Logical Pluralism

Logical Pluralism

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Logical Pluralism JC Beall and Greg Restall

CLAREND ON PRESS · OXFORD

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Great Clarendon Street, Oxford ox2 6dp Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © JC Beall and Greg Restall 2006 The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published 2006 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by the authors Printed in Great Britain on acid-free paper by Biddles Ltd., King’s Lynn, Norfolk ISBN 0–19–928840–2 978–0–19–928840–3 ISBN 0–19–928841–0 (Pbk.) 978–0–19–928841–0 (Pbk.) 1 3 5 7 9 10 8 6 4 2

Contents I

Preliminaries

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Introduction

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Logical Consequence 2.1 Propositions . . . . . 2.2 Arguments . . . . . . 2.3 Necessity . . . . . . . 2.4 Normativity . . . . . 2.5 Formality . . . . . . . 2.6 Cases . . . . . . . . .

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Pluralism Defined 3.1 What Kind of Claim is Logical Pluralism? 3.2 Logical Pluralism in a Nutshell . . . . . . 3.3 The Case for Logical Pluralism . . . . . . 3.4 What Lies Ahead? . . . . . . . . . . . . .

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Logics

4 Classical Logic 4.1 Specification of Cases 4.2 Admissibility . . . . . . 4.3 Emergence of Plurality 4.4 Applications . . . . . . 4.5 Summary . . . . . . . . 5

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Relevant Logic 5.1 Specification of Cases 5.2 Relevant Consequence 5.3 Admissibility . . . . . . 5.4 Pluralism . . . . . . . . 5.5 Applications . . . . . . 5.6 Summary . . . . . . . .

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Contents

6 Constructive Logic 61 6.1 Cases as Stages . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.2 Admissibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 7

Variations & Loose Ends 7.1 Free Logics . . . . . . . . . . . . . . 7.2 Second- and Higher-Order Logics 7.3 Languages and Logics . . . . . . . . 7.4 Loose Ends . . . . . . . . . . . . .

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Objections, Replies, Other Directions

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General Objections

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9 Specific Objections

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10 Other Directions 123 10.1 Recognising Plurality . . . . . . . . . . . . . . . . . . . . . . . 123 10.2 Exploiting Plurality . . . . . . . . . . . . . . . . . . . . . . . . 125 10.3 Plurality and Proofs . . . . . . . . . . . . . . . . . . . . . . . . 127 Index

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Acknowledgements Books tend not to be written in isolation, and this one is no exception. It is our pleasure to acknowledge those who have contributed to its formation. We especially appreciated feedback and discussions with audiences at seminars at the Australian National University, the University of Edinburgh, the University of Glasgow, Indiana University, Macquarie University, the University of Massachusetts (Amherst), the University of Melbourne, Monash University, the University of St. Andrews, Stanford University, Pittsburgh University, the University of Tasmania, the University of Toronto, and the University of Waterloo, and presentations at conferences including meetings of the Australasian Association of Philosophy, the Australasian Association for Logic, and the Society for Exact Philosophy. In particular, for conversations, comments, and feedback we thank Dirk Baltzly, the late John Barwise, Nuel Belnap, Phillip Bricker, Otávio Bueno, Colin Cheyne, Roy Cook, Mark Colyvan, Dave DeVidi, John Etchemendy, Mike Dunn, Roy Dychoff, Hartry Field, Bas van Fraassen, Jay Garfield, Geoff Goddu, Karen Green, Gary Hardegree, Allen Hazen, Lloyd Humberstone, Dominic Hyde, Gary Kemp, Kevin Klement, Fred Kroon, Bruce Langtry, Jim Lennox, the late David Lewis, Bill Lycan, Ed Mares, Bob Meyer, Chris Mortensen, Daniel Nolan, John Perry, Graham Priest, Augustín Rayo, Stephen Read, David Ripley, Su Rogerson, Gill Russell, Jerry Seligman, Stewart Shapiro, Koji Tanaka, Barry Taylor, Alasdair Urquhart, Achillé Varzi, Tim Williamson, Crispin Wright, and Ed Zalta. We also acknowledge the editors and referees of the Australasian Journal of Philosophy, the Journal of Philosophy, and the Philosophical Quarterly for feedback on papers that formed an early part of the research on this project. This book is much better than it otherwise would have been without the help and support of all the community, our colleagues, and friends. Of course, all responsibility for errors and unclarity remains ours alone. jcb: Thanks to the Australasian logic community for providing a lively and friendly atmosphere of inquiry, and for graciously ‘taking me in’ despite my accent and the absence of martial arts skills. I gratefully acknowledge a 2003 University of Melbourne grant that enabled me to work in Melbourne not only on restricted quantification, but also on the final stages of this book. Thanks especially to Graham Priest for ongoing support, both as colleague and friend. Beyond Australasia: paraconsistency and pluralism are now available in Storrs, thanks to the University of Connecticut Phivii

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Acknowledgements

losophy Department, to which I am grateful for support, friendship, and a delightfully productive working environment. (Never underestimate the power of cow pastures to promote philosophical productivity!) Thanks also to Charles and Bev Beall for long-standing support, and to Max and Agnes for entertainment. Most of all, I am grateful to Katrina Higgins, without whose love, support, friendship, and curry I would not have finished this project. gr: Thanks are due to the Australian Research Council, who supported this research through grant number a00000348. I warmly acknowledge my friends and colleagues in the Philosophy Department at Macquarie University, who helped me host the 1998 Australasian Association for Logic conference, where I met my co-author, and who supplied me with a supportive environment in which to teach and to write. I could not have asked for a more congenial environment in which to learn how to combine a teaching career with an active research programme. Thanks, too, to my new colleagues and friends at the University of Melbourne Philosophy Department, who have helped me refine these ideas and see them through to publication. You cannot ask for a better environment in which to teach and research philosophy. My greatest debt of gratitude is to Christine Parker and Zachary Luke Parker Restall, whose love, humour, and good sense kept me sane during the writing of the manuscript. I look forward to repaying that debt in the months and years ahead. jcb and gr: We would also like to thank Peter Momtchiloff of Oxford University Press, whose encouragement, efficiency, humour, and sage advice made the publishing process enjoyable, and Sue Hughes who thoroughly read and corrected many infelicities in expression and spelling at the final stages of production of this manuscript. JC Beall Storrs Greg Restall Melbourne March, 2004

Part I

Preliminaries

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Chapter 1

Introduction In considerations of a general theoretical nature the proper concept of consequence must be placed in the foreground. — Alfred Tarski, “On the Concept of Logical Consequence,” [127, page 413]

Logic is about consequence. Logical consequence is the heart of logic; it is also at the centre of philosophy and many theoretical and practical pursuits besides. Logical consequence is a relation among claims (sentences, statements, propositions) expressed in a language. An account of logical consequence is an account of what follows from what—of what claims follow from what claims (in a given language, whether it is formal or natural). An account of logical consequence yields a way of evaluating the connections between a series of claims—or, more specifically, of evaluating arguments. Since philosophy (like other theoretical disciplines) proceeds by way of argument and inference, an account of what logical consequence amounts to is both a central issue in philosophy, and vitally important to philosophy. The focus of the book: In this book we present and defend what we call logical pluralism, the view that there is more than one genuine deductive consequence relation, and that this plurality arises not merely because there are different languages, but rather arises even within the kinds of claims expressed in the one language. The structure of the book: Throughout the book we attempt to be as concise as possible. Part I, which comprises §1–§3, does three things: » Introduction (§1), this chapter, sketches the plan of book, and gives you some hints for how to read it. » Logical Consequence (§2) sets the stage by discussing the target notion: logical consequence. We briefly set logical consequence in its historical context, explain why it is important, and clarify what we take to be the settled core of this notion. » Pluralism Defined (§3) defines logical pluralism, as we use the term, and indicates the type of arguments that we take to support the position. 3

4

Introduction

Part II, which comprises §4–§7, gives concrete examples of the various consequence relations that we, qua pluralists, endorse. » Classical Logic (§4) presents two different ways to explain and motivate the mainstream account of logical consequence: the classical logic of Frege and Russell. » Relevant Logic (§5) motivates and defends relevant (and, in general, paraconsistent) consequence. » Constructive Logic (§6) presents intuitionistic logic, and motivates and defends it in terms of constructibility. » Variations & Loose Ends (§7) touches on other dimensions of pluralism (free logic, second-order quantification), and also emphasises a crucial difference between our pluralism and that of Rudolf Carnap, namely, that our pluralism is not merely a pluralism about languages. Part III, comprising the remaining chapters, presents and responds to a variety of objections. Prerequisites of the book: Familiarity with formal logic is a prerequisite, as is exposure to philosophy of logic. Familiarity with philosophy of language will be helpful. Sufficient background may be obtained from numerous places. We recommend: » Stephen Read, Thinking about Logic (Oxford University Press, 1995) » Stewart Shapiro, Thinking about Mathematics (Oxford University Press, 2000) » JC Beall and Bas van Fraassen, Possibilities and Paradox: An Introduction to Modal and Many-Valued Logic (Oxford University Press, 2003) » Graham Priest, An Introduction to Non-Classical Logic (Cambridge University Press, 2001) » Susan Haack, Philosophy of Logics (Cambridge University Press, 1978)

History of the book: The view expressed here in this book is a development of the position taken in our articles ‘Logical Pluralism’ [18], ‘Defending Logical Pluralism’ [19] and ‘Carnap’s Tolerence, Meaning and Logical Pluralism’ [109]. The position is roughly the same, though we hope that it has been expressed much more clearly. In particular, we have taken the time and space to fix the notion of logical consequence more precisely in Chapter 2, to more sharply define the kind of claim that is logical pluralism in Chapter 3, to more clearly defend classical, intuitionistic, and relevant logics as accounts of logical consequence in Part II. Finally, our discussion of objections in Part III is much more developed than in our earlier work. Conventions of the book: This book does not contain a lot of symbols (not counting English words as ‘symbols’); however, in those places where symbols are used, we simply trust context to steer the reader around confusion as to when expressions are used and when they are mentioned.

Introduction

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As a guide, we almost always talk about the formulas in this book rather than using them. Almost all of what we assert is in our native English. Other conventions are these: » Chapters, Sections: We use ‘§n’ to denote Chapter n and ‘§n.m’ to denote Section m of Chapter n. » Quotation marks: We follow the Analysis convention: single quotation marks throughout, both for mentioning an expression and for ‘shuddering’ that expression. Example: ‘Cartesian mind’ contains thirteen letters and one space, but your ‘Cartesian mind’ is devoid of letters or space (at least according to Descartes). Again, context is your guide. » Names: We use ‘we’ to denote the authors, and ‘you’ to denote you (a reader). On some rare occasions we use ‘jcb’ to refer to JC Beall, and ‘gr’ to refer to Greg Restall. » Symbols: We use the following vocabulary for the formal languages of propositional and predicate logic. The symbols ∼

∧

∨

⊃

∀

∃




♦

express negation, conjunction, disjunction, conditionality, universal and existential quantification, necessity and possibility, respectively. The binary connectives of conjunction, disjunction, and conditionality are written , and the unary ‘connectives’ (or operators) of negation, the quantifiers, and the modalities are written prefix. Parentheses disambiguate compound expressions in the usual manner. » Citations: We use ‘[n]’ to refer to the nth work in the Bibliography. The bibliography reference is used always semantically neutrally, as is a footnote marker. » Spelling: We have tried to employ consistent spelling, and in particular Australian spelling. The trouble is that, while one of us (gr) is an Australian and the other (jcb) is almost Australian (a permanent resident), the almost Australian spends a lot of time in the usa, and his spelling sometimes reflects that. We have enjoyed the stirling assistance of Sue Hughes, our copy-editor, and hope that we have found (and fixed) all of the misspellings, but we apologise in advance if some remain.

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Chapter 2

Logical Consequence There is accordingly as much difference of opinion in regard to the definition of logic as there is in the treatment of the science itself. This was only to be expected in the case of a subject, in regard to which most writers have only employed the same words to express different thoughts. — J. S. Mill, Logic [80, §1]

This chapter has but one purpose: to explore the concept of logical consequence. This book is an exposition and defence of pluralism about logical consequence, so our first port of call is the concept of logical consequence itself. Only when we have some understanding of what logical consequence is can we see why it might make sense to be pluralist about it. On our way through this chapter, we will encounter a number of debates about the concept of logical consequence. For some of these debates, we will take a stand on one side of the disagreement. On others, it will be enough to explain the disagreement, and to indicate that this is an issue upon which we need not take a partisan stance. Logical pluralism is a comprehensive position in the philosophy of logic, but it is not implicated in every debate. In some of these debates, a pluralist may take one or other of the competing sides, without calling her pluralism into question, and in some of these debates, a pluralist may take both sides. But this is to get ahead of ourselves. In this chapter, pluralism is not at issue. Here, we wish to examine the concept of logical consequence, to see what we can fruitfully say about it. Our job will not be easy. As Tarski noted, the concept of logical consequence is not sharp. Any theorist proposing a precise account of logical consequence is engaged in a kind of conceptual revision. The concept of logical consequence is one of those whose introduction into the field of strict formal investigation was not a matter of arbitrary decision on the part of this or that investigator; in defining this concept, efforts were made to adhere to the common usage of the language of everyday life. But these efforts have been confronted with the difficulties which usually present themselves in such cases. With respect to the clarity of its content the common concept of consequence is in no way superior to other concepts of everyday language. Its extension is not sharply bounded and its usage fluctuates. Any attempt to bring into harmony all

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Logical Consequence

possible vague, sometimes contradictory, tendencies which are connected with the use of this concept, is certainly doomed to failure. We must reconcile ourselves from the start to the fact that every precise definition of this concept will show arbitrary features to a greater or less degree. [127, page 409]

As Tarski was well aware, no account of logical consequence will respect every claim made on its behalf throughout its centuries of use. That is too much to expect from any investigation of a much-used concept. However, we can hope in this case to be reasonably comprehensive. There are some patterns, common features and recurring themes in presentations of logical consequence, and any investigation into logical consequence would do well to respect the general structure of the terrain. Many decades have passed since John Stuart Mill introduced his valuable work on logic with these sentences, and important thinkers here and beyond the Channel have devoted their best powers to logic and have enriched its literature with ever new presentations. But even today these sentences could serve as a suitable expression of the state of logical science, even today we are very far from complete agreement as to the definition of logic and the content of its essential doctrines. — Edmund Husserl, Logical Investigations [59, §1]

2.1

Propositions

Logic does not study formal languages for their own sake. That discipline is formal grammar. Techniques of logic are useful in the study of electric circuits, through the representation of circuits in Boolean algebras [134], or linguistic types, through the Lambek calculus [65] or computation, through the λ-calculus and linear logic [4, 55, 119], but each of these endeavours are algebra and not logic. They are tributaries off the main stream of logic— strong, interesting and important tributaries, of course, but they are not logic in the important narrow sense which concerns us. Logic is to do with the evaluation of arguments. Of course, if it could be shown that arguments are genuinely a matter of computation, or linguistic syntactic classification, or electric circuitry, then these applications of logic could play a role in logic proper; but in the absence of any convincing arguments to that effect, it is wisest to proceed on the basis that these formal pursuits count as logic only in the sense that they have a family resemblance to the core tradition. Logic, in the core tradition, involves the study of formal languages, of course, but the primary aim is to consider such languages as interpreted: languages which may be used either directly to make assertions and denials, or to analyse natural languages. Logic, whatever it is, must be a tool useful for the analysis of the inferential relationships between premises and conclusions expressed in arguments we actually employ. If a discipline does not manage this much, it cannot be logic in its traditional sense.

Propositions

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Many different basic units of evaluation have been taken to be the components of arguments: assertions, judgements, claims, utterances, interpreted sentences, sentences in contexts-of-use and propositions have all been singled out as the right kinds of things for logic to take as its raw material. We will not be forced at any stage to take sides in this debate, except to say that, whatever logic evaluates, the sentences which might be used to express arguments are not the fundamental unit of evaluation. Sentence types, in isolation from circumstances of use, are too thin to bear any kind of explanatory weight. For example, it appears that the one sentence » Tired children and parents can be stressed at the end of the day. may be used to express quite different claims. In one, the adjective ‘tired’ modifies ‘children’ but not ‘parents’ and in another, it modifies the conjunction ‘children and parents’. What we may say about the correct inferences one might draw from an assertion of this sentence depends on more than just the string of words that was uttered. If this one sentence may be used in these two different ways, then logical consequence cannot be a relation holding between sentences alone. If bare sentences do not suffice, what might? It seems that there are three broad options. Regimented Sentences: One might keep sentences as the fundamental unit of evaluation, but add to sentences whatever is needed to draw all of the distinctions we need. This may be achieved by taking logic to analyse arguments involving sentences in an appropriately regimented language. This regimentation might merely distinguish ambiguous expressions, or it might do more theoretical work by exposing a particular form or structure of the sentences being analysed. In this case, the sentences relevant to logical analysis might be sentences in (a disambiguation of English) or in some language designed to represent ‘deep structure’ hidden by the surface structure of natural languages. If we take English
to be the target, then a relevant item of logical analysis might be one of the following two sentences. » [[Tired children] and parents] can be stressed at the end of the day. » [Tired [children and parents]] can be stressed at the end of the day. A related option, still taking sentences to be the primary bearer of logical analysis, involves keeping the sentences of a natural language as the target, but adding to each sentence whatever is required to interpret it fully. So now, the unit of evaluation is a sentence-with-interpretation, rather than a sentence in a more regimented language. Judgements: Another natural response to the ambiguity between two different uses of a sentence is to conclude that logic is fundamentally the analysis of these individual uses of sentences, and not of the sentences themselves. Another way of expressing this point is to take the premises and

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Logical Consequence

conclusions of reasoning to be the judgements made. Not judgement in the sense of content (as what is judged) but judgement in the sense of act (the judging). In the case under discussion, for example, the judgement involving an inclusive is simply a different action (and a different kind of action) from the judgement using an exclusive disjunction. Much more must be said about what a judgement might actually be before this is to do much theoretical work. It is clear that the theoretical weight of the concept of logical consequence applies more fundamentally to types of judgement rather than individual tokens. After all, the very same premise repeated in reasoning, in the way that utterance tokens are literally un-repeatable. The conclusion I accept might be the very same thing as the premise you question or reject. In these cases the individual token acts of judgement are not repeated, but the types are repeatedly instantiated. It appears that logic, if it is to concern itself with judgement, speaks first of the forms or types of judgement, and through this, we can conclude things about individual acts of inference involving token judgements. However, given this reading, we still must decide whether logic applies to judgement types which are instantiated, or all possible kinds of judgement—what we might be able to judge were we to think of it, or what we would be able to say were our language radically different. Contents: Some philosophers—Frege and Geach come to mind—have pointed out that an analysis of inference purely at the level of judgement is quite difficult to maintain. Consider the inference of modus ponens: » If Zack sleeps well tonight, he will be happy tomorrow. Zack will sleep well tonight. Therefore, Zack will be happy tomorrow. It certainly appears that this is a valid inference, and that its validity is guaranteed by the fact that the second premise is the antecedent of the first premise. The second premise is an assertion or a record of the judgement that Zack will sleep well tonight. But in the conditional in the first premise, this appears as an antecedent and it is not asserted at all. What is shared by the two premises is not a judgement or an assertion, but something else. (Notice that it is not a sentence either, as the form of words differs in the two premises.) For Frege, what is shared in the two premises is a content. The content of a judgement—what is judged—may be shared with other judgements, or denials, or questions, etc., and it may appear as a part of another judgement, as in this case. The antecedent of a conditional is not asserted in that conditional construction, but it may well be asserted elsewhere in reasoning. Frege analyses this as the sharing of content. Another word often used for this notion is ‘proposition’. The proposition Zack will sleep well tonight is asserted in the second premise, but it appears unasserted as the antecedent of the first premise.

Propositions

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Introducing terms of art like ‘content’ or ‘proposition’ opens more questions than it answers.1 If contents exist, what sorts of things are they? Are they abstract, like numbers or shapes, or are they concrete? What kinds of features do they have? If they have structure, what kind of structure? Are they akin to sentences in a language, or do they have other kinds of structure? For example, are they sets of possible worlds, or are they computations from circumstances returning truth values, or something very different again? What is their connection with the so-called propositional attitudes, such as belief? Does ‘x believes that p’ express a relation holding between a believer and a content? If not, how is it to be analysed? These questions deserve some kind of response, but it is to our great fortune that we are not required to give one at this point. It is enough to sketch this option and continue on our way, noting that talk of content must presuppose some kind of answer to these questions. Those of differing philosophical temperaments will prefer different options here. It is not our place to judge between any of these options, nor even to recommend that an ultimate choice must be made. For clearly, any moderately realistic account of logic must have something useful to say at all three levels of analysis. Logic must apply to judgement: both to actual judgements we make and hypothetical or possible judgements left unmade. We can evaluate as valid or invalid particular instances of forms made at times and places. Furthermore, the validity of these inferences must be a matter of what is claimed (or questioned or denied), and not how it is claimed. It should not matter, when it comes to logic at least, if the premises of an argument are said slowly or quickly, or in what typeface they are written, or whether they are expressed in order to inspire or to insult. These features of utterances are irrelevant to their logic. The validity of reasoning must depend only on the content of what is at question, and not on its mode of presentation. Furthermore, this content must, to some degree, be public and shared, if logic is to have any role in the analysis and evaluation of discourse. None of this is to say, of course, that logic is the only tool for the analysis and evaluation of discourse. We are only committed to the claim that it is an important tool in the analysis of the content, or what is said in discourse. Finally, it is clear that the linguistic structures used to express contents bear importantly on the kinds of inferential relationships between those contents. The history of logic makes clear that formal, regimented languages have their role to play in logical theory. The development of logic took markedly different forms as logicians moved from considering the structure of subject and predicate, to propositional connectives such as conjunction, disjunction, and conditionality, to variable-binding quantifiers. 1 However, perhaps these questions need no answers. An intriguing paper by Joseph Moore argues that proposition talk makes sense even if these questions have no settled answers [82].

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Logical Consequence

It is one thing to say that a comprehensive theory of logic must have something to say about judgement, content and regimented or interpreted languages. It is another to say that we must look for particular privileged accounts of these phenomena. Perhaps there are many different, equally acceptable, ways to regiment a language.2 It could be that different assignments of contents to judgement are acceptable—in much the same way as we can measure using metres and millimetres or feet and inches. For the rest of this book, we will neither presuppose any particular answer to these questions, nor take a stand on the priority (if any) in analysis of content, judgement or language. We will endeavour to pay due respect to each of the three approaches to understanding the subject matter of logical theory. Our talk of propositions expresses commitment to the general notion of content, without any corresponding commitment to any particular way of spelling this out. Though as the work progresses, it will become clear that some options for characterising content, such as taking a proposition to be a set of possible worlds, will not be satisfactory. The reasons for this and the options for pluralism will be clearer when we get to the places where we discuss non-classical logics. We also require that talk of content is applicable to individual acts of judgement—otherwise logic would hang in thin air without application to actual reasoning—and that it may be fruitfully described in terms of regimented languages, as the tradition from Aristotle and the Stoics to the present day presumes. In particular, we take it that the language of first order logic—of conjunction, disjunction, negation, and the universal and existential quantifier—may be used to regiment at least some of the structure of the premises and conclusions in our reasoning. Logical consequence is the central concept in logic. The aim of logic is to clarify what follows from what. — Stephen Read, Thinking about Logic [99]

2.2

Arguments

In the tradition stemming from Aristotle, logical consequence is the fundamental topic in logic. Logic concerns itself with the evaluation of arguments. According to some, such as John Etchemendy, this has not always been the case: Throughout much of this century, the predominant conception of logic was one inherited from Frege and Russell, a conception according to which the primary subject of logic, like the primary subject of arithmetic or geometry, was a particular body of truths: logical truths in the former case, arithmetical or geometric in the latter . . . This conception of logic now strikes us as rather odd, indeed as something of an anomaly in the history of logic. We no longer view logic as having a body of 2 Eli Dresner has done some interesting work taking the analogy of ‘measurement’ very seriously [35, 36].

Arguments

13

truths, the logical truths, as its principal concern; we do not, in this respect, think of it as parallel to other mathematical disciplines. If anything, we think of the consequence relation itself as the primary subject of logic, and view logical truth as simply the degenerate instance of this relation: logical truths are those that follow from any set of assumptions whatsoever, or alternatively, from no assumptions at all. [46, page 74]

This is a tendentious issue. There is no doubt that, both for Frege and for Russell, logic delivered a special collection of truths, and it was of great import to know what could be proved on the basis of these truths alone. It is unsurprising that someone who undertook to reduce mathematical truth to logical truth spent a great deal of time considering exactly what would count as logically true. This might also explain the pendulum swing back to the centrality of logical consequence. If logicism is not a going concern, then the body of purely logical truths will be less interesting. Whatever we think of why Frege and Russell took logical truth to be important, it is by no means clear that the distinction between taking logical truth as primary and taking logical consequence as primary does a great deal of work. Etchemendy himself notes that logical truth reduces to a degenerate case of logical consequence. However, logical consequence also reduces straightforwardly to logical truth. An argument is logically valid just when conditional, with the conjunction of the premises as antecedent and the conclusion as consequent, is logically true. We can readily slip from one notion to the other, at least in the case of arguments with finitely many premises. In the case of arguments with infinitely many premises, the putative reduction is less straightforward. In an argument from A1 , A2 , . . . to a conclusion B, the conditional cannot use each of the premises Ai (lest it be infinitely long itself, and hence not a straightforward sentence of any natural language) or we must mention them and use something like a truth predicate: ‘If each premise Ai  is true then B.’ Granting that logic is to be used in the study of arguments, what is an argument? Almost everyone agrees that arguments involve a collection of premises (maybe a collection of just one, and maybe even none) and a single conclusion. The argument is valid if the conclusion follows from the premises. However, perhaps surprisingly, this view of the structure of arguments is not a unanimous position. Some take seriously the idea that arguments might have multiple conclusions [122]. This idea has one overwhelming vice and one overwhelming virtue. The vice of the idea of multiple conclusion arguments is that it seems completely foreign to the evidence of the arguments we see in practice.3 Consider an argument in text or in speech. It is most often signalled by the presence of ‘warranting connectives’ such as ‘therefore’, ‘since’, or ‘so’. In each case, the natural reading links multiple premises together with one conclusion. If, in a text, I read a proposition following on from ‘therefore’, 3 Relating the structure of proofs to the structures of texts in which they reside is a very interesting research project [132].

14

Logical Consequence

I will take the statements until the ‘therefore’ to be the premises of the conclusion which follows the word. Similarly, if I make some claim, following it with ‘since’ and then proceed with other claims, those other claims will be taken to be the reasons for my first claim. In each of the examples we might consider, warranting connectives naturally tie a collection of premises to a single conclusion. The virtue of the idea is its elegance. A profound symmetry emerges when one considers arguments involving multiple premises and multiple conclusions. The natural reading of validity for arguments with this structure takes an argument to be valid when the truth of all of the premises guarantees the truth of some of the conclusions. The symmetry emerges when one notices that valid arguments may be inverted: the untruth of all of the conclusions guarantees the untruth of some of the premises. Gerhard Gentzen’s insight [53] was that a natural proof theory for all of classical predicate logic can be found when one makes room for multiple conclusions. At this stage, there will be no reason for us to commit to a view of logic which allows for multiple conclusions, or, on the other hand, to reject this view. The crucial point for us is that arguments at least include premises and one conclusion, and this is enough for the bulk of our work to proceed. We think that, in general, it is insightful to take arguments to have have multiple premises and multiple conclusions—and this choice will be important later on in the discussion of anti-realist arguments against classical logic in §4.4.2. However, if you disagree with us at this point, we may continue unhindered. Taking arguments to be multiple premise and single conclusion will suffice for the presentation and defence of pluralism. We have so far seen talking of logical consequence is talk of a relation— the relata of which are collections of propositions (or of contents or regimented sentences or judgements). To narrow our search further, we must ask what we can know about this relation. What kind of relation is logical consequence? Trying to characterize logic is pointless unless one has a view point from which it matters what logic is. — Steven Wagner, ‘The Rationalist Conception of Logic’ [133, page 7]

2.3

Necessity

We now turn to the core features of logical consequence. These features are central to the tradition, and any account of logic must take account of them. One of the oldest features determining properly logical consequence is its necessity. The truth of the premises of a valid argument necessitates the truth of the conclusion of that argument. (Or, given a multiple conclusion argument, the truth of all of the premises necessitates the truth of some of the conclusions.) If you have an argument in which it could be that the premises were true and the conclusion false, then the argument is

Necessity

15

not deductively valid. This distinguishes the deductive from the inductive. Premises such as » Zack has been sick five times in the last two months. might give us some reason to believe that Zack will get sick again in the near future, but any guarantee the premise provides for the conclusion falls far short of necessitation. Zack could well have a run of good health.4 We speak crudely when we talk of the conclusion being necessary on the basis of the premises—such talk is too easily taken the wrong way. The conclusion of a valid argument might, of course, be contingent, and it might even be false, if the premises of that argument are not all true. The necessity in a valid argument does not attach to the conclusion. Necessity is borne by the transition from the premises to the conclusion. What is necessary, in an argument from A to B, is not the conclusion B but the connection between A and B. The conditional if A then B is true of necessity. (If we have an argument with two or more premises, then the requirement is that, if each of the premises is true, then of necessity the conclusion is true too.) All of this says something, but it does not say much if we cannot say much more about necessity. This is an obstacle, because it is notoriously difficult to characterise the notion of necessity at work here. This difficulty is not made any easier by the fact that many people have attempted to reduce necessity to a notion defined in terms of logic—such as inconsistency (something is necessary just when its negation is inconsistent). If this is all we can say about necessity, we are working in a very tight circle indeed. Luckily, a little more can be said about necessity. Leibniz’s vision of necessity as truth in all possible worlds is compelling. The characterisation of the necessary as that which is true in all possible worlds strikes many as a natural way to begin to put some flesh on these bones. It might seem that this trades one undefined notion (necessity) for another (possible worlds), but even if this is the case, something is gained along the way. For example, we can explain the equivalence of
(A ∧ B) with
A ∧
B. (We reason as follows. If
A ∧
B is true, then, since A is true in all possible words and B is true in all possible worlds, A∧B is true in all possible worlds, and hence
(A∧B) is true. Conversely, if A∧B is true in all possible worlds, so are A and B, and hence
A and
B are true.) To talk about possible worlds is to invite the question of what such worlds might be. Some, such as David Lewis [70], are realists about possible worlds—others are not. The distinction is not particularly important at this point. What is required is an indication of the cash value of talk of necessity or possible worlds when it comes to logic. It seems to us that at least some of the value of this talk of necessity can be spelled out in terms of hypothetical reasoning. The fact that logical 4 Parents

hope that this kind of inductive argument fails at some time.

16

Logical Consequence

consequence is necessary means that logical consequence applies under any conditions whatsoever. If we consider what might happen if A were the case, and we reason from the premise that A, validly to a conclusion B, we ought, by rights, be able to conclude that if A were the case then B would be the case too. The applicability of logic is not a contingent matter; it works come what may, whatever hypotheses we care to entertain. Talk of necessity sometimes runs into talk of certainty. Such a slide is to be resisted. None of our commitment to the universal applicability of logical consequence thereby commits us to any special epistemic status for logical consequence. It is another thing entirely to say that the premises of a valid argument must make the conclusion epistemically certain, or that the connection between premises and conclusions must be certain. Those are matters for epistemology, and are not merely the concerns of conditionality, possibility and necessity. Taking logical consequence to be necessary does not entail that we ought to take our access to it to be certain. Necessity need not entail infallibility in logic, any more than it need do so in mathematics. We can make mistakes in our judgements of what follows from what.

2.4

Normativity

Logical consequence is normative. In an important sense, if an argument is valid, then you somehow go wrong if you accept the premises but reject the conclusion. That is, we use arguments we take to be valid to judge inferences. All of this seems straightforward. It undergirds the use of deductive inference in the rational assessment of beliefs and theories, arguments and hypotheses. It is tempting to think that the normativity of logical consequence is fundamental: that it can never be rational to violate these norms. This seems to us not to be the case, and it will be helpful to spell out why. The paradox of the preface is a well known problem for anyone who takes logical consequence to be a normative constraint on belief [74]. Here is how the paradox arises: We are committed to each of the propositions we assert in the course of this book. However, we also know that we have not got everything right. We are as prone to error as anyone, so we agree that not everything we say is correct. It seems that, as a result, we are committed to each proposition in this list: p1 , . . . , pn , ∼(p1 ∧ · · · ∧ pn )

We accept p1 on some grounds, p2 on other grounds, etc., and we also believe that not all of p1 to pn are right, on the grounds of a healthy realism about our capacities to get things right. But a logical consequence of this collection of commitments is the inconsistent conjunction (p1 ∧ · · · ∧ pn ) ∧ ∼(p1 ∧ · · · ∧ pn )

and we certainly do not believe this. (The inference is to a conjunction from its conjuncts. The point holds for classical, intuitionistic, and rel-

Normativity

17

evant logic.) In fact, we do not accept the first conjunct p1 ∧ · · · ∧ pn . No, even though we accept each of the conjuncts p1 , . . . , pn , we do not accept the conjunction. On the contrary, we explicitly reject it. Our commitments in this case are not closed under logical consequence. Instead, we accept all of the premises of a valid argument and we reject the conclusion. Furthermore, this is not an argument whose validity we dispute. No, we acknowledge the argument to be valid. We accept the premises. Nonetheless, we reject the conclusion. We have knowingly violated the norm we have espoused: the norm forbidding us to accept the premises of a valid argument while simultaneously rejecting the conclusion. Does this mean that logic is not normative after all? Clearly not. What we have here is a case of a necessary evil, or an epistemic dilemma. Epistemic criteria need not always agree, any more than moral criteria. The inference we use to elucidate our own incoherence (from a collection of conjuncts to their conjunction) applies and the fact that it applies helps elucidate an important feature of our beliefs. It shows that our commitments are implicitly inconsistent. Even though we do not endorse any particular inconsistent proposition (we hope!) in the course of this book, our commitments, considered together, clash.5 They cannot be true together, and the inference shows this. In an important sense, we ought not accept p1 , . . . , pn , while simultaneously rejecting their conjunction. However, if one has good grounds to reject that conjunction, and one has good grounds for each of the conjuncts, it seems that one has good grounds for having an incoherent collection of beliefs. The normativity of logical consequence remains, even if in this circumstance it is trumped by other norms. Before leaving this topic, we should say something about another putative ‘solution’ to the preface paradox: the solution which enjoins us to replace the categorical term ‘believes’ (according to which something is either believed or it is not) with a degree of belief. Degrees of belief range between 0 (for things rejected totally) and 1 (for things accepted totally), taking intermediate values (say 12 for statements we are purely indifferent towards).6 Degrees of belief are then taken to be rationally constrained by the axioms of orthodox probability theory. For example, you one believes A to degree p then you should believe ∼A to degree 1 − p; the degree to which you believe A ∧ B added to the degree to which you believe A ∨ B should equal the degree to which you believe A added to the degree to which you believe B, and so on. In fact, so-called ‘Dutch book arguments’ are proposed to show that anyone whose degrees of belief (measured by betting rates) do not satisfy the traditional axioms of probability theory is 5 As far as we can tell, the picture we paint here is consistent with many different approaches in epistemology, both internalist and externalist, both contextualist and noncontextualist. 6 According to some, degrees of belief can be ‘measured’ by taking betting rates. If one is equally happy to be on either side of a bet to win $1 if A is true, paying a price of $p, then my degree of belief in A is p.

18

Logical Consequence

subject to a ‘sure loss contract’, a system of bets according to which it is sure that they will lose. What does this have to do with the paradox of the preface? The proposed ‘solution’ using degrees of belief is straightforward. Instead of talking of believing each p1 to pn in the book and believing the negation of the conjunction ∼(p1 ∧ · · · ∧ pn ) we assign a high degree of belief (but not 1) to each pi , and this is completely consistent with assigning a very low degree of belief to the conjunction p1 ∧ · · · ∧ pn and a high degree to the negation ∼(p1 ∧ · · · ∧ pn ). The axioms of probability theory are not violated. Does this ‘solution’ conflict with anything we have said about normativity and logical consequence? It seems to us that there is no conflict. The degree-theoretic analysis of the preface paradox asks us to consider the concept of degrees of belief rather than categorical belief. This seems sensible enough: moving to a degree-based picture allows us to find a kind of coherence (cohering with the axioms of probability) where there was none (our commitments were simply inconsistent). So much seems like a good thing. However, changing the topic of discussion does not mean that the original phenomenon disappears. While degrees of belief are important and clarifying, they cannot tell the whole story about belief, commitment, assertion, and normativity. In the preface case as we discussed it, the book makes a number of assertions p1 to pn , and the negation ∼(p1 ∧ · · · ∧ pn ). These claims are asserted, and whether or not something is asserted is not a matter of degree. When we make assertions we do not tag them with numbers between 0 and 1 indicating the degree to which an assertion is made, and rightly so. An assertion commits the asserter to taking things to be thus and so, and we can rightly ask what follows from things being thus and so, what is consistent with this, and what contradicts it. To conclude: probability theory might provide a canon for evaluating degrees of belief, and this might give us helpful insight in the case of the preface paradox. Nonetheless, probability theory cannot be a complete answer here, for we also make assertions and denials (and hypotheses and many other things besides), and these may also be evaluated for coherence, using the norms of deductive logic. In particular, we hold that it is a mistake to assert the premises of a valid argument while denying the conclusion. This norm, however, may be trumped, and we may have reason to violate it.

2.5

Formality

Formality is one feature of deductive logic that is almost invariably taken as distinctive. Determining exactly what might be meant in this description is another thing entirely. What does it mean to say that logic is distinctively formal? The eminent Polish logician, Jan Łukasiewicz, addressed this question in Aristotle’s Syllogistic: ‘It is usual to say that logic is formal, in so far as it is concerned merely with the

Formality

19

form of thought, that is with our manner of thinking irrespective of the particular objects about which we are thinking.’ This is a quotation from the well-known text-book of formal logic by Keynes [61, page 2]. And here is another quotation, from the History of Philosophy by Father Copelston: ‘The Aristotelian Logic is often termed formal logic. Inasmuch as the Logic of Aristotle is an analysis of the forms of thought—this is an apt characterisation.’ [52, page 277] In both quotations I read the expression ‘form of thought’, which I do not understand. Thought is a psychical phenomenon and psychical phenomena have no extension. What is meant by the form of an object which has no extension? The expression ‘form of thought’ is inexact and it seems to me that this inexactitude arose from a wrong conception of logic. If you believe indeed that logic is the science of the laws of thought, you will be disposed to think that formal logic is an investigation of the forms of thought. [72, pages 12–13]

For Łukasiewicz, the historical characterisation of logic as the science of the forms of thought is not an appealing direction of investigation. It would make logic beholden to cognitive psychology in a way that it does not seem warranted by the tradition. Łukasiewicz goes on to explain what we might say about the formality of logic, without resorting to psychologism. . . . an argument of the Peripatetics, preserved by Ammonius in his commentary on the Prior Analytics, deserves our attention. Ammonius . . . says: If you take syllo-

gisms with concrete terms, as Plato does in proving syllogistically that the soul is immortal, then you treat logic as a part of philosophy; but if you take syllogisms as pure rules stated in letters, e.g. ‘A is predicated of all B, B of all C, therefore A is predicated of all C’, as do the Peripatetics following Aristotle, then you treat logic as an instrument of philosophy. It is important to learn from this passage that according to the Peripatetics, who followed Aristotle, only syllogistic laws stated in variables belong to logic, and not their applications to concrete terms. The concrete terms, i.e. the values of the ´ , of the syllogism. If you remove all concrete variables, are called the matter, νλη terms from the syllogism, replacing them by letters, you have removed the matter of the syllogism and what remains is called its form. Let us see of what elements this form consists. To the form of the syllogism belong, besides the number and the disposition of the variables, the so-called logical constants. Two of them, the conjunctions ‘and’ and ‘if ’, are auxiliary expressions and form part, as we shall see later, of a logical system which is more fundamental than that of Aristotle. The remaining four constants, viz. ‘to belong to all’, ‘to belong to none’, ‘to belong to some’ and ‘to not-belong to some’, are characteristic of Aristotelian logic. These constants represent relations between universal terms. The medieval logicians denoted them by A, E, I and O respectively. The whole Aristotelian theory of the syllogism is built up on these four expressions with the help of the conjunctions ‘and’ and ‘if ’. We may say therefore: The logic of Aristotle is a theory of the relations A, E, I, and O in the field of universal terms. [72, pages 13–14]

Łukasiewicz speaks for many in the 20th century: it has been a commonplace to characterise the formality of logic in terms of its being schematic.

20

Logical Consequence

Logic does not speak at first of individual concrete arguments. Instead, it categorises forms. Perhaps this is all there is to say about the formality of logic. If this is the case, it turns out that the constraint of formality (as Łukasiewicz himself noted) says very little about what kind of theory is distinctively logical and what is not. Consider the following nine argument forms: a1 a2 a3 a4 a5 a6 a7 a8 a9

some F are G, all G are H; therefore some F are H. p, if p then q; therefore q. x is red, if x is red then x is coloured; so x is coloured. x knows that p; therefore p. x is more F than y is, y is F; therefore x is F. x is larger than y, y is larger than z; so x is larger than z. x is red; therefore x is coloured. x is water; therefore x is H2 O. x is a baby; therefore x cries.

Each is presented schematically, so each is schematic to some degree. However, no-one accepts each argument form as valid: argument a9, strictly speaking, could have a true premise and a false conclusion—as unlikely as this circumstance might be—so it fails on the test of necessity. Argument a8 cannot be known a priori to be valid, it seems, but if Kripke is right about natural kinds, this argument is necessarily truth preserving. The others, from a1 to a7 are necessarily truth preserving, but very few in the history of logic would endorse all seven as deductively valid. Argument a1 is the only syllogistically valid form, and only forms a1 to a3 are valid by the lights of classical predicate logic. The other forms ‘work’ (if they work in any sense at all) in virtue of something other than the connectives and quantifiers of predicate logic: a4 trades on knowledge; a5 on comparatives; a6 on the transitivity of ‘larger than’ and a6 on connections between colour terms. Now, some of these arguments have been taken to fail to be deductively valid on the grounds of their informality. But what could this mean? Each form is schematic in a similar way. Each contains variables which can be filled by items of the appropriate categories: x by names, p and q by propositions, F, G and H by predicates. If any difference is to be found between these forms, it must come down to something else. One possibility is that the relevant distinction is to be found in what is left over in those forms, unanalysed. We will try to do more with formality than Łukasiewicz and schematicformality have managed. This will require a more sympathetic attention to the notion of the ‘form of thought.’ We agree with Łukasiewicz (and Kant, Husserl, and almost everyone else) that psychologism is to be avoided, so let us read ‘thought’ not as designating a particular psychological item or event, but rather as its content, broadly construed. Then, the form of

Formality

21

thought is perhaps best construed as the structural features intrinsic to propositional content, rather than any of its accidental features. Here is an example of how this is to be spelled out. All judgement, for Kant, is essentially the predication of some property to some subject. It is clear, why on this picture the rules of Aristotle’s syllogism are naturally viewed as formal while the other inference schemata are not. Aristotle’s rules are plausibly thought to say something about the intrinsic nature of judgement as such. Every judgement has the form of a predication, and Aristotle’s syllogistic provide rules governing such predications. On the other hand, an inference like a4 works only in virtue of the concept of knowledge. Someone can judge independently of possessing the concept of knowledge, it seems, so the inference a4 fails to be formal in this sense. This Aristotelian explanation is one of a number of different ways to put flesh on the bones of the form–content distinction. Three distinct characterisations of formality are given by MacFarlane [73], in his thesis What Does it Mean to Say that Logic is Formal? f1 logic provides constitutive norms for thought as such. f2 logic is indifferent to the particular identities of objects. f3 logic abstracts entirely from the semantic content of thought. Each characterisation of formality is an attempt to elucidate the concept in a way which might enable us to draw sharper distinctions. 1-formality: To say that logic is 1-formal is to say that it provides constitutive norms for thought as such, as opposed to norms which might apply to particular kinds of propositional content as opposed to others. In earlier discussion we have noticed, in effect, that 1-formality can be used to defend Aristotle’s syllogism. Or rather, this defence can be made out if we are satisfied with a subject–predicate analysis of content. If another analysis is to be preferred, then perhaps other principles might be found to be logical. It seems to us that a plausible case can be made that rules governing the behaviour of conjunction, disjunction, and negation satisfy the criteria of 1-formality, because all propositional content can be operated on by means of these propositional connectives. 2-formality: To say that logic is indifferent to the identities of objects is to say that, if some argument is valid, then it will remain valid when you vary the reference of denoting terms in that argument.7 One way to enforce this criterion is to hold that all referring terms be schematic. It follows, then, that 2-formality does not rule out any of the inferences in our list, unless we have a particularly generous account of what counts as an object or a term. None of the inference forms listed depends on the identities of any objects whatsoever. However, if properties are to count as 7 This kind of criterion is spelled out in Tarski’s work on characterising logical notions as those invariant under permutations [76].

22

Logical Consequence

objects, or predicates are to count as referring terms, then inferences a7–a9 fail to be logical, since they trade on relations between properties. If these are to be taken as variable, we are left with the form a1 x is F; therefore x is G which fails to be valid on even the most generous account of logical consequence. The inferences a5 and a6 are difficult cases.8 While neither is valid by the lights of classical predicate logic, it is not clear whether the predicate comparison more F than ought to be treated as expressing a higher order property (and so, is fit for variation) or in some other way (in which case it might be kept fixed). Regardless, it is much clearer that the propositional connectives of conjunction, disjunction, and negation count as logical constants on this criterion. Connectives are not, on the standard picture at least, referring terms. 3-formality: This notion of formality requires a distinction between semantic and non-semantic (or pre-semantic) features of statements. The idea is straightforward: statements (or interpreted sentences or contents or thoughts) contain expressions with semantic content, and then there is the remainder—not to be thought of as expressions without content, but perhaps instead the way those contents are combined: The form of the sentences. On this picture, a statement such as Zack is tired and he should go to bed features ‘Zack’ as a name (with content) ‘is tired’ as a predicate, and so on. If the conjunction in the expression is to be thought of in the same way (that ‘and’ ‘refers to’ or ‘picks out’ the connective of conjunction) then it is not properly a logical constant. If, on the other hand, it indicates a way that the two thoughts Zack is tired and Zack should go to bed are combined, then conjunction is a notion which is 3-formal. These three characterisations of formality are often confused or identified, but as we have seen, this would be a mistake. MacFarlane shows— convincingly, it seems to us—that Kant held to all three versions of formality, and that on a Kantian understanding all three coincide. However, as we have also seen, they do not coincide for everyone. An important historical example is Frege. For Frege, logic is 1-formal without being 2-formal. Frege’s logic is not indifferent to the identities of objects at all. Logic, in Frege’s eyes, can tell you a great deal about numbers, and these are genuine objects. Frege’s logic is not indifferent to the particular identities of objects: it enables you to prove a great deal about them. This section ends inconclusively. We do not have a favoured notion of formality to constrain our investigations from this point onward. This is perhaps an appropriate conclusion, for, as MacFarlane shows, there is no settled notion of formality within the tradition. Furthermore, given a 8 See

the literature on comparative logic for more [31, 48, 88].

Cases

23

particular notion of formality, we do not have a very sharp boundary between the formal and the non-formal. We can commit to two things, however. First, the degree to which our accounts of logical consequence can be shown to be formal (beyond mere schematic-formality) will contribute to showing that our accounts of logical consequence belong to the core tradition. Second, on each of the ways of making formality precise, the propositional logical constants of conjunction, disjunction, and negation, and the laws governing them are plausibly taken to be formal. The world is all that is the case. — Ludwig Wittgenstein, Tractatus Logico-Philosophicus [136, §1]

2.6

Cases

Now we turn to what we take to be the most powerful and productive idea, constraining the concept of logical consequence. This idea is available in nascent form throughout the tradition, and it has surfaced explicitly in Leibniz’s possible worlds, Bolzano’s variable terms, Husserl’s manifolds, and most clearly, Tarski’s models. This idea is powerful, not only in providing a dividing line between the logical and the non-logical, but also in giving us a principle undergirding the other distinctive properties of logic. Let us introduce this principle by giving an example in which it has been used to introduce deductive logic in Richard Jeffrey’s textbook Formal Logic: Formal logic is the science of deduction. It aims to provide systematic means for telling whether or not given conclusions follow from given premises, i.e., whether arguments are valid or invalid . . . Validity is easily defined: A valid argument is one whose conclusion is true in every case in which all its premises are true. Then the mark of validity is the absence of counterexamples, that is, the absence of cases in which all premises are true but the conclusion is false. Difficulties in applying this definition arise from difficulties in canvassing the cases mentioned in it . . . [60, page 1]

This analysis of validity will form the centre of our book, and it is at the heart of our pluralism about logical consequence. » A valid argument is one whose conclusion is true in every case in which all its premises are true. We hold that deductive validity is a matter of the preservation of truth in all cases. An argument is valid when there is no counterexample to it: that is, there is no case in which the premises are true and in which the conclusion is not true.

24

Logical Consequence

This analysis of validity owes a great deal to the tradition, as we have already seen. It is also connected intimately with the constraints on consequence that we have already seen. Consequence is necessary in the following sense—it applies in all cases. Consequence is normative because rejecting a valid argument involves cutting the ground from under your own feet. To endorse the premises of a valid argument but to reject the conclusion is to contradict yourself in the following sense: There is no case in which those claims could hold true. Your commitments undercut themselves. Finally, the notions of formality which we find useful are notions of independence and abstraction. How is this to be analysed, except in terms of what is kept true in different cases, while we allow other things to vary? The insightful analyses of these notions are to be found by considering classes of models or interpretations, and these are at the heart of the analysis of logical consequence. So, the analysis of logical consequence as preservation of truth in all cases goes some way to explaining how a relation of logical consequence is necessary, normative and formal. This is not to say that this analysis answers every question you might wish to ask. Not at all. As Jeffrey makes explicit, we have much still to do. It is not always clear how the analysis is to be applied, because we do not yet know how the term ‘case’ is to be understood. That question, and the many different consequences of the different kinds of answers to that question, will occupy us for the rest of the book.

Chapter 3

Pluralism Defined Logical pluralism is a thesis about logical consequence. In this chapter we give an abstract sketch of pluralism and the kinds of arguments we shall advance on its behalf. With sketch in hand, we outline what lies ahead.

3.1

What Kind of Claim is Logical Pluralism?

Logical pluralism is a pluralism about logical consequence. Crudely put, a pluralist maintains that there is more than one relation of logical consequence. By way of illustrating the kind of claim involved in logical pluralism, we turn to a few analogies. 3.1.1 Example: Computability One prominent analogy is the Church–Turing Thesis: recursive.

A function is computable if and only if it is

The Church–Turing Thesis aims to make precise a common but imprecise notion. Specifically, the thesis enjoins us to understand the informal notion of computability in terms of the class of recursive functions, which enjoys a precise definition. The Church–Turing Thesis is not the kind of thing which may be conclusively proved; the everyday concept of computability is not amenable to such conclusive proof. Instead, the thesis may be justified by showing that the formal, precise notion of recursiveness plays the role filled by the informal notion of computability. The thesis is a biconditional, and evidence for it is found in two parts. First, we want evidence that all recursive functions are computable. For this, one might take an inductive definition of the class of recursive functions.1 In turn, one verifies that the basic functions in the definition are computable (perhaps by finding some computing procedure or other which does the trick), and then verifies that if some functions are computable, so too are others—the ones constructible by using the inductive procedures (from the definition of the class of recursive functions) on the ‘verified’ functions. 1 Boolos

and Jeffrey’s Computability and Logic provides two such definitions [25].

25

26

Pluralism Defined

For the converse, the evidence is more difficult to collect, but the orthodoxy holds that it is none the less true. There is no easy way to regiment intuitions about arbitrary computable functions to show that they are recursive—that was the point of introducing the thesis in the first place, which replaces an imprecisely defined notion by a precisely defined one. That said, people have shown that different ways of defining computable functions (say, by way of Turing machines, register machines, or other abstract and general computational devices) always produce recursive functions. Given that no non-recursive function can be computed by such devices, orthodoxy concludes that the Church–Turing Thesis is true. (However, it is true that every orthodoxy has a heterodoxy. Some proponents of hyper-computation hold that more than the recursive may be computed, by way of computational devices with extraordinary capacities: for example, the ability to compute an infinite number of operations in a finite duration [32].) 3.1.2

Example: Necessity

Another analogy involves the notion of necessity. One might think (reasonably, in our opinion) that the notion of necessary truth is constrained but not completely so. Sometimes we use the term to pick out what must be the case no matter what unconstrained by circumstance, and at other times we may talk of what is necessary, given some constraint or other. It is a great insight that the notion of necessity may be made more precise and amenable to analysis: Generalised Leibniz Thesis (glt): and only if it is true in every casex .

A claim is necessarilyx true if

This ‘thesis’ yields only the basic structure for particular accounts of necessary truth; an instance of the thesis (specifying necessityx ) is found by specifying the placeholder casex (which perhaps in turn may be made more precise in various ways). We may take casex to range over any ‘possible world’ whatsoever; this yields an account of what some have called bare or metaphysical necessity. (To be sure, ‘possible world’ requires more precision, but for present purposes the reader may insert her favourite account.) If, on the other hand, we take a casex to be a world like ours, satisfying the same physical laws as ours, then the thesis yields an account of what is physically necessary. Similarly, an account of historical necessity arises when we attend to histories which are like ours until the present moment. Other applications follow the same pattern. As we proffer an instance of glt, we provide a way of precisifying the general notion of necessity. Each ‘necessityx ’ is a kind of necessity. Even without detail with respect to casex , glt yields fruit. If, as seems to be true for ‘possible worlds’, a conjunction is true in a casex if and only if both conjuncts are true in a casex , glt yields that a conjunction is necessaryx if and only if both conjuncts are. A conjunction is necessaryx

What Kind of Claim is Logical Pluralism?

27

if and only if the conjunction is true in every casex ; the conjunction is true in every casex if and only if the conjuncts are true in every casex . This is equivalent to the necessityx of both conjuncts. So, glt illustrates a kind of constrained freedom: necessity may be precisified in a number of ways, but each such way is constrained by the general schema (glt). In this way glt serves as (at least a part of ) the ‘settled core’ of what we might call the pre-theoretic notion of necessity; it regiments the various ways of making the otherwise imprecise notion precise. 3.1.3 Example: Vagueness and other unsettledness Another analogy may be useful. English seems settled with respect to some terms, unsettled with respect to others.2 Paradigm examples involve socalled vague expressions. Some persons are such that ‘is bald’ determinately applies to them; some are such that ‘is bald’ determinately does not apply to them; some persons are such that, given the unsettled nature of the language, nothing determines whether ‘is bald’ applies or does not. Freedom is found in the unsettled region: Within the constraints imposed by the settled parts of language, one is free to make ‘is bald’ precise in numerous ways. If ‘is bald’ is made precise in different ways, the result is a class of different senses of ‘is bald’. Such senses of ‘is bald’ are admissible provided that each agrees with the originally settled fragment. Suppose that F is a predicate exhibiting unsettledness. Suppose that, for one reason or another, different (admissible) precisifications are formulated, thereby yielding different senses of F which, in turn, might be explicitly recorded by indexed predicates: Fi , etc. There is no sense in asking which of the various Fi is the correct precisification of F. Given that each Fi is admissible, each is ‘correct’ if any is. The important question concerns the extent to which the Fi work, the extent to which they serve the (settled) role of F. On this question, the question of utility, some precisifications may be better than others. On the question of which precisification is right or accurate, there simply is no answer—provided, as above, that each Fi is admissible. While it typically serves as a paradigm, vagueness is not the only example of unsettledness. This issue depends on the important and difficult question of how to characterise vagueness. A simple characterisation treats vague expressions as those giving rise to sorites arguments. Here is an example. A person with $10, 000, 000, 000 is rich; and if a person with $n is rich, then a person with $(n − 1) is rich. While this is neither precise nor uncontroversial, we will assume some such demarcation of vagueness for present purposes. Indeed, the situation with necessity (see §3.1.2) reflects this kind of phenomenon. The language is settled enough to reflect the ‘core’ of necessity, which is at least in part recorded in the glt; however, the basic core fails to determine a unique ‘correct’ precisification of the notion. As 2 So-called epistemicists [135] may object to what we say in this section; we note this only to set it aside.

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Pluralism Defined

with vague predicates, the question is not whether an admissible precisification is correct: the question is whether it is useful, particularly with respect to the settled role of the original (and otherwise imprecise) notion. To be sure, some instances of the glt may be dull, seemingly useless, or perhaps even silly, but this has nothing to do with whether they are admissible precisifications of the original notion. 3.1.4 Example: Consequence It is a commonplace that the informal notion of following from, as it applies to reasoning, may be made precise in at least two distinct ways. One way fixes on deductive consequence. This is the chief focus of our enquiry: Deductive consequence: A conclusion follows from premises if and only if the premises necessitate the conclusion (in some sense to be determined). Another route fixes on inductive consequence: Inductive consequence: A conclusion follows from premises if and only if the premises make the conclusion more likely. With respect to the latter, one might (and often does) go further by giving different accounts of inductive strength, where each such account purports to capture features of our pre-theoretic use of ‘follows from’. The extent to which such an analysis is deemed useful or fruitful is usually the extent to which it both incorporates the given pre-theoretic features and provides them with connections to other related concepts. Almost everyone agrees that the ‘goodness’ of an argument can be settled in these two broad ways. This will be a useful analogy in what follows, in the exposition of logical pluralism. Our pluralism about deductive logical consequence is of a piece with pluralism about consequence in general. We hold that deductive logical consequence can be (and has in fact been) settled in more than one way, just as consequence in general can be, and has been. 3.1.5 Summary §3.1.1–§3.1.4 illustrate the kind of claim involved in logical pluralism, which points to a number of different ways to make the pre-theoretic notion of (deductive) logical consequence precise. Each such precisification purports to incorporate the core features involved in the use of ‘follows from’ or ‘logical consequence’ (e.g. necessity, formality); however, none will do such a good job that it renders the others useless or otherwise undeserving of the role. This endeavour contains a degree of freedom: the pre-theoretic notion of logical consequence is not formally defined, and it does not have sharp edges. That said, the notion is not so free that any relation will do. We cannot simply stipulate that relation R is a relation of logical consequence. The degree of success of any account of logical consequence will be the degree

Logical Pluralism in a Nutshell

29

to which it works, which is measured in terms of the concepts uncovered in §2. Any settling of the relation of logical consequence must be a necessary, normative and formal relation on propositions.

3.2

Logical Pluralism in a Nutshell

Having discussed the kind of claim involved in Logical Pluralism, we turn to the claim itself, which essentially involves the following. Generalised Tarski Thesis (gtt): An argument is validx if and only if, in every casex in which the premises are true, so is the conclusion. Tarski puts Tarski’s Thesis, what we call the restricted thesis, thus: ‘The sentence X follows logically from the sentences of the class K if and only if every model of the class K is also a model of the sentence X’ [127, page 417]. As will be clear in subsequent chapters, we take this to be an instance of gtt given that ‘model’ is understood in a particular way (i.e., along Tarskian lines). Adopting the thesis in the analysis of logical consequence means that truth-preservation in cases does the work required of logical consequence. Like glt, this ‘thesis’ is only a recipe for specific accounts of consequence; particular precisifications are gained only when casex is specified. Endorsing one such instance commits one to gtt; however, it does not commit one to logical pluralism. Logical pluralism is the claim that at least two different instances of gtt provide admissible precisifications of logical consequence. Unlike the restricted Tarski Thesis, which admits only one instance of casex (Tarski’s models), the pluralist endorses at least two instances, giving rise to two different accounts of deductive logical consequence (for the same language), two different senses of ‘follows from’. As with unsettledness in general (see §3.1.3), so too with the current topic: the question is ultimately one of utility. Provided that the various gtt accounts of consequence are admissible, there is no sense in asking which is the correct account. To the question ‘Which account is the right account of consequence?’ there is no answer—provided, again, that the relevant candidates are admissible. Whether candidates are admissible turns on whether they agree with the settled parts of language, on whether they exhibit the features required by the (settled) notion of logical consequence. We hold that the notion settles some but not all features of any candidate relation of logical consequence; the unsettled features leave room for plurality. The foregoing sketch of logical pluralism is rather abstract; the details are canvassed in Part II. For now, we remain at the abstract level but briefly turn to the question of disagreement: How might one fail to be a logical pluralist?

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Pluralism Defined

3.2.1

How Not to be a Pluralist

There are at least two ways to not be a logical pluralist, in our sense of the term. Reject the Generalised Tarski Thesis: One quick route around pluralism involves rejecting gtt. One might think that, irrespective of any specification of casex , truth-preservation over cases yields an inadequate account of logical consequence. The objection, for example, might be that logical consequence is not transitive, or perhaps reflexive, in which case the recipe of the thesis is inadequate.3 On the other hand, you might take logical consequence to be reflexive and transitive (as is given by gtt), but you might none the less reject gtt as an account of logical consequence. Why might you do this? One reason could be ontological squeamishness about the different options for casex . Whatever the objection might be, one fails to be a logical pluralist (in our sense) if one thinks that no instance of gtt provides an accurate account of logical consequence. Endorse Exactly One Instance: Another, more likely, route around pluralism insists that there is only one admissible instance of gtt. On this line one recognises exactly one admissible specification of casex , deeming all others to yield something other than logical consequence. For example, one might insist that only Tarski’s models be assigned to ‘casex ’ in gtt, in which case the ‘Generalised Thesis’ collapses into the original restricted version, Tarski’s Thesis.

3.3

The Case for Logical Pluralism

The case for pluralism is two-fold: a direct argument from appearance, and a defence of the given appearances. While we think that the two-fold case is sufficient, we none the less add more: we discuss virtues of logical pluralism. The Direct Argument: There appear to be at least two senses of ‘validity’ or ‘follows from’ that correspond to admissible instances of gtt. (We will eventually point to more than two, but for now we are only sketching, and two senses are sufficient for pluralism.) For example, there appears to be a sense in which the argument from the contradictory A ∧ ∼A to arbitrary B is valid, and a sense in which it is invalid. On one hand, there is no consistent case in which A ∧ ∼A is true but B untrue; hence the argument is valid, at least in one (seemingly admissible) sense. On the other hand, there is also a sense in which B does not follow from A ∧ ∼A, as the former has ‘nothing to do with’ the latter. Hence, focusing on the ‘from’ in follows from appears to yield a sense of ‘validity’ according to which the argument from A ∧ ∼A to B is invalid. 3 Challenges

along these lines are taken up in Part III.

What Lies Ahead?

31

The upshot: provided that each of the noted senses of ‘validity’ corresponds to an admissible instance of gtt, there are at least two relations of logical consequence (in English), and so logical pluralism follows. In Defence of Appearance: That is how things appear, and appearances, we grant, sometimes deceive. Perhaps the appearance suffices to shift the burden of proof to the opponent of pluralism; we think it does, but we also think that more can be said. We will defend the given appearance by arguing that there are indeed different admissible senses of validity. How can we do this? The case will be similar to the defence of the Church–Turing Thesis. For each relation of logical consequence (corresponding to different specifications of casex ) we will show that it satisfies each of the desiderata of the core notion: anything that it is reasonable to take as a settled feature of the notion of logical consequence is satisfied by each of the candidate relations (or specifications) of logical consequence. Virtues of Pluralism: In addition to the given direct argument and corresponding defence, we will also take note of the many virtues of pluralism. One virtue is that the plurality of consequence relations comes at little or no cost. Another is that pluralism affords a more charitable interpretation of many important (but difficult) debates in philosophical logic than is otherwise available; we will argue that pluralism does more justice to the mix of insight and perplexity found in many of the debates in logic in the last century. Logical pluralism has many more virtues besides, which we will observe as we present the position in detail.

3.4

What Lies Ahead?

We have sketched the kind of claim made by Pluralism, the claim itself, and the sort of argument advanced on its behalf. In this section we briefly sketch the roles of subsequent chapters. 3.4.1 Part II: Logics Part II discusses various senses of ‘validity’ that correspond to (admissible) instances of gtt. We canvass various specifications of casex , showing how each gives rise to an instance of gtt which is admissible—satisfies the settled core of consequence and serves the settled role. Specifically, we discuss the following. Classical Logic (§4): By taking casecl to be complete and consistent—they are Tarski’s models—one gets a specification of ‘follows from’ corresponding to classical predicate consequence. (We will explain the requisite senses of (in-)complete and (in-)consistent in the target chapters.) Relevant Logic (§5): Restricting attention to consistent and complete models yields many benefits, but it has costs as well. Classical consequence fudges all distinctions among inconsistencies and among tautologies. By allowing cases to be inconsistent and incomplete (in a sense to be spelled

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Pluralism Defined

out in some detail, later), we motivate a specification of ‘follows from’ corresponding to relevant consequence. Constructive Logic (§6): We argue that constructive consequence, modelled in intuitionistic logic, may be motivated as a specification of the original sense of logical consequence, given a choice of cases as constructions. Variations & Loose Ends (§7): Furthermore, even when we grant these choices of some of the properties of cases to be used in the specification of consequence, more variations are possible. We will briefly discuss a number of such in §7. For each of the given specifications of logical consequence, we show that the settled core of the informal (and otherwise imprecise) notion of consequence is exhibited, and hence that each such relation is an admissible instance of gtt. 3.4.2 Part III: Objections, Replies, Other Directions Part III of the book proceeds without much pause: we explain requisite terminology, give the respective specifications of casex , and indicate the admissibility of each gtt instance. Accordingly, we pause little (if at all) to anticipate objections or address questions. Objections and questions are the focus of Part III, together with suggestions for future research and other directions that a pluralistic philosophy of logic might lead.

Part II

Logics

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Chapter 4

Classical Logic Pluralism, stripped down, is captured in the following list of conditions. 1 The settled core of consequence is given in gtt (see §3). 2 An instance of gtt is obtained by a specification of the casesx in gtt, and a specification of the relation is true in a case. Such a specification can be seen as a way of spelling out truth conditions. 3 An instance of gtt is admissible if it satisfies the settled role of consequence, and if its judgements about consequence are necessary, normative, and formal (in one sense or other discussed in §2). 4 A logic is given by an admissible instance of gtt. 5 There are at least two different admissible instances of gtt. For us, logic is a matter of preservation of truth in all cases. We take that to be the heart of logical consequence, the settled core of follows from. That is the upshot of (1). The import of (2)–(4) is that a logic is constructed by giving an account of the casesx over which gtt quantifies; the resulting instance of gtt specifies a logic if it is an admissible instance of gtt, an account of consequence the judgements of which are necessary, normative, and formal. Specifying the casesx over which gtt quantifies is not enough: one must also give some kind of story about which kinds of claims are true in what sorts of casesx . For example, you might give an account in which casesx are possible worlds. (You might go on to tell a metaphysical story about what sorts of entities possible worlds might be [70, 71, 123, 139], but then again, you might not.) Alternatively, you might specify casesx as settheoretic constructions, models of some sort, or you might specify casesx in some other way. Whatever one’s casesx may be, more work is required in giving a logic; one must also give an account of truth in a case. For example, your account of casesx and truth in casesx might include the condition » A ∧ B is true in x if and only if A is true in x and B is true in x where A and B are claims and x is a case. (Read our neutral term ‘claim’ as picking out sentences, propositions, utterances, statements, or anything else you think might feature in the premises and conclusions of arguments.) 35

36

Classical Logic

Such an assertion tells us that a conjunction is true in a case if and only if both conjuncts are true in that case. That gives us an account of truth in cases that not only specifies the behaviour of conjunction (in casesx ) but also gives some data about validity: for any casex , if A ∧ B is true in x then A is true in x, by the condition given above. Hence, by gtt, the argument from A ∧ B to A is valid. So goes one example of how you might begin systematically to spell out the conditions under which claims are true in gtt’s casesx . To do that is to do logic. Conditions (1)–(4) are not particularly controversial. The controversy in our position comes at point (5), according to which there are different ways to specify casesx in that each yields an admissible instance of gtt, and each yields a logic (on the same language). There is no canonical account of casesx to which gtt appeals. Parenthetical remark: The claim that (1)–(4) are not controversial is a small fib. One part is controversial. We have unashamedly privileged the modeltheoretic or semantic account of logical consequence over the proof-theoretic account. We think that a version of pluralism can be defended which does not privilege ‘truth in a case’ to the same extent. However, since most of the current debates with which we are interacting lie firmly within this model-theoretic tradition, and since we are comfortable with that tradition, we are developing pluralism in this way. End of parenthetical remark. The chief question in Logic is: What follows from what? What arguments are valid? The upshot of (5) is that, given a particular argument, there is more than one correct answer to Logic’s chief question. That is the heart of logical pluralism. In the next three chapters we will elaborate point (5) by discussing different ways that gtt can be—and has been—filled out. In this chapter we cover two well-known specifications of casesx : possible worlds and models for classical predicate logic. We will also elaborate (1)–(4) by noting the extent to which the canvassed accounts of consequence are admissible: the extent to which their respective judgements are necessary, normative, and formal. Along the way we will also mention applications of the various logics, for it is only at that level, the level of application, that rivalry between logics arises.

4.1

Specification of Cases

There are various ways of specifying the casesx in gtt that render valid all of the theses of classical logic. In this chapter we cover two such specifications: possible worlds and Tarskian models. We treat each in turn. 4.1.1 Possible Worlds as Cases A consequence relation that validates all theses of classical logic results by letting the casesx of gtt be possible worlds, or casesw . Let ‘w  A’

Specification of Cases

37

abbreviate ‘A is true in world w’. Then clauses for truth in a casew , or truth in a world, run as follows:1 » w  A ∧ B if and only if w  A and w  B. » w  A ∨ B if and only if w  A or w  B. » w  ∼A if and only if w  A. It is a little harder to account for the truth of quantified claims in possible worlds; however, if we allow each object in each world to have a name in our language, then the clauses are trivial: » w  ∀xA(x) if and only if for each object b in w, w  A(b). » w  ∃xA(x) if and only if for some object b in w, w  A(b). With no further analysis of what a world w might be, or how many there might be, gtt already yields a story of consequence. The account, as we mentioned above, already validates the inference from A ∧ B to A. The account also validates the inference from A to A ∨ B, from A ∧ (B ∨ C) to (A ∧ B) ∨ C, from ∀x(A ∨ B) to ∀xA ∨ ∃xB, and many more besides. If casesw encompass all possible worlds, then gtt is read thus: » An argument is valid if and only if, in any world in which the premises are true, so is the conclusion. (Equivalently: . . . if and only if it is impossible for each premise to be true but for the conclusion to not be true.) Call that the necessary truth-preservation (ntp) account of validity. 4.1.2 Tarskian Models as Cases The ntp account is one instance of gtt, but it is not the only one. In fact, the ntp account is not at all the orthodox account of logical consequence. The ntp account is too liberal for orthodoxy; it is not appropriately formal because it makes no essential use of the forms of the sentences analysed. To be sure, the ntp elucidation has picked out conjunctions, disjunctions, negations, and quantifiers, but none of that is essential to the account. As far as the ntp account goes, we could just as well have given clauses for colour terms: » w  ‘b is red’ if and only if b is red in w. » w  ‘b is coloured’ if and only if b is coloured in w. That explains why the ntp account of validity renders the argument from ‘b is red’ to ‘b is coloured’ valid; it is valid because, in any casew (that is, 1 We focus on propositional connectives for a number of reasons. First, laws concerning them are plausibly thought of as formal. Second, it is straightforward to present truth conditions for the connectives in possible worlds. Third, different accounts of the behaviour of the propositional connectives will be the focus of the difference between classical, relevant and constructive logic.

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in any possible world) in which something is red, it is also coloured; it is impossible that something be red and for it to fail to be coloured. According to logical orthodoxy, the argument from the premise ‘b is red’ to the conclusion ‘b is coloured’ is invalid, because it is not formal. It does not exploit any logical form: it has the form Fb  Gb, which is invalid. What orthodoxy requires goes beyond necessary truth-preservation; orthodoxy requires that such truth-preservation be in virtue of logical form. (We have already granted that any admissible instance of gtt is ‘formal’ in some sense or other; the point is that orthodoxy requires a special sort, a sort that requires validity in virtue of form, given a particular understanding of what constitutes the form of a sentence.) An account of classical consequence that requires the given ‘formality’ results from varying the casesx over which gtt quantifies. On the target approach, validity is a matter of form, and casesx interpret formal languages, in the current case the languages of first-order logic, in which we have simple predicates, names, variables, quantifiers, and propositional connectives. Sentences in such a formal language are interpreted in a model, specifically, Tarskian models of first-order logic. A Tarskian model, M, is a structure that comprises a non-empty set D, the domain, and an interpretation, which is a function I that satisfies the conditions » I(E) is an element of D when E is a name (in the given language). » I(E) is a set of n-tuples of D-elements when E is an n-place predicate. The extreme cases of 0- and 1-tuples are worth mentioning. There is one 0tuple: , so there are two sets of 0-tuples, the empty set {} and the singleton set {}. These do service for the truth values ‘false’ and ‘true’. For every element a of the domain D there is a single 1-tuple {a}. It does no harm to think of 1-tuples as domain elements, and 0-tuples as truth values. In turn, we use a model to interpret sentences of the language. (We use assignments of values to variables, in order to interpret sentences with free variables. If α is an assignment of values to variables, α(x) is the value of the variable x. Furthermore, an x-variant of α is an assignment which agrees with α in the values of all variables except possibly x.) » If α is an assignment of D-elements to variables, then Iα (x) = α(x). If a is a name, Iα (a) = I(a). » M, α  Ft1 · · · tn if and only if Iα (t1 ), . . . Iα (tn ) ∈ I(F). » M, α  A ∧ B if and only if M, α  A and M, α  B. » M, α  A ∨ B if and only if M, α  A or M, α  B. » M, α  ∼A if and only if M, α  A. » M, α  ∀xA if and only if M, α
 A for each x-variant α
of α. » M, α  ∃xA if and only if M, α
 A for some x-variant α
of α.

Specification of Cases

39

We now take Tarskian models to be casesx (casesm ). Via the standard recursive clauses, we have defined truth in a model for sentences of a formal language. By way of gtt the resulting account tells us about validity for arguments in the formal language: » An argument is valid if and only if, in every model in which the premises are true, so is the conclusion. For arguments of English (or any other natural language), validity is inherited by way of formalisation. We define truth-in-a-model for sentences of English by the standard processes of regimentation of those sentences, thereby achieving an account of formal validity for natural language arguments. Call that account the Tarskian account of validity of arguments in natural language (‘tm’, for Tarskian models). Technical remarks: What we have presented is a straightforward account of Tarski’s model theory for classical predicate logic, and a simple account of truth conditions in a possible worlds semantics. We have claimed that such accounts deliver classical logic. Is this indeed the case? It is commonly thought that this is the case, but in present company we may have reason to question this thought. Upon an inspection of the usual soundness and completeness proofs, we shall see that the full power of classical logic is required to complete the proof. To show, for example, that in every model A ∨ ∼A is satisfied, we need to show, for each model M, that M  A or M  A. But this is an instance of the law of the excluded middle! An intuitionist (for example) who rejects the law of the excluded middle will not endorse this reasoning. What can we say about this? We do not dodge this issue. We do not think that there is some constructively acceptable semantics for the whole of classical predicate logic. (Of course, there are constructively acceptable semantics for fragments of a full language: a fragment in which all predicates are decidable is an obvious example.) Our aim is not to provide novel, direct justifications or defences for logical systems, at least in the traditional sense. Any new justification we will provide will, of necessity, be roundabout. We will make the way open for the fan of constructive logic to endorse classical logic (and hence the classical semantics with its—classically proved—soundness and completeness proof ) without abandoning her endorsement of a non-classical logic like intuitionistic logic. We aim to undercut the argument from the virtues of logic X to the desirability of rejecting logic Y (where X = Y ). A defence of pluralism undercuts this kind of justification. This failure of this argument does not, of course, constitute an argument for the proponents of X to also accept Y . For that, however, we can appeal to whatever independent justification one might have for endorsing Y itself. The appeal of the X does not in itself constitute a reason for rejecting Y , for the pluralist option is to have one’s cake and eat it too. There are two more things we can say in response to the non-classical logician who questions our reasoning. First, we are not terribly troubled

40

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by the attacks on classical reasoning here. After all, the default position is surely that the semantics works, as the kind of classical reasoning we have employed is the default position in mathematics and metamathematics. The onus is on the non-classical (monist) logician to give a convincing case of where we have made a mistake. Second, even if we do not like the ‘model theory first’ account of classical logic, the proof theory mentioned in §4.4.2 can give an independent grounding of classical reasoning, acceptable to an anti-realist. End of technical remarks.

4.2

Admissibility

We now have two instances of gtt. The question is whether the given instances are admissible, whether they satisfy the settled features associated with the role of consequence: necessity, normativity, and formality. By way of answering that question, we address each core feature in turn. 4.2.1

Necessity

ntp validity is designed at its very heart to satisfy the necessity constraint. Recall that the necessity constraint holds that, for an argument to be valid, it must be necessary that if the premises are all true so is the conclusion (or, in the case of a multiple conclusion argument, if the premises are all true then so are some of the conclusions). An argument is invalid if (and here we mean ‘if ’ and not ‘if and only if ’), it is possible for the premises to be true and the conclusion to be false. The ntp account of validity takes this ‘if ’ to be an ‘if and only if ’. All that is required is that the necessity constraint be satisfied. What of tm validity? This also satisfies the necessity constraint if, for any argument, if it is possible that the premises be true and the conclusion be false, then there is some model in which the premises are true and the conclusion is false.2 Are such models always to be found? On a reasonably orthodox view of tm-valid arguments, the answer is ‘yes’. If there are enough abstract objects out of which to construct models, then, whenever there is a possibility invalidating an argument, anything done by that possibility can be done by some actual model. Abstract objects provide enough raw materials to show that tm validity is necessary. If, on the other hand, there could be more objects than there actually are (including abstract objects, if there are any), then it is possible that some tm-valid arguments are valid only contingently. That is, they are not necessarily valid. Etchemendy [47] discusses this issue at length. Instead of pursuing it any further here, we will note that the model theory of classical predicate logic is mathematics if anything is, and that the presumption of necessity seems to be well founded. After all, if there is a possible world invalidating an argument, then there is some actual (but abstract) model also invalidating that argument. Of course, there are issues 2 You may make the usual modifications here for multiple-conclusion arguments, of course.

Admissibility

41

to be broached here with the size of the model required. Perhaps, for certain applications, we need to admit models with a proper class as a domain. (Think of an ‘intended model’ of all of set theory.) We prefer to avoid exploring this issue just now, as it is an issue for everyone, pluralist or not. 4.2.2

Formality

Are the ntp and tm accounts of consequence formal in any of the relevant senses (see §2)? There are four relevant senses of ‘formal’. We treat each sense in turn. Schematic: The ntp account is not schematic. The tm account (classical predicate logic) is schematic, almost by default; indeed, it serves as a paradigm of schematic formality. 1-Formality: Does the ntp account provide laws applicable to content as such? Given that any subject matter may be evaluated at possible worlds, the answer is ‘yes’. For any statement at all, it makes sense to ask whether it is true or not in any other possible world. Wrinkles arise with non-denoting names. In particular, where a designates something in some worlds but not in others, choices need to be made about how to interpret Fa and ∼Fa in worlds where a fails to denote. Different choices are available, but we need not decide that issue now, save to say that, whatever choice we take, it makes sense to evaluate each statement in each possible world. The analysis does not restrict itself to only one form of statement; it applies to any kind of truth-apt content at all. For anything which we can consider, it makes sense to consider it were things to have been different. A similar but slightly more qualified answer holds for the tm account. Like the ntp account, non-denoting terms call for decisions, but the decisions are similar to those in the ntp account. The difference between the two accounts is that tm is tied to particular ‘forms’ of statement, whereas (as above) the ntp account does not restrict itself in this way at all. So whether tm is applicable to content as such depends on whether the analysis of forms of statement proffered in the language of predicate logic is applicable to all kinds of propositional content. We have already acknowledged that all propositional content may be conjoined, disjoined, negated, etc., without introducing new subject matter or restricting our attention to particular kinds of content. So the presence of the propositional connectives in the language does not set limits on the applicability of predicate logical consequence. It is not stretching the matter too much to acknowledge this point for the quantifiers as well. It seems uncontroversial that, if we have a content appropriately formalised as φ(a) (where a is a name), then there are also some contents appropriately formalised as ∀xφ(x) and ∃xφ(x). It would be a controversial position indeed according to which some statements simply cannot be embedded inside first-order quantifiers. This is nearly enough to assure the 1-formality of tm validity, but we are not quite done. We have maintained that all content seems susceptible

42

Classical Logic

to combination by the machinery of the connectives and quantifiers of first-order logic. However, it is consistent with this that there is some kind of propositional content completely foreign to the language of firstorder logic and unable to be formalised by it. Could it be that there is a thought (a sentence, a proposition, whatever) which cannot be represented as a sentence in the language of predicate logic at all? The answer here is, thankfully, negative. At the very least, any thought (sentence, proposition, whatever) can be translated as an atomic sentence in the language of predicate logic. Atomic sentences have the form Fa1 · · · an where a1 · · · an are names, and F is an n-place predicate. Of course, not all statements are predications of some property to some object(s). But this is no restriction to translating every statement as an atomic sentence, because we are free to use the extreme case of an atomic sentence, where n = 0. In this case, the form F is an atomic sentence, where F is a zero-place predicate. What role does this play in the model theory? A zero-place predicate is interpreted in a model by a set of 0-tuples, that is, either by {} (in which case it is false in that model) or {} (in which case it is true in that model). In other words, these atomic sentences are interpreted as true or false in a model. This is the only assumption that is made for zero-place predicates, and it is one that may be applied to all truth-apt claims whatsoever. We do no injustice to a truth-apt content to say (relative to a model) that it is true, or to say that it is not. As a result, the analysis afforded by the tm account of validity may be appropriately said to be 1-formal. 2-Formality: Is the ntp account indifferent to the particular identities of objects? The answer seems to depend on the connection between existence and necessity. For a logic to be 2-formal, we require that, if an argument, featuring a name a, is valid, the argument will also be valid if that name is uniformly substituted by another name, b. If we take there to be necessary existents, then it seems that this constraint is violated. The zero-premise argument to the conclusion ‘a exists’ will be valid for some choices for a, and not for others. If, on the other hand, we take there to be no necessary existents (perhaps by taking the relevant sense of necessity to be particularly strong), then perhaps the constraint for 2-formality might be satisfied. Whatever you might conclude here, at the very least, on some accounts ntp validity is sensitive to the particular identities of objects, and in that respect it may well not be 2-formal. The tm account, unlike the ntp account, is clearly indifferent to the particular identities of objects.3 (This might be seen as one of the motivations for a formal model-theoretic account of validity.) There are no constraints in Tarskian models that are peculiar to particular objects. It follows that tm, then, is 2-formal. 3 That is not to say that particular classical theories cannot impose constraints on some but not all objects.

Admissibility

43

If we wish 2-formality to act as a constraint on predicates as well as on names, then our answer must be more nuanced. Whether tm validity counts as 2-formal for predicates depends on the status of special predicates in the language. If, for example, identity is counted as a primitive two-place predicate, then the status of 2-formality seems to be lost. For example, treating identity as a constant, interpreted by the set of all pairs a, a where a ∈ D, then (∀x)x = x is tm-valid, as it is true in every model. However, the statement (∀x)xRx is tm-invalid. So, some predicates are treated differently from others, and 2-formality (if it is to be taken to apply to predicates) is violated. 3-Formality: Does the ntp account abstract from the semantic content of thought? No, the ntp account relies on the semantic content of ‘thoughts’ to determine their truth value at each possible world. The content of a sentence is what determines its truth value in a world, and there is no requirement in ntp validity to abstract away from content in the process of assigning truth values. This is in stark contrast to the tm account of validity. Classical predicate logic serves as a paradigm of 3-formality, because the content of propositions is drained away in the formalisation into the language of predicate logic. Provided that the structural components of sentences in the language of predicate logic (connectives, quantifiers) are properly form and not content, then the tm validity counts as 3-formal. 4.2.3

Normativity

We have indicated that, while not both are formal in every available sense, the ntp and tm accounts of consequence are formal in one of the settled senses or other. Both accounts, plausibly, also satisfy the necessity constraint. The remaining question is whether such judgements are normative. The question is this: What, if any, mistakes are made by an ntp- or tm-invalid argument? With respect to ntp the answer is plain: an invalid argument is one that affords the possibility of stepping from a true premise to a false conclusion; one can step from truth to untruth. That, even by the lights of logical orthodoxy, is a significant mistake. ntp validity is normative. What of tm invalidity? The answer is similar but slightly more stringent than the ntp answer: an invalid argument is one the form of which affords the possibility of going from truth to untruth. (Here, of course, possibility is taken to be modelled by Tarskian models, thus bringing the ntp and tm answers in line.) tm validity, too, is normative. tm invalidity is a failing of an argument, because the structure of the argument (that is, the structure elucidated by the language of predicate logic) allows the transition from truth to falsity, even if that transition is not actual, or not even possible.

44

4.3

Classical Logic

Emergence of Plurality

We now have our first dimension of plurality. We have two admissible instances of gtt. That is sufficient for pluralism. (This is not a very deep pluralism, we agree, but it is a start on pluralism.) The argument from ‘b is red’ to ‘b is coloured’ has the form Rb  Cb. Consider the question: Is that argument valid? The answer is yes for ntp validity and no for the Tarskian account of validity. There is no possible world in which b is red but not coloured; however, there are (many) many models in which Rb is true but Cb not true. The two accounts give different answers; but each account specifies consequence (in English) none the less. These two accounts of consequence are different but, with respect to the chief question of Logic (what arguments are valid?), they are not rivals. There is no sense in calling the two accounts rivals with respect to whether such and so argument is valid. Qua answers to Logic’s chief question, the two accounts do not compete. Is there, then, no rivalry among logics? Rivalry enters at the level of application. Such rivalry will become clearer in the next two chapters, where further dimensions of plurality are discussed. For now, we briefly mention a few applications of tm and ntp accounts of consequence.

4.4

Applications

An important feature of the relation of classical logical consequence is that it affords various useful applications beyond the explication of the Generalised Tarski Thesis. 4.4.1 The Analysis of (Classical) Mathematics Any passing exposure to the historical development of the formal system of classical predicate logic is sufficient to make clear that the formalisation of classical mathematical proof was a central target of Peano, Frege, and Russell. It is no coincidence that classical predicate logic is well suited to drawing out the mathematician’s sense of ‘valid argument’, for this was one of the design criteria in the development of classical logic. The case for this claim is made clearly and convincingly by Burgess [29]. If classical logic does not have something to say about the informal account of validity of arguments in classical mathematics, then it is difficult, if not impossible, to explain why formal independence results have the explanatory power that they in fact have. Cohen’s proof that the continuum hypothesis is independent of the axioms of zfc (Zermelo–Fraenkel set theory with the axiom of choice) is informative because we gain insight from the fact that there is a classical model (in Tarski’s sense) of the axioms of zfc that refutes the continuum hypothesis. By our lights, the application to the classical mathematician’s judgements about good and bad arguments is an important feature of classical logic, but it does not uniquely single out the classical consequence relation as being uniquely deserving of the title ‘Logic’. We have already explained

Applications

45

what it takes for a relation to be Logic: it is an admissible precisification of the settled core of the notion of logical consequence. The application of classical predicate logic to the analysis of the validity (in the mathematician’s sense) of classical mathematical reasoning is a further point. This application helps explain why classical validity is a useful notion. The applicability of classical consequence to classical mathematics can be simply explained. The validity of mathematical argument is not helpfully explained merely in terms of necessary truth-preservation, according to which any argument to a true mathematical conclusion is valid (since mathematical truths are necessary truths); rather, the validity of mathematical argument is helpfully explained in terms of universality. The validity of mathematical arguments seems best explained by their holding in all mathematical structures. Mathematical structures seem like good candidates to define Tarskian models. A mathematical structure, appropriately defined, will divide all (mathematical) claims about it into the true and the false. No vagueness or indeterminacy seems to arise. So Tarskian validity appears to model the kind of universality appropriate to the classical mathematician’s sense of ‘validity’. 4.4.2

Semantics

Another possible application of the classical notion of logical consequence is more general than the application to classical mathematics. It can be argued that classical logical validity is useful in regulating all of what we think or say, because of its role in a theory of semantics. It would take us too far away from our core purposes to spend much time on this issue, but it will be helpful to explain how the relation of classical logical consequence can play a role in different kinds of semantic theory: both those that are traditionally described as realist or truth-conditional, and those described as anti-realist, and which eschew talk of truth for talk of assertibility or some other such notion. Classical logic sits well with semantic realism. However, it does not require a realist conception of the project of semantics. We could, if we liked, pursue a semantic project in terms of conditions under which assertions and denials are appropriate, rather than conditions under which sentences are true or false, and still find that classical logic played a role in our semantic theory. This position is markedly less orthodox than the appropriation of classical logic to realist semantics; for the traditional approach for antirealists has been to endorse some other formal logical system, such as intuitionistic logic, as a rival account of logical consequence to the classical account. Our main goal in this book is to develop and defend a conception of logical consequence according to which classical, intuitionistic, and other accounts of logical consequence need not be seen as rivals. It is not the stronger (and, we think, totally indefensible) position that there is no rivalry among those who endorse one or other system of logic. The debate between realists and anti-realists in meaning theory seems to us to be

46

Classical Logic

important and interesting, but it is a different issue from our target, of pluralism about logical consequence. In this section we sketch how the debate between the realist and anti-realist in semantics is orthogonal to considerations salient to pluralism about logical consequence. We will sketch how both realist and anti-realist pictures of the semantic project can find an application for classical logic. For the semantic realist, it is quite straightforward to see how classical logical consequence can play an important role. For the semantic realist, the ‘content’ of an expression is to be explained in terms of its truth conditions (for declarative sentences) or in terms of its role in determining truth conditions (for other parts of speech). The possible worlds of the ntp account of validity, and the models of the tm account are different ways of explicating conditions under which a sentence is true. For the ntp account, the truth of a sentence can vary from one possible world to another. For the tm account, the truth of a sentence is explained in terms of the extensions of the predicates involved in that sentence, the referents of the names, and the objects in the domain. On either picture of the conditions under which sentences are true, the connectives of the language of predicate logic have distinctive features (on the standard view, at least). In the case of the connectives, the truth or otherwise of a complex sentence depends only on the truth or otherwise of its constituent sentences. For quantifiers the story is a little more complicated, but not very. On the tm picture, the truth (or satisfaction) of a quantified sentence depends on the satisfaction of its constituent open sentence. The traditional semantic clauses of Tarski play a theoretically important role as an explication of the truth conditions of sentences. This distinctive role for classical logic does not go unchallenged, of course, and we will examine some criticisms of the classical approach later. It suffices for our purpose to note that, for the traditional realist truth-conditional semantic picture, the underlying machinery of the tm account of classical logical consequence plays an important theoretical role. On the other hand, we might prefer an account of meaning in terms of norms governing assertion and denial, instead of the conditions under which claims are true. There are many different ways one might do this, and it is not our task to evaluate them all here. Suffice to say, there are some ways of accomplishing this task, according to which classical logic plays a central role.4 We will consider one here, in terms of conditions under which it is coherent to assert and deny sentences. We will call a pair [X : Y] of sets of sentences a package, and we will evaluate packages for coherence. (We appropriate the usual proof-theoretic shorthands: [A : Y] is the package with only one element A in the left set; [X : B] is the package with only one element B in the right set; [X, A : B, Y] is the package [X∪{A} : Y ∪ {B}].) The guiding idea is this: in the package [X : Y] we will think of 4 The

details of this account are to be found in gr’s ‘Multiple Conclusions’ [111].

Applications

47

the upshot of accepting each member of X and rejecting each member of Y . A package [X : Y] is coherent if accepting X and rejecting Y has no internal defects of an appropriately ‘logical’ kind. First, it seems that on this picture coherence has the following structural properties: » [A : A] is not coherent. » If [X : Y] is coherent, then one of [X, A : Y] and [X : A, Y] is coherent. » If [X : Y] is coherent, then so is [X
: Y
] where X
⊆ X and Y
⊆ Y . Just as for the realist, the truth of complex sentences formed using the connectives and quantifiers is transparent, so for the anti-realist the coherence assertions or denials using the connectives and quantifiers is parasitic on the coherence of assertions and denials using their constituents. » » » » » »

[X, ∼A : Y] is coherent iff [X : A, Y] is coherent. [X : ∼A, Y] is coherent iff [X, A : Y] is coherent. [X, A ∧ B : Y] is coherent iff [X, A : Y] and [X, B : Y] are coherent. [X : A ∧ B, Y] is coherent iff either [X : A, Y] or [X : B, Y] is coherent. [X, ∀xA(x) : Y] is coherent iff [X, A(c) : Y] is coherent for an arbitrary c. [X : ∀xA(x), Y] is coherent iff [X : A(c), Y] is coherent for some c.

These clauses could be taken as definitional of the import of asserting and denying complex claims involving the vocabulary of predicate logic.5 Note that this kind of semantic picture says nothing about truth, as such. None the less, it does provide an account of classical logic as an account of what it is coherent to assert and deny. Since [A : A] is incoherent, so is [∅ : A, ∼A], so it is incoherent to deny both A and ∼A, and hence [∼∼A : A] is incoherent. It is incoherent to assert ∼∼A but to deny A. This is but one example of a ‘deduction’ of a classically valid inference. This kind of approach is a reinterpretation of the classical multiple-premise, multiple-conclusion sequent calculus of Gentzen [53]. The X  Y is provable if and only if the package [X : Y] incoherent. On this account, a classically valid argument is one for which it is never coherent to assert the premises and deny the conclusion. It seems that we have a reading of the consequence relation of classical predicate logic which is understandable to the anti-realist, as well as to the realist. In this book we will continue to use the realist talk of worlds, models and cases, but the foregoing discussion should make clear that anti-realist readings are available too. Even if talk of truth in a model is not taken to be illuminating or explanatory, this talk can be construed as elliptical talk of maximal packages of assertions and denials. Nothing in what we say requires the realist reading of the core notions of ‘case’ or ‘model’. 5 Of course, these ‘definitions’ could also be resisted. We give no argument for these clauses other than to state them. It is not our business to defend the anti-realist reading of classical logic. It is simply to make it available.

48

4.5

Classical Logic

Summary

Our logical pluralism recognises at least two admissible instances of gtt. In this chapter we have found pluralism: possible worlds and Tarskian models both yield admissible precisifications of ‘follows from’, the ntp and tm accounts. ntp and tm compete at the level of application; the former, as above, performs rather badly as an explication of ‘follows from’ in (classical) mathematics. That said, the two accounts do not compete as accounts of consequence. Given a particular argument, the two accounts do not give rival answers on whether the argument is valid: they give different answers, and that is all. One might now wonder: Is there any basis upon which to choose between the two accounts? Is there any reason you might prefer one to the other? Tarskian validity is formal in ways that ntp-validity is not. Tarskian validity can be known a priori, but necessary truth-preservation (probably) cannot. If Kripke is correct [64], the argument from ‘a is water’ to ‘a is H2 O’ is necessarily truth-preserving, but that cannot be known a priori. If one wants one’s logical consequence relation to be a priori knowable (and this seems like a reasonable constraint to many), then epistemic considerations may well yield a platform for preference. There may be other platforms as well. At this stage, readers may well have many questions and objections; however, we will leave those until Part III. For now, we turn to yet another consequence relation that falls out of gtt, namely, relevant consequence.

Chapter 5

Relevant Logic I believe that there is a sense of ‘entails’ (or ‘implies’) in which it simply is not true that a contradiction entails (or implies) any old sentence whatsoever. — J. M. Dunn ‘Natural Language Versus Formal Language,’ [39, page 9]

gtt yields classical consequence when its casesx are taken to be possible worlds, where possible worlds are complete and consistent with respect to negation. That (classical) precisification of ‘follows from’ is familiar and useful, however, it is not the only sense of ‘follows from’ apparent in English. Another strongly apparent sense of ‘follows from’ takes ‘from’ seriously. In that sense of ‘follows from’ arbitrary B does not follow from arbitrary A ∧ ∼A; the former is not implicit in the latter. Likewise, arbitrary A ∨ ∼A does not follow from arbitrary B; the former is not implicit in the latter. In short, there is a sense of ‘follows from’ that is more restrictive than the classical sense; it demands that premises be ‘relevant’ to conclusions. Relevant consequence is our concern in this chapter.1 As above, that sense of ‘follows from’ is more restrictive than the classical one: it imposes constraints that go beyond truth-preservation over possible worlds. The constraints, in effect, concern the behaviour of negation. What is required is not only truth-preservation over possible worlds, but truth-preservation over casesx that go beyond the constraints of worlds, beyond the constraints of completeness and consistency. The task is to specify such casesx , thereby cashing out relevant consequence.

5.1

Specification of Cases

Relevant consequence arises from gtt by dropping the completeness and consistency of classical cases. We will pursue that option by letting casesx be situations along the lines of Barwise and Perry [6, 7, 8, 34]. 5.1.1 Informal Characterisation For situation theorists, the world is made up of situations. Situations are simply parts of the world. Claims are true not only of the world at large; 1 Calling it relevant consequence, rather than relevance consequence, follows the Australasian tradition rather than the American tradition. Consult the history for an explanation [45, 110].

49

50

Relevant Logic

some claims are true of situations, proper parts of the world. We need not (and will not) spend time on the theory of situations and their individuations, we will rely on a few examples. Example 1: In the situation s1 involving Greg’s household as he grades exams, it is true in s1 that Christine is reading a paper and Zack is sleeping; and it is also true in s1 that the stereo is playing; however, it is false in s1 that the television is on; and, given that the television is in fact an inhabitant of the (given) situation, it follows that it is true in s1 that the TV is off. Is it true in s1 that JC is feeding the cats? No. Is it false in s1 that JC is feeding the cats? No. JC doesn’t figure in that situation at all. Example 2: In the situation s2 in which JC is feeding the cats, it is true that Max and Agnes are purring; and it is true in s2 that Bob Dylan’s music is playing; and it is true in s2 that Katrina is sleeping. Accordingly, it is false in s2 that Katrina is awake, and false in s2 that Max or Agnes is hissing. Is it true in s that the TV is on? No. Is it true in s2 that Greg is writing? No. Is it false in s2 that Greg is writing? No, and it is not false in s that the stereo is on. Given that neither Greg nor the stereo inhabits s2 , such claims are simply indeterminate in s2 . The idea, then, is that situations, unlike possible worlds, may be incomplete. Situations ‘make’ some claims true, and they ‘make’ others false. But, by virtue of being restricted parts of the world, situations may fail to make a claim true or make it false. (We do not use shudder quotes around ‘make’ because we wish to avoid the use of ‘truthmaking’ terminology; to the contrary, we value the recent revival of this terminology and the analysis of the connections between claims and parts of the world that make them true [11, 51, 84, 102]. We know, however, that that terminology is not used by situation theorists, and that it would be a mistake to impute it to them.) To be sure, possible worlds, as we use the term, are situations; they are ‘special’ situations, the ones that are complete and consistent. Situations, however, are not possible worlds; the former may be incomplete. A natural generalisation of the incompleteness idea is the dual idea of inconsistency, inconsistent situations or ways things could not be [78, 104, 106, 118, 138]. According to our account of worlds as consistent, complete situations, such impossibilia cannot be a part of any world. Worlds are consistent, and hence have no inconsistent parts. That does not mean, of course, that there are no ways that things could not be; it simply means that the worlds are not (and could not be) among them. Examples of inconsistent situations include (among others) fictions, especially ones that are intentionally written to involve inconsistent situations, such as ‘Sylvan’s Box’ [93]. The important point is that such inconsistent situations need not be trivial: A and ∼A may both be true in such situations while some B is not true in the situation. What should be clear is that, given the foregoing sketch of situations, negation behaves differently at restricted situations than it does at possible

Specification of Cases

51

worlds. The negation of A need not be true in a situation if A is not true; and arbitrary B need not be true in a situation if A and its negation are true in that situation. 5.1.2 Further Details: Conjunction and Disjunction While the behaviour of negation marks a salient difference between possible worlds and (restricted) situations, conjunction and disjunction seem to behave precisely as per the classical clauses: » A ∧ B is true in s if and only if A is true in s and B is true in s. » A ∨ B is true in s if and only if A is true in s or B is true in s. The conjunction clause is almost never disputed. The disjunction clause is sometimes disputed, though we think that its rejection is, by and large, ill-motivated in this context. If in this situation the milk is on the table or in the fridge, then either in this situation the milk is on the table or in this situation it is in the fridge. By our lights, the behaviour of disjunction at situations need not deviate from its behaviour at possible worlds. Accordingly, we will retain the ‘classical clauses’ for conjunction and disjunction. Negation is a different matter. 5.1.3 Further Details: Negation Negation, as above, behaves differently at situations than it does at possible worlds. In particular, the classical negation clause (more generally and accurately, the classical negation clause for casex , since the content of the clause essentially depends on the value of casex ), namely » ∼A is true in a casex if and only if A is not true in that casex does not fully characterise the behaviour of negation; it characterises negation only over complete and consistent values of casex . Since situations are not always complete and consistent, a different clause is required to cover the behaviour of negation at situations. How is the behaviour of negation at situations to be characterised? We will turn to that in §5.1.4. What must be emphasised is that, in giving the target account of negation, we are not thereby rejecting classical negation or interpreting a different kind of negation. Quite the contrary. Our treatment of negation’s behaviour at situations is not the traditional account of negation because the traditional account ‘ignores’ situations, and hence it ignores the behaviour of negation at situations. Formally modelling claims of the form ‘A is true in situation x’ has not been a traditional task, but once that task is taken seriously, one soon sees that the classical negation clause is inadequate. That is not to say that the traditional, classical equivalence according to which ∼A is true if and only if A is not true is in question; it is not in question. What situations require is a deeper, further account of negation: ∼A is true if and only if ∼A is true in some (actual) situation or other; and A is not true if and only if A is not true in any (actual) situation whatsoever.

52

Relevant Logic

The traditional, classical equivalence is ensured provided that A is not true in any (actual) situation if ∼A is true in some (actual) situation; and that is simple to provide with three minimal theses, each of which is plausible: » There is a situation w of which every actual situation is a part. » If A is true in s and s is a part of s
, then A is true in s
. » If s is an actual situation and ∼A is true in s, then A is not true in s. The above theses connect negation and situations in such a way that the noted traditional, classical equivalence is ensured. In that respect, negation remains classical. The question still remains: How, then, does negation behave at situations? 5.1.4 Compatibility: Negation at Situations We have not yet given a systematic treatment of the truth or falsity of negations in situations. That can be done in various ways. One way of doing as much takes satisfaction and dissatisfaction of relations (in situations) as primitive, and then inductively builds up truth and falsity conditions of complex sentences. For example, let a conjunction be true in s when both conjuncts are true, but also (dually) let a conjunction be false when one conjunct is false. That approach is standard in situation theory, and it is also used in varieties of semantics for non-classical logics [7, 21, 40, 86, 101]. We favour a different approach: Dunn’s compatibility analysis of the semantics of negation [43, 44, 106]. On this compatibility approach, negation behaves in situations the way necessity (or possibility) behaves in possible worlds. Accordingly, our semantics admits non-actual situations (or models of non-actual situations) that are connected by a binary relation of compatibility, which we write ‘C’. A virtue of the apparatus is that we need not treat truth and falsity in parallel: negation is definable. » ∼A is true in s if and only if A is not true in s
for any s
where sCs
. The negation ∼A is true in s just when any situations in which A is true are incompatible with s. That clause follows fairly immediately from the intended meaning of ‘compatibility’ and features of negation that we have no reason to question. If ∼A is true in s and A is true in s
, then s is not compatible with s
. Conversely, if A is not true in any s
compatible with s, then s has ruled out A; that is, ∼A is true in s. Such a reading is entirely compatible with a classical view of negation; it does not rely on a ‘funny’ negation. Note, too, that the three minimal conditions cited above for a classical treatment of negation have their ‘compatibility’ readings: » Any actual s is part of a world w (this is as before). » w is a world if and only if wCw; and if wCs then s is part of w (in other words, worlds are maximal, self-compatible situations).

Relevant Consequence

53

» If sCt, s
is part of s and t
is part of t, then s
Ct
too (compatibility of wholes leads to compatibility of parts). These conditions on C are plausible given the account of compatibility provided above.

5.2

Relevant Consequence

The foregoing account of situations yields a natural reading of gtt, a situated reading, which precisifies a relevant sense of ‘follows from’. » The argument from P to A is relevantly valid if and only if, for every situation in which each premise in P is true, so is A. To speak loosely but suggestively, by making the premises true you make the conclusion true too. The relevance of that reading of consequence— that situational instance of gtt—is immediate. For example, the inference from A to B ∨ ∼B fails, since a situation in which A is true need not be one in which B ∨ ∼B is true; likewise, the inference from A ∧ ∼A to B fails, since a situation in which the former is true need not be one in which the latter is true. Relevant consequence is not widely understood, so it is worth our while to allay some misconceptions before we proceed. As we have seen, the inference from A to B ∨ ∼B is relevantly invalid. There are situations in which B ∨ ∼B fails. So, B ∨ ∼B is not true in every situation. However, relevant logicians typically endorse the law of the excluded middle. For example, the law of the excluded middle is a tautology in the principal relevant logics R and E of Anderson and Belnap [1, 2]. The way we view this apparent conflict will shape how we understand relevant logic. Our view retains both arms of the dilemma. The inference from A to B ∨ ∼B invalid; B∨∼B is not true in every situation, and B∨∼B necessarily true. The apparent dilemma is solved by denying that necessary truths must be true in every situation. Situations may be incomplete, and instances of the law of the excluded middle may fail. To slip into the language of truthmaking, what makes B ∨ ∼B true will be a situation that settles the matter of B. It is necessary that there is some such situation, but not every situation does the job. The necessary truths are not those which are true in every situation, rather, they are those for which it is necessary that there is a situation making them true.

5.3

Admissibility

That situations afford a natural instance of gtt is not surprising. The question is whether the given instance is admissible, whether it satisfies each of the desiderata of the core notion: necessity, formality, and normativity. By way of answering that question, we address each core feature in turn.

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5.3.1 Necessity Are the relevant judgements about validity in any sense necessary? The answer is ‘yes’ if the classical judgements about validity are necessary and any relevantly valid argument is classically valid. We have already agreed with standard opinion that the classical judgements about validity are necessary. The question, then, is whether any relevantly valid argument is classically valid. The question, in other words, is whether truth-preservation over all situations guarantees truth-preservation over all possible worlds. The answer is ‘yes’ if possible worlds count as (perhaps special) situations. They do; they are the complete and consistent situations. Hence relevant judgements about validity are necessary in just the sense in which classical judgements are necessary. Relevant consequence, which takes casesx to be situations, satisfies at least one of the core (settled) constraints on admissible instances of gtt. That is the upshot of the necessity of relevant judgements of validity. What of other constraints? 5.3.2 Formal Is relevant consequence formal in any of the senses discussed in §2? There are four such notions of formality; we treat each in turn. Schematic: Though we have focused only on the propositional level, expanding to the predicate level reveals that relevant consequence is indeed schematic in just the usual way in which classical consequence is schematic: relevant logic does not invoke any linguistic features beyond those invoked by classical predicate logic. 1-Formality: Is relevant consequence applicable to content as such? Inasmuch as any subject matter (even if inconsistent) may be evaluated at situations, relevant consequence is indeed applicable to content. Relevant consequence is just as 1-formal as classical logic is. There is nothing in the notion of a situation which means that some kinds of content are somehow disbarred from evaluation at situations. It may well be difficult to determine when a claim is true in a situation (for example: when is an arithmetic truth true in a particular situation? Always, or merely sometime?), but this does not mean that it is impossible. 2-Formality: Is relevant consequence indifferent to the particular identities of objects? Again, the answer is obviously affirmative. At the predicate level there are no constraints in relevant models that are peculiar to some but not all objects.2 Relevant consequence is 2-formal. 3-Formality: Does relevant consequence abstract from the semantic content of thought? Yes. Relevant consequence (as we will discuss further in §5.5), in virtue of recognising incomplete or inconsistent situations, 2 Of course, that is not to say that particular (relevant) theories cannot impose special constraints on some but not all objects.

Pluralism

55

affords more distinctions than does classical consequence; however, such flexibility is not achieved by imposing constraints beyond the sort involved in classical logic, nothing beyond fixing the interpretations of names on particular objects. In that respect, relevant consequence abstracts away from all semantic matters, leaving nothing more than the extension (or anti-extension) of a predicate at a situation. Relevant consequence is 3formal. 5.3.3 Normative We have indicated that relevant consequence is both necessary and formal. The remaining question concerns normativity. The question is: What, if any, sort of mistake is made by a relevantly invalid argument? Relevant consequence gives a plain answer: the conclusion of a relevantly invalid argument is not ‘carried in’ the premises; the conclusion of a relevantly invalid argument does not follow from the premises. Consider, again, the argument from A to B ∨ ∼B, which is relevantly invalid. The conclusion of that argument is true — indeed, necessarily true. The mistake is that the truth of the conclusion does not come from the truth of the premise. That is what the situational analysis reveals; there are situations—though no worlds—in which A is true but B ∨ ∼B is not true. The mistake, as it is often put, is that the truth of B ∨ ∼B is ‘irrelevant’ to that of A, and vice versa. There are mistakes made by relevantly invalid arguments, and so relevant consequence is indeed normative, and normative in just the way in which classical consequence is normative. To be sure, the mistakes involved in classically invalid and relevantly invalid arguments are different; however, they are both mistakes that are deemed as such by the corresponding senses of ‘follows from’.

5.4

Pluralism

The upshot of §5.3 is that the situated instance of gtt—the instance that arises from letting casesx be situations—is admissible. As we mentioned in §3, there may be admissible instances of gtt that have few (if any) interesting applications. One might wonder whether relevant consequence affords any interesting applications. We will briefly address that question in §5.5. Before turning to that question, it is important to pause for further pluralism. As above, the situated instance of gtt is an admissible one. But a classical instance, which takes casesx to be possible worlds, is also admissible. There are at least two admissible instances of gtt, each yielding a precisification of ‘follows from’. We have pluralism. What should be (re-)emphasised is that we have not abandoned classical logic (or classical negation) in our acceptance of relevant logic. Let relevant tautologies be those claims that are true in every situation. Then B ∨ ∼B is not a relevant tautology. That does not mean that we have adopted a strange, non-classical account of negation. We agree with the

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classical theorists that B ∨ ∼B is true, that it is true in every possible world. Our negation is classical. The invalidity of the argument from A to B ∨ ∼B is a relevant invalidity. (And that is as it ought to be. B ∨ ∼B is true, after all, but it need not follow from the truth of an arbitrary A, at least when we take the ‘from’ seriously.) That the argument from A to B ∨ ∼B is relevantly invalid does not impugn its classical validity; it is still classically valid, which is to say that any world in which A is true is one in which B ∨ ∼B is true. Similarly, the argument from A ∧ ∼A to B fails the relevant test: a situation in which A ∧ ∼A is true need not be one in which B is true. A situation might well be inconsistent about A without involving everything. That same situation gives us a counterexample to the relevant validity of disjunctive syllogism, the argument from A ∨ B and ∼A to B. A situation inconsistent about A but not judging B as true is a counterexample; in such a situation A ∨ B is true, as is ∼A, but B fails to be true. While a great deal of ink has been spilled on the ‘failure’ of Disjunctive Syllogism [77, 98, 115], we need spill none or no more, anyway. In the light of pluralism, the ‘debate’ is one in which both parties can agree: Disjunctive Syllogism is valid when gtt’s casesx are taken to be possible worlds, and it is invalid when those casesx are taken to be situations. That is that. To whether Disjunctive Syllogism is valid, the classical and relevant accounts give different but not rival answers. Accordingly, there is little to be gained in ‘debating’ whether Disjunctive Syllogism (or the like) is valid. (We have restricted our attention to the conjunction, disjunction, and negation fragment of relevant logics. More can be done to bring the notion of relevant entailment into the language at hand.3 ) That the relevant and classical accounts of consequence are not rivals as accounts of consequence is not to say that they never compete. Competition arises at the level of application, a topic to which we now turn.

5.5

Applications

Like classical consequence, relevant consequence affords various useful applications. In this section we list a few such applications. 5.5.1

Inconsistent, Non-trivial Theories

A theory, for our purposes, is a set of sentences (in some language) closed under some relation of logical consequence. A trivial theory is one in which every sentence (of the given language) is true. If classical consequence underwrites a theory, then the theory is trivial if it is inconsistent. One attractive application of relevant consequence is that it affords the option of modelling interesting inconsistent, non-trivial theories.4 A familiar ex3 For another approach to relevant logics which motivates two varieties of consequence, but from a very different perspective to ours, we refer the reader to Mark Lance’s ‘Two Concepts of Entailment’ [68]. 4 This attractive feature is shared by all paraconsistent consequence relations.

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57

ample of such a theory is naïve truth theory.5 Given Liar-like paradox (or, similarly, Russell-like paradox) [12], the naïve theory of truth seems to be inconsistent; however, the theory does not seem to have it that everything is true, in which case the argument from A ∧ ∼A to B must be invalid (according to the theory). Relevant consequence judges that argument—ex falso quodlibet or, more dramatically, explosion—to be invalid. In that respect, relevant consequence finds a useful application in modelling inconsistent but non-trivial theories. (We are not suggesting that all relevant logics are sufficient for modelling any inconsistent theory; the point is simply that relevant consequence opens more options for such modelling than classical consequence.) 5.5.2 Fictional Discourse Fictions are frequently inconsistent. That much is uncontroversial. The issue concerns modelling such discourse. As with theories, so too with fictional discourse: relevant consequence finds an immediate role where classical consequence does not. After all, while A and ∼A may (for one reason or another) each be true in a story, it is usually not the case that everything is true in the story. While there are ways of forcing classical consequence into service for such modelling, a natural response is to use relevant consequence for that purpose. 5.5.3 Truthmaking Of the sentences that are true, it has seemed plausible to many that there are things which make them true. Talk of truthmakers has played an important role in metaphysics [51, 84], and here, surprisingly, relevant consequence might play an interesting role [100, 102]. The idea is straightforward. The truthmaker thesis is that an object a is a truthmaker for the sentence A if and only if the existence of a entails that A, that is, if E!a ⇒ A. The notion of entailment apposite for such an analysis is almost certainly going to be relevant in its features, rather than simply classical. For if ⇒ were a classical kind of entailment, then we would have E!a ⇒ (B ∨ ∼B) for any B, and as a result, any and every thing would be a truthmaker for each sentence of the form B ∨ ∼B, and, indeed, for every tautology. The situation worsens if we take it that truthmaking respects disjunction: that is, if a truthmaker for A ∨ B is always a truthmaker for A or a truthmaker for B.6 If this is the case, and if the salient sense of entailment ⇒ is broadly classical, then every existing truthmaker makes true every truth! Suppose B is true. Since E!a ⇒ B ∨ ∼B, then either E!a ⇒ B or E!a ⇒ ∼B. Contraposing this second claim, we get B ⇒ ∼E!a, so either a doesn’t exist, or if a does, it is a truthmaker for B. 5 Priest and Sylvan [96] argue that there are many inconsistent but non-trivial theories, including physical theories, mathematical theories, and ‘theories’ of law. 6 This disjunction thesis seems plausible [102] but it is not without its critics [100]. Nonetheless, it is demonstrably consistent with the analysis of truthmaking with relevant entailment [108].

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So, the truthmaker thesis and the disjunction thesis spell trouble for truthmaking if ⇒ is classical entailment. If ⇒ is relevant, then trouble is averted [102, 108]. Furthermore, this trouble is avoided not by the imposition of an ad hoc device, but rather by conceptual resources appropriate to the task at hand. A relevant account of entailment and consequence is appropriate for the analysis of truthmaking. A relevantly grounded entailment is one where the consequent is true in virtue of the truth of the antecedent. This is what is required in any notion of entailment appropriate for the truthmaker thesis. We mean for the existence of a to underwrite the truth whatever it makes true. If a sentence A is necessarily true (or tautologous), independently of the existence of a, then the existence of a classically entails A (everything classically entails A); but that is not enough for the object a to make A true. For that connection, we need something stronger, and relevant entailment is well suited to do that work. 5.5.4

Situation Semantics

In §4.4.2 we saw how classical logic might play an important role in semantic theory, whether realist or anti-realist. The same is the case for relevant logic. One good example of this is to be found in Barwise and Perry’s Situation Semantics [5, 7, 8]. The details of situation semantics are too intricate to discuss here in any detail, but it is sufficient for us to note that situation semantics can be construed as a variation of the traditional truth-conditional approach to meaning, which takes the salient unit of analysis to not be the possible world or any other kind of complete, consistent index of evaluation, but rather, the more modest situation. This allows them to distinguish the meaning (construed as truth conditions) of different tautologies. Classical tautologies are sentences which simply must be true in virtue of their form. However, they need to be true in every situation. For something to be true, it must be true in some actual situation or other. So classical tautologies are claims which, by virtue of their form, must be true in some situation or other. This does not mean that these claims are true in any and every situation, so their truth conditions (construed as the situations in which the claims are true) may differ. One might take situations to be the ‘right’ units of evaluation for developing a general semantics. If so, the logic of preservation of truth in all situations (possible or impossible) is broadly relevant. If one does this, of course, classical logic can still have a place, as the logic of merely necessary preservation of truth. A proponent of situation semantics may say that a classically valid argument preserves truth (it is impossible for a premise to be true and the conclusion to be untrue), but it does not preserve truth-in-a-situation. Situation semantics provides a home for the both relevant and classical logic.

5.6

Summary

Pluralism, in our sense of the term, recognises at least two admissible instances of gtt. In this chapter we have again found pluralism: like possible

Summary

59

worlds, situations serve to yield an admissible instance of gtt, which specifies a relevant sense of ‘follows from’. Given a particular argument, classical and relevant consequence are not competing answers to whether that argument is valid: they are different answers, and that is all. That said, classical and relevant consequence compete at the level of application. Relevant consequence seems better for modelling inconsistent or incomplete theories; classical consequence seems to perform downright badly in that role. There is competition but only at the level of application; the two accounts of consequence do not compete as accounts of consequence. At this stage, readers may well have many questions and objections; however, we will leave those until Part III. For now, we turn to yet another consequence relation that falls out of gtt.

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Chapter 6

Constructive Logic An incorrect theory does not become less incorrect because it cannot be refuted by contradiction, in the same way that a criminal policy is not less criminal because it cannot be stopped by a court of justice.’ — L. E. J. Brouwer, ‘On the significance of the principle of the excluded middle in mathematics, especially in function theory.’ [28]

Intuitionistic logic, also called ‘constructive’ logic, is a particular account of logical consequence, at variance with both classical and relevant logical consequence. One way to introduce intuitionistic logic is by means of constructions. We give an account of constructive validity by first indicating what it is to construct a statement, and then instantiating gtt’s cases with constructions: an argument is constructively valid if and only if a construction for the premises provides a construction for the conclusion. Before turning to constructions, it will be useful to sketch the relationship between intuitionistic logic and intuitionism. First and foremost, intuitionism is a philosophical view of the foundations of mathematics; it was introduced by Brouwer [58] in the early 20th century and formalised by Heyting [57] some decades later. For intuitionists, mathematical proofs are correct to the extent that they encode the constructions of a creating mathematical reasoner: reasoning is a function of the intuition of the creating subject. To that extent, intuitionism is a kind of constructivism. But intuitionism goes further than other varieties of constructivism: it maintains that constructive reasoning is required by the nature of mathematical entities themselves. The entities in question are constructions of the reasoner in intuition, and such entities have only the properties bestowed upon them by their construction. Michael Dummett’s work has helped champion the cause of intuitionistic logic not only in the representation of mathematical reasoning, but in semantics more generally [37, 38]. But to call Dummett an intuitionist would be wrong, since for Dummett intuition plays no crucial role in representation. Better, we think, to defer to his own label for his position: he is a semantic anti-realist [38]. Intuitionism about mathematical objects is a kind of anti-realism, of a 61

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piece with Dummett’s general semantic anti-realism. Truths about mathematical entities cannot outstrip what we can truly say about them, and what we can truly say about them cannot outstrip our capacities to describe them. Constructivity seems to be a fitting partner for such anti-realism. All of that is fairly straightforward. None the less, the marriage of anti-realism and constructive reasoning is often taken to be much more intimate than what we have so far indicated. Many have taken the two to go hand in hand everywhere, to be inseparable partners. But, as discussed in §4.4.2, we think that anti-realism and intuitionistic logic are separable in at least one important sense: there is a reading of classical logic that is entirely acceptable to the anti-realist. So, anti-realism and intuitionistic logic can be separated in at least one direction. In this chapter we will argue that the separation can also go in the reverse direction: Anyone, regardless of metaphysical commitments, can reason constructively; and, in a sense we hope to clarify, anyone ought to reason constructively. Intuitionistic logic is to be freed from its ties to intuitionism and other anti-realist philosophical views. A logical pluralist is free to add intuitionistic logic to her toolkit of accounts of logical consequence, irrespective of her commitment to realism or anti-realism.

6.1

Cases as Stages

We will start by formalising the picture of ‘constructions’ mentioned at the start. Just as worlds stand to classical logic, and systems of situations stand to relevant logic, so too systems of stages stand to intuitionistic logic. Stages can be thought of as steps in a process of construction or verification. 6.1.1 Definitions The crucial features of stages are as follows. » Stages are potentially incomplete. A stage might neither verify a claim nor its negation; we need not possess total information at each step of our enquiry. » Stages are potentially extensible. A stage s might be followed by (extended by) another stage s
: s s
. » Stages are partially ordered by the relation of inclusion. Inclusion is reflexive (s s for each stage s), transitive (if s s
and s
s

, then s s

) and anti-symmetric (if s s
and s
s, then s = s
). All of this seems plausible on the intended interpretation of inclusion. As is the case in the semantics for classical and relevant logics, the cash value of stages is the relationship between stages and propositions. Before we can talk of the behaviour of the logical constants, we must lay a little more groundwork. First, there is the relationship between truth according to a stage and the relation of inclusion: » If s s
and s  A, then s
 A too.

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63

That tells us that information provided at stages is cumulative. There are two ways we can consider such a constraint. First, we could take it to be a condition on the simple propositions evaluated at stages, and then show that it holds also for complex propositions by a procedure of induction. Second, we could take it to be a global condition, governing all propositions, and then check that any definition for the truth conditions of complex propositions is consistent with the constraint. For our purposes, nothing hangs on favouring one interpretation over another. For the quantifiers, we need a notion of when an object is available for evaluation at a stage. Just as stages are not omniscient with respect to the information they warrant, they are not omniscient with respect to the objects they ‘know about’. To use the language of construction, each stage provides the wherewithal to construct certain objects, but not all of them. The important constraint is that objects available at a stage are also available at all later stages: » If s s
and an object a is available at s, it is available at s
too. We will return to the quantifiers below. For now, we move to the propositional connectives. For conjunction and disjunction, we proceed in the traditional manner. Nothing in talk of stages as incomplete forces us to change the interpretation. » s  A ∧ B if and only if s  A and s  B. » s  A ∨ B if and only if s  A or s  B. (Note that, if A and B are known to be preserved upward along the relation  of inclusion, so are their conjunction and disjunction.) For negation and implication, the story becomes more interesting. Fix, for a moment, the interpretation of implication according to which A ⊃ B is the weakest of all propositions that, taken together with A, suffice to entail B. (Note that that is ambiguous, if ‘entail’ is ambiguous. Take ‘entail’ to be constructive entailment in what follows.) Then taking A ⊃ B to be ∼A ∨ B is no longer sufficient, because it will be too strong. Presumably A ⊃ A is the weakest of all propositions (on this reading at least) because no additional information beyond A is required to infer A. On the other hand, ∼A ∨ A is now a substantive claim: it is not true at all stages, but only at those where A is decided one way or another. More, then, needs to be said about implication. At the very least, we need that if s  A ⊃ B, then s  B if s  A. But if s  A ⊃ B and s
 s, then, since everything true at s is alsotrue at s
, we need that if s
 A then s
 B too. That turns out to be exactly what is required for implication: » s  A ⊃ B if and only if, for each s
 s, if s
 A then s
 B. Negation, in turn, is similar. It is too much to say that s  ∼A if and only if s  A, for s might well be incomplete with respect to negation. On the

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other hand, given that stages are consistent, we can say that if s  ∼A then s  A, and, furthermore, that if s
 s then s
 A. And that suffices for negation. ∼A is the weakest proposition incompatible with A: » s  ∼A if and only if, for each s
 s, s  A. It follows from the clauses on implication and negation that, if ⊥ is a proposition true at no stage at all, then ∼A is equivalent (in the sense of being true at exactly the same stages as) A ⊃ ⊥. Conditionals and negations constrain further stages. A stage warrants ∼A by ruling out descendent stages at which A is true. A stage warrants A ⊃ B by constraining descendent stages to include B when they include A. Back to the quantifiers. Recall that stages are incomplete with respect to the objects available at them, as well as the information they warrant. For existential quantification, that means that a claim of the form (∃x)A(x) requires an available witness: » s  (∃x)A(x) if and only if s  A(a) for some a available at s. For universal quantification, we are likewise constrained by having only available witnesses; however, now we cannot simply say that, for all available a at stage s, (∀x)A(x) is true at s just when A(a) is true at s, for perhaps later we will find an object as a counterexample. Universal quantification, then, must constrain later stages in just the same way as the conditional and negation: » s  (∀x)A(x) if and only if, for each s
 s and for each a available at s
, s
 A(a). Given a system of stages, we say that an argument is valid (with respect to that system of stages) if and only if, at any stage at which the premises are true, so is the conclusion. An argument is intuitionistically or constructively valid if and only if it is valid in all systems of stages. 6.1.2 Example Inferences Some of the distinctive behaviour of intuitionistic logic is already apparent. For example, the inference of distribution (∀x)(A ∨ B)  (∀x)A ∨ (∃x)B

(6.1)

is valid in classical logic, but it is invalid intuitionistically. It is straightforward to construct a system of stages containing a stage s at which (∀x)(A ∨ B) is true, but at which (∀x)A ∨ (∃x)B is not. Here is a simple example. Take two stages s1 and s2 , where s1 s2 and at s1 we have the object a available (and s1  A(a) but s1  B(a)), but at s2 we also have b available (and s2  B(b) but s2  A(b)). We can reason as follows. s1  (∀x)(A ∨ B) since at stage s1 only a is available and s1  A(a), so s1  A(a) ∨ B(a). On the other hand, at s2 , b is also available, and s2  A(b) ∨ B(b), since

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65

s2  B(b). However, s1  (∃x)B since s1  B(a) and a is the only object available at s1 . Furthermore, s1  (∀x)A, since, even though every object at s1 has property A, the object b available at s2 will not have property A, so s1  (∀x)A ∨ (∃x)B. That was a formal counterexample to the inference. It is another thing to understand what is going on in that counterexample. Consider the following scenario, which makes the counterexample concrete. You are teaching a class (say, in philosophical logic) for the first time, and your class list contains twenty names. You don’t know who, among your students, has previously done logic or who has not. However, everyone who has enrolled has either done logic before or has got in with special permission. That is one of the rules of enrolment. So, as you sit in the classroom awaiting their arrival, you are at a stage at which (∀x)(A ∨ B) is warranted, where A(x) says of x that she has done the prerequisite course in logic, B(x) says of x that she has special permission to enrol, and the quantifier (implicitly) ranges over students enrolled in the course. Now, as the students come in, one by one, you ask them whether or not they have completed the prerequisite course, so that you can assign extra reading, if need be. The first student, a1 , has done introductory logic, and so, by being ‘available’ at that stage—call it ‘s’—warrants A(a1 ). But s warrants neither (∀x)A (since you don’t know the status of all of the remaining students) nor (∃x)B (since you have not yet encountered a student who has had her prerequisite waived). So s does not warrant (∀x)A ∨ (∃x)B. But that, of course, is not to say that we have no warrant for (∀x)A ∨ (∃x)B. That is a separate issue. All that is required, for purposes of making the constructive distinctions, is that the given stage s does not warrant (∀x)A ∨ (∃x)B. Given that stages are local, and that s does not provide a witness for (∃x)B, and that s does not provide any grounds for asserting (∀x)A, the information provided by s does not include (∀x)A ∨ (∃x)B. We should emphasise that the ‘local’ nature of stages does not mean that all inferences break down. The inference, for example, from (∀x)(A ⊃ B) to (∃x)A ⊃ (∃x)B can be shown to be constructively valid. Take any stage s for which s  (∀x)(A ⊃ B). We can verify that s  (∃x)A ⊃ (∃x)B as follows. Take s
such that s
 s, and suppose that s
 (∃x)A. We wish to verify that s
 (∃x)B. By the conditions for the existential quantifier, we have some a available at s
for which s
 A(a). Since s  (∀x)(A ⊃ B) and s
s, we have s
 A(a) ⊃ B(a). It follows that s
 B(a) and hence that s
 (∃x)B, as we desired. Note that none of the arguments considered so far used negation. Incompleteness is a distinctive feature of intuitionistic semantics, but that does not mean that its distinctive features essentially involve negation. That said, incompleteness results in distinctive features of negation too:  A ∨ ∼A

(6.2)

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Constructive Logic

The law of the excluded middle can be thought of as an inference with no premises: it is valid in intuitionistic logic if every stage warrants A ∨ ∼A. But not all stages will warrant A or warrant ∼A. Stages may well be incomplete. To think otherwise is to take constructions to provide answers to all questions, and that is against the spirit of the enterprise. Stages provide justification for some things and not for others. As a result, we also have the failure of double negation elimination: ∼∼A  A

(6.3)

A stage may warrant ∼∼A without also being a warrant for A. Double negation elimination fails because, even though A ∨ ∼A might fail at a stage, every stage warrants its double negation: ∼∼(A ∨ ∼A)

It is worthwhile seeing why this is the case. To show that each stage warrants ∼∼(A ∨ ∼A), it suffices to show that no stage supports ∼(A ∨ ∼A). Now, suppose to the contrary that we have s  ∼(A ∨ ∼A) for some s. That would mean that, for each s
 s, s
 A ∨ ∼A. As a result, for each s
 s, s
 A, and that would mean that s  ∼A, contrary to our assumption that s
 A ∨ ∼A. So, our initial supposition fails, and for each s, s  ∼(A ∨ ∼A) and we have our result. A double negation, then, is an interesting operator from the point of view of intuitionistic consequence. ∼∼A is true at a stage just when ∼A has been ruled out: that is, s  ∼∼A just when, for each s
 s, there is some s

s
where s

 A. If A is always a ‘live possibility’ no matter how things turn out after stage s, then ∼∼A is warranted at s. An immediate upshot is the special behaviour of what we might call final stages: if a stage s has no proper descendants (no s
 s), then s is straightforwardly classical. If s is open to no further development, then s cannot be incomplete: ∼∼A is equivalent (at s) to A, and ∼A is true at s if and only if A is not true at s. Final stages are omniscient. 6.1.3 Interpretation It is one thing to use the formal semantics for the evaluation of inference, quite another to say that the distinctions drawn by the semantics might serve some function. If we wish to employ the semantics in order to draw distinctions with some kind of application, we need to say more about the apparatus. What are these stages? States of Knowledge: Whatever we might say about stages, they are not arbitrary states of knowledge. Even the most hard-bitten constructivist must agree that we can know that A ∨ B without knowing that A or knowing that B. Stages are prime with respect to disjunction, while states of knowledge need not be prime. We cannot identify stages with states of knowledge.

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Warrants: Neither can a pluralist about logical consequence admit that stages are (or model) what is warranted by proof or total evidence at a stage of inquiry. Such a line might be thought to be a more palatable alternative to arbitrary states of knowledge, because at least a case might be made that warrants, conceived of as special kinds of bodies of evidence, are prime. But if we are pluralists about logical consequence, then it seems that any body of information will provide more than one kind of warrant. Given any collection of propositions explicitly warranted by some ‘warrant’ (whatever that is taken to be), we can take the consequences of that set of propositions to be those implicitly warranted by that ‘warrant’. That makes sense even if the notion of consequence in play is classical, and so a logical pluralist who endorses classical logic must allow warrant to outstrip constructive consequence, at least in part. The reasoner who endorses classical reasoning must give some credence to the claim that instances of the law of the excluded middle have some kind of warrant, even if it is not supplied by constructive reasoning alone. Constructive Warrants: Warrant is a good idea none the less. In what follows, we will think of stages as a special kind of warrant: a constructive warrant. The body of information constructively warranted at a stage is not necessarily all that might be appropriately inferred with all of the logical machinery one has at hand; it is only that which can be inferred while observing constructivist scruples. In particular, warranted disjunctions must have witnessing disjuncts, and warranted existentially quantified claims must have available witnessing objects. These restrictions mean that not all classical reasoning survives, but these strictures come with their own benefits, as we will attempt to explain in the next section, when we approach an application of constructive restrictions on reasoning. Just as in classical mathematics, the goal of constructive mathematics is to gain understanding of mathematical structures, and to prove theorems about them; however, the goal is to prove mathematical theorems with constructive content. If a statement asserting the existence of some mathematical object is proved in a constructive manner (using the rules of intuitionistic logic), then the proof will contain the means of specifying the object or structure in question. Wittgenstein illustrates the advantages of constructive proof over its classical cousin with respect to understanding: A proof convinces you that there is a root of an equation (without giving you any idea where)—how do you know that you understand the proposition that there is a root? [137, page 146]

This feature of constructive mathematics is guaranteed by the structure of constructive proofs and the nature of constructive warrant. Any intuitionistically valid proof of a disjunction (from no premises, of course) will prove a disjunct. Any intuitionistically valid proof of an existentially quantified statement (from no premises, of course) will prove an instance of that quantifier. Such proofs track mathematical constructions. Constructive

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warrant is local and modest (it allows for gaps and incompleteness), but it is at the same time totally specified (disjunctions are not left dangling without witnessing disjuncts, and existential quantifications also have witnesses).

6.2

Admissibility

It is one thing to have a formal system that has some use in modelling reasoning, and that can help draw the distinction between what can be constructed and what cannot; it is quite another thing for the formal system to deserve the name logic. For that, we have argued, we need to show that the distinctions drawn by intuitionistic logic fit as a way of making the pre-theoretic idea of logical consequence precise. We must show how intuitionistic logic fits the Generalised Tarski Thesis, and that the resulting notion of logical consequence can count as necessary, formal, and normative. Only when we have done this will intuitionistic logic count towards a pluralism about logical consequence. 6.2.1

Stages as Cases

First we need to show how intuitionistic logic stands as an instance of the schema provided by the Generalised Tarski Thesis. We do this by indicating how the stages of the semantics for intuitionistic logic can serve as the precisifications of ‘case’. A stage in an intuitionistic model serves as the right unit of evaluation. An argument is intuitionistically valid if and only if, for any stage (in any model) at which the premises are satisfied, so is the conclusion. We have a straightforward instance of the gtt. Note that stages here are very little like the complete and consistent models of Tarski’s semantics for classical predicate logic, or the possible worlds they resemble. Stages are much more like the situations in relevant semantics, but they have their own unique properties. The unique modesty of constructive stages ensures that the information preserved in such stages satisfies constructive constraints. However, the semantics achieves this goal of tracking what may be constructed not by an extra ‘filter’ added to a more orthodox semantics (as might be achieved by classical computability theory), but it arises out of the nature of stages themselves. As in any logical system, the collection of stages at which a set of sentences Σ is satisfied provides the model for the information (constructively) represented in Σ. The fact that there are stages at which classical tautologies (such as A ∨ ∼A) fail does not address, by itself, the issue of whether or not such tautologies are in fact true, any more than the fact that there are classical models where contingent truths fail means that these truths are somehow in doubt. No, the failure of A ∨ ∼A at certain stages is a sign of the incompleteness of those stages, and a sign of the constructive content of these classical tautologies which fail to be intuitionistic tautologies. A ∨ ∼A is true only at those stages at which A is decided. It is consistent with the general constructive picture that the actual world might in fact decide in

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favour of all classical tautologies (and hence that the truths in the actual world are true at some ‘final stage’). It follows merely that in doing this the world goes beyond what might be constructively proved. As we have seen in the case for relevant consequence, what might be actually true (and even necessarily true) might well go beyond what this or that logic might be able to tell us. This is no slur against the logic, but rather a recognition of its virtues. 6.2.2 Necessity Are the judgements of validity of intuitionistic logic in any sense necessary? We can give a straightforwardly affirmative answer to this question by noting that we have already given an affirmative answer in the case of classical logic, and that all arguments which are intuitionistically arguments are classically valid. But this is not all we can say; we may also argue directly. What is necessary is true in all possible circumstances. We need to know that whatever holds in all stages also holds in all possible circumstances. To do this, we need simply run through the clauses for truth at a stage, and verify that each of these conditions are satisfied by possible circumstances. But this is clear. Any Tarski model is a stage model of intuitionistic logic where there is only one stage. So no possibility is ruled out by the requirements of intuitionistic logic. 6.2.3 Formality Is intuitionistic consequence formal in any of the senses of formality we have discussed? Schematic: Intuitionistic consequence is clearly schematic in just the same way as any notion of logical consequence on the language of predicate logic. Intuitionistic logic does not pick out any different linguistic features from those picked out by classical predicate logic. Intuitionistic predicate logic is schematic in just the same way. 1-formality: Is the distinction between validity and invalidity in intuitionistic logic constitutive for the norms for thought as such? We will leave the discussion of the normativity of the distinction to the next section. Is intuitionistic consequence applicable to content as such? The answer to this is also affirmative. Intuitionistic logic picks out structural features relevant to any kind of content at all, provided that it makes sense to evaluate it relevant to stages. But stages conceived of as constructive warrants are applicable to any possible subject matter. Most examples of constructive reasoning are mathematical, but this is because the kinds of infinitary constructions where intuitionistic reasoning is most starkly salient arise in a mathematical context, and in these contexts we can distill the essence of such kinds of reasoning. However, all of this mathematical reasoning can be paralleled in reasoning about non-mathematical objects. Instead of talking of a sequence of numbers, we could talk of a sequence of points in space, instants, or periods of time or other more concrete entities.

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Nothing here requires that we pay attention merely to the subject matter of mathematics. 2-formality: Is intuitionistic logic indifferent to the particular identities of objects? Here the answer is obviously affirmative. Constructive models make no constraints on particular objects that are not constraints about all objects. In no case are objects treated specially. Of course, objects might be treated specially in models of particular theories (say, models of arithmetic, or some other mathematical theory), but what is true in all models of a theory X is not necessarily what is true on the basis of logic alone. Intuitionistic logic is 2-formal. 3-formality: Does intuitionistic logic abstract from the semantic content of thought? Here again, the answer is affirmative, for the same kind of reason as given for 2-formality. Intuitionistic logic constrains extensions of predicates no more than it fixes the interpretations of names on particular objects. Intuitionistic logic abstracts away from all semantic matters. All that is left for a predicate is its extension in each stage. This is more than is reflected in the semantics for classical logic, where an extension is an all-or-nothing affair. Intuitionistic logic allows for more flexibility and for the drawing of more distinctions than does classical logic (here the case is directly parallel to that of relevant logic and its situation semantics), but it provides no more constraints than does classical logic. Intuitionistic logic is 3-formal. 6.2.4 Normativity What kinds of mistakes are made by constructively invalid arguments? They are mistakes of constructivity. These norms apply whether or not we are engaged in mathematical reasoning or other kinds of reasoning, as we have already seen. In an intuitionistically invalid argument, the conclusion goes beyond what it said in the premises. Let’s consider two examples, first a classically valid argument which is both intuitionistically and relevantly invalid: A I B ∨ ∼B

Here, we have already seen the relevant analysis of the mistake in the inference from A to an (unrelated) B ∨ ∼B. It is relevantly invalid because the conclusion B ∨ ∼B is not implicit in the premises. The conclusion, although true, and true of necessity, is not true in every situation in which A is true. The truth of A does not bring with it the truth of B ∨ ∼B: it comes along in some other way. The constructive analysis of the invalidity of this argument is similar, but different. The argument from A to B ∨ ∼B fails not because B ∨ ∼B is irrelevant to the premise (the analogous argument from A ∧ ∼A to B is valid), but because the constructive content of B ∨ ∼B is non-trivial, and not true everywhere. B ∨ ∼B is true at stages that decide B in the affirmative of negative; and so to infer it from the premise A is to cut down on the number

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of possible stages available for evaluation and to fudge some of the distinctions we can draw with constructive logic. The inference is constructively invalid because not all stages decide B. Consider now the argument of distribution: (∀x)(A ∨ B) I (∀x)A ∨ (∃x)B

As we have argued before, this is constructively invalid, because there are stages at which (∀x)(A ∨ B) is true but (∀x)A ∨ (∃x)B is not true. The constructive content of (∀x)(A∨B) does not include all of the constructive content of (∀x)A ∨ (∃x)B. The argument is constructively invalid, and to infer the conclusion from the premise is to make a kind of mistake.

6.3

Applications

We have argued that intuitionistic logic provides another admissible instance of gtt. 6.3.1 Mathematical Reasoning We will examine closely a particular instance of mathematical reasoning which will help us understand the relationship between constructive considerations and the metaphysics of mathematics. The lessons are more general than simply mathematical, and we will then go on to see how constructive considerations may apply globally, wherever reasoning is applicable. Here is a scenario from Shapiro’s Philosophy of Mathematics [121]. It is a fairly simple piece of reasoning, which will illustrate the role of construction in reasoning and proof. Professor A: Next we will prove the Bolzano–Weierstrass Theorem: every bounded infinite set has at least one cluster point. Let C0 be an arbitrary, bounded infinite set.1 To prove the theorem, we must produce a point p with the property that every neighbourhood of p contains infinitely many points in C0 . . . We divide C0 into four equal squares by intersecting lines. One of these smaller squares must contain infinitely many points of C0 . . . choosing such a sub-square, label it C1 . We have C0 ⊆ C1 and both are closed and bounded. Now repeat this process. Divide C1 into four squares . . . By continuing this, we generate a sequence of closed squares Cn . . . appealing to the nested set property, there must be a point p that lies in all the sets Cn . This is the point that will turn out to be a cluster point for S. At this moment, a student with a double major in mathematics and philosophy raises her hand and is recognised. Student: You are using a constructional language in this lecture. You do not actually mean that you or some ideal mathematician has done this construction, 1 In this quote we have changed Shapiro’s ‘S’ to ‘C ’ to clarify the reasoning. Shapiro starts 0 with the set S then calls it C0 without explaining the transition.

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do you? How can anyone do an infinite number of things, and then after all of them—on the basis of them—do something else, in this case pick the point p? Professor A: Do not take this lecture literally. Of course, there is no such constructional process. I am describing a property of the plane. From principles of cardinality, I infer the existence of infinitely many points in some square C1 , and then in C2 . The axiom of replacement implies the existence of the whole sequence Cn . Finally, from the nested-set property, I deduce the existence of a point in all of the Ci s. I let p be the name of one such point. Now this is what I mean by this lecture; the constructional language makes it easier for you to see. [121, page 184]

There are a number of things going on in this example. Firstly, it shows how images of construction are used to clarify the processes of reasoning leading to a mathematical proof. As Professor A indicates, this language need not be taken literally; it may be used as a heuristic device. However, the language is not merely heuristic. If you are presented with an infinite set of points and are asked to find a cluster point for that set, the process you are given in this proof is one you might think to use. However, to use it, you must, as the student points out, perform an infinite task. The professor is wrong in thinking that the infinite task is impossible. Sometimes such an infinite task is feasible: it all depends on the infinite set you are given at the start. For example, if one is given the unit square [0, 1] × [0, 1], then one can choose each Cn as [0, 1/n] × [0, 1/n] and one’s limit point will be the origin, (0, 0). One can perform infinitely many choices if the choices are ‘easy’ enough; however, the choices are not always that simple, and, indeed, there is no algorithm providing a cluster point for each infinite set of points. Shapiro uses this example to motivate an understanding of the difference between intuitionist and classical mathematics. His explanation is a good example of the orthodox view of the relationship between classical and intuitionistic mathematics, so it is worth quoting at some length. One crucial difference between the classical constructive mode of thought [by this, Shapiro means the ‘constructing’ language used in mathematical demonstration] and the intuitionistic mode is that the former seems to presuppose that there is a (static) external mathematical world that mirrors the constructs. A traditional Platonist (such as Proclus) might claim that the existence of the mathematical world is what justifies or grounds the constructs . . . Whatever its metaphysical status, the supposition of an external world suggests certain inferences, some of which are the nonconstructive parts of mathematical practice rejected by intuitionism. For example, if a classical mathematician proves that not all natural numbers lack a certain property (i.e., she proves a sentence in the form ∼∀x∼Φ), she can infer the existence of a natural number with this property (∃xΦ). Following existential instantiation, she can give a ‘name’ to some such number and do further constructional operations on it. In the proof of the Bolzano–Weierstrass theorem . . . we have a similar instance of excluded middle at work. At each stage, the constructor knows that at least one of four squares has infinitely many points from the given set, but he may not know which one. Nevertheless the constructor can pick one such square, and go on from there.

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Intuitionists demur from such inferences and constructions because they understand the principles as relying on the independent, objective existence of the domain of discourse. For them, every assertion must report (or correspond to) a construction. In the present example, an intuitionist cannot assert the existence of a natural number with the said property, because such a number was not constructed. The intuitionist cannot choose a square with infinitely many points from the given sets, because such a square was not identified. Bishop [23] understands the law of the excluded middle as a principle of omniscience. [121, page 187]

Here the orthodox view is clear. Classical reasoners accept such infinitary constructions because they do not step from truth to untruth. If not everything is not Φ then something is Φ, regardless of the availability of a construction of this object. For intuitionists, this is unacceptable, for there may well be no such object. What can we say about the person who does not accept the intuitionistic analysis of mathematical objects? Is constructive logic irrelevant to her purposes? This issue arrives even if we do not think that there is such an objective mind-independent realm; it is enough that mathematical practice proceeds as if there is such a realm. The details of the debate between realist and fictionalist analyses of the metaphysics of mathematics is irrelevant for our purposes, even though it is an important live issue in the philosophy of mathematics [3, 49, 50, 121]. Mathematics that proceeds on the pre-supposition of an existing sphere of mathematical objects (whether realist or fictionalist) seems to use the whole panoply of classically valid arguments. Can constructivist considerations have any force for those who reject the intuitionist ontology? We think that the answer to this is still positive. Although a non-intuitionist may hold that there actually is a cluster point of any bounded infinite set on the plane, she may still agree with the intuitionist that it will be impossible to find a cluster point. Being presented with an infinite set does not in and of itself give one the means to find a cluster point for that set. All are agreed on this point, at least in cases where we are unable to complete the infinite division-and-choice task discussed in the proof. This consideration has force whatever your view of the nature of the mathematical universe, and this distinction, between what can be constructed and what cannot be constructed, is modelled in intuitionistic reasoning. In the idealised stages in intuitionistic semantics we have a model of a kind of process of mathematical construction, and these distinctions have force whether or not we take it that mathematical existence is tied to construction. Intuitionistic logic has application in the analysis of mathematical proof for the realist and the fictionalist, as well as for the intuitionist. 6.3.2

Constructive Warrant

We will do well to note that all of the considerations Shapiro brings in favour of a constructive understanding of proof apply independently of the nature of the objects described in the scope of that proof. What we take

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our ideal constructors to be able to do is a feature of our understanding of the process of construction; it is not a feature determined by the nature of the things constructed. We can observe constructive limits in our reasoning about anything at all. In doing so, we stay within the bounds of what might be constructively warranted. For this to count as an important application of the relation of classical logical consequence, we must find a use for the notion of a constructive warrant for claims in general. It is not a part of our job description to do this here. It is clear how this might go. Semantic anti-realists, like Dummett [38] and Tennant [128, 130], have given detailed explanations of how an anti-realist meaning theory might be made out. For our purposes, it is enough to point in that direction. Some have thought that semantics involving stages, interpreted as constructive warrants, can play an important role in giving the semantics of natural languages. If this is the case, then constructive logic has an important application in semantics writ large. It might be thought, then, that this application is in tension with the application of classical logic to semantics, as discussed in §4.4.2. This need not be the case for at least two possible reasons. On the one hand, one might defend a comprehensive semantic theory, in which both classical and constructive features are present, and so classical and constructive logics are motivated by semantic features. On the other hand, one might be a semantic pluralist as well as a logical pluralist: one might think that classical and constructive semantic theories are equally illuminating and acceptable (but different) ways to understand the semantic features of our language. Instead of discussing such options here, we will leave them to Part III, where we consider an objection to logical pluralism along these lines.

Chapter 7

Variations & Loose Ends This chapter presents variations on the theme we have developed. We have already presented logical pluralism. If our arguments are sound, we have demonstrated that there are different ways to settle the notion of logical consequence, each equally deserving the title ‘logic’. In this chapter, we will sketch some other degrees of variation possible for the logical pluralist. We examine an issue raised by such variations, as well as by variations considered in earlier chapters: does plurality in logic come only by way of plurality of languages? Finally, we close the chapter by tying up some loose ends, especially with respect to paraconsistency, dialetheism, and the notion of endorsing a logic.

7.1

Free Logics

A logic is free to the degree that it refrains from making assumptions of existential import with respect to its singular (and general) terms. Traditional classical predicate logic is not particularly free: the domain of quantification must be non-empty, and each name must have a denotation in the domain of quantification. Such assumptions yield existential import, and some have argued that those assumptions should not be made [66, 67]. The issue is not restricted to classical logic; relevant and intuitionistic logic have existential import in exactly the same manner. Free logics proceed without existential assumptions. You can define a (universally) free analogue of classical predicate logic by allowing the domain of quantification to be empty, and by allowing names to fail to denote. Parenthetical remark: A free logic is said to be universally free if it allows the domain of quantification to be empty. We will concentrate on universal freedom. We should note that, while they are not united with respect to requiring empty domains, free logics are united in allowing singular terms to be non-denoting; hence, the inference—universal instantiation—from ∀xGx to Gt is rejected, since ‘t’ may not denote anything in the domain of ‘∀’ (t may not exist, like Santa). End of parenthetical remark. More needs to be said about the detail in order to define the system precisely, of course. Furthermore, other modifications are possible, depending on what one takes to be the right behaviour of the quantifiers and non75

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denoting terms. (In Karel Lambert’s original definition of ‘free logic’, quantifiers have to be ‘treated standardly’, which meant that ‘∃’ is to be read there exists an object while ‘∀’ is read every existent object. We retain that stricture for purposes of this section.) A singular term ‘t’ has existential import if t exists, if something exists that is t. (Similarly for a general term ‘F’: there exists something that is F.) We do not need to go into any detail on this matter, the points we need to make are independent of any way of spelling this out. A (universally) free variation on classical predicate logic differs from the unfree traditional version in the arguments that are taken to be valid. A canonical case is the inference from an (unrestrictedly) universally quantified claim to an existential claim: ∀xFx  ∃xFx

That is valid in unfree classical predicate logic, and invalid in (universally) free classical predicate logic. Models with an empty domain can validate ∀xFx without bringing ∃xFx along. If domains have to be non-empty, as they are in unfree classical logic, this is impossible. The same kinds of considerations hold with respect to the ntp reading of classical consequence. Nothing in what we have said so far has settled the issue of whether it is possible for nothing to exist. This is a matter of some significant metaphysical debate, over which we pass in dignified silence. More interesting, for our purposes, is an approach that takes the matter to be importantly flexible. In particular, Philip Bricker [26] has argued that, on a realist account of possible worlds, two options are possible for evaluating modal discourse. On the first option, one takes each possible world as the unit of evaluation, when considering whether or not A is possible: A is possible if and only if there is some world in which A is true. But on Bricker’s favoured view of possible worlds (modal realism), the first option rules out some intuitively acceptable ‘possibilities’. According to Lewisian realism, possible worlds are maximally connected spacetimes, and so it is impossible that there be disconnected spacetime regions; that is, there is no possible world that includes disconnected spacetime regions. Similarly, there is no possible world in which nothing exists, because such a world would not (could not) exist. To deal with these issues, Bricker proposes another option for evaluating modal discourse: A is possible if and only if there is a collection of worlds in which (considered together) A is true. On this account, we can have a possibility (a collection of more than one possible world) in which spacetime is disconnected. The account also yields the ‘possibility’ (now, the empty collection of possible worlds) in which nothing exists. Bricker’s view is interesting for our purposes, because there is no difference between the metaphysical commitments of the two options towards possibility. The given options are simply different ways to evaluate the modal discourse of possibility and necessity. Such plurality is also available

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to the ntp analysis of validity. Taking possibilities to be concrete possible worlds, our ntp consequence relation is unfree, because it makes the bare existential commitment to some object or other; taking possibilities to be collections of worlds, ntp consequence is free, at least to the extent that it allows for the empty collection of worlds. Is there any fact of the matter as to whether one of the options yields genuine possibilities and the other not (or that one fails to provide enough of the possibilities)? There seems to be little to choose between them. To say that there is something conclusive for one option over the other is to say that the notion of possibility (and of ntp consequence) is settled enough to decide the matter conclusively. If that notion is not so settled, then the matter of existential commitment provides another avenue for plurality. For present purposes, we leave that avenue open for further debate.

7.2

Second- and Higher-Order Logics

We have said nothing of quantification over anything other than objects. Second-order logic allows for quantification over properties or relations or collections of objects, as well as over the objects themselves. Higher-order logics allow for quantification over properties of properties (or collections of collections) and much more. On some views of this matter, second-order quantification counts as genuinely logical, because we can define second-order consequence just as formally and precisely as we can first-order consequence [120]. But that is not a majority position. Second-order logic, defined in the standard manner, cannot be axiomatised; there is no list (whether finite, or infinite, provided that there is some recipe for specifying that list) of axioms and rules such that any second-order valid argument may be shown to be valid by way of these axioms and rules. Does second-order quantification count as genuinely logical, or should quantification over properties or sets be relegated to a substantial theory such as Zermelo–Fraenkel set theory, and not be taken as a part of logic? Here, again, there is scope for plurality. Nothing in the absolutely settled core of the notion of logical consequence dictates that the relation of logical consequence be axiomatisable. It might be thought that the properly logical must be formal and analytic, and perhaps such criteria, when held very tightly, rule out much of what has been considered as ‘logic’ in the past. (If, for example, you take it that anything which is logically valid can be verified to be logically valid by way of some finitary proof, then logic must indeed be axiomatisable, and unrestricted second-order logic is beyond the pale.) Still, it is far from clear that the settled core of the notion asks so much of logical consequence. If the meanings (construed as the truth conditions, for the moment) of the second-order quantifiers are given by the traditional rules, then it may well be that those meanings may be coherently specified without being effectively axiomatised.

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As far as we can see, second-order logic satisfies enough of the core criteria to count as ‘properly logical’, at least pending arguments to the effect that, contrary to our position, axiomatisation is required of any admissible precisification of ‘follows from’. As with other admissible precisifications, second-order logic affords useful applications. To be sure, in the development of classical mathematics, wherein we want our proofs to be surveyable and checkable, the restriction to first-order consequence is desirable; however, when we want to consider the limits of expressibility in formal languages, second-order quantification shows (at least one) of its virtues: extending one’s formal language to second-order quantification allows one to specify much more than what can be expressed in a first-order language [120]. Both accounts of consequence are admissible, each enjoying its own applications.

7.3

Languages and Logics

That last variation of pluralism—especially the choice of the formal languages of first-order and second-order logic, seems to take us very close to the kind of logical pluralism of Rudolf Carnap. The first attempts to cast the ship of logic off from the terra firma of the classical forms were certainly bold ones, considered from the historical point of view. But they were hampered by the striving after ‘correctness’. Now, however, that impediment has been overcome, and before us lies the boundless ocean of unlimited possibilities. [30, page xv] Principle of Tolerance: It is not our business to set up prohibitions, but to arrive at conclusions. [30, §17]

For Carnap, plurality in logic arose because we were free to choose the kinds of languages we might use to conduct theoretical enquiry. The choice between first-order and second-order logic as formal languages is but one choice of a limitless number. In logic, there are no morals. Everyone is at liberty to build his own logic, i.e. his own form of language, as he wishes. All that is required of him is that, if he wishes to discuss it, he must state his methods clearly, and give syntactical rules instead of philosophical arguments. [30, §17]

For Carnap, the choice of a language is unconstrained. We may construct a language with rules like those of classical logic, or we can restrict our inferential machinery. Carnap does this in the Logical Syntax of Language [30] by developing Language I, with constructive constraints, and Language II, a classical type theory. The resulting logics are different because the languages are different. What we want to emphasise is that Carnap’s pluralism is not our kind of logical pluralism. (A detailed account of the differences between our pluralism and Carnap’s is beyond the scope of this chapter. For detailed

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arguments and discussion, we refer the reader elsewhere [109].) For us, pluralism can arise within a language as well as between languages. Considered as formal languages, the language of first-order predicate logic and the language of second-order logic are indeed different, and the consequence relations differ. However, when used as an account of the form of claims expressed in a natural language, such as English, the different formal languages give different answers to the validity of arguments in the one language. Take the sentence » If two objects have the same properties, they are identical. If we formalise this using the language of second-order logic, we might find ∀x∀y(∀X(Xx ≡ Xy) ⊃ x = y)

which is valid in classical second-order logic. If we do not have the machinery of second-order logic at our disposal, our original sentence is not taken to be valid. In this case, the plurality between first-order and second-order logic gives rise to a plurality of verdicts about the one claim in the one language. The case is starker, of course, when it comes to classical, relevant, and intuitionistic logic, where arguments in the one formal language (the language of conjunction, disjunction and negation, for example) yield different verdicts of validity. That, too, is a case where we take there to be pluralism about logical consequence, a pluralism that is independent of any plurality of languages.

7.4

Loose Ends

In this section we tie up a few loose ends before turning to objections in Part III. After briefly discussing the (sometimes conflated) distinction between paraconsistency and dialetheism, we distinguish two notions of endorsing a logic, one appropriate for any dialetheist, the other for nondialethic paraconsistentists. Dialetheism is the view that some truths have true negations. Trivialism is the view that all sentences are true. On pain of trivialism, any dialetheist is a paraconsistentist, one who endorses a paraconsistent consequence relation. The question is: Is every paraconsistentist a dialetheist? The answer is ‘no’. Discussing that answer will help to clarify a few issues, especially with respect to pluralism, paraconsistency, and dialetheism. 7.4.1

Four Grades of Paraconsistent Involvement

A paraconsistent consequence relation is one according to which explosion (traditionally, ex falso quodlibet) fails, where explosion is an argument from (perhaps among other premises) A and ∼A to arbitrary B. There are four grades of paraconsistent involvement.

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» Gentle-strength paraconsistency: The first level of paraconsistent involvement is dissatisfaction with explosion as a valid principle of inference. Rejecting explosion commits you to gentle-strength paraconsistency. » Full-strength paraconsistency: The second level takes there to be interesting and important inconsistent but non-trivial theories (e.g. naïve truth theory, naïve set theory). If you think that there are interesting, inconsistent, but non-trivial theories, then you are a full-strength paraconsistentist. » Industrial-strength paraconsistency: The third level holds that some of the given inconsistent, non-trivial theories may be true. If you’re an industrialstrength paraconsistentist, you think that it is possible that inconsistent but non-trivial theories are true. Parenthetical remark: If you endorse ♦(A ∧ ∼A), then it is hard to avoid commitment to the contradiction ♦(A ∧ ∼A) ∧ ∼♦(A ∧ ∼A), given 
(A ∨ ∼A),
B  ∼♦∼B, and de Morgan principles for negation, each of which is valid in Priest’s semantics [92, 103]. The upshot is that industrial-strength paraconsistists avoid dialetheism by rejecting either de Morgan features of negation or the interdefinability of ♦ (the possibility operator) and
(necessity). For present purposes, we set this issue aside. End of parenthetical remark. » Dialetheism: The fourth level is dialetheism, according to which some inconsistent but non-trivial theories are true. Each successive level presupposes its predecessors, but, at least on the surface, weaker levels do not seem to require their stronger successors; in particular, one need not accept dialetheism in order to enjoy one of the ‘lower’ grades of paraconsistency. That, at least, is the way things appear; however, Graham Priest [95] argues that there is a slippery slope from the ‘lowest’ to the ‘highest’ grade of paraconsistent involvement. We turn to Priest’s argument, tying our response to pluralism. 7.4.2

Slippery Slope?

Priest’s argument goes as follows. There is a difference between the various grades of paraconsistent involvement, as it is possible to belong to any weaker level without endorsing a stronger level; however, given the endorsement of a weak level of paraconsistency, the next stronger level is much more enticing, and there may be very little reason to resist it. Suppose that you are a gentle-strength paraconsistentist, in which case you reject explosion. In that case, the second grade appears to be quite reasonable: if a consequence relation is paraconsistent, then there are indeed theories, closed under that relation, that are inconsistent but non-trivial. But, then, it is a small step towards taking some of those theories to be interesting and important, especially if you are a philosopher of language (or of logic) with interests in naïve truth theory, naïve extension theory, and other such theories in the literature [13, 83].

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We are happy to accept the ‘slide’ from gentle- to full-strength paraconsistency; however, the slide through industrial-strength paraconsistency, and on to dialetheism, we think, may be arrested. Priest argues that a full-strength paraconsistentist has no good reason to resist the jump to industrial-strength and, in turn, to dialetheism. Priest propounds a challenge: if you take there to be interesting and important inconsistent theories, then what reason could you have for ruling out the truth of each and every one of them? Priest maintains that no good reason is available. We are not sure whether monistic (non-pluralistic) full- or industrialstrength paraconsistentists can arrest the slide that Priest advances; however, we think that pluralists can arrest the slide. The simplest way to arrest the slide is to invoke explosion, or at least the so-called rejection-converse of explosion: » If B follows from A ∧ ∼A, then if one rejects B, one ought (rationally) to reject A ∧ ∼A. With that principle in hand, plus the validity of explosion, one has reason— namely, the pain of trivialism—to rule out the truth (or even the possible truth) of dialetheism. As Priest points out, the rejection-converse of explosion is the most powerful, direct argument against the slide towards dialetheism; however, Priest also points out that, by accepting full- or industrial-strength paraconsistency, one has renounced one’s rights (as it were) to that argument. On that score Priest is correct—unless one is a pluralist. 7.4.3

Arresting the Slide: Pluralism

Pluralism affords an arrest of Priest’s slide. After all, suppose that you are a full-strength paraconsistentist but also a pluralist. Suppose, further, that you think that what is possible is represented by complete and, in particular, consistent situations. Then you have an immediate reply to Priest’s challenge: you do have a ‘right’ to the rejection-converse of explosion, since you never renounced it! Qua pluralist (of the sort under supposition), you use paraconsistent logic to evaluate inferences, produce theories, prove things, and perhaps more; however, you also use classical logic for similar purposes (though probably not the same theories). While you use paraconsistent consequence to model naïve truth theory (or some such inconsistent but non-trivial theory), you do not thereby take all of its cases to represent possibility; possibility, you think, is represented by a proper class of such (paraconsistent) cases, namely, the complete and, in particular, consistent ones. Accordingly, you take it to be impossible for A ∧ ∼A to be true and B untrue; you accept that explosion is valid. Are you, then, a full-strength paraconsistentist? Yes: full strength (even industrial strength) does not require that you take all the situations of your given (paraconsistent) consequence relation to represent possibility.

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A pluralist, then, may arrest Priest’s slide by invoking the validity of explosion while staying true to her full- or industrial-strength paraconsistency. Whether a monist may do likewise is something we leave to the reader. For present purposes, the important upshot is that initial appearances remain accurate: one can endorse a paraconsistent consequence relation without thereby endorsing—or even sliding into—dialetheism. At this stage, a question might arise: What happens if one is a dialetheist? To what extent can a dialetheist be a pluralist? To what extent, that is, can a dialetheist endorse an explosive consequence relation (like classical or constructive), in addition to her paraconsistent relation? 7.4.4 Weak vs Strong Endorsement Some (not many) contemporary philosophers are dialetheists, including one of us, namely jcb [9, 14, 15, 16]. We need to clarify the sense in which a dialetheist can be a logical pluralist. Towards that end we introduce the following definitions: » An instance of gtt satisfies the actuality constraint if and only if the actual case is in the domain of its quantifier. » One strongly endorses a consequence relation if one takes it to be an instance of gtt and accepts that it satisfies the actuality constraint. Suppose that you take possibility—ways things could be, etc.—to be represented by complete and consistent situations. (You will then be with the majority of our readers, and also squarely with one of us, namely gr.) As we have indicated in Parts I and II, you may endorse relevant (paraconsistent) and constructive accounts of consequence, in addition to classical. In what sense of ‘endorse’ may you so endorse those logics? The answer, which we hope is clear from the foregoing chapters, is that you may strongly endorse each such logic; after all, your ‘actual case’ is a special sort of your other cases—it is a complete and consistent one. That much, we hope, is plain from the previous parts of the book. Parenthetical remark: There are some subtleties to be noted here. One could still be a pluralist, and still get the spirit of strongly endorsing a logic, without holding that there is an actual case at all. For example, you might not take there to be a ‘world’ comprising all of the actual situations and circumstances, perhaps because of cardinality constraints. None the less, we might hold that a class of cases satisfies the actuality constraint (in an attenuated sense) as long as, for every collection of truths X and untruths Y , there is a case in that class in which each member of X is true and each member of Y is true. This class of cases, then, will still suffice to ensure that every argument from true premises to untrue conclusions is invalid, even if there is not necessarily one case (the actual case) that does the job. Of course, if one takes there to be a set X∞ of all of the truths, and a set Y ∞ of all of the untruths, then any case satisfying all of X∞ and dissatisfying all

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of Y ∞ will count as the actual case. However, if one were not happy with such comprehensive objects as worlds, one would probably not be happy with such comprehensive collections as the set of all truths either. End of parenthetical remark. Dialetheists are different. Can a dialetheist strongly endorse classical consequence, in addition to a paraconsistent consequence relation? The answer is ‘no’, because the actual case, according to the dialetheist, is not consistent. So, dialetheists cannot strongly endorse classical consequence. The same applies to constructive consequence and, in general, to any instance of gtt that takes cases to be (one and all) consistent: a dialetheist cannot strongly endorse the given consequence relation, as such a relation will be explosive (resulting in triviality). That said, a dialetheist may weakly endorse an explosive consequence relation, as long as she (with us) does not impose the ‘actuality constraint’ as a criterion of the admissibility of a consequence relation. So, we will say: » One weakly endorses a consequence relation if one takes it to be an admissible instance of gtt. Anyone who strongly endorses an instance of gtt thereby weakly endorses it, but not necessarily vice versa. A dialetheist is free to weakly endorse an explosive consequence relation, but she cannot strongly endorse it. Similarly, an intuitionist who rejects some instances of the law of the excluded middle cannot strongly endorse a consequence relation according to which A entails B ∨ ∼B, though she can weakly endorse it. That is not to say that a dialetheist is ‘less pluralistic’ than her non-dialethic colleague. A dialetheist can weakly endorse all those logics endorsed by her colleague, and she can strongly endorse a wide range as well. For example, the dialetheist might be convinced (as we are) that the logic of truth-preservation in all worlds is an admissible instance of gtt; however, she will take worlds to afford inconsistency (be they complete or not). Moreover, the dialetheist can endorse the logic of truth-preservation in all situations, but she may do so word-for-word, entirely agreeing with §5. The dialetheist can also strongly endorse a logic of truth-preservation over constructions, provided that the account of constructions is modified to include inconsistent constructions. A dialetheist can be just as pluralist as a non-dialetheist. The debate over the truth of dialetheism may continue; however, it need not be and, we think, should not be a debate about which instances of gtt are ‘correct’. On that ‘issue’, both parties may (and, we think, should) agree. Disagreement, as always, comes at the level of application; and one disagreement between dialethic and non-dialethic pluralists concerns which consequence relations, out of the plurality of such relations, one ought to strongly endorse. But that is an issue on which we disagree, and we leave it for another time.

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Part III

Objections, Replies, Other Directions

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Chapter 8

General Objections We hope that the initial parts of this book have charted a coherent and viable pluralism about logical consequence, and one that is well supported. By way of further clarifying logical pluralism, we now turn to some general objections. Parenthetical remark: We will write the objections and replies as if the objector were directing the comments towards us, so ‘you’ in the objections denotes us, not you (the reader), and ‘I’ denotes the objector, not us (the authors). End of parenthetical remark. *

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Objection: Isn’t your victory pyhrric? You argue for a pluralism about logical consequence which is wide enough to encompass classical, relevant, and intuitionistic logic. Yet in doing so you propose a view which the adherents of relevant and intuitionistic logic by and large would find repugnant. If you endorse classical logic, then you take all instances of the law of the excluded middle to be necessary. But intuitionists, almost to a person, take the law of excluded middle to have counterexamples. Similarly, if you endorse classical logic you take it to be necessary that if A ∧ ∼A is true then B is true. But relevantists deny this. Your pluralism is a hollow position: it is far less interesting to hold a view according to which intuitionistic logic and relevance logic are acceptable if it is a view which few advocates of such logics could endorse.1 Reply. The premises of this objection are correct, but the conclusion, that pluralism is uninteresting, should be resisted. It is true, the combination of endorsing classical, relevant and intuitionistic logic involves a position that is not endorsed by the majority of relevant or intuitionistic logicians (or classical logicians, for that matter). But this is not surprising; since most intuitionists or relevantists are (at least when it comes to their explicit theory) monists rather than pluralists, it is to be expected that something in their view of logic will be inconsistent with the kind of pluralism we have endorsed. An intuitionistic logician is a practising monist when 1 We

thank an anonymous reader of this volume for putting the issue so starkly.

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she takes it that intuitionistically invalid inferences are to be rejected uniformly, and that all deductively valid arguments are intuitionistically valid. However, this position is not an essential component to endorsing intuitionistic logic. One is free to endorse intuitionistic logical consequence, and to agree that the law of the excluded middle is not an intuitionistic tautology and is not without any constructively significant content—without thereby concluding that the law of the excluded middle is possibly untrue. Intuitionistic logicians do not typically invoke such freedom, but the freedom is none the less available. A pluralism cannot endorse everything in a number of different positions when those positions are inconsistent. Something will need to be modified in order to restore coherence. We will discuss some specific worries in §9. *

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Objection: You are not a pluralist about logic. I grant pluralism about logical consequence.2 I see that there are many different consequence relations, each determining a different kind of logical connection between premises and conclusions. However, it does not follow that you are a pluralist about logic, because logic is the study of all inferential relations between premises and conclusions. You have given us no reason to be pluralist about logic, only about logical consequence. Yours is not a genuine pluralism. There is no rivalry between different logical consequence relations, because they are all aspects of one great ‘logical reality’. Reply. We grant the point that we have defended pluralism about logical consequence, but not pluralism about ‘logic’ understood as the study of consequence relations. We are not pluralists about logic, understood in that sense. In fact, it is difficult to see how one could sensibly be a pluralist about logic in that sense, other than by individuating the discipline very narrowly. Recall that we are not relativists about logical consequence, or about logic as such. We do not take logical consequence to be relative to languages, communities of inquiry, contexts, or anything else. We do not take logic to be relative in this way. We are pluralists about logical consequence because we take there to be a number of different consequence relations, each reflecting different precisifications of the pre-theoretic notion of deductive logical consequence. This is a pluralism, not a relativism. We do not take different logics to be rival analyses of the one fundamental notion (of logical consequence) because we think that the one fundamental notion of logical consequence can be made precise in different ways, each of which sharpens and disentangles that notion in different ways. These different relations are not in competition and they are not rivals. (What could it mean for two binary relations to be rivals?) Pluralism about logical consequence requires rivalry no more than pluralism about ethics or 2 Thanks to Koji Tanaka and Kevin Klement for pressing this point in person and in correspondence.

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about electrons. (Ethical pluralism is the claim that there are different, irreducible, ethically relevant considerations or grounds for ethical norms, that there is not one unique ground of ethical normativity—a controversial meta-ethical claim [124]. Electron pluralism is the claim that there is a plurality of different electrons—an uncontroversial physical claim.) To require rivalry in a pluralism is to mischaracterise it. *

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Objection: What is a case? You say that Tarskian models count as cases. You say that possible worlds (construed as complete and consistent) count as cases. You say that situations (be they incomplete, inconsistent, or both) count as cases. You say the same about constructions; they count as cases. But what, exactly, is a case?3 Reply. Cases, whatever else they are, are ‘things’ in which claims may be true. By specifying ‘truth conditions’ for claims, you thereby specify cases. Whether your resulting class of cases affords an admissible instance of gtt is a separate, important question, but it is independent of whether you have specified a case. We do not pretend to have given precise individuation conditions for cases. We are not sure that such conditions can be given. However, we have said enough to indicate that we are moderately liberal with respect to what counts as a case: for any set of sentences, there is a case in which those sentences are true. If those sentences are compossible, then this is a possible case, or a way things could be, and if those sentences are not compossible, then this is an impossible case, or a way things couldn’t be. Now, one might be extremely liberal with respect to cases, and hold that for every class of claims there is a case in which those and only those claims are true [87]. We are not fully liberal in that sense. Our moderate liberalism allows for a lot, but it need not be committed to all manner of ‘cases’ for arbitrary gerrymandered sets of sentences. Given our moderate liberalist view of cases (as above), it is straightforward to respond to the following kind of objection, which aims to exploit our liberal position. Call a c-case a case in which A → B is true if and only if A is true. Define c-validity using the class of all c-cases. The argument from A → B to A is c-valid, yet nobody thinks that this argument is valid in any sense.4 It is important to understand how a pluralist can respond to this objection, so we will cover it in some detail. First, we must attend to the conditional operator →. Either it is to be understood as a conditional or it is not. If it is not, then there is no problem in endorsing the validity of the argument from A → B to A. The validity clashes with our common sense if we take ‘→’ to have some predetermined meaning. So, ‘→’ is to 3 Geoff

Goddu raises this point [56]. to Koji Tanaka for presenting this objection.

4 Thanks

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be understood as some kind of conditional. If it is a conditional, then we are not committed to there being c-cases as is defined in this objection. The trivial case, in which every sentence is true, is a c-case, and so is the empty case, in which no sentence is true, but it is hard to find any others unless we are committed to extreme liberalism. For example, provided that all identity conditionals A → A are necessarily true, in any possible world some claim is not true, then no c-case is a possible world. (For any world w, let B be a claim not true in w, then B → B is not true in w, which is a counterexample to the c-case condition for the arrow.) Given this fact, c-consequence cannot be strongly endorsed (none of the c-cases can be the actual world), so it is unclear why c-consequence might have any interest at all, and it is completely unclear as to how c-consequence could be an admissible instance of gtt. It takes more than simply specifying a class of cases to provide a genuine consequence relation, so c-consequence does not provide a counterexample to logical pluralism. *

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Objection: Is logical pluralism true but uninterestingly so? If cases are as you say, then everyone ought to be a logical pluralist merely in virtue of the propositional and predicate divide.5 After all, let casesp be propositional (classical, Boolean) valuations. Such casesp blot out any quantificational structure, but they count as cases by the minimal criteria you’ve specified. But, then, the gtt account in terms of casesp yields a different consequence relation than the tm account (in terms of Tarskian models); for some arguments, the two accounts give different answers. But surely, logical pluralism is not that easy! Surely, logical pluralism purports to be more interesting than that! Reply. We agree that even the rather boring example between propositional and predicate models is sufficient for pluralism; however, we are not sure that everyone will be happy to acknowledge that each of the given instances of gtt gives a true but different answer to whether, say, » All cats are happy. Max is a cat. Therefore, Max is happy is valid. Moreover, we do not want to push that example too far, as we do not want to suggest that pluralism turns on controversial issues concerning level of (formal) analysis. (It doesn’t so turn.) That said, we are happy to accept the example as a rather uninteresting example of pluralism, but we leave aside the question of whether it is an ‘easy’ route towards pluralism. Suppose, though, that you do take the given ‘easy’ route towards pluralism. Well and good. By our lights, the more interesting aspect of pluralism goes 5 Geoff

Goddu [56] raises this point.

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beyond the boring cases; it accepts the range of cases that we have canvassed in earlier chapters, yielding consequence relations that go beyond classical logic (or beyond relevance and back to classical logic, as the case may be), and so on. *

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Objection: Transitivity and reflexivity. You parade your pluralism not only as the best approach to logical consequence, but also as a framework in which to understand contemporary logic. The trouble is that your pluralism does not clearly afford understanding of the whole scene. For example, what, if anything, does pluralism say about ‘logics’ in which consequence is not transitive? Presumably, gtt yields only transitive accounts of consequence; if so, then the ‘logics’ in question are not real logics after all. But if they aren’t logics, what are they? And if they are logics, how can they be, given that, again, gtt seems to yield only transitive accounts? Regardless of how gtt’s cases are spelled out, gtt still yields only preservation-type accounts of consequence. In this context, the requisite sort of preservation is essentially transitive. Reply. Logic is in the business of characterising consequence relations, and such relations arise from specifications of gtt’s cases. As the objection notes, gtt yields accounts in which consequence amounts to preservation, truth-preservation over worlds, situations, or constructions. In a significant sense, varying the logic is a matter of varying how much is to be preserved—or, to put things differently, varying the range over which truth is to be preserved. The point of the objection concerns an apparent corollary: pluralism, given in terms of gtt, seems to have it that consequence must be reflexive and transitive, no amount of varying kinds of cases can vary this. But, then, as the objection states, our pluralism rules out some activity in contemporary logic. For example, the Martin–Meyer system S-for-syllogism, which rejects the inference from A to itself (on grounds of circularity), is ruled out given the lack of reflexivity [75, 79]. Moreover, Tennant-style ‘relevant’ logics, which reject transitivity, likewise fail to fall under the banner of logical consequence given in gtt [129]. What can we say? We hold the line. The given kinds of non-transitive or irreflexive systems of ‘logical consequence’ are logics by courtesy and by family resemblance, where the courtesy is granted via analogy with logics properly so called. Non-transitive or non-reflexive systems of ‘entailment’ may well model interesting phenomena, but they are not accounts of logical consequence. One must draw the line somewhere and, pending further argument, we (defeasibly) draw it where we have. We require transitivity and reflexivity in logical consequence. We are pluralists. It does not follow that absolutely anything goes. *

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Objection: One true logic after all? You have indicated that there are different ways that ‘case’ can be interpreted in gtt. But gtt has a universal quantifier in the front; it says that an argument is valid if and only if in all cases in which the premises are true, so is the conclusion. Is not real validity, then, preservation of truth across all cases? Will that not mean that the true logic is the intersection of all logical systems given by gtt? You have one true logic after all. Reply. We think that that conclusion should be resisted. One reason is that we see no place to stop the process of generalisation and broadening of accounts of cases. For all we know, the only inference left in the intersection of (unrestricted) all logics might be the identity inference: From A to infer A. That identity is the only really valid argument is implausible and, we think, an unmotivated conclusion. Consider what ‘universalism’ about logic involves. To say that an argument is invalid according to logicx but valid according to logicy is to say that there is a casex according to which the premises are true and the conclusion is not, but that there is no casey in which the premises are true and the conclusion is false. The universalist says that the argument is really invalid. If this is simply a matter of preference for one logical consequence relation over another, then this is no case against pluralism. If, on the other hand, it is to say that the inference is mistaken, and that anyone who endorses it as valid makes a mistake, then the claim presumes too much without any further detail. Take an example where logicx is classical first-order predicate logic and logicy is classical second-order logic. There are arguments which are second-order-valid but first-order invalid. If ‘universalism’ is correct, then the position of a pluralist about first-order and second-order logic (who endorses both second-order and first-order validity as ways of making the pre-theoretic notion of consequence precise, both with their uses in the evaluation of the validity and invalidity of arguments) is mistaken. But it seems to us that simply asserting that ‘the right’ consequence relation is the one with the largest collection of cases has no grip in this case. There are first-order counterexamples to second-order valid arguments; those counterexamples tell us something important about the transitions made from premises to conclusions in those arguments; and the first-order invalidity of those arguments is a genuine invalidity. Yet, those arguments are also second-order valid, and this sense of validity necessarily preserves truth, is formal, has a use in regulating reasoning, and bears all of the marks essential to consequence relations, and many other theoretically important features besides. That another consequence relation, with its own virtues, more tightly draws the line between valid and invalid arguments seems to cut no ice against the fact that is also a consequence relation. The only way that this objection can have any force is if the presumption against pluralism has already been made. *

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Objection: Validity appropriate to situations. How do you avoid Priest’s argument that pluralism narrows down to monism in a radically different way? The argument runs as follows. We often reason about some sitation or other; call it s; suppose that s is in different classes of situations, say, K1 and K2 . Should one use the notion of validity appropriate for K1 or for K2 ? We cannot give the answer ‘both’ here. Take some inference that is valid in K1 but not K2 , α  β, and suppose that we know (or assume) α; are we, or are we not entitled to accept β? Either we are or we are not. [Footnote: It could, I suppose, be maintained that there is no fact of this matter; that both answers to the question are equally correct. But this is relativism (about truth, and so about validity). And b&r6 maintain that their pluralism is not a relativism.] A natural reply is that we should use the notion of validity appropriate to the smallest class of situations that s is in: in this case, presumably K1 ∩ K2 . But if we should, indeed, apply the notion of validity appropriate to the smallest class that s is in, then we should apply the notion appropriate to {s}. Thus, the valid inferences are those that have a premise false in s, or whose conclusion is true in s. In other words, it is now pluralism that has become vacuous. [94]

Reply. Priest’s argument proposes to reduce pluralism to vacuity. Take the situation as Priest describes, in which α is true in the situation s, and the inference from α to β is K1 -valid (without loss of generality, let K1 be the class of all worlds and the resulting logic, classical) but not K2 -valid (take K2 to be the class of all situations and the resulting logic, relevant). If α is true in s, and if s is a member of K1 , then, by the K1 validity of the inference from α to β, it follows that β is true in s. That is not at issue. The pluralism in our position comes from the plurality of relations of logical consequence, not any plurality about what is true in a case. Priest’s way of construing the role of logical consequence in the evaluation of arguments seems misconceived. If we are reasoning about what is true in some situation s, then we are, presumably, entitled to use any information about that situation that we have at hand. Suppose that all we have is an argument from α to β, and a situation s such that we know that α is true in s. If K is a class of situations such that we know that s is in K, then if we apply K-consequence, then any sentence following from α is also true in s. That applies even if we take K to be {s}, in which case the {s}-consequences of α are exactly the sentences true in s. But if we do not know exactly what is true in s in the first place, then we do not know what {s}-consequence is! The more we know about the situation s, the more we can say about {s}-consequence. All that Priest’s argument shows is that the more we know about s, the more we know about what is true in s. But that, we think, is not altogether surprising. * 6 In

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this quotation ‘b&r’ denotes Beall and Restall.

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Objection: Warrant and entitlement. There is another argument that pushes towards monism. After all, there is a single relation of preservation of warrant or entitlement, and that singles out a unique consequence relation. Take the premises of an argument. If one has warrant to believe those premises, is the conclusion warranted too? Even if there’s an ambiguity in the notion of ‘follows from’ which can be made precise in a number of ways, there is a single relation which tracks the preservation of entitlement, and you can’t be a pluralist about that.7 Reply. By way of answering the objection, we will invoke Priest’s example (above). An example of our pluralism about the evaluation of a particular argument goes thus: the inference from α to β is not K2 -valid, and as a result, there is some situation in K2 in which α is true and in which β fails. For concreteness, let us suppose that β is γ ∨ ∼γ for some γ irrelevant to α. Then, indeed, there is a situation in which α is true but in which γ ∨ ∼γ fails. The inference from α to γ ∨ ∼γ is valid in the usual classical sense: if α is true, then of necessity γ ∨ ∼γ is true. There is no possibility (that is, no possible world) in which α is true and in which γ ∨ ∼γ fails. So, we are classically entitled to infer γ ∨ ∼γ from α. But, of course, that inference is not as ‘good’ as others in which inference is relevant. In the given inference, α has done nothing to contribute to β, a fact recorded in our semantics by the existence of a situation in which α holds but in which γ ∨ ∼γ fails. That situation is not our original s, of course, and it is not in the class K1 of worlds, but it none the less contributes to the relevant invalidity of the argument at hand as applied to s. The argument is invalid; so, we are not relevantly entitled to infer γ ∨ ∼γ from α. We conclude, then, that in an important sense, whether we are entitled to infer β is ambiguous; it has more than one answer. This is not a pluralism or a relativism about truth, since it is the plurality in the notion of entitlement that is doing the work. We will say more below about whether our pluralism is in any sense a relativism about truth. Here, it is enough to note that the situation is no more relativist than in the case of a deductively invalid but inductively strong argument. Suppose that we believe the premises of an arbitrary argument. Are we entitled to infer the given conclusion? That depends on how high the bar for entitlement is placed. In some reasoning contexts (mathematical reasoning, for example), it makes sense to raise the bar for entitlement very high, but in others a much looser standard for entitlement seems appropriate. As far as we can see, there is nothing in the notion of entitlement that will help draw the distinction between inductive and deductive logic, or serve to dictate a unique relation of deductive consequence that is properly favoured for capturing entitlement. 7 Thanks to Gary Kemp and Stephen Read for both putting this objection in discussion in February 2000.

General Objections *

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Objection: Warrant and disjunctive syllogism. Even if I grant that the bar for the presevation of warrant can be set at various levels, it doesn’t seem to vary as much as the pluralist requires.8 Consider the use of disjunctive syllogism in the famous case of the dog. Once there was a dog, pursuing a man down a bush track. The dog follows the scent until she gets to a fork in the track. The dog believes that the man went down either the left fork (call this proposition A) or the right fork (call this proposition B). So the dog believes A ∨ B. The dog sniffs for a while down the left fork, and detects no scent. The dog thus also believes ∼A: the man did not go down the left fork. Without hesitating to check for the scent down the right fork (and thus, without trying to find any independent evidence or warrant for B), the dog continues her pursuit down the right fork in the track, inferring B, the claim that the man has gone that way [1, 2, 22]. On this account of the dog’s behaviour, she is reasoning—and, indeed, she seems to be reasoning well. She has used the inference disjunctive syllogism, which has the following form: A∨B

∼A

B

Apparently, and independently of rejecting it in the case of inconsistent premises, disjunctive syllogism seems to preserve warrant in the reasoning of the dog. If you think that there is another sense in which disjunctive syllogism is valid, then you must admit that relevant logic is too weak, and the preferred logic is irrelevant. Reply. We seem to have a genuine problem. It seems that we do have a classical inference which preserves warrant in this important sense. Will that not single out one logic as the logic that preserves warrant in all reasoning situations? Will that not deserve to be called the one true logic, since it seems best placed to guide reasoning? If warrant is preserved in cases such as the dog’s reasoning, then this seems to point to a logic stronger than relevant logics as appropriate for guiding and analysing reasoning. The initial appeal of pluralism threatens to wear thin. Relevant excursus: It might be thought that this is not really a worry for the proponent of relevant logics, because, even though disjunctive syllogism isn’t relevantly valid, disjunctive syllogism is not all we have at play here. After all, the dog does not merely know that A ∨ B and know that ∼A: the dog also knows that it’s not the case that A and ∼A. This might be the case. However, it’s not at all obvious how this is going to help, because the argument A∨B

∼A

∼(A ∧ ∼A)

B 8 We

thank Stephen Read, especially, for pursuing this point.

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is no more relevantly valid than is disjunctive syllogism. (After all, the third premise ∼(A ∧ ∼A) is relevantly entailed by the second premise ∼A, at least in most mainstream of relevant logics, so adding it really does not do much for the argument.) The problem, of course, is that, in inconsistent circumstances in which A and ∼A are both true, so is ∼(A ∧ ∼A). So the truth of ∼(A ∧ ∼A) in a circumstance is not enough to ensure that this circumstance is a consistent one. To patch disjunctive syllogism into a relevantly valid argument, you must do one of two things. First, you could do what you do with any invalid argument to make it valid. Add an extra premise dictating that if the premises are true, so is the conclusion. In this case, (A ∨ B) ∧ ∼A → B, where this ‘→’ is a relevant conditional satisfying the relevant validity of modus ponens. Perhaps one could say that the dog endorses this conditional, though saying so is perhaps pushing the bounds of canine cognitive capacities. On the other hand, one could add a second conclusion to the argument, giving this: A∨B B

∼A

A ∧ ∼A

which is relevantly valid. Any situation in which A ∨ B and ∼A are true is one in which either B or A ∧ ∼A are true. If we have, therefore, a warrant for the dog to reject A ∧ ∼A (as opposed to a warrant for the dog to accept ∼(A ∧ ∼A)), then, perhaps, one has a relevantly valid argument that goes some way towards representing the dog’s reasoning. Still, taking such an analysis is not without its costs. It seems quite plausible that there be cognitive agents with reasoning and representing capacities sufficient for representing A ∨ B and ∼A and B, without their being able to consider more complicated claims (such as conjunctions of a claim and its negation A ∧ ∼A, or relevant conditionals). In this case, the cognitive state of the agent making the inference is best modelled using the original argument (which was disjunctive syllogism) rather than any modification of it. If we have such an agent, then it seems as if there is an appropriate sense in which the original argument was a good one, even though it was classically (and not relevantly) valid. End of relevant excursus. By way of responding to the current objection (and with the given excursus out of the way), let us suppose, for the sake of discussion, that there is a single level at which the bar of preservation of warrant ought to be set. Perhaps, contrary to appearance, there is a single standard for the preservation of entitlement in arguments. Whatever the standard may be, it still insufficiently settles a relation of logical consequence resulting from the distinctive features of entitlement and warrant. Preservation of entitlement is not guaranteed by logical consequence, no matter how logical consequence is distinguished, because there are logically valid arguments (at the very least, according to classical predicate logic, and other consequence relations we take to be widely accepted in the community) that are

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not known to be valid. There are many open questions in number theory, for example, and it would be a great advance to know whether or not they are consequences of accepted theories such as Peano Arithmetic. There are statements to which we have no entitlement but that are none the less consequences of theories to which we are entitled. The only way that logical consequence preserves entitlement is when one is entitled to accept that such and so is a consequence, and when we are entitled to strongly endorse that consequence (see §7.4.4). But those conditions do not single out a particular logic. In general (but not always),9 if we are entitled to the conditional ‘if α then β’, then if we are entitled to α we will be entitled to infer β. That will be the case even when the conditional is contingent and not ‘logical’ in any sense at all. The transfer of entitlement does not single out a consequence relation, as entitlement can be transferred in many ways from premises to conclusions, not only (and not always) though relations of logical consequence. But, then, considerations of warrant preservation fail to determine a consequence relation at all, and so the fact that warrant preservation reaches beyond relevant consequence cuts no ice. Warrant preservation is important, and consequence relations have a great deal to tell us about the ways in which warrant is preserved. None of that, however, singles out a unique consequence relation. *

*

*

Objection: Meaning theory. If there is more than one kind of case in which things are true, then there is more than one kind of condition in which things are true. But truth conditions determine meanings. Pluralism about logical consequence then becomes pluralism about meaning. If you are not pluralists about meaning, you should not be pluralists about logical consequence. Graham Priest puts the problem this way: [Beall and Restall argue that] we can give the truth conditions for the connectives in different ways. Thus, we may give either intuitionist truth conditions or classical truth conditions. If we do the former, the result is a notion of validity that is constructive, that is, tighter than classical validity, but which it is perfectly legitimate to use for certain ends . . . We can indeed give different truth conditions. But the results are not equally legitimate. The two give us, in effect, different theories of vernacular connectives: they cannot both be right. [94]

Reply. We will examine the case of negation in constructive and classical logic, as it will illustrate the matter clearly. We take both of the following clauses to be explications of the behaviour of negation: » ∼A is true in a world w if and only if A is not true in w. 9 We say ‘in general’ because there are counterexamples. One’s entitlement to α might conflict with one’s entitlement to the conditional ‘if α then β’. Recall the discussion of the preface paradox (§2.4).

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» ∼A is true in a construction c if and only if there is no construction d extending c in which A is true. Without doubt, the clauses provide different accounts of conditions under which a negation is true. For Priest, it follows that we have different theories of the one connective, and that they cannot be correct. At most, one—no more than one—can be a true account of the behaviour of negation. That would be a fatal objection to pluralism if it were the case. How can each clause for negation be equally accurate? The clauses can both be equally accurate in exactly the same way as different claims about a thing can be equally true: they can be equally true of one and the same object simply in virtue of being incomplete claims about the object. What is required is that such incomplete claims do not conflict, but the clauses governing negation do not conflict. The classical clause gives an account of when a negation is true in a world, and the constructive clause of when a negation is true in a construction. Each clause picks out a different feature of negation.10 A model with both worlds and constructions must address the relationship between them. We may as well take worlds to be a special kind of construction: worlds decide every statement as true or false, so they do the job of final constructions, those constructions that are not extended by any other constructions. So, our model will contain a family of constructions, some of which are worlds. The constructions are ordered by the partial order of extension, which we represent by ‘ ’. Worlds are endpoints in the ordering: if w is a world, there is no c = w where w c. We could say more about the relationship between constructions and worlds, but we need not. The story, irrespective of further details, is consistent. In all constructions we use the one clause for negation: ∼A is true in c iff A is not true in d whenever c d, in which case worlds behave as we expected them to. If w is a world, then ∼A is true in w if and only if, for each construction c  w, A is not true in c. But w is the only construction extending w, so the condition is simply that A is not true in w. Choosing worlds to be final constructions gets the condition exactly right. The point is that the different semantic clauses for negation are compatible. Do considerations of meaning theory make us choose one kind of clause over another? The natural choice might be to fix on the most universal clause possible, the behaviour of negation in all cases whatsoever. Only then, perhaps, is a comprehensive story of meaning told. Even if we grant that point, it does not follow that the universal consequence relation is the only admissible instance of gtt, as we have already discussed. *

*

*

10 Or one could just as truly say that each clause picks out a feature of the objects described, be they worlds, constructions, or situations.

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Objection: Desert island. Suppose you were to be stranded on a desert island and you could take only one logic with you. Which of the plurality of your logics would that be?11 Reply. Pluralism is not inconsistent with there being favourites among the plurality of logics. We could simply say: classical first-order logic is the answer to each of your questions, and that would not touch pluralism. Or we could say that intuitionistic higher-order logic, or some other system, is the logic we would take to an island. The mere fact that a single logic has a particular role to play (in a particular context) does not mean that other things are not also logic. To cause problems for pluralism, one needs to show that a given job (use in presentation of fundamental theory, or something akin to it) is an essential characteristic of consequence; that is, one needs to show that a given application is required of any admissible instance of gtt. As above (and throughout), we see no reason for imposing such a constraint. *

*

*

Objection: What logic are you using now? You take all of your arguments to be valid, in some sense or other. Which sense might that be? In other words, your fundamental theory will presumably be based on a logic. Which logic? Any answer to such questions seems to lead you to monism.12 Reply. The current objection is in many ways similar to the desert island objection, with which we have dealt. We will take the current objection as a question about the validity or otherwise of our own reasoning. As anyone who applies formal logics knows, the fit between deductive validity and analysis of actual reasoning is not always an easy one; however, some useful things can be said about the connection. The pluralist claim is that, given a body of informal reasoning (that is, reasoning not produced in a particular system of logic), you can use different consequence relations in order to analyse the reasoning. As to which relation we wish our own reasoning to be evaluated by, we are happy to say: any and all (admissible) ones! Our arguments might be valid by some and invalid by others, good in some senses and bad in others. But that is not the end of the story. Once we learn that our argument is bad in some sense—for example that a verification of some premises will not itself be a verification of the conclusion—we will not necessarily thereby reject the usefulness of the argument. It depends, of course, on whether the given kind of verification preservation is important to the task at hand. Similarly, one might say ‘Look, here is a Tarskian model of first-order logic which makes your premises true but conclusion false.’ What should we do in response? Well, it depends. We may well 11 Thanks 12 Thanks

to Gary Hardegree for suggesting that we be stranded on a desert island. to Mike Dunn and Ed Zalta for raising this objection.

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revise our given argument if the model brings to light some equivocation of terms in our reasoning, or some other flaw that indicates that the reasoning does not set out what we wished it to do. Alternatively, we might be happy with our reasoning; perhaps the classical invalidity is merely an artefact of the expressive weakness of classical first-order logic. *

*

*

Objection: Pluralism about logical truth. Pluralism about consequence is one step away from pluralism about logical truth.13 Does it make any sense at all to say of a particular statement, for example one of the form A ∨ ∼A, that it is a logical truth according to one logic but not according to another? Is the pluralist about logical consequence a pluralist about logical truth? Doesn’t that make you a pluralist about truth too? After all, are not logical truths true? And what would it mean to be a pluralist about truth? Reply. We are pluralists about logical truth, but not necessarily pluralists about truth.14 It will be useful to explain why that is the case. There are many different things one could mean by ‘logical truth’ but, given our account of consequence, a sensible thing to mean by ‘logical truth’ is straightforward. Given the logical consequence relation defined on the class of casesx , the logicalx truths are those that are true in all casesx . If you like, they are the sentences that are x-consequences of the empty set of premises. The logicalx truths are those whose truth is yielded by the class of casesx alone. Since we are pluralists about classes of cases, we are pluralists about logical truth. Some statements (like A ∨ ∼A) are true in all worlds but not true in all situations. On that reading, A ∨ ∼A is a classical logical truth, but not a relevant logical truth. Parenthetical remark: That account of logical truth for relevant logics is not the only one. For example, in standard relevant logic, A ∨ ∼A is a logical truth, despite the fact that in the semantics A ∨ ∼A is not true in every situation. A ∨ ∼A is true in every normal situation, where normal situations satisfy conditions not required of arbitrary ones. In the standard semantics for relevant logic, logical truth is defined as truth in all normal situations. In our use of the term ‘relevant logical truth’ above, we do not make that assumption. None of this is to say that we reject the standard notion of logical truth in relevant semantics as interesting and important. Instead, all we claim is that the notion of logical truth as truth in all situations whatsoever is also a useful notion. Truth in all situations differs much more drastically from truth in all worlds than does truth in all normal situations. In relevant semantics, normal situations are much more like worlds, as they are 13 Thanks

to Stephen Read for pressing this objection. of us (jcb [10]) is sympathetic with versions of what Crispin Wright calls truth pluralism, which is simply that there are different notions of truth at play in English; however, that issue is entirely orthogonal to our logical pluralism and the current issue of logical truth. 14 One

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complete and, in some models, consistent [107, 116, 117]. This conception of logical truth (for relevant logic) is much closer to the standard classical conception than is the concept we favour. Far fewer things are true in all situations, so truth in all situations is much harder to come by than truth in all normal situations or truth in all models. So, if we can understand an account of pluralism about logical truth in which different kinds of logical truth can vary so widely, then so much the better. Anything we say for our account of logical truth for relevant logics will apply in the more conservative case where relevant logical truth is much more like classical logical truth. End of parenthetical remark. We take it that what is true is what’s true in the actual case.15 We are not pluralists about truth in any controversial sense. (Of course, we are pluralists about truth in a straightforward and non-controversial sense; we take it that there are many different truths! But that is not a controversial claim at all.) So, what is logicallyx true is actually true if the actual case is one of the casesx . The logicalx truths are actually true in just those cases where logicx can be correctly (strongly) endorsed, which is when the actual case is in the class of cases. None of that, we think, is very surprising. Of course, the foregoing sense of ‘logically true’ is not the only one that is worth our attention. One might think that what is logically true is what can be found to be true on the basis of logic alone. In that case, we (qua pluralists) have another way we could read the criterion for logical truth: a claim A is logically true if and only if, for some class X of cases determining a logic, A is true in every case in X. This alternative definition differs from the plural reading of the term; a logical truth (no subscript for a specific logic) is one that is a logical truth for some logic or other. Given this sense of ‘logical truth’, there is no pluralism about logical truths due to the variation of logics, because the variation among logics is ‘factored out’. It seems to us that this, too, is a sensible notion of logical truth (even though determining the precise boundary of properly logical consequence relations is very difficult), and this notion seems univocal. Parenthetical remark: This might be thought to raise a worry for us. If we are to accept this univocal notion of logical truth, does this not parallel a univocal notion of logical consequence? An argument is logically valid (in this new sense under discussion) if and only if it is logically validx for some choice of x. This is a univocal sense of logical consequence, and perhaps this makes us a monist. Does it? We don’t think it does. The fact that one can find one particular class out of a plural collection (their union) does not make one a monist about that plurality. Yes, a perfectly singular sense for ‘logically valid’ can be found by taking the union of the plural senses ‘logically validx ’, but for that to entail monism, we would need to 15 As discussed before, we take there to be a case in which all and only those things which are true are true. This is not absolutely required, but it certainly makes things more straightforward.

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ensure that none of the other validx s count as logical consequence. This consideration provides no new argument to this end. End of parenthetical remark. It follows that we can be doubly pluralist about logical truth. We can think of logical truth plurally in a straightforward sense: A is a logicalx truth if and only if A is true in all casesx , and x may be cashed out in various ways. On the other hand, we can say that A is a logical truth (no subscript) if and only if A is a logicalx truth for some logic x. This pluralism about logical truth is due to the plurality of logics and in no way comes down to a pluralism about truth simpliciter. *

*

*

Objection: Carnapian pluralism. I admit that there are many different consequence relations, but this is due to the fact that there are many different languages. If you wish to use a language without classical negation, or with a relevant conditional, that’s your prerogative. Define your language, and you have a notion of logical consequence. There is nothing more and nothing less to logical pluralism than the freedom we have to construct a language. Reply. We need not spend long on this objection because we have seen it before, in §7.3. A broadly ‘Carnapian’ approach surely does motivate a kind of logical pluralism, but it is not our target [109]. A more robust pluralism takes there to be different relations of logical consequence because these relations can differ in their evaluation about the one and the same argument, not merely because different relations relate different things (consequence 1 evaluates arguments in language 1, consequence 2 evaluates arguments in language 2, etc.), but because there are different relations relating the same things. In addition to the discussion in §7.3, we will add just one further point. The segregation of logics to individual languages is never going to suffice for a general picture of logical consequence because of the scope for crosslinguistic evaluations. We can take a premise from language 1 and a premise from language 2 and see how they lead to a conclusion from language 3. Consequence relations, while they are used to evaluate connections between claims expressed in some language or other, cannot be restricted to just that language. *

*

*

Objection: Pluralism
. I am happy to say that there is a general notion of consequence that can be instantiated in different ways depending on how you understand ‘case’. However, suppose that someone wants to say that one instantiation of case is special in that only by using it does one define Real Consequence, and instantiation on the others yields only

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consequence-like notions: Consequence
, Consequence

, and so on. The dispute between this kind of monist and a pluralist appears to be merely verbal, and as a result pluralism does not seem to be a distinctive position.16 Reply. Suppose that someone (call her Ms X) acknowledges that there are different instantiations of case that give rise to various consequence-like relations, but thinks that only one of those yields Real Consequence. Where is the disagreement, if it’s not merely verbal? The disagreement, we think, surfaces along one of the following lines. Either » Ms X rejects that the core of follows from (or consequence) is settled only up to gtt (that is, is unsettled to the degree that it fails to determine a unique relation) or » Ms X objects to the admissibility criteria on instances of gtt (§3). In short: if one agrees that the core of consequence is captured by gtt— that it is settled only up to gtt—and that gtt does not uniquely determine a particular instantiation of case, then one will take Ms X’s line only if one imposes further constraints (or different constraints) on admissible instances of gtt. As the objection is given, Ms X agrees that the pretheoretic notion affords different consequence-like relations. But why only consequence-like? What, in other words, motivates the ‘stars’? As far as we can see, there is no clear motivation at hand, other than perhaps a desire to use the term ‘logical consequence’ univocally and to not disambiguate it in a number of different ways. Pending such motivation, we see no reason to ‘star’. That said, we are not terribly troubled by Ms X’s ‘monism’. She seems to agree with us up to the ‘stars’, and in that sense the ‘disagreement’ may well be merely verbal. It may well be that there is a simple translation manual (involving the insertion or deletion of stars) between her vocabulary and ours. But there is a difference between our positions: the difference, as above, is that in her vocabulary the ‘stars’ reflect a distinction without a difference. Furthermore, we can resist the only motivation we can find for ramifying the vocabulary in this way: the simple desire for a univocal precisification of the notion of logical consequence. The picture we have in mind is one in which the pre-theoretic notion of consequence is both overand under-determined. It is over-determined in calling for, considerations which cannot be satisfied together. Logical consequence calls for features of relevance between premises and conclusions, and, for the possibility of counterexamples, that an argument is invalid only if it is possible that the premises be true and the conclusion false. Logical consequence calls for 16 Thanks to Hartry Field and Graham Priest for pressing the issue, and to Allen Hazen for helpful discussion.

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both the axiomatisability of the consequence relation, and the expressibility of full second-order consequence. In this way, logical consequence is over-determined. It is under-determined because there is no unique relation which maximises all of these desiderata. Different choices can be made to define different relations. The best course through this under- and over-determination, we think, is to be pluralist. Admit that the unsettledness of logical consequence affords different, legitimate precisifications of the core notion. Ms X agrees with that, but none the less she wants the pre-theoretic notion to specify a unique relation, and for that ‘reason’ proceeds to ‘star’. But what she needs are simply the relations that fall out of gtt. gtt won’t always give you what you want; but it gives you what you need, and there is no additional need to ‘star’. * Objection: Logical form. logical consequence:

*

*

‘The following, I take it, is a truism about

(lc) A sentence S is a logical consequence of a set Γ just in case the logical forms of S and the sentences in Γ suffice to guarantee that either S is true or some sentence in Γ is not. . . . Each of the predicates (1) ‘. . . is true’, (2) ‘. . . is the logical form of . . . ’, and (3) ‘. . . suffices to guarantee . . . ’ has a unique (i.e. parameter-free) and determinate extension, as used in (lc). Therefore, since (1)–(3) are the only potentially problematic expressions in (lc), (lc) delivers a unique, determinate class of logically valid arguments.’ [97]17 Reply. This argument, we think, breaks down in a number of different ways. We will briefly sketch all of the options for a pluralist in resisting the conclusion. First, and foremost, the pluralist can deny (lc) as both an analysis of the notion of logical consequence and as a statement determining its extension. The condition (lc) rules out any relevant account of logical consequence, because any statement true on the basis of its form alone will be the conclusion of any and every argument. Relevance is not a recent matter, even though the contemporary theory of relevant consequence is quite new. The earliest developed example of logical theory, Aristotle’s syllogistic, itself is sensitive to conditions of relevance. For example, syllogisms with inconsistent premises or with tautologous conclusions are not always valid. » All Fs are Gs Some Gs are Hs Therefore, all Is are Is 17 Thanks

to Augustín Rayo for sending us the manuscript containing this objection.

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is not a valid syllogism (on Aristotle’s analysis), despite being deemed as correct by (lc) if the ‘all . . . are . . . ’ construction is preserved in logical form. Similarly, » All Fs are Gs Some Fs are not Gs Therefore, all Hs are Is are also not valid. None the less, on the basis of the form of this argument we can see that the premises are never jointly true, and as a result either the conclusion is true or one of the premises is false. Now, Aristotelian syllogistic is not a straightforward theory of logical consequence. For one thing, it is merely a categorisation of validity and invalidity of arguments in syllogistic form. None the less, it is good evidence that relevance is to be found at the earliest juncture in the development of logic. One need not think that Aristotle is entirely right on the matter, in order to acknowledge that there is at least a strong, initial appearance that valid syllogisms share subject matter from premises to conclusions. In accepting (lc) as the recipe for logical consequence, one not only ignores the apparent criterion of relevance; one defines it away at the stroke of a pen. There are further problems with (lc) as it stands. Relevantists like Aristotle and Anderson and Belnap are not the only ones who are disenfranchised by (lc). So too are intuitionists, since if the argument from A to A is valid, then (lc) has it that either A is true or it is not. (lc), we think, far from being a ‘truism’, is to be rejected as an account of logical consequence, at least as one that purports to capture—or even provide a recipe for—the pre-theoretic notion. Better, we think, to see (lc) either as a limited attempt to capture various features of the pre-theoretic notion, or, more charitably, as an ‘account’ that cares little about the pre-theoretic notion. More, we think, can be said. Even if (lc) is used to determine a subclass among the collection of logical consequence relations, the subclass is not narrowed down to one. Even if we grant that (lc) fixes the meaning of ‘logical consequence’, there is still room for plurality. We think that room for variation is to be found among the terms (2) ‘. . . is the logical form of . . . ’ and (3) ‘. . . suffices to guarantee . . . ’, and not in (1) ‘. . . is true’. Consider the case of second-order quantification. If logical consequence, as determined by (lc), is determinate, then either second-order quantification is (determinately) to be found in logical form or it is (determinately) not. Neither disjunct seems appealing to us. Second-order quantification can be usefully thought of as a logical notion [120], and useful formal accounts of logical consequence rely on taking second-order quantification as non-logical [133]. It is hard to see what considerations could be brought to bear to decide that matter once for all, and pluralism about logical form is at the very least a live option, allowing for room for variation in (lc). Even if we are wrong about (2), it is hard to see that enough sense has

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been given to the term ‘guarantee’ to fix a determinate extension. Rayo [97] attempts to do as much when he says ‘it is natural to think that the logical form of S suffices to guarantee its truth just in case our theory of truth yields the result that S is true, and does so without making use of [an account of how the world is or the meanings of the non-logical terms].’ But that is at most a sketch, and upon examination it does not yield the required precision. Consider, for example, the sentence S of the language of secondorder logic, which is true if and only if the continuum hypothesis is correct. Does the logical form of S guarantee its truth? Well, does our theory of truth yield the result that S is true without making use of an account of how the world is or the meanings of the non-logical terms in S? S has no non-logical terms, and it is hard to see how ‘the way the world is’ might enter into this matter. What we are left with is whether or not S is ‘yielded’ by our theory of truth. That, presumably, is to say that S is a logical consequence of our theory of truth. (What else could it be? We cannot think of anything else that our objector might mean by ‘yields’ here.) It follows that we can be pluralists about ‘yielding’ or ‘guaranteeing’ if we are pluralists about logical consequence. And if we can be pluralists about ‘guaranteeing’, we can be pluralists about logical consequence. A tight circle indeed. A friend of both first-order and second-order logic can say that, for example, the theory of truth does not first-order-yield the continuum hypothesis (or its negation), but that it does second-order-yield the continuum hypothesis (or its negation). Even (lc) yields room for pluralism, contrary to the thrust of the objection.

Chapter 9

Specific Objections We now turn from objections to pluralism as such (the topic of §8) to objections against the particular directions in which we have taken logical pluralism.1 There are a number of objections to classical, intuitionistic and relevant logic which are specific to those logics, and not to the scheme of pluralism as a whole. Since we have argued that it makes sense not only to be a pluralist about logical consequence, but to be a pluralist who endorses each of classical, constructive, and relevant reasoning, we thereby have an obligation to say something about such logic-specific objections. *

*

*

Objection: Coherence and relevant models. There is no sense to be found in relevant models. It makes no sense to talk of a situation in which A and ∼A both hold, because to conjoin A with ∼A is unimaginable; it makes no sense, and it is incoherent. Recall Aristotle: And the most certain principle of all is that about which it is impossible to be mistaken . . . It is clear, then, that such a principle is the most certain of all and we can state it thus: ‘It is impossible for the same thing at the same time to belong and not belong to the same thing at the same time and in the same respect.’ — Aristotle, Metaphysics 1005b12–20

The law of non-contradiction is so thoroughly basic that, if we reason without it, we have no principles for reasoning at all [69]. We have no way of understanding so-called ‘inconsistent situations’, and as a result, relevant or paraconsistent semantics is incoherent and worthless. Reply. This is an objection in which the difference between dialetheists (those who assert that some truths have true negations) and non-dialethic paraconsistentists (those who endorse a paraconsistent logic but who deny that there are any true contradictions) comes into play (see further, §7.4). The current objection arises from a non-dialethic, non-paraconsistentist position. By way of defusing the objector’s worry, we will reply from a nondialethic paraconsistentist perspective, as this is closest to the objector’s own position. 1 We

retain the style of §8, putting the objector in the first person, and so on.

107

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Our first move, of course, is to deny the premise. One need not think that statements of the form A ∧ ∼A are (or even could be) true in order to acknowledge that such statements make sense. At the very least, they are not senseless in the sense of ‘saying nothing’, like unintelligible statements. The problem with inconsistent statements is that they are never true, they are not possibly true. But that they are never true is achieved not by being without sense, but rather by being replete with sense. An inconsistent statement (say, of the form p ∧ ∼p) is inconsistent because it says too much, not too little. Inconsistent claims, then, have ‘sense’ at least in the minimal sense of saying something. But another sense of ‘sense’ is at play in the objection, one in which making sense is tied to imagining. The worry is that claims of the form A ∧ ∼A are unimaginable. (That someone might make such a claim, of course, is not at issue.) Imaginability is a very difficult issue to judge, but perhaps we can none the less say something useful. We can agree with the objector that it is impossible to picture what is said in an inconsistent claim.2 If picturing is required for imagining, then we concede that we cannot imagine the kind of inconsistent claims at issue. But that kind of imagining is not required for understanding what has been said in an inconsistent claim. If we understand what it is for something to be a square and we understand what it is for it to be a triangle, then we have what we need to understand what is required for something to be a triangular square: it must be both. It is because we do understand what is said that we see the difficulty in making the claim true. Our very understanding of inconsistent claims is precisely what convinces us that such claims are ‘unimaginable’ in the sense that requires a picture [17]. Another step in the response is to disambiguate the term ‘coherent’. Paraconsistents have often described their position as holding that theories can contain contradictions without being ‘incoherent’, meaning that inconsistent theories can be non-trivial, in the sense of not containing every proposition, as not every proposition (paraconsistently) follows from a given inconsistency. As a term of art, of course, it is acceptable to use the term ‘incoherent’ in that way—as synonymous with ‘non-trivial’—and to hold that inconsistent theories or statements can be coherent. But that is not the only sense of ‘coherence’ worth the name, and it is by no means clear that it is the sense used by the current objector. Another sense of ‘coherence’ is fit together, and the inconsistent statement is incoherent (in the ‘fit together’ sense) because it does not fit together. Whatever is required to make A true conflicts with, does not ‘fit with’, what is required to make ∼A true. We would do better to acknowledge, rather than ignore, the claim that contradictions are incoherent in that sense. Once we have acknowledged it, we can go on to explain it, to show how paraconsistent semantics 2 We can picture a four-sided triangle viewed from its side (as a line), but it seems harder to picture it in such a way that we see that it is four-sided and see that it is a triangle.

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respect that insight. So, how does the paraconsistent semantics discussed in §5 respect the foregoing claims about contradictions, while at the same time resisting the conclusion that contradictions cannot be understood? Recall the evaluation condition for negation at situations: » ∼A is true in s if and only if A is not true in s
for any s
where sCs
. A negation is true in s if and only if it is not true in any compatible situation s
. Inconsistent claims can be true at a situation s when (and only when) the situation s is not self-compatible. If we do not have sCs, then we can have s  A and s  ∼A. What are situations that are not self-compatible? More to the point, are there any such situations? Here, of course, is where the debate lies. We need not provide a conclusive proof that there are non-self-compatible situations (though that would be nice, were one available). For present purposes, it is sufficient to explain how, within the confines of the foregoing discussion of ‘coherence’, ‘understanding’, and so on, such situations are understandable, well-motivated, and coherent. To begin, note that to admit non-self-compatible situations is one way of agreeing with the objector that inconsistencies are incoherent. The compatibility relation on situations models coherence. Situations s and s
cohere if they are compatible, if sCs
. This is to say that there is nothing true in s
is ruled out by s. To say that s is non-self-compatible is to say that s is incoherent in itself, in the very sense of incoherence at hand (failure of ‘fit’). The situation s rules out something that s itself requires. s, qua nonself-compatible (incoherent), clashes with itself; hence, according to the non-dialethic paraconsistentist, s cannot be actual, since what it requires cannot be made true. The paraconsistent semantics does not ignore the claim of incoherence; instead, the semantics provides a way to model that notion of incoherence. By admitting that there are inconsistent situations, we do not thereby admit that they are possible (in the sense of being possibly actual): we maintain (qua non-dialethic paraconsistentists) that they are impossible. Why would we want to admit such situations into our semantics? We have already answered that question in §5, and in §8 we indicated our modest liberalism about cases. That is enough to place the notion of an inconsistent case on the agenda. Still, it may be useful to see how the introduction of non-self-compatible cases can be motivated without falling foul of the worries about intelligibility. The requirement of relevance in consequence is natural and present even in the earliest understandings of logical consequence. The difference between the following two arguments illustrates the point: All footballers are bipeds. Some footballers are not bipeds. Therefore, some footballers are not footballers.

All footballers are bipeds. Some footballers are not bipeds. Therefore, all men are mortal.

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Specific Objections

For Aristotle’s Syllogistic, the first argument is valid, because it has the valid form » All As are Bs » Some As are not Cs » Therefore, some Bs are not Cs in the special case where B = C. The second argument is invalid (on Aristotle’s Syllogistic), because the conclusion terms are not mentioned in the premises. A simple and natural way of clarifying the distinction, one motivated by gtt, is to look for a case in which the premises of the second argument are true but the conclusion is not. The only such case that might be suitable is an impossible situation, one in which all footballers are bipeds and some are not. That situation is inconsistent either about footballers or about bipeds (or, perhaps, the quantifiers, but we will ignore that here), but, so the story goes, it need not be inconsistent about men and mortality. Paraconsistency does not require that such inconsistent situations be possible, only that they be not all identical. It is one thing to require some inconsistency about footballers, quite another to require an inconsistency about mortality, or about numbers, or about something else. The first argument is valid (on Aristotelian grounds) because, even in the inconsistent situation under discussion, since all footballers are bipeds, all non-bipeds are non-footballers; but since some footballers are not bipeds, it follows that some footballers (those who are non-bipeds) are non-footballers too. The situation, then, is inconsistent about footballers: it sets at least two standards about being a footballer, and some creatures are required (by the lights of this situation) to be both in and out of the extension of ‘footballers’. Further reassurance is achieved by noting that the idea of incoherent cases finds a home even in classical model theory. The claim that something is red and colourless can be satisfied in a classical model by assigning overlapping extensions to the predicates ‘red’ and ‘colourless’. It was never thought that, in this case, the model theory rendered it possible that some red thing could be colourless! So, the presence of models in which inconsistent statements are true does not, by itself, mean that we must take inconsistencies to be possible. We are still to reply to the final charge in the objection: to the effect that the law of non-contradiction is basic and that we cannot reason without it. The non-dialethic paraconsistentist can endorse the law of noncontradiction in a number of strong forms, especially (although not exclusively) if he or she is a pluralist. We can agree to take ∼(A ∧ ∼A) to be necessarily true—that it is not the case that a contradiction is true—and, furthermore, that it is always ‘wrong’ to assert a contradiction. Although inconsistencies are true in some situations, none of those situations are possibly actual. A pluralist has even more to say on the matter (as discussed in §7.4): she can strongly endorse the consequence relation that quantifies

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only over all possible worlds and not over all situations, and thereby can strongly endorse the inference from a contradiction to an arbitrary conclusion. In those senses, then, a non-dialethic paraconsistentist endorses the law of non-contradiction; she does not countenance inconsistencies as possible. The given semantics merely involve different ways things can’t be [104, 106]. We have replied to the ‘coherence’ objection only from the vantage point of a non-dialethic paraconsistentist. What of the dialetheist? A great deal of the foregoing response still holds water and can be used as it stands. Dialetheists can agree that inconsistencies are incoherent in the sense of containing a conflict. Non-self-compatible situations are admitted just as they are for the non-dialethic paraconsistentist, and the same story can be told about what can be imagined and made intelligible. The significant difference is that the dialetheist takes some of these inconsistent situations to be possible. As a result, the law of non-contradiction, in most of the forms we discussed above, is rejected. (On pain of trivialism, the dialetheist, as per §7.4.4, cannot strongly endorse classical logic, or any consequence relation that unrestrictedly contains the inference from A ∧ ∼A to B.) But, again, the dialetheist can give the main gist of the foregoing reply, short of rejecting that some inconsistent claims are possibly true. *

*

*

Objection: Relevance and explosion People who take relevance to be an important constraint on logical consequence ought to reject your formulation of pluralism, including the gtt, because it leads to failures of relevance.3 We can define the behaviour of a new sentence ⊥ in the following way: ⊥ is true at no case. Then, by the gtt, the inference from ⊥ to B is valid for any B. But ⊥ need not be relevant to B, and so, gtt does not respect criteria of relevance, contrary to your claim that it does. Reply. Our reply takes two parts. First, we deny that relevance, in any strong sense, is a core constraint on the notion of logical consequence. The lessons of classical logic ought to be enough to teach us that one can have a sensible notion of logical consequence without a rigorous constraint of relevance between premises and conclusions. Of course, given a consequence relation, we can use it to define a sense of relevance, if we wished to do so. You could say that the classically valid argument from a contradiction to an arbitrary conclusion shows us that a contradiction is, contrary to first appearences, relevant to that conclusion. We would not quibble with someone who wanted to use the term ‘relevant to’ in that way, but we would argue that there is another way to use the term which cuts more narrowly, and which takes relevance to be a little harder to come by. Any such stronger notion of relevance, however, is not central to the notion of logical conse3 Thanks

to David Ripley for formulating this objection.

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quence, so the relevantist premise of this objection—that all consequence relations must be relevant in the strong sense—is not one that we endorse. None the less, an argument remains, and we must respond to that in the second part of the reply. For even if, qua pluralist, we claim that not all logics need be relevant, we are committed to the view that all relevant logics are relevant. This objection attempts to show that our relevant logics (those motivated by the gtt) are not relevant, and so we do not motivate or defend or endorse relevant logics, contrary to what we have advertised. Our response to this part of the objection is also two-fold. Here, there are two different responses one could make, and we leave the reader to choose between them (or to accept both). Response 1: Deny the premise. We could first deny the premise used in the objection, that there is a claim ⊥ which is true in no cases whatever, at least in the class of cases used to provide the semantics for a relevant consequence relation. After all, it is not adequate simply to assert that there is a claim true in such and such situations, because perhaps the class of situations does not admit this. We admit that there are incomplete situations and that situations can be filled out to more comprehensive situations across the relation of inclusion such that, if A is true at s and if s s
, then A is true at s
. It follows, for example, that the putative ‘definition’ of so-called ‘Boolean negation’4 (−A is true in s if and only if A is not true in s) is illegitimate. There is no such operation definable on the class of all statements, because if there were then situations could not be partially ordered by inclusion, for −A is preserved along only if the ordering relation is trivial, and we never have s s
unless s and s
agree entirely on what is true. (The proof is simple. If s s
, then, by definition, if s  A then s
 A. Conversely, if s
 A, then we must have s  A, since if s  A then s  −A, and hence s
 −A, and so s
 A, contrary to what we had assumed. So, s and s
agree on what is true.) Perhaps, we say, ⊥ is just as illegitimate as ‘Boolean negation’. The falsum ⊥ does not offend against preservation in the way that ‘−’ does, but perhaps it offends against some other constraint on situations. A plausible constraint (§8) is weak liberalism about situations: the view that for any set of sentences there is some case or other in which those sentences are true. This, clearly, rules out the definition of ⊥ as it stands. Response 2: Deny the conclusion. On the other hand, we could admit the definition of ⊥. Once we do this, we need to argue that endorsing ⊥ does not involve a genuine failure of relevance. This is not a hard claim to make; for ⊥, given the definition, is a logical constant (its extension is determined by the logic, not by the extralogical features of the language), and the issue 4 We put ‘scare quotes’ around ‘Boolean negation’ because we think that the negation in relevant logics could be classical Boolean negation, at least if the world is consistent and complete. ‘Boolean’ negation is negation which makes even incomplete or inconsistent situations consistent and complete. Clearly, there is no such thing.

Specific Objections

113

of the relevance or otherwise of a claim like ⊥ to other claims in the language is an open question. The effect of a false constant like ⊥ in relevant logics is well known. Once you notice that it is not equivalent to an arbitrary contradiction (if it were, then we would accept A ∧ ∼A  B ∧ ∼B as relevantly valid, but we don’t), the worries about ⊥ dissipate. If ⊥ has an analogue in a standard language, it is much more akin to the universal claim (∀p)p in a language with propositional quantification. This, clearly, is relevant to all claims, as the inference (∀p)p  A is underwritten by a universal instantiation (of p by A). If relevantists can swallow this limited kind of explosion—and we can see no reason why they must resist it—then the limited explosion in endorsing ⊥  A is no worse. We can take ⊥ to be a much less revealing abbreviation for (∀p)p, and constraints of relevance are maintained. We need not worry that this has diluted our relevantist credentials; for, recall, relevance was founded on the rejection of explosion and other like inferences. It endorses a sharpened sense of relevance which takes contradictions to not entail everything, and tautologies like excluded middles to not be entailed by everything. This may be maintained even if we agree that there are some things which entail everything (and (∀p)p is a good candidate for that), and equally, there are some things (say,  or (∃p)p) that are entailed by anything at all. *

*

*

Objection: Intensionality and the slingshot. There is no problem with using a formal semantics that utilises models to interpret a language. But once you accept a theory that uses supposedly intensional objects—such as facts or situations—you are subject to ‘slingshot’ arguments, due to such figures as Davidson and Gödel. Take situations, for example. Consider two different true sentences A and B and an object d. A slingshot argument will show that A and B are true in exactly the same situations. Consider the following four statements: » » » »

A ιx(x = d ∧ A) ιx(x = d ∧ B) B

Whatever situation suffices to make A true will suffice to make the next statement true. The second statement is that d is the object which is d and (by the way) A is true. But the only way for that to be true is for A to be true, and, conversely, if A is true then the second statement is true. Likewise with respect to B and the third statement. Now consider the middle pair of sentences. If A is true (as per supposition) then ιx(x = d ∧ A) denotes the object d, and if B is true (as per supposition) then ιx(x = d ∧ B) denotes the object d. So, one can move from one statement (ιx(x = d) = ιx(x = d∧A)) to another (ιx(x = d) = ιx(x = d∧B))

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by substituting co-referring terms, and hence the statements express the same fact, or are true in the same situations. The upshot is that all true statements are true in exactly the same situations: true statements collapse into the one world. Situation theory, and all other intensional theories like it, are incoherent [85]. Reply. It would be very surprising if that argument actually worked; if it did, it would be successful against models as well as against situations. Yet it is not the case that, for any two truths, they are true in exactly the same models. So, the argument must break down in the case of models. But if the argument breaks down in that case, then perhaps it will break down in the application to situations in exactly the same manner. So let us first consider the slingshot as it might be applied to models. First, we agree that, if the sentence A is true in a model M, then (provided that the term d has a denotation) d = ιx(x = d ∧ A) is true in that model. Similarly, we can agree that, if d = ιx(x = d ∧ B) is true in a model, then so is B; but that is more controversial because it fails for certain analyses of definite descriptions. For example, according to those theories in which all descriptions suffering what we might think of as ‘reference failure’ actually refer to a special object (call it ‘∗’), then the very special case ∗ = ιx(x = ∗ ∧ B) could be true even if B were false. But that counterexample can be avoided by choosing another denotation for the name ‘d’ rather than ∗. So, the first step, from A to d = ιx(x = d ∧ A), and the third step, from d = ιx(x = d ∧ B) to B, can plausibly be thought to preserve truth-ina-model, on most accounts of the semantics of descriptions. That leaves the middle step, from d = ιx(x = d ∧ A) to d = ιx(x = d ∧ B). The middle step is precisely where the argument breaks down for models. A model in which d = ιx(x = d ∧ A) is true is not necessarily one in which d = ιx(x = d ∧ B) is true, for d = ιx(x = d ∧ A) requires the truth of A and d = ιx(x = d ∧ B) requires the truth of B. In the case of a failure of one and not the other, one of these identities fails. The same thing, with a little more subtlety, can be maintained in the case of situations. Here, we have the task of explaining why d = ιx(x = d ∧ A) and d = ιx(x = d ∧ B) can be true in different situations, even if A and B are both (as a matter of fact) true, and we proceed in the same way as before. Take a situation s in which A is true but B is not. In this case, we may well have d = ιx(x = d ∧ A) true in s without d = ιx(x = d ∧ B) true in s. We must attend, then, to the argument that purports to move from the one statement to the other by substitution of co-referring expressions; for since A and B are both true (but not true in exactly the same situations) it seems that ιx(x = d ∧ A) and ιx(x = d ∧ B) are co-referring. A number of plausible responses are open to the friend of situations. The simplest is to follow Neale and others [85] in taking the definite description operator to not be a referring expression. If it is treated as a quantifier in the usual Russellian way, then d = ιx(x = d ∧ A) and d =

Specific Objections

115

ιx(x = d ∧ B) are to be understood as something like (∃x)(x = d ∧ (∀y)(y = d ⊃ y = x) ∧ A) and (∃x)(x = d ∧ (∀y)(y = d ⊃ y = x) ∧ B), and no

friend of situations worth the label is going to take those sentences to be true in the same situations. If A and B contain no free variable x, then it is plausible to take the statements to be relevantly equivalent to (∃x)(x = d ∧ (∀y)(y = d ⊃ y = x)) ∧ A (∃x)(x = d ∧ (∀y)(y = d ⊃ y = x)) ∧ B

and

respectively, and those are certainly true in different situations if, as we have it, A and B are true in different situations. So, the foregoing slingshot is no problem to the friend of situations if she takes descriptions to be analysed in a broadly Russellian manner. If she does not take descriptions to be Russellian, but prefers to treat them as properly referring expressions (at least, when they refer!), then it is still possible to resist the slingshot if she takes the substitution of co-referring expressions to hold in only a limited sense. Such restriction need not be ad hoc. After all, it is well known that the substitution of identities involving descriptions into modal contexts must be restricted—for reasons entirely independent of slingshots. For example, it is the case that John Howard is the prime minister of Australia, jh = ιx(PMx)

and that John Howard is necessarily John Howard,
(jh = jh)

but it is not the case that John Howard is necessarily the prime minister of Australia, ∼
(jh = ιx(PMx)). In just the same way, substitutions must be restricted in situation-contexts too. Some situation s might suffice for John Howard’s being male, s  Mjh

but s might not suffice for the Prime Minister’s being male, s  Mιx(PMx)

since, despite its truth, the identity jh = ιx(PMx) need not be true in situation s. The situation s might suffice to ensure the truth of the claim that John Howard is male but fail to ensure ensure that he is prime minister. That is just one example; and for the general approach to be a coherent account of the referential and substitutional features of descriptions, more would need to be said. Still, it is clear that, if the account simply restricts

116

Specific Objections

description-involving substitutions to those contexts in which such substitution would be licenced on a Russellian account, then no special slingshot problems emerge that are not already present in the vocabulary without descriptions. While we have replied to the slingshot only with gestures, we hope that enough has been said to deflect it: friends of worlds and situations have sufficient resources to resist the alleged collapse.5 *

*

*

Objection: Constructivity and semantics. If you endorse classical consequence, then you endorse (in some sense) the inference from ∼∼A to A. However, anti-realists have argued that that inference—‘doublenegation elimination’—is unwarranted. How can you endorse classical consequence? It appears that an anti-realist pluralist cannot hold much by way of your logical pluralism. Replies. There are in fact three different responses the pluralist can give, so we give them in turn. Reply 1. The pluralist can deny the anti-realist presupposition of the objection. Classical consequence is acceptable because we can give it a realist semantics, and so the anti-realist objections are beside the point. To be sure, the anti-realist pluralist might have a problem, but this is not shared by the realist pluralist, whose position concerning intuitionistically unacceptable reasoning is no worse off than that of the realist monist. Reply 2. The anti-realist pluralist can take a leaf out of the dialetheist’s book (see §7.4.4). It is an option for the anti-realist pluralist to merely weakly endorse classical logic, without thinking that the inference of double negation elimination and other constructively invalid inferences always and everywhere preserves truth. None the less, this kind of pluralist can weakly endorse classical consequence, not by taking all classical worlds to be genuinely possible, but in maintaining that classical worlds none the less afford a genuine consequence relation, which may prove of great use in evaluating particular kinds of reasoning. For example, it might be thought that in certain contexts, where completeness is assured (when all predicates are decidable), the restriction to complete and consistent cases makes sense. That is one option for the anti-realist pluralist, but in our opinion it is a much weaker response. The kind of ‘pluralism’ that results is not particularly distinctive or enlightening. We can do much more with a more robust response. 5 One of us (gr) has written more on the failure of slingshot arguments against situations [112].

Specific Objections

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Reply 3. Instead of granting that an anti-realist cannot strongly endorse classical logic the pluralist can reject this. The pluralist can show that classical logic in its full range of application is still an option for an antirealist, even one who takes warrant to be the basis of a meaning-theory [38, 91, 128, 130]. For example, the anti-realist pluralist can provide a justification for classical proof theory on anti-realistically acceptable grounds. We have already seen one way in which that can be done, using the traditional multiple-premise, multiple-conclusion sequent calculus (§4.4.2). Considerations of warrant may be used to justify that kind of proof theory, provided that the sequent X  Y is to be read as the claim that there is no possible warrant for accepting X and denying Y . Objection: Constructivity and inconsistency. A pluralist employing constructive and classical reasoning risks inconsistency. Not only is intuitionistic mathematics famous for being weaker than classical mathematics (by rejecting certain classical validities); it is also famous (infamous?) for taking certain classical mathematical results to be outright false. That seems to commit you to an inconsistency, inasmuch as you endorse both classical and constructive consequence. If you, qua classical reasoner, can prove A but, qua intuitionistic reasoner, can prove ∼A, then as a pluralist you seem to be committed to A and ∼A (or, because both consequence relations are adjunctive, A ∧ ∼A). Reply. Here is our general strategy by way of reply. We must closely examine the proofs in question. If there is a genuine intuitionistic proof of ∼A, where A is provable in classical mathematics, then there must be a premise used that is not true in classical mathematics. It is certainly not the logic that gives us ∼A, as intuitionistic logic is weaker than classical logic. So, if we genuinely have a proof of ∼A, we have used premises that are false in classical mathematics. Upon examination, of course, we may see that we have not really proved ∼A, but rather have proved that we cannot prove A. And that is another way to resolve the conflict. That we cannot (constructively) prove A is not in conflict with the claim that we can (classically) prove A. The situation differs no more than that with such trivialities as A ∨ ∼A (where A is constructively undecidable). A ∨ ∼A is classically provable but constructively unprovable (with constructively undecidable A); however, it does not follow that ∼(A ∨ ∼A)! That said, there are intuitionistic theories in radical variance with classical theories. There are theories in which one can prove the negations of classical theorems. We must understand what to say about these if our position is to be consistent. The first line of defence, then, is deference to an important tradition in constructive mathematics. The constructivism of Errett Bishop [23, 24], Douglas Bridges [27], Fred Richman [81, 114], and others can best be

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described as mathematics pursued in the context of intuitionistic logic.6 This brand of constructive mathematics is explicitly consistent with classical mathematics. Bishop-style constructivists reject any inference in conflict with classical reasoning [114]. That is the approach we too must take. But what does all that mean in practice? By way of illustration, we must examine cases of conflict in some detail. There are two major ways in which intuitionistic theories conflict with classical theories. The conflicts arise from notions of choice sequences and realisability. We will consider each in turn. Parenthetical remark. The remainder of this section is quite technical, and may be ommitted without cost if you believe that we can develop constructive theories as consistent sub-theories of classical theories. End of parenthetical remark. Choice sequences: A choice sequence is a sequence α(0), α(1), α(2), . . . of natural numbers. (Our presentation closely follows van Dalen and Troelstra’s helpful short expositions [33, 131].) We let α(k) be the sequence α(0), α(1), . . . , α(k − 1), the initial segment of length k of α. These sequences are taken to encode the choices of a creating mathematical subject. A choice sequence may be completely freely constructed (by the analogue of tossing a coin at each stage) or it may be completely determined by law (such as the law defining the decimal expansion of π), or it may be somewhat constrained but somewhat free (each step may be free within certain constraints, such as taking α(k) to be the kth digit in the expansion of π if the toss is heads, and 9 minus that digit if the toss is tails). A typical intuitionistic thesis about choice sequences is this: if F is a function on choice sequences, then, given that choice sequences may be free creations, F must depend on some initial segment of the choice sequences accepted as inputs. There is no way to assume more about the choice sequence, since at any stage of reasoning not all of the sequence has been constructed. So, there is another function f on segments such
that ∀α∃k F(α) = f(α(k)) . A consequence of this is Brouwer’s continuity principle for functions:
 ∀α∃k∀β α(k) = β(k) ⊃ F(α) = F(β)

(9.1)

Given the continuity principle, we have conflict with classical analysis. We must reject one classical principle, the limited principle of omniscience:
 ∀α ∃x(α(x) = 0) ∨ ∀x(α(x) = 0)

(9.2)

It is instructive to see why we cannot constructively prove (9.2). It is not a problem with the law of the excluded middle. Identity for natural numbers 6 Tait provides a more explicitly philosophical account that draws very similar distinctions to the work of constructive mathematicians [125, 126].

Specific Objections

119

is decidable when, given two natural numbers, we have a simple routine for determining if they are identical. So we can prove
 ∀α ∀x(α(x) = 0 ∨ α(x) = 0)

as, for any value the choice sequence presents to us, we can determine if it is a zero or not. The crucial move not allowed in intuitionistic logic is the following inference of distribution referred to before (6.1): ∀x(A ∨ B)  ∃xA ∨ ∀xB

As we have seen, distribution is constructively undesirable, because a routine showing that every x is either A or B does not necessarily provide a routine to find an A or to show that all objects in the domain are B. For this we need to survey the domain. So, our routine for verifying ∃xA ∨ ∀xB outstrips routines for verifying ∀x(A ∨ B). Now, if (9.2) were true—as we take it to be, since we endorse the law of the excluded middle as true, even though it has constructive content and is not a constructive tautology—we would have a function F such that  F(α) =

0 1

if ∃x(α(x) = 0) if ∀x(α(x) = 0)

Applying the continuity condition, F must be determined by an initial segment of its input. In particular, since F applied to the constant 1 choice sequence (β where β(x) = 1 for each x) gives 1, continuity tells us that there is some sequence 1, 1, . . . , 1 such that every continuation γ yields 1: that is F(γ) = 1. However, there are many continuations of the series 1, 1, . . . , 1 that contain zeros. It follows that such a function F cannot exist. But the existence of F is a consequence of the classical tautology:
 ∀α ∃x(α(x) = 0) ∨ ∀x(α(x) = 0)

What do we do? The intuitionistic response is to assert the negation of the thesis:
 ∼∀α ∃x(α(x) = 0) ∨ ∀x(α(x) = 0)

A pluralist response cannot follow the intuitionist orthodoxy. We must look elsewhere if we are to maintain consistency. Is there any well motivated option open to us? One option is this: reject the continuity principle. Once we reject intuitionism we have no reason to agree that functions on choice sequences must be determined by initial segments of those sequences. The function F is a case in point. For a pluralist, (9.2) is true without being constructively provable. Functions such as F may exist without being constructed. Constructive considerations give us no reason to endorse (9.1).

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Specific Objections

So, one route to inconsistency fails. Choice sequences are unacceptable to the pluralist, for they make illegitimate assumptions. They rule out of existence functions like F that are classically demonstrable. If we have reason to allow the existence of such functions (as we think we do), then we have reason to reject choice sequences. This means that we reject certain branches of constructive mathematics, not the whole study. The constructive mathematics of Bishop, Bridges, and Richman [23, 24, 27, 81, 114] makes no use of choice sequences, and it makes no counter-classical claims. Realisability: Take an enumeration of all partial recursive functions, with {n} the function with index n, so {m}(n) is the function with index m applied to the natural number m. (Details of how we deal with partiality and undefined results we leave to elsewhere [20].) Similarly, we encode pairing, so that (n)0 and (n)1 are the first and second item in the pair n. so (n, m)0 = n and (n, m)1 = m, and (n)0 , (n)1  = n. With this technology, we define a relation between (codes of ) functions and sentences of the language of arithmetic: Case

Condition

n r A (A atomic) nrA∧B nrA∨B

A is true (n)0 r A and (n)1 r B If (n)0 = 0 then (n)1 r A, and if n1 = 0 then (n)1 r B For all m, if m r A then {n}m r B (n)1 r A((n)0 ) For all m, {n}(m) r A(m)

nrA⊃B n r ∃xA(x) n r ∀xA(x)

The justifications for these clauses are straightforward: » Atomic sentences are self-justifying. We take everything to be a realisation of an atomic sentence. » A realisation for a conjunction is a pair of realisations for each conjunct. » A realisation for a disjunction is a realisation for a disjunct, combined with an indication of which disjunct has been realised. » A realisation for a conditional is a function transforming realisations for the antecedent to realisations for the consequent. » A realisation for an existential quantifier is an object together with the realisation that that object satisfies the formula under the quantifier. » A realisation for a universal quantifier is a function sending objects to realisations that the object satisfies the formula under quantifier. A nice result is that every thesis of Heyting Arithmetic (Peano Arithmetic using intuitionistic predicate logic: we write this as ‘HA’) is realisable. That is, if HA  A, then for some n, n r A. However, more is realised than simply

Specific Objections

121

the theses of HA. Consider what counts as a realisation of a ∀∃ formula. If n r ∀x∃yA(x, y), then for each m, {n}(m) r ∃yA(m, y); and, given the definition of a realiser for an existentially quantified formula, we have that for each m
 ({n}(m))1 r A m, ({n}(m))0 But this has consequences of its own. Abstracting out the m, we have a realiser l such that 
l r ∀xA x, ({n}(x))0 where l is the code of the recursive function sending m to ({n}(m))1 . As a result, we have
 n, l r ∃e∀xA x, {e}(x) Now, the function that sends n (the realiser of ∀x∃yA(x, y)) to n, l is itself recursive, so the code of this function is a realiser for the following claim:
 ∀x∃yA(x, y) ⊃ ∃e∀xA x, {e}(x)

(9.3)

That claim is known as Church’s Thesis.7 The thesis states that, given any true ∀∃ formula, there is a recursive choice function choosing the appropriate instance of the existential quantifier for each input into the universal quantifier. The thesis
is false.  Given that there is one non-recursive
function, f, we have ∀x∃y f(x) = y , but we do not have ∃e∀x f(x) = {e}(x) , which states that f is recursive (as it is identical in extension to the recursive function {e}). If Church’s thesis is false, then why is it realisable? Look back to the verification that (9.3) is realisable. We argued that, if n is a realiser for ∀x∃yA(x,
y), then  there is a realiser (recursively constructible from n) for ∃e∀xA
x, {e}(x). Does it follow that if ∀x∃yA(x, y) is true, then so is ∃e∀xA x, {e}(x) ? We can safely deny that. After all, there may be a sentence of the form ∀x∃yA(x, y) which is true without having a realiser. It is true that all theorems of HA have realisers, but it may not be the case that all arithmetic truths have realisers. If truth in arithmetic outstrips truth in HA (with Church’s thesis), then we have no reason to think that, simply because of ∀x∃yA(x, y) transforms into the realisability
the realisability  of ∃e∀xA x, {e}(x) , the truth of the former also gives us the truth of the latter. Realisability semantics have only recursive realisations in play. It is little surprise that in this ‘universe’ all functions are recursive. If we wish to reason constructively about the mathematical universe studied by classical mathematicians, this realisability semantics will not do. We will need more constructions than those provided by recursive functions. There is 7 Beware: This is not the Church–Turing thesis to the effect that every computable function is recursive. This Church’s thesis is much stronger, to the effect that every function is recursive.

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Specific Objections

no force in the argument that realisability semantics motivates a departure from classical arithmetic. We can reason constructively without fear of contradiction with classical theories.

Chapter 10

Other Directions This book is not the end of the story of logical pluralism. There are many questions we have left unanswered, and many possible pathways for fruitful research. In this chapter we will sketch just a few of the ways in which we think a pluralistic view of logical consequence could be developed. In three short sections, we will indicate » the upshot of recognising the plurality of consequence relations; » how the plurality can be exploited; » ways that pluralism might be further developed in the theory of proofs.

10.1

Recognising Plurality

Pluralism is not the dominant tradition in the logical community. It is not a completely foreign tradition, to be sure, but it is probably the case that most ‘practising logicians’ (however we are to carve out that class) are monists and not pluralists. That makes a difference: how a monist ‘does logic’ differs from how a pluralist does it, in important ways. Recognising a plurality of logical consequence relations, we think, is a liberating insight that makes a difference in how theory might be developed. In this section, we make a conjecture. Conjecture: Monism artifically restricts the development of logic, either by requiring that each consequence relation do too much, or by ignoring some of the important tasks for which we require consequence relations. The moral of logical pluralism is that not every consequence relation needs to do everything that must be done by consequence relations. Failure to learn that (pluralistic) moral unnecessarily hinders logical theorising. Here are some examples of what we mean. Case 1: Axioms for relevant logics. The axioms for traditional relevant consequence are a mixed bag. In traditional relevant logics, the inference of double negation elimination, from ∼∼A to A, is taken to be relevantly valid. We can ask why that inference is taken to be relevantly valid. If the only answer is that there is no possible world in which ∼∼A is true and A is not, then the answer is not good enough, because—at least from 123

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Other Directions

a non-dialethic perspective—that answer would also justify the inference from A ∧ ∼A to B, or from A to B ∨ ∼B, both of which are relevantly invalid. So some other explanation must be given. A similar question can be raised about the tautology A ∨ ∼A. Why is that a relevant tautology in standard systems such as r and e? Because it is true? That is hardly an answer. Many true things are not axioms of logical systems. Is the answer, then, that it is true in all possible worlds? Or true in virtue of the meanings of the connectives? That doesn’t suffice, unless you tell a story of how that is connected to relevant consequence, in general. A final example is the relevantist argument over the validity of disjunctive syllogism. Clearly, disjunctive syllogism is valid in some sense—at least given consistency assumptions—but it is not relevantly valid. The natural goal for monist relevant logicians is to attempt to recover it inside the vocabulary of traditional relevant logics, or to add it explicitly as a new rule, or to bite the bullet and say that it is invalid [22, 98, 115]. So, a clear reason for a monist to either add or recover the foregoing principles is that we use them in argumentation, and it is clear that in some sense those arguments are valid. The pluralist has a ready response: of course such principles are valid in some sense. The important question in the development of relevant logics is whether they are valid in the relevant sense. For a pluralist, who has attempted to make relevant consequence more precise by way of a formal semantics involving situations (see §5), such inferences should stand or fall on that semantics, and not on general intuitions on what might or might not be logically true in an unspecified sense. In the ‘situation’ semantics we presented, the inference from ∼∼A to A may well fail to be relevantly valid, and A∨∼A is certainly not true in every situation, and disjunctive syllogism fails. But the inference ∼∼A  A is ‘world-valid’, and A ∨ ∼A is true in all worlds, and disjunctive syllogism works at worlds, and so such inferences are preserved in their classical sense. We leave it to the reader to judge whether our account or the standard one is more natural. It is enough to grant that our approach is an option worth pursuing, and that making relevant logics look excessively classical is not the only way to develop the theory. Case 2: Strong axioms in constructive mathematics. One kind of monist development of non-classical logic, as we have seen, requires the logic to become quite ‘classical’, since it must be seen to achieve everything that classical logic (the dominant logical tradition) can manage. Another option, as we have seen in the previous chapter in the discussion of choice sequences and realisability, is to develop strong principles that are in opposition to those in classical logic. The reasoning in this case is similar: we need strong principles to prove (this time, using constructive logic) strong results in mathematics. For the pluralist who endorses classical logic, such endeavours are interesting, important, and useful, but in the way that nonstandard models are useful. It is possible for constructive mathematics to

Exploiting Plurality

125

be developed as properly weaker than classical mathematics. The fact that there is a classical proof of an important theorem but no constructive proof does not mean that we should attempt to add new axioms to the constructive theory so that it will be provable (though this might be an interesting endeavour in its own right). We could, as pluralists, just acknowledge that this is one place where constructive mathematics is properly weaker than classical mathematics. The above examples show that merely recognising the plurality of consequence relations will go some way towards shaping the way in which logical consequence is studied and theories are constructed.

10.2

Exploiting Plurality

It is one thing to recognise plurality in the development of a logical system, but something more interesting to let pluralism go somewhere. We can exploit the fact that we have more than one consequence relation, and allow the presence of a number of distinct consequence relations to do some work for us. There are a number of examples in the current literature. Relevant predication: Dunn’s account of relevant predication, according to which an object a has a property F just in case (∀x)(x = a → Fx) is true, where ‘→’ is a relevant conditional [41, 42, 63], is a good example of a theory, underwritten by a relevant consequence relation, that sets no store on the idea that relevant consequence is the only form of consequence worth the name. The application is completely independent of the kind of dialetheism that takes a paraconsistent consequence relation to be the only kind one can strongly endorse, or from monism that takes endorsing one consequence relation to involve rejecting the others. Instead, all that is required is that the language contain the relevant conditional, and hence that there be a consequence tighter than classical consequence. The necessity of a relevant consequence relation for the enterprise, once we admit a relevant conditional, is simple: The relevant conditional discriminates between contradictory antecedents, and between tautologous consequents. The ‘paradoxes’ of implication, A ∧ ∼A → B, and A → B ∨ ∼B, are not always true for a relevant conditional →. So, if the conditional is to have a simple semantics, there must be points in the model structure at which inconsistent claims are sometimes true, and at which tautologies can sometimes fail. Once we have such points, we can define relevant consequence as the preservation of truth in such situations, in the usual way. None of that is to say, of course, that consequence defined over worlds is not also logical consequence. Constructive and classical mathematics: Mainstream constructive mathematics is an example of pluralism at work. The kind of constructive mathematics we have in mind is the kind that takes there to be interesting and important things one can say about ordinary mathematical

126

Other Directions

theories when one takes the time to examine what is constructively provable within them [113, 114]. It is a common misconception that intuitionistic mathematics deals with a special class of mathematical objects that are, in some sense, constructive . . . But when an intuitionist does group theory, he is developing a constructive theory of groups, not a theory of constructive groups. [113]

This kind of mathematical practice is common, we prefer constructive proofs when we can get them, and we note when we have them. We don’t take classical proofs to be worthless, and we don’t take constructive proofs to really be in a different vocabulary (utilising intuitionistic negation instead of Boolean negation) or to be about different things. (We have in mind the epistemic interpretation according to which constructive proofs are merely about what can be constructively proved, the negation operator being read to mean there is a proof that it is not . . . .) Constructive mathematics, pursued in this simple and straightforward way, is amenable to pluralism. The constructive mathematician who utilises classical reasoning, but who also notes when she departs the strictures of the constructive high road to the broad and comfortable classical low road, is a perfect example of a logical pluralist at work. Notice that this is not necessarily the same distinction as one sees with set theorists who like proofs which do not use the axiom of choice, but are prepared to use the axiom of choice when it is required. Such set theorists can be plausibly thought of as monists who have scruples about certain principles which are none the less true of the single set theoretical universe. (This is to phrase the monism/pluralism distinction in set theory in a realist fashion. An anti-realist reading could be found by talking of theories rather than the ‘set-theoretical universe’.) The pluralism/monism distinction in set theory and in logic is richer than this. Endorsing constructive logic does not merely mean refraining from the use of classical principles like double negation elimination or the law of the excluded middle when one can so refrain. Endorsing constructive logic and finding a use for it involves more: it is to find the distinction between constructively valid and invalid arguments important, that is, to take constructive counterexamples as marking an important distinction. The correct analogy with set theory would be (speaking with a ‘realist’ voice for the moment) to take there to be at least two set-theoretic universes, one governed by the axiom of choice and one not. Proofs not utilising choice deliver conclusions true in both ‘universes’. Proofs utilising choice deliver conclusions true in only the more restrictive universe. This is a kind of set-theoretic pluralism. It is not to say that the two ‘universes’ are independent, that neither is reducible to the other or that they bear no interesting relationships to each other: it is to say that they are both appropriately called universes of sets and that theories describing each universe are genuine set theories. The same holds for the constructive mathematician who also endorses intuitionistic logic.

Plurality and Proofs

127

This mathematician takes constructive counterexamples to play an important mathematical role, and the tighter boundary drawn by constructive consequence (when compared with classical consequence) is marked by an admissible logical consequence relation. First- and second-order logic: Stewart Shapiro’s work on the applications of second-order logic in the philosophy of mathematics is an excellent example of how one can exploit pluralism about logical consequence [120]. Shapiro’s case is that there is space in our conceptual landscape for a genuine, non-effective, second-order consequence relation, and that we can exploit such a logic to provide the resources required to express strong principles of induction, the categoricity of mathematical theories, and so on. None of this turns on taking second-order logic to be the only logic worth having. Shapiro does not need to think, for example, that mathematical proof is always and everywhere second-order, and that axiomatisable consequence relations have no place in our armoury or in our account of logical consequence. He requires only that second-order consequence is legitimate, not that it is the only consequence relation. The foregoing offers but three examples of how a broadly pluralist temperament can make space for interesting, useful, and creative logical work. There is much more work to be done. We can look for other interesting applications of classical, constructive, relevant, and paraconsistent consequence, unconstrained by the thought that, if we use a consequence relation for this job, it must be the consequence relation we must use for every job. Such applications of pluralism—and the general philosophical position itself—cries out for some more general theorems. There is very little work done on a general theory of formal languages with more than one consequence relation. We have no idea of the range and depth of the kinds of results which can be proved. In what cases do facts about one consequence relation transfer to another? In what ways can consequence relations be independent? What kinds of connectives behave in the ‘same way’ with respect to a range of consequence relations, and how is it that some connectives ‘fit better’ with one consequence relation rather than another? There are many more questions like these which would benefit clarification and further study. What, on a purely formal level, can be said about languages with more than one consequence relation? What interesting theorems are there to be proved? We have no idea, but we look forward to seeing whatever is found.

10.3

Plurality and Proofs

Finally, we must note that we have said almost nothing about proof theory, which, it must be acknowledged, is a central plank in any view about logical consequence [62, 90]. Some of the work required to understanding a pluralist account of deductive proof has already been done in the work on

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Other Directions

classical and constructive proof theory. It can be seen in these proof theories that classical and constructive connectives do not have to be thought of as different connectives at all—the same proof rules govern the connectives in both classical and constructive logics—the only difference is found in the suite of structural rules governing the shape of proofs [53, 89]. This proof-theoretical perspective is amenable to pluralism. On this picture, the rules governing connectives stay constant, but the external considerations (in this case, the structure of proof ) govern the way these rules interact and can be applied. Two further avenues of inquiry must be explored to bring this work into the fold so that it can help underwrite a genuine pluralist proof theory. First, we need to connect the structural conditions on proof to the semantic considerations we have used to explain and to justify the distinctive features of each consequence relation. This connection will go some way to providing an alternate justification or explanation of the kinds of distinctions allowed in each logical system. The general story about the structure of proof and the connection between structural rules and inferential principles is known to some extent [105, 107], but it is none the less at a very rudimentary stage of development.1 Second, we should examine how contemporary theories that take proof and inference to do some genuine foundational work can apply in a pluralistic setting. Do these theories pick out privileged consequence relations amenable to a particular goal (in developing a meaning theory, for example), or are these requirements able to be met by a range of different consequence relations? We have sketched the way that this second response might be developed, at least in the case of classical logic and an anti-realist foundation for proof theory (§4.4.2). Much more can and will be done to explain how this story could be made comprehensive.

1 One of us (gr) is working out the details of this ‘further direction’. For more on this, see http://consequently.org/edit/page/Proof_and_Counterexample.

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Index −, see negation, Boolean
, see necessity ♦, see possibility α, assignment of values to variables,

assertion, 8–10, 18, 46 attitude propositional, 11 baldness, 27 Barwise, Jon, vii, 49, 58 Beall, JC, 4, 5, 50, 93, 97, 100 belief, 11, 16, 17 degree of, 17 degrees of, 18 Belnap, Nuel, vii, 53, 105 betting rates, 17 Bishop, Errett, 117 Bolzano, Bernhard, 23 Bolzano–Weierstrass Theorem, 71 ‘Boolean negation’, 112 Boolos, George, 25 brackets, see parentheses Bricker, Phillip, vii, 76 Bridges, Douglas, 117 Brouwer, L. E. J., 61 Bueno, Otávio, vii Burgess, John, 44

38 ⊥, see absurd proposition ∃, see existential quantifier ∀, see universal quantifier ⊃, see conditional, material ι, see definite description λ-calculus, 8 ∧, see conjunction , the 0-tuple, 38 ∨, see disjunction ∼, see negation

, see stage, inclusion , see truth, in a world

a priori knowledge, 20 abstract, 11 absurd proposition, 113 absurd proposition, ⊥, 64 adjective, 9 admissible instance, 35 Agnes, JC’s cat, 50 algebra, 8 Boolean, 8 Ammonius, 19 Anderson, Alan Ross, 53, 105 antecedent, 10 anti-realism, 40, 45, 47, 58 arguments against classical logic, 14 semantic, 61, 74 argument, 8, 12, 13, 15, 16 arguments, 3 Aristotle, 12, 18, 21, 105, 107 Syllogistic, 110 arithmetic, 12, 54 assertibility, 45

C, see compatibility Carnap, Rudolf, 4, 78 Carnapian pluralism, 78 case, 23 casecl , 31 casem , 39 casesx , 49, 54 casew , 36, 37 casex , 26, 29–32, 35, 36, 49, 55 casex s, 51 category, 20 cats, 50 certainty, 16 Cheyne, Colin, vii children tired, 9

137

138 Church’s Thesis, 121 Church–Turing Thesis, 25, 31 circuit electrical, 8 claim, 9, 11, 35 class proper, 41 classical logic, see logic, classical coherence, 46 colour, 20, 37, 44 Colyvan, Mark, vii commitment, 18 comparatives, 20 compatibility, 52, 64 complete, 31 computable function, 25 computation, 8 conclusion, 8, 13–15, 23 conclusions multiple, 13, 14 concrete, 11 conditional, 5, 10, 11, 13, 15 intuitionistic, 63 conditionality, 16 conjunction, 5, 9, 11, 12, 15–17, 21–23, 26, 35, 37, 51, 52, 63 connective, 11 binary, 5 unary, 5 warranting, 13 consequence, 3, 12, 17, 91 as normative, 16 as not reflexive, 30 as not transitive, 30 classical, 31 concept of, 7 as vague, 7 constructive, 32 deductive, 3, 28 defined by proof, 36 definition of, 7 inductive, 28 reflexivity, 91 relation, 3, 13, 14 relevant, 31, 49, 53 transitivity, 91 universal applicability, 16 consequence relation paraconsistent, 79

Index strongly endorse, 82 weakly endorse, 83 consistent, 31 as self-compatible, 52 construction, 32, 61, 62, 67, 72, 73 constructivism, 61 content, 10–12, 22 contingency, 15 continuum hypothesis, 44 contradiction, 49 Cook, Roy, vii counterexample, 23 creating subject, 61 criminal policy, 61 Davidson, Donald, 113 deduction, 23 definite description, 113 denial, 8, 11, 18, 46 description definite, 113 DeVidi, David, vii dialetheism, 79, 82 dialetheist, 83 dilemma epistemic, 17 disjunction, 5, 10–12, 21–23, 37, 51, 57, 63, 67 disjunction thesis for truthmakers, 57 disjunctive syllogism relevantly invalid, 56 distribution of universal quantifier over disjunction, 64, 71 Dog, The, 95 domain, 38, 41 Dresner, Eli, 12 Dummett, Michael, 61, 74 Dunn, J. Michael, 99 Dunn, J. Michael, vii, 49, 52, 125 Dutch book, 17 Dychoff, Roy, vii Dylan, Bob, 50 E, the logic, 53 empty set, 38 English
, 9 entailment, 63

Index relevant, 57 epistemicism, 27 epistemology, 16, 17 Etchemendy, John, vii, 12, 13, 40 ex falso quodlibet, 79 excluded middle law of the, 39, 53, 65, 100 not a relevant tautology, 55 existential quantifier, 5, 12, 37, 64, 67 explosion, 30, 57, 79, 81 fiction, 57 fictionalism about mathematical objects, 73 Field, Hartry, vii, 103 first-order logic, see logic, predicate form, 43 formal schematic, 20 formalisation, 39 formality, 18, 19, 24, 35, 36, 38, 41 1-formality, 21, 22, 41, 54, 69 2-formality, 21, 42, 54, 70 3-formality, 22, 43, 54, 70 schematic, 41, 54, 69 Frege, Gottlob, 10, 12, 13, 22, 44 function computable, 25 recursive, 25 Gödel, Kurt, 113 Garfield, Jay, vii Geach, Peter, 10 Generalised Leibniz Thesis, 26–29 Generalised Tarski Thesis, 29–32, 35–40, 44, 48, 49, 53–56, 58, 59, 61, 68, 71, 82, 83, 89–92, 98, 99, 103, 104, 110–112 accepting only one instance, 30 actuality constraint, 82 rejecting it, 30 Gentzen, Gerhard, 14, 47 geometry, 12 glt, see Generalised Leibniz Thesis Goddu, Geoff, vii, 89, 90 gr, see Restall, Greg grammar formal, 8

139 Green, Karen, vii gtt, see Generalised Tarski Thesis Haack, Susan, 4 Hardegree, Gary, vii, 99 Hazen, Allen, vii, 103 Heyting Arithmetic, 120 Heyting, Arend, 61 Higgins, Katrina, 50 history, 26 Howard, John, 115 Hughes, Sue, viii, 5 Humberstone, Lloyd, vii Husserl, Edmund, 8, 20, 23 Hyde, Dominic, vii hyper-computation, 26 hypothesis, 18 I, see interpretation function

impossibilia, 50 incoherence, 17 incomplete, 31, 50 inconsistent, 15, 31, 50 but not trivial, 50 indentity, 91 inductive strength, 28 infallibility, 16 inference, 9, 10 judged, 16 valid, 10 infinite task, 72 infix, 5 inspiration, 11 insult, 11 interpretation, 24 interpretation function, 38 intuition, 61 intuitionism, 61, 73 intuitionist, 39 intuitionistic logic, see logic, intuitionistic jcb, see Beall, JC Jeffrey, Richard, 23–25 judgement, 9, 11, 12, 21 as act, 10 as content, 10 record of, 10 Kant, Immanuel, 20, 22

140 Kemp, Gary, vii, 94 kinds natural, 20 Klement, Kevin, vii, 88 knowledge, 20 Kripke, Saul, 20 Kroon, Fred, vii Lambek calculus, 8 Lambert, Karel, 76 Lance, Mark, 56 Langtry, Bruce, vii language, 12 formal, 8 interpreted, 8 natural, 8 regimented, 9, 12 Leibniz, Gottfried, 15, 23 Lennox, Jim, vii Lewis, David, vii, 15 limit point, 72 lnc, see non-contradiction, law of logic, 3, 8, 35 classical, 14, 16, 39, 49 anti-realist arguments against, 14 justification of, 39 no constructively acceptable semantics for, 39 comparative, 22 constructive, 61 definition of, 7 formal, 18 free, 75, 76 universally, 75 history of, 12 intuitionistic, 16, 32, 61, 73 non-classical, 12, 39 predicate, 14, 20, 22, 36, 39 relevant, 17, 49, 53 second-order, 77–79 logical consequence, see consequence logical pluralism, 29 logical pluralist how to not be one, 30 logical truth, see truth, logical Łukasiewicz, Jan, 18–20 Lycan, William, vii

Index MacFarlane, John, 21, 22 Man, The, 95 manifold, 23 Mares, Ed, vii Martin, Errol, 91 mathematics, 13, 16, 40, 44, 57, 71 and construction, 73 classical, 67, 72 constructive, 67, 72 foundations of, 61, 73 Max, JC’s cat, 50, 90 measurement, 12 Meyer, Robert K., vii, 91 milk, 51 Mill, John Stuart, 7, 8 mind Cartesian, 5 modal realism, 76 model, 23, 24, 31, 35, 39 intended, 41 model theory, 40 modus ponens, 10 Momtchiloff, Peter, viii Moore, Joseph, 11 Mortensen, Chris, vii name, 20, 37, 38, 42 non-denoting, 41 natural kinds, 20 Neale, Stephen, 114 necessary, 16 necessary truth-preservation, 37, 40–44, 46, 48, 76, 77 necessity, 5, 14–16, 20, 24, 26, 27, 35, 36, 40, 54, 69, 80 circularity of the definition?, 15 historical, 26 metaphysical, 26 physical, 26 necessityx , 26 negation, 5, 12, 15, 17, 18, 21–23, 37, 49, 63–65 ‘Boolean’, 112 behaving differently at situations, 50, 51 classical behaviour, 52 compatibility treatment of, 52 de Morgan features of, 80

Index non-classical not a different kind, 51, 55 Nolan, Daniel, vii non-contradiction law of, 107 normativity, 16–18, 24, 35, 36, 43, 55, 70 ntp, see necessary truth-preservation number, 22 object, 42, 57 available for evaluation at a stage, 63 identity of, 21, 22 operator, 5 modal, 5 opinion, 7 package, 46 paraconsistency, 56, 79 dialetheism, 80 full-strength, 80 gentle-strength, 80 industrial-strength, 80 paradox, 16 liar, 57 parentheses, 5 parents, 15 tired, 9 Parker, Christine, 50 Peano Arithmetic, 97, 120 Peano, Guiseppe, 44 Peripatetics, 19 permutation invariance, 21 Perry, John, vii, 49, 58 Platonism, 72 pluralism Carnapian, 102 electron, 89 ethical, 89 logical, see logical pluralism pluralism, the nature of the claim, 25 possibility, 5, 16, 40, 80 possible world, 12, 15, 23, 26, 35, 36, 38, 40, 49 realism about, 15 precisification, 27

141 predicate, 11, 20, 22, 43 predicate logic, 14 preface paradox of, 16, 18 prefix, 5 premise, 8, 11, 13, 14, 23 premises multiple, 14 Priest, Graham, vii, 4, 57, 80, 81, 93, 97, 103 Prior Analytics, 19 probability, 17, 18 proof constructive, 67 mathematical, 44, 61, 72 structure of, 13 proof theory, 36, 127 proper class, 41 property, 21 proposition, 8–11, 20 propositional attitude, 11 propositional quantification, 113 psychologism, 20 psychology, 19 quantifier, 5, 11 existential, see existential quantifier universal, see universal quantifier question, 11 R, the logic, 53 Rayo, Augustín, vii, 104, 106 Read, Stephen, vii, 4, 12, 94, 95, 100 realism, 45, 58, 62, 73 reasoning hypothetical, 15 recursive function, 25 red, 37, 44 regimented sentence, see sentence, regimented register machine, 26 relation consequence, 3 relevance, 53 relevant logic, see logic, relevant ‘relevant’ vs ‘relevance’, 49 Restall, Greg, 46, 50, 93, 97, 116

142 Restall, Zachary, viii, 10, 15, 22, 50 riches, 27 Richman, Fred, 117 Ripley, David, vii, 111 rivalry, 89 rivalry between logics, 36 Rogerson, Susan, vii Routley, Richard, see Sylvan, Richard Russell, Bertrand, 12, 13, 44 Russell, Gillian, vii schema, 19, 20 Seligman, Jerry, vii semantics, 45 sentence, 9, 10 form of, 9 in context, 9 infinitely long, 13 interpreted, 9 regimented, 9 type of, 9 use of, 9 sequent calculus, 47 set theory, 41 ‘settled core’, 27 Shapiro, Stewart, vii, 4, 71, 73, 127 sickness, 15 since, 13 situation, 49, 53, 55 actual, 52, 58 incomplete, 50, 53 inconsistent, 50, 56 non-actual, 52 part of, 53 weak liberalism about, 112 situation semantics, 58 size, 20 sleep, 10 slingshot arguments, 113 so, 13 spelling, 5 stage, 62, 73 as a case, 68 as incomplete, 62 final, 66, 69 inclusion, 62 not a state of knowledge, 66 not a warrant, 67 statement

Index pre-semantic features of, 22 semantic features of, 22 stereo, 50 Stoics, 12 stress, 9 stressed, 9 subject, 11 sure loss contract, see Dutch book syllogistic, 18, 20, 21 Sylvan’s Box, 50 Sylvan, Richard, 50, 57 symmetry, 14 Tait, William, 118 Tanaka, Koji, vii, 88, 89 Tarski models, 69 Tarski’s Thesis, 29, see Generalised Tarski Thesis, 30 Tarski, Alfred, 3, 7, 8, 21, 23, 29 Tarskian models, 39–44, 46, 48, 90 task infinite, 72 tautology, 31 relevant, 55 Taylor, Barry, vii television, 50 temperament philosophical, 11 Tennant, Neil, 74, 91 term, 21 settled, 27 unsettled, 27 variable, 23 theory, 16, 42, 56 inconsistent, 56 trivial, 56 therefore, 13 thought, 20, 22 as action, 20 content, 20 form of, 19, 21 tm, see Tarskian models tolerance principle of, 78 trivialism, 79 Troelstra, A. S., 118 truth, 45, 47 arithmetic, 54 at a stage, 62

Index conditions, 39, 46, 58 in a model, 39 in a situation, 52 in a world, 37 logical, 12, 13 naïve theory of, 57 necessary, 26, 53 pluralism about, 100 predicate, 13 preservation of, 23, 48, 49, 54 truth value, 38 truthmaking, 50, 57 tuple, 38 Turing machine, 26 type linguistic, 8, 11 typeface, 11 unit square, 72 universal quantifier, 5, 12, 37, 64, 92 untruth, 14 Urquhart, Alasdair, vii utterance, 9, 11 vagueness, 27 validity, 23, 31, 46 analysis of, 23 relevant, 53

143 validx , 29 van Dalen, Dirk, 118 van Fraassen, Bas, vii, 4 variable, 20, 38 Varzi, Achillé, vii Wagner, Steven, 14 warrant, 67 constructive, 67, 74 ways things can’t be, 50 Williamson, Timothy, vii Wittgenstein, Ludwig, 23, 67 world actual, 52, 68 as consistent, 52 as maximal, 52 part of, 49 possible, 12, 15, 26, 35, 36, 38, 40, 49, 50 Wright, Crispin, vii, 100 x-variant, 38

Zack, see Restall, Zachary Zalta, Edward, vii, 99 Zermelo–Fraenkel set theory, 77 zfc, Zermelo–Fraenkel set theory with choice, 44