. Still on the subject of the preceding exercise, intermediate logics, we conclude this brief glance at Pahi’s work by noting an aspect of it which is close in spirit to the “special in a given respect” ideas of 9.26 above. As we recalled there, the connective ¬ is special in the intermediate logic KC not just in the respect in which it is special in all subclassical intermediate logics – lacking ¬¬p → p although this is what we have now learnt to call a restricted generalization of the provable ¬¬¬p → ¬p – but also, unlike IL itself, being in this same position with regard to the provable ¬p∨¬¬p and its unprovable restricted generalization p ∨ ¬p. In this case, we can actually axiomatize KC over IL by taking as an axiom the formula whose restricted generalization is not provable. Pahi ([1971a], Remark 6) asks whether the other Dummett–Lemmon intermediate logic, LC (discussed in 2.32 above) can similarly be axiomatized by a formula meeting that description, and notes that of necessity (the positive fragments all coinciding with their Pahi completions), any such candidate axiom will need to involve ¬. To answer this question, consider the formula mentioned after 4.21.3 above and used by Bull [1962] to axiomatize the implicational fragment of LC (a formula already mentioned in Dummett [1959b]): ((p → q) → r) → (((q → p) → r) → r). (This formula, in our ‘deductive disjunction’ notation from 1.11.3 (p. 50), is (p → q) (q → p), and is interdeducible with the disjunction proper, of p → q and its converse, in the presence of the IL axioms governing ∨. See the discussion preceding 4.22.10 above.) Thus to obtain an LC axiom of the kind Pahi asks for, it suffices to consider the following restricted substitution instance of Bull’s axiom: ((p → q) → ¬r) → (((q → p) → ¬r) → ¬r). Of course for this to work, we do need to check one thing: Exercise 9.27.13 Show that the formula inset above is IL-provable. (Use either the Kripke semantics to check for validity, or else a pinch of Glivenko – e.g., in the shape of 2.32.2(ii), at p. 306 above.)

Notes and References for §9.2 The references given for §9.1 continue to be relevant as further reading here. The fact that the converse of a theorem in a range of extensions of BCI logic is what is called a Mingler in Exercise 9.23.1 (which asks for a proof of this

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fact) was noted (for the case of BCI logic itself) in Proposition 4.2 of Humberstone [2006b]. The points about negation as special in IL and universally representative in CL are all well known; see Koslow [1992], §12.4, for example. Head-linkage and head-implication are discussed (with special reference to IL) in §7 of Humberstone [2001] as well as in Humberstone [2004b]. (The several occurrences of the phrase “law of triple consequences” in the former paper should be “law of triple consequents”. Some questions left open in the latter paper were settled in Połacik [2005].) In fact the idea of head-linkage is just a minor variation on a theme of Urquhart [1974c]: provable equivalence (in the →-fragment of IL) to implicational formulas with the same rightmost variable. (Cf. Exercise 9.25.4(ii).) I am grateful to Mike Dunn for alerting me, after a seminar at Melbourne University in the 1980s in which much of the other material in this chapter was presented, to the work by Pahi discussed in 9.27.

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Index #-boolean valuation (# = ∧, ∨, ¬, etc.), 65–71 #-classical (# = ∧, ∨, ¬, etc.), see classicality #b (truth-function associated with # = ∧, ¬, etc., on boolean valuations), 387, 403, 620 & (special conjunction-like notions), 346, 662, 669, 702, 906 (modal operator), 276 , see Church negation Λ, 1182–1185 Ω Porte’s constant for the Ł-modal system, 484 temporary use as a 1-ary connective, 1080 closure relation in the semantics of orthologic, 919 gcr (for a contrast with consequence relations), 392, 844 propositional entailment in Gärdenfors semantics, 643 structural completion of a consequence relation, 163, 179 supervenience determined consequence relation, 1142 !A! (set of valuations, or set of points in a model, verifying A), 138, 279, 643, 811, 895 (sequent separator), 103, 188 “” notation, 355 (IL nand ), 1241 component switching operation on formulas, 716, 736 tuple splicing operation, 40 ◦ composition of functions, 9

fusion (or multiplicative conjunction), 147, 345, 347 hybrid connective (conjunction and disjunction), 469 relative product, 501, 732 various other connectives, 256, 479, 1148 (deductive disjunction), 50, 420, 555, 892 notation, 63 ... ∨ (Church disjunction), 235 ¨ (pseudo-disjunction, q.v.), 235, 555, ∨ 1320 δ Łukasiewicz’s variable functor notation, 1157 Blok–Pigozzi δ(t), ε(t) notation, 258 superscripted to denote the dual of a truth function, 405 ∪˙ (closure of union), 10 κ, 1112–1114 connective to form contraries, 850 converse-forming connective in BCIA logic, 1112 λ, see lambda notation, see lambda ∧b , ∨b , etc. (#b for boolean #), 403 ↔, 83 biconditional connective (see biconditional, equivalence), 48 ↔m (multiplicative biconditional), 1130 -based algebraic semantics, 246– 250 arbitrary partial order (with converse ), 2 ordering of truth-values, 621 partial ordering of valuations, 138, 436, 750, 914, 1173 1440

INDEX ∨* (theorem disjunction or upper bound for theorems), 573 |= |=M with M a matrix, 203 “|=” used for semantic consequence relations, 57 truth relation for equational logic, 32 truth relation for first order logic, 36 truth relation in Kripke semantics for IL, 307 truth relation in Kripke semantics for modal logic, 279 truth relation in Urquhart semantics for relevant logic, 337 μ μBCI (monothetic BCI ), 1108 multiplicity function, 373 temporary use in connection with probability, 657 (linear implication), 345, 349, 669 ¬c , ¬i (cohabiting intuitionistic and classical negation), 585 ¬ complementation, 21 negation connective (see negation), xvi, 48 ¬d (dual intuitionistic negation), 92, 1222 ω, xv ⊕ (temporary notation for a variant disjunction), 835 ⊗ combined matrix evaluations, 216 direct products of algebras or matrices, 28, 212 Girard’s notion for multiplicative conjunction, 345 π (binary connective for IL), 1068, 1076 (pre-order), 2 L , 14, 15 R , 14 σ connective to form subcontraries, 850 variable over sequents, 120 ( (weak disjunction), 887

1441 and , xiv, 275 , 276 ⊃ additive implication in linear logic, 349 enthymematic implication defined in relevant logic, 1098 material implication as a new primitive in relevant logic, 1260 material implication defined in relevant logic, 327 Meredith’s simulation of classical implication in terms of intuitionistic implication, 335, 1079, 1272 Sasaki hook, 302, 1192 • (ternary connective for IL), 1068 →, 83 Gentzen’s use of, as a sequent separator, 103 implication connective (see conditionals, implication), xvi, 48 relative pseudocomplement in Heyting algebras, 22 →-intuitionistic (consequence relations), see intuitionistic logic →d (dual intuitionistic implication), 546 L (disjunctive combination on the left), 12 R (disjunction combination on the right), 12 L (conjunctive combination on the left), 12 R (conjunctive combination on the right), 12 consequence relation, 55 generalized consequence relation, 73 · (special consequence relation defined in 5.34), 744 DD (Double Disjunction), 64, 600 IL , CL , etc., see intuitionistic logic, classical logic, etc. P P (probability-preserving consequence relation), 652 P V (pair-validity consequence), 747

1442 SCI , see SCI KK (minimal bimodal inferential consequence relation), 744 K , S4 , etc. (inferential consequence relations associated with K, S4, etc.), 294 Suszko (Suszko consequence relation), 204, 208 ∨ disjunction connective (see disjunction), xvi, 48 lattice join, 7 ∨Kr , ∨Be (cohabiting Kripke and Beth disjunction), 897 ∨c , ∨q (cohabiting classical and quantum disjunction), 587 ∧ conjunction connective (see conjunction), xvi, 48 lattice meet, 7 ∧∨, ∨∧, etc. (product connectives), 464–468 ℘, 2 (exclusive disjunction), 398, 780 (∧ Left), (¬ Right), (→E) etc., see rules, (∧ Left) etc. .2 (modal principle), 277, 298 .3 (modal principle), 277 → (would counterfactual), 1008 →(might counterfactual), 1008 “!” notation (modal logic), see also exponentials, 278 § (demi-negation), 536, 576 1, 2 (two constants for pair validity), 760 4 (modal principle), 277, 284 5 (modal principle), 277, 298 A (alternator), 890 a (anticipation connective), 625 AAL, see abstract algebraic logic Abbott, J. C., 237, 238, 241, 274, 275 Abelian groups, see groups, Abelian logic (see also BCIA logic), 335, 1098, 1101 Abell, C., 635 Abraham, W., 675

INDEX Abramsky, S., 371 absolutely free (algebras), 30, 49, 219, 419, 497, 681 absorption — laws, 7, 469, 1050, 1125, 1296 ‘absorption’ (unfortunate word for contraction), 1125 abstract — vs. concrete conception of languages, 48 — algebraic logic (AAL), 97, 257, 273, 275, 874, 1158, 1311 — completeness theorem, see completeness (semantic) — conception of rules, 628 — logics, 218, 268 accessibility relations, 279 Ackermann constants, see t, f Ackermann, R., 271 Ackermann, W., 342, 345, 371, 1110 “actually”, 489, 930, 1289, 1301 Jackson’s “actually” argument, 931, 932, 936–937, 1040– 1043 rhetorical use, 931, 1041 Aczel, P., 103, 180, 181, 1162 Adams, E. W., 655, 928, 931, 938, 1056 Adams, M. E., 1301 Adams, M. M., 829 Adams, M. P., 970 additive and multiplicative rules or connectives, 143, 147, 342, 346–355, 475, 664, 667 Adjunction (rule), 171, 340 adjunctive vs. connective interpretation of truth tables (Reichenbach), 633 admissible (rules), see rules, admissible ‘affine’ (logic), 372 agrees consequence relation agreeing with a gcr, 844 consequence relation agreeing with a set of formulas, 158 Ajdukiewicz, K., 676 Akama, S., 1251 Akatsuka, N., 969, 1054

INDEX Alessi, F., 673 alethic (modal logic), 276, 471 algebraic functions, see derived operations, algebraically derived semantics, 195, 219–268, 273– 275 vs. model-theoretic semantics with models whose frames are algebras, 336 algebraizable (logics), 257, 261, 1309– 1312 algebras, 8, 17 Allen, W., 193, 631 Allwein, G., 373 Almeida, J., 43 Almukdad, A., 1251 Alonso-Ovalle, L., 1022 alterjection, 1179 ‘alternative denial’ (see also nand, Sheffer stroke), 607 alternator, 890 amalgamating conditionals ‘would’ and ‘might’ counterfactuals, 1009 consequential and nonconsequential, 943 helping and otherwise, 947 implicit conditionals subjunctive and indicative, 1010 matrix evaluations, 204, 1330 ambi-assertion, 1179 ambi-medial (law), 1133, 1148–1149, 1161 ambiguity claimed as a response to differences between logics, 585, 592 of ‘and’, 667 of ‘but’, 675 of ‘or’, 769, 780, 789, 798 structural vs. lexical, 667, 803, 805, 1017 Amis, K., 676 analogous (in the sense of Zolin), 463 analogue intuitionistic analogues of classical connectives, 405, 617

1443 analytic implication, 772, 925 “And” (see also conjunction), 631– 677 and -like (∧-like) connectives, 708– 714, 716–720 and -representable, see representable, ∧-representable Anderson, A. R. (see also Anderson– Belnap), 930, 1251 Anderson, C. A., 501, 1162 Anderson, L., 816 Anderson–Belnap (A. R. and N. D.), 126, 158, 335, 342, 345, 371, 475, 659, 992, 1053, 1056, 1091, 1092, 1095, 1098, 1115, 1119, 1121, 1195, 1200, 1217, 1295–1297, 1303 ‘fallacy of equivocation’ response to deriving (EFQ) using disjunction, 789, 790 RM0, RM and relevance, 334, 368, 369 different versions of matrix validity, 249 on first degree entailments, 341 on modal relevant logic, 296 on natural deduction, 190 on Smiley’s four element matrix for relevant logic, 431 on Sugihara, 566 on ticket entailment, 229 terminology of consecutions, 190 the prefix ‘co-’, 546 variations on the Deduction Theorem, 193 Andreoli, J.-M., 147, 188, 352, 371 Angell, R. B., 1056 Anisfeld, M., 1301 Anna Karenina, 767 Anscombe, G. E. M., 675, 948, 1034 Anscombre, J. C., 675 anti-chains, 320 anti-commutativity, 239, 241, 738 anticipation (relation between formulas), 625–626, 1129 Antilogism (rule), 1205 antisymmetric (relations), 2, 231, 583, 894 uniqueness by antisymmetry, 583, 584

1444 antitone (± ‘with side formulas’), 490, 492, 609, 991, 996, 1049, 1076, 1190, 1259, 1264 “any”, 808 apodosis, 933 Appiah, A., 271, 1055 Applebee, R. C., 1299, 1330 approaches (to logic), 109 Åqvist, L., 192, 640, 803, 1017, 1020, 1021, 1051 arbitrary consequences, method of, 387 Ardeshir, M., 189, 370 Areces, C., 111 argument by cases, 820–843 of a function, 165, 400 premisses-&-conclusions sense vs. course-of-reasoning sense, 117, 188, 1044 Arieli, O., 1198 Aristotelian logic, 443, 1165 arity (of an operation or relation), 2 ‘Arizonan-Minnesotan’ example (D. Miller), 1138 Armour-Garb, B., 1195 (AS) – ‘antisymmetry’ quasi-identity, 231 “as if”, 948 assertion (as a speech act), 209, 303, 306, 337, 512, 634, 648, 651, 773, 804, 893, 938, 940, 953, 979, 1019, 1039, 1175, 1177 conditional, see conditional assertion Assertion (formula, axiom, schema, rule), 159, 244, 332, 672, 1067, 1092, 1097, 1103, 1108, 1110, 1113, 1117 asseverative (conditionals), 942 associated with consequence relation associated with a consequence operation (and vice versa), 56 consequence relation or gcr associated with a proof system in (Set-Fmla or SetSet), 114, 268 truth-function associated with a connective on a valuation

INDEX (or over a class of valuations), 376, 393 associativity, 7, 498, 640, 663, 749, 755, 759, 782 assumption classes, 977 assumption-rigging, 120, 124, 125, 189, 190, 341, 514, 665, 789, 1256 assumption-rules (Schroeder-Heister), 522, 527, 1077 assumptions Hilpinen’s — vs. statements, 1020 in natural deduction, 114–117 asynchronous connectives (in linear logic), 352 Athanasiadou, A., 1054 atheorematic (consequence relation or gcr), 201, 205, 208, 264, 266, 459, 1070, 1077 atomic (formulas), 48 atoms (in a lattice), 21 Austin, J. L., 941 ‘Australian Plan’ vs. ‘American Plan’, 1194, 1198, 1200, 1203 automorphisms, 27 Avron, A., 354, 371, 373, 442, 475, 525, 665, 1095–1097, 1185, 1198 fusion as extensional conjunction, 662 logical frameworks, 105 on RMI, 329 on additive and multiplicative rules or connectives, 346 on an enthymematic implication definable in RM, 1069 on hypersequents, 111, 315 on invertible rules, 150 on linear logic and the relevance tradition, 342, 345 on relevant logic, 336 on the external and internal consequence relations associated with a proof system, 268 on three-valued logics, 273 unwise use of the term ‘uniform’, 206 axiom-schemata, 156, 161 axiomatic

INDEX approach to logic, 104, 157 extension of a consequence relation, 180 axioms, 156, 160, 161 initial sequents sometimes called —, 374 Ayer, A. J. on emotive meaning, 1051 on fatalism, 827 B (implicational principle), 159, 164, 229 B (modal principle), 277, 298 B (relevant logic), 328 b subscript (→b , etc.), see #b B (implicational principle), 229 Baaz delta, 372 Baaz, M., 111, 372 Bach, K., 634, 673, 675 backward (as opposed to forward) along R-chains, 903, 1328 Bacon, J., 992 Badesa, C., 788 Balbes, R., 30, 43, 44 bands (idempotent semigroups), 737, 759 rectangular, 737–738, 752–757 varieties of, 765 Banfield, A., 767 Bar-Hillel, M., 820, 842 Bar-Hillel, Y., 1301 Barbanera, F., 673 Barendregt, H., 166, 1126 Barense, J., 483 Barker, J. A., 928, 944, 992, 1054, 1055 Barker, S. J., 812, 813, 815, 940, 944, 954, 959, 1054–1056, 1163 Barnes, J., 674, 783 Barnes, R. F., 592 Barres, P. E., 814 Barrett, R. B., 781, 816 barring (in Beth’s semantics for IL), 894 Barwise, J., 931, 1023 ‘Basic Logic’ of Sambin et al., 103, 151, 301, 370, 522 other, 370

1445 Batens, D., 1097 Battilotti, G., 103, 151, 370, 522 Bayart, A., 297 bc(·) (BCIA-algebra induced by an Abelian group), 1114 BCI logic, 164, 166, 1119–1121 monothetic version of, 1108, 1122 BCI -algebras, 231, 242 BCIA logic, 1098, 1101–1106, 1108– 1116 converses in (see also Conversion, rule of), 1326 BCIA-algebras, 1108–1116, 1122 BCIW logic, 169, 235 BCK logic, 98, 168, 235, 1121 BCK -algebras, 231, 236–242 implicative and positive implicative, 274, 1084 Beall, J. C., 298, 537, 846, 1195 Beeson, M. J., 370 “before”, 799 Belding, W. R., 1250 belief belief vs. assertion, 656 belief revision, 642, 937, 938 degrees of belief, 652 logic of, see doxastic logic Bell, J. L., 369, 924 Bell, J. M., 951 Bellin, H., 633 Belnap, N. D. (see also Anderson– Belnap), xvi, 107, 192, 334, 916, 983, 1046 his criterion of relevance (variable sharing), 243, 327, 353, 355, 361, 362, 567, 1094, 1106, 1205, 1257 his reply to Prior on Tonk, 537, 566, 569, 576 on Boolean negation, 567, 1205 on conditional assertion, 940, 1052, 1054 on conservative extension, 577, 578, 1206 on display logic, 107, 108, 605 on paraconsistent logic, 1194, 1199 on rule completeness, 129, 131 on the Two Property, 1127

1446 on the use/mention distinction, 507 on truth-value gaps and supervaluations, 842 on unique characterization, 575, 586, 627, 628 second order implicit definability (as what is involved in unique characterization), 627 Strong vs. Weak Claim, 101, 389, 915, 917 tableaux for linear logic, 345 Belnap–Thomason formulas, 983–985 Bencivenga, E., 81 Bendall normal form, see normal forms Bendall, K., 189, 578, 1175, 1176, 1178–1180, 1183 Bender, E. D., 412 Bennett, J., 927, 932, 954, 958, 1007, 1012, 1023, 1054, 1055 Benton, R. A., 283 Berger, A., 628 Bernays, P., 156, 1131, 1285 Berry, R., 970, 1055 Bertolotti, G., 875 Bessonov, A. V., 578, 623–625 Beth, E. W., 189, 304, 495, 839, 843 Beth Semantics for IL, 893–899, 902, 924 between with or, 807 Bezhanishvili, M. N., 578 Béziau, J.-Y., xvii on many-valued logic, 442 on negation, 1211 on Suszko on many-valued logic, 210 on translations, 874 Bhatt, R., 949, 951 BHK interpretation of intuitionistic logic, 304, 308, 370, 512, 893 BI (logic of bunched implications), 349 Białynicki-Birula, A., 1197, 1251 Bianchi, C., 635 biconditional, 83, 1127–1161 Bierman, G., 346 Bigelow, J., 1012, 1300 Bignall, R. J., 641 bijection, 5

INDEX bilattices, 1198 bimodal logics, 287 Birkhoff, G., 16, 29, 30, 32, 36, 44, 236, 298, 369 bisemilattices, 1050 bivalence, 195, 196, 210, 270, 810, 830, 831, 833, 975, 1164 Black, M., 688, 707, 819, 1138 Blackburn, P., 111, 297 Blakemore, D., 675 Blamey, S. R., 107, 111, 188, 190, 249, 594, 605, 1001, 1005, 1052 Blok, W. J., 97, 168, 220, 240, 245, 257–268, 275, 420, 483, 546, 574, 1069, 1307, 1309, 1311 Blok and Pigozzi’s unfortunate use of ‘normal’, 293 Bloom, Claire, 795 Bloom, S. L., 101, 218, 442, 1161 Blum, A., 1049 Blyth, T. S., 44 BN, 1200 BN4, 1200 Bochvar, D., 201, 1050 Boër, S. E., 955 Bolc, L., 111, 210, 272 Boldrin, L., 673 Bolinger, D., 951 Bonini, N., 1164 Bonnay, D., 576 Boole, G., 788 boolean (see also negation, Boolean and groups, boolean) algebras, 21, 31, 200, 223 connectives, 65 formula, 380 representability, 709 valuations (∧-boolean, ¬-boolean, etc.), 65, 82, 394 Boolos, G., 280, 283, 297, 1214 Booth, D., 100 Boričić, B. R., 140, 370 Borkowski, L., 471, 1067 Borowik, P., 111, 210, 272 Borowski, E. J., 630, 783 Bosley, R., 1055 Bostock, D., 107, 190 Boudriga, N., 732, 765 bounded lattices, 20

INDEX Bowen, K. A., 605–623, 1242 Bowie, G. L., 1007 Božić, M., 296, 323 Brée, D. S., 942, 949, 968, 1055 Bradley, F. H., 816 Brady, R. T., 925, 1197 logics without contraction, 1124 on depth relevance, 328 on RM3 and BN4, 207, 1200 on rules in relevant logic, 1097, 1205 Braine, M., 960 branching quantifiers, 53 Brandom, R., 101, 535, 1170 Breitbart, J. J., 443 Brennan, A., 1164 Brink, C., 105 Brogaard, B., 1038 Brouwer, L. E. J., 298, 302, 304, 370, 893, 1226 Brown, D. J., 101, 218 Brown, M. A., 499 Brown, R., 43 Browne, A. C., 816 Brunner, A., 546, 1250 Brünnler, K., 108, 112, 860 Bryson, Bill, 649 Bull, R. A. logics without contraction, 1124 on contraction, 671 on Galois connections, 101 on implicational LC, 555, 1335 on intuitionistic modal logic, 296 on modal logic, 277, 283, 297 on OIC, 542 on the implicational fragment of LC, 370 on Urquhart semantics and variations, 371 Bull, T., xvi, 1032, 1056 bunched implications, see BI Bunder, M. W. a logic not closed under Uniform Substitution, 192 logics without contraction, 1124 on BCI - and BCK - algebras, 231, 235, 243, 274, 1107 on ‘only’, 1055 on Aczel and Feferman, 1162 on extending BCI, 1106, 1107

1447 on intersection types, 673 on paraconsistent logic, 1285 on the Curry–Howard isomorphism, 166 on the Two Property, 1053, 1127 Burgess, J. A., 188, 192, 269, 271, 707, 1056 Burgess, J. P., 577, 772 ‘for that reason’: a conditional reading for relevant implication, 992 logics of conditionals, 1007, 1027, 1030, 1032, 1055 on CL and IL, 305 on relevant logic, 371, 667, 790 on semantics for IL, 370 on tense logic, 288 Burks, A. W., 988, 992, 1019 Burris, S., 43, 44, 407, 443, 784 Buss, S. R., 191, 192 Buszkowski, W., 109, 191 “But”, 633, 674–676 Butchart, S., xvi, 578, 1108, 1125, 1127, 1213 BV (class of boolean valuations for some language clear from the context), 65 Byrd, M., 202, 483, 701 Byrne, R. M. J., 814, 1055 C (implicational principle), 164, 229 Cahn, S., 826 Caicedo, X., 447, 448, 616, 628 Calabrese, P., 443 Caleiro, C., 181, 210, 647, 874 Campbell, R., 826 cancellation — conditions for logical subtraction, 684– 687, 1156 Shoesmith–Smiley, 206, 214, 380, 863, 923, 1261 — laws (see also semigroups, cancellation), 19, 740 left and right cancellation, 415, 741, 1111 cancelling of implicatures, 633 Cancelling-Out Fallacy (Geach), 707

1448 negation as cancellation, 540 of intuitionistic negation, 540 Cantrall, W. R., 935 Cantwell, J., 842 Carden, G., 942 Cariani, F., 811 Carnap, R., 72, 101, 102, 105, 190, 192, 483, 691, 915 Carnielli, W., 210, 1250 Carpenter, B., 271 Carroll, L., 525, 528 Carston, R., 651, 674, 1163 Casari, E., 346, 1102, 1112 Castañeda, H.-N., 960, 965 categorial grammar, 191, 636–638, 676, 810 and multigrade connectives, 783 and treating connectives as operations, 507 order-sensitive, noncommutative, 1251 Celani, S., 370 Celce-Murcia, M., 970 Cellucci, C., 140 ceteris paribus conditionals, 1017, 1019 Chagrov, A., 191, 272, 297, 630, 863, 874, 887, 922–924, 1159 chains (see also linear ordering, Rchains), 8, 17, 30, 35, 198, 269 Chalmers, D., 1041 Chandler, M., 965, 1055 Chang, C. C., 33, 44, 197, 242 Chapin, E. W., 874 characteristic functions, see functions, characteristic matrix, 272 preservation characteristics, see local/global, local vs. global preservation characteristics Chellas, B. F., 277, 278, 294, 298, 492, 497, 508, 870, 994, 995, 997, 998, 1055, 1203 Cherniak, C., 529 Chierchia, G., 816, 819 Chisholm, R. M., 930 Chiswell, I., 403 choice of meta-logic: the logic used in the metalanguage, 243,

INDEX 311, 370, 1096, 1242 Church disjunction, see disjunction Church, A., 101, 102, 168, 169, 214, 235, 330, 342, 371, 406, 630, 784, 1209, 1285 confused use of ‘commutative’, 499 Chytil, M. K., 783 Ciabattoni, A., 111, 146, 150, 191, 374 Cignoli, R., 242, 269, 616, 628 Čimev, K. N., 443 Cintula, P., 372, 673 CL, see classical logic Clark, A., 782 Clark, B., 674 Clark, D. M., 1301 Clark, R., 1012 Clarke, M. R. B., 100, 593 classical logic, 61–102, 114–163 classicality (∧-classical, ¬-classical, etc., consequence relations or gcr’s), 62, 76 clear formulas, 983–987 Cleave, J. P., 207, 249, 371, 629 Clifford, A. H., 416 Clifford, J. E., 640 CLL (classical linear logic), 351 clones, 407, 409, 443 Close, R. A., 1054 closed sets, 10 closure operations, 9 relations, 268, 919 systems, 10 Cn (consequence operation), 54 “co-” prefix, ambiguity of, 546 co-atoms, see dual atoms Coates, J., 1054 Coburn, B., 189, 843 Cocchiarella, N. B., 192 Cohen, D. H., 1054 Cohen, L. J., 270, 655, 673 Cohn, P., 16, 443 “coimplication”: a potentially confusing term, 546 collectively equivalent, 646–649, 1068 Collins, J. D., 937

INDEX combinators, combinatory logic, 165, 166, 237, 274, 673, 1099, 1126 commas as distinct from semicolons within sequents, 665 comma connectives, see connectives, ‘structure connectives’ in connection with gcr’s, 76 Lemmon’s subderivation commas replaced by dashes, 116, 125, 1255 on left and right of “”, 151, 318 additive vs. multiplicative, 342, 349, 664, 665, 1196 no uniform connectival reading, 537 on the right of “” and “”, 843– 860 replaced by semicolons in listing sequents, 123 common consequences, method of, 97, 386–388, 454, 476, 784, 845, 1167 commutativity, 7, 498, 663, 755, 782 compactness, 133 comparatives, 806–808, 1102 complementizers, 972 complements, complementation, 14, 21, 22, 274 complemented lattices, 21 completeness (other than semantic) functional, see functional completeness Halldén, see Halldén completeness Kuznetsov, see Kuznetsov completeness Post, see Post completeness structural, see structural completeness completeness (semantic), 127 abstract (‘instant’) completeness theorem (for gcr’s), 75 abstract completeness theorem (for consequence relations), 59 completeness of a natural deduction system for classical logic, 128

1449 of a modal logic (in Fmla) w.r.t. a class of frames, 283 of a modal logic (in Fmla) w.r.t. a class of models, 280 rule completeness, 129, 182, 187 in modal logic, 876 complexity degree of complexity of a logic, 272 of a formula, 48 composition (of functions), 9, 19, 413 compositionality, 208, 209, 211, 271, 636 of translations, 536 compositionally derived, see derived operations Comrie, B., 929, 943, 1054 ‘conceptivism’, 772, 925 concessive (clauses, conditionals), 676, 957 conditional assertion, 938–940, 959, 1052, 1054 Conditional Excluded Middle, Law of, 959, 1008, 1010, 1013, 1045 Conditional Proof (see also rules, (→I), 81, 115 four forms of, 975 conditionals, 925–1056 consequential, 943–944 indicative, 927–947, 1038–1044 as material implication, 926 projective (in Dudman’s sense: see also hypotheticals), 929, 932–934, 941 ‘sideboard’, 941, 942, 949 subjunctive, 802, 927–932, 935– 937 ‘counterfactual fallacies’, 1034 semantics for, 987–998, 1007– 1034 conditions induced by a determinant (on a gcr or consequence relation), see determinantinduced conditions congruence (relations) congruence connective, 1153 congruences on an algebra, 27, 220, 254, 498 formula-definable (Porte), 223

1450 matrix congruences, 259 congruentiality, 175, 246, 422, 484– 507 ↔-congruential, 484 ‘with side formulas’, 455 congruential modal logics (in Fmla), 877 Coniglio, M., 210 conjunction, 15, 62, 631–677 additive vs. multiplicative, 347 deductive, 51, 773 generalized conjunction, 97 intensional (see also fusion), 658– 661 probabilistic, 653 strong conjunction, 632, 673 temporal, 639–641 ‘theorem conjunction’, 575 conjunctions (in the grammarians’ sense), 635 conjunctive — combinations (on the left or right of a relational connection), 12, 134, 286 — normal form, see normal forms connected (relations or frames), 855 weakly, 856 connection, relational, see relational connection connectival, see non-connectival operations on formulas connectives abstract vs. concrete conceptions of, 47–53 hybrids of, 461–484, 576, 750– 765, 1168, 1277 individuation of (logical role vs. syntactic operation), 53, 82, 87–90, 324, 376, 379, 461, 539, 590, 617, 1206 multigrade, 53, 630, 783, 1150 ‘structure connectives’, 107, 577 subordinating vs. coordinating, 636, 933–934, 965 connexive (logic or implication), 659, 668, 925, 959, 1048, 1056 ‘connexivism’, 925 consequence operations, 54 consequence relations, 55

INDEX generalized, see generalized consequence relations maximally non-trivial, 397 the phrase “(generalized) consequence relations”, xv consequent-distributive (connectives), 173 consequent-relative (versions of connectives in general), 572 Consequentia Mirabilis, 1258 consequential, see conditionals, consequential conservation of synonymy, 1233 conservative (operation), 479, 480 conservative extension, 307, 335, 363, 368, 369, 439, 536–556, 566– 569, 576, 577, 607, 615, 623, 624, 626, 683, 685, 687, 713, 723, 743–745, 748, 785, 838, 871, 891, 1080, 1112, 1137, 1191, 1204, 1206–1208, 1225, 1232, 1236, 1252, 1319 nonconservative extension involving quantifiers, 547, 626, 722 of a proof system vs. of a consequence relation, 539 of theories, 721 consistent proof system, consequence relation or gcr, 248, 575 set of formulas, 205, 281, 310 valuation consistent with a consequence relation, 58 valuation consistent with a gcr, 74 ‘constant-valued’ logic, 91, 382, 742 constants individual, 33, 36, 443 ‘logical constants’, 180, 193, 511, 533 sentential or propositional, 48, 100, 341, 350, 371, 380, 484, 566, 622, 760, 865, 1097, 1181, 1260, 1266 constructive (logics), 875 content domain (Sweetser), 674, 794, 941, 970 contexts, 424 n-ary connectives vs. n-ary contexts, 49, 393, 637, 709, 995

INDEX 1-ary — in CL and IL, 451 an unrelated use of the term, 347 contingency operator, 490 contra-classical logics, 1048, 1101, 1113, 1325 profoundly, 536 Contrac (Contraction axiom), see also W, 331 contraction ∼-Contrac, 1205 ¬-Contrac, 355, 1098, 1187, 1189, 1242, 1259, 1264, 1268 ¬-Contracres , 1190 Contrac (see also: W ), 331, 333, 353, 356, 359, 908 Contrac (for → ), 1033 hidden, 143, 1249 in belief revision, 645 structural rule, see rules, structural contractionless logics, 1098–1121 contradictories, 815, 1165, 1167 ctd operation, 49, 52, 680 contradictoriness as a property of formulas or propositions, rather than a relation between them, 21, 1217 contradictory disagreement (Grice), 790 contradictory negation, 1169 ‘the contradictory function’, 817 contraposition for conditionals, 960, 1034 schema, 178, 340 selective, 1259 simple, 430, 431, 1049, 1190, 1223 simultaneous, 207, 429, 430, 1259 contraries/contrariety, 93, 505, 583, 849–850, 1112, 1163–1172, 1223–1225 compositional contrariety determinant, 438, 1165 contrary determinants, 382, 385 conventional implicature, see implicature convergent (frames), see also piecewise convergence, 312, 856

1451 conversational implicature, see implicature converse — proposition fallacy, 72–73, 102, 278, 284, 914 ‘converse effect’, 1113 Converse Subj(unctive) Dilemma, 1016–1022 implicational converses, 1325 Conversion (rule of), 1105, 1109 Cook, R. T., 576 Cooper, D. E., 270, 1056 Cooper, W. S., 192, 210, 812, 813, 940, 1044–1053, 1056, 1439 coordination (see also connectives, subordinating vs. coordinating), 674 Copeland, B. J., 269, 297, 336, 371, 1203 Copi, I., 1019 Corcoran, J., 520, 1182 Cornish, W. H., 235, 239, 274 correctness functions, 1176 Corsi, G., 300 coset-validity, 254 Costello, F., 1164 Cotard’s syndrome, 1164 counterfactuals, see conditionals, subjunctive creative (definitions), 721 Cresswell, M. J. (see also Hughes– Cresswell), 283, 296, 604, 924, 1046, 1154 Cresswell, W. D’Arcy, 634 Crolard, T., 189, 546 cross-over property, 1, 13, 15, 16, 69, 136, 286, 730, 732, 733, 736, 1050 Crossley, J. N., 489, 1041 Crossman, V., xvii Crupi, V., 1164 ctd, see contradictories Čubrić, D., 405 Cunningham-Green, R. A., 443 Curley, E. M., 928 Curry’s Paradox, 317, 1098, 1123– 1127, 1129 biconditional variant of, 1213 Curry, H. B. (see also Curry’s Paradox), 142, 169, 274, 318,

1452 319, 636, 780, 860, 1121, 1126, 1181, 1211, 1252, 1274, 1275, 1285 Curry–Howard isomorphism, 166, 513 Cut Elimination, 146, 147, 191, 351, 363, 364, 366, 515, 605, 608, 614, 861, 1125, 1219 cut product (of a pair of sequents), 135, 387 cut rule, see rules, structural cut-inductive (rules, connectives), 191, 365–368, 374, 614, 1246 Cutland, N. J., 301 Cuvalay, M., 1054 CV (constant-valued gcr), 91, 382 Czelakowski, J., 97, 100, 167, 172, 175, 186, 187, 190, 215, 218, 220, 265, 275, 456, 483, 485, 1136, 1158 D (modal principle), 277 D’Ottaviano, I., 242 Dale, A. J. a subtlety concerning independence of axioms, 185 on Dudman on conditionals, 932 on Post complete extensions of implicational IL, 1101, 1217 on Smiley’s matrix for relevant logic, 1197 on transitivity and indicative conditionals, 992–993 Dalla Chiara, M., 300, 369 Dancy, J., 1055 Dancygier, B., 943, 944, 949, 950, 968, 971, 993, 1054, 1055 Daoji, M., 1108, 1110, 1115 Ðapić, P., 1330 Dardžaniá, G. K., 1124 Dascal, M., 675 Davey, B. A., 16 Davies, E. C., 929, 1054 Davies, M. K., 192, 707, 935, 1289 Davis, W., 529, 536, 634, 673, 936 Davoren, J., 371 DD (Double Disjunction logic), 64, 599 de Bruijn, N. G., 227, 556 de Cornulier, B., 955 de Jongh, D., 297, 578, 875

INDEX de Lavalette, G. R. R., 425 de Mey, S., 1055 De Morgan De Morgan algebras (or lattices), 22, 44, 1198 De Morgan Logic (KC), 319 De Morgan’s Laws, 14, 306, 310, 319, 355, 356, 364, 475, 553, 556, 1045, 1049, 1179, 1196 Jennings on, 809, 810 negation, see negation, De Morgan de Oliveira, A. G., 189 de Paiva, V., 346 de Queiroz, R., 111, 189 de Rijke, M., 297 decidable (formulas), 319 Declerck, R., 1054 Decontraposition (schema), 178 Dedekind, R., 19 deduction Deduction Theorem, 157, 160, 164–180, 193, 229, 330, 1279 for BCIW, BCKW and BCK (1.29.10), 168 for BCI (1.29.9), 167 local, 167 of a formula from a set of formulas, 157 deductive disjunction, 50, 335, 420, 555, 773, 892, 1061, 1222 ‘deep inference’, 108 definability implicit second order — and unique characterization, 627 modal — of a class of frames, 284, 286, 565, 854, 856, 877 of connectives, 418–423, 608 in IL and LC, 419–422 strict (Umezawa), 421 strong (Prawitz), 421 sequent — of a class of valuations, see sequents defined connectives: object-linguistic vs. metalinguistic view, 423– 426, 443 definition implicit, 627 of connectives, 423–426

INDEX of non-logical vocabulary, 720– 729 generalized, 725 demi-negation, 576 Denecessitation (rule), 289, 505, 853, 873 Dénes, J., 443, 1161 denial, see rejection dense (relations), 281 dependence (of function on argument), 411, 413, 718 Depossibilitation (rule), 876 depth relevance, 328 derivable (rules), see rules, derivable derived objects and relations (tuple systems), 37–43 derived operations algebraically derived, 25 compositionally derived, 25, 43 -compositionally derived, 24, 26, 43, 403, 404, 406–408, 418, 419, 443 s-compositionally derived, 407 designated values (in a matrix), 199, 273 designation-functionality, 208 detachment (Detachment-Deduction Theorem, Rule of Detachment), 179 determinant-induced conditions, 211, 377–402 determinants (of a truth-function), 377–378, 442 notion extended to a many-valued setting, 939 simple, 389 “determinately”, 196, 658, 832 determined consequence relation determined by a class of valuations, 57 fully determined, partially determined, completely undetermined (connective, according to a gcr), 378 logic determined by a class of frames, 283 logic determined by a class of models, 280 Deutsch, M., 1164

1453 di-propositional (constants), 1260 dialetheism, 1194 Diaz, M. R., 168, 192, 336, 371 Diego, A., 227, 231, 234, 235, 274, 1058 Dienes. Z. P., 273 Dietrich, J., 227, 1210 DINat (proof system for dual intuitionistic logic), 1224 direct product of algebras, 28, 29, 33, 255, 412 of matrices, 212–216, 271, 464 of models, 33, 35, 37, 44 direct sum (of matrices), 213, 272 Dirven, R., 1054 discourse connectives, 638 disjoint unions (of frames), 285, 788 disjunction, 15, 62, 65, 767–924 additive vs. multiplicative, 349 Church disjunction, 235, 1069, 1129 connected (subsumable), 1301 consequent relative, 99, 572 Cornish disjunction, 235 deductive, see deductive disjunction Dyirbal disjunction, 641, 795– 798, 808, 817 exclusive, see exclusive disjunction generalized disjunction, 97, 99 intensional, 789–790, 798, 1205 ‘model-disjunction’, 855 probabilistic, 826 pseudo-, see pseudo-disjunction rule of, 873 ‘strong disjunction’ (Grice), 790 ‘theorem disjunction’, 575 weak disjunction, 887–888 whether -disjunction, 560, 770 Disjunction Property, 311, 312, 314, 861–862, 875, 877–893, 922, 1223, 1236, 1263, 1273, 1276 n-ary -Disjunction Property, 874 disjunctive — combinations (on the left or right of a relational connection), 12, 286 — normal form, see normal forms

1454 — syllogism, 341, 789, 1196, 1205, 1223 display logic, 107, 522, 577, 1251 distinctness implicature, 780 ‘distributes over’: ambiguity of this phrase, 557 Distribution Law (see also lattices, distributive), 299–301, 340, 341, 351–353 Divers, J., 296 Dixon, R., 796, 797, 817 DLat (proof system for distributive lattice logic), 248 dominance reasoning, 826, 842 Došen, K., xvi, 444, 646 axiomatics of negation, 1185 history of substructural logics, 1124 intuitionistic double negation as a single operator, 1180 on category theory and cut elimination, 191 on category theory and logic, 193 on intuitionistic double negation as a single operator, 323, 462, 1250 on intuitionistic modal logic, 296 on logical constants, 193 on logics weaker than IL, 370 on modal translations of IL, 874 on modal translations of substructural logics, 874 on semantics for substructural logics, 1093, 1121, 1124 on sequent to sequent rules, 191 on special logical frameworks for modal logic, 605, 860 on strict negation, 1169 on the history of relevant logic, 371 on two-way rules, 151 on unique characterization of connectives, 577, 584, 586, 600, 627 semantics for substructural logics, 371, 374 Dowty, D., 641 doxastic logic, 276, 277, 297, 483, 656

INDEX Dragalin, A. G., 321, 370, 1125 Driberg, T., 793 DS, see disjunctive syllogism dual intuitionistic implication, 546 logic, 546, 1250 negation, see negation, dual intuitionistic duality dual atoms (in a lattice), 21 dual of a consequence relation, 102 dual of a gcr, 93 dual of a rule, 348 Galois duality, 3 generalized Post duality, 410 lattice duality, 8 poset duality, 3 Post duality, 405 Dudek, W. A., 245 Dudman, V., 781, 782, 790, 816, 929– 935, 941, 942, 960, 993, 1041, 1054, 1056 Dugundji, J., 891, 1159 Dummett, M. A., 163, 186, 188, 196, 209, 210, 270, 303, 370, 940, 1174, 1175, 1250 ‘oblique’ rules, 521 ‘pure and simple’ rules, 312, 519, 1182 a sense in which the classical truth-tables are correct for IL, 1174 comparison of semantics for IL, 370, 897, 924 defining disjunction in Heyting arithmetic, 1182 LC and KC (intermediate logics), 312, 313, 319, 370, 420, 555, 874, 892 long-winded definition of a binary connective, 1068 on ‘but’, 674, 675 on ‘smoothness’, 882 on bivalence, 209, 270, 1044, 1164 on conditional assertion, 939, 940 on cut elimination, 191 on dual intuitionistic logic, 1226 on fatalism, 822, 827 on harmony, 525, 528

INDEX on matrix methodology, 272 on philosophical proof theory, 511, 519, 520, 535, 586 on quantum logic, 299, 301, 369, 826 on rejective negation, 1211 on rules derived from truth-tables, 1175 on the BHK interpretation of the language of IL, 303, 543 on the business of logic, 189, 882 on the rationale for many-valued logic, 209–211, 1230 on vagueness, 271 sympathy for IL, 303 Duncan, H. F., 32 Dunn, J. M., 223, 245, 263, 272– 274, 342, 344, 371, 567, 665, 1046, 1054, 1097, 1098, 1110, 1121, 1194, 1197, 1200, 1201, 1207, 1208, 1210, 1336 mistaken characterization of algebraizability, 275 on ‘dummying in’, 1296 on Boolean negation, 567, 1205 on conditional assertion, 1052 on Curry’s Paradox, 1125 on distribution types, 551 on Galois connections, 101 on harmony, 528 on LC, 319, 370 on negation, 1186, 1211 on residuals, 544 on RM, 339, 362 on the Urquhart semantics for R, 907, 1093, 1210 propositions as equivalence classes, 222 relevant logic, 332, 341, 344, 371, 668, 1094, 1203 semantics for linear logic, 373 the ‘American Plan’, 1195, 1198, 1199 Durieux, J. L., 732, 765 Dwinger, P., 30, 43, 44 Dyckhoff, R., 321, 1125 Dyirbal, see disjunction, Dyirbal dynamic

1455 dynamics of belief, see belief revision logic (modal logic of programs), 276, 297 semantics for conjunction, etc., 642 E (modal principle), 298 E (relevant logic), 342, 663, 1091, 1092, 1095, 1195, 1296, 1297, 1303 Edgington, D., 673, 927, 931, 940, 1055 Eells, E., 1055 (EFQ), 119, 313, 580, 1252–1254, 1259 (EFQ)# , 1166 and assumption rigging, 341 Anderson and Belnap on, 789 “either”, 817 El-Zekey, M. S., 53 eliminability (and definitions), 721– 725, 1288 uniform, 418 elimination rules, see rules, elimination major premiss, 513 Ellis, B., 846, 936, 938, 980, 982, 1012, 1022 emotive meaning, 1051 Endicott, J. G., 676 endomorphisms, 27 Enoch, M. D., 1164 entailment A. Avron on, 1095 as a binary relation, 108 as opposed to implicature, 952, 954 as strict implication, 276 E. J. Nelson on, 658 in the Gärdenfors semantics, 643 in the relevance tradition (see also E), 328, 663, 1091, 1195, 1304 tautological (or first degree), 341, 1156, 1194 ‘entropic’ (groupoids), 1161 epistemic logic, 276, 471, 483 Epstein, G., 272 Epstein, R. L., 371, 874

1456 equational logic, 29, 35, 43, 258, 486, 737 generalized, 255 equivalence, 82–87, 1127–1161 equivalence relations, 3 frames with accessibility an equivalence relation, 286 generalized equivalence (in the sense of McKee), 1150 logical equivalence, 83 set of equivalence formulas, 98 equivalential algebras, 1130 combinations on the left/right, 16, 1131 double meaning of the term ‘equivalential’, 170, 1128 fragment of CL, 1103, 1116, 1128– 1135 fragment of classical predicate logic, 723 fragment of IL, 1130 logics, 98, 170, 223, 264, 265, 1153 Ernst, Z., 335 Erteschik-Shir, N., 1055 Ertola, R. C., 628, 1240 Esakia, L., 321 essentialism (about connectives), 427, 1192 essentially n-ary (function), 411 Esteva, F., 269 euclidean (relations), see also generalized euclidean, 281 evaluations (matrix evaluations), 199 as opposed to valuations, 199, 206 valuations induced by, 207 “even”, 953, 957 “even if”, 946, 949 Everett, C. J., 16 Evnine, S., 529 Ewald, W. B., 296 Ex Falso Quodlibet, see EFQ “except”, 678 Exchange (structural rule), 143 Excluded Middle Law of, 119, 145, 199, 303, 306, 309, 356, 585, 827, 829, 958, 1169

INDEX added to ML yields LD, 1274 multiplicative form, 357 not requiring bivalence, 831 Weak Law of, 312, 1222, 1329 exclusionary disjunctions, 781, 788 exclusive conjunction, 779 exclusive disjunction, 398, 709, 780– 788 existence of connectives with prescribed logical properties, 536–578 existential existential formulas in model theory, 33, 44 propositional quantifiers, 1325 provability as an existential notion, 128 quantifier ambiguous according to Paoli, 789 quantifier constructively interpreted, 306 quantifier elimination rule, 528 quantifier in dynamic semantics, 642 quantifiers and Skolem functions, 734 expansion in belief revision, 645 of an algebra, 18 structural rule, see rules, structural exponentials, 345–346, 351, 372 extensional (consequence relations, gcr’s, connectives w.r.t.), 444– 484 extensionality on the left/right (of a relational connection), 2, 12 extractibility (of a variable from a formula), 727–728, 765 extraposed version (of a given connective), 554, 556 F (Church falsity constant ‘Big F ’), 342, 343, 351, 355, 358, 566, 567, 569, 1193, 1208 F: 1-ary constant False truth-function, 398, 406 F: truth-value (falsity), 57

INDEX f (Ackermann falsity constant ‘little f ’), 342–344, 350, 351, 357, 368, 567, 1102, 1193, 1203 f (temporary notation for a contravalid formula), 434 F (tense operator), 287, 832, 1295 F, see frames Faber, R. J., 842 Faggian, C., 103, 151, 301, 370, 522 faithful (translations), 259 Falk, R., 820, 842 fallacies ∧-Elimination as fallacious, 660 affirming the consequent, 939 conditional fallacy, 1164 conjunction fallacy, 1164 counterfactual fallacies (strengthening the antecedent etc.), 1034–1038 denying the antecedent, 939 disjunction fallacy, 1164 fallacies of modality, 1091, 1304 fallacies of relevance, 1091, 1304 fallacy of suppression, 345, 1094, 1095 fatalism, 196, 822, 826, 832, 833, 842 Fauconnier, G., 941, 954 Feferman, S., 193, 1162 Fennemore, C., 765 Fermüller, C., 111 Ferreira, F., 820 Ferreira, G., 820 Field, H., 656 Fillenbaum, S. (see also Geis–Fillenbaum Equivalence), 673, 770, 771, 950, 951, 955, 964, 966, 968, 1055 Fillmore, C. J., 1054 filters, 27 Fine, K. his treatment of disjunction, 903, 910 on incomplete modal logics, 283 on many-valued logic, 269, 270 on negation as failure, 1164 on semantics for relevant logic, 899, 1202 on supervaluations, 842

1457 on vagueness, 830 finite model property, 212, 228, 273, 280, 624, 884 finitely approximable (logics), 273 Fischer-Servi, G., 296 Fisk, M., 779, 791, 798 fission (see also additive and multiplicative rules or connectives and disjunction, intensional), 143, 340, 789, 798 Fitch, F. B., 116, 190, 296, 508, 525, 546, 1000, 1077, 1251 Fitelson, B., 335, 1102, 1128, 1164 Fitting, M., 44, 101, 290, 298 fixed point equivalence, 1126 “fixedly”, 1289 ‘flat’ (conditions), 713, 720 Fleischer, I., 291 fm(σ) formula corresponding to a sequent for classical logic, 127 for relevant logic, 331 Fmla, Fmla-Fmla, see logical frameworks focus, 955, 1055 Fogelin, R., 630, 1056 Føllesdal, D., 297 Font, J. M., 97, 179, 218, 242, 264, 265, 268, 296, 484, 1197 Forbes, G., 296, 812 Forder, H. G., 275, 1122 formulas, see languages Forster, T., 291, 1008 Foulkes, P., 659 Fox, J., 1197 fragment (of a language), 52 fragments, 52 frame consequence, 289 frames expanded — in modified Urquhart semantics, 907 for intuitionistic and intermediate logics, 307 for modal logic, 282–288 Gärdenfors —, 642 semilattices as — in Urquhart’s semantics, 906 framework(s), logical, see logical frameworks

1458 Francescotti, R. M., 954 Francez, N., 676 Frank, W., 1139 Franke, M., 674 Fraser, B., 949, 953, 957, 1055 free algebras (see also absolutely free), 30, 31, 224 Freeman, J., 1157 Frege, G., 103, 104, 169, 271, 499, 501, 530, 675, 1177 axiom named after, 169 Fregean (consequence relations), 456, 1158 French, R., xvi Friedman, K. S., 842 Fuhrmann, A., 296, 707 Fujii, S. Y., 948 Fujita, K.-E., 140 ‘full model’, 1299 fully modalized (formula), 295 functional — completeness, 403–409, 442 strong, 403 — dependence (Smiley), 628 functionally free (algebras), 31, 45, 756 functions characteristic, 11, 72, 373, 500 injective, 5, 417 surjective, 5 of more than one argument, 417, 1299 fundamental operations, see operations, fundamental tuples in a tuple system, 40 Funk, W.-P., 943, 1054 Furmanowski, T., 255, 874 fusion, 143, 340, 661–671 and fission in RM, 368 as strongest formula successively implied by a pair of formulas, 344 consequent relative, 572, 671 idempotent in RM, 663 fuzzy logic, 197, 268–269, 370, 372, 667 G (Gärdenfors frame), 643 G (Gödel connective), 888

INDEX G (modal principle), 278 G (tense operator), 287, 832, 1295 G3 (Kleene sequent calculus), 152– 153, 321 Gabbay, D. M., 884, 924, 1211 “A B” notation, 995 completeness for KC, 884 definability of ∨ in LC, 555 gcr’s for IL using the Beth semantics and the Kripke semantics, 899 hypersequents, 111 intermediate logics with the disjunction property, 875 many-dimensional modal logic, 489 nonmonotonic logic, 100, 593 on a ‘tense logical’ intuitionistic connective, 1250 on gcr’s, 100, 102 on IL, 370, 846 on KP, 923 on negation, 1164 on new intuitionistic connectives, 575, 607, 614–618, 898, 1223, 1236, 1238 on rules, 161 on unique characterization, 586 sequent frameworks with labels, 111, 188 weak vs. strong classicality, 101, 392–399 Gahringer, R., 660 Galatos, N., 275, 546, 1214 Galli, A., 628, 1102, 1240 Galois connections, 3, 9, 101, 283 antitone vs. monotone, 16 gaps (truth-value gaps), 830, 1044, 1199 García Olmedo, F. M., 213 Gärdenfors, P., 641–645, 936, 938, 1054, 1055, 1300 Gardner, M., 842 Gardner, S., xvii, 771 Gargov, G., 1260 Garson, J. W. analysis of disjunction rules, 138– 140, 913, 914, 924 commenting on Belnap, 576

INDEX Garson analysis of some rules for →, 1065 incorrectly formulated condition, 140 on disjunction in the Beth semantics for IL, 924 on modularity in semantics, 919 Weak vs. Strong Claim properties, 101 Gasking, D., 772 Gauker, C., 1055, 1056 Gaukroger, S., 707 Gazdar, G., 411, 649, 673, 674, 782, 783, 790, 816, 933 GCn (global consequences of a set of rules), 1062 gcr, see generalized consequence relations Geach, P. T., 14 modal principle named after, 298 on a non-transitive implication relation, 371 on bare particulars, 779 on cancelling out, 707 on categorial grammar, 637, 676 on contrariety, 1167–1168 on Curry’s Paradox, 1125, 1206 on many-valued logic, 270 on the ceteris paribus reading of conditionals, 1019 on the role of conjunction, 645, 676 on the syntax of negation, 1163 on Wittgenstein’s N , 630 use of ‘symmetrical’, 508 Geis, M. L. (see also Geis–Fillenbaum Equivalence), 934, 936, 946, 949, 955, 971, 973, 974, 1055 Geis–Fillenbaum Equivalence, 964– 970, 974 Gen (Set-Set version of Gentzen’s LK ), 141–150, 152, 164, 184, 313, 315, 349, 369, 521 general frames, 291 generality in respect of side formulas, see rules, general in respect of side formulas validity (on a frame), 489 generalized

1459 — conjunction, see conjunction, generalized — consequence relations, xv, 55, 72–82, 843–850, 854–860 — disjunction, see disjunction, generalized — euclidean (relations), 857 ‘generalized equations’, 255 ‘generalized piecewise’, see gpwgenerated clone generated by a set of functions, 407 freely generated by, see free algebras point-generated subframe or submodel, 285–286, 309, 313, 320, 602, 745, 852–854, 861, 872, 875, 879, 884, 1221, 1276, 1280, 1313, 1328 subalgebra generated by a set of elements, 28, 1057 ‘generic’, see functionally free (algebras) Gentzen, G., xvi, 41, 90, 102–105, 109, 114, 141, 142, 145, 146, 151, 153, 189, 190, 304, 306, 313, 314, 320, 345, 427, 511– 513, 515, 516, 524, 535, 588, 594, 977, 1125 Georgacarakos, G. N., 1284 George, H. V., 1054 Gerhard, J. A., 765 Ghilezan, S., 166 Giambrone, S., 371, 1124, 1197 Gibbard, A., 1054 Gibbard, P., 1211 Gibbins, P., 301, 369 Gil, A., 200, 268 Gil, D., 649 Gillon, B. S., 771 Gindikin, S. G., 405, 407, 442, 443 Ginsberg, M., 100, 1198 Girard, J.-Y., 107, 319, 352, 613 his unprovoked attack on (∨E), 820 linear logic, 229, 313, 327, 342, 345–346, 351, 371, 669 on (legitimate) connectives, 323 on contraction and infinity, 1124

1460 on cut elimination, 191, 351 on the significance of Identity and Cut, 613 proof-nets, 189 semantics of linear logic, 373 unified logic, 107 Girle, R., 817 Giuntini, R., 300, 369 Glivenko’s Theorem, 305, 306, 316, 317, 618, 626, 743, 881, 1214– 1222, 1335 fails for intuitionistic predicate logic, 305, 626 fails for ML, 1271, 1317 weakest supraminimal logic satisfying, 1272 Glivenko, V., 306 Glo(·), 136, 1062 global, see local/global gluts (truth-value gluts), 1199 Goad, C. A., 578 Goddard, L., 273, 442, 788, 816, 1051, 1228 Gödel, K., 202, 211, 227, 298, 305, 874, 888, 891–893, 901, 902, 922, 1159, 1216, 1218 Gödel connective, 888 Gödel matrix, 171, 1220 Godo, L., 269 Goguen, J. A., 269 Goldberg, S., 842 Goldblatt, R., 189, 283, 297, 298, 300, 302, 874, 919–921, 1186, 1188, 1190–1192, 1197, 1285 Goldstein, L., 818, 1125, 1164 Goldstick, D., 1138, 1300 Gonçalves, R., 181, 647, 874 Gonseth, F., 270 Goodman, Nelson, 678, 818, 991, 996, 1022, 1024, 1138 Goodman, Nicolas, 304, 1226, 1250 Goodstein, R. L., 405 Goré, R., 108, 112, 551, 1250, 1251 Gottschalk, W. H., 443 Gottwald, S., 269, 1169 Governatori, G., xvi gp(·) (Abelian group induced by a BCIA-algebra), 1114 GPD, see duality (generalized Post)

INDEX gpw -connected, convergent (frames), 856–859 Graczyńska, E., 371 Grandy, R., 577 Grätzer, G., 16, 23, 43, 924 greatest lower bound, 7 Greechie, R., 300 Green, G. M., 674 Green, K., 270 Greenbaum, S., 949, 969 Grice, H. P., 632–634, 639, 673, 675, 677, 772, 782, 790, 793, 955, 980, 1038 Griggs, R., 771, 817 Grišin, V. N., 1124 Groenendijk, J., 560, 641, 642, 770 groupoids, 18, 43 groups, 18 Abelian, 19, 1114–1115, 1122 boolean, 413, 741, 782, 1116 grue, 818, 1138 Grzegorczyk, A., 304, 526, 775 Guessarian, I., 442 Guglielmi, A., 108, 1130 Gunter, R., 674, 675 Gurevich, Y., 1229, 1251 Guzmán, F., 1084, 1086, 1325 H(·) or HV (·) – set of verifying valuations (in V ), 495, 774 H (tense operator), 287, 832, 866, 1295 Haack, S., 269, 628, 842 Hacking, I., 148, 193, 317, 536, 577, 614, 619, 990, 1169 Hackstaff, L., 157 Haegeman, L., 941, 1054 Hähnle, R., 271, 272 Haiman, J., 674, 933, 1054, 1055 Hájek, A., 653 Hájek, P., 269, 484 Halbasch, K., 630, 783 Hall, P., 443 Halldén, S., 863 Halldén-completeness, 51, 205, 763, 861–872, 923, 1276, 1278, 1281 global, 871 Kracht’s sense vs. ours, 923 Hallett, M., 369

INDEX Halmos, P., 222, 273, 498, 819 Halpern, J. Y., 192, 298 Halpin, T., 1055 Hamblin, C. L., 641, 655, 805, 866 Hampton, J. A., 660, 1301 Hanazawa, M., 313 Hand, M., 629 Handfield, T., xvi Hanson, W. H., 489 Hansson, S. O., 645 Hardegree, G. M., 101, 263, 273, 275, 300, 302 Hare, R. M., 978 Harman, G., 535, 951, 972 harmony, 525–528, 533, 576, 614 Harris, J. H., 627 Harris, K., 335 Harris, M. B., 1054 Harrison, J., 930, 935, 1011–1013 Harrop, R., 100, 179, 218, 272, 432, 862, 878, 884 Hart, A. M., 419 Hart, W. D., 576, 627, 1174 Hartline, A., 783 Hartonas, C., 1164 Haspelmath, M., 674 Hawranek, J., 272, 1278, 1279 Hayakawa, S. I., 1164 Hazen, A. P., xvi, 101, 1121 modal application of barring à la Beth, 924 modal translations of IL, 874 on B KW , 193 on “actually”, 489 on ‘assumption rules’ in Fitch, 525 on a need to invoke the Axiom of Choice, 45 on admissible propositions, 593 on Belnap–Thomason formulas, 983 on Curry’s Paradox, 1125 on even if, 958, 1055 on generalized natural deduction, 1075, 1077 on new intuitionistic connectives, 898 on strong negation, 1251 on subminimal negation, 1185, 1268, 1269

1461 semantics of KC, 370 head-implication (relation), 1080, 1322– 1325 head-linked (formulas), 556, 1319– 1322 Heasley, B., 673 Hegarty, M., 949 Heintz, J., 768 Hellman, G., 370 ‘helping’ (and ‘non-hindering’) conditionals, 944–947, 949, 992 Hendriks, L., 625 Hendry, H. E., 419, 444, 783, 1129 Henkin, L., 43, 53, 274, 297, 448 Herrlich, H., 16 Herzberger, H., 484, 842, 1015 heterogeneous frame or class of frames, 866 logics, 111, 1185 sequents with labelled formulas, 977 strongly heterogeneous (frames), 868 Heyting algebras, 22, 23, 203, 224, 227, 257, 320, 1137, 1225, 1226 Heyting, A., 202, 302–304, 370, 1080, 1226 Hickman, R. C., 924 Higginbotham, J., 271, 802 Hil and Hil : axiom system and consequence relation named after Hilbert, 157–161 Hilbert algebras, 227, 231–235, 274, 1084 Hilbert, D., 103, 104, 156 Hilpinen, R., 297, 677, 799–804, 817, 934, 973, 1011, 1019–1022 Hinckfuss, I., 1056 Hindley, J. R., xvi, 165, 166, 274, 335, 673, 1117, 1126, 1127 Hinnion, R., 1212 Hintikka, J., 53, 271, 297, 471, 483, 774, 793, 808 Hinton, J. M., 797 Hirokawa, S., 237, 354, 358–361 Hiż, H., 177, 575, 577, 628, 1139 Hoare, C. A. R., 442 Hobbes, T., 707 Hocutt, M., 471

1462 Hodes, H., 111, 193, 273, 530, 629 Hodges, W., 34, 44, 53, 271, 403, 814 Hoeksema, J., 674, 969 Holdcroft, D., 940, 1054 holds equation holds in an algebra, 189 formula holds on a boolean homomorphism, 223 sequent holding vs. ‘being true’, 189 sequent holds at a point in a model for intuitionistic logic, 307 for modal logic, 293 sequent holds in a model Gärdenfors semantics, 644 for intuitionistic logic, 307 for modal logic, 293 Urquhart semantics, 337 sequent holds on a matrix evaluation, 199 sequent holds on a valuation for Set-Fmla, 127 for Set-Set, 134 sequent m holds in a model, 850 Hollenberg, M., 1229 Holton, R., xvii Homič, V. I., 192, 577 homogeneous frame or class of frames, 866 relational connection, 2 homomorphism, 26 matrix — (various notions), 597 Hoo, C. S., 1116 Hopper, P., 635 Hori, R., 372 Horn formulas, 32–37 metalinguistic Horn sentences of the first and second type, 64 strict, 35 Horn, A., 44 Horn, L. R., 675, 791, 808, 809, 816, 953, 1055, 1163, 1211 Hornsby, J., 707 Horsten, L., 781 Hösli, B., 475 Hosoi, T., 370, 420, 577

INDEX Hossack, K. G., 1181, 1250 Hu, Q. P., 1116 Huddleston, R., 674, 950, 968 Hudson, J. L., 678, 689, 690, 707 Huet, G., 188 Hughes, D., 189 Hughes–Cresswell (G. E. and M. J.), 278, 280, 294, 297, 298, 462, 871 on Halldén incompleteness, 865 Hugly, P., 1156–1157 Humberstone, B. D., xvi, 1161 Humberstone, J. A., 842 Hunter, G., 946 Hurford, J. R., 673, 817 Hušek, M., 16 Hutchins, E., 649, 797 Hutchinson, L. G., 674 hybrid logics, 111 hybrids, see connectives, hybrids of Hyland, M., 346 hypersequents, 111, 315, 860 hypothetical Hypothetical Syllogism (rule), 503, 992, 993, 1035 hypotheticals (in Dudman’s sense), 931–934, 941 Hyttinen, T., 53 I (implicational principle), 164, 229 I: identity truth-function, 406, 412 I (variable-identifier), 889 Iatridou, S., 934, 949 Ichii, T., 1161 Ichikawa, S., 842 (id ) condition on models for conditional logic, 995 Id (Identity axiom), see also I, 331, 340, 344 ‘Id-inductive’, 366 ideas, individuation of, 649 idempotence, 7, 476, 498, 501, 663, 755 in Menger’s sense, 501 strong, 758, 760 identity of indiscernibles, 259 identities of an algebra, 7 ‘identity connective’, 393, 709

INDEX identity element (in a group or groupoid), 18 Identity Rule, see rules, structural ‘identity-inductive’, 374 propositional, see propositional identity truth-function, see I Idziak, P. M., 233 Iemhoff, R., 879, 924 “If”, see conditionals “if only”, 949 “If you can’t see my mirrors”, 961 if–then–else, 378, 442, 670 IGen (Set-Fmla0 version of Gentzen’s LJ ), 314–316, 605 IL, see intuitionistic logic ILL (intuitionistic linear logic), 351 Imai, Y., 274 IML (intuitionistic and minimal logic), 1282 implication, 82–87, 925 additive, 349 enthymematic, 1098 in BCI, BCK, etc., see BCI, BCK logic intuitionistic (see also intuitionistic logic), 1057–1088 linear, 345, 346 relevant (see also relevant logic), 328 strict, 276, 502, 503, 552, 817, 987–990, 1169 variably strict, 990–998, 1034 implication formulas, set of, 97 implicative, see BCK algebras, implicative implicature, 673 conventional, 633, 954 conversational, 633, 954 criticisms of Grice, 634 in cinematography, 634 implicit (implicational) converse, 1325 connectives, 628 definability, see definability, implicit importation/exportation, 269, 661 for subjunctive conditionals, 1034 impurity, see rules, pure and simple

1463 INat, 114, 304 Inc (inconsistent gcr), 91, 92, 383 incomplete (modal logics), 283 independence definitional, 420, 1129 functional, see functional dependence of rules or axioms, 183–185, 195, 202 probabilistic, 657 relations of, 685–686 indiscriminate validity (in an algebra), 250–257, 475, 752, 754, 755, 757 individuation of logics, 180–188 ‘induction loading’, 1100 inevitable informal sense: ‘now unpreventable’, 827, 833 strongly inevitable (formula at a point), 896, 902 weakly inevitable (formula at a point), 896, 902 ‘inference ticket’, 979, 1092 inference-determined (consequence relation), 1142, 1161 inferential consequence, 289 ‘inferential semantics’, 535 inferentialism, 535 infinite model property, 568 initial sequents, 148 injective, see functions, injective injectivity rule, 877 ‘insertion’ vs. ‘introduction’ (rules), 143, 144 insertion (as opposed to introduction) rules, 143 intermediate logics, 172, 180, 297, 312–313, 318–321, 354, 370, 420, 542, 556, 623, 1217 fragments of, 629 those not finitely axiomatizable, 625 with the Disjunction Property, 862, 877 internal/external comparisons of connectives, 462, 483 consequence relations, 268 intervals, 641

1464 intraposed version (of a given connective), 556 introduction rules, see rules, introduction introductive (entailment), 516, 680 intuitionistic logic, 302–326 ¬-intuitionistic consequence relations, 1254 →-intuitionistic consequence relations, 84, 178, 326, 329, 441, 594 intuitionistic restriction (sequent calculus), 314, 1246, 1248 liberal vs. restrictive versions, 1248–1250 Inverse — Congruentiality (rule), 877 — Monotony (rule), 876 inverses in groups/groupoids, 18 left and right, 18 left and right, for 1-ary connectives, 541, 564, 685 of rules, see rules, invertible Inversion Principle, 513 invertible — rules, see rules, invertible — substitutions, see substitution, invertible ‘invisible contraction’, 316, 317 ‘invited inferences’, 955 involution, 44, 1195 Ippolito, M., 929, 954 Iséki, K., 239, 274, 1116 Isaac, R., 842 Ishii, T., 1155 isomorphisms, 27 isotopy, 443 Jackson, F. C., 529, 657, 658, 673, 675, 770, 772, 791, 811, 842, 928, 930, 932, 978, 979, 981, 983, 1022, 1035, 1055 logical subtraction example, 679 Jacobs, B., 372 Jaeger, R. A., 679, 684, 686, 690, 707 James, F., 773, 934, 1054 Jankov, V. A., 51, 313, 320, 370, 1298

INDEX Jankowski, A. W., 101 Jansana, R., 97, 179, 218, 265, 268, 370 Janssen, T., 271 Jaoua, A., 732, 765 Japaridze, G., 297 Jaśkowski, S., 190, 202, 211, 272, 339, 1053, 1117, 1127 Jay, N., 1164 Jayez, J., 804, 808 Jeffrey, R. C. on conditional assertion, 939, 940, 1047, 1054 on dominance, 821, 822 proof annotations, 1184 tableaux, 189 Jeffrey, W. P., 968 Jennings, R. E., 536, 769, 779, 781, 783, 796, 798, 799, 806–810, 812–814, 816 Jespersen, O., 951 Ježek, J., 765, 1330 Jipsen, P., 275, 546, 1214 Jirků, P., 273 Johansson, I., 189, 303, 313, 546, 690, 1181, 1257, 1263, 1272, 1284 Johnson Wu, K., 483 Johnson, D. L., 274 Johnson, W. E., 981 Johnson-Laird, P. N., 676, 814, 1055 Johnston, D. K., 951 join (in a lattice), 7 join-irreducible, 11 join-prime, 11 ‘joint denial’, 607 Jun, Y. B., 274 K (implicational principle), 229 K (Kleene-inspired four-element matrix), 431 K (modal logic), 277, 288, 294, 990, 1305 K (universe of a Gärdenfors frame), 643 K.2 (modal logic), 855, 875 K.3 (modal logic), 855, 875 K1 (Kleene matrix), 201, 249, 250, 271, 272, 429, 431, 459, 487, 1050, 1230

INDEX K1,2 (Kleene matrix), 201, 249, 250, 271, 272, 429, 431, 459, 487, 1200 K4 (modal logic), 293, 294, 370, 873, 990, 1305 K4.3 (modal logic), 856 K4! (modal logic), 553 K4c (modal logic), 559 Kt (tense logic), 581, 1314 Kabziński, J., 243, 245, 274, 1077, 1102, 1128, 1136, 1161, 1162 Kahneman, D., 1164 Kalicki, J., 32, 213, 272, 754, 1116 Kalman, J. A., 274, 275, 1115, 1122, 1127 Kalmbach, G., 44 Kaminski, M., 148, 524, 608, 609, 613, 615, 618, 628, 630, 898 Kamp, H., 269, 270, 635, 799, 804, 808, 830 Kanger, S., 297, 467, 1278 Kanovich, M., 345, 358 Kaplan, D., 192, 297, 298 Kaplan, J., 674 Kapron, B. M., 859, 860 Karpenko, A. S., 271, 336 Karttunen, L., 560, 953 Kashima, R., 1125, 1127 Kato, Y., 808 Katriel, T., 675 Kawaguchi, M., 296 Kay, P., 815, 942, 956, 1055 KB (modal logic), 298, 1191 KC (intermediate logic), 312, 313, 318, 320, 370, 608, 624, 874, 875, 1335 KD (modal logic), 800, 865, 868, 869, 891, 1305, 1313, 1314 KD4 (modal logic), 1305, 1314 KD! (modal logic), 288 Kearns, J. T., 207, 1056 Keedwell, A., 443, 1161 Keefe, R., 818 Keisler, H. J., 33 Kennedy, R., 506–507 Ketonen, J., 1124 Ketonen, O., 142, 148, 314, 1249 Khomich, V. I., see Homič, V. I. Kijania-Placek, K., 202, 249 Kim, J., 1161

1465 Kimura, N., 765 Kiriyama, E., 1124 Kirk, R. E., 370, 542, 875, 990 Kjellmer, G., 1055 Kle (proof system related to Kle 1 and Kle 1,2 ), 207, 249, 429, 430 and relevant logic, 340, 1195 matrix characterization of, 430 structural incompleteness of, 430 Kle 1 and Kle 1,2 (proof systems for Kleene matrices), 207, 429 atheorematic nature of the former, 208 inadmissibility of contraposition for, 430 intersection of associated gcr’s, 249 Kleene, S. C., 143, 151, 153, 193, 874, 1050, 1193 Kleene matrix, see K1 , K1,2 Kneale, M., 140 Kneale, W., 140, 151, 537 Kolmogorov, A. N., 303, 304, 370, 893, 1215, 1272, 1285 Komori, Y., 197, 233, 234, 236, 237, 242, 274, 374, 863, 924, 1124 König, E., 1055 Koopman, B. O., 842 Korzybski, A., 1164 Koslow, A., 94, 100, 108, 587, 1301, 1336 simple or restricted treatment of ∧, 629 Kowalski, R., 1055 Kowalski, T., 275, 546, 578, 1108, 1120, 1214 KP (intermediate logic), 313, 630, 875, 877, 883–888, 923 Krabbe, E. C. W., 1007 Kracht, M., 297 his variant notion of Halldén completeness, 923 terminology of global Halldén completeness, 871 Kratzer, A., 805, 817, 956, 972, 1054 Kreisel, G., 304, 311, 313, 578, 616, 875, 877, 883, 923 Kremer, M., 190, 1054 Kremer, P., 49

1466 Krifka, M., 674, 808 Kripke semantics for intuitionistic logic, 307 for modal logic, 278, 291 Kripke, S. A., 297, 304, 307, 924 on Halldén completeness, 864, 867 on supraminimal logics, 1277, 1285 on the ‘rule of disjunction’, 872, 923 relevant logic treated substructurally by disallowing Weakening, 371 Krolikoski, S. J., 270, 484 KT (modal logic), 277, 283, 294, 990 KT4 (modal logic), see S4 KT5 (modal logic), see S5 KT! (modal logic), 278, 286, 472, 605, 1168, 1305 KTB (modal logic), 298, 1191 KTc (modal logic), 294, 472, 868, 1305 Kuhn, S., 53, 296, 491, 810, 1185, 1260 Kurucz, A., 489 Kuznetsov completeness, 423, 444, 619 Kuznetsov, A. V., 370, 422, 423, 444 KVer (modal logic), 278, 286, 472, 602, 605, 865, 868, 869, 1305 Ł (implicational principle), 197, 242 Ł3 (see also Łukasiewicz, J., threevalued logic), 196 Łω , see Łukasiewicz J., infinite-valued logic l’Abbé, M., 448 Ladd-Franklin, C., 964 Ladusaw, W., 808 Lafont, Y., 52, 371 Lahr, C. D., 274 Lakoff, R., 673–675, 770, 771 lambda — calculus, 165, 535, 673 — notation (λ), 165, 406 with propositional variables, 425 Lambek Calculus, 229, 372, 637, 1251

INDEX Lambek, J., 109, 193, 372, 637, 1124, 1251 Lance, M., 937 Lane, D., 193 Langton, R., 1301 language-dependence objections, 1138– 1139, 1298 languages, 47–54 Larsen-Freeman, D., 970 Lat (proof system for lattice logic), 246 Latocha, P., 191 lattices, 7, 16 distributive, 8 orthomodular, 302 Lau, J., 1164 Lavers, P., 645, 707 Law of Excluded Middle, see Excluded Middle, Law of Law of Triple Consequents, see Triple Consequents, Law of LC (intermediate logic), 180, 312, 313, 318–320, 322, 335, 362, 370, 420, 540, 542, 555, 624, 874, 875, 892, 1050, 1130, 1267, 1321, 1335 converses in, 1326 implicational fragment of, 555 LCn (local consequences of a set of rules), 1062 LD (Curry’s supraminimal logic), 1274 LE Curry’s supraminimal LE, 320, 1274, 1275 four-valued in Fmla but not in Set-Fmla, 1277 Halldén-incompleteness of, 1276 Left Expansion (LE), 361 Left Extensionality condition (LE), 454 least — common thinning (of a pair of sequents), 135, 369 — upper bound, 7 Leblanc, H., 101, 508, 543, 577, 656, 677, 1131 Leech, G. N., 936, 1054 left-prime consequence relation or gcr, 110, 588, 755, 835, 917, 1074

INDEX validity property, 252 left-reductive (operations), 416 LeGrand, J. E., 820 Lehmke, S., 667, 673 Lehrer, K., 819 Leibniz, G. W., 279, 296, 731 Leivant, D., 977 Lejewski, C., 420 Lemmon, E. J., 116, 119, 121, 125, 142, 181, 183, 189, 190, 277, 297, 298, 304, 306, 312, 313, 332, 336, 370, 515, 520, 543, 603, 626, 665, 722, 723, 843, 860, 872–875, 923, 976, 977, 1056, 1083, 1186, 1256, 1276, 1278, 1284 commas vs. dashes, 125 Lemmon-style proofs, 116 on “therefore”, 103 on Halldén completeness, 864 on logical relations, 503, 1166 source for Nat, 114 Lenzen, W., 297, 1229, 1250 Leśniewski, S., 676, 721, 1128, 1133, 1134 Lev, I., 206, 442 Levesque, H., 529, 1301 Levi, I., 938 Levin, H. D., 676 Lewin, R. A., 1102 Lewis, C. I., 276, 278, 330, 341, 707, 789, 1007, 1205 Lewis, D., xvii, 707, 979, 1007, 1300 a problem in representing some conditional constructions in his language, 1009 inner and outer modalities, 1026 on ‘counterfactual fallacies’, 1034 on ‘would’ vs. ‘might’ counterfactuals, 1008 on a statement’s being about a certain subject matter, 829 on categorial grammar, 676 on Centering, 1011 on completeness for logics of conditionals, 1007 on compositional semantics, 271, 636 on conditional obligation, 826

1467 on conditional probabilities, 567 on counterpart theory, 1023 on disjunctive antecedents, 1022 on implicature, 935, 1012 on natural properties and genuine resemblance, 1301 on permissibility statements, 804 on possible worlds, 296 on propositionally indexed modalities, 994 on Stalnaker’s Assumption, 1008, 1009, 1013, 1039 on subjunctive conditionals, 802, 934, 936, 962, 987, 994, 996, 1011, 1017, 1027, 1054, 1055 on the Limit Assumption, 1014, 1015 on the semantics of questions, 560, 770 on two-dimensional modal logic, 489 on vagueness and supervaluations, 830 on whether worlds must be comparable in respect of similarity to a given world, 1014 respects of similarity in the semantics of counterfactuals, 1023 treatment of context for counterfactuals differs from context for knowledge ascriptions, 1038 Leśniewski–Mihailescu Theorem, 1136 Libert, T., 1212 Lindenbaum — algebras, see Tarski–Lindenbaum algebras — completeness, 1217 — matrix (for a logic), 204, 205, 220, 1057 — monoid, 557 — propositions, see propositions, as equivalence classes of formulas —’s Lemma, 60, 76, 78, 281, 310, 311, 386, 1065 Lindenbaum, A., 204 linear linear formula, 1330

1468 linear logic, 52, 143, 229, 299, 313, 331, 345–358, 371, 665, 667, 1124, 1295, 1323 linear orderings, 8, 288, 312, 639, 856, 866, 867 Litak, T., 297, 871 Lloyd, J. W., 34, 1164 Loc(·), 136, 1062 local simulation, 1077 local/global global range (of a rule or set of rules), 136, 1062 local vs. global preservation characteristics, 129, 160, 290, 428, 431, 432, 435, 436, 831, 860, 895, 913, 1064 local range (of a rule or set of rules), 136, 1062 other distinctions, 325, 499 syntactically local conjunctions, disjunctions, etc., 94, 649, 650, 1068 valuationally local conjunctions, disjunctions, etc., 650 locally — based (rules), 604 — finite classes of algebras, 227 logics, 227, 1160, 1161, 1298 ‘locally tabular’ (logics), 227 finite classes of algebras, 238 logics, 238 Lock, A. J., 810, 1164, 1169, 1192 Loewer, B., 1022 Log(·) consequence relation or gcr determined by a class of valuations, 57, 74 log(·), logical transfer of a property of sets of valuations, 457 Logfin (·), finitary version Log(·), 81 logical frameworks, 103–109, 188 Fmla-Fmla, 108, 113, 189, 246– 248, 251, 584, 587, 655, 754, 757, 1190 Fmla-Set, 214 Mset-Fmla, 105 Mset-Fmla0 , 351 Mset-Mset, 105, 189 Mset-Mset1 , 359

INDEX Seq-Fmla, 105, 977 Seq-Fmla0 , 314 Set1 -Fmla, 104, 108, 266 Set1 -Fmla, 629 Set-Fmla0 , 104 Set-Fmla, 82, 103, 757 Set-Fmla0 , 314, 1184 Set-Set, 82, 102, 757 an ambiguity in this terminology, 106 duality of, 1193 plural conception (Oliver–Smiley), 106 logical pluralism, 328, 537 logical relations Lemmon on, 503–507 traditional account of, 1166 logical subtraction, 645, 677–708, 1138 logics and consequence relations CL, CL (classical logic), 66, 320 IL, IL (intuitionistic logic), 51, 174, 304, 320, 983 ML, M L (Minimal Logic), 320, 1257 PL, P L (Positive Logic), 320, 983 QL, QL (quantum logic), 921 logics vs. theories defining connectives in, 428, 1181, 1211, 1212 different ways of drawing the distinction, 810 Dummett on, 189 Meyer on Uniform Substitution and, 188 variable for variable substitution and, 192 loop, see r López-Escobar, E. G. K., 370, 546, 673, 1250, 1251 Lorenzen, P., 513 Łoś, J., 123, 205, 206, 214, 923, 951 Lotfallah, W., 53 lottery paradox, 658, 677 Lowe, E. J., 930, 1036–1037, 1042 on a would/might analogue for indicative conditionals, 1009 Lucas, J. R. on intuitionistic logic, 540

INDEX on truth for future directed statements, 833 Łukasiewicz, J., 196, 1061, 1230 axioms for CL due to, 185 choice of primitives, and Polish notation, 269 early attempt at modal logic using three values, 270, 470 effects of his concentration on Fmla, 200 his conjecture on intermediate logics and the Disjunction Property, 875 his three-valued matrix not suitably monotone, 250 hybrids, products and the ‘twins’ analogy, 468, 471 IL as an extension of CL (a bad idea), 305 infinite-valued logic (Łω ), 197, 336, 1128, 1129 on bivalence, 209, 827 on classical implication simulated in IL, 1080 on early natural deduction, 190 on many-valued logics, 168, 242, 271, 272 on Peirce’s Law, 1067 on variable functors, 463 products, 484 shortest axioms, etc., 1128 the Ł-modal system, 471–472, 484 three-valued logic (Ł3 ), 196–199, 209, 220, 242, 456, 598, 1130, 1158 Lust, B., 633 Lycan, W. G., 933, 936, 954, 955, 962, 966, 1055 Lyndon, R. C., 44, 178, 274, 442 Lyngholm, C., 451 (M): structural rule, 59, 112 (M): condition for being a consequence relation or gcr, 55, 73 M (modal principle), 278 ma(·) (modal algebra induced by a frame), 290, 863 Mabbott, J. D., 778, 779 MacFarlane, J.

1469 on logical constants, 193, 326 on truth for future directed statements, 833 Mackie, J. L., 957, 1054, 1056 Maduch, M., 273 Maehara, S., 100 Magari, R., 283 majority (truth-function maj and connective maj), 916 Makinson, D., xvii, 163, 645, 707 canonical model proofs in modal logic, 297 his use of the term ‘congruential’, 508 nonmonotonic logic, 100 on and and or, 817 on choice of primitives, 1139 on conditional probability, 653, 842 on De Morgan negation, 1199 on disjunctive permission, 799, 801, 802, 804–806 on intuitionistic logic, 986, 1167 on language dependence, 1138 on probabilistic consequence relations, 656 on rules, 520 on structural completeness, 163, 191, 882 on Uniform Substitution, 192 on Fmla-Fmla, 189 special notions of implication, 772 Maksimova, L. L., 320, 370, 577, 630, 863, 875, 922 Malinowski, G., 100, 210, 270–272, 306, 484, 1155, 1158, 1161 Malinowski, J., 275 Malmnäs, P.-E., 422, 874, 1258, 1285 Mancosu, P., 203 Manes, E. G., 43, 442 many-sorted (logics), 53 many-valued logic(s), 65, 168, 195– 219, 249, 269–272, 380, 403, 443, 836, 1169, 1239 Marcos, J., 120, 210, 491 Marcus, P. S., 982 Marcus, S. L., 1056 Mares, E. D., 296, 371, 667, 924, 1186, 1201–1203, 1207, 1278

1470 Markov, A. A., 303, 1228 Martin, E. P., 371, 1203 Martin, J. A., 442 Martin, J. N., 207 Martin, N. M., 9, 71, 100, 436, 442, 629, 1179 Martin-Löf, P., 535 Martini, S., 673 Massey, G. J., 101, 389, 409, 442, 783, 915–917 ‘material’ implication, equivalence, etc., 83 ‘material disjunction’, 791 Mates, B., 189, 296 Matoušek, M., 273 matrices, 82, 199–245 full, 1299 generalized, 217 simple (or reduced), 259 unital, 220, 273, 458, 459, 483, 1094 Matsumoto, K., 369, 1090 maximal — avoiders, 1065 — consistent (set of formulas), 281, 1065 maximum formula occurrence, 126, 334, 513 Mayer, J. C., 1032 Mayo, B., 951 ‘MCC’, 1278 McCall, S., 242, 1056, 1122, 1166– 1168 McCarthy, J., 641 McCarty, C., 370 McCawley, J. D., 101, 508, 673, 783, 791, 949, 958–961, 966, 975, 1055, 1300 McCullough, D., 572, 578 McCune, W. M., 1122 McDermott, M., 1055 McGee, V., 193, 529, 1009 McKay, C. G., 227, 568, 569, 577, 629, 889 McKay, T., 1017, 1022 McKee, T. A., 1150 McKenzie, R., 16, 23, 43 McKinney, A., 492, 508 McKinsey, J. C. C., 1159

INDEX definitional independence of standard primitive connectives in IL, 419 Disjunction Property for IL, 922 fragments of IL, 227 Halldén completeness for modal logics, 868, 923 IL and topology, 620, 622 IL not locally finite, 227 modal axiom named after, 278 on Horn formulas, 44 on reducibility, 1151, 1158–1161 “quasi” terminology, 488 translation embedding IL in S4, 874 McNulty, G. F., 16, 23, 43, 765 McRobbie, M. A., 105, 107, 345 medial (law), 1133, 1148, 1161 Medlin, B., 500 Medvedev’s Logic (of finite problems), 625, 630, 888 Medvedev, Y., 630, 888 meet (in a lattice), 7 meet-irreducible, 11, 872 in a lattice of logics, 864 in a Lindenbaum algebra, 1295 in the lattice of valuations, 71 Meinke, K., 44 Meng, J., 274 Menger, K., 407, 408, 443, 501 ‘mental models’, 814 ‘mere followers’, 459, 597 Meredith, C. A., 1079–1081, 1084 axiomatics, 1122 first axiomatizes BCIA logic, 1098, 1101 his cousin David, 274 implicational axioms named after combinators, 166, 274 on BCI and BCK, 274 on a redundant axiom of Łukasiewicz (Łω ), 197 translation embedding implicational CL into implicational IL, 335, 874, 1068, 1081– 1087 Meredith, D., 274, 1278 mereological analogies, 659, 899 Merin, A., 804 Meseguer, J., 442

INDEX Meskhi, V., 321 meta-ethics, 1051 meta-logic, see choice of — meta-schematic letters, 155, 156, 523 Metcalfe, G., 111, 269, 1102 Mey, D., 1124 Meyer, J. J. C., 676 Meyer, R. K., xvii, 235, 245, 334– 336, 345, 373, 907, 910, 925, 1056, 1101, 1121, 1122, 1124, 1196, 1197, 1200, 1201, 1203, 1207, 1210 axiomatizing Łukasiewicz logics, 242 defining connectives with propositional quantification, 421 disjunction property, 862, 922 enthymematic implication, 345, 346, 1098 implicational theorem of RM not provable in RM0, 335, 363 local finiteness and implicational R, 228 on Ponens Modus in Abelian logic, 1109 on Abelian logic (q.v.), 1098, 1122, 1258 on Boolean negation, 1205 on contraction, 671 on Curry’s Paradox, 1125 on ‘intensional implication’ in RM, 1080 on intersections of logics, 923 on LC, 319, 370 on logics in Fmla vs. theories, 188, 192 on modal relevant logic, 296 on multisets, 105 on Peirce’s Law, 1067 on ‘strict substitutions’, 192 on structural completeness, 173 on the infinite model property, 568 on the rule of Conversion in Abelian logic, 1105 on upper bounds for pairs of BCI theorems, 573 on upper bounds for pairs of BCI -theorems, 578

1471 relevant equivalence property, 1296 semantics for BCI, 245 sentential constants in R, 371 separation property, 577 two views of definition, 443 Urquhart semantics for contractionless logics, 371 Michael, M., 1167 Michaels, A., 1161 Michalski, R. S., 968 Miglioli, P., 630, 875 Mihailescu, E., see Leśniewski–M. Theorem Milberger, M., 1139 Mili, A., 732, 765 Mill, J. S., 647, 659 Miller, Dale, 371 Miller, David, 189, 688, 707, 843, 1138, 1226, 1298 Milne, P., 317, 371, 528 Minari, P., 320, 370, 629, 879, 923 Mingle (schema), 329, 331, 332, 335, 339, 361, 363, 369, 663, 664, 907, 1091, 1093 Minglers, 1307, 1335 Minimal Logic, 272, 303, 304, 307, 320, 381, 1257–1263, 1284, 1317 Mints, G. E., 120, 179, 370, 673, 860, 878, 881–883, 924, 1121 Mioduszewska, E., 943, 944, 950, 993, 1054 Miura, S., 923 Miyakoshi, M., 296 ML, see Minimal Logic ML* (variant of ML with biconditionally defined negation), 1271 MNat, 114, 1257–1262 MNat 1 , 1264 modal algebras, 290, 863 modal logic(s), 275–298 intuitionistic, relevant, substructural, 296 normal (see also normal (modal logic)), 277 modalities, 462 modalized, see fully modalized ‘modally defines’, 284 model-consequence

1472 in intuitionistic logic, 309 in modal logic, 289, 291, 488, 850–854 generalized, 854–860 models characteristic, 280 full, see full model in the sense of validating matrices, 272 Kripke models, see Kripke semantics of an equational theory, 29 of Horn theories, 34, 35 Modus Ponens, 115, 123, 161, 529 horizontal form, 155 Modus Tollens, 121, 521, 942, 960, 1055, 1181, 1258 Moh, S.-K., 1125 Moisil, G., 312, 546, 1240 Moktefi, A., 525, 1019 monadically representable (relations), 506, 729 “monkey’s uncle”, 942, 1181 monoids, 18, 19 commutative, 1120 monomodal logics, 288 monothetic (logics), 221, 244, 245, 458, 493–495, 573, 1104, 1106, 1108, 1312 not necessary for algebraizability, 263 monotone vs. monotonic, 490 connectives, 490 partial functions, 249 truth-functions, see truth-functions, monotone with side formulas, 491, 621 Monteiro, A., 274, 1240 Moor, J., 1056 Morel, A. C., 44 Morgan, C. G., 656 Morgan, J. L., 954 Morgenbesser, S., 819 Moro, G., 930, 935 Morrill, G., 271 Morris, C., 635 Morriss, P., 1056 Morsi, N. N., 53 Mortensen, C., 1056

INDEX Morton, A., 628 (MP) – quasi-identity inspired by Modus Ponens, 231 (mp) condition on models for conditional logic, 995 Mset-Fmla, Mset-Mset, etc., see logical frameworks “much”, 966 multigrade, see connectives, multigrade multimodal logics, 287 multiplicative, see additive and multiplicative rules or connectives multiplicity functions, 373 multisets, 105, 108, 373 Mulvey, C. J., 641 Mundici, D., 242 Muravitskii, A., 370 Murphy, R., 942, 970 MV -algebras, 242 Myhill, J. R., 1098, 1125, 1251 N, xv N: negation truth-function, 398, 406, 412 n (converse non-implication), 605 Naess, A., 817 Nagayama, M., 236 Nakatogawa, K., 296 nand, 388, 607 intuitionistic (), 1241 NAT (Set-Set natural deduction system), 140–141 Nat (Set-Fmla natural deduction system for CL), 66, 114– 121, 124–132 natural deduction, 89, 114–121 ‘natural semantics’ (Garson), 915 necessary and sufficient conditions informal idea, 680, 961 separate clauses in the definition of truth ((N) and (S)), 610 Necessitation (rule), 277 negation, 49, 52, 62, 1163–1285 — normal form, see normal forms Boolean, 1192, 1204, 1205 ‘Brouwerian negation’, 1250 canonical (in Abelian logic), 1113

INDEX Church negation ( ), 1209 classical, 585, 591, 1172 De Morgan, 1192 dual intuitionistic, 92, 583, 1172 ‘Heyting negation’, 1250 in Minimal Logic, 1258–1263 intuitionistic, 583, 585, 591, 1214– 1252 left —, right —, 1185 metalinguistic, 791 Moisil, 1240 strict negation, 1169, 1188, 1241 strong negation, 540, 1200, 1228 paraconsistent, 1230, 1235, 1251 negative — existential (claims), 1164, 1298 — formula (in a special sense), 1271 — objects on the left/right, 14, 752, 1172 — polarity items, 799, 806, 808, 941, 966, 967 Negri, S., 111, 191, 371, 535 neighbourhood semantics, 497 Nelson, D., 619, 1228, 1229, 1240, 1251 Nelson, E. J., 658–660, 668 Nemesszeghy, E. A., 1272 Nerlich, G., 925, 961 neutral element, see identity element (in a group or groupoid) Newstead, S. E., 771, 817 Nicod, J., 630 Nieuwint, P., 935, 1054 Nishimura, H., 301 Nishimura, I., 227, 623 No (gcr), 91, 92, 201, 382, 383 Nolan, D., 1041 non-connectival operations on formulas, 49, 50, 53, 393, 681, 716 non-conservative extension, see conservative extension non-creativity (as a condition on definitions), 721 nor, 607 normal (modal logic), 187, 276–296 general reading on, 297 normal or -normal consequence relation, 291, 490, 562, 832

1473 normal forms Bendall normal form, 1180 conjunctive normal form, 35, 324 disjunctive normal form, 404 for natural deduction proofs, 146, 535 negation normal form, 1179 prenex normal form, 33 normalization of proofs, 334, 515 “Not” (see also negation), 585, 1163– 1252 Novák, V., 269, 667, 673 Noveck, I., 816, 819 Nowak, M., 177 nullary, see zero-place Nute, D., 994, 1007, 1012, 1017, 1056 O’Hearn, P., 349, 669 O’Neill, B., 1138, 1300 O-system (McKay), 568 oblique, see rules, oblique Ockham algebras, 44 Ockham, William of, 829 Odintsov, S. P., 1251, 1275, 1278, 1285 Ohlsson, S., 1056 Ohnishi, M., 369, 1090 OIC (Bull’s intermediate implicational logic), 370, 542, 555 Okada, M., 345, 358, 371, 670 OL and OL (orthologic), 301 Oliver, A., 106, 192 Olivetti, N., 111 OML and OM L (orthomodular logic), 302 One,Two-Property (1,2-Property), 1053, 1099, 1116, 1127 “only”, 950, 954–963, 1055 “only if”, 950–964 Ono, H., 147, 245, 275, 296, 370, 372, 374, 546, 924, 1121, 1124, 1125, 1214 Ono, K., 1181, 1278 operations derived, see derived operations fundamental, 17 nullary or 0-ary, 17 Oppy, G., xvii “Or” (see also disjunction), 767–843

1474 or -like (∨-like) connectives, see and like Ord, Ord (W. S. Cooper), 1044– 1053 ordered pairs, 738–750 ‘Ordinary logic’ (Cooper), see Ord Ore, O., 16 ortholattices, 22, 301, 1186 orthologic, 298–302 orthomodular, see lattices, orthomodular Orłowska, E., 924 Osgood, C. E., 675 Osherson, D., 411, 1164 P (implicational principle): see also Peirce’s Law, 237 P (tense operator), 287, 832, 1295 Pahi completion (of a logic), 1334 Pahi, B., 313, 1299, 1330, 1332–1336 pair-validity, 746, 751 Pałasiński, M., 242 Palmer, F. R., 932, 1054 Pancheva, R., 949, 951 Paoli, F. ambiguity claim for existential quantifier, 789 on Abelian logic, 268, 1102, 1112, 1162 on linear logic, 327 on paraconsistent logic, 93 on Quine on deviant logics, 628 on substructural logics, 319, 371, 374 on tautological entailments, 341, 772 terminology for the distinction between additive and multiplicative connectives, 346 par (fission), 372, 798 paraconsistent logics, 92, 201, 1048, 1123, 1194–1206, 1257 paradoxes of implication material, 475, 926, 980 strict, 658 Pargetter, R., 811, 1022 Parigot, M., 140, 166 Paris, J. B., 677 Paris, S., 927 Parks, Z., 335, 368, 1053

INDEX Parry, W. T., 772, 925 Parsons, C., 467, 1277, 1278 Partee, B., 271 partial logic, 249 partial order, 2 Pascal, B., 818 Passy, S., 1260 ‘pathological’, 384 Peacocke, C., 193, 528, 530, 532– 536, 576, 978, 1170, 1171 Pearce, D., 642, 1235, 1251 Peetz, V., 792 Peirce’s Law, 177, 178, 197, 237, 238, 307, 310, 313, 427, 517, 521, 530, 534, 540, 543, 547, 556, 618, 695, 883, 986, 1057, 1059, 1061, 1068, 1080, 1083, 1085, 1117, 1127, 1217, 1258, 1272, 1275, 1276, 1278, 1317, 1326 (Peirce)# , (Peirce)¬ , 1166 and →-subcontrariety, 1060–1067 for head-linked formulas, 1320 Peirce, C. S., 630 use of the phrase “logical subtraction”, 707 Pelletier, F. J., xvii, 269, 271, 442, 781, 783, 816 Pendlebury, M., 1054 perfect (Galois connections), 4 Perm (Permuting antecedents), see also C, 159, 340, 663, 1091 Perrin, N. A., 970 persistence, 203 — and QPers for ML models, 1261 — relations, 914 a problem concerning, 611 a similar condition on Routley– Meyer models, 1202 and topoboolean conditions, 620– 621 as a requirement for intuitionistic intelligibility, 593 condition and lemma (P0 , P) in semilattice semantics, 337 condition on Kripke models for IL, 307 in ‘plus’ (modified Urquhart) semantics, 909

INDEX in modal logic, 309, 873 in the Beth semantics for IL, 894 in the possibilities semantics, 900 inversely persistent formulas, 1328 lemma for IL (2.32.3), 308 one candidate for exclusive disjunction not persistent, 786 positive and negative for IL with strong negation, 1230 relaxing this condition on IL models, 370 perspectives on many-valued logic, two different but both valuable, 206–207 Peters, S., 953 Petrus Hispanus, 809 Pfeiffer, K., 843 phase space semantics (for linear logic), 345 philosophical proof theory, 371, 511– 536 phrasal (conjunction), 667, 674 Piatelli-Palmarini, M., 842, 1164 Piazza, M., 372 piecewise connectedness, 855 convergence, 313, 855 definability, 725 generalized, see gpw weak connectedness, 856 weak convergence, 856 Pietarinen, A., 53 Pigeonhole Principle, 735, 742, 743 Pigozzi, D., 97, 218, 220, 257–268, 275, 293, 456, 483, 1136, 1158, 1307, 1309, 1311 Pizzi, C., 948, 1055 PL (positive logic), 36, 320, 512–534 and ML, 1258–1259 clear formulas in, 983 Curry’s name LA for, 320 extension by McKay of, 568 extension by Moisil of, 1240 Plantinga, A., 1300 Plisko, V. E., 304, 1285 Plotkin, G., 188, 296 Plumwood, V. (formerly V. Routley), 925, 1164 pluralism, logical, 537

1475 PNat, 114, 512 Poggiolesi, F., 112 Pogorzelski, W. A., 129, 163, 169, 177, 191, 193, 775 translations from CL into Łukasiewicz many valued logics, 1238 point-consequence, 289 Połacik, T., 1336 Polish notation, 269, 672, 745, 1122 Pollard, S., 9, 71, 100, 436–442, 1179, 1213 Pollock, J. L., 944, 958, 1007, 1015, 1032, 1054 ‘polynomial(ly)’, 23, 43 ‘Ponens Modus’, 1109 Popov, V. M., 336 Popper, K. R., 151, 526, 535, 537, 585, 626, 691 Porębska, M, 1136 portation, see importation/exportation Porte, J., 172, 175, 178, 184, 193, 223, 485, 876, 1101, 1161, 1270, 1276 on D- vs. T-independence, 184 on the Ł-modal system, 484 posets, 2, 7–9, 15, 16, 307, 583, 836, 900 Positive Logic, see PL Posner, R., 676 Possibilitation (rule), 877 Post duality, see duality, Post Post, E. L. (see also Post-completeness), 272, 405, 407, 409, 414, 442, 492 Post-completeness, 248, 278, 286, 602, 1099, 1100, 1103 Postal, P., 931, 1041 Pottinger, G., 111, 673 Potts, C., 634, 675 Pr ‘primification’, 1066 class of prime valuations, 916 probability function, 652 Pra (Prawitz style natural deduction system), 515 pragmatics, 634 Prawitz, D., 115, 125, 126, 142, 146, 189, 190, 295, 420–422, 511– 516, 519, 522, 534, 535, 874,

1476 976, 977, 1058, 1183, 1253, 1258, 1285 pre-order(ing)s, 2, 3, 15, 240, 246, 549, 1050 Predelli, S., 941, 949 Pref (Prefixing axiom), see also B, 328, 340, 1091 premiss/conclusion contrasts: vertical vs. horizontal, 104 prenex normal form, see normal forms preservation characteristics, 129 Prešić, M. D., 31, 739 Preston, G. B., 416 pretabular (logics), 320, 370, 892 Price, H., 1177, 1179, 1250 Price, R., 630 Priest, G., 189, 1206 logic of conditionals, 1055 on Boolean negation, 567, 1206 on dialetheism, 1194 on paraconsistent logic, 1195, 1200 semantics for relevant logic, 1201 Priestley, H. A., 16 prime filters, 248 sets of formulas (esp. theories), 310 two-sidedly (see also left-prime, right-prime), 254, 256, 755 valuations, 847 ‘primitively compelling’ (Peacocke), 529 principal formula, 321 Principle of Predication, 1300 Prinz, J., 660 Prior, A. N., 157, 274, 1080, 1087, 1122 a logic not closed under Uniform Substitution, 192 nicknames for axioms, 169 on BCI and BCK, 274 on a tensed notion of necessity, 827 on axiomatics, 50, 224 on Bull’s OIC, 370, 542 on contraction and Curry’s Paradox, 1125 on many-valued logic, 270 on possible worlds semantics, 297

INDEX on propositional identity, 1157 on supra-minimal logics, 1276, 1278 on tense logic, 287, 297, 639, 866 on the equivalential fragment of CL, 1103 on the Ł-modal system, 484 on three-valued logic, 269 on Tonk, 91, 532, 536, 537, 576, 614 probability, 677 ‘probabilistic disjunction’, 826 and ∧-introduction, 650–658 conditional Lewis on, 568 conditional (and (∨E)), 821–822, 824–826 conditional (and indicative conditionals), 770, 938, 978, 979, 1035, 1055 conjunction fallacy, 1164 probabilistic semantics, 656 product, see cut product, see direct product projection functions (projkm ), 24, 407 hybridizing the logics of, 465, 750–765 projection-conjunction connective or condition, 392–394, 398–402 promises, see threats (and promises) propositional – logic, 47 — attitudes affective, 971 linked to force, 1179 — identity (as a binary connective), 274, 306, 1150–1158 — quantifiers, see quantifiers, propositional — variables, 47 propositions as equivalence classes of formulas, 22, 73, 222, 279, 525 as mappings, 642 as sets of worlds, 279, 291, 1260 protasis, 933 protoalgebraic (logics), 99, 264, 265 Prucnal, T., 172, 173, 191, 275, 630, 874, 879, 1084, 1122

INDEX pseudo-boolean algebras, 22 ¨ ), 555, pseudo-disjunction (see also ∨ 839, 1068, 1072–1076, 1272 pseudo-imperatives, 674 ‘pseudo-subcontraries’, 556, 1272 pseudo-truth-functional, 442, 451 pseudocomplementation, 22, 583 Pugmire, J. M., 442 Pullum, G. K., 411, 649, 674, 783, 933, 950, 968 Pulman, S. G., 1055 ‘punctuationism’, 806, 810 purity, see rules, pure and simple Putnam, H., 313, 369, 823, 875, 877, 883, 923 Puttock, T., 819 Pym, D., 147, 166, 349, 669, 1125 Pynko, A. P., 1197 Q (implicational principle), 237 QL, see Quantum Logic QNat, 299 quantifiers branching, see branching quantifiers generalized, 447 in intuitionistic predicate logic, 276, 306, 543 in natural deduction, 189, 543 nonconservativity and new —, 722–723 propositional (see also second order propositional logic), 363, 420, 615, 1101, 1325 quantum logic, 299, 369 disjunction in, 820, 918–922 quasi-boolean algebras, 22, 249, 1186, 1198 representation theorem for, 1197 quasi-commutative (BCK -algebra), 242 ‘quasi-connectives’ (R. Shock), 100 quasi-identities (quasi-equations), 35, 231, 232, 259 quasi-normal (modal logics), 488 ‘quasi-truth-functional’, 442 quasivarieties, 36, 44, 259 questions (see also ‘whether’), 770, 951

1477 Quine, W. V., 508, 585, 603, 607, 628, 630, 938, 964, 965, 1023, 1054 mixing up terminology for binary relations and binary operations, 499 quotient algebra, 27, 221, 241, 498 (R): structural rule, 59, 112 (R): condition for being a consequence relation or gcr, 55, 73 R (relevant logic), 52, 169, 235, 331, 663 ↔ special in, 1307–1312 R (requirement structure), 693 R (accessibility constant), 1260 r (reflexivity constant), 1259–1260 R-chains, 308, 309, 320, 903 (RAA), 114 (RAA)# , 1166 for NAT, 141 purified, 520, 580 rigging assumption dependencies for, 341 Rabin, M. O., 44 Rabinowicz, W., 303 Radford, C., 1054 Raftery, J. G., xvii, 188, 240, 245, 260, 263, 273, 546, 1122 on an enthymematic implication definable in RM, 1069 on definability of disjunction in RM, 420 on RM, 168 order algebraizability, 263 the abbreviation |A|, 574 Ramachandran, M., 1037 Ramanamurty, P. V., 1116 Ramsey Test (for conditionals), 937, 938, 978 Ramsey, F. P., 52, 937, 938, 1055 Rasiowa, H., 200, 273, 275, 484, 1164, 1197, 1251 Ratsa, M. F., 444 Rauszer, C., 546, 1224, 1225, 1250 Rautenberg, W., 172, 893 {∧, ¬}-fragment of IL is 3-valued, 1221, 1245

1478 calls weakly extensional consequence relations congruential, 483 consequence relation for hybridizing ∧ and ∨, 470, 472–474, 476 Deduction Theorem not automatically inherited by extensions, 193 example of ∨-classicality not preserved in extending a consequence relation, 64, 601 extending Jankov’s result (on additions which classicize IL) to Set-Fmla, 313 mistaken idea that adding “either” makes or exclusive, 817 no detachment-deduction theorem for {∧, ∨, ¬}-fragment of IL, 179 notions related to Rautenbergvalidity, 220, 254, 257, 1161 on clones of truth-functions, 442 on Gärdenfors style semantics, 642 on matrix methodology, 259 products vs. hybrids, 466, 759 strengthenings of ∨ , 466 two-valued consequence relations maximally non-trivial, 396, 1245 when M1 ⊗M2 =M1 ∩ M2 (consequence relations), 467 Rautenberg-validity, 254 RE Right Expansion (RE), 361 right extensionality conditions (RE) and (RE ), 454–455 Read, S., xvii, 112, 140, 336, 341, 342, 371, 528, 576, 665, 667, 668, 789, 798, 817 real-world validity (on a frame), 489 realizability, 304, 1251 Rebagliato, J., 260 (Recip) and (Recip)ni , 1140–1142 reciprocal (nth -argument — function), 415, 684, 1111, 1137 rectangular

INDEX bands, see bands, rectangular relations, 729 Red, see reducibility (of sequents to sets of sequents) reducibility of sequents to sets of sequents, 149, 325 reducible and n-reducible (logics), 1158–1161 reduct (of an algebra), 18 Reductio ad Absurdum, see (RAA) refinability, 900 reflexive (relations), 2, 281 reformulation, or of, 795 refutability ‘classical refutability’ (Curry’s LE), 1275 ‘complete refutability’ (Curry’s LD), 1274 of formulas, 1216 ‘regeneration’ (of boolean algebras), 1137 regular modal operators, modal logics, 492 regular relations, 765 regular semigroup elements, 765 rules, see rules, regular Reichenbach, H., 633, 783 rejection à la Łukasiewicz, 484, 1179 as denial or dissent, 1178, 1211 as distinct from assent to the negation, 303 relational — connection, 1 between sequents and formulas, 133 between valuations and formulas, 69 — structure, 2 — system, 38 relevant logic, 326–345, 371 bad response to the problem of logical subtraction, 683 conjunction and fusion in, 661– 671 intensional and extensional disjunction in, 790 negation(s) in, 1192–1210

INDEX relevant implication in Fmla, 1091–1098 relevant implication in Set-Fmla, 1088–1090 semantics for disjunction in, 905– 910 representable (see also boolean representability) ∧-representable connective or truth-function, 709 relations, 505, 729 ↔-representable connective or truth-function, 709 relations, 729 ∨-representable connective or truth-function, 709 relations, 729 representative — instance of a schema, 160 universally —, see universally representative requirement semantics (for logical subtraction), 693–707 Rescher, N., 101, 201, 202, 272, 442, 443, 484, 655, 874, 992, 1051 residuals, 269 residuated (semigroups, monoids, lattices), 545 “respectively”, 44, 744 Restall, G., xvii, 243, 508 on substructural logics for categorial grammar, 638 mixing up terminology for binary relations and binary operations, 499 motivating Set-Set, 846 on ‘coformulas’, 100 on ‘structure connectives’, 577 on BN4, 1200 on contraction and Curry’s Paradox, 1123 on Curry’s Paradox, 1125 on cut elimination, 191 on display logic, 108 on graph proofs, 189 on Hinnion and Libert’s version of Curry’s Paradox, 1211–

1479 1214 on intersection types in substructural logic, 673 on logical pluralism, 298, 537 on logics weaker than IL, 370 on logics without contraction, 1124 on modal sequents, 860 on negation in relevant logic, 1200 on relevant logic, 332, 371 on split negation, 1164 on substructural logics, 371, 374 on three-valued logic and supervaluations, 842 on tonicity and the calculus of structures, 1130 on truth-makers and disjunction, 817 semantics for relevant logic, 1201 restricted — ∧-classicality, 629 — ∨-Elimination, see rules, (∨E)res — ∧ introduction, 587, 629 — cut (for quantum logic), 301 — generalization (Pahi), 1332 Richard, M., 1179 Richards, M. M., 675 Richards, T. J., 781, 816, 1055 Rieger, L., 273 Rieger–Nishimura lattice, 227, 623 right-prime (gcr), 79, 110, 755, 845 Riguet, J., 765 Rips, L. J., 661, 1056 Riser, J., 149, 619, 1249 Ritter, E., 147, 166, 1125 (RM): structural rule, 147, 149, 166 RM (R-Mingle), 331, 334, 335, 354, 361–364, 367–369, 475, 663, 664, 667, 1053, 1058, 1069, 1080, 1091, 1119 converses in, 1326 deduction theorem, 168 RM0, 329, 331, 334, 363, 368, 1118, 1309 RMNat, a natural deduction system for relevant implication with Mingle, 329–334 RNat, a natural deduction system for relevant implication, 330,

1480 332–339 Robin, N., 1056 Robinson, T. T., 1101, 1278 robustness implicature, 980, 983 with disjunctions, 770 Rodenburg, P. H., 883 Rodríguez Salas, A. J., 213, 242 Roëlofs, H. D., 959, 964 Roeper, P., 656, 676, 677 Rogerson, S., xvii, 358, 1125, 1127, 1213 Rose, A., 442 Rose, G. F., 304, 889 Rose, T. A., 817 Rosen, G., xvii Rosenberg, I., 443 Roth, Philip, 795 Rott, H., 645 Rousseau, G. F., 107, 111, 210, 444, 577, 619, 987 Routley, R. (see also Sylvan, R.), 273, 345, 772, 910, 923, 925, 1051, 1056, 1094, 1125, 1158, 1195, 1197, 1198, 1200, 1201, 1203, 1207, 1229, 1251 Routley, V. (see also Plumwood, V.), 1158, 1195, 1197, 1198, 1201, 1203 Routley–Meyer semantics, 1094, 1196, 1201, 1202, 1210 Roxbee Cox, J. W., 818 Royse, J. R., 731 RPref (Prefixing Rule), 159 RSuff (Suffixing Rule), 159 RTrans what relation is transitive?, 503 RTrans (Transitivity Rule), 159 Ruitenburg, W., 189, 370, 451 rules (+ Left)ms , 349 (+ Right)ms , 349 (F Left)ms , 351 (T Right)ms , 351 (⊥E), 1254 (◦ Left)ms , 347 (◦ Right)ms , 347 (¬¬I), 121 (¬ Left), 142 (¬ Left)ms , 349 (¬ Right), 142

INDEX (¬ Right)ms , 349 (¬¬E), 114 (→I)d ((→I) with discharge restriction), 332–334, 336, 339, 427, 518, 975, 1088, 1118 (→ Left), 142 (→ Left)ms , 349 (→ Right), 142 (→ Right)ms , 349 (→E), 114 (→I), 114 (→I)P ra , 514, 518 (→I)ms , 1118 (∨ Left), 142 (∨ Left)ms , 349 (∨ Right), 142 (∨ Right)ms , 348 (∨E), 114 (∨E)res (restricted (∨E)), 299, 527, 586, 776, 835, 893, 918– 922, 1191 motivation for, 823 with (∨I) uniquely characterizes ∨, 586–587 (∨I), 114 (∧ Left), 142 (∧ Left)ms , 347 (∧ Right), 142 (∧ Right)ms , 347 (∧E), 114 (∧I), 114 (∧I)rel (relevant (∧I)), 665 (∧I)res (restricted (∧I)), 587, 629 (t Left)ms , 351 (t Right)ms , 351 admissible, 38, 126, 146, 148, 160, 164, 167, 177, 178, 181, 242, 289, 322, 430, 436, 516, 556, 661, 876, 878, 881, 924, 986, 1094, 1105, 1124, 1125, 1159, 1254 contralateral, 605 cut-inductive, see cut-inductive (separate entry) derivable, 38, 126, 181, 289, 430, 516, 876, 878, 1105 elimination rules, 114, 522, 1284 general in respect of constituent formulas, 521, 746

INDEX general in respect of side formulas, 521, 527, 536, 546, 587, 1170, 1226 Harrop’s Rule, 872, 878–883 introduction rules, 114, 521 invertible, 146–151, 322–326, 349, 352 ipsilateral, 605 Modus Ponens, see Modus Ponens (separate entry) Modus Tollens, see Modus Tollens (separate entry) oblique, 521, 544, 1167 pure and simple, 312, 321, 397, 519–523, 581, 586, 785, 809, 962, 998, 1131, 1178, 1182, 1246 (RAA), see (RAA) (separate entry) regular (in the sense of Kaminski), 148–150, 366, 374, 613 RPref, RSuff, RTrans, see RPref etc. (separate entries) rules of proof vs. rules of inference, 161, 163, 170, 187, 290, 503, 882 sequential, 123 structural, 120 a different use of the terminology, 129 Contraction, 143, 152, 316, 321, 326, 345, 353, 354, 356, 358, 361–364, 372, 571, 664, 1119, 1124–1126 Cut Rule (see also (T)), 112, 347 Exchange, 230, 347, 354, 372 Expansion, 268, 332, 354, 361– 364, 664, 1118, 1119 Girard’s vs. Gentzen’s terminology, 345, 347 Identity (see also (R)), 347 rule of symmetry, 253, 482 weakening (see also (M)), 112, 316, 354 two-way, 151, 545, 661, 1001, 1004, 1005, 1126 Uniform Substitution, see separate entry for Uniform Substitution

1481 zero-premiss, 112, 115, 123, 129, 144, 153, 155, 156, 183, 526, 527, 533, 590, 595, 597–601, 605, 752, 847 Rumfitt, I., 774, 846, 1211 Rundle, B., 634, 650, 651, 675, 946 Russell’s Paradox, 1123 Russell, B., 420, 555, 707, 731, 1157 Russell, G., xvi Rybakov V. V., 924 Ryle, G., 772, 779, 793, 979 S (implicational principle), 169, 229 S1 (modal logic), 278 S13 (modal logic), 192 S2 (modal logic), 278 S3 (modal logic), 278 S4 (modal logic), 278, 284, 295, 307, 320, 330, 370, 552–554, 601, 604, 615, 867, 873, 874, 891, 988, 991, 1080, 1151, 1168, 1283, 1305 S4.2 (modal logic), 278, 855, 874, 875 S4.3 (modal logic), 278, 855, 856, 874, 875 S5 (modal logic), 278, 286, 553, 855 Sadock, J., 942, 1055 Sagastume, M., 628, 1102, 1240 Sahlqvist, H., 298, 564, 1313 Salerno, J., 1038 Salomaa, A., 443 Sambin, G., 103, 151, 283, 370, 522 Sanchis, L. E., 135, 191, 442 Sandewall, E., 968 Sandqvist, T., 535 Sandu, G., 53, 1229 Sanford, D. H., 442, 959, 964, 1055, 1300 Sankappanavar, H. P., 43, 44, 784 Santa Claus (is coming to town), 767 Sasaki hook, 300, 1192 Sauerland, U., 781 Saul, J. M., 634 Savile, A., 768 Sayward, C., 635, 1156–1157 Scedrov, A., 371 Scharle, T. W., 412 Schechter, E., 1102

1482 Schellinx, H., 346, 349, 351, 358, 372 Schiffrin, D., 638, 674, 949 Schmerling, S., 673 Schnieder, B., 442 Schock, R., 100 Scholz, B., 649 Schönfinkel, M., 636 Schroeder-Heister, P., 193, 522, 525, 535, 577, 584, 600, 627, 646 Schultz, M., 443 Schumm, G. on Halldén completeness, 863, 865, 923 on intersections of modal logics, 923 Schurz, G., 192 Schütte, K., 100, 106, 425, 613, 1331 Schwartz, N., 635, 1164 Schwichtenberg, H., 105, 107, 153, 191, 350, 371, 535, 1271 SCI (sentential calculus with identity), 1151 Scott, D. S., 32, 74, 76, 100, 102, 134, 151, 190, 210, 277, 293, 297, 298, 304, 550, 594, 630, 754, 844, 860, 872, 873, 875, 923, 1001, 1116 on rules, 123 Scroggs, S. J., 321, 488 second-order predicate logic, 730, 870, 895, 923 propositional logic, 420, 615, 627, 1284, 1325 Seeskin, K. R., 250, 270 Segerberg, K., 45, 100, 102, 157, 191, 273, 277, 283–285, 288, 293, 296–298, 311, 319, 386, 419, 442, 488, 489, 492, 508, 630, 640, 856, 870, 998, 1018, 1139, 1233, 1261, 1263, 1272, 1278, 1281, 1284, 1285, 1294 Seiler, H., 1054 Seki, T., 296 Seldin, J. P., 165, 1126 ‘self-extensional’, 508 Seligman, J., 111 ‘semantic pollution’, 112 semantics (see also validity)

INDEX algebraic, see algebraic semantics and pragmatics: a contested boundary, 635 Kripke, see Kripke semantics proof-theoretic, 535 terminological problem, 512 validity in, 516–519 Routley–Meyer, see Routley–Meyer semantics Urquhart, see Urquhart semantics valuational, 57–102 Semenenko, M. I., 906 semicolon confused reference to, 188 distinct from comma within sequents, 665 in listing sequents, 123, 266 in specifying algebras, 17 semicomplementation, 1164 semigroups, 18, 19, 416, 501, 578, 678, 683, 737, 738 as frames in Urquhart semantics, 1092 cancellation semigroups, 20, 30, 36 left-zero/right-zero, 753, 755, 756 semilattice semantics (for relevant logic), see Urquhart semantics semilattices, 18 seminegation, 1164 Sendlewski, A., 1251 Senft, G., 797 sentence letters, 47 separated, see separation separation separated (condition), 714, 730 ‘separation of variables’ (Maksimova), 863 separation property (for proof systems), 547, 577, 1113, 1131, 1178, 1278 Seq-Fmla, Seq-Seq, etc., see logical frameworks sequent calculus, 142 approach to logic, 109 classical Gentzen system Gen, 141–150

INDEX for linear logic, 347–369 intuitionistic Gentzen system IGen, 314 misuse of the term ‘sequent calculus’, 190 terminating, 321 sequents (see also logical frameworks), 103–114 sequent definable (classes of valuations), 134 sequent separator, misconceptions concerning, 103 serial (relations), 281 Sesotho, conditionals and negation in, 941 Set-Fmla, Set-Set, etc., see logical frameworks Setlur, R. V., 131, 132, 432, 434, 484, 761, 1121 Seuren, P., 799, 1163 Shafaat, A., 1146 Shafir, E., 842 Shalack, V., 1113 Shapiro, S., 1139 Sharp, W. D., 442 Sharvy, R., 959, 1055 Sheffer functions, 405 Sheffer stroke, 149, 388, 405, 408 intuitionistic analogues of, 607– 615, 1241–1250 multiplicative and additive versions of in linear logic, 1250 references on, 630 Sheffer, H. M., 630 Shekhtman, V. B., 370, 630 Shepherdson, J. C., 1164 Sher, G., 193 Shields, C., 968 Shimojo, S., 842 Shoesmith, D. J., see Shoesmith– Smiley (D. J. and T. J.) Shoesmith–Smiley (D. J. and T. J.), 206 cancellation condition, 205, 206, 923 consequence relations not determined by a matrix, 1260 gcr’s agreeing with a given consequence relation, 844, 845 matrix methodology, 260, 272

1483 on arguments, 188 on generalized consequence relations, 102 on graph proofs, 189 on multiple conclusion arguments, 846 the authors complain about certain sentential constants, 380– 381 Weak vs. Strong Claim properties, 101 Shope, R., 1164 Shramko, Y., 1250 Shuford, H. R., 948 side formulas, 144, 147, 154, 178, 347, 522, 775, 834, 991, 1249 generality in respect of, see rules, general in respect of side formulas Siemens, D. F., 630 signed — formulas, 112, 1175 — sequents, 1176 Sikorski, R., 273 Silvestre, R., 192 Silvestrini, D., 875 similarity type (of an algebra), 17 Simmenauer, B., 576 Simons, L., 977 Simons, M., 642, 781, 794, 805, 806, 810–812, 814, 816 Simons, P. M., 500 Simpson’s Paradox, 842 Simpson, A. K., 296 singular (sequent), 358 ‘singulary’ vs. ‘unary’, 14 Sinnott-Armstrong, W., 1056 Skala, H., 443 Skolem functions, 734 Skolem, T., 227, 234, 1058 Skura, T., 875 Skvortsov, D. P., 630, 887 Skyrms, B., 937, 1055, 1056 Slaney, J. K., 327, 578, 707, 1098, 1101 his program MaGIC, 193 on Ponens Modus in Abelian logic, 1109 on ‘intensional implication’ in RM, 1080

1484 on a redundant axiom of Łukasiewicz (Łω ), 197 on Abelian logic, 335, 373, 568, 1098, 1122, 1258 on BN4, 1200 on different modes of premiss combination, 668 on motivating Abelian logic, 1102 on sentential constants in relevant logic, 371 on structural completeness, 173 on the rule of Conversion in Abelian logic, 1105 semantics for logics without contraction, 1124 Slater, B. H. on counterfactual fallacies, 1037 on harmony, 528 on ‘material disjunction’, 791– 792 on paraconsistent logic, 93 Słomczyńska, K., 1131, 1162 Słupecki, J., 1161 Smetanič, Y. S., 623 Smiley models, Smiley matrices, 218 Smiley, T. J. (see also Shoesmith– Smiley), 305, 443, 874, 1166, 1176 abstract logics (Smiley models), 217 consequence relations not determined by a matrix, 204 four-element matrix for relevant logic, 431, 1197 on ‘suppression’, 1094 on analytic implication, 925 on Dana Scott’s use of manyvalued logic, 210, 270 on denial, 1177 on functional dependence, 595, 597, 628 on signed formulas, 1175, 1178, 1183 on the Ł-modal system, 484 on uniform substitution, 192 problem with McKinsey’s argument for definitional independence in IL, 419 relative necessity, 1260

INDEX relevant logic treated substructurally by disallowing Weakening, 371 rules of inference vs. rules of proof, 161, 290 synonymy, 173, 222 Weak vs. Strong Claim properties, 101 Smith, A., 193 Smoryński, C., 297, 1214 (SMP) – strengthened (MP), 231 Smullyan, R., 133, 189 Sobel, J. H., 1013, 1034 Sobociński, B., 340, 442, 472, 474, 475, 1053, 1238 Sørensen, K., 942 Sørensen, M. H., 166 Sorensen, R., 846 Sorites paradox, 658 Sotirov, V., 484 soundness, 59, 127 rule soundness, 129 in modal logic, 876 Sowden, L., 826 Spasowski, M., 102 Spielman, S., 842 Spinks, M., xvii, 268, 275, 641, 677, 1102, 1162, 1240, 1251 Square of Opposition, 443, 1166 Staal, J. F., 676 stable formulas, 319 stability in philosophical proof theory Dummett, 525 Zucker and Tragesser, 523– 524 Stalnaker’s Assumption, 1008, 1009, 1013, 1039, 1043 Stalnaker, R., 931, 936, 937, 987, 991, 996, 1007, 1008, 1010, 1011, 1015, 1034, 1035, 1038– 1040, 1043, 1054, 1055 on epistemic logic, 297 on fatalism, 827 on uniform substitution, 187, 191, 682 Steedman, M., 676 Stenius, E., 769, 805, 809 Stenner, A. J., 781, 816

INDEX Stevenson, C. L., 1049 Stevenson, J., 576 Steward, H., 948 Stirling, C., 296 Stirton, W. R., 535 Stokhof, M., 560, 641, 770 Stouppa, P., 108 Stove, D., 1164 Strassburger, L., 108, 1130 Strawson, P. F., 14, 509, 633, 651, 930, 992 Strengthening the Antecedent, or (Str. Ant.), 803, 991, 1021, 1034, 1036 confusedly called Weakening the Antecedent, 1056 strict definition, see definability of connectives, strict implication, see implication, strict negation, see negation, strict strong vs. weak Kleene matrices, 1050 negation, see negation, strong Strong Claim Property (of a truthfunction), 402 Strong/Weak Claim, 64, 67, 70, 71, 77, 84, 101, 390, 391, 394, 402, 533, 632, 778, 1174, 1178 strongest, see superlative characterizations strongly connected, 8 consequence relations, 70, 549 structural — completeness, 191, 882–883 of a consequence relation, 162 of a proof system, 129 of consequence relations vs. of proof systems, 163, 882 — nonconservativity, 369 — rules, see rules, structural structures as a liberalization of Kripke models, 1001 ‘calculus of structures’, 108, 1130 groupoids for use in model theory, 1092 in the sense of display logic, 107

1485 models for first order languages, 33 Stuart, D., 942 subalgebra, 28 subconnective (relation), 397, 461, 481–483, 590, 625, 666, 701, 1207, 1245 subcontraries/subcontrariety, 93, 503, 505, 583, 686, 690, 697, 815, 848–850, 1060, 1061, 1112, 1163–1172, 1224, 1225, 1227 (Subc)# # 1-ary, 1166 # binary, 1064 (Subc)→ and (Peirce), 695, 1060– 1067 compositional subcontrariety determinant, 437, 1060, 1165 subdisjunction, 918 subformula property, 141, 145, 363, 547, 1124, 1193 Subj(unctive) Dilemma, 1015, 1027, 1028 subjunctive conditionals, see conditionals, subjunctive mood, 800 submatrix, 598 subminimal (negation), 1185, 1268 subordination, see connectives, subordinating vs. coordinating substitution as endomorphism, 49 invertible, 192, 1303 variable-for-variable, 188, 192, 688, 1303 substitution-invariant consequence relations or gcr’s, 60, 120, 203, 426 rules, 120, 122, 332 substructural (logics), 101, 111, 143, 146, 191, 229, 326, 370, 373, 1125, 1240, 1250 noncommutative, 372 subtraction as a name for dual intuitionistic implication, 546 logical, see logical subtraction succedent, 129, 190, 191

1486 successively (imply), 662 Sugihara matrices, 339 Sugihara, T., 566 Sundholm, G., 370, 535, 577, 614 supercover (Simons), 769, 810 superdependence (of function on argument), 413, 1141 strengthened version, 414, 416 superintuitionistic logics, 1217 superlative characterizations (strongest, weakest), 344, 526, 527, 583, 584 superposition (of functions), 25 supervalid (sequent), 832 supervaluations, 101, 269, 830–833, 842, 1011 supervenience, 1144, 1161 — determined (consequence relation), 1142–1147 Troelstra–McKay supervenience, 590, 629 Suppes, P., 116, 142, 189, 677, 720– 722 supposition supposing vs. updating, 937 suppositional accounts of conditionals, 977, 998–1007, 1056 suppression, see fallacies supraminimal logics, 1274 Surarso, B., 372 surjective, see functions, surjective Surma, S. J., 420, 1121 axiomatizing equivalential CL, 1128 Galois connections, 101 on a variant of the Deduction Theorem for equivalential CL, 170, 1132 on Henkin style recipes for axiomatizations, 448 on Jaśkowski matrices for IL, 211 Suszko matrices, 1153 Suszko, R., 100, 101, 120, 123, 192, 204–206, 210, 214, 218, 271, 274, 306, 951, 1151, 1153, 1155, 1161 Sweetser, E., 674, 675, 771, 791, 794, 795, 807, 808, 941, 949, 960, 970, 993, 1054

INDEX Swenson, D. F., 780 Sylvan, R. (formerly R. Routley), 305, 1020, 1175, 1201, 1250 symmetric vs. commutative, 499, 500, 508, 640 vs. symmetrical, 76, 102, 508 gcr’s, 256, 384 and indiscriminate validity, 253 and pair validity, 752 hybrid of ∧ with ∨, 482 pure negation fragment of CL, 1172 relations, 3, 14, 281 symmetric-intuitionistic logic, see negation, Moisil synchronous connectives (in linear logic), 352 synonymy, 173, 220 syntax, 635 Szatkowski, M., 630 Szostek, B., 1250 T (Church truth constant ‘Big T ’), 342–344, 351, 355, 546, 566, 910, 1097, 1205, 1296 semantic treatment of, 1097 (T): structural rule, 59, 112 (T+ ): condition for being a consequence relation or gcr, 55, 73 (T): condition on consequence relations or gcr’s, 55, 73 T (modal principle), 277, 278, 284, 289 T (relevant logic), 229, 328, 1091, 1092 T: truth-value (truth), 57 t (Ackermann truth constant ‘little t’), 342–344, 351, 364, 367– 369, 546, 567, 906, 910, 1097, 1112 t (temporary notation for a valid formula), 434 t-norms, 269 T! (modal principle), 278 Tc (modal principle), 278 tableaux, 189 tabular (logics), 211, 228, 272, 313, 320, 475, 888, 889, 891, 1159

INDEX Taglicht, J., 957, 1055 tail-linked (formulas), 1319 Tait, W., 106 Takahashi, M., 210 Takeuti, I., 237 Tanaka, K., 1251 Tanaka, S., 274 Tannen, D., 675 Taoripi, 797 Tarski algebras, see BCK algebras, implicative Tarski, A. (see also Tarski–Lindenbaum algebras), 43, 45, 57, 100, 132, 193, 227, 443, 620, 622, 775, 843, 874, 922, 1061, 1099–1101, 1151, 1158–1161 Tarski–Lindenbaum algebras, 31, 221, 224, 238, 256, 267, 497, 741, 1137, 1299 Tarski-style conditions (on connectives), 63, 526, 775 tautological consequence, 66 entailment, see entailment, tautological tautologous (sequent), 127 tautology, 66, 433 Tax, R. E., 1131 Taylor, J. R., 948 Taylor, W., 16, 23, 43, 45, 1330 tb subscript (see also topoboolean conditions), 620 ten Cate, B., 111 Tennant, N., 535, 536, 576, 577, 586, 619, 630 mistaken claim about ML, 1271 on ⊥ as exclamatory, 1182–1185 on harmony, 527, 533 on his intuitionistic relevant logic, 371, 528 on natural deduction, 126, 189 on philosophical proof theory, 519 on rules derived from truth-tables, 1175 on sequent calculus, 1184 on simple rules for the Sheffer stroke, 1182 on the Sheffer stroke, 630

1487 strongest or weakest formulas satisfying given conditions, 525–526 tense logic, 279, 287, 288, 297, 483, 562, 563, 581, 639, 641, 676, 832, 858, 866, 901, 1294 Tentori, K., 1164 terms and term functions, 24 Terui, K., 146, 191, 345, 358, 374 Terwijn, S., 924 Thatcher, J. W., 16 “then”, 933, 934, 948, 949, 981 theories (Cn-theories, -theories), 56, 647 Thomas, I., 448, 1238 Thomason, R. H., 129, 131, 192, 543, 679, 832, 983, 1000, 1011, 1229, 1251 Thomason, S. K., 283, 288, 291, 295, 370 Thompson, S. A., 674 threats (and promises), 771 ticket entailment, see T (relevant logic) times (fusion), 372 Tinchev, T., 1260 Tindell, R., 442 Tokarz, M., 163, 205, 420, 628, 874, 1238 Tomaszewicz, A., 405, 443 ‘tone’, 675, 770 ‘Tonk’, 86, 90, 532, 537, 538, 575– 576 sequent calculus rules for, 614 tonoids, 263 topoboolean conditions, 308, 324, 620, 621, 787, 986 formulas (in IL), 986, 987 topology, 9, 100, 622 Torrens, A., 200, 242, 268, 269, 274, 1069 Tovena, L., 804, 808 Townsend, A. V., 188, 249 Tr (·) – set of formula true on a valuation or over a class of valuations, 775 Tragesser, R. S., 322, 522–524, 527, 572, 579, 584, 586, 627, 1077 transitive (relations), 2, 281

1488 translations between interpreted formal languages, 688, 1138, 1298 between natural languages, 796 compositional, 536 ‘translation lore’, 1009 translational embeddings, 181, 536, 873–875, 1081–1088, 1191, 1238, 1258 transplication (Blamey), 1052 Traugott, E. C., 635, 675, 1055 Trethowan, W. H., 1164 triangular norms, see t-norms Triple Consequents Law of, 239, 672, 1081, 1316, 1320, 1324, 1336 Veiled Law of, 672, 677 Triple Negation, Law of, 305, 452, 540, 672, 1180, 1216, 1263, 1273, 1299, 1316 trivial ‘Trivial’ modal logic, 278, 553 algebra, 20 consequence relation or gcr, 70, 92, 208, 383, 397, 460 equational theory, 32 Troelstra, A. S., 107, 153, 191 his notation in linear logic, 352 his terminology for multiplicative and additive rules, 347 on ‘Additive Cut’, 350 on ‘negative’ formulas in ML, 1271 on contexts, 425 on cut elimination, 351 on defining a connective in a non-logical theory, 1182 on intuitionism, 370 on linear logic, 346, 349, 350, 371, 665, 1127 on new intuitionistic connectives, 578 on normalization and λ-calculus, 535 on the history of IL, 1285 on Troelstra–McKay supervenience, 629 prefixing rather than infixing use of “” with sequents, 105 semantics for linear logic, 373

INDEX Trojan horse classical disjunction as a, 587 classical negation as a, 586 truth (see also bivalence) inductive truth-definitions, 279 valedictory (Lucas), 833 truth-functions, 376 ‘alternating’, 414 ‘linear’, 414 monotone, 250, 414, 621 truth-set as set of points in a model, 279 as set of valuations, 495 truth-set functionality, 411, 495 Tsitkin, A. I., 924 Tucker, J. V., 44 Tulenheimo, T., 53 tuple system, 40 “turn(ed) out”, 928 Turquette, A. R., 202, 273, 312, 546, 1241 Tversky, A., 842, 1164 ‘Twin Ace’ paradox, 826, 842 ‘twins’ (Łukasiewicz), 471, 484 two-dimensional (modal logic), 489, 940, 1017, 1043, 1052, 1260 Two-Property (2-Property), 1053, 1099, 1116–1118, 1127, 1135 U.S., see Uniform Substitution ubiquitous (formulas), 1095 Uchii, S., 1012 Ueno, T., 296 “uh” (Stenius), 806 Ulrich, D., 884, 990, 1108, 1128 Umbach, C., 676 Umezawa, T., 312, 421 Ungar, A., 140, 191, 513, 515, 535 Uniform Substitution, see also substitution invariant, 191–193, 612 admissibile vs. derivable, 160 as a rule of proof, 161 for sequents, 120, 203 not substitution invariant, 123 variable for variable, 188 ‘uniformity’ as a name for cancellation à la Shoesmith and Smiley, 206 unipotence, 740

INDEX unique — characterization, 88, 186, 545, 575, 578–626 of negation in Minimal Logic, 1193, 1262, 1263 of negation in relevant and linear logic, 1192 — predecessor condition, 563, 1313 — readability, 48, 100, 393, 681 uniqueness by antisymmetry, 583, 584, 627 of identity element in a group, 18 of inverse in a group, 20 unital, see matrices, unital universal — algebra, 17, 23, 28, 43, 49, 407, 443 — decision elements, 442 — relations, 280, 286 universally representative (connectives), 566, 1287–1301 universally representative all by itself, 1289 universe (of an algebra), 17 “unless”, 964–975 “unless if”, 975 does not mean if not, 966–968 treated as 1-ary, 968 unravelled (frames), 1313 Urbach, P., 102, 1138 Urbas, I., 1250 Urquhart semantics, 336–339, 343, 345, 906–911, 1092–1093, 1260 Urquhart, A., 189, 197, 201, 206, 210, 215, 227, 269, 271, 272, 334, 336–338, 342, 343, 371, 442, 637, 663, 664, 668, 906, 908, 1089, 1093, 1121, 1124, 1200, 1201, 1208, 1210, 1336 Urzyczyn, P., 166 USHil (Hil with Uniform Substitution), 160 Uspensky, V. A., 1285 V (imagined Venusian connective), 593

1489 V: 1-ary constant True truth-function, 406 vF (constant-false valuation), 81, 90, 205, 383 vT (constant-true valuation), 59, 68, 70, 90 vh valuation induced by the evaluation h, 207 Väänänen, J., 53 Vakarelov, D., 189, 1211, 1236, 1238, 1251 Val (·) class of valuations consistent with a consequence relation or gcr, 58, 74 val (·), class of valuations on which a set of sequents hold, 133 validity 1-validity vs. -validity, 247 at a point (in a frame), 289 in a matrix, 200 indiscriminate, see indiscriminate validity (in an algebra) on a general frame, 291 on a Kripke frame intuitionistic logic, 308 modal logic, 282 on a Routley–Meyer frame, 1202 proof-theoretic, 517–519 V -validity, for V a class of valuations, 136, 396 valuations, 11, 57 Valuations (with a capital ‘V’), 278 van Alten, C. J., 273, 546, 1122 van Bendegem, J. P., 1097 van Benthem, J., 109, 111, 192, 193, 283, 288, 290, 297, 298, 338, 637, 638, 676, 801, 866, 870, 883, 887, 923, 924, 1033, 1034 van Dalen, D., 370, 425, 1182 Van der Auwera, J., 793, 797, 940, 951, 955 van der Hoek, W., 676 van Dijk, T., 638, 675, 808 van Fraassen, B., 101, 133, 272, 305, 369, 593, 598, 656, 700, 707, 708, 817, 830, 839, 842, 843, 939, 1011, 1052, 1056 van Inwagen, P., 1017, 1022

1490 van Oirsouw, R. R., 674 van Polanen Petel, H., xvii Vardi, M., 298 variable propositional, see propositional variables ‘variable functor’, 463, 1157 variable-identifier, 889 ‘variable-like’ (formulas), 1331 variety (of algebras), 29 equationally complete or minimal, 31 Varlet, J. C., 44 Varzi. A., 842 Vaughan, H. E., 1161 VB (modal principle), 871 veiled, see Triple Consequents, Law of Veldman, W., 311 Veltman, F., 192, 336, 641, 936, 942, 1232, 1251 Venema, Y., 297, 372, 489 Venneri, B., 673 Ver (modal principle), 277 Verdú, V., 200, 264, 268 Verhoeven, L., 781 verisimilitude, 1138 Veroff, R., 268, 1102, 1162, 1240, 1251 Vickers, J., 1007 ‘visibility’ (in Basic Logic), 522 Visser, A., 370 von Fintel, K., 958, 969, 1055 von Kutschera, F., 192, 525, 1251 von Neumann, J., 369 von Plato, J., 191, 535 von Wright, G. H., 297, 471, 639– 641, 676, 709, 817, 930, 1054 Vorob’ev, N. N., 1251 Vučković, V., 1238 W (implicational principle), 229 Wagner, E. G., 16 Wagner, G., 1235, 1251 Wagner, S., 530, 576 Wajsberg’s Law, 1059, 1064, 1067, 1121, 1273, 1317 Wajsberg, M., 197, 271, 578, 1128 defining connectives using propositional quantifiers, 420

INDEX Wajszczyk, J., 640 Wakker, G., 942, 1054 Wälchli, B., 674, 797 Walker, R., 673 Walton, D., 792, 798 Wang, H., 146, 184 Wansing, H. on (modal) display logic, 108 on dual intuitionistic implication, 546 on dual intuitionistic negation, 1250 on information based semantics, 370 on intuitionistic modal logic, 578 on negation, 1211 on numerous conceptions of sequent, 111, 860 on proof-theoretic semantics, 525 on the notion of constructivity, 875 on Tonk, 576 Warmbr¯od, K., 193, 1015, 1022 Warner, R. G., 638 warnings v(Δ) = F not the negation of v(Δ) = T, 74 ambiguity in the notion of logical framework, 106 atomic formulas do not correspond to atoms in the Lindenbaum algebra, 225 Boolean and De Morgan negation notations reversed, 1192 ‘danger: hidden contraction’, 1249 different meanings of ‘contrary’, 1165 double horizontal lines in proof figures – two meanings, 151 double use of ‘v(Γ)’, 255 double use of “V ”, xvi, 279 eliminability – not to be confused with elimination rules, 326 non-standard notation for De Morgan and Boolean negation, 1192 notions of homogeneity for frames, 866

INDEX on different classifications of Uniform Substitution w.r.t. the local/global preservation contrast, 290 on the label .3, 278 on the terms ‘trivial’ and ‘inconsistent’, 92 other uses of the phrase logical framework, 188 two notions of global Halldén completeness, 923 Wasilewska, A., 1161 Wason, P. C., 939 Wasserman, H. C., 45 Watari, O., 296 Waugh, A., 926 Weak Claim, see Strong/Weak Claim Weakening (see rules, structural) →-Weakening, 1082 ‘Weakening the Antecedent’, 1056 weakest, see superlative characterizations weakly — connected, see connected, weakly — extensional (connective, consequence relation), 455, 456, 1158 — left-prime (consequence relation or gcr), 110 Weatherson, B., xvii, 243, 275, 1056, 1107 anticipation rules nonconservative over intuitionistic predicate logic, 626 on ‘only’, 954, 956, 962 on conditionals, 529, 949, 954, 1041, 1043–1044 Webb, P., 630, 1182 Wechler, W., 44 Weinstein, S., 370 Weintraub, R., 842 Weir, A., 528, 1194 Wekker, H., 1054 Welding, S. O., 500 Wen-li, K., 948 Wertheimer, R., 959, 961 West, M., 676 Westerståhl, D., 193, 271 Wheeler, R. F., 443

1491 “whether”, 560, 770, 807, 945, 948, 950, 951 supervenience and knowing whether, 1144 Whitaker, S. F., 964, 968–970, 1055 White, A. R., 829, 842 White, W. H., 937 Whitlock, H., 443 Wideker, D., 829 Wierzbicka, A., 674, 947, 1054 Wijesekera, D., 296 WIL (weak intuitionistic logic), 1279 Wille, R., 16 Williamson, C., 443 Williamson, T., xvii, 660, 1301 admissible modal rules, 876, 923 ‘no litmus test for understanding’, 528, 529, 536 on conditionals, 1043 on contexts, 425 on identity, 603 on inverses, 565, 566, 685, 877 on unique characterization, 627 on vagueness, 269, 842 rule of disjunction, 923 Winston, P. H., 968 Wiredu, J. E., 604, 1154 Wittgenstein, L., 630 Wójcicki, R., 98, 100, 120, 196, 217, 220, 265, 271, 272, 483, 508, 578, 1164 on definitional equivalence, 1238 on degree of complexity, 272 on duals of consequence relations, 102 on generalized matrices, 217 on meet irreducibles and maximal avoiders, 71 on referential semantics, 497 on Suszko and SCI, 1152, 1161 on translations between logics, 874, 1238 on Łoś and Suszko, 206 ‘purely inferential’ terminology, 205 Wojtylak, P., 173, 184, 191, 775, 874, 1068, 1131 Wolniewicz, B., 707, 817 Wolter, F., 546, 871, 874 Wood, M., 676

1492 Woodruff, P., 1272, 1285 Woods, J., 660, 792, 798 Woods, Jack, xvii Woods, M., 1055 Woolf, V., 768 Wos, L., 335, 1128 Wright, C., 1171 Wright, J. B., 16 Wroński, A. class of BCK -algebras not a variety, 236, 238 on BCK -algebras, 242 on disjunctive consequents in intermediate logics, 879 on equivalential algebras, 1077, 1162 on equivalential logics, 275 on fragments of intermediate logics, 629 on Halldén completeness, 923 on intersections of logics, 923 on Jaśkowski matrices for IL, 211 on matrices for IL , 371 on matrix methodology (errors in Urquhart), 272 on Tax’s conjecture (concerning the ↔ fragment of IL), 1131 on the {↔, ¬} fragment of IL, 1136 on the Deduction Theorem for intermediate logics, 173 on the disjunction property in intermediate logics, 922 on the Veiled Law of Triple Consequents, 677 WWIL (doubly weak intuitionistic logic), 1279 Yablo, S., 708 Yamanashi, M.-A., 942, 1054 Yashin, A. D., 578, 623 Yes (gcr), 91, 201, 382, 383 Yourgrau, W., 451 z (Zucker–Tragesser ternary connective), 322–324, 524–525, 1077– 1079

INDEX z (nullary connective for a ‘mere follower’, q.v.), 596 Zach, R., 111 Zachorowski, S., 211 Zadeh, L., 269 Zaefferer, D., 945, 951 Zakharyaschev, M., 191, 297, 630, 863, 874, 887, 922–924, 1159 Zamansky, A., 150 Zeman, J. J., 923, 1168, 1169, 1227– 1229 Zepp, R. A., 941 zero element (two-sided, left, right), 20 zero-place connectives, see constants, sentential or propositional operations or functions, 17, 24, 403 zero-premiss rules, see rules, zeropremiss Zielonka, W., 109, 372, 1251 Zimmer, L. E., 826 Zimmermann, T. E., 793–794, 797, 817 Zolin, E., 462, 483, 765, 874 Zucker, J. I., 322, 522–524, 527, 572, 579, 584, 586, 627, 1077 Zwicky, A., 955 Zygmunt, J., 44, 217, 272, 1278

The Connectives

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The The Awakening

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The Connectives

Connectives As Discourse Landmarks (Pragmatics and Beyond New Series)

Connectives in the History of English (Amsterdam Studies in the Theory and History of Linguistic Science)

The

The

The Cutie (The Mercenaries)

The Defector [The Delivery]

The Dreamseller: The Calling

The Memnoch the Devil

The Sky's the Limit

The The Awakening

The The Battle Master

The Beyond the Grave

The The Grey King

The The Witching Hour

The Lady, the Lion

The Demon, the City

The The Struggle

The

The

The Defector [The Delivery]

The More the Terrier

The Carpenter, The Fairy

The Temple and the

The Sky's the Limit

The Deader the Better

The The Kingless Land

The The Wizard Heir

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