Relevant Logics and Their Rivals 1 (Western Philosophy Series)

The basic philosophical and semantical theory Richard Routley with Robert K. Meyer Val Plumwood and Ross T. Brady

RELEVANT LOGICS AND THEIR RIVALS

PART I.

THE BASIC PHILOSOPHICAL AND SEMANTICAL THEORY

Richard Routley with Val Plurawood

Robert K. Meyer and Ross T. Brady

1982

©

R. Routley 1982

ISBN

0-917930-66-5 Paper 0-917930-80-0 Cloth

SO W E M U S T F L Y A R E B E L F L A G AS OTHERS DID BEFORE US, AND W E MUST SING A REBEL SONG A N D J O I N IN R E B E L C H O R U S . WE'LL MAKE T H E TYRANTS FEEL T H E STING 0' THOSE THAT THEY WOULD THROTTLE;... (from H . L a w s o n , " f r e e d o m ON THE WALLABY")

Cover design by Ian Sharpe, Graphic Design Unit, Australian National University. Published in the United States of America by R I D G E V I E W Publishing Company P.O.Box 686, Atascadero, C A 93423

Printed in the United States of America by R I D G E V I E W Letterpress & Offset Inc. Independence, Ohio 44131

PREFACE TO PART I (hying to the burgeoning length of the book3 it will be published in two parts. A tentative list of contents for Part II is included. The full text was to have appeared as number 4 in the Monograph Series of the Philosophy Department of the Research School of Social Sciences, Australian national University. In• its wisdom3 however, the School terminated this series.

OTHER PUBLICATIONS OF THE PHILOSOPHY DEPARTMENT, RESEARCH SCHOOL OF SOCIAL SCIENCES, AUSTRALIAN NATIONAL UNIVERSITY School Publications: R. and V. Routley, The Fight for the Forests, First edition, 1973, Second edition 1974, Third edition 1975. Departmental Monographs: M.K. Rennle, Some Pses of Type Theory In the Analysis of Language, 1974. D. Mannison, M. McRobbie and R. Routley, editors, Environmental Philosophy. 1980. R. Routley, Exploring Melnong'a Jungle and Beyond, 1980. Yellow Series (Research Papers of the Logic Group): Why I am not a Relevant1st. Sentential Constants in R. Firesets and Relevant Implication. A Boolean-Valued Semantics for R. Almost Skolem Forms for Relevant (and other) Logics. A Note on

Matrices.

Abellan Logic (from A to Z). Relevant Algebras and Relevant Model Structures. Relevantly Interpolating in HM+. Paraconsistency and Cj. Research in Logic In Australia, New Zealand and Oceania. Green Series (Discussion Papers In Environmental Philosophy): Roles and limits of paradigms In environmental thought and action. In defence of cannibalism I. Types of admissible and inadmissible cannibalism. Unravelling the meanings of life? Semantical foundations for value theory. Nlhlllsms and nihilist logics. Nuclear power - ethical, social and political dimensions. Disappearing species and vanishing rainforests. The irrefutability of anarchism.

I

CONTENTS

Page INTRODUCTION Relevant and irrelevant logics

x

World semantics for relevant logics

xii

Notes on citation, notation, etc.

xiii

The order of topics and the use of the text

xiii

Acknowledgements and origins

xiv

A message to critics and readers CHAPTER 1.

§2.

xv

THE IMPLICATION CONNECTION, AND THE ENSUING INADEQUACY OF IRRELEVANT LOGICS SUCH AS CLASSICAL AND MODAL LOGICS

The main rival positions on implication

1 1

§2.

The inadequacy of material-implication

§3.

The trouble with the metalinguistic repair

17

%4.

What is wrong with strict implication

21

§5.

Conditionals: the theory sought in contrast to extensional and modal attempts

41

§G.

The classical hang-up; how one can go wrong with material implication, and why no classically-based logic can be adequate

50

§?.

Dialectical logic and the repudiation of the dogma that the world is consistent

CHAPTER 2. §2. §2.

5

58

DERIVABILITY, DEDUCIBILITY3

AND THE CORE OF ENTAILMENT

Semantical analysis of the deducibility relation |}The axiomatisation of the relation by a derivability relation (- 3 and the objection from non-transitivism

69 69 72

§3.

Adding conjunction: scmi-lattice logic and holism

78

I4.

The varieties of connexivism

83

§5.

Adding disjunction: conceptivism

95

§5.

Negation:

classical

distributive lattice logic and

De Morgan lattice logic and the parting of

and relevant

ways

110

§7.

On the axiomatisation and semantic analysis of algebra

117

§5.

Systems with non-normal negations and disjunctions: orthologics and relevant quantum logics, basic contraposition logic and its extensions;

120

Rival negations and rival connectives3 and the determinable theory of connectives such as negation and implication: what negation is3 and derivation of the normal negation rule

127

§0.

Jii

CONTENTS

Page §10. Suppression and -its connections with insufficiency and irrelevance; positive and negative suppression characterised, the disastrous effects of suppression, and Lewy 's arguments that the negative paradoxes and Disjunctive Syllogism do not involve suppression refuted

140

§11. Relevant logics and normal relevant logics characterised3 the case against Disjunctive Syllogism and Rule Antilogism elaborated; logical tradition and intuition further considered3 and the coherence of refined intuition; and relevant resolution of the Lewy paradoxes

153

§12. Intensional lattice logic3 and one way of eliminating Modus Ponens

166

CHAPTER 3.

§1.

THE SHAPE OF THE FIRST DEGREE LOGICAL AND SEMANTICAL THEORY3 AND COMPETING PROFILES FOR HIGHER DEGREE LOGICS

The first degree of entailment and implication3 alternative semantics for the first degree theory3 and how to dispose of yet other objections to the semantics Deducibility formulation of FD Entailmental formulation of FD Semantical formulation of FD Adequacy theorem for FD The finite model property and decidability of FD Matrices for FD, including the Belnap 8-valued matrices Suppression freedom and paradox freedom of FD Algebraic analysis of FD

§2.

§3.

§4.

Alternative semantics for the first degree3 and disposing of various objections to the first degree theory

170

170 171 172 173 173 175 176 180 182 190

1.

The elimination of worlds

190

2.

The Australian plan and the American plan; the elimination of *, and four-valued semantics for FD Yet further semantics for FD Alternative tableaux and proof-theoretic methods for FD

192 199 200

Harmonious consequences ofs and applications of the first degree theory; additional connectives and further constants under the theory3 and alternative formulations of the theory

206

The need for, and shape of3 higher degree investigations. An initial classification of types of systems and classes of questionable principles

219

§5.

The matter of choice of logic; and why E is not entailment3 nor R (the) system of relevant implication 230

§6.

The criterion of strength3 and relevant (and otherwise appropriate) extensions of E and of R Connexive extensions of E Normal modal strengthenings of E, and non-normal modal strengthenings of E XaJL

240 242 246

CONTENTS

Page Extramodal strengthenings of E Strong distributional strengthenings of E and of R Minglish extensions of R and E §7.

Other - sometimes questionable or vacuous - criteria for the choice of system

§fl. Counterexamples and counterarguments to excessive principles of T3 E and R; and the case against E and R completed §3.

Contraction and Reductio principles3 and the failure of Contraction in paradoxical situations and theories and where self-reference features

CHAPTER 4.

THE SEMANTICS OF ENTAILMENT AND SUFFICIENCY CONDITIONALS3 RELEVANT AFFIXING SYSTEMS WITH NORMAL CONJUNCTION3 DISJUNCTION AND NEGATION3 AND THEIR EXTENSIONS

%1.

Syntax of the sentential systems

§2.

The preferred formulation of B:

247 247 252 255 263

278

284 285

the removal of affixing

rules and distribution

292

§3.

Formal semantics for B and extensions

294

§4.

Valuations, interpretations3

301

§5.

Semantic implication3

§5.

Semantic completeness: synopsis of the argument and the detailed argument

305

Alternative semantics for affixing systems: the American plan3 with classical-style negation and without stars3 completed

319

§7.

1. 2. 3. §5.

§3.

validity

truth and soundness

Classical-style semantics for B and their adequacy Some positive extensions of B and the problem with negative extensions Reduced classical-style semantics for system C, and extensions and complications thereof

325 328

334

Semantical analysis of various principles of superstrong relevant logics3 and present limits of the semantical methods

340

CHAPTER 5.

§2.

319

The irreducibility of models for sublogics of C3 the addition of disjunctive rules in unreduced modellings and their role in reducing modellings

%10. Synopsis of certain frequently referred to systems and their model structures

%1.

302

FURTHER INVESTIGATION OF RELEVANT AFFIXING SYSTEMS AND THEIR PARTS.

347

348

The conservative addition of constants3 the logic of t3 and inadequate definitions of routine modalities

348

Alternative and extra connectives3 including intensional disjunctions3 intentional conjunction3 fusion3 consistency3 and'independence; and Anderson and Belnap's "case" against Disjunctive Syllogism

356

b)

CONTENTS

Page §3.

Semantically closed relevant systems, dialectical resolution of systemic semantical antinomies, inconsistency and incompleteness

367

14.

Classical-type connectives and crypto-relevant logics; so-called classical (C) and superclassical (K) relevant logics

370

Nondegeneracy more explicitly, null and universal worlds, classical negation redone, and the truth-perseroation connective

379

§5.

§5.

What have been called normal model structures:

y and its

limits

387

§7.

Hallden reasonableness and normal characteristic matrices

392

§5.

Parts, initial separation results, and modelling simplifications and variations

394

§9.

Decidability and undecidability. model structures

399

I. Filtration of relational

APPENDIX 1.

The semantics of entailment IV:

E, II' and II"

APPENDIX 2.

The pure calculus of entailment is the pure calculus of entailment

407 425

POSTSCRIPT TO THE APPENDICES

430

REFERENCES

435

INDEX (by Jean Norman)

452

T E N T A T I V E CONTENTS O F PART II INTRODUCTION TO FURTHER TOPICS CHAPTER 6.

RELEVANT LOGICS WITH NON-NORMAL NEGATIONS OR DISJUNCTIONS, AND THEIR MORE FAMOUS IRRELEVANT EXTENSIONS: INCOMPLETENESS AND INCONSISTENCY CONTINUED

%1.

Relevant De Morgan minimal logics and their extensions.

§2.

Relevant minimal logics and their extensions, and recovery of orthodox relational semantics for minimal and intuitionist logics

§3.

Weakening positive logic strictly and relevantly; Lewy and related systems, and relevant da Costa logics and their extensions

§4.

Reformulating stronger relevant logics in terms of the falsum x, and more on the role of constants in such logics

§5.

Relevant orthologics, and their elegant Gentzenisations and interpolation properties, and inelegant semantics

§6.

Options for relevant quantum logics

17.

Other alterations of disjunction, and Urquhart systems

v

TENTATIVE CONTENTS OF PART II

§8.

Contractionless relevant logics: properties of RW3 EW3 TW and their relatives

§9.

The accommodation of contradictions and paradoxes in relevant logics: paraconsistent and dialethic logics

CHAPTER 7.

NON-NORMAL RELEVANT SYSTEMS AND THEIR MODAL EXTENSIONS AND IMPLICATIONAL SYSTEMS CONTAINING THE PRINCIPLES OF FACTOR AND SUMMATION

%1.

Variously occupied and overlapping situations:

§2.

Situation occupation more generally, and further bizarre systems

§3.

Relevant systems related to modal systems, and improved completeness arguments for non-normal affixing systems

%4.

Why Factor and Summation are tempting

§5.

How Factor and Summation go bad in the presence of exported syllogistic principles

§£.

The affixing I systems

§7.

The first degree of I systems and of strict implications

§5.

Alternative proof methods for I systems: subscripted natural deduction methods and tableaux analyses, and metavaluations

CHAPTER 8. §1.

non-normal systems

MULTIPLYING CONNECTIVES3 AND MULTIPLY INTENSIONAL LOGICS I. SYSTEMIC CONNECTIVES

Internal and external negation: logics

a prelude to Meinongian sentential

%2. Multiplying systemic connectives:

relevant alethic and tense logics

§3.

Reductions of relevant deontic and imperative logics3 through sanctions

%4.

Permissivist ethics3 nihilist logics3 and relevant deontic logic

§5.

Non-normal modal superstructures3 conventionalist modal logics relevantly based

§6. Applications of relevant logic in the philosophy of science: relevant explanation3 probabilityverisimilitude3 vagueness3 etc. CHAPTER 9.

MULTIPLY INTENSIONAL LOGICS II.

RELEVANT NEIGHBOURHOOD CONNECTIVES

%1.

Semantical analysis of functors which do not distribute over conjunction or disjunction

§2.

One-place non-systemic connectives

§3.

Relevant preference and value theory: neighbourhood semantics

%4.

A general relevant semantical theory for n-place connectives; the route to the universal semantics

§5.

Double and multiple implication systems: theory of conditionals

%6.

Other applications of the logics: the theory of evidence3 confirmation3 aboutness3 weight of argument

VA.

examples of two-place and

a major example is the

TENTATIVE CONTENTS OF PART II

CHAPTER 10. §2.

IMPLICATION, ENTAILMENT, AND NECESSITY

Entailment is_ logical implication

§2. Bounds of the logics of implication and necessity; logical necessity §3.

Logical implication and reduced modellings: many others; the theory of UL

the features of

system

(i.e. NR) and

§4. Semantic entailment, truth, soundness and completeness in the reduced case §5.

More comprehensive relational semantics for logical implication: unreduced modellings for affixing systems; and non-normal semantics for E, R, etc.

56.

Negated entailments: a problem of Alan Ross Anderson, and other connected problems

%7.

E = N what?, and the connections of entailment and necessitated implicational systems more generally

§5.

The choice of modal systems: S4 vs S5 as explicating logical modalities and resolution in favour of S5, and genuine fallacies of necessity

CHAPTER 11.

THE INTEGRATION OF LOGICAL METHODS. OPERATIONAL SEMANTICS AND ALTERNATIVE PROOF-THEORETIC AND SEMANTICAL FORMULATIONS OF RELEVANT AFFIXING LOGICS AND AFFIXING RIVALS

§2.

Theoremless logics, and semantics for pure implicational logics with a prefixing or suffixing rule

§2.

Peripatetic logic, reasoning and progressive reasoning, and the ancient and modern rejection of the law of Identity

§3.

Adding normal connectives operationally: and conjunction, disjunction and negation more generally and the basic Peripatetic logics

%4. Reduced operational semantics where conjunction is normal and negation De Morgan: operational modellings for CM and extensions such as the big-time systems T and R §5. Reduced suffixing semantics for Ackermann-Anderson-Belnap system E and for immediate neighbours §6. Reduced operational semantics for logical implication, for S4C and extensions such as NR §7.

Unreduced operational semantics for affixing systems, especially non-exportative systems, and intermapping of operational and relational semantics

%8. More comprehensive operational semantics for logical implication, and operational modal semantics §5.

Multiplying modalities operationally

§20. The derivation of semantical tableaux methods for relevant logics §22. The derivability of subscripted natural deduction methods for relevant logics §22. The derivability of Gentzen methods for relevant logics §23. Other Gentzen formulations

TENTATIVE CONTENTS OF PART II %14.

Decidability and undecidability II. Filtration of operational model structures, and undecidability of Pittsburgh systems

§25. Other interpretations of relevant logics: geometric and categorytheoretiCj and applications thereof CHAPTER 12.

THE ALGEBRAIC ANALYSIS OF RELEVANT AFFIXING SYSTEMS

§1. Ackermann and De Morgan groupoids §2.

De Morgan groupoids algebraize B

§3.

Extensions of B and its parts have algebraic counterparts

%4.

Representation of the algebras in model structures

§5.

Semantical analyses connected with algebraic analyses

§5.

Embeddings and conservative extensions

§7.

Some finite algebras of importance in the relevant enterprise

§5.

Finite relevant logics3 and axiomatisations of relevance-establishing matrices

§9.

Consequence and relevant consequence relations and operations, and some applications

§20. Relevant logic and combinators3 CHAPTER IS.

and relevant combinatory logic

THE MORE GENERAL SEMANTICAL THEORY OF IMPLICATION AND CONDITIONALLY I. NONAFFIXING REPLACEMENT SYSTEMS AND NONSUFFICIENCY CONNECTIONS

§2.

The general semantical rule for evaluating implication

§2.

The basic replacement system F3 some of its extensions3 and some of its subsystems; and semantics for these systems

§3.

Adequacy of semantics for replacement systems

%4.

Simplifying the semantics:

§5.

The general rule for evaluating negation3 and obtaining specialised rules for certain negation determinates

%6.

Classical negation3 and weak strict implicational systems

§7.

Combining affixing and replacement systems

§8.

Negation and contradiction

§3.

General rules for evaluating conjunction and disjunction3 and varying the normal rules for connexive and nonadjunctive logics

removing V

§20. Semantical analyses of connexive logics and their problems §22. Applications to the general theory of conditionals: counterfactuals3 conditionals, dispositionals

causation3

§22. An impossible theory of conditionals3 and relevant logics of conditionals CHAPTER 14.

§2.

THE MORE GENERAL THEORY OF IMPLICATION AND CONDITIONALLY II. NONREPLACEMENT RELEVANT SYSTEMS AND THE LOGIC OF NOTIONS OF THE ORDER OF PROPOSITIONAL IDENTITY

Replacement failure: analysis

propositional identity and the paradox of

vUJL

TENTATIVE CONTENTS OF PART II

§2.

Avruda-da Costa P systems and adjacent non-replacement relevant systems

§3.

Semantical analysis of basic positive system P+

§4.

Semantics for extensions of P3 especially the important H systems

§5.

Substitutional extension of FD3 and Priest's "sense" system

§6. Anomalous non-replacement systems: Ackermann's minimal system §7.

Smiley's curious systems and

Use of the semantical theory to establish such results as decidability

§8. Modal analogues of non-replacement systems §5.

The logics of independence and propositional identity: "metatheory" accomplished systemically

more

§10. Integrating non-replacement with affixing systems §11. Other issues generated3 and open problems CHAPTER IS.

THROWING AWAY THE CLASSICAL LADDER

§1.

Intensional semantics for intensional logics: toilless semantics3 general metavaluational semantics3 and other assorted semantics

§2.

Relevant logic semantics done relevantly: classical-type metalogic

§3.

Classical and nonclassical rule and derivability structures

dispensing with a

§4.

Towards a nonclassical theory of inference and derivability

§5.

Further conditions for adequacy on choice of logics of inference and deducibility: depth relevance

§7. Meeting earlier conditions of adequacy3 such as those concerning recovery §5.

On the choice of logical systems for deducibility and conditionality; and arguments to affixing systems for entailment

§5.

Tentative selection of ranges of systems3 and of specific systems3 for entailment3 implicational and conditionality

§10. Relevant logic as universal? CONCLUSION REFERENCES INDEX

(by Jean Norman)

0.1

RELEVANCE NOT OF THE ESSENCE, BUT A BV-PKOVUCT

INTRODUCTION This volume is primarily a logical and semantical investigation of an extensive class of zero-order intensional logics, i.e. of intensional logics which do not include variable binding devices such as abstraction operators, descriptors, quantifiers or their equivalents. The effect of adding variable binding devices will have to be reserved for another volume. Many of the philosophical investigations and issues which are presupposed by or arise from this predominantly formal study will, we still hope, appear in yet other publications (e.g. Beyond the Possible3 long in preparation). The separation of these matters is admittedly deplorable (whether the proposed multiplication of book-entities is also deplorable will be left for readers to decide). The exclusion of quantifiers and descriptors deprives the logics of some of their interest and usefulness in the analysis of natural languages and philosophical and other argumentation, and the partial exclusion of intimately connected and motivating philosophical issues is artificial and weakens the case for such a detailed study of particular intensional logics. However this volume is evidently long enough already. Relevant and irrelevant logics. We focus on those intensional logics that, satisfying weak relevance principles, have become known as relevant logics. The class of sentential logics that satisfy weak principles of relevance is however wide and includes many logics which are, in principle, rivals to the position(s) we shall be advancing. We want it to emerge with stark clarity, however, that our main concern is not really relevance at all - the appropriate sort of relevance is a byproduct of any good implication relation, which comes out in the wash. Only one weak necessary condition for relevance features in what follows: that is all 1 . A study of relevance, of the sorts of relevance, of sufficient conditions for relevance, ... - all these matters are philosophically interesting, and some of them are important, especially for the logics of evidence and probability - but they are not our present concern. For this reason the name 'relevant logics', or 'relevance logic', is not entirely satisfactory - perhaps even, to lodge a much stronger claim, unfortunate - since the name tends to suggest, wrongly, that relevance is of the essence, instead of being a peripheral concern. Nonetheless the name has a point, and it is a little late to change it. What our concern is with is implication and its varieties, and in particular with genuine implication in the sense that amounts to total sufficiency. Thus our concern is, in the first place, with sufficiency, or, as it is otherwise equivalently put in the logical case, with complete logical dependence, with total inclusion of logical content, and so on. Implication is not confined however to logical implication or deducibility; we are very much interested in having our systems apply to other sorts of sufficiency, physical or law-like sufficiency in particular, and to provide the bases, in enthymematic ways, for analyses of partial sufficiency, for instance for insufficiency conditionals for conditionals, for example, which are obtained from genuine sufficiency conditionals by suppression of true or necessary antecedents (or, symmetrically, of false or impossible consequents). This will take us back through the usual logics of the textbooks, to intuitionistic logic and modal logics, and, in the extreme case, to classical two-valued logic. 1

This condition is the variable-sharing requirement, WR, introduced in 1.1. Beyond the sentential stage, however, further considerations are bound to enter (see UL, §7, and Routley 77).

X

0.2

THE MAWIF0LP DEFICIENCIES OF CLASSICAL LOGIC

Although we shall try to persuade the gentle reader that a goodly number of the stronger relevant logics, including some of the better known ones such as system E and its neighbours, are misguided as analyses both of entailment and of associated notions to be investigated such as implication and conditionality, nonetheless the main enemies have been, and remain, irrelevant logics, most notably classical logic with its material implication, but more recently modal logic with its variety of strict implications. Nor are these all: there are other though less important irrelevant logics, such as positive logics, intuitionistic and minimal logics, inconsistent logics and fuzzy logics, even productive logics and unproductive logics, and so forth. But the main attack will be upon classical logic and its extensions and elaborations. The contemporary state of complacency with respect to the manifold deficiences of classical logic and classical theories reflects, not only the usual (if deplorable) scientific process of entrenchment whereby once revolutionary young theories become, as they age, conservative members of the establishment, but also the fact that classical logic is not greatly subject to the testing process; for it has come to be taught in a way which largely insulates it from application in real and live argument. Students of logic in philosophy courses may be taught to pick out a few of the grosser fallacies which are classically invalid, but rarely learn to marry their philosophical and logical training, and the fact that the bulk of the defective reasoning they encounter cannot be handled classically passes unmarked. (It is symptomatic of the situation that when a need is felt for logical tools applicable to real reasoning practice, there is often a turning back to traditional logic and traditional fallacies, despite the comparatively unsystematic and impoverished nature of that theory.) If more attempt were made to apply the classical theory in reasoning, its inadequacy, and especially that of its treatment of such central notions as deducibility, valid argument, and sufficiency, would be much more widely noticed. But the very limitations of the classical theory set up a vicious circle, since the more these limitations themselves discourage application, the more they encourage protection of the theory from testing against real reasoning, and the easier it becomes for its very failures to seem unquestionable fact. Only this sheltering from the application of classical theory to live reasoning could have made it possible for so many classical logicians to maintain so vigorously that a logic which allowed one to deduce any necessary proposition from any other, for example, thus laying waste the prime field for the use of reasoning - the a priori - is harmless, or even desirable, and to contend that alternative non-classical logics which avoided such results were not worth investigating, except as symbolic exercises. The fact that classical logic has so many limitations and is simply inadequate for many basic reasoning tasks is one of the reasons why it has failed to live up to its early promise as a tool for clarifying and in some cases resolving philosophical and methodological issues. It is for similar reasons that classical logic bears a large measure of responsibility for the growing separation between philosophy and logic which there is today, to the detriment of each. It seems so obvious that an implication relation which makes any necessary truth imply any other must fail as a tool for philosophical and other sorts of analysis that it is, on face of it, surprising that so many philosophers have shown indifference or hostility to acquiring, or even investigating acquiring, better tools, since logics without such drawbacks are available or not impossibly difficult to devise. If classical logic is a modern tool inadequate for its job, m o d e m philosophers have shown a classically stoic resignation in the face of this inadequacy. They have behaved like people who, faced with a device, designed to lift stream water, but which is so badly designed that it spills most of its freight, do not set themselves to the design of a better model, but

0.3

WORLV SEMANTICS

rather devote much of their energy to constructing ingenious arguments to convince themselves that the device is admirable, that they do not need or want the device to deliver more water; that there is nothing wrong with wasting water and that it may even be desirable; and that in order to "improve" the device they would have to change some features of the design, a thing which goes totally against their engineering intuitions and which they could not possibly consider doing. Some point to the beautiful, clean, simple lines along which the device is constructed, and the fact that when performance and function are overlooked, this devicer considered as an abstract construction, is really just as good as any other; in no sense can it be claimed then to be inadequate, or any other device more adequate. Meanwhile, most of what little water the community has continues to be obtained by the traditional, tedious, limiting, hand-carrying methods. In the investigation which follows the emphasis will be on designing better tools, and, ideally, working towards selecting the best tools1 - locating even if only approximately, the optimal systems for the implication notions analysed rather than reconciling oneself prematurely to inferior ones. World semantics for relevant logics. The main methods adopted for investigating these superior logical systems, relevant logics, are world semantics. Appeal to worlds, and realisation of the importance of considering alternative worlds, goes back to the beginning of philosophy (as we know it). Anaximander (c.570 BP), successor of Thales in the School of Miletus, described the fundamental stuff out of which the world is made as 'the boundless', since out of it are formed countless worlds. Democritus and others put the appeal to worlds to theoretical work. But a history of the extensive role of worlds in philosophy (not really our present concern, for which it is enough that worlds have such a role) remains to be written. The emphasis is on semantical analysis in terms of worlds, not merely because such methods yield a wealth of logical results, often in easy and intelligible ways, but because worlds semantics affords the prime example of a philosophically significant semantics. World semantics are, in contrast for instance with algebraic semantics, intuitively and philosophically significant in themselves, because the notion of world, in contrast to that of (say) monoid or pseudo-Boolean algebra, is historically-rich and intuitively accessible, because the notion of other worlds is easily and often ordinarily grasped. World semantics are also philosophically significant because they fit the relevant systems for which semantics are provided into a framework of explanation which enables illuminating comparisons and connections to be made with other systems which also have world semantics; for example, they enable one to see, in the case of a relevant logic, how it stands in relation to other systems which also fit into the same framework, e.g. various strict implications. Thus the framework enables illuminating comparison of relevant logic with material and strict implication, and in general enables assessment and comparison of relevant logics and their rivals, in terms of the types of worlds, considered, conditions on the worlds involved, and so on. In addition, world semantics are rich in connections with important philosophical and pre-formal notions - such as information, inclusion of content and propositional identity, etc. which may be defined in much the way familiar from strict implication. Important and very natural analogues of the defective strict notions can be defined for relevant logics, in effect showing how those important notions have been seriously and systematically distorted by the restrictive assumptions of strict and 1

Similarly Kielkopf 77 aims to define 'the best possible formal reasoning'. The matter of optimisation modellings for best choice of entailment system is broached in 3.7, and the matter of determining a superior logical technology in chapter 15. xLL

0.4

REFERENCES ABBREVIATIONS,

SYMBOLIC NOTATION

material implication. Nor is this all: world semantics sharply illuminate many other intensional notions, and indeed enable a general treatment of intensionality (as later chapters will reveal). Of more immediate importance, world semantics facilitate the elaborations of fundamental arguments concerning such notions as entailment, valid argument, and so on (as chapter 2 will show). It is for all these sorts of reasons that world semantics will occupy a central place in what follows. Moreover - this is yet another virtue of world semantics - other semantical and analytical methods fall out of world semantics: chapters 11 and 12 are designed to establish just such a unification of logical methods. Notes on citations, notation, etc. Two forms of reference are used. Texts and papers to which we refer frequently are assigned special abbreviations (e.g., GR,ABE, SE V) and otherwise works are cited by giving the author's name and the year of publication (with the century deleted in the case of the twentieth century). Where the author has published in the one year several papers that we want to refer to, we order the papers alphabetically (e.g. Meyer 74a, Meyer 74c). We have included in the bibliography only items that are actually cited in the text (for much more extensive bibliographies of entailment and relevant logics see ABE and Wolf 76). We use some standard abbreviations, such as 'iff' for 'if and only if' and 'wrt' for 'with respect to'. The metalanguage is logicians' ordinary English enriched by a few symbols, most notably '>' read 'if ... then ...' or 'that ... implies that ...', '&' for 'and', 'v' for 'or', '-' for 'not', 'P' for 'some' and 'U' for 'every'. We do not always use these abbreviations however, and sometimes write expressions out in English. The quantifiers are the familiar non-ontological ones explained in GR; why we insist on these we explain in chapter 4, §3. Bracketing conventions are the same adaption of those of Church used in GR and also in ABE. The relative strength of the main connectives used is explained in chapter 4, §1. We have abbreviated most cross references in obvious ways, e.g. 'see 3.3' means 'see chapter 3, section 3' and 'in §4' means 'in section 4 (of the same chapter)'. The labelling of theorems and lemmata is also chapter relativised. For labelling of systems and subsystems we have frequently been forced for obvious and often substantial reasons into using notation resembling that of ABE. The notation, commonly adaptions of that of ABE, is, for the most part, explained as we go. As we explore, however, a great many systems not investigated at all in ABE or, often enough, anywhere else, we have invented many system labels. But generally the systemic labelling is thoroughly unsystematic and there will be, when we have a better idea of the range and types of systems, a need for a systematic theory-bound relabelling of systems. In quoting other authors we have taken some liberties; most significantly we have adjusted symbolism within quotations to match that adopted in the text. The order of topics and use of the text. The order of chapters in the text often represents the order in which semantical problems were solved rather than some more rational ordering. A superior ordering would perhaps have transposed the last chapters of the text on replacement systems and proposltional identity systems so that they followed chapter 3. But the organisation into a monograph of papers that had become too large to publish was already well-advanced before the analyses of the last chapters were devised.

0.5

ACKNOWLEDGEMENTS, ANV COMPOSITION OF THE TEXT

It is some compensation, then, for the questionable ordering of topics, that many of the chapters are relatively independent and can be read without substantial reference to preceding chapters. In particular, chapter 2, a major organisational and scene-setting chapter, can be studied without reference back to chapter I, and chapter 4, which perhaps contains the fundamental material on formal semantics for entailment logics, is self-contained. Such later chapters as chapter II, which develops the earliest written material in the volume and which synthesizes many logical techniques, and chapter 12, which incorporates algebraic studies of relevant logic, are also largely self-contained and can be read, we hope, without much outside reference. The volume is not intended as a popular work; in fact we shall be well satisfied if it is sufficiently unpopular. Nor is the volume intended as a textbook, though we should be flattered If someone were to find such a use for parts of it. Until recently very very little of the material had ever been presented in seminars or lectures, and a great deal of it has not been published anywhere else, though some parts of the work have been in limited circulation for several years. We do not believe that it is any the worse for its limited exposure. Acknowledgements and origins. Practically all of the material represents the results of independent research (in the usual, questionable, sense), though it will be evident that parts of it build on, or are indebted to, the work of other investigators. In particular, we have taken advantage of the work of (what is sometimes known as) the Pittsburgh School, as so far mainly consolidated in ABE. At the same time we are highly critical of much of the underlying philosophical motivation and theory of the Pittsburgh group 1 (see especially 3.4-3.8 and 5.2). In general we have tried to record our main debts to other investigators at the relevant points in the text. But we have several debts which fall outside this class, and there are some special debts we should like to record separately. Firstly, we have included in the text other workers' material. For example, we have made heavy use early in chapter 1 of Geoffrey Hunter's unpublished notes on if; and we have incorporated material worked out by others or in collaboration with others, most notably Andrea Loparic and Graham Priest. Secondly we are much indebted for several textual points to Michael Dunn and Newton da Costa, who acted as referees for the volume, and to Malcolm Rennie. On the production side we have been generously helped, in almost every aspect from initial research to final proofing, by Jean Norman, without whose assistance the volume would have been much slower still in appearing and much inferior in final quality. We are indebted to many typists of the Philosophy Department, RSSS, Australian National University, who, over many years, have typed many drafts and final copy of a very difficult typescript. Many other people helped in lesser ways with the organistation, printing, financing and distribution of the text. This text has been a long time in the writing, and a long time in completion (with the result that it has been listed as forthcoming every year since 1977, and that some jokers do not expect it to appear in their lifetimes). The earliest material included, based on V. Routley 67 and R. Routley's work on I systems dates back to about 1966. The main semantical ideas the book elaborates were worked out first between about 1969 and 1972, and the limited monograph of Routley and Meyer from which the text grew was first drafted in 1972. As noted, some of the later material has been in circulation for some time, for example 1

None of the authors, except Meyer, can be accounted a member of that group. The Australian relevant logic group, though it used to profitably liaise and cooperate with the (now largely defunct) Pittsburgh group, is an independent movement with its own life, momentum and direction.

0.6 MESSAGE TO CRITICS AMP REAPERS

chapter 1 which was completed in 1976, and chapter 4 (formerly chapter 3) which has been cited from about the same time. The text developed from joint work of Routley and Meyer (mainly included in Appendix 1 and chapters 4 and 12) which grew beyond article form. Progressively, as the publication of the text was delayed from year to year - largely due to dissatisfaction with earlier forms, a thoroughly misguided attempt to be definitive in a rapidly advancing field,1 and other commitments of the first author more and more unpublished work of the authors was absorbed into the text (as recorded subsequently in notes). The text is based squarely on this joint work. There was a difficult policy decision as to where to draw the line as regards other authors of the volume. There was no doubt that Brady and Mortensen2 and V. Plumwood (who formerly called herself V. Routley3) should certainly be included: each wrote pieces specifically for the book and contributed in substantial ways to the logical theory developed. It was with potential authors not included as authors that there was a problem, for example, Hunter, Loparic and Priest who contributed nothing specifically for the book but whose work we have absorbed in the text (with due acknowledgement). Had they been included as authors several other people whose work we also made use of would in fairness have had to be listed also, and a very slippery slide would have been initiated. Despite the arbitrarily limited authorship, the authors are well aware of the extent to which their work, although containing much co-operative but often individual labour, is a social product which has relied heavily on the work of many others, especially all those who contributed to the elaboration of connectedness logics and worlds semantical theory. A message to critics and readers. The text has been almost entirely written by the first author, contributions from other authors have been, to varying degrees, overwritten by him. Accordingly, the philosophical positions adopted are primarily though not exclusively his insofar as they are anyone's, and should not be automatically attributed to other authors. Since every view presented in the text is held by some author or authors, a critic who attacks any single view held in the text will accordingly not be attacking a strawman (since Strawman is not among the authors); but a critic who assembles views from different sections runs some risk of attacking only a composite author, so to speak. However the responsibility for mistakes remaining rests primarily upon the first author. Finally we should appreciate feedback from critics and readers, in particular solutions to open problems, ways of strengthening or improving arguments, details of (what will be inevitable) errors and misprints, and constructive criticism. To any readers and even to critics: Good Luck! but Take Care! J

The idea of a solution to all the main problems of relevant logics, at least at the sentential level, which was one of the factors delaying the text, has had to be given up - like the idea of some sort of ephemeral definitiveness. Time alone has done for the latter illusion given the rapid increase of work in the area - which has not however come into its own yet, remaining very much a minority pursuit. Instead we now tend to regard the text as the first of an ongoing series, though we have not presented it as such, since some of impetus and results needed for a worthwhile sequel have yet to eventuate, and may not (as well as because promised second volumes have a poor appearance record).

2

C. Mortensen's contributions are to Part II.

3

Thus for V. Routley read V. Plumwood, and for Routley2 read R. Routley and V. Plumwood. XV

1.0 IMPLICATION

THE FUNDAMENTAL LOGICAL NOTION

CHAPTER 1 THE IMPLICATION CONNECTION3 AND THE ENSUING INADEQUACY OF IRRELEVANT LOGICS SUCH AS CLASSICAL AND MODAL LOGICS Implication, the main relation studied in this work, is fundamental in reasoning, particularly in deductive reasoning. Hence its central importance in philosophy, logic, and mathematics, where such notions as entailment and valid argument are central. The importance of implication in mathematics, if not entirely obvious, emerges clearly from Russell's initial definition of pure mathematics (37, p.3): Pure mathematics is the class of all propositions of the form "p implies q", where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. The definition - whatever its defects, and Russell himself was later to argue that they are manifold - at least succeeds in revealing the fundamental role of the implication connection in mathematics (and incidentally, in the sharing of variables requirement, the importance in mathematics of relevant connections). Implication is likewise basic in logic, where it (along with its derivatives such as consequence) is one of the most important relations to be explicated: ... implication seems to be the most important connective in any logical system (Rasiowa 74, p.167). It is, moreover, sometimes said to account for the importance of logic: The chief importance of logic lies in implication ... (Quine 51, p.xvi). Lastly, but first in importance in this group, philosophy: since the essence of philosophy, and especially of dialectics, lies in argumentation and reasoning, implication is once again a crucial connection. It is fundamental in the formalisation of philosophical discourse and arguments, where key features of concern are conditionality (represented by 'if ... then ...'), entailment, and valid argument. None of the main going systems are at all adequate for these purposes. New systematisation is essential, not only for adequate accounts of implication and valid arguments, but also for derivative reasons. A great many important philosophical notions depend, in one way or another, on satisfactory implication relations, for their semantical analysis, for a satisfactory explication; for example such diverse notions as aboutness, empiricalness, moral commitment, evaluativeness, semantic information, degree of conclusiveness of arguments, confirmation and rationality. (Exactly how improved philosophical explications can be provided, which avoid many standard difficulties, is explained in detail in BP and, for certain cases, in UL.) On the whole there has been far too much effort expended on trying to accommodate philosophical clarifications to going logical systems, and especially to prevailing classical-style accounts of implication and valid argument - rather than on trying to develop logical systems to handle the evident data and to deal with going philosophical problems. Thus prevailing Classical-style logics, and classical logic in particular, have distorted

1

1.1 RIVAL POSITIONS AS TO IMPLICATION

the initially-obvious resolutions of a number of philosophical problems, and by their distortions generated several gratuitous philosophical problems (see UL). Classical logic, although once and briefly an instrument of liberation and clarification in philosophy and mathematics, has, in becoming entrenched, become rigid, resistant to change and highly conservative, and so has become an oppressive and stultifying influence. For instance, classical logic has exerted a very conservative influence on philosophical problem-solving, especially paradox resolution, and has hamstrung much enterprise. Classical logic is, as now enforced, a reactionary doctrine. Nowhere is the oppressive effect of classical logic more evident than in its treatment of implication. §1. The main vixial positions on isnvliaation. In view of the philosophical and logical importance of implication and its derivatives, it is hardly surprising that questions as to the properties of implication have been a major source of dispute, at least from Greek times, through the Middle Ages, and into modern times. In the modern dispute (which, surprisingly, had a main point of origin at Harvard: see Parry 74) several main schools can be discerned, many of which have their roots in ancient or scholastic positions. Briefly the positions are as follows (they will be characterised in more detail later, when variations and other positions will also be distinguished): (1) The PhiIonian position (held by hookers) according to which implication is adequately (enough) captured by the material conditional of classical logic. (2) Metalinguistic positions which claim that implication is a metalogical relation not to be confused with the material conditional. Both these first positions adhere to extensional languages, and depend ultimately on the mistaken assumption that the extensional way of doing things is the only clear and unmuddled way of proceeding. (3) Strict positions (maintained by stricters, i.e. strict hookers) according to which logical implication at least is to be explicated through some brand of (C.I. Lewis or E.J. Lemmon) strict implication. Though positions under (3) are non-extensional they are closely connected with those under (2) (which simulate intensionality), the divergence beginning only with issues as to the iteration of intensional connectives and their combination with quantifiers. (4) Positive, intuitionistic and constructive positions which characteristically base their theory on some conservative extension of Hilbert's positive logic. (5) Similarity theories (like those of Stalnaker and D. Lewis, developing Ramsey's position) according to which an important class of conditionals, at least, is to be explicated semantically through similarity relations on the possible worlds supplied in furnishing semantics for (3).

1.1 RELEVANT, C0NNEX1V1ST AMD C0NCEVTTV1ST

THEORIES

(6) Relevant theories such as those of Ackermann, and of Anderson and Belnap, which reject the paradoxes of implication and, most characteristically, the classical principles of Disjunctive Syllogism, i.e. in symbols to be explained shortly, A &(~A v B) B, and Antilogism, if A & B -* C then A & ~C ->• ~B. Such relevant positions will be the main theories investigated and defended in the text. (7) Connexivist positions which reject or qualify such principles of classical logic as Conjunctive Simplification, A & B -»• B in favour of theses which imply Aristotle's principle, ~(A ~A) . (8) Conceptivist theories, developing from Parry's theory of analytic implication, which qualify Addition, A -*• (A v B) , and more generally require that when A -»• B holds the concepts of B (construed, e.g., in terms of the variables of B in the case of sentential logic) are included in those of A. The positions sketched are not exhaustive. There is as well a range of patch-up positions which try to define a new connective for implication in terms of the rest of apparatus supplied by (3), there are other positions which try to characterise conditionals in probabilistic terms, and so on. None of the first five positions is satisfactory. For no implication (or conditional) is going to be adequate for the analysis of natural language and the logic of discourse, or adequate for most philosophical purposes, which is not relevant, in which the consequent of a true implication is not somehow connected with the antecedent, or the antecedent relevant to the consequent.1 A necessary, but not sufficient, formal condition for such relevance at the sentential level is furnished by Belnap's weak relevance criterion: WR. That A implies that B is a theorem, in symbols A -*• B, only if A and B share a sentential variable, where A and B are well-formed formulae built up from sentential variables p, q, r, p', ... using just sentential connectives such as -»-, & ('and'), v ('or'), ~ ('not'), etc.2 It is immediate, then, that logics like strict, positive and similarity theories which include as theorems paradoxes of implication (such as, in the first cases, p & ~p q, q -»• (p -»• p) and q -»-.p v ~p) and so violate WR, are irrelevant, and hence are inadequate for important purposes. The emphasis in the formal work of this volume is on relevant logics: irrelevant logics on which formal investigations of implication have in the past been excessively concentrated will be studied only insofar as they appear as extensions or by-products of relevant systems. 1

Though weak relevance is not a fundamental matter for entailment (see FD and BP) but a derivative feature of a good sufficiency relation, it provides an extremely important formal test of adequacy.

2

There is accumulating linguistic evidence that natural language implications and conditionals are relevant, see e.g. T. van Dijk 74. In fact van Dijk goes further and advances the interesting, if false, thesis that all natural language connectives are relevant.

3

1.1

STRATEGIES IN

THE

FACE OF 1MPLJCATJONAL

PARADOX.

There are at least as many sorts of relevant sentential logics as there are resolutions of C. I. Lewis's "independent proof" of the paradoxical scheme A & ~A -*• B. The argument for this scheme - which has always seemed the hardest obstacle proponents of relevant logics have to surmount - is as follows:(1)

(2) (3) (4) (5) (6) (7)

A & ~A A A & ~A ~A ~A -»• ~A v B A & ~A -*• ~A v B A & ~A A & (~A V B) A & (~A v B) + B A & ~A -»- B

The distinctive principles used in this argument are:Conjunctive Simplification: A & B -»• A and A & B -*• B, used in lines (1) and (2). Addition: C -* C v B, used in line (3). Disjunctive Syllogism: A & (~A v B) -»• B, used in line (6). Rule Syllogism: A B, B ->- C A C, i.e. where A -»- B and B C are theorems (i.e. provable) so is A -»• C, applied in obtaining lines (4) and (7). Rule Composition: A B, A -»• C A (B & C ) , applied in obtaining line (5). Alternatively Rule Factor: A - * B - * C & A - > - . C & B, may be applied to obtain line (5) from line (3) thereby eliminating lines (1) and (4) and use of Rule Composition. Strategies abandoning each of the first four (uneliminated) principles have been investigated to varying extents, and rather evidently (given recent semantical insights) further strategies could be devised which avoid the paradox but give up less than tried moves do. The main tried strategies, each of which can lead to weakly relevant logics, are these:Connexivism, which rejects Conjunctive Simplification. Since the position keeps classical connections between &- and v-principles, it is also obliged to reject Adjunction. However by weakening contraposition principles connexivism could retain Addition. Connexivism, apparently found in Aristotle and Boethius and rampant among the Stoics, was reintroduced in modern times by Nelson and developed by Angell and McCall. That the position has attained a new popularity can be seen from the number of recent journal articles on implication espousing connexivist principles (see further 2.). Conceptivism, which retains all the principles employed except Adjunction. Obviously since the position retains Conjunctive Simplification it has to weaken contraposition principles. Conceptivism, as so far developed, through the so-called logics of analytic implication, satisfies an (over-) strong relevance condition, namely: SR. |-A -*• B only if all sentential variables occurring in B occur in A. Analytic implication, due to Parry, is supposed to explicate a Kantian insight. The logical theory has been elaborated by Dunn, Urquhart, Fine and others. Non-transitivism, which rejects only Rule Syllogism, blocks the move from (5) and (6) to (7). This desperate strategy, proposed by von Wright, has been elaborated and defended by Geach. It has also found favour with Prior 67a. Relevant positions, which reject only Disjunctive Syllogism, and which retain full contraposition principles. This strategy, which appears to have no historical roots, is discernible in Duncan-Jones and was applied by Hallden. But it owes its clear and formal development to Ackermann and to Anderson and Belnap, the earliest work dating back to 1956 only. However the strategy of defeating the paradox argument by rejecting just Disjunctive

4

7.2 MATERIAL-IMPLICATION, ITS INADEQUACIES

Syllogism was adopted (implicitly) in the minimal logic of Kolmogorov and of Johannson, which goes back to 1926. Minimal logic rejects lines (6) and (7), in effect by throwing out the principle A B, that a false (or absurd) statement implies anything, as can be seen upon using Kolmogorov's definition of negation: ~A = A ->• A > which converts (7), e.g., to A & (A -»• A ) -*• B. Sadly "minimal" logic made the mistake of building on Hilbert's positive logic, and thereby was automatically committed to the paradox that anything implies a true statement, i.e. B A for any detachable A, and hence to weakening contraposition principles in order to save its insights concerning negative paradoxes. (The correct course, which minimal logic could historically have taken, would have been to correspondingly weaken Hilbert's positive logic, as Church went on to do.) We have reserved the fashionable label 'relevant logics' for logics which are paradox-free and which reject Disjunctive Syllogism (and its mates, such as Antilogism) , though these logics are by no means the only ones that satisfy formal relevance criteria (or for that matter the only ones that satisfy criteria such as those of relevance, necessity, and strength which were supposed to distinguish the system E of Anderson-Belnap entailment). The case for concentrating on relevant logics among the various positions meeting the condition WR - and as to what is wrong with the alternatives - will be picked up subsequently (in chapter 3). Here we begin on making out the case for studying unorthodox and little known logical systems in preference to such well-marketed alternatives as classical and strict logics - essentially the well-marketed goods are shoddy and won't perform many of the tasks they are supposed to. §2. The inadequacy of material-implication. Since the implication connection is fundamental, any adequate logic should not fail outright in capturing its leading properties: classical and modal logics in particular flunk the test, and accordingly have to be condemned as inadequate. To begin, the prevailing logic, classical logic (i.e. two valued sentential logic and its quantificational and set-theoretical extensions) offers, as is well-known, only pathetically inadequate accounts of implication and of conditionality. Material-implication or hook, symbolised '=>', has only a slightly better claim to representing a genuine implication than materialequivalence, '='. C.I. Lewis, in his long campaign against material-implication as an account of logical implication, i.e. entailment or the converse of systemic deducibility, has explained in detail why material-implication gives nothing like an adequate account of logical implication (see the Lewis references cited). What Lewis did not realise (and would have disputed, cf. 18 pp.224-6) - something the counterexamples to be presented bring out is that material-implication fails almost as dismally as an account of conditionality. There are at leaist three statement relations that an adequate theory of implication and conditionality has to explicate, entailment, law-like implication, and conditionality,1 and material-implication fails for every one of them. The most striking manifestations of breakdown are of course the para1

Subsequently we shall find that there are more than three. The conflation of entailment and the conditional would involve that piece of Spinozist rationalism, according to which every truth is a necessary truth, which Lewis in 14 thought he detected in PM and was concerned to argue against.

5

1.2 COUNTEREXAMPLES TO

MATERIAL

PRINCIPLES

doxes of implication and conditionality. These not only lead to amazing transgressions of relevance, validating such entailments as "Canberra is located at the North Pole -*• 2 + 22 = 24" and collapsing theories containing any falsehood, i.e. most theories, to triviality; but they also validate an enormous range of relevant implications and conditionals which are intuitively invalid, e.g. the following examples (from Adams 65, p.166 and Stevenson 70, p.28): John will arrive on the 10 o'clock plane. Therefore, if John does not arrive on the 10 o'clock plane, he will arrive on the 11 o'clock plane (illustrating A entails that if ~A then B). John will arrive on the 10 o'clock plane. Therefore, if John misses his plane in New York, he will arrive on the 10 o'clock plane. If he dies tonight he will visit us tomorrow; because he will visit us tomorrow. (The latter examples both illustrate B entails that if A then B.) But it is not simply that material-implication, construed as an implication, admits of such paradoxes as that a false statement implies any statement at all - which ruins any direct application of the logic to the investigation of false hypotheses - but that there is a large class of principles that hold for material-implication but fail for implication and conditionality. Consider, for example, the following set of arguments which would be valid if implication were material-implication, if 'if... then...' were adequately rendered by ' (most of these examples are assembled in Hunter 72). As Hunter remarks, none of these arguments is valid, for each either has, or could easily have, true premisses but a false conclusion. (a) (Switches paradox): If you throw both switch A and switch B then the motor will start. Therefore, either if you throw switch A the motor will start, or if you throw switch B the motor will start. That is, throwing one switch suffices: joint sufficiency is eroded to separate sufficiency. The material principle on which this paradox - which may be illustrated by any case where joint premisses are both used in the argument - rests is the following: A & B 3 C.

Therefore (A => C) v (B => C).

(b) It is not true that if she is over forty she is still young. if she is still young she is over forty.

Therefore

~(A ^ B). :. (B d A). (c) If he won't propose to her unless he finds that she's wealthy, then he's mercenary. He will find that she's wealthy, but won't propose to her. So he's mercenary. ((~A a ~B) ^ C) & A & ~B.

So C.

(d) If this figure is rectangular and equal sided it is a square. Therefore, EITHER if this figure is rectangular and not equal sided it is a square, OR if this figure is not rectangular and is equal sided it is a square. (A & B) => C.

Therefore [(A & ~B) => C] v [ (~A & B) ^ C].

(e) If John is in Paris, then he is in France. If he is in Istanbul, then he is in Turkey. Therefore, if John is in Paris he is in Turkey, or if he is in Istanbul he is in France. (A ^ B) & (C

d

D).

So (A ^ D) v (C ^ B).

6

7.2 FURTHER ELEMENTARY COUNTEREXAMPLES TO CLASSICAL

PRINCIPLES

(f) If Jones comes from Georgia, then he is a southerner. Therefore, if Jones rides a bicycle to work he is a southerner, or if he comes from Georgia he owns a bicycle. A 3 B.

Therefore (C => B) v (A => D) .

(g) It is not the case that if we follow this road we shall reach the city. Therefore we shall not reach the city. ~(A = B).

So ~B.

(h) If Goldbach's conjecture is correct, then it is false that if the mayor's telephone number is an even number, it cannot be represented as the sum of two primes. Therefore, if the mayor's telephone number is not an even number, Goldbach's conjecture is not correct. A 3 ~(B 3 C).

Therefore ~B => ~A.

(i) It is false that if we can sensibly assert that we are sound asleep then Malcolm on Dreaming is correct in all his contentions; therefore it follows that we can sensibly assert that we are sound asleep. ~(A ^ B).

Therefore A.

Consider also (j):- The tax-collector says: 'If you made a mistake in your income-tax return and owe the government money, you made the mistake deliberately, with the intent to cheat the government.' If 'if' is equivalent to '3', then I cannot deny this allegation without admitting that I made a mistake and owe the government money. (g) and (i) may be combined as in the following example of Stevenson (70, p.28): This is false: if God exists then the prayers of evil men will be answered. So we may conclude that God exists, and (as a bonus) that the prayers will not be answered. ~(A 3 B). .*. A & ~B. (k) Not both Hunter is a bachelor and Hunter is not married; Hunter is a bachelor then Hunter is married. ~(A & ~B) .

therefore if

A 3 B.

(1) Either Dr. A or Dr. B will attend the patient. Dr. B will not attend the patient. Therefore, if Dr. A does not attend the patient Dr. B will. (A v B) & ~B.

~A 3 B.

(m) If John will graduate only if he passes history, then he won't graduate. Therefore if John passes history he won't graduate. (A 3 B) 3 ~A.

B 3 ~A.

(n) It isn't true that if he breaks a mirror he will have bad luck. he doesn't break a mirror he will have bad luck. ~(A 3 B).

So if

~A 3 B.

(o) If Albert's age is greater than twenty and less than twenty-three, then Albert is either twenty-one or twenty-two. Therefore if Albert's age is greater than twenty then Albert is twenty-two, or if his age is less than twenty-three then he is twenty-one. A & B 3. c v D.

(A 3 D) v (B 3 D).

(p) He is a former communist. So if he is smiling he has a bomb in his pocket or else if he has a bomb in his pocket he is smiling.

7

7.2 T H E METHOV OF INTUITIVE

A.

COUNTEREXAMPLES VEFENVEV

(B 3 C) v (C 3 B).

Worse still, the premiss of (p) can be omitted.1 Several of these counterexamples - which destroy any claim materialimplication may have had to credibility either as a conditional or as a general implication (upon substituting 'implies' for 'if...then') or as an explication of entailment - also apply, after but minor modification, against other accounts of implication, e.g. against many-valued implications, against positive and intuitionistic implication, and against several of the numerous syntactical adjustments of material-implication that have been rigged up in an effort to avoid some of the worst features of material-implication. Although we shall make heavy use of the method of presenting intuitive counterexamples in establishing the bounds on adequate theories of main topics we investigate, the method has to be handled with considerable caution and is far from foolproof (as Simons 65, splendidly, if somewhat unintentionally, reveals). Since we are trying to logically capture unformalised notions,2 the method is unavoidable; and intuitive (or preanalytic) countercases form an important part of the data to be taken account of in designing logical theories. Major problems for the use of intuitive counterexamples are caused by the common occurrence of enthymematic reasoning, where understood or presupposed assumptions or premisses are omitted, and by the common use of arguments which are correct for a restricted class of statements or in restricted classes of situations and contexts or in a presupposed uniform context. Our methodological strategy in the face of these problems is to set our sights on obtaining first of all accounts of non-enthymematic argumentation which is not statementally or situationally restricted, but which is universally valid, that is correct for all (reasoning) situations.3 While this strategy enables us to avoid the problems mentioned, other problems remain, and the strategy has an important bearing on the use of examples in trying to establish principles as correct. For the asymmetry between falsification and verification - already present since correct principles as universally true are simple universal statements - is accentuated, because in establishing that a principle holds good it has to be contended on the basis of intuition that all examples of it hold in an appropriate nonenthymematic and contextually unrestricted way. But intuition but seldom provides such general information. Defences that have been given of the connexive principle, if A -*• ~B then ~(A -*• B) (one form of the principle called Boethius), offer a good illustration of failure to meet the methodological requirement that the principle must hold for all statements. A common procedure (used, for example by Cooper 68 and Stevenson 70) is to cite 1

The examples have the following sources: (a), (1), (m) Adams 65; (c), (d), (n), (p) Stevenson 70; (b), (e)-(h), (o) Cooper 68; (i), (j) J. Nelson 66; (k) Hunter 72. Hunter goes on from the examples to neatly demolish various defences of the interderivability of If A then B and A ^ B.

2

This clause illustrates nicely a virtue of the split infinitive; for our original sentence 'we are trying to capture logically unformalised notions' contained an unfortunate ambiguity.

3

We are left of course with the residual problem of trying to demarcate reasoning or deductive situations; see UL.

8

7.2 MINOR PROBLEMS WITH SURFACE FORMS OF

ARGUMENTS

one or two instances of the principle and to conclude on the basis of the acceptability of the instances of that sort that the general principle is correct, e.g. Boethius is vindicated on the basis of cases like that where A is 'This is gold' and B is 'This is soluble'. But the cases considered, and perhaps envisaged, are ones where both A and B are contingent statements; so the examples establish no more than what stricters, for instance, would readily concede, that Boethius is correct where A is possible. That is, the examples only vindicate Boethius for a restricted class of statements, those where A is assumed to be possible; they do not vindicate it generally, as is required for validity.1 A residual problem with the use of intuitive counterexamples in confining valid argument arises from omissions, truncations, ambiguities, and contextual variation in natural language discourse from which the examples are drawn. The omission of an intended particle such as 'even' or 'still', sanctioned in English, can have a drastic effect on the validity of an argument, because meaning is altered: thus the surface form of an argument cannot be relied upon in a completely unqualified way.2 This is an ancient point, backed up by examples of invalid syllogisms with ambiguous middle terms and by examples of the folly of taking surface structure at its face value (e.g. examples forcing a distinction between the 'is' of predication and that of identity* such as: The class of Apostles is twelve and twelve is a number so the class of Apostles is a number); but it is still a serious point of contemporary relevance. For instance, some counterexamples that have been presented to transitivity of the conditional are less than conclusive because they can be written off as having an ambiguous middle statement. One of Stevenson's examples is of this sort, namely the argument: If I pass her I'll be ashamed of myself. If she gets down to work I'll pass her. So if she £ets down to work I'll be ashamed of myself - the accusation being that 'I pass her' means 'I pass her under the present circumstances' in the first premiss but not in the second, something reflected by the shift from 'I' to 'I'll'. Stalnaker's famous example (in 68) is also of the sort, the argument being: If J. Edgar Hoover were today a communist then he would be a traitor. If J. Edgar Hoover had been born a Russian then he would today be a communist. Therefore if J. Edgar Hoover had been born a Russian he would be a traitor. The argument concerns only conditionality and not of course law-like implication or entailment: for being born in Russia is hardly sufficient for later being a communist. The allegation is that for the premisses to be true 'communist' has to be read differently in each premiss, roughly speaking as 'Russian communist' and 'American communist'. The retort might be that it suffices to take 'communist' as meaning 'communist of some sort'; but the further allegation would be that either this takes us to a traditional fallacy of a particularly quantified middle term or else the premisses are no longer both true. In any case, the case is stalemated, and the direct intuitive counterexample lost. It is more difficult to apply this technique against examples like the following provided by Adams (65, p.166): If Brown wins the election, Smith will retire to private life. If Smith dies before the election, Brown will win it. Therefore, if Smith dies before the election, then he will retire to private life. For here at least there appears to be no ambiguous middle term to fall back upon. What can be said to have happened however is that the context is changed from the first premiss to the remaining premiss and the conclusion, and that the antecedent of the X

Practically all of Cooper's examples of classically invalid arguments which are intuitively acceptable (68, pp.298-9) can be faulted on this ground.

2

For the most part, however, deviation from surface structure should be minimised.

9

1.2 A CLASSICAL

RESCUE

ATTEMPT:

THE

DEFINITIONAL RETREAT

latter two conditionals undermines the grounds for accepting the first.1 Adams himself makes a not dissimilar but very general and important point, that classical logic cannot be safely used in inferences whose conclusions are conditionals whose antecedents are incompatible with the premisses in the sense that if the antecedents became known, some of the previously asserted premisses would have to be withdrawn. Whether such conventionalist stratagems as those designed to defend Conjunctive Syllogism succeed or not - we maintain they should be avoided as far as possible in obtaining an applicable theory - they are enough to indicate that the method of counterexamples is fraught with difficulties. No simple rescue operation, of the sort just sketched (we do not need to claim that it succeeded) for transitivity of the conditional, can be successfully mounted on behalf of material-implication. No removal of ambiguities, restoration of context, or change of syntax to reveal intended structure, is going to save material-implication against damaging intuitive counterexamples. There are, however, other strategies that can, and have, been tried in retaining the entrenched logic's theory of "implication". What these strategies have in common is that they all try to weaken in one way or another the account taken of intuition in delivering counterexamples. A favourite strategy is what might be dubbed the "definitional retreat", according to which none of the alleged counterexamples are such when material-implication is definitionally expanded (e.g. in terms of negation and conjunction or negation and disjunction). The move means abandoning the claim that material-implication explicates implication or conditionality - and so concedes the main point at issue - but this some of its exponents are prepared to do, substituting only such claims as that it is safe, reliable, is all that is required for the deductive sciences and provides the core of any correct account of conditionality and entailment (contentions which we try to show are false, beginning in 1.6). Other defenders of material-implication are less happy to abandon the thesis that it provides some sort of explication of the conditional that works for more than isolated cases. The main alternative strategy has been to claim that intuition is not a reliable guide, that it is corrigible and sometimes misguided or confused (points we willingly concede), and that it can be shown, or even conclusively proved, on the basis of principles that intuition itself supplies, that material-implication answers exactly to the ordinary conditional. It is argued, for example (Simons 65, p.82), that while intuition may supply a set of principles it cannot ascertain what all the consequences of those principles are; and among the consequences of intuitive deliverances concerning 'if... then' are that this connective is none other than the truth-functional conditional. But, firstly, intuition characteristically rejects cases as well as accepting cases: indeed intuition tends to work at the level of particular cases as opposed to that of general principles, so rejection of universal l Objections of this general type, that the assumed context or background is changed from premiss to conclusion, can also be made against Stevenson's other countercase to Conjunctive Syllogism, to D. Lewis's countercases (in 73) and to Bennett's countercase (in 74): see further p.46.

10

1.2 INTUITION

VOES NOT SUPPORT THE PARADOXES O F

IMPLICATION

principles is more central than acceptance. It may happen, as a result, that intuition occasionally yields contradictory statements: consider, in particular logical and semantical antinomies. In the case of implication and conditionality, however, neither uncorrupted nor refined intuition yields, or need yield, inconsistent results. Only if the logically uncorrupted are gulled into accepting general principles (such as Exportation) which extend far beyond the reaches of intuition, are the unwary led, or bullied, into both rejecting and accepting paradoxes of material-implication as logical features of implication or the conditional.1 The procedure of Simons and others is to represent intuition as accepting the premisses of independent arguments for the paradoxes (in Simons' case just variants of the arguments Lewis had considered); and then, naturally, it is a straightforward matter to demonstrate that ordinary 'if...then' is really, surprising as the paradoxes and other counterexamples may be, material— implication. Unfortunately for Simons the arguments used in his 'fundamental observation' (65, p.79) - 'that arguments intuitively acceptable taken one by one collaborate to entail that the conditional which appears in them is truth functional' - do not rest on generally intuitively acceptable principles, whatever he may claim. The main arguments turn on the alleged intuitive acceptability of a version of Exportation: if A and B entail C then A entails that if B then C. But since a paradox of conditionality is an immediate upshot of this principle using Simplification and Detachment, i.e. it is immediate that A entails that if B then A, the principle can hardly claim intuitive acceptability. A little more generally, identify C with A in the Exportation,let A be some true statement, and choose B as irrelevant to A, or in fact as any statement such that "if B then A" is false. For example, take A as 'There are still a few tigers in India' and B as 'India is an island'. In this way the principle is intuitively falsified. And that is not all that is wrong with Simons' "fundamental observation". For the exportation principle, which is smuggled in as intuitive conditional proof, is applied to a version of Disjunctive Syllogism, ~(A & ~B) & A -*• B (a principle flawed in 2.9) to yield the principle that ~(A & ~B) entails that if A then B, an argument form already intuitively faulted as invalid under example (k) above. A less devious attempt - this time to prove that A ^ B is equivalent to "if A then B", where => is the material-conditional - can be found in C.I. Lewis (18, p.226). Lewis's proof is as follows (we set it down verbatim, in our notation however, omitting only references back to already proven or assumed premisses, in particular to the crucial postulate that p • (p = 1) for every proposition p): A 3 And B = B =

B gives "If A - 1, then B • 1", and hence "if A then B". "If A, then B" gives A = B, for it gives "If A - 1, then 1", and (a) Suppose as a fact A - 1. Then, by hypothesis 1, and A => B. (b) Suppose that A ^ 1. Then A • 0, and A => B.

l The situation with respect to the paradoxes of strict-implication is less straightforward owing to the widespread (though far from universal) intuitive acceptance of Disjunctive Syllogism. The question of the intuitive acceptability of Disjunctive Syllogism, Lewy's charge that the (?) Intuitive concept of entailment is inconsistent, and the general question of the role of intuition in settling the (?) logic of entailment, are taken up in chapters 2, and 4, especially in 2.9 ff.

11

7.2 "PROOFS" THAT I F . . . T H E N IS TANTAMONT TO HOOK

The main trouble with this "proof" may be located in the very first step. For A => B does not give "If A = 1 then B = 1", i.e. in effect "if A is true then B is true", unless the unformalised (metalinguistic) 'if-then' is already assumed to have properties of the material-conditional which are in question. Lewis has, in effect, assumed the already rejected principle that ~(A & ~B) gives "if A [is true] then B [is true]", when all he is entitled to from the truth-table for => is the principle that ~(A & ~B) gives "either A is not true or B is true" where the 'either-or' is extensional inclusive disjunction. Thus Lewis has established little more than that A = B is equivalent to "If A then B" where 'if-then' is understood as the material-implication. This is hardly news, and it assumes the point at issue as far as linking materialimplication with 'if-then' is concerned. A more subtle attempt to establish that A B and "if A then B" are genuinely interderivable (i.e. entail each other) rests on the assumption that conditionality is syntactically enthymematic, that is (in modal form) (a) a necessary and sufficient condition for the truth of "if A then B" is (Faris's condition E) that there is a set S of true propositions such that B is deducible from A together with S. Thesis (a) is defended in Faris 62 (pp.117-8), and is the basis of his argument that "If A then B" is derivable from A = B. In fact (a) could also have served as a basis of a proof that A = B is derivable from "If A then B", a claim which Faris takes as generally conceded and hardly in need of proof. For (apparently unknown to Faris) Myhill 53 proved, in effect, that where (8) "If A then B" is defined as, or taken as logical equivalent to: for some true propositions p, A and p strictly imply B, i.e. in symbolism (Pp)(p &. A & p B), then A => B and "if A then B" are indeed interderivable. The proof depends essentially on certain (but not all) properties of strict implication, in particular, as Faris brings out in his informal argument (p.117), on the principle of Disjunctive Syllogism, that B is deducible from A and A B. That Disjunctive Syllogism is no correct principle of entailment is a main negative thesis of this text: but that is not enough to dispose of the interderivability thesis. For it would be sufficiently embarrassing if A => B and "if A then B" were strictly equivalent; and more important, the use of Disjunctive Syllogism can be removed by way of the following relevant characterisation of enthymematic implication: (y) A then B is taken as interdeducible with: for some true proposition p and false proposition q, A and p entail B or q, i.e. in symbols (Pp,q) p & ~q &. A & p B v q. Then, where -*• is the implication of a relevant logic, such as systems E and R, A s B is interdeducible with ij^ A then B, so characterised (the result is proved in Meyer 73). The adequacy of this attempt to equate the conditional with the materialconditional accordingly reduces to the question of the adequacy of (a) and ($) or, more exactly, of (y). But we already have a more than sufficient basis for rejecting (a) - (y) in the shape of examples like (a) to (p) above.

12

7.2 OTHER EXPEDIENTS

IN DEFENCE OF HOOK

Moreover we can see directly that (a) - (y) are false (when 'if-then' is construed ordinarily); for according to them (read as sufficient conditions), whenever B is true ±f_ A then B, for any A. (In (a) take S as {B}; in (y) let p be B and q be ~B). That is, (a) - (y) at once reinstate the counterintuitive paradoxes of conditionality. (For the negative paradox let p in (y) be ~A and q be A; then whenever A is false, if A then B, whatever B.) This serves to refute Faris, for the only argument he offers for the sufficient condition in (a) which is crucial to his case is: 'I think that in any case in which we believed that a set S existed as specified we should be prepared to assert if_ A then B' (62, p.118). The paradoxes reveal, however, that for most of us the claim is false: we are not prepared to assert if A then B on the strength of the truth of B. The more general lesson emerging from such attempts is that enthymematic explications of 'if...then' of the syntactical brand, which impose no restrictions on the class of truths (or falsehoods) that may be tacked on to entailments, are bound to be defective. Neither 'if...then' nor 'implies' are enthymematic implications of such a simple syntactical kind. There remain other more desperate expedients that hookers may resort to in defence of the material-conditional. A favourite is this:- Hook is at least a necessary condition for 'if...then', but you say 'if...then' amounts to more than =>. The analysis of conditionality must then take the form: if A then B is logically equivalent to A B and J, where J represents some as yet unspecified condition. But what can J be? It can't be this and it can't be that, so it must be nothing. For example, Thomson. 63 argues that J can't be of the form 'because ...' where a statement of the grounds for the claim 'if A then B' are given, since this would inject the grounds for a statement into the meaning or analysis of a statement. While this is correct, only a rather blatant commission of a false dichotomy fallacy would enable the conclusion that J is null to be reached. Nor is it particularly difficult to indicate the lines along which J should be filled out (especially when Hunter 72 has done this for us in rebutting Thomson). Clause J will read 'there is some not merely truth-functional connection, of an appropriate sort, between antecedent A and consequent B'. In fact, in the light of metavaluational semantical methods, it is possible to be quite precise about J: J can be the clause j- jlf A then B, where |- is the provability functor of the system which formulates the logic of the conditional. (For the fullest exposition of metavaluational methods, see Meyer 72). The endemic circularity of such analyses makes it plain that there is no easy escape by way of the material-conditional from logical field work, from the messy and commonly inconclusive business of trying to determine the logics (or formal connections) of conditionality. In the quest for satisfactory theories of conditionality and implication or perhaps more realistically for systems still in the running for these esteemed positions - it is salutory to record the negative results obtained from failed attempts, for these place worthwhile bounds on satisfactory accounts. There are, in short, some lessons to learn, as Lewis realised, from the failure of material-implication, about the character of satisfactory accounts, about what entailment is not, and likewise what conditionality is not. Lewis in fact intended his wider argument to show that no merely many-valued connective would serve as entailment (see Lewis and Langford 32, p.237 ff; but the argument is not completely general). We will try to record in a more systematic fashion some of the lessons that emerge given a little reflection; we present them as a set of fallacies that correct

13

1.2 FALLACY

EXCLUSION

ANV FALLACIES

EXCLUVEV

accounts of conditionality and entailment must avoid. For the avoidance of fallacies is an overarching requirement of adequacy that should be imposed on accounts of entailment and conditionality. We call this general requirement, the first of many conditions of adequacy we shall impose, the requirement of fallacy exclusion. Under it, as determinate cases, fall: 1. The truth functional (or matrix-assessibility) fallacy. The fallacy is that of assuming that whether A B holds is a function of the truth-value, truth or falsity (or more generally, in the finitely many-valued case, of the value) of one or other or both of the components A and B, But the implications one statement has are never a matter just of its truth-value, and so knowing whether it is true or false has no bearing on what it implies. Similarly what implies a statement is not a matter solely of the statement's truth value. Implication and conditionality are matters rather of the intensionality, or meaning, of statements, not of their extensional values. 2. The finite-valued fallacy. The fallacy is that of supposing that logics of implication or conditionality can be merely finitely valued logics. That it is indeed a fallacy can be shown by arguments of the sort that demonstrate that no rational logic is finitely many-valued.1 An n-valued logic can only distinguish n statements; yet entailmentally there are infinitely many nonequivalent statements. In any sequence of n + 1 variables Pi'-,''Pn+i a t least two of these miTst be assigned the same value in an n-valued logic. So given that p -*• p, since a correct entailment (and conditional), takes a designated value, and that a disjunction with one designated disjunct takes a designated value - the disjunction (p.. -> p„) v ( p -»• p )v...v (p p ) v (p p )v i z ± j i n z l ...v(p^ Pn_-^) must hold good, for the reason that one component at least will have the same value as p + p in an n-valued logic. This chain is obviously a generalisation to n values of the cycle condition (p^ p^)

v

P^) that material-implication satisfies, and equally fallacious. It is not difficult to devise counterexamples to such n cycles. Consider the positive integral line (or the natural numbers) and suppose each integral place is occupied by, or has, a randomly selected colour. Consider the set of true elementary statements of colours, e.g. 1 is blue, 2 is black, 3 is vermillion, etc. By construction of the example, no one of the elementary statements entails any other; hence no finite disjunctive chain condition can be correct. Avoidance of the first two fallacies alone takes the quest for adequate analyses of entailment and conditionality beyond the confines of many-valued logic. We now make a beginning on listing fallacies that narrow the remaining class of logics (and further fallacies will be adduced as we proceed). 3. The truth-copulation fallacy. The fallacy is that of taking A -*• B as true [valid, a theorem] where both A and B are. Thus any system of conditionals in which the rule A, B A -*• B is admissible is fallacious: this rules out among other things, the system of Stalnaker and Thomason 70, and of Cooper 68, and the main systems of D. Lewis 73. Lewis and Langford base their critique (p.238 ff.) of classical and many-valued accounts of deducibility essentially on the commission of this fallacy; and their case applies also against accounts of conditionality which incorporate truth-copulation. 1

On this point, which is a further reason for avoiding standard many-valued logics in any quest for an analysis of implication and of conditionality, see Routley and Wolf 74.

14

7.2 FALLACIES

OF TRUTH-COPULATION

ANV

MODALITY

They present essentially three points - none of them absolutely inescapable by a hardened truth-copulator whose logical intuition has become warped in arguing that truth copulation is indeed a fallacy. Firstly 'two propositions could be independent (one not deducible from the other) only if one of them be true and the other false. As a statement about deducibility, this is quite surely an absurdity' (p.238). For otherwise, to progress to the second point, any 'system which... could represent the truth about something or other, would be deducible in toto from any single proposition of it taken as a postulate ' (p.2381). Thirdly, 'the falsity of the assertion "Every true proposition is deducible from every other" hardly requires proof1 (p.239). Nonetheless this Moorean appeal can be supported by intuitive counterexamples. Consider any two (independent) irrelevant statements, e.g. a necessary statement q^ (e.g. 23 = 8) and a contingent one q^ (e.g. Manaus is situated on the Rio Negro). Then none of the following statements are true: "if q^ then q2"» "that q^ implies that q2", "q^ if, and only if, q2". Moreover they commit fallacies of other types, namely fallacies of relevance (on which see ABE2) and of modality. 4. Fallacies of modality. There is really a cluster of traditional fallacies falling under the heading 'fallacy of modality', and there is dispute about the extent of the cluster, e.g. in which cases, if any, it is fallacious to have a contingent statement entailing a necessary one (see FM and ABE). A commonly-committed modal fallacy, of importance in refuting one form of epistemological scepticism, consists in applications of the principle: if it is necessary that A implies B then if A then necessarily B. For example, the sceptic argues that if one knows that p then p must be the case; but since contingent statements such as that the cheese is on the table in front of us never have to be the case (by contraposition and legitimate modal distribution), one can never really know that the cheese is there. However the important fallacy of modality for present discussion is that of necessity-transmissionfailure, which is simply that of violating the necessity transmission principle that if A entails B then DA entails []B, that necessary statements do not entail non-necessary ones, contingent ones in particular. This fallacy, like most of those concerning modality, only applies where a logical implication or entailment occurs. The statement "that q^ entails that q^" commits a fallacy of this type, since it would have a necessary statement entailing a contingent one. Several systems, both relevant and irrelevant, are ruled out as systems of entailment because they contain theses which commit this fallacy of modality. (It is assumed, what we later argue for, that entailment is closed under Modus ponens.) For example, so ruled out are systems which contain any of the following principles: A -*•. B '•*• A (as B may be necessary and A merely true); A (A ->- B) B i.e. Assertion (since A and B may both be contingent though A -»• B is necessary); A -*• (B -> C) B -*•. A C, 1

Lewis and Langford consider an attempt to escape from this consequence, but in insufficient detail, considering that they attempt a rather similar escape from a parallel objection to strict implication (p.252ff.).

2 To have much bite in natural language cases, more than the formal weak relevance requirement is needed in explaining fallacies of relevance.

15

7.2 AW AVEQUATE THEORY THAT AVOWS

FALLACIES MUST BE INTENS10NAL

i.e. Commutation; A & B C A B-+C, i.e. Exportation. Given further principles, subsequently defended as correct, systems containing the following principles are also ruled out- A & B & (A B) ~A ~B; (A ->• B) v (A ~B) , i.e. Stalnaker. As to the last let A be some necessary truth, e.g. q^ and B some independent contingent truth, e.g. Then since ~B is false A cannot imply B, by the fundamental counterexample principle. But neither can A entail B, without a fallacy of modality, and this exhausts the disjunctive cases. Although failure of necessity transmission only shows that the principles rejected cannot be correct entailment principles, it can be argued by way of direct counterexamples that none of the principles rejected are correct (analytic) conditionals either. The example already given reveals, for example, that Stalnaker cannot be a correct conditional principle, since the conditional "if q^ then q 2 " is not true, and the conditional "if q^ then ^ 2 " is not true either, by the counterexample principle. Adoption of material-implication as an account of entailment [conditionally] would lead to commission of [practically] all the fallacies discussed. Given the manifold deficiencies of material-implication as an analysis of logical implication - and there are but few these days who really think that hook serves to analyse entailment, since it does not even supply a logical relation - there are two directions in which to proceed:- To put it roughly, there is an extensional direction, encouraged by the observation that objections are somewhat lessened by equating logical implication with tautological material implication rather than ungarnished material implication, which takes shape in a metalinguistic analysis of entailment. And there is an intensional direction, which endeavours to analyse entailment systemically by way of intensional or modal notions in addition to purely extensional notions. The first direction, that of a metalinguistic analysis, is, so some of its exponents have tried to argue, compulsory; attempts at systemic analyses all rest on serious use/mention confusions. Our next main task will be to defuse this criticism, and to reveal some of the deficiencies of metalinguistic analysis of logical implication and deducibility. V The need to go beyond classical logic in order to obtain an (even halfway) adequate account of deducibility points up a basic flaw in classical logic conceived as a general all-purpose logic; for such a logic should be rich enough to express its own deducibility relation. Deducibility (i.e. the converse of entailment) is after all the fundamental logical relation. Classical logic is inadequate because, as an extensional logic, it cannot satisfactorily express either overtly or covertly (as in metalinguistic constructions) intensional notions such as deducibility. And of the intensionality of the entailment and implication connections there can be no doubt: A -*• B and A = C do not ensure C -*• B and neither do A ->• B and B H D ensure A -*• D.1 An adequate theory of entailment must be intensional, i.e. nonextensional, and more generally non-value-functional. Thus too no extensional patch-up of material-implication, by defining a new extensional relation in terms drawn from classical logic or of a merely extensional cast, can be adequate. More generally, no value-functional patch-up can suffice (where 1 = symbolises as usual material equivalence with A = B •**. (A => B) & (B a A). The various readings of A -*- B, and A «• B (where A -»• B = f (A B) & (B A)), will be a matter taken up in the next section.

7.3 THE mPLJCATlOUAL-COmriONAL

TRANSFORMATION

value-functionality is characterised as in GR) This is the next of the many conditions of adequacy that we will impose on a correct theory of entailment, what we call the intensionality requirement. It is evident that, for similar reasons, an adequate theory of conditionality must also be intensional. The metalinguistic proposals do not fail the intensionality requirement because the metalinguistic procedure represents a backdoor, and somewhat covert way, of introducing intensionality - allegedly in an extensionally acceptable form (because it's not too high a grade!). §3. The trouble with the metalinguistic repair.1 Like most going theories, especially defective theories, classical logic has a wall of defences against objections to its theory of logical implication. The metalinguistic defence (pioneered by Carnap and popularised by Quine) is to separate implication which is claimed to be a metalinguistic notion combining names of statements from hook, relabelled as material-conditional, which compounds statements, i.e. declarative sentences on Quine*s account. Quine's argument turns on the point that construing hook as any sort of implication involves a usemention confusion (51, pp. 31-3; 60, p.196). Thus Whitehead and Russell are accused of being careless of the use-mention distinction in writing (i)

p implies q

interchangeably with (ii)

if p then q

;

and Lewis is alleged to have followed suit in his account of strict implication.2 Even granting this, the distinction does not rule out the interchangeability of (ii) with (iii) that p implies that q of what we call the implication-conditional (IC) transformation. Both (ii) and (iii) are of the form (iv)

... p ... q ...

of statement connectives in Quine's sense, which we may write in symbols (v)

p -»• q.

In Quine's terms (51, pp. 32-3) 'the relation of implication produces a derivative mode of composition of ... statements themselves - namely a mode which consists notationally of compounding the statements by means of 'implies' and two occurrences of 'that'.' Quine however only considers the case of compounding 'by means of 'implies' and the two pairs of quotation marks', as would occur if (ii) were equated with (vi) 1

'p' implies 'q'.

This section was written before the valuable Appendix - Grammatical Propaedeutic of ABE became available. The section can profitably, we suggest, be read in conjunction with the ABE Appendix (about which we have only a few minor reservations).

2

Lewis and Langford's very suggestive comparison of strict implication with tautologous material-implication in their defence of strict implication as an analysis of deducibility (32, p.238 ff.) no doubt encouraged the metalinguistic defence, but at the same time opened the way for Quine's allegation (especially p.242) - which would stand had Lewis and Langford been using quotation marks in the fashion of the current narrow orthodoxy.

77

1.3 PULLING VOWN THE METALINGUISTIC ACCOUNT Of IMPLICATION

Quine's principal objection to this procedure - of putting quotation or 'that' into the sentence connective to gain interchangeability with the conditional (iii) - is that it involves abuse of quotation. This is true if quotation is narrowly construed (as forming a constant name even from variables), but the objection fails entirely for the operator 'that' or if quotation marks are construed as quotation functions (a not uncommon construal whose liberating effects are explained in GR). For the statement variables buried in (iii) (or in (vi) if the marks are taken as quotation functions) can be treated as constituents of (iii), just as those in (ii) can. Thus Quine's argument rests on a false dichotomy, obtained by conveniently dropping the method of putting 'that' into the functor. The method also escapes (in a way Quine is happy to adopt elsewhere) Quine's objection to introducing 'that..,' as a substantive clause, namely the obscurity of the items designated; for 'that' need figure only as an integral part of the functor. Quine may try to object that the connective 'that ... implies that ' is (implicitly) metalinguistic, and would have to be so rephrased in any adequate reconstruction of language. This we deny (for reasons sketched in GR); and so should Quine. For firstly he is prepared to admit intensional predicates containing 'that' into canonical notation (e.g. 60, p.147), and secondly he claims, in proposing his maxim of shallow analysis, that 'embedded in canonical notation in the role of logically simple components there may be terms of ordinary language without limit of verbal complexity' (60, p.160). The logically simple statement connective 'that — implies that ' can be one of these components. Thus an implication relation can be introduced into a sentential object language without any use-mention confusion.1 And this is of course the way things are in natural languages: the idea that implication is a metalinguistic notion does not stand up to much linguistic investigation. The implication-conditional transformation integrates, then, implicational and conditional statements, and, as a corollary, precisely the examples which show that material-implication is completely inadequate as an account of implication (expressed in form (iii)) reveal that the material-conditional, hook, is inadequate as an explication of conditional statements in mathematics as well as elsewhere. In fact the inadequacy of the material-conditional is obvious from one of the most important tests of adequacy of explication of conditionals, namely substitutivity tests. Material-equivalence of statements, i.e. sameness of truth-values, warrants intersubstitutivity in materialconditional statements, but not in the conditionals asserted in any of the sciences. The IC transformation also provides a criterion for when an 'if ... (then) ...' statement is a (genuine) implication. A declarative sentence of form (ii) is an implication if it is interchangeable (preserving values, and sense) with one of form: that p implies that q. The requirement rules out as implications such statements as 'if you want nuts there are some on the the sideboard', since there are familiar circumstances where this is true but 'that you want nuts implies that there are some on the sideboard' is false. -j

Quine's proclaimed policy of admitting none but truth-functional connectives (51, p.33) can be seen to be quite arbitrary as soon as the further desirable step of admitting a that-operator (or quotation functions) into the object language is taken. For then logical Implication as defined in the metatheory can be mapped into the object language. Quine's policy is extraordinarily narrow, and apparently based on little but extensional prejudice.

18

1.3 ENTAILMENT, VEVUCIBILITY AMP LOGICALLY NECESSARY IMPLICATION INTERRELATE!?

The criterion also excludes non-conditional uses of 'if (discussed in 1.5), e.g. 'I rather like your paper, if I may say so', and a variety of counterfactual and non-contraposable occurrences of 'if', e.g. 'If he had telephoned he would have got me', 'If he's here I didn't see him'. Central among implication relations, and crucial for logical reasoning is the relation of entailment. Entailment here has the meaning assigned to it by Moore, namely the converse of deducibility.1 That is (vii)

that p entails that q iff that q is deducible from that p.

Is there any way of picking out what sort of implication entailment is? A familiar answer which we shall argue for is that entailment is logically necessary (or analytic) implication, where furthermore the implication is a sufficiency relation and logical necessity satisfies the postulates of a normal Lewis modal system (specifically those of system S5). Thus in particular (viii) that p entails that q iff it is logically necessary that that p implies that q. The argument for part of our claim is straightforward. It is generally conceded that entailment is a sort of implication and that where it holds it holds necessarily. What is more likely to be disputed is that where it fails it fails necessarily, and more generally that logical necessity meets not just S4 postulates but those of S5 as well, in particular that (ix)

~Q]p

CHlp,

i.e. that it is not logically necessary that p implies that it is logically necessary that it is not logically necessary that p. We have defended the correctness of S5 as an analysis of logical necessity before (Routley 69), and later (in chapter 10) we take the argument further. Later too we shall expand the case for characterising entailment as in (viii), i.e. for defining A =» B as D(A •> B) . 2 Given the standard connection forged between deducibility and valid argument, namely (x)

that q is deducible from that p iff there is a logically valid argument from p to q,

it follows (xi)

that p entails that q iff there is a logically valid argument from p to q.

With the introduction of these connections the metalinguist will return to the attack. He will argue that such notions as validity and deducibility are metalinguistic, not systemic, and what is more they are relative to 1

There are also, of course, several analytically-linked analyses of entailment that we will pick up subsequently, e.g. those of valid argument and of the availability of a valid derivation, of logical sufficiency and of inclusion of logical content.

2

The arrow symbol -*• is the way we represent the determinable, 'implies' and 'if-then'; but when we need to distinguish implicational determinates such as entailment, implication and the conditional, we use the double arrow,5*, for entailment and the single arrow for implication. Similarly where we want to specifically show that the relation is material-implication or strict-implication we use the symbols => and -i respectively.

19

7.3 EXPLICATION VOES NOT HAVE TO BE METALINGUISTIC

particular logical systems. Moreover the resources of classical logic in providing an adequate implication relation are by no means exhausted by material implication. Indeed implication can be defined metalinguistically along the lines of Tarski's classical definition of logical consequence. For example, for classical quantification logic Q, A logically implies B wrt Q iff B is a logical consequence wrt Q of the class consisting just of A. There are several different objections to unscramble here. To begin with, there are stronger and weaker versions of the thesis that logical implication, like validity, has to be defined metalinguistically relative to a given system. The weaker claim, which we concede, is that there are implication-type relations which can be defined metalinguistically relative to given systems. These metalinguistic relations (which characteristically correspond to firstdegree strict implication) have however decided peculiarities, especially once matters of quantification and iteration are considered (as explained in Routley 74) - peculiarities which rule them out as worthwhile explications of the pre-analytic implication relations of philosophy and mathematics. Moreover these system-relative relations do not exclude either absolute implication relations (any more than language-relative notions of truth rule out an absolute notion) or systemic explications of implicational relations and corresponding conditionals. The stronger claims say that the explication has to be metalinguistic. One basis for this claim, that a systemic explication of implication involves a usemention confusion we have already met, essentially through the IC transformation. Another ground for the claim appeals to limitative theorems which are alleged to show that semantical notions, of which validity and so deducibility are specimens, are not fully explicable systemically. It would take us too far afield to take issue with this dogma here. It is enough to point out that the limitations are quite remote and will certainly not be felt at sentential or even quantificational levels; and accordingly do not preclude complete systemic investigations of deducibility at these levels. Indeed the most that this objection shows is that a systemic investigation of deducibility is bound to be incomplete at the edges, something that would be unsurprising on quite independent grounds (such as failure of explications to take account of quirks of context and so on). Furthermore if the objection were right metalinguistic explications would in due course suffer precisely the same fate. In fact a purely metalinguistic explication of implication is bound to be more inadequate than a systemic explication since it provides only part of the requisite story. Roughly speaking it accomplishes only a somewhat anomalous first degree explication without iterated occurrences of implications,1 and everything it does can be systemically reflected - in a system of quantified strict implication in the case of the classical account of logical implication according to Q. But it is quite evident, for example, that implications can be iterated and that variables in implication statements can be quantified, that the fact that p ->• q and q -*• r can, and does, imply that q -»• r, that for every x f(x) -*• g(x) does imply that for some x,fx -*• gx, and so on. These quite ordinary truths the metalinguistic account cannot accommodate - without 1

Iteration of a sort can be achieved by the artifice of ascending to a sufficiently high level of language. It is then transparent, however, how far removed the metalinguistic account is from the familiar notions we are trying to isolate and explicate, and that it is a fantasy to suppose that the metalinguistic construction can serve as any sort of replacement for the originals.

20

1.4 THE THESIS THAT ENTAILMENT IS STRICT-1MPLI CATION

enlarging the usually admitted metalinguistic apparatus (e.g. by the introduction of unstructured quotation functions) in a way that facilitates an elementary systemic mapping of metalinguistic relations such as implication anyway.1 In short, the proposal that investigations of implication should be, or have to be, metalinguistic is bound to lead to only quite inadequate explications of implication. The reasons advanced so far are not the only reasons for dissatisfaction with the metalinguistic accounts we have been offered. Firstly, standard accounts only characterise logical implication. As in the case of strict implication we are left with n£ analyses of those very common brand of implications, non-logical implications, better than the quite inadequate material connection or less than satisfactory enthymematic accounts (criticised below). Secondly, standard accounts correspond to a part of the theory of strict implication (in a way that Carnap 56, p.l73ff., makes clear)2, and -accordingly suffer from the main defects of strict implication as a theory of deducibility. §4. What is wrong with strict-implication. Strict-implication, or fishhook, symbolised and defined: A B & ~B) or equivalently as D(A ^ B), is admittedly much superior to material-implication as an analysis of entailment. For example, Lewis strict-implicational systems avoid the modal fallacies that material-implication would admit, such as that statements which are logically necessary can entail merely contingent truths. Nor does strict implication have to be given up for the petty grammatical reasons Quine elicits - 'p q' can be read unproblematically 'that p strictly-implies that q'. But, like its mate material-implication of which it is the necessitation, strict-implication tolerates not merely a large class of paradoxical principles but also very many other principles to which there are intuitive counterexamples. And the peculiar properties of it are neither important logical discoveries nor absurdities; they are merely the inevitable consequences of a novel denotation for an old and familiar word, long used in common parlance in a different meaning. (Lewis 30, p.34, but referring to material-implication, not strict-implication.) Despite all this the thesis that entailment is strict-implication boasts many advocates. Before we begin to detail the case against strict-implication as the analysis (it is substantial and central parts of the argument extend far into chapter 2), let us hear from the friends of strict implication, and from the stricters themselves. For these people think that they have an 1

We should really like to propose, what is highly disputable these days, that the metalanguage should be reflectable in a sufficiently rich (object) language (which includes its own semantics). This is one step in the direction of abolishing the metalinguistic hierarchical structure altogether, a plan we begin to outline in chapter 15. As to how we live with the semantical paradoxes and avoid Tarski's strictures, see 1.8, 5.3 and 6.5.

2

This is not an essential feature of metalinguistic accounts as Routley 74 shows. In fact, a variety of first-degree theories of implication have metalinguistic analogues; for instance every semantical account in chapter 2 can be represented metalinguistically.

8

1.4 NEUTRALITY OF THE VEFINITION OF ENTAILMENT THROUGH INCONSISTENCY

easy time, that they can remove rival analyses with a few of the following worn-out charges:A first charge is that the strict definition of entailment is the only one that approaches real clarity (suggested in the case Hughes and Cresswell 68 make for strict-implication, p.336). This is simply false: for if the strict definition is clear then the Philonian definition of material-implication is at least as clear (and many would say decidedly clearer since no modal notions are involved). But of course the Philonian definition is patently inadequate and does not even make entailment a logical matter. So the stricter may contend that his definition is the only definition which is not patently inadequate which has real clarity. To start with this begs one of the questions at issue, as to whether the strict analysis is or not patently inadequate. We think it is patently inadequate, for reasons we will detail. But letting this pass, and conceding, as we believe, that some modal and intensional notions have requisite clarity, it is still the case that practically every rival analysis can produce a definition of entailment that is at least as satisfactory as that of strict implication. Some connexivists have in fact used the very definition stricters propose, only they offer a non-extensional account of conjunction (see Routley and Montgomery 78). More important, Nelson, another connexivist, long ago revived the following definition of entailment that Lewis had once used:1 (xii) that p entails that q iff that p is inconsistent with that ~q, or, in symbols and notation that we will subsequently prefer: A -»• B

~(A o ~B),

where o is our transcription of the two-place consistency connective first introduced by Lewis. This definition can be agreed on by most parties, in particular by strict, relevant, connexive and conceptive positions; and it is just as clear as the strict reduction - which other perceptive parties reject - of consistency to a one-place modal notion. From this perspective the strict analysis is distinguished by its insistence that A « B be analysed as 0(A & B), that the consistency of A with B is nothing but the conjoint possibility of A and B. But the strict analysis of consistency is mistaken in the same way as the strict analysis of entailment; it would make the impossibility of one component A, for example, suffice for its inconsistency with any other component B, quite neglecting the connection there should be linking B with A (cf. 5.2). Syntactical definitions of entailment, whether of the relevant or strict kind, though important, are in the end of limited merit. For they amount to trading one notion in for another that either raises practically as many problems, e.g. consistency, or if it doesn't, as in the material case, is wrong. For where the properties of entailment are in doubt so are those of consistency or possibility, and while the properties of consistency or possibility remain in dispute, e.g. the iterative properties of possibility, so correspondingly do those of entailment. In fact the strict analysis of entailment no more leads to a unique theory of entailment even when conjunction and negation are extensionally battened down than does the relevant analysis in terms of an irreducible consistency relation. There are as many strict analyses of entailment as there are systems of strict-implication, that is, infinitely many. It is surprising, 1

But the account, like so much in the implicational area, is much older, and may be found stated in Sextus Empiricus: ... those who introduce the notion of a connexion say that a conditional is sound when the contradictory of its consequent is incompatible with its antecedent. (in Kneale and Kneale 62, p.129)

22

1.4 THE REQUIREMENT ON ENTAILMENT OF "MEANING CONNECTION"

in view of the supposedly luminous nature of strict implication, how reluctant most stricters are to let it be known which system of strict implication represents entailment, and why that particular one.1 Accounts which reject the strict analysis of entailment characteristically do so, it is alleged, in a second group of charges, on the ground that a further condition of q's being deducible from p is that there should be some connection of 'content' or 'meaning' between p and q. It is, however, extremely difficult, if not impossible, to state this additional requirement in precise terms; and to insist on it seems to introduce into an otherwise clear and workable account of deducibility a gratuitously vague element which will make it impossible to determine whether a given formal system is a correct logic of entailment or not (Hughes and Cresswell 68, pp. 336-337). Similar charges are levelled in many other places (e.g. in Bennett 54, p.460 ff. and, in a rather better substantiated way, in Bennett 69, p.214 ff.). But the fact is that accounts of entailment as clear and workable as strict analyses - relevant analyses in terms of inconsistency - already yield formally precise accounts of content linkage, which there is no difficulty in stating, e.g. the sharing of variables requirement WR. Nor is there any difficulty in defining an appropriate exact notion of content - it is defined in the classical way - or the appropriate connection of content - it is that of inclusion.2 That is, A entails B iff the content of B is included in the content of A, in symbols c(B) £ c(A) (for details see UL). This is just one of several, interconnected, analyses which meets Bennett's requirements on an appropriate notion of "meaning-connection", namely The failure in more than 60 years of research to resolve Lewis's early puzzle as to whether S2 or S3 or some other strict system really captured entailment, or to advance the issue of which system of strict implication, if any, represents entailment at the statemental level, is only one of the surprises in the history of strict-implication. Another is the fact that strict-implication has rarely been applied in theory construction, even in such initial enterprises as the formulation of set theory and number theory. There is also another stricter inclusion-of-meaning relation - that tied to sense as distinct from informational content - which diverges from entailment. The stricter relation, important in assessing the paradox of analysis and in helping to clear up some similar paradoxes of Lewy 58, yields an implication relation which does not satisfy intersubstitutivity of coentailments. The "paradoxes" and the implication relation are studied in chapter 14; the notion of sense and the various meanings of "meaning" concerned are examined in Routley 77. Lewy's "paradoxes of entailment" (76, p.117 ff.)» which depend primarily on illegitimate suppression steps, are dealt with in 2.9 ff. Note well that it does not matter for those who say that entailment requires a connection of meaning that there are accounts of meaning inclusion which entailment relations violate: it is enough that there is some satisfactory account of connection of meaning which goes with entailment (and there is, as UL shows for content inclusion). This point is enough to rebutt Pollock's case (66, pp. 186-191) against requiring that an adequate account of entailment meet a connection of meaning condition.

23

1.4 RELEVANT VALUES OF BENNETT'S FUNCTOR . A -»• A and such principles as A A A), it fails to observe that there are likewise situations where other laws of thought fail to hold, where in particular Excluded Middle, A v ~A, breaks down. Yet surely if A A can fail in some situations so can A v ~A. We shall find that it is easy to locate situations where Excluded Middle does fail, incomplete ones - and these serve to falsify Expansion. Unlike the negative paradox where the various relevant approaches are divided, several approaches join forces against Lewis's positive paradox argument in rejecting Expansion; relevant logics, conceptivism and connexivism all reject Expansion. Indeed the positive paradox argument - though according to Bennett (69, p.229) simply the contrapositive of the negative argument (it is not) - has not enjoyed anything like the reputation or plausibility of the negative argument. Lewis's independent argument is as follows:B -*•. B & A v. B & ~A (B & A) v (B & ~A) ->-. B & (A v ~A) B & (A v ~A) A v ~A B -»-. A v ~A

Expansion Converse Distribution by Simplification applying Rule Syllogism repeatedly.

The commonplace objection to this argument is that Expansion is false and should be corrected to B & (A v ~A)

B & A v. B & ~A

Corrected Expansion,

i.e. to a form of Distribution. This is a correction upon which conceptivism and connexivism agree with the relevant position; and it reduces the paradox conclusion to the principle B & (A v ~A) A v ~A, to nothing but a form of Simplification. Bennett, in his apologia for strict-implication, has endeavoured to come down heavily upon the commonplace objection. He charges (in 69, p.229) that resulting positions deny the widely accepted principle A & C + B, DC-OA + B

Necessary Premiss Suppression.

This is not true of connexivism. But it is true of relevant logics, which view necessity suppression as a widespread fallacy, as an extremely damaging principle which leads directly to paradoxes. For in virtue of Simplification, •B -* A -»• B at once results. So likewise the Lewis paradox and much more result, shortcircuiting the "independent" argument. Bennett contends, how(footnote 1 continued from previous page) And Bennett, though he first suggests (p.230), without any evidence at all, that Expansion is part of 'the common concept of entailment', then concedes that it may look odd. We suggest that Expansion is not part of the common concept - to the limited extent that there is such a common concept. Our evidence is circumstantial: part is the consensus among a variety of nonclassical positions that Expansion is invalid, and part is the fact Lewis's argument for the positive paradox has never carried the conviction that the argument for the negative paradox has, and that the step in the argument students regularly baulk at is the first expansion step.

29

1.4 DEFEATING THE LEWIS CARROLL ARGUMENT FOR SUPPRESSION OF NECESSARV TRUTHS

ever, that anyone who rejects Necessity Suppression totally is caught in an infinite regress, as Lewis Carroll found (and Lewy 76 makes the same, invalid, point)• For if Suppression is rejected every necessary assumption used in an argument must be stated and conjoined to the given premisses; but to this procedure there is no end. If this picture were right there could be no deduction in relevant logics since these prohibit necessary premiss suppression: but as there are deductions, the picture cannot be right. It is very far from right. The myth that deduction not merely permits, but requires for its very operation, the deletion of (conjoined) premisses when they are logically necessary, is embedded deep in the modal Weltanschauung; it is worth some detail to try to dispell it. Let us begin with the exportation-importation confusion that is one of the muddles underlying the assumption that conjoined necessary antecedents must be sometimes 'deletable without loss of validity' (Bennett 69, p.230). According to Bennett, As Lewis Carroll once showed, we cannot automatically declare Q -*• S to be false just because the derivation of S from Q rests upon, presupposes, or requires a necessary truth which is not a conjunct in Q; for this would commit us to denying every entailment statement. For example, the derivation of (2) [i.e. (Q & £) V (Q & ~P)] from A [i.e. Q & (P v ~p)] rests upon B: A D (2) ; and the derivation of (2) from A and B rests in turn upon C:

(A & B) 3 (2)

and so on backwards and outwards. The argument confuses 'resting upon' a premiss in the sense of being implied by a premiss, with its imported form, being conjoined with a premiss. Consider the example. The derivation of (2) from A applies Modus Ponens, not Material Detachment, and rests upon B' : A

(2) .

The derivation of (2) from B' and A "rests", for what it is worth, on C* : B' -»•. A + (2)

,

i.e. on a case of Identity: A -*• (2) A -»• (2), not on its imported form A & (A (2)) (2). Thus with C*, i.e. B' -*• B', the regress effectively terminates. All steps 'backwards and outwards' are cases of Identity, C' -»• C', C' -»• C' -»-. C' -* C', etc. There is thus no vicious regress, and no basis for conceding that Necessary Premiss Suppression is "sometimes valid". The Carroll argument does not show, what Bennett claims (69, p.230), that someone who denies that (1) -*• (2) [i.e. Expansion, using the commonplace objection] must say that necessary premisses are sometimes but not always deletable without loss of validity. But non-classical logicians will in any case admit that antecedents, whether necessary or not, are sometimes deletable without loss of validity, notably where the resulting implication is independently valid; e.g. B is deletable in A & B -*• A because A A is valid. This admission naturally does nothing to reinstate Expansion.

30

1.4 ADMISSIBLE SUPPRESSION, ACCORDING TO HOOKERS AND TO STRICTERS

Relevant logicians agree with stricters that Rule Exportation is inadmissible, that true (or even proven true1) antecedent conjuncts cannot be omitted from entailments without invalidity. Exactly the same holds for necessarily true components; but this the stricter denies: in stating a strict implication one cannot omit a merely true premise which is one of a set of premises which together give the conclusion; but one can omit a necessarily true premise. The omission of a premise which is a priori or logically undeniable does not affect the validity of a deduction; but the omission of a required premise which is true but not a priori leaves the deduction incomplete and, as it stands, invalid (Lewis and Langford 32, p. 165).2 Such an omission would, prima facie, leave the deduction incomplete, and so invalid, irrespective of the modal status of the omitted antecedent, and irrespective of whether it was known to be necessary or not. There is nothing special about necessarily true premisses that permits their omission. The stricter thinks there is, but much of his case relies on circularly appealing back to what is at issue, the positive paradox and the assumption that all statements entail ones that are necessarily true. Indeed the latter, e.g. DC -fi> D C, together with Disjunctive Syllogism, yields (in the framework of basic relevant logic) Necessary Premiss Suppression, as follows: A & C - > B - * ~ B - > ~(A H C) A & C B, DC -F ~B C, ~B -> ~(A & C) ~B ->. C & ~(A & C) -o ~B -> ~A A -> B

Rule Contraposition Paradox principle Rule Composition Disjunctive Syllogism, Rule Syllogism Rule Contraposition.

The hooker can similarly argue, using the positive material paradox C -*»D -»• C, that true premisses can be suppressed; and the stricter who mistakenly accepts Necessitation, C-^QC, everywhere can -likewise argue in this way to the (exportation) rule A & C - > B , C-fA^-B The rule is defective, and is defeated by any case where A and C jointly entail B but do not do so separately and C happens to be true (e.g. by the switches paradox presented above). So the stricter is right as against the hooker; but he is only part-way right, for what he says against the hooker's omission of merely true premisses can be duplicated for the omission of necessarily true premisses. That is, counterexamples and countermodels invalidating the Necessity Suppression Rule are readily devised. Inversely, what the stricter urges against those who reject the suppression of necessarily true premisses can be recast as castigation of those who reject the suppression of true premisses, whereupon the inadequacy of the case becomes evident, even to some stricters. Consider, for instance, Bennett's assertion (69, p.233) so adapted: Because in S3 and subsystems the Necessitation Rule: A DA is inadmissible. Entailments, though true, are not according to the systems necessarily true. Thus the stricter is committed to one of Quine's two dogmas of empiricism, that the necessary/contingent distinction can be made good. The relevant position need not depend on this distinction at all.

31

1.4 HOW STRICT SYSTEMS ERASE A FUNDAMENTAL DISTINCTION

Someone who thinks that ... validity may be lost by the deletion of - - - true premisses must presumably be muddling validity with educational or heuristic satisfactoriness. That is certainly not a claim Bennett would agree to, though it differs from his remarkable assertion only by deletion of the word 'necessarily' (where the three dashes occur). Furthermore it points out the way in which it would be argued, against Bennett, that in the necessary case there is no muddle: it would take almost precisely the lines along which Bennett should argue that in the truth case there is no muddle. To put it briefly, there is no muddle because validity requires a universal guarantee over all deductive situations: the deletion of a true, or necessarily true, premiss can easily undermine this guarantee (as the semantical analyses of chapters 2 and 4 will make very clear). What Bennett really wants to insist upon is not this warm-up point but that the suppression of necessary truths (and the corresponding negative suppression of logically impossible statements) which the paradoxes facilitate, is an inevitable feature of deductive reasoning, and ipso facto of applications of entailment within philosophy. It is on this sort of ground that the paradoxes are often claimed to be essential, and desirable, outcomes of a correct analysis of entailment (and this very likely covers what Bennett intended, in saying that the paradoxes are part of the case for Lewis's analysis, 69, p.198). The critical issue here, and also underlying Lewis's independent argument for the positive paradoxes, comes down to whether a distinction can be made between two roles for necessary entailment statements, an inference facilitating role and a conjoined premiss role (cf. Bennett 69, p.230 ff). Despite Bennett's doubts, such a distinction of roles can be drawn very easily, in both syntactical and semantical fashion. Syntactically the distinction is that between A -*-. B C and A & B -»• C with A an entailment (i.e. of the form D -*• E) ; and what permits inference is Modus Ponens, not Necessary Premiss Suppression. The requisite distinction is obliterated by all strict systems that have ever been seriously considered as offering analyses of entailment; for even in system S2° A -»•. B •*• C and A & B -*• C are interderivable where A is necessitated. Nonetheless the distinction between exported and imported forms is syntactically quite obvious and is clear from elsewhere, indeed not just from relevant systems but from weaker modal systems such as SO.5 which still maintains Lewis's definition of strict implication. Semantically, the distinction is that between situations conforming to an entailment (or principle of valid inference) and an entailment's holding in situations (see FD and UL). Once again strict systems erase this fundamental distinction in the case of necessary truths (so it is not altogether surprising that Bennett fails to see it); for necessary truths are supposed to hold everywhere.1 We can usefully adapt the distinctions between premiss roles in dealing with attempts that have been made to reinstate the material-conditional as the conditional and strict-implication as entailment. Consider, for example, the material case. Simons argues for intuitive recognition of that version of Exportation he calls 'conditional proof', namely: if A and B entail C then A entails that if B then C, by way of examples like the following:1

In S2 this result is only achieved by not asserting D(A A). That is, A -»• A can fail in some situations because the theory does not require that it is a necessary truth - and really cannot so require.

32

1.4 HOW WHEW EXPORTATION VIELVS A CORRECT RESULT IT IS OTIOSE

From the theory of universal gravitation (G) together with the initial condition that x is unsupported near the earth (U) it follows deductively that x moves towards the centre of the earth (C). This is to say (29) G, U therefore C is a valid argument. But "if U then C", the singular consequence of the lawlike generalisation is thereby established as a consequence. But if so, it seems to be established because (30) G, therefore if U then C is taken to be valid if (29) is valid (Simons 65, p.83). The mistake in this argument can be pinpointed in the word 'thereby1; for the point at issue is thereby assumed. Less obliquely, (30) is not established from (29), except incorrectly; but (30) may be established along with (29). The argument, which illustrates a method of avoiding "conditional proof" of wide applicability, namely an Instantiation method, is as follows:Firstly, G entails that, for every y, if y is unsupported near the earth then y moves towards the centre of the earth. Then (30) follows by Instantiation and Syllogism. Simon's more general argument for conditional proof in the sciences, from which he argues to the result that the conditional of the sciences is truth-functional, may be dealt with in the same way. The correct argument is not: T (Theory), A (antecedent) .*. C (consequent), so by "conditional proof": T if A then C; but T so (lawlike) generalisation: always if Ax then Cx, and so T if A then C. The incorrect exportation step is otiose. The Simons example also illustrates in a very elementary way more general features of the way in which relevant theories, such as relevant arithmetic and set theory, can operate without suppression of necessary truths; namely it is very often possible to prove relevantly what classically or modally is proved by suppression means or using paradoxes. For example, in standard relevant logics and many relevant theories it can be shown that Rule y of Material Detachment is admissible, that whenever both A and ~A v B are provable then B is also relevantly provable. Modally, of course, B results immediately by detachment from Disjunctive Syllogism, but this illegitimate step does not furnish a relevant proof of B. A relevant proof can however always be obtained by different, and legitimate means. The situation is familiar from mathematics and the theory of constructive proofs: often a theorem proved by indirect means can be proved (in a less controversial way) constructively. For instance, the whole of classical Peano arithmetic can be reworked intuitionistically. Relevant logics require not that valid arguments be constructive1 but that for an argument to be valid it conform to relevant standards. An argument which proceeds by suppression of necessary premisses will be admissible only if it can in principle be reconstructed relevantly or replaced by a relevant argument. A great many classical arguments are so reconstructible or replaceable (cf. investigations of relevant arithmetic). This is more 1

Constructive relevant logics where proofs conform both to relevant and to constructive standards can, of course, be designed: sentential parts of such logics are studied in chapter 6.

33

1.4 BENNETT'S CASE FOR THE POSITIVE LEWIS PARADOX IS UNCONVWCINGLV

CIRCULAR

than enough to refute the view that necessary premiss suppression is an inevitable feature of deductive reasoning. The strategy of replacing the Necessary Premiss Suppression, A & C

B, DC-0A->B

by the case of Modus Ponens, C ->•. A

B, C - P A + B

,

is just a very simple example of one way in which replacements can sometimes be effected. Bennett has, he believes, a case for the positive paradox 'which rests neither on Lewis's analysis nor on the second Lewis argument* (69, p.232). It is based on the feature of derivation and proof that we may at any stage in a proof introduce a previously proved statement, or, in the parallel informal case, that 'a necessary proposition is entitled to crop up anywhere in an argument without being explicitly related to anything that has gone before' (p.233). But contrary to what Bennett assumes, this is not a 'stronger truth' than 'that a necessary proposition is entailed by every proposition': it does not imply the paradox at all without further assumptions, such as that the procedure can be extended to hypothetical proof and a deduction theorem applied. Derivation and arguments of the line-by-line type introducing established statements are relevantly acceptable and do not lead to paradox (as chapter 4 will make plain). It is only if classically designed inference rules are erroneously reconstrued as representing valid arguments - they do not - that anything like Lewis's positive paradox is produced: the derived rule B-» A v ~A

,

of relevant logic E, for example, does not however represent valid argument in any sense. There are derivation rules, especially derived rules, which correspond to valid argument and derivation rules that do not. The logic of entailment distinguishes the cases, but there are other ways of doing so, e.g. natural deduction explications of proof from hypotheses, and explications of formal deducibility as distinct from derivation. Now Bennett does realise that it can be objected that the introducibility of a necessary proposition in an argument without loss of validity is not the same as the deducibility of a necessary proposition from any premisses (69, p.234); but he does nothing effective about meeting the objection. Instead, to erase the introducibility/ deducibility distinction, he appeals back to the strict analysis he is supposed to be defending: accordingly the "case" for the positive paradox reduces to an unconvincing circular one. The case for the strict paradoxes is, despite initial protestations, the strict theory itself. For Bennett simply accepts the Lewis thesis that arguments purporting to prove necessary propositions cannot be sorted out into valid and invalid simply on the basis of whether each nonpremise line is entailed by earlier lines (p.234). And this is simply to accept from the outset, what was to be defended, that necessary propositions are entailed by every proposition. There is no argument, but instead another piece of tournament philosophy: no argument, but another challenge. Bennett invites those who do not accept the amazing Lewis thesis to defend a rival position. But each different theory of entailment can furnish such a sorting of the arguments in question into valid and invalid, at least for sufficient range of cases, e.g. by use of canonical systems the challenge can readily be met (later chapters show how in detail).

34

7.4 THE MISTAKEN SUBSTITUTION PRINCIPLES STRICT-IMPLICATION SUPPLIES

Just as the framework of the material-conditional, classical logic, is inadequate to define a decent conditional, so, to turn now to the attack, the framework of strict-implication systems is inadequate to entailment. The basic trouble with strict-implication can be located in the substitution conditions it admits. (The key to the logical behaviour of a connective lies in its substitutivity conditions.) The equivalence which guarantees intersubstitutivity within strict implication sentence frames is strict equivalence, defined as strict coimplication thus: A 6-4 B =

(A -» B) & (B

A) .

For systems which include Feys' system S2° or Lewis's system S2 - the only strict systems ever seriously considered as offering an analysis of entailment - coimplication can be redefined: A M B = D f d(A = B). The upshot is that provable material equivalents are everywhere intersubstitutible in these systems, so that a great many undesirable implicational theses are immediately derivable. In particular, no discrimination in deductive power can be made between theorems, since any two have exactly the same set of consequences - something that is obviously false. Similarly no discrimination can be made between logically false statements: they all have the same set of consequences, namely everything. These disastrous results are magnified in theories based on entailment, such as theories of logical content and semantic information (as FD and UL explain in detail), e.g. all necessary truths have the same logical content, namely none, and convey the same information, again none. The worst manifestations, in purely entailmental terms, appear in the paradoxes - that a contradiction entails every statement and that any statement whatsoever entails each necessary truth. For any contradiction is strictly equivalent to B & ~B, and B & ~B entails B for arbitrary B; hence, by substitutivity, any contradiction entails B. Similarly using substitutivity, B entails, on the strict account, every necessary truth. These results can be traced back of course, to the fact that strict implication, as an analysis of entailment, amounts to necessary material implication. As in the case of material implication, however, it is not just that the analysis admits these paradoxes. There are several other principles admitted which (though analytic of course for strict implication) fail to hold when -3 is construed as entailment. The stronger the strict implicational system the more of these unwarranted and unwanted principles there are as theses. Because of these significant differences it is important that stricters tell us which system of strict implication it is that really captures entailment. But this most stricters are extraordinarily reluctant to do, even when they are informationally equipped to make a choice1 - perhaps for good, if rarely divulged, reasons. 1

This applies in part even to Lewis, who wrote in 1932, at a stage when he was still informationally unequipped to distinguish S2 and S3: Prevailing good use in logical inference - the practice in mathematical deductions, for example - is not sufficiently precise ... to determine clearly which of these five systems expresses the acceptable principles of deduction. (Lewis and Langford 32, pp. 501-2). (footnote continued on next page)

35

1.4 IMPLAUSIBLE "ENTAILMENT" PRINCIPLES OF SB

The few who do choose often plump for S51 which, despite its merits as an analysis of modal notions such as logical necessity, is hopeless as an analysis of entailment, containing some singularly undesirable principles. For example, Peirce*s "law" ((A ^ B) o A) => A, rightly rejected by all logics (such as intuitionistic logic) built on positive logic as an unsatisfactory implicational principle, reappears in an unconvincing modalised form in system S5 (but not in S4 or weaker systems) as ((A -f B) H A) A, where A is strict, i.e. of the form C -3 D. In fact S5 is distinguished as a strict implicational system from its rival S4 by the modalised Peirce "law" or one of its complications such as:(((A-?B)-JC)-JB)-i. A -J B (((A -J B) -3 C) B) B-iD-J. A W D (((A-3 B) C) B) -5. B -i D -?. E -i. A—? D (((AH ( ( (A

B ) - I C ) - J D ) -4. B D -3. A -3 D B) C)T3 D ) B - » D -3T. E . A-* D

Although none of these principles is evident2 or indeed at all plausible as an (footnote 1 continued from previous page) Apart from the limitation to the five systems S1-S5, this is a claim with which we have a good deal of sympathy (cf.chapter 3). Lewis's indecision, such as it was, was not to last. Already in Lewis and Langford (32, p.178) it is said: It is this System 2 which we desire to indicate as the System of Strict Implication; and by 1960 Lewis wanted to insist: I wish the system S2 ... to be regarded as the definitive form of Strict Implication (Preface to Dover Edition of Lewis 18). l The choice is often made on such insufficient grounds as strength, mathematical elegance and simplicity: cf. Lewis's astute observation (32. p.502): Those interested in the merely mathematical properties of such systems of symbolic logic tend to prefer the more comprehensive and less 'strict' systems, such as S5 and Material Implication. The interests of logical study would probably be best" served by an exactly opposite tendency. Occasionally however the choice is based, more satisfactorily, on the semantical structure of S5 and its logical theory of modalities. This points up the unfortunate historical conflation of entailment with modality, and how a very likely account of logical modality can lead to a most unlikely account of logical implication. 2

AS Prior (62, p.49) remarks of Peirce, p -»• q -*• p -»• p, itself: Its truth is not evident at a glance, but we might look at it this way: it means If p does not imply q without being true, then it i:s true. That is, it means Either p implies q without being true, or it is true. If p is true the second of these alternatives holds. And if p is false, then it does imply q (for COq = 1, whatever q may be) and of course does so without being true, so in this case the first alternative holds. (footnote continued on next page)

36

1.4 LEWIS SYSTEM S2 AMD OTHER MODAL AFFIXING SYSTEMS

like the system S3 that Lewis once favoured, has to be made. It is easily demonstrated (using matrices or a semantical analysis of non-normal modal systems, e.g. Kripke 65) that S3 has no theses of the form A -5 B. The focussing by Lewis of the choice of a correct (or as he would have said, acceptable) logic of deducibility on the systems S3 and S2 undoubtedly represented a vast improvement over most of what had gone before him, especially in the direction of the formalisation of the logic of entailment. Since S2 is, according to its author, 'to be regarded as the definitive form of Strict Implication' (Lewis 60) and since it provides the locus of much subsequent discussion of entailment, we exhibit the system in explicitly entailmental form, not in the usual modal form. Connectives -*• (symbolising Lewis entailment, i.e. Strict Implication), &, and ~ are taken as primitive, and further extensional and modal connectives are defined thus: A v B ~(~A & ~B); A a B - d £ ~A v B; A = B « D f (A = B) & (B => A) ; A B » D f (A B) & (B -*• A); DA ~A A; 0A » D f ~0~A. (A detailed discussion of the morphological structure of sentential entailmental systems may be found in 4.1.) The postulates of S2 in Lewis entailmental form are these:- A •* A (Identity) (A -*• B) & (B -*• C) A -»• C (Conjunctive Syllogism) A&B+A A&B+B (Conjunctive Simplification) (A -*• B) & (A C) A -»» B & C (Composition) A & (B v c ) (A & B) v C (Distribution: the scheme is not independent in S2) ~~A -»• A (Double Negation) A •*• ~B •*•. B -»• ~A (Contraposition) (A ~A) ~A (Reductio) A&B-+C->-. A&~C->-~B (Antilogism) A, A -*• B B (Modus Fonens) A, B A & B (Adjunction) A -*• B, C D-»B C •*•. A D (Affixing, or Becker rules). Other strict implicational systems of entailmental interest are these (the classification anticipates that we subsequently adopt in 3 for entailmental systems):Other Affixing Systems: These result by varying the axiom schemes for S2; e.g. Feys' system S2° results by deleting Reductio, or what perhaps surprisingly (and wrongly in the larger relevant context) is axiomatically equivalent: A & (A B) -»• A, (Conjunctive) Assertion. Another example is the relevant logic DL (of DCL, formulated without sentential constants) which results from S2 by deleting Antilogism; and yet another example is the relevant logic DK (of UL) which results from S2° by deleting Antilogism, i.e. from S2 by deleting both Antilogism and Reductio. The formulation of S2 given thus brings out sharply the key separation point between strict and relevant accounts of entailment, a point to which we will repeatedly allude (especially 2.6 ff), namely the principle of Antilogism and its outcome, Disjunctive Syllogism. It was this one major obstacle that kept Lewis forever out of the relevant paradise. (But really the unquestioned loyalty to Disjunctive Syllogism and its mates, such as Rule Antilogism, is part of the larger classical-modal psychosis which we subsequently analyse.) Hitherto the main concentration in relevant logic studies has been on analogues of the exportative affixing system S3, obtained by addition to S2 of one form of Exported Syllogism, i.e. A -*• B C -*• A C -*• B (Prefixing) , or A B B -»• C A -»• C (Suffixing), or of one of its modal equivalents, e.g. A-*B-*-. DA-*- []B (Necessity Distribution), or A -»• B -»-. OA -*• 0B (Possibility Distribution). Plainly both the Affixing rule and Conjunctive Syllogism are otiose in such exportative affixing systems. Other Exportative Affixing Systems. These include S30 - S2° + Exported Syllogism; S4 which adds to S3 any of the following schemes: B A -*• A, •(A -»• A), DA -*• CDA, OCA B as true where B is necessary. 7. The modal reduction fallacy. The fallacy is that of defining A -»• B as i(f>(A,B) where i is a (one-place) modal connective and 4>(A,B) is a truthfunction of A and B. More generally, neither entailment nor conditionality are modal. Any adequate account will meet this strong intensionality requirement; and this is the basic reason why the investigation of implication generally has to be in the area of the intensional beyond the modal, why - to state a central thesis soon to be advanced - it has to be ultramodal. § 5. Conditionals: the theory sought in contrast to extensional and modal attempts. The orthodox attempts at gaining an analysis of entailment - the metalinguist ic shift to avoid overt systemic introduction of modal and intensional notions, and the strict implication account in terms of modality - leave one with no satisfactory account of non-logical implication. Thus, given that our case against material-implication stands up, the problems of a natural implication, and of conditionals, are not satisfactorily resolved at all within the usual metalinguistic and strict implicational frameworks. Yet a great many

41

7.5 'IF' STATEMENTS, AMD WHV SOME NEW CLASSIFICATION OF THEM IS ESSENTIAL

implicational statements are not analytic, for instance the lawlike statements of science and everyday conditionals, and many conditionals are, as we have noticed, not implications: but such statements are commonly deployed in reasoning, e.g. in the empirical sciences, and in legal and ordinary arguments, and they have a logic, distinct from material-implication, which it should be the business of a full theory of implication to capture. In an effort to fill these lacunae there have recently appeared on the market various theories of conditionals, which are supposed to supplement the modal view and thereby overcome its acknowledged weaknesses on the conditionals front. Logical space is made for these theories, not by abandonment of the thesis that entailment is necessary implication but by the distinction of nonnecessary implication from the conditionality relation. The conditional connections these theories are intended to capture are supposed to be neither sufficiency relations nor extensional connections such as material-implication (cf. Stevenson 70, pp.28-29). So much is not in dispute: There certainly is a class of non-sufficiency conditionals, in common usage in natural languages, whose intensional analysis is an important logical matter. Although some have hoped that these conditionals could be defined enthymematically in terms of a satisfactory sufficiency relation, there is nothing to stop, and reason to encourage, their independent logical investigation. For, among other things, if enthymematic reducibility is to be established as distinct from just claimed, an independent account of -the class to be reduced needs to be given. There is not only a strong case for independent logical investigation of (non-sufficiency) conditionals - and conditionals are after all central in much everyday reasoning - but there is also a good case for jettisoning many former divisions and exclusions built into or presupposed by theories of conditionals. For example, there are semantical reasons for not trying to make the artificial and never satisfactorily determined or vindicated division of English conditionals into "counterfactuals" and "noncounterfactuals", where a "counterfactual" usage is not simply (a counterfactual) one where the antecedent is false, but 'one in which its antecedent is uttered in the knowledge, belief, or temporary supposition that it is false' (Cooper 68, p.291). Part of the argument is that there is no evidence that 'if' differs in semantical meaning when contextual conditions are so varied, and indeed evidence that it does not - just as there is evidence that the correctness of an implication cannot be determined from the truth- or modal-value of its consequence. Likewise there is a prima facie case for not separating out counterlogical or counterphysical conditionals, i.e. those with logically or physically impossible antecedents. Nonetheless it seems that any non-vacuous logical theory of conditionals is bound to exclude some natural language occurrences of if ; that is, it seems that there is no non-vacuous way of encompassing all occurrences of 'if' within a single uniform analysis. The argument, such as it is, is that any all-encompassing theory would be vacuous because any proposed thesis it contained could be defeated by a counterexample of sorts. Remember that in English 'if' can do duty for such connectives as 'whether', 'even if* and 'as if', as well as for 'if ... then'. The claim is that the intersection of the logical properties of these diverse connectives is null. Consider, to illustrate the inconclusive case-by-case method of confirming the claim, a few of the more likely or desired principles of an i^ of argumentative value. Firstly, consider detachment, i.e. Modus Ponens. This appears to fail in cases where 'if' can be replaced by 'whether', e.g. from "you see if you can turn the wheel" and "you can turn the wheel" we cannot validly infer "you see". Next, consider Identity, and let 'if' do duty for 'even if* in the intended

42

1.5 GENUINE CONDITIONALS SURVIVE THEN-TRANSFORMATION

sense where this makes a contrast, as in 'he is intelligent if crazy*. Then "he is intelligent (even) if he is intelligent" is not just weird but not true. Differently "I am wondering if 1 am wondering" is not always true. In the same way, by taking examples where 'whether* or 'even if' can be substituted for 'if', countercases to such basic principles as Conjunctive Simplification can be devised. No principles, it would seem, survive this sort of procedure. The procedure does indicate, however, the first kind of condition that should be used to restrict the class of genuine conditionals, namely that uses of 'if' where 'if' does duty for 'whether', 'even if' or 'as if' should be excluded (at least from the core theory). There is a simple linguistic test which shakes out many of the undesired cases, namely reordering and then insertion. The test, which we call then-transformation, is this: given 'A if B' reorder to 'if B, A' and insert then to obtain 'if B, then A'; finally ask whether the result makes sense or has the same sense as the original. Examples: 'if you can turn the wheel then you see' does not have the same sense (as qualified salva veritate tests will show) as 'you see if you can turn the wheel', and 'he is sound if unimaginative' does not have the same sense as 'if he is unimaginative then he is sound'. A necessary condition on a genuine conditional is that it should survive then-transformation. Genuine conditionals do appear to have a logic, even if the pure conditional logic itself is not of vast logical interest, consisting, as it seems to at bottom of Modus Ponens and Identity (such a logic is reviewed in 2.1 and subsequently in chapter 5: the minimal B conditional logic is in fact the pure implicational fragment of system B of chapter 4, and all its theorems are instances of Identity). But, rather like the parallel logic of deducibility, the interest and difficulty of a general conditional logic quickens when further connectives, such as those of the orthodox set {&, v, are introduced. Many principles that appear correct for an implicational logic fail for a general conditional logic - something to be expected since the I-C transformation (of 1.3) fails for many genuine conditionals. For example Contraposition which appears correct at least in some forms for implication (e.g. in the form: when A ->- B then ~B -»• ~A) fails for conditionals taken generally. Counterexamples to Contraposition abound: while "if you are tired then we will sit down" may be true "if we won't sit down then you are not tired" is commonly false; "if it rains then I'll take my umbrella" rarely ensures "if 1 don't take my umbrella then it doesn't rain"; and so on. The failure of Contraposition (especially for "temporal" if-thens) is indicative of the fact that conditionals do not in general meet sufficiency requirements: that the antecedent A of a true (or acceptable) conditional A •*• B is commonly not logically or naturally sufficient for the consequent B. When A is not sufficient for B, further background assumptions are involved in getting B from A, so that when ~B holds it may not be A that is negated but some of the further assumptions that lie in the background.1 But not all negation principles fail: 1

Syntactical enthymematic representation of the conditional A 3 B in the relevant framework as A & t -»• B, where t represents some background class of truths, shows up a technical reason for Contraposition failure. ~B D ~A would require ~B & t -»• ~A, that is it would depend on the rejected Antilogism. But the syntactical representation is of only limited value, since it sanctions irrelevance; it also happens to validate Augmentation.

43

7.5 THE PROBLEM OF THE WATERSHED PRINCIPLE OF AUGMENTATION

Double Negation, for example, appears to hold in both directions. Contraposition is not the only implication principle that is faulted when ordinary conditionals which do not pretend to offer sufficient conditions are countenanced. With conditionals, as distinct perhaps from implications and entailments, very many commonly accepted logical principles are up for reconsideration, and rejection. A critical issue in determining the general logic of conditionals is whether the rule Affixing holds; for the rule is a watershed principle in semantical analyses of implicational-conditional logics (as will become evident in subsequent chapters). An important test case for Affixing is provided by the principle A -*• C -*•. A & B -*• C

((Antecedent) Augmentation)

in combination with A & B -*• A; for were Affixing correct, Augmentation would result from Conjunctive Simplification. But according to recent logical wisdom Augmentation fails for the conditional relation. The case against Augmentation is two-fold; it tends to be based on intuitive counterexamples, these being backed up by various, incompatible, theoretical explanations drawn from the different theories that have emerged. A familiar counterexample (Stalnaker 68, p.106; Stevenson 70, p.27) runs as follows: there are cases where it is true that if you strike that match it will light without it being true that if you wet that match and strike it it will light.1 Another example is this: "If I walked on the ice, it would remain firm" could be true, while "if I walked on the ice and I wore 60 lb. boots, it would remain firm" is not (cf. Bennett 74, p.384). There are ways of explaining away examples like these; but there are, so Lewis, Bennett and others contend, too many examples for the dismissals to continue to look convincing. As almost always, the arguments for this kind of claim are not decisive. Bennett's argument, for example, makes the assumption that moves to explain away apparent counterexamples to Augmentation must treat conditionals as elliptical (or enthymematic), to be expanded, in determining the logic of conditionals, by further clauses conjoined to the antecedents. And it is true that there are then well-known difficulties about ever completing such clauses satisfactorily, since the background conditions to be incorporated are open-textured in several ways which defy complete linguistic description. But the strategems aimed at explaining away the counterexamples do not have to be of this sort; they can legitimately refuse to take up background or contextual conditions syntactically in antecedents.2 It is enough, presumably, to meet the "counterexamples" to contend that the conditions which falsify the conclusion of a deduction of A & B C from A C (to take an entailmental form of Augmentation), that is typically which guarantee A & B but falsify C, undermine the basis for, or are incompatible with, or background conditions for, A -*• C. Thus in Bennett's example, my wearing 60 lb. boots undermines (or in Adams' revealing terms, is incompatible with) the background or contextual conditions under which the conditional "If I walked on the ice, it would remain firm" is held true. This strategem, which we call the A-stratagem (after Adams 65), is of very general application, defeating, in its fashion not merely all counterexamples to Augmentation but also those to Conjunctive Syllogism and to other conditional principles that initial intuitive considerations might seem to reject (cf. p.10 above). 1

It is enough that there be appropriate cases to rebut an entailment or conditional. The fact that the Greenlite matches that Routley often has to use at Plumwood Mountain will light after being dipped in water does not count against familiar cases where non-waterproof matches are used. The fact does help expose however one of the background assumptions - that that match is non-waterproof - thereby emphasizing the non-sufficiency character of the connection. (footnote 2 on next page) 44

7.5 THE SEPARATION OF RULES FROM THESES IN THE LOGIC OF CONDITIONALS

On the other hand, there are, rather obviously, various difficulties with this sort of stratagem. For one, it is no longer possible to assess on their own untreated conditionals, conditionals that have been subject to no analysis or investigation1: to know whether a conditional holds one needs to know, so to speak, what is supposed to follow from it. And this makes conditionals virtually unassessible, and a great many ordinarily acceptable conditionals false - because otherwise, were they true, we could, by defeating background conditions, counter Augmentation. Given that Augmentation is to be rejected for untreated conditionals, and also for important subclasses of conditionals such as counterfactuals, given indeed that Augmentation is a "fallacy" (as D. Lewis puts it), an important question for formalisation arises: What forms of the principle are fallacious? The same difficult question arises also for other rejected principles, Contraposition and Syllogism. Granted that both conditional and entailmfental forms of Augmentation fail for conditionally, i.e. neither Antecedent Augmentation as formulated above nor its entailmental reformulation, A -* C A & B C (where distinguishes entailment), are correct, what of the rule form: A C —t>A & B -*• C? The rule is evidently separable from theses forms, and does not succumb to the counterexamples given to theses forms. It is noteworthy, for instance, that, despite Stalnaker's emphasis on the breakdown of Augmentation, the formal system Stalnaker arrives at admits the rule form. Similarly, though Hunter and Graves 73 consider Augmentation to be a principle which differentiates conditionality from entailment, they adopt principles (including Conjunctive Syllogism) which lead to the rule form of Augmentation.2 Thus an easy answer to the question about the rule form is that we can leave the matter open for the present: what fallacies there are, if any, are avoided by avoiding the theses forms. The distinction of rules and theses is important not merely with respect to the axiomatisation of the logic of conditionals, but also as regards the common claim that the failure of Transitivity (in conjunctive form) follows from the failure of Augmentation (Stalnaker 68, p.106; Lewis 73, p.32ff; Bennett 74, p.385). Unless entailment is erroneously identified with strict implication, this common claim is false. The assumption is that Simplification, A & B ->• A, being logically necessary, can be suppressed; so given Conjunctive Syllogism in the form (A & B A) & (A C) A & B C, Augmentation results (and similarly where the main connective is strengthened to an entailment). But such suppression of conjoined necessary premisses is inadmissible, and fallacious. All that the failure of Augmentation establishes against Transitivity is the failure of Suffixing (in thesis and rule forms), not the failure of Conjunctive Syllogism. Only the failure of the rule form (footnote 2 from previous page) Bennett has, it may be alleged, made the similar error - in assuming that background conditions can be taken up as antecedent conjuncts - to that he made in his "case" for the general suppression of necessary truths. 1

Thus heavy use of the A-strategem would violate our earlier methodological strategy.

2

The collapse of the original Hunter-Graves system does not detract from this point. For the point is correct for subsystems of the HunterGraves one.

45

7.5 THE FURTHER CASE AGAINST CONJUNCTIVE SYLLOGISM, AND ITS DISSOLUTION

of Augmentation would undermine Conjunctive Syllogism, as it would undermine even Rule Syllogism, A B, B C A -»• C. Whether Rule Syllogism holds or not makes a substantial difference to the semantics of a logic (as will become evident in chapter 13); so the determination of exactly what is supposed to be rejected in conditional and counterfactual logics is not a merely idle matter. From this perspective the way the common rejection of transitivity is stated is rather sloppy, not to say misleading; thus, for instance, Stalnaker writes: '... the conditional corner is a non-transitive connective. That is, from A > B and B > C, one cannot infer A > C* (68. p.106). But in the usual way in which inference rules are stated (e.g. in Church 56) one is entitled to infer, in Stalnaker*s system C2, A > C from A > B and B > C (for when A > B and B > C are valid so is A > C). The claim intended is that A > B and B > C do not entail A > C; that there is, in this sense, no valid inference from the premisses to conclusion. The remaining case, directed against Conjunctive Syllogism is again twofold; arguments based on direct counterexamples, and more theoretical arguments from the analyses of conditionals proposed. We have already glanced at the sorts of direct counterexamples offered, and observed that there are conventionalist strategems by which they can be disposed of; and one stratagem will do, the A-strategem. Consider how this disposes of a counterexample to Syllogism from Bennett: If there were snow on the valley floor, I would be skiing along it; and if there were an avalanche just here, there would be snow on the valley floor; but it is false that if there were an avalanche I would be skiing on the valley floor — (74, p.385). According to the stratagem, the antecedent of the conclusion, that there be an avalanche, is incompatible with the truth of the initial premiss, because if there were snow from an avalanche I (or Bennett) would not be skiing along it. Insufficiency implications, such as conditionals are, are contextually-bound: they are true at best under present conditions, and when the conditions are changed, as by an avalanche, they fail. The apparent counterexample emerges only, so the A-strategem complains, by changing the contextual conditions as we go from premisses to conclusion. As we have observed, however, in the case of Augmentation, the A-strategem, may do too much; and there is, moreover, a good methodological case for a theory of (relatively) untreated conditionals. Furthermore if a general theory of untreated conditionals can be devised, we can then check the adequacy of reductions and likewise of strategems rigorously, and we can also investigate the particular logics of conditionals which satisfy given classes of conditions, e.g. contraposable conditionals, transitive conditionals, augmentable conditionals, to begin on a classification by logical principles of conditionals (cf. Rennie's procedure in the case of adverbial modifiers, in 74).1 The idea is that the logic of augmentable, contraposable conditionals, for example, will emerge as a special case of a more general conditional logic, in something the same way that the' theory of commutative groupoids comes out as a special case of the theory of groupoids. Similarly the logic of counterfactual conditionals, as a case of the logic of lawlike conditionals where the antecedent is false, will be a subcase of the logic of l The classification of conditionals by way of logical principles they satisfy is only one, among many, of the classifications that might be tried. Other classifications, most of them less than fully satisfactory, already occur in the literature, e.g. Goodman's classification in 55.

46

7.5 FAILINGS OF RECENT (IRRELEVANT) THEORIES OF CONDITIONALS

transitive augmentable contraposable conditionals, assuming (as we shall argue in chapter 8) that these properties hold for lawlike connections. The acceptance of Conjunctive Simplification separates main modal theories of conditionality from connexivism, a position with which, because of the rejection of Augmentation, they otherwise have a good deal in common. For the deletion and defeating features of the added antecedent in Augmentation is exactly what lies behind the connexivist critique of Augmentation, a critique which extends automatically to apply against Simplification (see chapter 2). It is not too surprising, then, that modern theories of conditionality are split by Conjunctive Simplification into two factions, a connexivist group (which includes Cooper 68, Stevenson 70, Downing, e.g. 61) and a non-connexivist group (which includes not only resemblance theorists such as Woods 67, Stalnaker 68 and Lewis 73 but many others, e.g. Hunter and Graves 73, Xqvist 71, Gabbay 73). Many of the recently proposed theories of untreated conditionals, whether connexivist or not, have the alleged virtue that they do not retain the faulty principles of Augmentation, Contraposition and Transitivity. But most of these theories are nonetheless defective, in that they incorporate principles which do not hold for conditionality, in particular relevance-violating principles. Thus they succumb, exactly like the theories they reject, to intuitive counterexamples. For example, the most publicised systems (American ones of course), those of Stalnaker and Thomason 70 and of D. Lewis 73 are badly irrelevant. Both the Stalnaker-Thomason system and Lewis's official logic of counterfactuals have as theses (and in rule form) the paradoxes: (~A -*• A) (B A), i.e. anything implies a conditionally necessary statement, and A & B A -»• B, thus any truth conditionally implies any other truth.1 In short, the theories provide no requisite connection between true antecedents and true consequents in a conditional or between any arbitrary antecedent and any necessary consequent: in such cases the accounts are every bit as fallacious as material-implication and strict implication, committing truth-copulation and modal-copulation fallacies. These systems also contain several other bizarre or undesirable principles - arising in part from the fact that they were obtained principally by working back from a possible world modelling to an axiomatic basis. For example, Stalnaker's system (investigated formally by Stalnaker and Thomason) includes the principle (A B) v (A -»• ~B) already criticised, the paradoxes (A -*• ~A) A -*• B (a principle which also rules out proper treatment of counterlogical conditionals) and A = (B =. A •*• B), and the dubious principle ~(A -»• ~A) A -*• B =. ~(A -* ~B) . In rejecting these theories we also reject the particular semantical and philosophical analyses in terms of which the acceptances and rejections of logical principles are justified. The worst faults of the semantics, the modal framework, we shall criticise, but the question of what can be salvaged from the extraordinary theories of neighbouring worlds and similar worlds which are written into the intended construal of the semantics, we postpone until we elaborate our own (ultramodal) theory of conditionals (at which stage resemblance theories are severely criticised). 1

~A -»• A B -»- A holds in all three Lewis systems, and A, in two systems.

47

A ->• B

7.5 THE INADEQUACY OF MODAL THEORIES

Other recently proposed theories of conditionals, especially those that are properly equipped with semantical modellings, are equally defective, both in terms of the principles they contain and as regards the underlying philosophical analysis. For example, according to Aqvist's 71 system a principle of connectivity: (A -*• B) v (B -*• A) , where B is of the form C •> D, holds, while according to Gabbay's 73 system, if A s B is a theorem, so is A •*• B, whereupon the full force of the paradoxes is immediate. One trouble with all these theories of conditionals is that they take as entailment, as the underlying logical implication, strict-implication - something we have already seen to be inadequate. But the basic trouble with these theories of conditionals - of which the adoption of strict-implication as logical implication is symptomatic - is that, like strict-implication they operate essentially within the possible worlds framework of modal logics, and so are subject to the inevitable deficiencies of such approaches, namely modal substitutivity conditions and ensuing paradoxes and sundry other unpleasantness. As customary, we say that an n-place connective $ is modal in the ith place iff whenever A B B is true (or A = B is a theorem) $(... A ...) iff $(... B ...). A functor is modal iff it is modal in each place, and a sentential language is modal iff all its connectives are modal.1 Possible world modellings are only adequate for modal languages. Modal notions have been raised to a position of excessive prominence in recent philosophy, partly due to the mistaken idea that the entailment connection, propositional identity, and conditionality are modal matters, and partly owing to technical limitations, that known semantical analyses - possible world semantics, and their metalinguistic correlates involving state descriptions succeeded at best for modal notions. This elevation of modal notions has done an enormous amount of philosophical damage, not merely in the case of entailment and conditionality, but with a range of other notions of major philosophical importance and interest such as evidence, confirmation, belief, knowledge, perception and obligation (see UL). It has even fostered the grossly mistaken, but widespread, idea that all intensional notions are modal or, if not, merely sentential (admitting replacement of no logical equivalents of any sort but only of identical sentences). The fact is that a great many notions fundamental in philosophy (including all those mentioned above and many more) are of more than modal strength: they are ultramodal intensional notions, which cannot be forced into the modal straitjacket without grave distortion and the generation of many gratuitous philosophical problems (as UL and BP explain). Most important here, none of the main notions we are seeking to explicate, namely entailment or logically necessary implication, law-like implication, and conditionality, nor any of the associated notions, such as propositional identity, are modal. All are ultramodal. Take implication. It is not a modal matter; for the substitutivity conditions required are stronger than modal, i.e. formally the truth (or provability) of A H B guarantees neither A -»• C iff B C nor C -*• A iff C -*• B. The ground we have already traversed shows this, and more. More generally, there can be no reinstatement, by patch-work repairs and qualifications from strict-implication and material-implication and similar purely modal and extensional notions such as the modern probability relation, of strict-implication or material-implication as satisfactory accounts of entailment and conditionality, and similarly there can be no constructions of satisfactory accounts from these elements. For such accounts always satisfy modal substitutivity conditions. This disposes of many proposed analyses (e.g. those of Adams 65, Stevenson 70, Geach 74). 1

These technical senses take up one (the main) traditional sense of 'modal'.

48

1.5 CHARACTER OF THE SOUGHT THEORY OF CONDITIONALS

So results a further condition of adequacy on theories of entailment and conditionality, the strong intensionality, or ultramodal, requirement, that adequate theories must be ultramodal. Just as strict-implication is insufficiently intensional to provide a satisfactory account of entailment, so, as we have now seen, modal accounts of conditionality, such as modal similarity theories, are likewise insufficiently intensional. Actually, as compared with implication and conditionality, which are fundamental notions in argumentation and reasoning and so central in philosophy and also in such more practical matters as the law, modal notions are of relatively slight importance. Through modal muddlement, however, which has taken entailment and conditionality as modal notions, the importance of modal notions and modal logics has been much exaggerated - though not of course to the extent that the importance of classical logic has. Modal befuddlement is not the only factor at work in modal-style analyses of entailment and conditionality. There is also the older, associated, assumption to be found in C. I. Lewis and in Burks' analysis of causal implication 51, but reappearing in Stalnaker and Thomason 70 and D. Lewis 73, that conditionality can, like implication, be correctly defined in terms of a oneplace connective (characteristically but not necessarily, modal), such as physical necessity or conditionally-defined necessity, in combination with truth functional connectives. Meyer has put the kibosh on this assumption (in 74b; adapted in ABE under the misleading title 'Relevance is not reducible to modality', since neither relevance nor modality are to the point). There is no need to appeal to the outcome of modal substitutivity conditions to show that genuinely two-place connectives such as entailment and conditional relations cannot be defined in terms of some one-place connective, modal or~ not, applied to a combination of truth functional connectives. That is, A ->• B cannot be defined as $$(A,B), where $ is a one-place connective satisfying a fairly minimal set of conditions and \|»(A,B) is some truth-functional combination of A and B.1 To begin, then, from the assumption that conditionality can be captured by simply adding a one-place connective to the logic of truth-functions, is to make a false start. Some of the newer systems of conditionals have implicitly recognised this limitation, that conditionality will have to employ genuinely two-place connectives. However they have persisted with the assumption that conditionals are nonetheless modal. This is false, as relevance considerations alone reveal in the case of natural language conditionals. Assembling the conclusions reached sets the shape of the general theory of conditionality sought. The theory will contain an irreducibly two-place conditionality connective which is ultramodal. The theory will be relevant, and ideally will be coupled with a relevant theory of entailment - certainly not with a strict one. The theory will, in its most general form (of 13), fail such principles as Contraposition, Augmentation and Transitivity, but validate such principles as Simplification. The theory should be underpinned by a semantical analysis preferably of a suitable philosophical cast. There are several hitches to this ambitious undertaking, to some of which we now turn, the main intellectual obstacle being the mistaken idea that any satisfactory theory must include a classical sublogic. 1

For the conditions required, see ABE, p.466, p.471. Note that system E can be replaced by any subsystem of RM3 which includes the basic logic of chapter 13; and accordingly that the argument applies both to the desired system D of deducibility, whatever its precise final form, and to that sought for conditionality.

49

7.6 DISTINGUISHING FORMS OF THE CLASSICAL DISEASE

§6. The classical hang-up; how one can go wrong with material implication, and why no classically-based logic can be adequate. The trouble with all the attempts, so far criticised, to produce satisfactory theories of conditionals and logical implication is that they are far too classical in orientation features which emerge, firstly, in the way they all proceed by adding to classical logic and aim to incorporate classical logic as an integral part,1 and, secondly, in the semantics where only classical-like possible worlds are admitted. For no such classical-type account is going to provide a tight enough connection of antecedents and consequents in conditionals or therefore to preserve relevance. Strict-implication and irrelevant conditionals, are both classical in a worlds way, in that each world used in their semantical analysis conforms to classical canons. It is from this classicalness that their main inadequacies stem. Classicalness is a persistent and widespread disease (which affects even relevant logicians who like the logical ease and "smoothness" of classical semantical corditions which involve no new or less familiar functions). It has resulted in a vast overemphasis of modal notions, and underlies attempts to provide modal analyses of a great many - sometimes, in extreme cases, all - intensional notions, though it is evident that many of these notions are bound to resist modal, or extended-classical, analyses. But in contrast with implication - which is the basic argument relation* and hence fundamental in philosophy - modality is much less important; it has assumed an exaggerated importance largely through defective accounts of implication and conditionals, which reduce them to modal notions and reduce coimplicational substitutivity conditions to modal interchange conditions. A much weaker claim is sometimes advanced in favour of classical logic or modal logic, than that they are adequate. It is claimed that even if material-implication [strict-implication] is defective as an implication [entailment] and perhaps needs some augmentation, classical logic is nonetheless correct and classical logic has to be included in any adequate logic of implication. Moreover, since one can not go wrong using classical logic, it provides a minimal working base, which there is a good case for using in controversial areas. It is, so to speak, a core logic with which one can comfortably rest. This is all too familiar bunk, which we will try to expose. The core-correctness claim may be weakened still further to: one can't go wrong with material-implication so long as simply figures as the main connective in inferences - which is all it is ever needed for. Even this latter assertion is of pretty doubtful correctness unless only very limited purposes are envisaged - certainly not formalisation of much of discourse as it figures in commerce and in newspapers and occurs on television, nor formalisation of the intensionally less rich mathematical discourse, where nonetheless second degree arguments and inferences with iterated implications occur. We will examine these stronger and weaker forms of the claim that one can't go wrong with hook together. First of all, we do not of course dispute the truism that classical logic is correct in classical contexts, i.e. 1

Modal logic is incorrect for the same reason, because it incorporates classical logic as an integral part of its structure, whereas a correct modal logic would be built on a relevant base. Similarly for modal conditional logic.

50

7.6 WAV'S OF GOING WONG

WITH CLASSICAL LOGIC

in those contexts where it is correct. But these contexts provide only a quite proper subclass of the contexts where some logic is needed, and where classical logic is supposed and intended to apply. There are then logical contexts which are not classical, and within these classical logic is incorrect (see also UL). Moreover while we agree that the & - v — theorems of classical logic though not doxastically compulsory - are true (for the extensionally-refined connectives of natural language that they consider), and accordingly should be theorems of any adequate logic of implication, it does not follow that classical sentential logic is correct, or should be included in every theory, or can never go wrong. Classical logic is only correct if it is coccoonised, if it does not assert too much more than the & - v — tautologies. It goes wrong all too quickly under needed and intended applications (as the counterexamples of 1.2 reveal). One can go wrong with classical logic in two main sorts of ways. One can go wrong interpretationally, by interpreting => as amounting to some sort of implication rather than as simply short-hand for an extensional combination (e.g., not-or)1, and one can go wrong in a deeper way by making the consistency and completeness assumptions of classical logic and their worlds. We deal with these ways of going wrong in turn.2 A common view - summarised in the slogan 'You can't go wrong with material-implication' - is that because => is truth-preserving it is safe to use for valid argument. The common view is well-represented in Lemmon (65, p.60): While admitting that this discrepancy [between ^ and if-then] exists, we may continue safely to adopt 'A => B' as a rendering of 'if A then B' serviceable for reasoning purposes, since, as will emerge ... our rules at least have the property that they will never lead from true assumptions to a false conclusion. Indeed how can any truth-falsity counterexample (i.e., where the antecedent is true and conclusion false) to any of these laws be possible, when these laws are laws of material-implication, which is, as everyone knows, truthpreserving? The truth-falsity counterexamples are possible firstly because A ^ B is at best a necessary, and is not a sufficient, condition for A -*• B. The truth-preserving properties of material-implication mean that one cannot find truth-falsity counterexamples - of an extensionally-admissible kind (i.e. one world examples) - to A -* B when A ^ B holds; but if A ^ B is only a necessary condition for A -*• B, one can find a truth-falsity counterexample to the main connective in A => (B •* C), when A = (B => C) holds. If B C is only a necessary, and not a sufficient, condition for B -*• C, B C may be false when B a C is true. Hence one has only to find one of these cases to obtain a higher-degree truth-falsity counterexample to the main connective in A 3 (B -*• C) when A is true and A ^ (B ^ C) holds. (Exportation and Commutation may be countered in exactly this fashion: cf. 3.7 where counterexamples to Commutation are presented.) 1

The counterexamples of earlier sections all rely on intended interpretations of connectives, on the interpretation of extensional connectives in terms of natural language surrogates, and of and -3 as 'if-then* and 'entails' respectively.

2

In discussion of the first way we borrow heavily from V. Routley 6 7.

37

1.6 WHV HOOK IS MOT SAFE TO USE IN REPRESENTING CORRECT ARGUMENT

The second reason why counterexamples are possible - though not counterexamples of a one-world truth-falsity variety - is because, though A => B is true, A B is rendered false by a situation, distinct from the actual one which determines truth, where A holds but B fails to hold. Modal counterexamples to A -*• B, where A and B are both contingently true, are of this sort; a possible world is envisaged where A => B fails to hold. And the specific countercases to Disjunctive Syllogism (given in 2.9) are of this kind; though A & (~A v B) a B holds in the actual situation, deductive situations can be designed where A & (~A v B) holds but B does not. In the light of this, the common view, that because is truth-preserving it is safe to use for valid argument, or as a surrogate for correct conditionals, must be defective. First, the truth-preservation property of => does not show that it is safe unless it is assumed quite circularly that the only way an implication can be wrong is by having its premiss true and conclusion false in the actual situation, the real world, T. Second, o is not truth-preserving in the right sense to guarantee safety for valid argument. For it is not truth-preserving in the sense that if we substitute 'there is a valid argument from ... to ...' or is deducible from ...' or even 'only if' for all occurrences of '=>' in the laws of =>, what we obtain will continue to be true where the original => law was true. Similarly, once such a substitution is made, => can lead from true premisses to a false conclusion, as counterexamples show. Admittedly all the false conclusions are themselves about valid argument (or conditionals), but failure of this sort can not be discounted given the uses of deducibility in assessing responsibility. Nor is it an adequate defence against these counterexamples to claim that the supposedly false conclusions are always about valid argument and that since truth-preservation, as represented by =>, is all that is required for deducibility, they must not be false at all, but true, even if surprising. Such a defence is quite circular, for we can only decide whether a particular connective, say ij^, as claimed, adequate for valid argument, by substituting 'there is a valid argument from ... to — ' for '=>' and seeing whether the results continue to hold; hence we could not without circularity decide these results on the basis of what holds for the connective itself. (A similar objection applies to the claim that the material-implication paradoxes are not paradoxical for deducibility because they follow from the truth-table for =>.) A connective is only safe to use for valid argument if upon substitution of 'there is a valid argument from ... to ...' for all occurrences of the connective in its laws, the result continues to be true, i.e. if every law of the connective is also a correct law of valid argument. It must licence no methods which are not also methods of valid argument; but many of the principles discussed, which are => laws, do license methods which are unacceptable as methods of proof or valid argument.1 And thereby one is licensed to go wrong. 1

These points also destroy van Fraassen's curious claim that classical logic is correct for mathematical English (71, p.4). Insofar as mathematics relies on valid argument, its proper formalisation is not in terms of classical logic. While it is true that much of classical mathematics can be reformulated using classical logic, such a formalisation is hardly unique, and alternative formalisations using strict systems or even relevant systems can undoubtedly be devised. In fact classical mathematics uses only comparatively weak logical inference principles, few highly nested implications and nothing like the power of classical logic or stronger Lewis systems. Finally what 'the correct logic for a language' is remains a somewhat obscure matter; but what seems clear is that a language may have a variety of logics associated with it (see Routley 75, for further discussions of this issue).

52

1.6 THE PEEPER TROUBLE WITH CLASSICAL THEORIES: MATERIAL DETACHMENT

Even if the interpretation of ^ as any sort of implication or validargument indicator or conditional is given away entirely and = construed as no more than an extensional abbreviation (e.g. for not-or) classical logic and classically-based logics remain in real trouble. For there is a deeper way in which classical logic is wrong, and in which exponents can, and do, go wrong with classical logic. Even though all extensional tautologies are true, one can still go seriously wrong through assuming the unqualified correctness of the principle of Jfeterial Detachment: Y- Where A and A => B are theses, so is B, or, in application to theory c, through assuming its theory analogue: Yc. Where A and A => B hold in c, so does B. Classically, of course, yc is assumed not only where c is the class of all truths T but also for every theory (i.e. y is assumed to hold not just for the actual world but for mathematical and scientific theories). This is part of the (mistaken) assumption that classical logic is universally applicable, to every (scientific) theory, and that it is fundamental in reasoning. It has too often escaped attention that principle y is a fundamental component of classical logic and of all its patch-ups. Classical sentential logic, as ordinarily presented and understood, consists not just of and-ornot schemes, or some batch of these taken as axioms, but also of the rule of material detachment. The deeper trouble with classical systems lies in the unrestricted assumption of this rule - which would be correct were => indeed a sufficiency connective - in the assumption, that is, that y holds for any and every application of the logic, for every theory. Consider what happens when classical logic is applied as the underlying logic to a theory which is simply inconsistent but not trivial, in that not every proposition holds in it. Such theories are not just important, they are common: philosophical theories, logical systems, dialectical positions, and belief systems furnish a wealth of examples (detailed cases are set out in DCL and in Routley2 75). But the presence of material detachment excludes them, and trivialises every simply inconsistent theory, thus:- Since a simply inconsistent theory contains B and ~B for some sentence B, it also contains ~B v C, i.e. B C, for arbitrary sentence C. Hence by Material Detachment it contains C, and so it contains every sentence.1 That is, it is trivial. Exactly the same problem also arises from use of any implication, such as strict implication, which satisfies the principle of Disjunctive Syllogism, DS . A & (~A v B)

B.

For this principle yields, by Modus Ponens closure, y. Or, more directly, since ~B v C and B belong to the simply consistent theory, so, by DS and implication closure, does C. The limited applicability of the rule of Material Detachment emerges from the conditions necessary and sufficient for its correctness. We have already observed that simple consistency is a necessary condition for its correctness. For sufficiency, let c be a theory which is prime, i.e. whenever A v B e c either A e c or B e c, and simply (i.e. negation) consistent, i.e. for no A do both A and ~A belong to c. Then y holds at c, i.e. yc is correct. For suppose A e c and ~A v B e c. As A e c, ~A / c, by negation consistency. But, as c is prime, either ~A e c or B e c; so B e c.2 1

It is assumed that the theory meets certain quite minimal normality conditions - which will happen in main cases of interest. (footnote 2 on next page)

53

1.6 TYPES Of OBJECTIONS TO MATERIAL VETACHMENT

The first class of objections to Material Detachment, and derivatively to its thesis analogue Disjunctive Syllogism, centres, then, on the point that they obliterate a class of theories of considerable philosophical and logical interest. A second class of objections arises from this, namely that many other important theories may, for all we know, belong to this class (cf. Lukasiewicz 70). In particular, it may well be that some important mathematical theories fit into the category of simply inconsistent but non-trivial theories. Naive set theory is a prime candidate. It is certainly simply inconsistent, but it is only obvious that it is rendered trivial by paradoxes like Russell's when it is classically formalised. We now know that, on the contrary, naive set theory is non-trivial when given a suitable relevant formalisation - a far more appropriate formalisation of the intuitive theory than conventional classical ones (as EMJB explains; see also 4.2).1 (footnote 2 from previous page) The sufficiency conditions also indicate the line of a main strategy for proving that y is admissible for a given logic L, namely by showing that each L-theory c which keeps out D can be replaced by a prime simply consistent L-theory c' which keeps out D. The feat is accomplished, where it can be, by first expanding c, by Lindenbaum methods, to a prime L-theory c + . But c + will commonly be negation inconsistent, so the technique is to select a subtheory c' of c + with the desired properties. This can be done, in some cases, by splitting c + , or, what accomplishes the same, by metavaluation methods (as in Meyer 72). *Russell observed long ago (06) that any way of avoiding the logical paradoxes, as classically formulated, involves changing the logic (of set theory), and accordingly he went on to consider three important ways of amending the logic of sets. But what he did not consider was the obvious move of changing the underlying classical logic. Yet this was the change of logic that really should have been considered. The option opened up, of retaining all of highly appealing intuitive theories by adjusting the underlying, usually classical, logic presupposed for formalisation, raises some awkward questions for the dedicated pragmatist; for example as to whether to trade off little used sentential principles for the powerful unqualified abstraction thesis in a formalisation of set theory. But the classical hang-up most pragmatists have has blocked investigation of this attractive alternative. The abandonment of the unwarranted classical assumption that every simply inconsistent theory is trivial also raises other severe problems for pragmatism. For the pragmatist conception of logic which is supposed - in opposition to absolutist views on logic - to make the choice of underlying logic a pragmatic matter, in fact in the presentations of its leading exponents, assumes that classical consistency conditions must be met - otherwise revision in the light of recalcitrant experience could not play its assumed role. In short, the pragmatist view presupposes part of the absolutist position it is supposed to be rejecting. It is facile to try to meet this objection as Haack (74, p.36) and others do by trying to rule out simply inconsistent logics. Dialectical logics are formally viable, and are coming to be part of the logical scene: they cannot simply be ruled out of court as not 'logics'. Nor can they be dismissed on the grounds that they fail to discriminate valid arguments since an inconsistency entails everything. For such paradoxes of deducibility any worthwhile dialectical logic would repudiate.

54

1.6 GROUNDS FOR SCEPTICISM ABOUT MARKETED SET THEORIES

Set theory of adequate power may well be inconsistent. (No doubt consistent theories may be obtained by sufficient mutilation - simple type theory without an infinity axiom provides a familiar example - but so far these theories are far too underpowered to haul the classical mathematical train.1) Most important, any classical set theory that approximates closely enough to naive set theory to undertake its work is likely to be simply inconsistent. Marketed set theories don't approximate too well at all, and there is a chronic need for better set theories than the inferior goods we have so far been offered in the way of formalisations.2 The fact that Quine's theories appear in some respects better than most others is indicative of the situation. Marketed set theories don't however have to approximate very well for inconsistency to threaten. Indeed it seems to us that some marketed set theories - Quine's system ML is the most conspicuous example are quite likely inconsistent. So scepticism about set theory and set theoretical foundations for mathematics has some warrant. A trivial foundation for mathematics - which is all that classically formulated inconsistent set theory would provide - is hardly of much merit, and one does not have to go through the laborious circuit of set theory to provide it. The conventional classical reply is that if a theory such as set theory is simply inconsistent it might as well be trivial; simple inconsistency is hardly any better than triviality. This supposes that a theory which is simply inconsistent but not trivial is of no logical interest, an assumption we have already seen to be false. Abstract set theory, with abstract sets characterised in terms of the abstraction axiom, contains, as Cantor saw, inconsistent sets, and accordingly it provides a good prima facie3 candidate for a simply inconsistent theory; but that's no good reason for concluding that the theory is trivial. Thus universal use of classical logic in set theory is without justification. The situation with regard to various other parts of mathematics is scarcely better than the set theoretic situation (cf. DLS).U Other historical mathematical theories have also proved to be simply inconsistent, most conspicuously the theory of infinitesimals and the theories of calculus and analysis that this theory supported. It was not evident - it is still not obvious - that these theories were thereby trivialised - and under a dialectical formalisation the theories may well turn out to be viable, as historically they were thought to be. 1

With only a small increase in capacity, however, they are powerful enough to haul the main semantical train of subsequent chapters.

2

The classical story of what mathematicians do when they use intuitive set theory, of what they used to do before the paradoxes were discovered and continued doing afterwards, is largely mythology - there is the parable of the consistent subtheory, of the restricted domain of discourse where sets are well-behaved, and so on.

3

The case so provided is only prime facie because there can of course be consistent theories of inconsistent objects. Other considerations lead, however, in the case of set theory to the conclusion that the theory is simply inconsistent, especially the intuitive arguments for the logical paradoxes.

u

Furthermore, through classical set theoretic formalisation, much of the rest of modern mathematics has been contaminated by set theory. But really classical set-theoretic formalisation is too rough a device to capture subtler parts of mathematical argumentation and inference, and even at quite (footnote continued on next page)

55

1.6 LIVING WITH CLASSICAL LOGIC IS LIVING DANGEROUSLY AND IRRATIONALLY

Worse still, the consistency of a sufficiently rich classically formulated mathematical theory can never be classically assured, and so conventional use of Y can never be justified. As a result of the classical limitative theorems, and especially Godel's consistency theorem, classically based theories are forced into a completely untenable position. For let MT be any sufficiently rich classical theory, i.e. any classical theory in which every recursive function is representable. Most mathematical theories satisfy this requirement by virtue of including arithmetic. Then, by Godel's theorem, the consistency of MT cannot be proved using just methods of MT; any attempt to prove the consistency of MT in a non-question-begging way will lead to an endless regress through larger and larger, and increasingly suspect, theories. In short, a non-circular proof of the simple consistency of MT is classically impossible. But vindication of the use of y in MT requires a non-circular proof of the simple consistency of MT. Thus use of y can never be classically vindicated in the case of any sufficiently rich mathematical theory. Accordingly, too, use of classical logic in formulating such theories cannot be justified. There is indeed far too much complacency about formalisations, especially classical formalisations, of set theory and other substantial regions of the foundations of mathematics. The fact is that they are in dreadful philosophical and logical shape. A lot of glib and supposedly reassuring tripe is trotted out about how everything is alright - like the economic experts on an obviously ailing economy. But really the crisis in logic is far from over. It is simply that, like hyperinflation in some economies, people have come to live with it. The fact that one can live with crises, can live dangerously with classical logic (as van Fraassen is quite explicitly prepared to do: 71, p.5), is hardly a good reason for continuing to do so when the risks can be reduced as they can be giving up material detachment everywhere except where consistency can be established, that is in most places. There is no need to live in the shadow of possible disaster in the fashion of classical set theorists, and it is mostly irrational to do so. Nor can there be any doubt that allowing for the possibility that a set theory is a simply inconsistent non-trivial theory substantially reduces risks.1 Naturally where a version of Godel's theorem applies, risk cannot be entirely eliminated; but firstly, Godel's result has not been established for theories which are not classically based (see UL), and, secondly, there is good prospect that for such theories a non-Godelian strategy can succeed (for example, dialectical theories automatically escape the main impact of limitative theorems, since they admit the paradoxes, whereas the limitative theorems, and the scepticism they inject, are consequences of getting burnt by the paradoxes after their main fire has allegedly been extinguished). (footnote u continued from previous page) elementary levels creates distortion. (Consider, for example, the settheoretic representation of groups and semi-groups discussed in van Fraassen 71.) Too many logicians are prepared to tolerate this level of distortion, to accept isomorphisms for identities, etc. Probability considerations support this claim. Just as, to use a geometrical analogy, it is probable that isolated singularities remain isolated and do not connect with every point, so there is a considerable probability that isolated inconsistencies do not spread everywhere.

56

1.6 THE NEED FOR REASSESSMENT OF THE PLACE OF CLASSICAL LOGIC

In view of all this, and especially the severe limitations on the reliable applicability of the core classical rule of Material Detachment, and (as we shall see) the defensibility of dialectical theories, we believe that a reassessment of the place of classical logic in theory formulation is wanted. Classical logic was the first formalised logic of much scope on the scene and it remains by far the best developed. Practically all the extant formalisations of mathematical theories have been carried out on a classical basis not because classical logic was necessarily or obviously the best medium for all - or any - of these formalisations, but in part because of historical accident and vagaries and of conservatism of the educational process and in part because it alone was there. Nowadays classical logic, which has many powerful adherents, has consolidated its position and become entrenched,1 and much mathematical theory is being rewritten in terms of it. A comparison with the rewriting of history in terms that suit a new dominant ideology is not inappropriate. For it appears that the informal reasoning used in classical mathematics at least until the mid-century rarely2 or never deployed the excessive power of, or, for that matter, the paradoxical features of, material implication, and it also appears that most of the reasoning can be reset on alternative logical bases including relevant ones. It is our view that the importance of classical logic - which captured the market because first into it and because of its childish simplicity - has been grossly exaggerated. Given time and relevant effort it is our hope that history will in due course bear out our claim on this matter. What importance classical logic does have is in providing a simple model, and testing ground, for arguments that can then be re-engineered to extend to more adequate alternative systems.3 1

Thus rival logics are derisively dismissed as 'deviant' in (much of) U.S.A. and its intellectual satellites: elsewhere they are 'non-classical'.

2

If there were many such cases, which we doubt, they could be catered for in a relevant formalisation by additional mathematical postulates - the classical postulational bases having in any case a measure of indeterminacy.

3

We have already, impetuously, proposed relevant logics as candidates for the job from which we are arguing classical logic should be dismissed (see, e.g. UL). As a matter of history, relevant logics were not introduced merely for recreational or academic advancement, or demotion, purposes, but were intended as serious alternatives to classical logic and to such improvements on classical logic as modal logics. We certainly believe that relevant logics should, because of their unusual scope and range of other virtues, be very seriously considered as alternative logics. So it pained us to encounter Haack 74 - a whole book on alternative logics in which relevant logics get no mention. Of course recognition of relevant logics would have destroyed the organisational framework and several of the arguments of Haack's book. To begin with, relevant logics do not fit into S. Haack's oversimple classification (in 74, p.4ff) of alternative logics. For although all theorems of classical (sentential) logic are theorems of the main relevant logics studied (those including system G below), these relevant logics do not in general simply extend classical logic, but lack (except as an admissible extra in many cases) the characteristic classical rule of Material Detachment. Moreover non-trivial inconsistent extensions of the best relevant logics - the dialectical systems - no longer allow material detachment as an admissible addition. Thus relevant logics - with -*• considered, like the strict implication and necessity of modal logics, as a further connective beyond the standard set {&, v, ~} of classical logic - though they typically include classical logic as admissible, are not essential extensions of that logic. Doesn't that make relevant logics quasi-deviant logics on Haack's classifi(footnote continued on next page)

1.6 THE RATIONAL LOGICAL POLICY IS TO REDUCE RELIANCE ON CLASSICAL LOGIC

To sum up the main drift thus far:- Not even the residual claim made for classical logic - that classical logic will have to be part of any adequate logical system - is correct. On the contrary the residual claim is drastically mistaken. Admittedly the tautologies of classical logic are analytically true (though, as noticed, not compulsory, in that one doesn't have to believe them). Material Detachment is however an integral part of the classical scene, and this rule is formally unsatisfactory and becomes positively undesirable when investigating the consequences of philosophically and mathematically important theories which may be inconsistent. Then one can go seriously wrong using classical logic. One can, in short, go wrong with classical logic if one applies it in the wrong situations. And if one applies it to discourse concerning a range of worlds that are not classical such as inconsistent and paradoxical situations, then one is quite certain to go wrong: and so of course for any arbitrary situation there's a chance of going wrong. Furthermore for most theories of interest the requisite conditions for the application of classical logic cannot be classically established. Standard logic is not, ther, a device that one can apply with real confidence anywhere much beyond decidable theories. Only the completely unfounded, and classically unsupportable, presumption of correctness props up the universal application of classical logic, and a rapid - but impermissible - retreat towards pure logic is often attempted by exponents in the face of criticism. Rational applications of logical theories should include logical caution, which means dropping classical logic as a universal, or even extensively used, tool especially in investigations of consistency. For we do not check situations carefully to make sure they are not bugged by contradictions somewhere before applying logical reasoning - yet that is what the safe application of classical logic would appear to require. Moreover, even if, as is classically assumed, the world is consistent it is not generally safe to apply classical logic in investigating theories (which may diverge from the world). The rational course would seem to be to reduce risks and adopt a relevant logic (a thesis developed in detail in DCL). § 7. Dialectical logic, and the repudiation of the dogma that the world is consistent. So far we have been arguing against classically-based logics primarily on the grounds that there are non-trivial inconsistent theories which just cannot be discounted, and that, moreover, a great many of our other intuitive theories may, for all we know, be of this sort (and that classical theory prevents us from ever knowing otherwise). (footnote

continued from previous page)

cation? No, for the theorems of the common vocabulary with connectives &, v, ~ do not differ. But the matter of the choice of connectives of alternative logics and the comparison made with the connectives of classical logic is important, and, unfortunately for Haack's classification, the status of a logic is not invariant under choice of its formulation. For example, Lewis's systems are classed by Haack as extended logics. But should we consider Lewis's systems of strict-implication, as formulated in a quite standard way with connective set {&, v, =>} with = as a strict implication, as an alternative to classical logic (the way Lewis and others have thought of these logics), then Lewis systems are deviant logics on the Haack classification - not extended logics as Haack has it. Similarly relevant and intuitionist logics so considered are deviant logics, and so likewise are the very different connexive logics which contain decidedly non-classical theses and which should be separately classified. Accordingly the Haack classification is neither suitably stable under systemic formulation nor sufficiently revealing, and it is neither exhaustive or appropriately exclusive.

58

1.7 PARACONSISTENT LOGICS, ANV THE DIALECTICAL CRITICISM OF CLASSICAL LOGIC

The dialectical criticism of classical logic is much harsher. For according to it, there are true theories that are inconsistent, and thus the world T (considered as everything that is case, as the class of truths) is simply inconsistent. If so, classical logic cannot be reliably applied, even in its home (extensional) territory, to true theories: it is incorrect in a quite drastic way. The new dialectical criticism of classical logic and repudiation of the consistency of T is based on the development of paraconsistent logics. The new logical case reinforces the older intuitive case, in a quite remarkable way (as we shall see in 6.5), and furnishes many of its main theses. A paraconsistent logic is, at bottom, a logic which admits arbitrary contradictions without thereby being trivialised, i.e. a necessary condition for a paraconsistent logic is that it does not contain (as a derived principle) the spread rule: A, ~A -frB.1 Let us take this necessary condition to characterise weakly paraconsistent logics, so leaving it open what additional conditions have to be met by paraconsistent logics proper2 - just as we have characterised weakly relevant logics in terms of satisfaction of Belnap's weak relevance requirement and so far left open what additional conditions must be met by relevant logics (but this question is closed in chapter 3, where relevant logics are characterised as logics which are weakly relevant and conservatively extend distributive lattice logic). Since most positive logics would qualify as paraconsistent under the account we also require that the logics concerned should be sentential and contain as well as •*• a full stock of extensional connectives, typically &, v and 1

A less sweeping requirement that merits some investigation is that under which a logic admits some contradictory pair A and ~A of further theses without being trivialised.

2

Da Costa in a number of publications, particularly 74, has imposed further conditions on paraconsistent logics, and similarly Jaskowski has required that his discussive logics - which are paraconsistent logics - should meet further conditions. But some of these conditions are not formally tractable (and are too strong insofar as they are), e.g. that a paraconsistent logic should have an intuitive sense or that is should be sufficiently rich to formalise the bulk of useful sentential reasoning (Jaskowski 69, D'Ottaviano and da Costa 70), and others are undesirable, e.g. the condition that a paraconsistent logic should not contain the thesis ~(A & ~A). Some important dialectical logics have ~(A & ~A) as a thesis though for some B, B & ~B is also a thesis, e.g. the system DL of 6.5. Though the modern logical theory of formal inconsistent systems originated in Poland with Jaskowski - its roots go back to Lukasiewicz 70 - extensive development of theory has occurred in Brasil, and is to be found in the work of da Costa, Arruda and coinvestigators. The theory of formal inconsistent systems, or as they are now called, paraconsistent systems (the term 'paraconsistent* being coined by Quesada), has become very much a Latin American institution, and distinguishes Latin American logic in much the way that the identity theory of mind used to distinguish Australian philosophy. Important work on paraconsistent systems has also been done by Asenjo, Apostel, Priest, Routley and others (see DLS).

59

1.1 THE CLASSICAL FAITH (IN CH) CONTRASTED WITH DIALECTICAL HARDHEADEDNESS

It is evident that the classes of logics distinguished, weakly relevant and weakly paraconsistent logics, properly overlap. The relevant affixing systems (of chapter 3) generally fall into the overlap, but most of the systems studied in depth by da Costa are irrelevant (the P systems are an exception), while the original rigorous implication systems of Ackermann 56 are relevant but not paraconsistent since they contain the rule y.1 The usual irrelevant logics which all contain Disjunctive Syllogism are neither relevant nor paraconsistent. A dialectical logic is a paraconsistent logic that realises the potential that a paraconsistent logic allows for, that is, a dialectical logic is simply inconsistent as well as non-trivial, i.e. it contains contradictory theses.2 Thus dialectical logics are a subclass of paraconsistent logics, and relevant dialectical logics - those of main interest to us - of relevant paraconsistent logics. Let c be a situation that conforms to a dialectical logic. The semantics of relevant logic will reveal that such non-classical situations are easily supplied. Situation c upsets an application of y, namely yc. So much we have already seen. The additional dialectical thesis is that T is such a situation, and that so also are certain subsituations, such as c, of T. Hence, by the negation of the consistency hypothesis, yT is upset. There are three positions that can be taken with respect to the consistency hypothesis, CH, that the world T is consistent, and so three positions with respect to yT - positions which correspond to the three standard positions with respect to the existence of God. There are, firstly, theists or believers who accept CH, mostly as an article of faith. Such is the classical position, such also was Ackermann's position, and, so it appears, that of Anderson and Belnap. Secondly, there is the agnostic position, argued for as the rational position in DCL. Finally, there are atheists or infidels: such are thoroughgoing dialecticians (and the same dialectical position is argued for in UL and in DLS).3 l It is assumed in characterising paraconsistent logics that the extensions considered are closed under the rules of the logic: otherwise even classical logic could be paraconsistent. 2

0ur use of the term 'dialectical' has encountered a fair measure of criticism, partly because 'dialectical' and 'dialectic' have another important (related) sense, and partly because the links of dialectical logic with the enterprises of Hegel and the dialecticians have not been made sufficiently clear. But, firstly, our use of 'dialectic' falls squarely under the second part of the dictionary listings for the terms (see OED): dialectic, n (often in pi.) Art of investigating the truth of opinions, testing of truth by discussion, logical disputation; (Mod. Philos; not in pi.) criticism dealing with metaphysical contradictions 6 their solutions. For we are certainly concerned with the application of dialectical logic in resolving metaphysical contradictions (see below): this is the central reason for investigating them. (Jaskowski's discussive logics and paraconsistent logics are intended to tie in also with dialectics in the Greek meaning, i.e. as under the first part of the dictionary listing.) Secondly, Hegel and his followers do not have a monopoly on the use of the term 'dialectic' even if (footnote 2 and 3 continued on next page)

60

1.7 WHAT IS AT STAKE IS THE QUESTION OF THE CONSISTENCY OF THE WORLD

For present purposes we do not need to settle the issues between consistency agnostics and consistency atheists, even if we could; it is enough to knock out the classical, theistic, position. What we shall do is to set out the seemingly powerful, and pretty impregnable, case that the agnostic has (adapting the argument from DCL) and in the course of presentation, indicate how it can be biased in favour of the dialectical position. We have seen that what is at stake is nothing less than the question of the consistency of the world T, that if the world is simply consistent (and also, as against the intuitionists, complete) then the classical position is correct, whereas if the world is inconsistent then the dialectical position is correct.1 Whichever is the case, however, the relevance position does not go wrong. This provides a major reason for claiming that the relevance position is more rational than the other positions, should it turn out that the matter of the consistency of the world cannot be definitively settled. But the question of the consistency of the world cannot be conclusively settled in a non-question-begging way - so at least the agnostic argues hence the keeping-options-open strategy of (non-classical non-dialectical) relevant logic is the rational one from a decision-making viewpoint. The rival positions of course contend that the matter can be settled, in their way - not perhaps definitively, but in the way that other high level theoretical hypotheses are settled. Here the dialectician has a point, as we shall see. What is at stake is not the absolute consistency or non-triviality of the world, or the logic or theory whose truth it reflects. All the positions can agree that the world is absolutely consistent - which is as well for them, since the absolute consistency of the world can be empirically verified. The statement, q , "R. Routley is cutting firewood at Plumwood Mountain on the (footnotes

2

and

3

continued from previous page)

they were responsible for the modern usage; and though it would be very interesting to apply logical and semantical methods to the explication of Hegel's philosophy, we are under no obligation to do so. We do believe, however, that the dialectical logics we have so far investigated have a direct and important application in clarifying the logic of Soviet dialectical philosophy (see DCL). 3

0ne of the fringe benefits of going dialectic is that one can no longer be condemned quite so easily for the occasional inconsistency in one's work. But of course we can make the classical plea that our position has changed - improved we should like to say - over time.

x

The dialectician can admit with the intuitionist that the world is incomplete, but the intuitionist cannot admit with the dialectician that the world is inconsistent; for intuitionistically, since A -*• B, A -+A •*•. A -*• B, whence —iA A -*• B and A & —|A -*• B, i.e. intuitionistic inconsistency implies collapse into triviality. Of course a minimalist who rightly rejects A -»• B can go some way with the dialectician. In general, minimalism is a relevantly rather more congenial position than intuitionism: if only however "minimal" logic had been based on a relevant positive logic instead of on Hilbert's positive logic (the so-called "absolute calculus").

61

1.7 THE CHARACTER OF THE CONSISTENCY HYPOTHESIS

afternoon of August 1, 1976", for example, can be empirically falsified by observing Routley in Brasil at that date. Hence q Q is false and q^ does not belong T. Therefore, T is absolutely consistent. It is often supposed that given the absolute consistency of the world, the (simple) consistency of the world is also proved, and the classical position thereby established. However the "proof" of simple consistency must depend on the special rule n.

A, ~A -o B

which transforms simple inconsistency into absolute. But this paraconsistencyexcluding rule is tantamount to rule y, even in weak sublogics of classical logic (e.g. in DML of 2.8), and is semantically equivalent to the assumption that T is simply consistent (given absolute consistency or non-degeneracy of the world). Hence the question is begged by rule n> since precisely what is at issue is rule y and the consistency of T. The question of the consistency of the world is not, it seems, empirically decidable. Nor i~ it a high level scientific hypothesis. It is a metaphysical thesis, but a perfectly significant one. Since it involves a universal claim to the effect, as canonical semantical modellings will show, that for every statement A exactly one of A and ~A belongs to T, i.e. holds true - it would appear that the dialectician is in a much stronger position to establish his claim than the classicist, since universal claims can be falsified in principle by a single counterexample. To establish his thesis however the classicist has to establish a claim that extends over all true theories, whether scientific, mathematical, evaluative, or whatever. Should a contradiction lurk deep in some still unpenetrated mathematical or other theory then the classicist is undone. Only a priori "evidence" that such a contradiction or inconsistency cannot occur would seem to exclude the possibility: thus the classical position if true is not empirically so in any simple way, no endless search is contemplated, nor would its results be relied upon. The dialectician, on the other hand, has the chance of falsifying the consistency hypothesis by a solitary counterexample. But, as any dialectician soon finds out, a simple falsification of the consistency hypothesis which would satisfy the opposition is not so easily achieved, as each alleged contradiction can be avoided, though often none too convincingly when the wider theoretical framework is remembered, by theoretical shifts (coupled with conventionalist strategems). In this way too it quickly becomes apparent that the consistency hypothesis is a very highly theoretical claim. Characteristically dialecticians have appealed to paradoxes to establish the inconsistency thesis. Hegel was, it seems, convinced both by the Kantian antinomies and by Zeno's paradoxes of motion; Soviet philosophers are more impressed by Zeno's paradoxes and invest heavily in the contradictions they observe in motion (which generalise to development). None of these cases are however decisive, and even if the classicists do not have really convincing solutions, say to some of Zeno's paradoxes, they do have alternative classical resolutions which so far get by. Much more convincing examples may be drawn from the logical and semantical paradoxes (this is the basis of the argument in UL against CH), but dialecticians have not generally made much appeal to such examples. Our case however is based, in the first place, squarely upon the logical and semantical antinomies and their like (self-referential arguments, including those of the limitative theorems). The arguments for the Kantian antinomies are conspicuously fallacious, and Zeno's arguments generally fail to convince. The antinomies we depend on have a different character: the arguments are convincing and are not conspicuously fallacious, if falla-

62

7.7 THE ARGUMENT TO INCONSISTENCY FROM PARADOXES

cious at all. The treatment of these antinomies within the framework of classical logic leaves an enormous amount to be desired, philosophically, linguistically, and mathematically. The artificial hierarchies of languages or types that classical semanticists appeared forced into, to avoid the catastrophic effect of semantical antinomies in combination with classical logic, are frankly unbelievable.1 One basic problem is that classical logic has to exclude deductive reasoning within and surrounding the antinomies, though it undoubtedly occurs (cf. the Prior-Mackie exchange, especially the initial paper, Prior 61). What has seemed especially puzzling is that paradoxical situations seem to be perfectly possible, in the sense that they could quite easily occur, and perhaps sometimes do occur. And this would make T inconsistent - which is one reason why attempts to dispose of the antinomies have exercised philosophers and logicians so extensively. Consider, for example, the policeman-prisoner situation, where the prisoner states only that everything the policeman says is true while the policeman, whose statements are otherwise unquestionably true asserts that whatever the prisoner asserts is false. Nothing stops such a situation from occurring indeed it could be arranged in a real-life courtroom situation.2 But no catastrophical breakdown would occur: legal reasoning could go on as before. And in fact such a paradoxical situation could go unnoticed. Paradoxes just do not spread and affect everything else: paradoxical situations do not have the logical features that classical analyses ascribe to them. The case for dialectical logic and the inconsistency of the world is not confined to that based upon "paradoxes" of one sort or another, of logic and semantics, of mathematics and physics. There are inconsistent principles in many legal codes which are valid (even if they are not said to hold true), and within the shadows of which legal reasoning takes place. To put it bluntly and without the detailed argument really called for: the prevailing law is sometimes inconsistent, yet deductive reasoning, incorporating the principles of the law as postulates, continues both within and outside the courts without being trivialised. Legal logic would accordingly appear to be dialectical logic. The argument we have outlined (more detail appears in UL and DLS) does not pretend to be entirely conclusive against the opposition, though the case seems to us very damaging. The classicist will insist of any such theory which contains a contradiction that it is not true, that it is (if perchance absolutely consistent) a non-standard theory distinct from the actual one which must be consistent. Thus counterexamples lead to a stalemate: the consistency hypothesis is not straightforwardly falsifiable or, the relevance position tries to insist, falsifiable at all. A useful example of the theory-saving methods of the classicist in the face of apparent inconsistency in nature is provided by Rescher 73, who, confronted by the incompatible actual states assumed by the Everett-Wheeler 1

The manifold deficiencies of the hierarchial approaches are well enough known: they are brought together, e.g. in Routley 66. The hierarchical approaches are not of course completely unavoidable classically; for every consistent, classically-based set theory will, in principle, yield an analogous protothetic; but these will typically be even less satisfactory than the set theory from which they derive.

2

How it is described is another matter, of course.

63

1.7 REASONS FOR REJECTING CLASSICAL THEORY-SAVING STRATAGEMS

theory of wave packet reduction in quantum physics, casts around for a difference of respect between the incompatible states in order to save the law of non-contradiction, and "locates", i.e. postulates, such a suitable distinguishing respect in a further time dimension.1 Rescher's difference-of-respect procedure (a method that goes back to the Socratic dialogues) thus offers further confirmation for the popular claim that a logical or mathematical theory2 can always be saved - at varying costs - by making sufficient changes or complications in scientific theories, since logical principles rarely confront empirical data in isolation and generally only do so rather indirectly in combination with other theoretical assumptions (as pragmatists have long pointed out). But though a non-empirical principle, such as the consistency hypothesis, never directly encounters the hard empirical data and can always be saved in one way or another, with greater or less cost, by changes elsewhere, the costs may be too high, and it may be better to give up the principle. A convincingly microphysical theory based on dialectical logic might provide such a reason. For success with theories based on classical logic has always been at the finite macro-level (as Br^uwer would have said): there is, in principle at any rate, no reason why classical logic should not go the way of Euclidean geometry satisfactory locally in regions of consistency, but globally defective. It should now be obvious that such a change would be perfectly possible; and persuasive evidence supporting the call for a conceptual revolution in logic, which queries the consistency hypothesis, is fast accumulating. If the consistency hypothesis is not straightforwardly falsifiable, it is far less readily verifiable. There is no easy or obvious way of surveying or sampling the class of true theories, particularly those yet or never to be discovered. Moreover by virtue of a by-product of the logical paradoxes, the situation appears to look still blacker for the classical position. Consistency is a logical matter, and if it can be established it will presumably be by logical means. Suppose, with a view to proving consistency, that the theory of T is formalised as far as it can be classically, and suppose that the truths of T included provide the principles of Peano arithmetic: then, according to Godel, the consistency of T is not incontrovertibly provable classically except by means at least as demanding as the resources of T (nor can they exceed the resources of T without appealing to falsehoods?). As we have observed (in §6), the same problem arises each time a reformalisation is attempted 1

To what extent the Everett-Wheeler theory can be satisfactorily constructed on the basis of a dialectical logic is an interesting question that takes us beyond the scope of the present venture. The same goes for questions as to whether such discredited theories as the classical theory of infinitesimals can be "satisfactorily" reformulated in the framework of dialectical logic, and the connected question of the extent to which the admission of inconsistencies at the infinitesimal level would help to resolve Zeno's paradoxes. For each of these theories, each of which leads to at least isolated inconsistencies at micro-levels, would demand a substantial amount of technical development before a worthwhile discussion could get off the ground.

2

An analogous, equally unsatisfactory, method in legal theory is to insist that there is an implicit priority ranking on laws, which, should a contradiction become manifest, the courts will explicitly determine. But what such post-hoc determinations really indicate is that before the consistencizing step, legal reasoning had been operating with inconsistent premisses.

64

1.7 THE MODAL STATUS OF THE CONSISTENCY HYPOTHESIS

aimed at incorporating outstanding truths syntactically and thus approximating syntactically to T. In short, the classicist can never conclusively establish his position; for, any formal proof of his hypothesis will use means at least as powerful and accordingly as questionable as his claim. Although we can be sure that the world is locally consistent for some regions we work in, like locally Euclidean (more exactly, there are consistent subtheories, like pure quantification theory, that we commonly use),1 globally we can have no such confidence, thanks to the fine dose of scepticism Godel's classical results inject (cf. 1.6). Therefore the question of the consistency of the world cannot be conclusively resolved classically in favour of the classical position, which was one of the theses to be argued. This in turn suggests the uncharitable proposition that belief in the consistency of the world is a mere act of faith, a part, so to speak, of the classical religion. While the belief may be an act pf faith for many people, it doesn't have to be quite so obviously one; for, the consistency hypothesis may be represented as part of a correct scientific theory, and overarching principles of sufficiently comprehensive scientific theories are in somewhat the same kind of epistemological plight as the consistency hypothesis. There can be good evidence for such theories even when they cannot be conclusively verified. The uncharitable proposition then rests on a false dichotomy between conclusive resolution or else a mere act of faith. Even so the comparison of CH with an overarching scientific principle does not, as already observed, stand up to too much examination: rather the hypothesis functions classically as an unfalsifiable metaphysical hypothesis. This takes us straight into one remaining puzzle, namely the modal status of the consistency hypothesis: is it a contingent statement, or a purely logical one, or something else again? For example, what seems to us more likely, synthetic non-empirical? It certainly does not appear to be empirical in the way that the deeper assumptions of physics such as invariance principles are. The alternative, on the simplistic classification that positivism so obligingly imposes, seems even less satisfactory, namely that the hypothesis is logical, and so if true analytic and if false logically false. The alternative does however have the virtue of grouping the consistency hypothesis with a range of other principles which raise similar epistemological problems, in particular the axiom of choice and axioms of infinity. The consensus that seems to have been reached regarding these principles is that we reason as far as possible without them, note carefully where they are used, try to establish whether or not they are essential where they are used, mark out those cases where they are essential, and so on. And the consensus view reflects a rational course of action, namely to minimize questionable logical assumptions. The same method would incline one to the relevance position to eschew the consistency hypothesis and to work, where consistency cannot be established, i.e. in most classical places, with a relevant logic or, at worst (as in 3.3), with some restriction of CH as a further explicit assumption - since this seems the rational course of action. But there are further factors that are not to be forgotten, in particular that, in contrast to the axiom of choice, there is a substantial dialectical case against CH. In fact 1

The official Soviet view is not dissimiliar; e.g., classical logic does apply in the case of simple, relatively stable objects and relations: see the discussion of Kedrov's position in Kline 53 p.84, and compare and contrast the intuitionist position on the areas where classical logic is correct and may be safely applied.

65

7.7 COROLLARIES

OF REJECTION

OF THE CLASSICAL

FAITH

the comparison of CH with the axiom of choice is in some ways misleading, and a better comparison is that (emerging from Rescher 73) with the principle of the uniformity of nature or that, hinted at above, with the assumption of the existence of God. Admittedly there are differences between these cases but the similarities are striking. In each case the assumption underpins the application of an extensive theory, classical logic, theories of induction and religions respectively; in each case the assumption is not, in its standard senses at least, an analytic one1, and would fail in its intended role if it were; in each case, however, it is extremely doubtful that the assumption is an empirical one - the assumptions being impervious to falsifying empirical evidence - yet in each case these are considerations, metaphysical arguments for instance, which incline many people to conclude that they are false; and yet in each case the assumptions can be maintained, despite all counter-considerations, and they can be bolstered or supported by a variety of expedients, by faith, by metaphysical arguments (invariably of a questionbegging cast), by the purchase of Kantian glasses (the wearing of which makes the assumptions part of one's perceptual scheme), or by other familiar political means, e.g. exhortation, persuasion, propaganda, repression, force, and so on. But we don't have the classical faith, we are not persuaded by the questionbegging classical arguments for CH, we don't wear Kantian glasses and, in any case consider the whole Kantian business fraudulent, especially as false assumptions can be enforced in this fashion, and we hope we can avoid classical thuggery, for lesser political means are not going to move us. The viability of dialectical logic has more corollaries than just the unseating of the consistency dogma, and curbing of classical logical imperialism. Several other classical projects and accounts are also upset. Firstly, for example, the classical idea, deriving from Hilbert, of absolute, i.e. nonrelative, simple consistency proofs by way of a modelling in the world has to be adjusted. Non-triviality may be so established, but to establish negation consistency it has also to be shown that a consistent subsituation of the world is selected. (Alternatively such proofs could be reconstrued as relative consistency proofs, showing simple consistency relative to T, for what that is worth.) Secondly, the familiar account of rationality which makes simple consistency a necessary condition of rationality in beliefs, theories, or actions has to be discarded. There is nothing in principle to prevent a dialectician's beliefs from being perfectly rational and reasoned, and a dialectical theory may be completely logical. The two issues, of rational inconsistent theories and rationally-held inconsistent beliefs are of course intricately connected: 2 for (so Routley 75 argues) an animal's beliefs constitute its theory as to how things are, i.e. as to part of T. The "framework of rational inquiry" thus extends beyond the consistent to dialectical theories and practices. Consistency imposes no Kantian (or Strawsonian) bound on rational inquiries. The same holds for intelligibility: intelligibility likewise is not bounded by consistency (see further 5.6). The rational intelligent person is not one who adheres to the consistency hypothesis but one who allows for the possibility at least that the world, though hopefully rational and intelligible 1

Forcing CH to be analytic takes us around the meaning-of-negation circuit that we shall go around in 2.8 and 6.5. Briefly, however, natural language negation is not so restricted in sense as to render CH analytic, and does not conform to classical exclusion principles.

66

1.7 THE PARACONSISTENT REQUIREMENT ON AN AVEQUATE THEORY Of ENTAILMENT

but possibly neither, is inconsistent, and who realises that inconsistency implies neither irrationality nor unintelligibility nor total disorganisation. To cope with the wide range of dialectical reasoning, both everyday and technical, an adequate theory of entailment, and likewise an adequate theory of conditionality, should be paraconsistent. Let us call this the paraconsistency requirement. It is a corollary that adequate theories cannot be simply extensions of classical logic, e.g. modal logics; and it will emerge that the requirement wipes out various apparent alternatives to relevant logics, e.g. certain conceptivist and non-transitivist positions. Since classical and modal logics are rejected as inadequate, the question is: what are the new logics of entailment and conditionality to be like?

67

2.0 TAKING THE ROAV TO SIMPLE DEDUCIBILITY ANV TO FIRST DEGREE ENTAILMENT

CHAPTER 2. DERIVABILITY> DEDUCIBILITY, AND TEE CORE OF ENTAILMENT. In this chapter we begin logically at the beginning, with the notion of deducibility in the absence of any sentential connectives, and then gradually add on connectives and complicate the class of formulae admitted until we arrive at the theory of simple deducibility (tautological entailment) and (early in chapter 3) at the first degree theory of entailment. We hope that in this way the reader will be led, ineluctably, to the first degree theory of entailment, the common core of all the good entailment logics studied in later chapters. The first degree is remarkably stable; it is the same for a surprising span of systems, and it leaves, so we shall argue, but little room for choice of rival candidates. That it to say, while there is, as we shall see in detail in chapter 3, a great deal- of room for debate as to the specific postulates of the higher degree of entailment, there is no such room for debate, in our opinion, at the first degree. We will nonetheless take time off to display some of the rival lower degree systems, to present semantics for them and to aid their case or, occasionally, to pour scorn on them. The chapter puts together in a self-contained and very elementary way results about deducibility and entailment many of which are known to the cognoscenti but which are not readily accessible or assembled elsewhere. §2. Semantical analysis of the deducibility relation j-f. In the absence of connectives there is a remarkably simple account of deducibility that wins almost universal acceptance among competing theories of implication: it is in fact precisely the classical theory of deducibility, which goes wrong only in its analysis of connectives and in its misguided attempt to reduce logical truth to deducibility (from null premisses). In order to facilitate comparison with the classical theory of deducibility, and for simplicity, we concentrate upon the case where the deducibility relation [•} is singular on the right. That is we, we consider primarily the form A 14^®, where A is a non-null sequence of wff of the logic L and B is a wff of L.

ff^ is

the deducibility relation with respect to logic L. For the most part the background logic L will be taken for granted, and ' |4j/ written simply ' [4 '. 'A (4 B' reads 'B is deducible from sequence (subsequently, set of wff) A' or 'A entails B'. Where A is a finite sequence, (A^,...,An) say, A f| B is equated with A^,...,An |4 B, and this sequence is read 'A^ and ... and A q entail Bf (cf. Smiley 59, p.235). So long as A is non-null the account of relevant deducibility - in the absence of specific sentential connectives - need not diverge at all from the classical (modal) account (see LC). That is, A |4 B is true in model M = 1 with K a set of worlds or situations and I a two-valued interpretation on K, iff whenever I(A,a) = 1 for every wff A in A then I(B,a) = 1 for every a in K. Then A |4 B iff A (4 B is true in every model. Thus the relevant account is exactly the classical account. 1

Allowance is made for additional items to go in M.

69

2.1 LOGICAL TRUTH DISTINGUISHED, AND THE PROPERTIES OF VEVUCIBIL1TY

It is evident that [4 can easily be extended to handle multiple expressions on the right - and we shall take advantage on occasions of the symmetrical form A |4 T. Then A ff T is true in M iff whenever I(A,a) = 1 for every wff A in A then I(B,a) = 1 for some wff B in F, for every a e K. Furthermore the definition is easily recast in terms of satisfaction upon using the following interconnection: an assignment of values at worlds satisfies A at world a iff it assigns value 1 to A at a. Then, as classically, A^,..., A^ |f B iff every assignment of values at each world a which satisfies each of A,,...,A at a also satisfies B at a. 1 n Where A is however null, the account begins to diverge from the classical account. Classically |4 B, i.e. B is a logical truth or logically valid, is equated with A f f B, i.e. B is true at every world in every model. But this account leads either to paradox or error according as the class of worlds considered is restricted to the logically regular ones (i.e. to "possible worlds" in one sense) or not. If not logical truths may fail to get verified, whence error ensues. A superior approach is to equate ff B with t ff B, where t is the truth constant, already mentioned, which satisfies the following interpretation rule: I(t,a) = 1 iff a is regular. And this approach, which will be exploited later, then reveals that t is avoidable. ff B is true in M - with M now of the form , where the set of regular worlds 0 is a subset of K - iff I(B,a) = 1 for every a e 0. A separate definition, in place of the classical reduction of ff B to A [4 B, is enough, then, to avoid paradoxes of deducibility such as that anything entails a logical truth.1 Finally, f f B iff f | B is true in every model. Several properties of the relevant deducibility relation are now derivable from the semantical account quite independently of whether any particular connectives occur or not (Smiley's list of properties, 59, p.236, is varied): Order properties: Reflexivity:

A |-f A

Transitivity:

If A |4 B and B, T ff C then A , T f4 C.

Substitutivity:

If A f f B and B f-f A then when A, A |-f C also B, A |-f C.

The property of substitutivity is derivable from transitivity. The order properties of |-f are really those of a partial order, since when both A |4 B and B |4 A, A and B are deductively identical. Hence too the more explicit formalisation of deducibility in algebra using the partial order < in place of |4But the cost of the explicitness is that a theory of identity is thereby presupposed. Set properties: Permutation:

If A , A, B, T |4 C then A, B, A, T ff C.

Repetition Omission:

If A, A, A f f B then A, A |4 B

Repetition Introduction: If A, A ff B then A, A, A f f B

1

And the simple avoidance of these paradoxes is of course highly desirable. The whole issue is taken up again in chapter 12 with the further discussion of the consequence relation - a basic relation for the algebras of logics.

7Q

2.1 AUGMENTATION v-6 SUPPRESSION;

ANV THE REPUDIATION OF SUPPRESSION

Premiss addition and deletion properties: Augmentation: Modus Ponens:

Iff A |j- B then C, A |j- B.1 If If- B and B ||- C then |f- C.

The latter property is the relevant cut-down of the usual classical property of suppression of logical truths. Thus although Augmentation or the addition of irrelevant premisses is validated, its classical mate, the suppression of logically true premisses, is not. That is, the following ridiculous classical property gets repudiated:(Positive) Suppression:

If |f- B and B, A |J- C then A |}- C.

For even if, in a given model, B holds in all regular worlds, still in other worlds C may only be deducible from A given B, i.e. B is essential even though logically true. In multiple formulation Positive Suppression, expressed more generally as: If if- B and B, A If- V then A if- T, is matched by its dual Negative Suppression: If C If- and A |f- T, C then A \\- T, where C |f- if C is a logical falsehood, i.e. C fails to hold at every regular world (i.e. classically ~C is logically true, i.e. holds at every regular world). The rejection of Suppression marks out clearly which of the two evident main classes of routes has been taken in avoiding a classical deducibility paradox, with an arbitrary B entailing a logical truth A, derived as follows (cf. Smiley again 59, p.246):A |J- A A, B If- A B ||- A

Identity Augmentation (and Permutation) Suppression

The mechanism of the negative paradox is exactly parallel but requires multiple formulation for its expression: D |f- D D ||- B, D D If- B

Identity Augmentation Negative Suppression,

where D is a logical falsehood. The other main option to the rejection of Suppression, within the framework of the deducibility theory offered, is the connexivist position (criticised in §4 and discussed again in §7), that of abandoning Augmentation. However the route of rejecting Suppression is far from uniquely determined: it does not, for example, distinguish a relevant logic approach from a Parry analytic containment approach. The differentiation of these approaches will turn on the semantical rules given for specific connectives, especially &, v and There remains a variant way of blocking the deducibility paradoxes which lies outside the framework of the classical theory since it involves changing the semantics for If- , the method of rejecting not merely Suppression but more generally Transitivity (of which Suppression can be seen as a degenerate case). This is one of the important preliminaries to investigate before semantical rules for specific connectives are introduced. The other, to which we turn first, is the matter of the axiomatic characterisation of the semantic relation of deducibility. We consider these in turn:*In the computing science literature this principle is called Monotonicity. While deducibility is certainly monotonic, there are inferential relations (e.g. those of inductive, analogic and probable reasoning) which are not, where inferences may break down when further premisses are added (e.g. because initial premisses are undermined). Such is the proper basis for nonmonotonic logics, which include connexive and various conditional logics.

71

2.2 AXIOMATISATION OF THE DEDUCIBILITY RELATION

§2. The axiomatisation of the relation \\- by a derivability relation [- , and the objection from Non-transitivism. Let L be a formal logic with denumerably many wff A, B, C Let A, T etc. be non-null sets of wff of L: in short the set properties of the intended relation will be built into the initial formulation. But a Gentzen-style formulation in which set properties required are explicitly listed could easily be designed. Each set A is countable. The sole formation rule of logic ML based on L and with the vocabulary cited, together with the symbol is: Where A is a wff and A a set of wff of L then I^A and A f^A are wff of ML. Much as before the subscript L is usually omitted, and where A = {B^,....B^}, A (- A is written B1,...,Bn f- A. A is derivable from A. Axiom scheme: Rules:

'A

A' is read:

The postulates of ML are as follows:

A {- A

From A (- B and B, T |- C derive A, T (- C (Transitivity, Cut). From A |- B and A £ T derive T |- B (Augmentation, Weakening). From

B and B (- C derive

C

(Modus Fonens).

Theoremhood of ML is defined in the usual sequential way. It is perhaps worth observing that f- A is a theorem of ML amounts to l-^. (fr A) and that ML L A B is a theorem of ML amounts to f ^ (A which corresponds to

More important, we do not need to see the formal structure in the object language-metalanguage way the system notation was chosen to suggest. Alternatively one might contend that one really has just one system ML with further unspecified formal objects (called obs) A, B, C, etc. and sets of these. Then the inner logic L drops out of the picture. (This type of approach, due to Curry 63, is adopted in the formalisation of distributive lattice logic below). Lemma 2.1 D € V iff T (- D. Proof. If D « r , then as Df-D, T, Df-D by augmentation, and so T |- D. For the converse it is shown by induction on proofs that if A [- D then D e A. Where the theorem is an axiom A = {D} so D e A. Moreover the rules that apply (Modus Ponens does not) preserve the property. That Augmentation does is immediate. For Cut, if A, T |- D then by one premiss D e T or D = B whence by the other premiss if D = B D e A. Hence D e A or D e T as required. Corollary ML is decidable. Proof. To decide whether T (- D is a theorem or not, enumerate T and begin checking through the sequences to determine in which D occurs. finitely many steps D will appear in one or other of T and T. Then r a theorem or not according as D occurs in F or in T.

_ T, and After [• D is

An ML model M, to be more explicit, is a structure & = where K is a set of worlds, 0 is a subset of K, the regular worlds, and I is a bivalent interpretation function which assigns to each wff A at each world a in K just one of {1,0}, i.e. I(A,a) = 1 or = 0. A |- A is true in M iff whenever I(B,a) = 1 for B e A, I(A,a) = 1, for every a e K; and |-A iff I(A,a) = 1 for every a e 0. Finally, A f- A is valid iff A |- A is true in every model, and |- A is valid iff |- A is true in every model. Moreover, to connect with the semantical notions, A ff A iff A (- A is valid and ||A iff |-A is valid.

72

2.2 THE FIRST ADEQUACY THEOREM - FOR PURE VERJVABTLJTY AS IN ML

Adequacy Theorem 2.1. A |- A Is a theorem of ML Iff A |-f A, and |-A is a theorem iff |-f A. Proof. Soundness is established by induction over the length of proofs in ML. A f-A, the only axiom, is plainly valid, since if I(A,a) = 1 then I(A,a) 38 1 always; and direct verification shows that the rules preserve validity. Completeness is proved by the routine method of providing a canonical model. Let a theory c be any class of wff closed under (- , i.e. if A c c and A |- B then B e e . A theory c is normal iff whenever |-B, B e c. Define a canonical model M = as follows:~c c c K is the class of all theories; 0 is the class of all normal theories: and c ' c I(B,c) = 1 iff B e e for each wff B and each theory c. Suppose A is not derivable from A, i.e. A J-A. In order to show A is not deducible from A, i.e. ~(A |4 A), a simple Lindenbaum construction is first used. Let Aq,A^,...,An,... be an enumeration of the wff of L. Define a sequence of sets Aq, A^,...An,... inductively as follows:-

A^ ® A; if A |~Aq then

=

A u {A }, and otherwise A .. = A . Let a = u A . Then n n n+1 n n <w n (i) if D e a then A f- D, for each wff D. This is almost immediate from the construction of a;

it may be established by induction.

by the axiom scheme and augmentation. for D to belong to

If D e A^, then A f- D

Otherwise D = A^, for some i and in order

A |- D.

(ii) If a |- D then A |- D. (i) A |- D.

For if a |- D then by a lemma D e a, and so by

(iii): a is a theory. Let T be some subset of a and D some wff such that T |- D. D will be some element of the enumeration, say A^. It has to be shown that A

for then A^ e a as required.

Since T £ a and T |- A^, by augmentation,

a (- A^, for then A^ e a as required. Hence by (ii) A |- A.. Since A

A, by (i) A f a though A c a with a e K . As I (A,a) + 1 A |- A

is not true in M ;

~c

and so ~ ( A

14 A).

''

Suppose, for the other case, |-B is not a theorem. It suffices to find a normal theory a to which B does not belong. For then I(B,a) ^ 1, and so ~ |-| B. Let c be the class of all wff B such that |- B is a theorem. (in the degenerate case we are concerned with here c is null). Then c is a (degenerate) normal theory to which B does not belong. Corollary. ML is completely axiomatised by the sole axiom scheme and the rule Augmentation. For of the rules, only Augmentation is used in the completeness proof. In fact, in the narrow framework of ML, Modus Ponens is vacuous, and Cut is a derived rule, obtained thus:Given A |- B and B, T |- C, then by the first premiss B e A , i.e. {b} c_ A and so {B} u T c A u T , whence by Augmentation and the second premiss A, T |- C. Though there is very extensive agreement among the implicational camps as to the correctness - if classical incompleteness - of such a core theory of deducibility as ML captures, the agreement is not universal. For Augmentation is found objectionable by connexivists (for reasons we will outline first of all when conjunction Is introduced, since connexivist objections to Augmentation are put in terms of the conjunction of drone antecedents); and

73

2.2 THE ATTEMPT TO FAIL TRANSITIVITY OF ENTAILMENT

Transitivity is rejected under the von Wright-Geach method of diverting the paradoxes of implication. According to von Wright (57, p.181) A entails B, if and only if, by means of logic, it is possible to come to know the truth of A => B without coming to know the falsehood of A or the truth of B. The account1 was taken by its exponents to rule out transitivity of entailment; for example,to admit the premisses A & B -»• A, A & (~A v B) •*• B, etc., of the classic Lewis paradox argument but to reject the conclusion A & ~A -»• B. But critics were to point out both the many alternative construals of coming to know, in particular how one might come to know A & ~A B by the paradox argument using transitivity of implication, and also that both informal and technical accounts by von Wright and Geach were defective and failed to block not merely the Lewis paradox arguments but other equally damaging paradoxes (see ABE, p.215ff., Strawson 58, Bennett 59 and especially Lewy 76, pp.141-8). There are, then, formal explications of the von Wright-Geach account (none of them bearing a very good resemblance to the original, since epistemological features drop out) some of which defectively bring out transitivity and others of which fail transitivity.2 The account does not settle the issue of transitivity, then, and in a good sense the question of the correctness of transitivity is something to be settled prior to such an account - as condition of adequacy on a notion of entailment, as distinct from other notions such as obvious entailment, immediate inference, etc., 1

The account has antecedents in Russell, Johnson and Broad (as Lewy 76 observes in the two latter cases). What Johnson has to say on the paradoxes of implication is especially relevant to our subsequent criticism of Lewy, that Lewy has illegitimately converted what is satisfactory as ? necessary condition into a sufficient condition. According to Johnson (21, p.47): I believe that they [the paradoxes of implication] can all be resolved by the consideration that we cannot without qualification apply a composite and (in particular) an implicative proposition to the further process of inference. Such application is possible only when the composite has been reached irrespectively of an assertion of the truth or falsity of its components. In other words, it is a necessary condition for further inference that the components should really have been entertained hypothetically when asserting that composite.

Although Johnson is right about the necessary condition, the condition does not on its own provide, or effectively delineate, a resolution of the paradoxes: for example, it leaves the choice of ways of defeating the negative Lewis paradox almost entirely open. 2

There are serious formal difficulties in the main accounts so far presented. For example, that of Geach 74, an adaption of Smiley 59, depends on the decidability of the underlying classical logic on which the account is parasitic, and so will fail to extend from sentential logic even to quantificational logic. For other defects in Geach's theory see Dale 73 and Kielkopf 75a (the latter shows that the theory violates the paraconsistency requirement), Kielkopf 77, Briskman 75 (which shows that Geach is committed to Lewis-style paradoxes), and Iseminger 78.

74

1.1 THE OVERWHELMING CASE FOR TRANSITIVITY OF ENTAILMENT

It is evident that there are non-transitive restrictions of entailment, such as obvious entailment, and also that given such non-transitive notions a transitive notion can be obtained by taking the transizive closure. That is, let B- be some immediate deducibility-type relation which however is not transitive; then a transitive deducibility relation ff can be formed as the ancestral of B- . But it is this closure relation, not the immediate relation, which is what is ordinarily understood by deducibility, and it is the closure relation that we are interested in explicating. Moreover without the ordinary closure relation ('ultimate deducibility1 as Parry 76 calls it), much of the power of deducibility, and of the point of logic in applications, is lost. As Smiley (59, p.242) put it, in an emphatic statement: .... the whole point of logic as an instrument, and the way in which it brings us new knowledge, lies in the contrast between the transitivity of 'entails' and the non-transitivity of 'obviously entails' and all this is lost if transitivity cannot be relied on. To put the central point in terms of 'proof' or 'valid argument': a very conspicuous feature of proof is that a proof may extend over a substantial sequence of steps. Proof is characteristically carried by an implication or analogous (derivability) relation. Without the transitivity of this relation central, and immensely important, features of proof and deductive argument would be lost. The overwhelming case for the transitivity of entailment derives, then, both from the sense of 'entails' and from the properties of the cluster of notions analytically tied to entailment. By entailment is meant the converse of deducibility: since deducibility is transitive, so is entailment (for when R is transitive so is RC). One argument for the transitivity of deducibility is as follows: B is deducible from A iff there is, in virtue of the meaning of deducibility, a (valid) deductive argument from A to B, i.e. a sequence of elements beginning with A and ending with B each element of which is validly obtained from predecessors; in short there is a valid proof sequence beginning with A and ending with B. But such a notion of deductive argument or proof is automatically transitive: for if there is an appropriate sequence beginning with A and ending with B and another beginning with B and ending with C then there is of course one beginning with A and ending with C. The analysis of deducibility in terms of logical consequence yields the same result; then B is deducible from A iff B logically follows from A, i.e. iff, syntactically, there is a valid derivation of B from A, and iff, semantically, B holds in every deductive situation in which A holds. Both syntactical and semantical analyses ensure transitivity (as has been shown by way of ML). The same applies to every analysis of entailment based on inclusion; e.g. A entails B iff the logical content of B is included in that of A, i.e. c(B) _£ c(A). But inclusion is transitive; hence entailment is also. Given that logically necessary conditionals are entailments, the arguments for transitivity vindicate Conjunctive Syllogism, (A B) & (B C) -»•. A -*• C: they do not vindicate, however, the exported forms of this principle which feature prominently in the system E of "entailment". Arguments along similar lines, that is arguments from the sense and analytic connections of entailment, may be likewise used to vindicate other entailment principles, e.g. Identity and Modus Ponens. Analyses of entailment through inclusion, for instance, at once yield Identity.1 This is not to say that there are not relations closely related to entailfootnote continued on next page)

75

2.2 "THE CAMBRIDGE CASE AGAINST UNRESTRICTED TRANSITIVITY "

Marshalling these points in favour of transitivity serves to emphasize that no explication of entailment that rejects transitivity - in the form: if A B and B ->• C then A C, as a matter of logical necessity - can be adequate to capture the preanalytic notion. Transitivity is, in short, a requirement of adequacy on analyses of entailment. There is no sense of entailment - as distinct from restrictions of entailment, and as distinct from conditionality - in which entailment is not transitive, and accordingly there is no resolution of paradoxes of entailment by replacement of entailment by some non-transitive relation. As it has been supposed that there is, especially by recent Cambridge logicians, it is perhaps worth glancing at the source of this illusion: ... it is this. We want our knowledge that A entails B not to rely on any antecedent knowledge of the propositions involved. The relation between A and B we think we have in mind when we say that A entails B must not be based on the modal value of A or that of B. ... But what the paradoxes show is that entailment which tries to capture this idea, if it is really to work, will lack unrestricted transitivity (Lewy 76, pp. 124-5; cf. also p.128 bottom). The premisses have been granted (in the fallacy of modal dependence, 1.5); but Lewy's conclusion should be resisted: The paradoxes do not show any such thing (unless 'really to work' is seriously overworked). Every analysis of entailment which is paradox-free in a sense avoids the modal dependence fallacy; in particular relevant analyses do. So transitivity need not be sacrificed. There are two reasons why Lewy supposes that such an entailment relation 'must involve a restriction on transitivity'. The first is that all the other moves in the proofs we have discussed, or at least all the other moves in Lewis's first proof, are entirely valid (76, p.130). This is a large claim, which happens to be false (see especially 2.9). The second reason is that Lewy is thinking of proposals like those of von Wright, Geach and Smiley which do not merely offer avoidance of modal dependence as a necessary condition for entailment, but present it as a necessary and sufficient condition. Thus Lewy goes on at once to say (p.125) that Smiley's criterion attempts - unsuccessfully it soon turns out (p.130) - to formalise 'the basic idea' he is talking about. Smiley's proposal (in 59, to which Smiley is not committed) does restrict transitivity. It is as follows, where represents entailment in the sense of Smiley: A^,...An H" g® i^f A^ & ... & A^ z> B is a substitution instance of a classical tautology A^' & ... & A r ' => B' such that neither B' nor ~(A ' & ... & A ') is a classical tautology. (A definition 1 n which goes beyond the confines of sentential logic results upon replacing (footnote

1

continued from previous page)

ment which fail Identity. Non-reflexive (or non-stuttering) entailment, defined through entailment and non-equivalence (as A + B & ~(A B)), and corresponding to proper inclusion, fails Identity. It has turned out that there is a real point in studying implicational-type relations which fail Identity (always), namely in solving technical problems such as the P - W problem (see Martin and Meyer 80), and also in the analysis of progressive reasoning (cf. 11.2).

76

2.2 SOME TROUBLES WITH SMILEY "ENTAILMENT"

classical tautologousness by logical necessity, as Lewy observes.) The Smiley proposal, while (like Lewy's premisses) unexceptional as a necessary condition on entailment, is quite unacceptable as a sufficient condition. But when only its acceptable, necessary half is taken, transitivity can be retained. The argument for these claims is, in outline, as follows:Firstly, where A -»• B is a first degree entailment (cf. system FD belov^ then A HgB. (For proof details, cf. ABE, p.219.) This shows that Smiley entailment , [f „, taken as a necessary condition, is unexceptional from a relevant viewpoint and need lead to no violation of transitivity. Taken as a sufficient, as well as necessary, condition for entailment, however, Smiley entailment underwrites many defective principles: where A and B are contingent, Disjunctive Syllogism is readily established for such an "entailment"; though A & ~A B is out, (A & ~A) v (A & B) -*• B is not; and though A -*- B v ~B is put, A -»•. A & (B v ~B) is not, yet surely the latter is just as paradoxical, and generally undesirable, as the positive paradox. Classical logic is no base from which to start in the quest for a satisfactory account of entailment; and patch-ups like Smiley entailment, which work at best for first degree entailments (where entailment is the main connective, and so never iterated), and which treat only the most obvious symptoms of the paradoxes, retain too much that is classical - thus many of the implicational counterexamples of 1.2 ff. apply against Smiley entailment - and diverge from the classical only at the wrong places, on matters such as transitivity which entailment itself must satisfy. These points provide evidence that Smiley entailment does not capture an intuitive notion of implication. And later (p.137) Lewy 'submits that the criterion "captures" no intuitive concept', though his evidence is hardly 1 impressive. Lewy contends that the considerations that there is something very compelling about Smiley's criterion and that yet there seems to be no intuitive concept which the criterion captures - lend additional weight to [the] claim that the intuitive concept of entailment is inconsistent (p.137). This is all decidedly dubious. Given a little reflection Smiley's criterion is (as we have tried to show) not at all compelling; and it is just false that 'we all have the feeling that it does "capture" some intuitive concept'. Many relevant logicians do not have this feeling (cf. ABE, p.215 ff.). What the criterion does is to approximate, as a necessary condition, to a condition that is intuitively appealing, namely the avoidance of modal dependence fallacies. There would have to be something especially compelling and satisfactory about Smiley's criterion as a sufficient condition for entailment for additional weight to be lent in this way to Lewy's inconsistency thesis; for only in this way would Smiley's criterion support the intuition that entailment is not transitive, in contradiction to the intuition, for which there is good evidential support, that entailment is transitive. *The fact that it does not answer to one intuitive notion that has been suggested, namely obvious entailment, is hardly evidence for the general claim. Lewy's very nice demolition of the suggestion (made, e.g., by Bennett 69) that Smiley entailment reconstructs the idea of obvious entailment is however worth drawing attention to: it is in 76, p.136.

77

2,2 "THE INTUITIVE CONCEPT OF ENTAILMENT IS NOT CONSISTENT"

In Slim, Levy's "additional" case for the inconsistency of the intuitive concept of entailment does not stand up to examination. One of his main arguments for inconsistency hardly differs, and accordingly fares no better: it is (76, p.133): ... consider the following three propositions (1) S-necessitation [i.e. Smiley entailment] is a sufficient condition for "A entails B"; (2) Entailment is not unrestrictedly transitive; (3) It is not the case that (A & ~A) entails B. This triad is patently inconsistent; yet we feel a strong inclination to accept all the three propositions. ... each proposition brings out an intuition we have in the area, but these intuitions are inconsistent. We feel no inclination at all to accept (1); nor could Lewy's argument for (1) persuade us otherwise, unless we granted him - what is fallacious - the conversion of an A-proposition. Furthermore (1) has considerably less claim than (2) and (3) to be intuitively based; indeed we suspect that for very many philosophers (1) is not intuitively assessable at all, in view of the character of Smiley entailment and the way all substitution-instances of complex entailments have to be grasped in assessing (1). Thus (1) does not bring out an intuition most of us have concerning entailment: if, however, 'sufficient' in (1) were replaced by 'necessary' it would yield a proposition that many of us would be lead, by a proof, to accept. (The rest of Lewy's case for his thesis that 'our intuitive, that is pre-formal, concept of entailment is inconsistent' will be examined in 2.9.) §3. Adding conjunction: semi-lattice logic and holim. Fundamental though a derivability relation is, its separate logic is hardly exciting. To obtain more interesting logics where distinctive theses really occur and where rules like Transitivity actually do some work, some operations subject to axiomatic constraints will have to figure in L. In the absence of specific connectives in the sentential case for instance, the powerful deducibility machine merely idles. We begin adding, one at a time, extensional connectives to ML. In the case of logics which lack the undesirable portation principles (which combine importation and exportation), namely in implicational form A

(B -*• C)

iff

A & B -f C

(Portation),

there is definitely - for reasons that will slowly emerge - a preferred order in which standard extensional connectives can be added to the basic theory of implication or deducibility: namely conjunction comes first, disjunction second and negation last. This order also coincides, to a fair extent, with the way in which rival accounts of implication and entailment diverge from one another. Thus there is substantial agreement about the logical behaviour of conjunction within the restricted (lower degree) framework at which we are isolating issues; there is less agreement over disjunction principles, and less agreement still over negation, and moreover some of the disagreement over disjunction principles can be put down to disagreement over negation principles. Indeed negation is the watershed connective.

78

2.3 SEMI-LATTICE LOGIC AMP ITS SEMANTICAL ANALYSIS

Semi-Lattice Logic (SLL) adds to ML the following postulates for the conjunction operation: Axiom schemes:

A & 6 |- A

A & B |- B

A, B (- A & B

(Simplification) (Adjunction)

Alternatively Adjunction may be replaced by the derived rule of &-Composition: r, A j- B, r, A h c-f r, A |- B & C, i.e. if r , A B and r , A |- C (are theses) then T, A |- B & C (is a thesis); otherwise read: from I", A |- B and T, A (- C to infer T, A |- B & C, etc. (The rendition of rules and the theory of inference rules will be issues taken up again later, in 4 and 15.) The rule of &-Composition follows from Adjunction as follows:1. Hence 2. 3.

T, A, B |- A & B

By Adjunction, Augmentation.

If T |- A then r , B |- A & B,

by Transitivity and 1.

If T |- A then T |- B then T |- A & B, by 2 and Transitivity.

Conversely Adjunction follows using &-Composition from Identity and Augmentation (see Tl, §5). The semantical analysis of SLL reveals very clearly the intended sense of the connective &. The analysis is obtained simply by adding to that for ML the normal conjunction rule: I(A & B,a) = 1 iff I(A,a) = 1

and I(B,a) = 1.

A rule stated in this form is intended to hold generally, i.e. for every situation a in K and all obs A and B. Truth, validity and so on are defined on the same plan as for ML. Adequacy Theorem for SLL 2.2. A |- A is a theorem of SLL iff A H A, i.e. iff A |- A is SLL valid. Proof builds on the analogous result for ML. Thus soundness is simply a matter of verification of the new postulates, which is done by applying the &-rule. Suppose, for Simplification, that for a in I(A & B,a) = 1. Then, by the &-rule, I(A,a) = 1 and I(B,a) = 1. Hence both A & B |- A and A & B |- B are SLL valid. Suppose, for Adjunction, that I(A,a) = 1 and I(B,a) = 1, for a e K; then I(A & B,a) = 1, validating A, B |- A & B. For completeness it is enough to show (given adequacy of ML) that for a e K c , I(A & B,a) = 1 iff A & B e a. And this is shown as follows:I(A & B,a) = 1 iff I(A,a) = 1 = I(B,a), by the &-rule iff A e a and B e a, by induction hypothesis iff A & B e a, as the following arguments show:Suppose A e a and B e a. Then as {A, B} c a and A, B |- A & B, by closure A & B e a. Suppose A & B e a. Then as Ta & B} £ a and A & B J- B, by closure of a, A e a and B e a. Completeness proofs for more comprehensive systems containing conjunction (e.g. the systems of chapter 3) require that the theories used be closed under adjunction as well as |- , i.e. that whenever A e a and B e a then A & B e a. This requirement is satisfied by the theories of SLL, but in general closure under entailment (i.e. -*•) does not ensure closure under adjunction. The semantical analysis opens the way for various arguments for the conjunction postulates of SLL, all of them aimed at vindicating the normal

79

2.3 ARGUMENTS FOR ANV AGAINST THE NORMAL CONJUNCTION «/LE

conjunction rule for all logical situations. Most of these arguments begin from the assumption, usually enforced as analytic, that logically to say that A & B holds in a situation just is to say, or means the same as, that A holds and B holds.1 (Such an argument is developed in Routley2 75.) Alternatively the conjunction postulates of SLL may be argued for on the basis of intended interpretations of entailment. Take the logical sufficiency interpretation for example: as A is sufficient for A (i.e. A -*• A) , A and anything else B must also be sufficient for A (i.e. A & B •*• A), for the addition of further information in the shape of B cannot undermine A's sufficiency. (This assumes that the addition of B does not interfere with A and undercut its sufficiency, an assumption - written into the ordinary use of 'and' - that the smart connexivist will dispute.) Similarly when A is sufficient for B and also sufficient for C then it is surely sufficient for both B and C, i.e. for B & C. In an analogous way inclusion interpretations can be used to argue for the conjunction postulates; for example, as the logical content of B is part of the logical content of A & B, A & B + B . Such interpretations also underwrite such characteristic conjunction principles as Commutation (A & B •*• B & A) and Associativity (e.g. A & (B & C) (A & B) & C). But none of these arguments is without assumptions, and accordingly none is completely conclusive. Like everything else, each of the conjunction postulates can be disputed.2 Before entering the dispute, let us record agreements and disagreements between the implication camps over the very ordinary and basic conjunction postulates of SLL. They are accepted by classical, intuitionist, relevant, orthological, non-transitivist and conceptivist positions. The axioms are however rejected by connexivism, for much the same reason that the principle of augmentation of premisses gets rejected, for example that a consequent only follows from antecedents if the joint force of the. antecedents is used, or if there is some cogent way of using all the premisses, certainly not if an added premiss is irrelevant to the derivation of a conclusion, or if an added antecedent somehow undermines the initial antecedent. Tiius connexivism rejects both forms of Simp but retains Adjunction. Connexivism is not alone in the rejection of normal conjunction postulates: Adjunction has also been questioned, but on quite different grounds, by non-adjunctivists. Consider some of reasons that have been given for rejecting Adjunction: there are at least two of importance. Firstly counterexamples to Commutation are easily devised, and these reflect back, not on Simplification, but on Adjunction, which yields A, B ]- B & A. The counterexamples derive from cases where 'and' clearly functions as 'and then', e.g. "he pulled out to pass and he hit an oncoming car", which does not imply "he hit an oncoming car and [then] he pulled out to pass" in the temporal sense of 'and', though it does imply both "he pulled out to pass" and also "he hit on oncoming car". And this already begins to reveal that the temporal conjunction determinable 'and then1, 1

As the English 'and' used in the explanation is supposed to conform to analogues of the semi-lattice postulates, there is an element of circularity in the semantical rule. It is this, undamaging circularity, that makes the analytic entrenchment possible.

2

Just as anything can be believed or disbelieved, so anything, including the correct laws of logic, can be disputed. An argument for this, premissed on relevant logic, is given in Routley2 75; and see further EMJB. Nonadjunctive systems are investigated in chapter 13, when "conjunction" rules are varied.

80

2.3 REASONS THAT ARE GIVEN TOR REJECTING ADJUNCTION

which certainly occurs in English, is a restriction (and/then, so to speak) of the more general, and logically important, commutative conjunction. Thus temporal conjunction is a notion that can be captured later, in the stage of logical analysis when temporal operations and tensing are taken account of (e.g. in 8.1). But the initial, and characteristically logical, objective concerns the logical behaviour of conjunction unrestricted by temporal ordering requirements. A second reason of importance for rejecting Adjunction also succeeds in bringing out significant features of the normal logical conjunction which are commonly overlooked. According to holism, which does not dispute Commutation, the whole is sometimes more than the mere sum or assemblage of the parts: applied to conjunction the message is that A & B may say more than A says and B says, that, more precisely, the logical content of A & B may exceed the logical content of A together with that of B, i.e. in symbols c(A & B) £ c(A)u c(B). The systemic whole may exceed the sum of the parts of a system because, so it is said, of the interrelations of parts that emerge on assemblage and hence because of emergent properties of the system. Whether this is so or not (it depends on how the whole and parts are characterised and whether systemic relations are appropriately incorporated in the characterisation of parts), it is a little fanciful to link conjunctions of statements with systemic wholes. A whole is a certain totality of things and their interrelations, whereas a conjunction is a mere result of an operation on statements. Explication of the notion of a whole is a logically important matter, especially in such disputes as those concerning methodological individualism, but it is not something to be accomplished by displacement of the general notion of conjunction. Rather it is a different and subsequent endeavour, which really requires elaboration of the theory of intensionality, including entailment, in order to proceed in a logically satisfactory fashion. Likewise intensional features, supplied semantically through the theory of worlds, are required to explicate the notion of conjunction which does result from holistic criticism of extensional conjunction a criticism which can stand even when the analogy of wholes with conjunctions is duly weakened. For according to the holistic criticism, and indeed to all the criticisms of the normal conjunction rule, whether A & B holds depends not just on whether A holds and B holds but also on the relation between A and B, and it is this relation that the extensional (intraworld) account leaves out.2 Granted there are intensional conjunctions which do take account of interrelations of components and which merit study (see, e.g. 5.2 for analyses of certain of these); the normal English conjunction is not, when unrestricted, one of these, and it is important for a variety of reasons to include the normal conjunction in a comprehensive theory of entailment and implication (the reasons are essentially those set out in SL, chapter 5, for the inclusion of classical connectives in significance logics). It may also be desirable for various purposes to include other conjunctions as well: it is enough for our argument that the normal conjunction be included, not that other conjunctions be excluded. And there is nothing to prevent the subsequent addition of a holistic conjunction to the connectives admitted 1

Thus too there are ways of recovering the more general operation from the restricted operations.

2

Interrelations of components come down in the semantic analysis to evaluations over related situations. Semantical rules for non-normal conjunctions will be introduced in subsequent chapters, in particular chapter 8.

SJ

2.4 THE LEAVING THESES OF CONNEXIVISM

(in this crucial respect classical negation turns out to differ: it cannot be added without inducing system distortion of an undesirable character). Our main point against holism is then that an English conjunction, the normal one it seems to us, does conform to &-Composition, and that such a connective should be included early on in logical investigations, especially as regards comparisons with going systems. A third ground for non-adjunctionism - really a special case of the second - points to the feature that someone may accept A and B separately where they are encountered, e.g. B is ~A, but not be prepared to accept the conjunction A & B, e.g. the explicit contradiction A & ~A. This ground affects Adjunction not as an entailmental principle however, but the question of the logical behaviour of 'and' when covered by such functors as 'It is accepted that', 'Kick believes that', etc. The latter issue is taken up again in chapter 8. §4. The varieties of connexivism. In contrast to holism, according to which a conjunction may say more than its conjuncts, connexivism maintains that a conjunction nay say less than its conjuncts, i.e. in more precise form c(A & B) ^ c(A) u c(B). Connexivism, that is, rejects Simplification but retains Adjunction. Addition, A A v B, is generally rejected along with Simplification, since Contraposition and De Morgan principles are generally accepted. Similarly Augmentation is rejected. The features do not however fully characterise connexivism1 which has the following two leading theses:1. Simplification (A & B -*• A, A & B B) fails to hold, and its use (and that of its outcome, Addition) is what is responsible for the paradoxes of implication, not the ancient and correct principles of Disjunctive Syllogism or Antilogism. 2. Every statement is self-consistent,2 symbolically A * A, where the relation of consistency with, symbolised o , is interconnected with implication in the standard fashion: A -o B ~(A -»• ~B). The doctrine of universal consistency appears in stronger and weaker forms: the stronger form, which reappears in modern times in Nelson's 1930 system, may be represented sententially by the principle A B ->•. A • B (called Boethius), and this implies the weaker form, ~(A •*• ~A) (i.e. Aristotle) using Identity. (In stronger systems which contain Exported Syllogism the principles are interderivable as we shall see.)

Evidently thesis 2 guarantees thesis 1 - on pain of negation inconsistency otherwise. For suppose Simplification were generally correct. Then, as in the negative paradox arguments, (A & ~A) -»• A and (A & ~A) -*• ~A; but accordingly, by strong consistency, ~((A & ~A) -*• A), contradicting (A & ~A) -»• A. A similar effect is had using A • A when Contraposition is applied. For A -*• ~(A & ~A) , whence (A & ~A) •*• ~(A & ~A), i.e. ~((A & ~A) • (A & ~A)). Thus connexivism avoids the usual counterexamples to Aristotle's principle by relinquishing Simplification. Put differently, Simplification plays a key role in establishing, what is the current orthodoxy, that contradictions are not selfconsistent. These arguments also show the decidedly non-classical cast of principles such as Boethius and Aristotle: their adoption renders inconsistent and so trivialises classical and modal logics. But the interconnections between theses 1 and 2 run much deeper than this; for 1 and 2 may be unified, as we shall subsequently try to show, through an underlying (so far unformalised) semantical view, a view which lies behind and informs much traditional logical thinking. (Footnotes

1

and

2

on the next page)

82

2.4 EXPLAINING THE HISTORICAL PERSISTENCE OF CONNEXIVISM

Connexivism has a much more ancient lineage than any of our other candidates for weakly relevant logics, most of which appear to be1 relative newcomers on the historical scene. Connexivism appears to go back at least to the Stoic dispute over implication and indeed, there is a good case for claiming, to Aristotle (see Lukasiewicz 51, pp.49-50).2 It was accepted, it appears, in the early middle ages by Boethius. In modern times its distinctive non-classical principles keep reappearing in quite diverse writers (some of them not informed by the logical literature on the topic), e.g. Strawson 52, Downing 61, Cooper 68, Stevenson 70, as well as Nelson 30, Angell 62, McCall 66. The persistence of connexivism is something that needs explaining, and which we shall try to explain. It is important to say, though not enough to explain the persistence of connexivism, that it represents one of the main options open to anyone who wants to reject the paradoxes of implication but not to reject Disjunctive Syllogism and Antilogism, and also to have a more or less coherent case for what one is doing. Connexivism may be a relatively minor doctrine but its elucidation and criticism is not just of isolated interest, but is important for the much larger issue of linguistically correct theories of negation and consistency. There are four main classes of objections to connexivism, the last of which ties up with our explanation of the persistence of connexivism; 1. Objections to the connexivist attempt to solve the implicational paradoxes by rejecting Conjunctive Simplification, and to the motivation for this rejection; 2. Objections based on the serious effects of these rejections, e.g. the way argument is hamstrung; 3. The difficulties engendered by these rejections, especially for basic models of entailment such as inclusion and containment; 4. Objections to the underlying motivation for the rejections, namely the theories of incompatibility, negation and conjunction presupposed. The way connexivism avoids the paradoxes of implication is, as we have seen, by throwing out Simplification, or (what is equivalent in the framework of SSL, but commonly thought to be more general) rejecting Augmentation, the addition of premisses. One paradox demonstration (much favoured by Lewis, e.g. 18, and subsequently by Nelson, e.g. 30) which emphasizes how it is that connexivism is a major option in resolving the paradoxes proceeds from just two premisses:(Footnotes from the previous page) 1

We have already remarked in 1.4 that Chrysippus's requirement that implication there be a connexion between antecedent and consequent is, in a sense, satisfied by any weakly relevant logic, but not by connexive logics, e.g. those of McCall - is no basis for calling a connexive in the modern sense.

2

On another familiar account of consistency A & ~A is not self-consistent but inconsistent; for it entails p & ~p for some p. The connexivist (e.g. Nelson 30) has split the usually coincident accounts of consistency, not entailing a contradiction, and not entailing its own negation.

in an - which some logic

Since most Stoic documents have been lost, including all those elaborating the view that implication requires a connection, there can be no certainty that the idea of a relevant implication is not an ancient one. 2

The case would be conclusive were it certain that the parameters Aristotle uses, 'A is white' and 'B is great', are completely general and not contingently restricted.

83

2.4 ATTEMPTS TO SOFTEN THE WHOLESALE CONNEXIVIST REJECTION OF SIMPLIFICATION

A & B |- A

Simplification

A&B

Rule Antilogism

|- C —• A & ~C |- ~B

Then as A & B |- A by the first, A & ~A j- B by the second. Thus either Simplification or rule Antilogism has to go: this is a fundamental choice. Connexivism alone chooses the first route. (It is a matter of history that all who choose the first route and try to justify their choice are tempted by the characteristic connexivist principles, for reasons we shall try to educe when we discuss the deeper motives for connexivism.) Other paradoxical derivations - indeed these and much more - are destroyed by the single expedient of dropping Simplification. Nonetheless the choice may be the wrong choice. For if a combination of two factors - premiss addition and suppression - lead to trouble, and one, suppression, is independently objectionable and the other is not, the choice of which to get rid of is obvious. Surely the Simplification of premisses is sometimes admissible? Surely the addition of irrelevant antecedents is not always objectionable? The wholesale rejection of Simplification - which is, as we shall see, and obviously anyway, too drastic - is not usually what is proposed by connexivists in their criticisms of the paradox arguments. There are three ways in which standard connexivists have endeavoured to qualify their outright rejection of Simplification, namely by admitting that Simplification is correct provided first that the added premiss B (in A & B -*• A) is relevant to A, or provided secondly that augmented premisses are actually used in the derivation (a move which strictly construed rules out all instances of Simplification1), or provided thirdly that the added premiss B is consistent or compatible with A. If the second move means 'has to be used in the derivation' then it is too strong, and no connectives of systems so far formulated satisfy it. Nor is it likely to be satisfied in an adequate system with substitution on variables, since other principles will lead to instances of Simplification. For example, Conjunctive Syllogism (A -»• B) & (B -»• C) -»-. A C has as a special case (A -*• A) & (A -*• A) -*•. A -»• A which violates the requirement. The restriction, which is like a requirement of minimal premisses, would be inoperable in other ways as well. On the other hand, if the second move simply comes down to 'can (or may) be used in a derivation' then the restriction is too permissive, and lets through versions of the paradoxes. The first restriction on Simplification proposed, the relevance restriction, seems to be based on the mistaken assumption that unless A and B are relevant or related they cannot have joint consequences (i.e. the first point boils down to the third).2 This is to assume that consequences result from 1

This is the conclusion that Nelson 30 in fact reaches, though invalidly. Nelson argues that A & B -»- A is not assertible because cases of it together with the correct principle of Antilogism lead to paradox. Then he goes on to claim (30, p.448): In view of the fact that p & q -*• p is not assertible, we must restate the Principle of the Syllogism (p -»• q & q -*• r -»•. p r) this way: p ^ q ^ r p-+-q&q->-r-»-. p + r ... Otherwise, if we substitute, for example q for r, we get p-»-q&q-*q-»-. p q, which is of the form p & q -»• p. But to say that A & B -»• A is not assertible is only to say that it fails in some cases - as the argument for its non-assertibility shows - it does not follow that there are no instances where it holds. ~ |- A & B •»• A is like ~(A)(B) A & B -»• A, i.e. (PA) (PB) ~(A & B -»• A), not. like (A) (B) ~(A & B -»- A) , which corresponds to |- ~(A & B -*• A). Put differently, Nelson's case rests on a scope confusion between ~ |- D and |- ~D.

Footnote 2 on the next page.

u

2 A THE RESTRICTIONS ON SIMPLIFICATION WOULV SABOTAGE VEVUCT1VE REASONING

the intersection, or interaction, of two premisses rather than by inclusion (cf. again Nelson 30). The connexivist seems to be endeavouring to replace the inclusion model of logical consequences by some sort of intersection account, where in order that premisses have consequences all propositions must be used in or are essential in the deduction. If the idea that all premisses must be used in order that the argument be valid, that all conjuncts of Simplification be applied, is carried out consistently it leads again to a demand for minimal premisses. For precisely the argument that counted against Simplification would show that any unused subpart or subconjunct of a used statement concerning conjoint antecedents would render the argument invalid; the conclusion would not follow unless the assumed subpart were removed and the antecedents pruned to the minimum. Thus we would count no argument as valid unless we could guarantee that the premiss set could not be further reduced. It is hardly necessary to emphasise the devastating effect such a requirement would have on the practice of reasoning. For in very few cases can we be satisfied that such a requirement is met and that there does not lurk within one of our statements some subconjunct which is unnecessary to obtain the consequent. Deductive reasoning would become unworkable. Consider, more generally, the effect of connexive constraints on the business of assessing someone's written work, or belief set, or theory. We could not extract subparts of any of these with the clear assurance that they were entailed by the whole set. Thus the practice of analysis would be destroyed, for we could not assess parts of the theories or texts.1 After all, not all theories are consistent, not all parts of texts are consistent with one another, and some authors are repetitious or redundant or insert irrelevant or otiose material.2 Even if the effect of such requirements in sabotaging deductive reasoning is overlooked, the requirement that premisses be constrained, e.g. as minimal, appears in any case quite misconceived and contrary to basic deductive practice. For why should anyone suppose that an argument the premisses of which include some unused information is thereby rendered invalid. If such otiose premisses or details are included, inadvertently or otherwise, no one would say that such an argument was invalid, certainly not in the same way as arguments such as affirming the consequent. Does the teacher send back the student's exercise in which an otiose premiss has been inadvertently included, marked 'Invalid', in the same way as if it included a faulty derivation, for example conversion of an A-proposition. And do we say that an axiom set, one axiom (Footnote 2 from the previous page) 2

The relevance restriction also runs into various technical difficulties. If relevance were all that was required such principles as A & A A should be connexively correct, since A is relevant to A. But then by Rule Antilogism A & ~A ~A, and similarly A & ~A A; and so, by an earlier argument negation inconsistency ensues. Other qualifications of Simplification, likewise suggested by relevance controls also lead to difficulties, e.g. A & (A & B) ->-. A & B.

1

Though this would hamstring proper discussion of authors' views and positions, this would no doubt suit some holists, who maintain that no assertion of a historical figure can be assessed in isolation from everything else he wrote. Connexivism could then provide (unintentional) logical assistance and sustenance for obscurantism.

2

Classical practice is not without its problems as regards the assessment of texts either. For classically a text is tantamount to the conjunction of all the assertions in it, but we do not readily concede that a text is false because just one (perhaps minor) statement in it is.

85

2.4 CONNEXIVISM IS BASED ON AN INADEQUATE DIAGNOSIS OF IUPLICATIONAL PARADOXES

of which is subsequently shown not to be independent, had after all no nontrivial consequences, and that the impression that it had was surely an illusion? If we did say this what sense would be given to independence and how would we establish it? Insofar as standard connexivism is an attempt to solve the implicational paradoxes, it is based on an inadequate explanation of how the paradoxes are caused and what is objectionable. It takes irrelevance as caused by the addition of irrelevant premisses as epitomised in A & B -*• A. But this is an incorrect analysis of what has gone wrong in the paradox arguments; it is not our freedom to introduce irrelevant or unused further premisses which is at fault (and objectionable), but our ability to suppress necessary truths, for example through Disjunctive Syllogism or Antilogism (or, what proves the semantical equivalent, the exclusion of worlds which admit inconsistent assumptions). It is possible to show independently of their roles in the paradox arguments how these principles warrant the suppression of necessary truths. It is also possible to show how the suppression of necessary truths leads to such paradoxes as that everything implies a necessary truth through intuitive models for deducibility such as containment. Irrelevance is simply a symptom of this more basic disease of suppression, which is at least as damaging to the entailment notion as irrelevance itself. For the suppression of logical truths and falsehoods is greatly at odds with the ordinary (nonmathematical ly corrupt) practice of deducibility. Thus the rejection of Simplification may be seen as an attempt to maintain this objectionable feature, the suppression of necessary premisses, but by a shallow formal trick to avoid the manifestation of its most obvious symptom, namely irrelevance. The trick is worked by an unwarranted and misconceived restriction on our freedom to form and augment premiss sets. Thus as a solution to the paradoxes, standard connexivism is quite unsatisfactory; it removes the more obvious symptoms but does nothing about the real disease. What is in effect being proposed by those who would solve the paradoxes by abandoning Simplification, is that in order to maintain one traditionally accepted argument, allegedly endorsed by Aristotle, which is however peripheral for many deductive purposes, we should abandon much of the basic application of deducibility in traditional deductive practice. Unlike their relevant rivals (for which (y) is provable) standard connexive logics do face the charge that they are too weak, and that crucial principles needed for deductive proof have in fact been abandoned. For the relevant logics can in fact show that strict and classical logics are enthymematically recoverable, but the argument depends crucially on the presence of Simplification. (The point and importance of enthymematic recovery will be explained subsequently). Deducibility is not simply a notion that floats free and which can be eliminated in any way that one chooses. Deducibility has to answer back to requirements imposed by associated models for deducibility, and its properties are in part constrained by such models, for example, such models for deducibility as sufficiency, inclusion and containment models. A fundamental feature of all these models is that they allow the addition of further premisses without sacrifice of deducibility, that is they validate Augmentation and Simplification. Consider the sufficiency picture for instance. If A is logically sufficient for B then the addition of a premiss C to A is not going to render it insufficient for B - unless C should somehow remove or weaken A: but how can it do that? For similar reasons Simplification must be part of any inclusion model, for if C is contained in A it must be contained in any augmentation of A. These important models and connections are lost if Simplification is abandoned. Such traditional models can, however, be maintained and improved by relevant logics which keep Simplification but reject Antilogism and Disjunctive Syllogism (see further 3.3). A conjunction for which Simplifi-

S6

2.4 DISTINGUISHING C0NNEXJV1SMS: STANDARD AD HOC AND INTUITIVE C0NNEXIV1SMS

cation holds good is an essential component of any adequate theory of deducibility (but this fact does not of course exclude other conjunction-like connectives for which Simplification does not hold good generally, any more than the necessary inclusion of a normal negation in an adequate theory excludes other negation-like connectives such as more intuitionistic negations). The standard connexivist view accordingly appears mistaken: adding to the antecedent of a conditional may interfere with consistency, as adding A to ~A does. But it does not affect the logical power of what it is added to or delete from its class of consequences - unless C somehow removes or deletes some of the content of A, something that the mere addition of C to a situation where A holds never does. But according to some varieties of connexivism it does: the addition of C may delete some of the content of A. It is apparent then that connexivism either violates Intuitive models such as sufficiency or else is bound to take a different view - the uncharitable would say a curious view - of either conjunction or negation or, as it will turn out, both. For, as to the first point, the sufficiency model just is that which is explicated (at the first degree) by way of a universal guarantee in all situations. If a connexivist accepts this, then he is obliged to vary his conjunction evaluation rule from the normal. It is important, at this stage, to distinguish two forms of connexivism ad hoc connexivism, and intuitive connexivism. Ad hoc connexivism gives no intuitive basis deriving from the character of conjunction for its rejection of Simplification; either it offers no basis for its rejection of Simplification or else argues for the rejection of Simplification simply on the basis that its incorporation would lead to inconsistency or paradoxes. The formal connexivist theories so far developed (those of Nelson 30, Angell 62, McCall 66) are all of the ad hoc variety. Nelson, for example, argues for the rejection of Simplification on the ground that it together with Antilogism leads to paradox (since Rule Antilogism takes A & ~B -*• A into A & ~A -*• — B ) . Ad hoc connexivisms are subject to the criticism - additional to those already made of connexivism which continue to hold - that they lack a proper motivational basis. Intuitive connexivism, in contrast to merely formalistic connexivism, tries to give some intuitive basis, in terms of the meaning of negation and conjunction, for its rejection of Simplification.1 According to this view of conjunction and negation, conjoining an antecedent as in Simplification is not, as is commonly thought, introducing a further premiss which holds in the antecedent situation: it is introducing an assumption which can interfere with what already does hold. In particular, it can do this if it negates or wipes out part of what is supposed already to. hold. Thus "conjunction" so construed - combined with negation, so construed as to reflect wipe-out - can undermine cases of Simplification, and this presumably is what happens, according to the intuitive version of connexivism, in some of the classic paradox arguments. In the Lewis argument for A & ~A -»• B, Simplification appears in the forms A & ~A -*• ~A and A & ~A •*• A. The connexivist is presumably bound to say that although A entails A the addition of ~A as an antecedent undercuts A so that the consequent A no longer ensues. Unless we can guarantee that a B added to A does not interfere with the deductive force of A we cannot assert A & B + A, a s A & B will not be sufficient for A. Since for an arbitrary B 1

The formalism of ad hoc connexivism can of course be underpinned by an intuitive basis. Attempts to offer semantics for Hilbert-style systems of connexive logic are characteristically attempts to do just this. On the other hand the intuitive bases so arrived at may show the need for changes in the formalism.

87

2.4 EQUIVALENCI ZING CONJUNCTION ANV VELET10NAL1Z1NG NEGATION

we have no such assurance, Simplification is connexively invalid. But the invalidity so claimed is only as good as the accounts of conjunction and negation underlying it. The connexivist owes us an account of these notions, because they are considerably removed from the orthodox notions. The character of the connexivist "conjunction" may be brought out through semantical modellings for connexive logics. Thus Routley and Montgomery 78 show that a number of connexive logics, including McCall's system CC1 which axiomatises the Angell's four-valued matrices for connexive logics, interpret conjunction as an equivalence relation in some situations. But a conjunction that sometimes turns into an equivalence is only an Alice-through-the-Looking-Glass sort of conjunction. Such a modelling is, however, only one among possible semantical analyses of connexive conjunction. The conjunction may be - and in the case of more satisfactory connexive systems apparently has to be - analysed in less damaging fashion through an intensional conjunction rule (see e.g. chapter 5), which does not - at least for satisfactory systems - reduce in some worlds to an equivalential rule. Unless implication is characterised using conjunction (either directly, or indirectly through consistency construed as joint possibility) negation as well as conjunction has to be given a non-normal interpretation. For just as conjunction has to be equivalencised in some worlds to invalidate A & B -»• A, so negation has to be "deletionised" to validate Aristotle's principle, ~(A -*• ~A) , in which conjunction does not explicitly enter. Sometimes connexivists have reduced the problems to one connective. For example, McCall 66 and Angell (in some unpublished writings) define '->•' in Lewis fashion with A B = D~(A & ~B). In this way the question of the interpretation of connexivism may be transferred to the question of the interpretation of conjunction alone (cf. Routley and Montgomery 78, where the semantics for connexive logics given differs from modal logic semantics in the evaluation rule for conjunction only, not on rules for necessity and negation). Alternatively the interpretational issues might be locatable in the interpretation of negation alone. Such reductions of the interpretational problem depend however on such mistaken accounts of entailment as the Lewis account, and tend to lead to paradoxes. In general, semantics of paradox-free connexivism is going to require revision in the interpretation of both conjunction and negation, and also of such interdependent notions as compatibility and inconsistency. There is a variety of connexivism, then, which is based on a formalistic and opportunistic resolution of the Lewis paradoxes and which offers a mistaken account of entailment in the context of truth functional connections. Typically this sort of connexivism rejects or restricts Simplification principles in an ad hoc way because it cannot see how Antilogism and other suppression principles can possibly be rejected. There is however another approach which is connexivist in character since it restricts Simplification and accepts Aristotle, which does have a sounder intuitive base and which can offer an important insight into problems of relevance and sufficiency. This sort of connexivism qualifies Simplification not because it in combination with Antilogism yields paradoxes, because it is «jased on a subtraction account of negation and incompatibility which is non-classical and which forces qualification of Simplification and related principles. This line of connexivistic development, though historically important and intuitively well motivated, has however as yet not received any clear formal expression (and there may be some difficulties in obtaining a formal analysis for it, for reasons we will come to).

88

2.4 NEGATION AS CANCELLATION, DELETION, ERASURE, SUBTRACTION

Subtraction or cancellation negation, if not, as Strawson (52, p.8), rashly asserts, 'the standard and primary use of not', is at least a basic and important negation in natural language. It takes up the basic notion of negation as a deletion or removal of a condition. Thus it is often pictured in simple and natural terms such as the following: p as running up a flag [which states the condition given in p], ~p as running it down again; p as writing something on a board, ~p as rubbing it out again, or putting a line through it, cancelling it out,1 p as recording a message, ~p as erasing it; p as stating something, ~p as withdrawing it. These sorts of pictures for the deletion notion, however appealing, are nonetheless defective, since they confuse statings with statements, the condition stated by p with the performance of asserting, in some mode, p. Thus the examples above are all examples of undoing or deleting the performance of asserting p, not the undoing or unmaking of the condition p itself. Thus for example, if the flag we run up the flag-pole states that the fox is in the hen-house, the lowering of the flag has only undone the performance of asserting this and has done nothing whatever about undoing the condition stated by the flag, that is done nothing about removing the fox from the henhouse . The notion of the negation of p as the deletion undoing or removal of the state or condition that p, is better viewed as a propositional analogue of the familiar operation of subtraction in mathematics. It is because the minus operation, of quantity subtraction, in mathematics has had such a powerful influence in logical thinking about negation that we have called connexive negation subtraction negation. Just as the negative integer -a is obtained by the subtraction operation from positive integer a, so on the subtraction view of negation the negative proposition ~A is obtained by negation from the positive proposition A, and similarly where B is negative (i.e. ultimately of the form ~C) the positive proposition ~B (ultimately C) is its negation. And just as a positive number added to its negative counterpart gives zero, so a positive proposition A conjoined with its negative ~A is seen as giving zero (A-)content. Thus the subtraction account contrasts sharply with the classical complementation view of negation according to which a proposition conjoined with its negation has total content, i.e. says everything. (The classical negation picture is explained in more detail in 2.9.) These results follow for subtraction negation because in A & ~A, ~A is not seen as adding a further condition to A but rather as the removal, deletion or subtraction of the condition first given, so one is left with nothing. Thus the whole A & ~A is, on the subtraction view less than the sum of each part since each part takes something from, and indeed may quite nullify, the other part. Accordingly 1

Thus Strawson (52, p.3): Contradicting oneself is like writing something down and then erasing it, or putting a line through it. A contradiction cancels itself and leaves nothing. The linkage of contradiction with negation is, according to Strawson (p.79), as follows:a standard and primary use of 'not' in a sentence is to assert the contradictory of the statement which would be made by the use, in the same context, of the same sentence without the word 'not'. A beginning is made in NC on tracing the long history of cancellation accounts of negation.

89

2.4 HOW THE SUBTRACTION ACCOUNT YIELDS DISTINCTIVE CONNEXIVE PRINCIPLES

Simplification, in the forms A & ~A ->• A and A & ~A ~A, fails. Since in A & ~A each of the conditions deletes the other the whole A & ~A contains neither of its apparent parts; thus A & ~A does not imply A, and A & ~A does not imply ~A. Thus also A & ~A does not imply A v ~A, i.e. using de Morgan principles A & ~A does not imply ~(A & -A). It is not difficult to see how such a subtraction notion of negation gives rise to Aristotle's law, ~(A ~A) . Suppose, on the contrary, A did entail ~A. Then ~A would be part (of the content of) A. But then, as in the case of A & ~A ~A, ~A would cancel A out, so that ~A could not be implied. Hence A does not imply its negation, and similarly ~A does not imply A. Another argument for Aristotle likewise starts from the assumption that A does entail ~A. Given this assumption, A must be impossible. Since A is impossible, A is equivalent to q & ~q for some q, i.e. A is equivalent to some contradiction. But then by replacement in ~(q & ~q -*• ~(q & ~q)), A + ~A, contradicting the assumption. Hence the principle A -»• ~A fails in the only case in which it could hold, namely where A is an inconsistent statement. The argument may be reformulated in a direct fashion. Either A is possible, in which case A ~A, or A is impossible, in which case by the argument given A -f- ~A; so A / ~A. Given Aristotle's principle, Strawson's principle,1 A -*• B ~(A -*• ~B), i s readily established, by essentially the argument Aristotle himself gave, namely:{A -*• B) & (A

-fi) -»-. (A •*• B) & (B

~A), using Contraposition

-»-. A •* ~A

, applying Syllogism.

Hence, contraposing, ~(A •*• B &.A -»• ~B). Strengthening Strawson's principle to an entailment, i.e. to Boethius's law, A -*• B -»-. ~(A -*• is less straightforward, though given certain principles correct for S4 for example, Aristotle does yield Boethius. The assumptions required are that provable statements (or negated entailments) are necessary, that necessary statements can be commuted out, and that syllogism holds in exported form. (A sufficient set of these principles holds in entailment system E). First, by Exported Syllogism and Contraposition, A -»• B -*-. A -»• ~B -»-. A

~A.

Hence, contraposing again,

A -»• B -*. ~(A ->• ~A)~(A-»- ~B). Since ~(A it can be commuted out, whence A -*• B -*• ~(A

~A) is necessary, since provable, ~B).

Boethius can also be argued for informally, and independently of Aristotle, on the subtraction account of negation. For it follows from the subtraction picture that a statement cannot include an inconsistent part. For using the principles given above, an inconsistent part would delete what it was inconsistent with, and the inconsistent parts would be apparent parts only. But precisely what Boethius asserts is that B's being part of A is logically sufficient for B's being consistent with A, i.e. A -»• B -*. A • B. For A ^ B, i.e. A is consistent with B, iff ~(A -»• ~B). (A similar argument is outlined in Nelson 30, p.447). To state the same argument a little differently:Suppose A entails B. Then as B is contained in A, B must, as a matter of 1

The attribution is based on the following passage (Strawson 52, p.87)5 As an example of a law which holds for 'if', but not for we may give the analytic formula '~[(if p, then q) & (if p then not ~q)].

2.4 THE CONTINUING IMPORTANCE Of DELETION ACCOUNTS OF NEGATION

logic, be consistent with A; for an inconsistent part would have deleted the argument in A that did the entailing.1 Another informal argument for Boethius appeals to the commonly accepted (but as we will see questionable) logical interconnection of A -*• B and A & B A. Now suppose B is incompatible with A, i.e. A ~B. Then, as a matter of logic, A & B does not imply A. For, as B incompatible with A, B subtracts from A, deletes part of A. That is, A ~B -*• ~(A & B ->A), whence Boethius results upon contraposing, and using A B A & B ->• A. (To make the argument connexively acceptable, the last implication should be replaced, e.g. by A -*• B A & B -*• B & B; the argument then relies on the fact that A/B implies that A & B does not imply B & B, because A subtracts from B,) The subtraction or deletion picture of negation and associated accounts of incompatibility have been extremely important historically in thinking on implication. Indeed it, subtraction negation, rather than classical negation, appears to be the notion of negation that underlies traditional logic. For it enables not only fuller and more coherent explanations of the intuitions of many traditional thinkers on implication principles, Aristotle in particular, but also of the central traditional theory of the syllogism. Thus the whole theory of the syllogism comes out on direct translation in connexive quantification theory (cf. McCall 67); whereas classical quantificational theory destroys syllogistic theory unless some very ad hoc assumptions are imposed - assumptions which furthermore destroy, without any warrant, the generality of the theory, which is not restricted in the terms it admits, e.g. to those which carry existential import. The subtraction account of negation continues to influence thinking about entailment in modern times. For example, it finds (as already noted) a modern expression in P.F. Strawson's Introduction to Logical Theory where , the notion of the negation of A as deleting or erasing A and A & ~A as yielding zero content is explicitly stated, without however its being clearly realised that the notion so expounded is quite distinct from classical negation.2 The fact that it has not been appreciated how distinct subtraction negation is from classical negation, helps to explain why a clear formal analysis of the subtraction picture of negation and the resulting, historically important, connexivist theory has never been properly worked out, and 1

2

This may suggest that A & B -»• A is correct subject to the qualification A O B. But this is a qualification that needs to be handled with care, as we have seen. For example, the qualification cannot be satisfactorily rendered as A • B A & B A where it is also conceded that A & B -*• A O B and that Contraction is correct. The question of the respective necessary and sufficient conditions under which Simplification holds is an important open question for connexivism. For these (and other, for example the account given of logical reasoning is neither classical nor connexivist but Peripatetic) reasons Strawson's logical theory is incoherent. It is worth noting that connexivist assumptions appear elsewhere in Strawson's text 52, for example in the proposals (p.24) for avoiding paradoxes of implication, that the inconsistency of a conjunctive assertion is not assured by the inconsistency of a conjunct. In short, in one way or another the principle ->• ~ ^(A & B) is to be avoided; and so, given distribution principles, ~A -*• ~(A & B) is to be rejected.

91

2.4 TROUBLES WITH NAIVE CONNEXIVISM, ANV INTERACTION CONNEXIVISM

also why contemporary efforts to formalise historical connexivism have fallen so far short. Thus, for example, McCall's attempt (in ABE, p.434 ff.) to formalise the first degree part of (an algebraic formulation of) connexive logic is based on, of all things, a Boolean algebra, which, of course, contains classical, not connexive, negation. The confusion between connexive negation and classical negation has, of course, been fostered by the classical logicians' insistence that classical negation is the unique intuitive account of negation found in natural language, an approach which confuses not only the connexivist, but also the classical logician, since the latter is often found providing a subtraction model of negation in the mistaken belief that it is an intuitive model of classical negation. But although the deletion model certainly seems to be coherent and is in many ways intuitively appealing (which helps explain why connexive principles keep appearing anew and being "rediscovered" by new workers on implication), it is not, we believe, the way negation must be explicated - or a correct account of the way negation must operate if we are to understand and to explain satisfactorily linguistic data and the functioning of a large part of language. It is only part of a larger story, and a part that needs to be told more carefully than connexivists have so far told it. Among intuitively based connexivisms, naive connexivisms should be distinguished from a more sophisticated position which we call interaction connexivism. Naive connexivisms are distinguished through their insistence on one, or usually more, of the following claims:- Firstly, as with classical logic, an absolutism about connectives is assumed: there is just one ordinary language negation, not the classical one, the connexivist one, and similarly just one conjunction. Furthermore this negation is the subtraction one modelled by the deletion account. Even so classical sentential logic should hold and does hold for connexive negation and conjunction (i.e. given ~ and & as primitives classical logic is forthcoming, a result written in, e.g., to Angell 62, by the addition of further & — axioms to ensure the full classical apparatus). Interactionist connexivism repudiates every one of these claims. According to interactionism there is, to begin with, more than one negation and more than one conjunction; but among the negation and conjunction determinates of natural language are those of connexivism. But connexive negation is not that given by the simple deletion account. The naive deletion account of negation if taken seriously would have a quite damaging effect on any theory of implication it was coupled with; for it would destroy relevance. If ~A really erased A without residue then A & ~A would have no content, say nothing. So it would not differ from B & ~B as regards content! That is, the deletion account would prevent us distinguishing distinct contradictions. As A & ~A = B & ~B, since they have the same content, namely none, A & ~A B & ~B, contravening relevance, and yielding paradoxes. The naive view of contradictions is hardly better than the classical and modal view. On the modal view inconsistent statements occur, though never in modal worlds and always in a degenerate way, but as far as naive connexivism is concerned there are none such. Such simple models can be powerful and persuasive but they do not accord with the facts of language. The naive deletion account of negation has accordingly to be abandoned, or rather adjusted, and with it the simple arithmetical model. The account can be adjusted, and the arithmetic generalised. The adjustment is by way of the relativisation already hinted at: ~A deletes A content, and A and ~A together say nothing A-wise, i.e. A & ~A has no A content. But to say A & ~A have no A content is not to say it has no B content, so the unfortunate outcome of Strawson's account that all contradictions say the same, namely

92

2.4 STANDARD CONNEXIVE LOGICS VIOLATE PARACONSISTENCY REQUIREMENTS

nothing, is avoided, and with it connexive paradoxes. Correpondingly the arithmetical model for connexivism is to be sought not in ordinary arithmetic with its single line through zero, but in generalised arithmetic (as outlined in Kleene 52) with many zeros - one for each distinct statement and many sequences - one by positive (and negative) extension from each zero. Connexive logics so modelled will presumably be weakly relevant: they would certainly be unsatisfactory if they were not, but relevance is as always not enough. Lastly, interaction connexivism will not incorporate classical logic; for not all classical tautologies are connexively valid, on pain of violation of the paraconsistency requirement (cf.1.7). The connexive logics of Angell 62 and McCall 66, for example, both violate the paraconsistency requirement. Since Disjunctive Syllogism is (claimed to be) connexively correct, the rule Y of Material Detachment holds. But also A & B ^ A , A s . A v B , etc., Hence all principles required to derive the hard Levis paradox in rule form are available, and A & ~A-*B results. Since too A =>. B =>. A & B, paraconsistency is violated. A similar criticism can be made of Nelson 30; for he concedes the principle: A & B-*A. The interactionist strategy for avoiding the problem should be obvious from the intuitive basis for connexivism already sketched: namely A & B —*A requires qualification in much the way, and for just the reasons, that A & B -*• A requires qualification. Thus, for instance, if ~A interacts vith A, so that A & ~A -+• A fails, then presumably ~A interacts with A so that A cannot be derived from A & ~A. Yet surely Angell and McCall are right in thinking that there is a conjunction - in ordinary use - for which all classical tautologies are correct. As this conjunction cannot, given the preceding argument, be connexive conjunction, interaction connexivism has, strictly, to carry two conjunctions, connexive conjunction and (something like) a normal conjunction. And this points the way to a reconciliation of relevant logic and interaction connexivism, with a theory which adds to the underlying common account of deducibility, both normal and connexivist connectives. The question as to how the connexive notions, and in particular connexive negation, are to be clarified formally and semantically is a matter that is best postponed until the larger relevant theory has been elaborated. There are several reasons for this procedure. One is that key connexivist theses such as Aristotle's law cannot be expressed within the logical framework - of first degree entailment - so far elaborated, even when negation is introduced. Negated deducibility statements such as ~(A ff ~A) are not well formed, at least at this stage in the development. (But, as we shall argue, any theory, such as Zinov'ev's in 73, which claims that such expressions are pimply not well formed at all, at any stage, is bound to be inadequate.) The second reason is more important, and more interesting. Whereas the usual views concerning the addition of connectives to the basic theory of implication or deducibility is what can, suggestively, be called separatist (or atomistic), the connexivist view tends to be an interactionist (or holist) one. Connectives interact with one another in a way that defeats natural separation of the role of connectives. The separatist ideal is best exhibited in formulations of systems, such as those of Gentzen, where each connective has its own rules without the appearance of other connectives. It is exhibited to a lesser extent in Hilbert-style formulations of systems where each connective - apart from implication which now assumes a fundamental position - has its own postulates, e.g. separate postulates for &, separate postulates for

2.4 INTERACTION CONNEXIVISM VERSUS THE SEPARATIST

WEAL

separate postulates for v, and for **.1 Connexivism questions, or repudiate this separatism. Connectives interact in a critical way on the connexivist view, partly because the polarity of statements (their positivity and negativity) is written in at the bottom. Thus negation interacts with everything else, with implication as the distinctive connexivist theses show, and in parti cular with conjunction, as the case against Simplification makes evident. The interaction of connectives not only makes the isolation of parts of connexive logic difficult (though so far artificial separation is not excluded): it also makes the determination of the first degree part of connexive logic difficult, and not a particularly natural starting point for the investigation of connexive logic (despite McCallfs proposal in ABE). For these reasons, then we postpone axiomatisation and semantical studies of connexive logic until much later (Part II). The interactionist features of connexivism have one other important outcome; namely that the semantical method (which is the main method of analysis in this text) is not a particularly appropriate method for investigating connexive logic: for the semantical method, at least in its pure forms, presupposes separatism. ^ " Interaction connexivism does not, in a good sense, offer a competing account of deducibility or a competing solution to the paradoxes to that provided by relevant logic. For the interaction connexivist can be seen as one who accepts essentially the same account of entailment as that of relevant logic, but introduces different conjunction and negation connectives, the difference appearing especially in the area where conjunction and negation interact. There need be no conflict unless it is claimed that either theory has captured the One True conjunction or negation of natural language. In our view any such claim by either party would be extremely rash. The accounts are in fact complementary rather than competing, since interaction connexivism has an important insight into the problems of sufficiency and relevance that complements that of relevant logic. Interaction connexivism diagnoses another important way in which loss of sufficiency of premisses can cause irrelevance. According to the theory already sketched in 2.1, principles which allow suppression, together with augmentation principles, may interact to produce irrelevance. The mechanism at the first degree is this. Augmentation princi pies allow the adding of an irrelevant assumption, either as a conjunct to the antecedent or as a disjunct to the consequent. So far sufficiency of antecedent for consequent is maintained. Suppression principles then however allow the dropping of the relevance-carrying part of the antecedent or consequent (depending on whether positive or negative suppression is involved), thus creating irrelevance of antecedent and consequent. Irrelevance is produced because suppression principles enable the destruction of the sufficiency of antecedent for consequent. The solution to irrelevance, then, lies not in rejecting the blameless augmentation principles which maintain sufficiency, but in rejecting the suppression principles which enable the destruction of sufficiency and the dropping of the relevance-carrying parts of antecedents or consequents. 1

Relevant logics do not quite conform to the separatist ideal since, as will be seen, in Hilbert formalisation, v is rather inextricably intertwined with &.

2.4 HOW SUBTRACTION WAV DESTROY SUFFICIENCY AND THE WAY TO SYNTHESIS

The interaction connexivist may appear to be opting for the first and wrong alternative, that of rejecting augmentation principles. But this is not so. What interaction connexivism has seen is that suppression is not the only way to destroy sufficiency. This can be done if we can apparently add to a premiss set a condition which in fact deletes some part of the original e.g. if starting from A ff B we proceed to A, C ff B where C deletes or subtracts a part of A which is necessary for B. This becomes possible when subtraction negation is used, and under these conditions unrestricted augmentation principles will enable the destruction of sufficiency of antecedent for consequent. Under certain conditions the destruction of sufficiency in this way will also produce irrelevance. Augmentation principles maintain sufficiency only provided that what is added is always a further condition which does not interact with and leaves unchanged the original antecedent or consequent, as it does with the conjunction and negation of relevant logic; but where such interaction can occur, as with the conjunction and negation of interaction connexivism, sufficiency can be destroyed by the addition of further conditions, and the interaction connxivist is right to reject augmentation principles. So though at first sight interaction connexivism and relevant logic may appear to be in disagreement, a more careful examination reveals the same insight at the source of each, that it is loss of sufficiency which is damaging, and productive of irrelevance. That connexive connectives are not the only connectives in ordinary use is brought out by ordinary counterexamples to connexive principles. Consider, e.g.., Hunter's counterexample to Strawson's principle. Let A symbolise "he caught the boat" and B "he caught the 2.15". Thus both A and B are contingent (so the example will also serve to counter theses of Stalnaker and Thomason and other proposed logics of conditionals, such as qualified Boethius; A...3. A -*• B = ~(A -*• ~B)). Then according to Hunter (and we agree) one can conceive circumstances in which it would be true both that if he didn't catch the 2.15 he didn't catch the boat, i.e. ~B -»• ~A, and if he did catch the 2.15 he didn't catch the boat, i.e. ~B -> Aj Contraposing, both A -*• B and A ~B, contradicting Strawson's principle. Interaction connexivism and relevant logic have then each discerned basic and important senses of negation and conjunction connectives. Explicitly introducing each sort of connective not only enables the development of a comprehensive logic in which all these basic notions are captured and their differences allowed for and treated, but thus helps avoidance of confusion between them. Failure to distinguish clearly between these different varieties of connectives, especially negation, is responsible for much confusion about which implicational laws hold, both for entailment and more especially in the case of physical sufficiency, causality and counterfactuals (as is argued in chapter 8). Both sorts of theories are essential to any full account of sufficiency, whether logical sufficiency or physical sufficiency. §5. Adding disjunction: distributive lattice logic and conoeptivism. The real issues concerning disjunction turn out to derive from issues concerning the analysis of negation and consistency. These issues are magnified in the case of extensional disjunction which is characteristically connected with conjunction through negation, by the coentailment: A v B •**• ~(~A & ~B). But new issues also enter to complicate matters, in particular 1) the question of constructive proof and truth, which emerges at the first degree sentential level with the introduction of disjunction, and the question 1

The train-then-boat example can be elaborated to remove the charge that one of the ifs involved is really an even if. 95

2.5 NEW ISSUES WITH DISJUNCTION, ANV THE C0NCEPTIV1ST THEME

as to whether A v B should be provable, or constructively true, in cases when neither A nor B is; 2) the question of the distribution of conjunction over disjunction - in such principles as A & (B v C) (A & B) v C - which orthologics and quantum logics reject; and 3) the question of the correctness of Addition (A -»• A v B, B A v B) , in the light of the criticism, especially by Parry, of this principle. The first of these new issues can be postponed (it will reappear with negation) since relevant logics coincide with constructive-intuitionistic logics on positive matters, at least at the first degree. Indeed distributive lattice theory, which we will investigate first, supplies just the sorts of rules for connectives & and v that positive logics ought, according to constructive insights, to have. The second of the new issues also involves negation - because the correctness of distribution is an issue as to the correctness of A & ~(~B & ~C) -> ~ (~(A & B) & ~C) - and we will postpone discussion of it until we have introduced negation. But, unlike constructivism, it provides a direct challenge to the account of disjunction we develop. For according to the proponents of quantum logics (that repudiate classical logic) distributive lattice logic is empirically incorrect. Such a logic no more applies to the quantum world than Euclidean geometry applies to the relativistic universe (see, e.g., Putnam 74). That distributive lattice logic is empirically incorrect is a matter we dispute (see UL, §13); and whether it would matter if it were so incorrect is something we should be prepared to take up if we were to lose the previous round (the issue is dealt with in Lewis 29). The third issue, as to the correctness of Addition, we will take up at once. Much of what we have argued against standard connexivism also applies here. For given a contraposable negation, which ties conjunction and disjunction, repudiation of Addition implies repudiation of Simplification. Parry, in his theory of analytic implication, breaks the link with connexivism by abandoning Contraposition, even in rule form, and so is able to retain Simplification while abandoning Addition. According to Parry, it is Addition that is at fault in the hard Lewis argument, there is nothing at all wrong with Disjunctive Syllogism. A similar position appears in various subsequent writers on implication, some of them unaware of Parry's pioneering work. Parry's position - for which we have coined the ugly term conceptivism is that no implication A -»• B is correct where B contains concepts which do not occur in A. Plainly this makes A -*• A v B incorrect since B may well, in an obvious sense, "contain concepts" not in A. Similarly Rule Contraposition is vitiated since even if A -*• B is provable, so that the concepts of B are indeed contained in those of A, the concepts of ~A will not in general be contained in those of ~B; that is, ~B ~A will be incorrect.1 Rule Antilogism is similarly rejected, and so presumably much of traditional logic theory that connexivism retains. 1

A two-way conceptivism, according to which the concepts of A must coincide with those of B, would restore Rule Contraposition and would render Simplification incorrect. Thus double conceptivism would lead to (weak) subsystems of connexive logic, and again to an equivalential-like "implication". But quite apart from having little to recommend it - double conceptivism would do nothing to motivate characteristic connexivist theses (like ~(A -*• ~A)) unless it also invalidated such theses as A -*• ~~A. Alternative concept and variable sharing requirements will be examined below.

96

2.5 CRITICAL APPRAISAL OF PARRY'S ARGUMENTS FOR CONCEPTIVISM

What is the appeal of conceptivism? Why have some logicians been tempted to say that when A entails B it is not enough that A and B are connected in meaning, that moreover B should be contained conceptwise in A, though A may outstrip B as to concepts contained? Parry's argument against the principle of Addition indicates part of the answer. The argument, presented in his thesis (31, p.119) and reproduced in recent revival work (68, 74, 76), is as follows:- To determine the correct principles of deducibility we should consider concretely a formal deductive system such as Euclidean geometry in a standard formulation. Then one f inds... that a system might contain the proposition "Two points determine a straight line", and yet not contain the proposition "Either two points determine a straight line or some angels have red wings". In fact, a mathematician would rightly consider it, not only ridiculous, but utterly erroneous, to infer the latter proposition from the former (74, p.5; the latter paragraph also appears in 31, p.119). But this argument, were it to succeed, would count equally against a number of principles, such as A i B + A and (A •*• B)&(B -*• C) -*•. A C, that Parry is concerned to retain, as against Addition. For if the system does not contain the proposition "Either two points determine a straight line or some angels have red wings" then it will not (unless extraordinarily formulated) contain such propositions as "Both two points determine a straight line and some angels have red wings" either. And presumably a mathematician's reaction (for what it is worth) to the use of the conjunction in Euclidean geometry would be similar to his reaction to the inferential use of Parry's disjunction. In short, if extra systematic propositions are to be excluded, they should be excluded systematically, in antecedents as well as consequents. But, of course, there is no reason for such exclusions from the point of view of deducibility. For deducibility is not and has never been system confined in the way Parry envisages. Even if it would be strange in a given context to assert A & B because B concerns notions remote from the topic of discourse, the validity of A & B + A is not impugned. The same considerations undercut Parry's other slightly different way of putting his case (68, p.151): The starting point [for analytic implication] is the view expressed in class by H.M. Sheffer: that the conjunction pq really implies p, but that not always does p imply p or q. What is wrong.with the latter? If a system contains the assertion that two points determine a straight line, does the theorem necessarily follow that either two points determine a straight line or the moon is made of green cheese? No, for the system may contain no terms from which 'moon', etc., can be defined. Again, if this is the reason that p -*• p v q fails, that q introduces new terms, then p & q p also fails. But one can see what Parry is getting at: one may want a systemic implication, such that for a system with a given stock C of terms one never derives a theorem containing a term not in C. And the rule A-»A v B would enable this, but A & B d o e s not, since the

97

2.5 REPUDIATION Of THE CONCEPTIVIST ARGUMENT AGAINST ADDITION

confinement of the terms of B to C must have already been established in the premiss A & B.1 A somewhat similar systemic restriction is often argued for in the logic of belief - that though x believes A he may not believe A v B since B introduces new concepts which x does not have. (This no-new-concept position, the initial appeal of which begins to evaporate as the logic of the position is examined, is discussed and criticised in R. and V. Routley 75.) But the systemic restriction so envisaged really applies to rules, not to implications where new unproved hypotheses can always be introduced, e.g. through Simplification. Moreover there is no need to repudiate Addition in order to design a systemic logic. Such a logic can be obtained more satisfactorily either through the formation rules of the system or by appropriate use of restricted variables. Consider the case of a systemic geometry: such constant subjects and predicates as 'the moon' and 'is made of green cheese' simply would not appear, any more than 'is happy' appears in Peano arithmetic. Thus 'the moon is made of green cheese' would not be a wellformed expression of such a systemic geometry and the implication Parry objects to could not be formulated in the system though Addition is maintained.2 Conceptivism, whatever its inital appeal then in the logical analysis of more highly intensional notions such as belief and propositional identity does not impugn Addition as an entailment principle relating statements. For deducibility, as modelled though sufficiency or inclusion of logical content, does not require concept preservation. Parry has claimed (in reply to criticism from Anderson and Belnap, criticism reproduced in ABE, p.432) that containment does require concept preservation, that A cannot contain A v B where B is totally irrelevant to A. On the familiar, and intuitive, account of containment in terms content inclusion, Parry's claim is mistaken. For the content of A v B, written c(A v B), just is c(A) ^ c(B) which is included in c(A), i.e. c(A v B) £ c(A).3 Moreover a cursory examination of paradigmatic entailments, such as "This is red entails that this is coloured" and "Ralph is a brother entails that Ralph is a sibling", would seem to confirm the point, that concept preservation is not required. But Parry wants to separate off such counterexamples to his position through a distinction of logical and structural entailments. Structural entailments lead back however to logical ones. Consider such examples as "Blue Blaze won the 4.30 entails that Blue Blaze was a place-getter" (said, e.g., in explaining to some new chum that he could collect on his bet); this is based on, and expands to, the entailment: Blue Blaze came in first in the 4.30 entails that Blue Blaze came in first or came in second or came in third. That is, the structural entailment is supported by a logical entailment - and surely the antecedent is sufficient for the disjunctive consequent - which conceptivism has to repudiate. Similarly in significance logic, A & B -pA is admissible because A & B, when proved, is significant; but A v B may well be non-significant when A is significant, so A -P A v B is not admissible. Significance issues can however be postponed by restricting consideration to sentences that yield statements: for the criticisms of Parry and others have not been based for the next part, on the significance difficulties A -*• A v B can cause. The question of satisfactory entailment logics with significance is very much an open one: for some of the problems see GR and Woodruff 74. Less charitably, Parry can be seen as forced into criticising Addition by the desire to remove the paradoxes but retain Disjunctive Syllogism. In view of his rejection of Antilogism, in even rule form, his case in favour of Disjunctive Syllogism is decidedly weak (the traditional case that Lewis relies on, for example, is sacrificed). ^ _ 3 ^ . fc (footnote on next page)

U

2.5 EMBARRASING ARBITRARINESS IN CONCEPTIVIST CONTAINMENT REQUIREMENTS

Conceptivism itself, as presented by Parry and others, satisfies only a limited containment of concepts requirement. For the systems proposed include such principles as A -*• ~~A and A -*(A & A) which introduce new concepts in the consequent, negation and conjunction respectively, which may not be contained in the antecedent, A liberal construal of terms or concepts would rule out many of Parry's axioms for analytic implication, e.g. A -*• — A , (A C) & (B -*• C) -»-, (A v B) C, A B ->-. ~A v B, A -»-. A A, each of which may introduce new logical notions in the consequent. Such a liberal construal which took conceptivism seriously and counted all concepts, would indeed cripple logic as an explicative device. Conceptivism depends then, for any plausibility, on a narrow, and rather arbitrary, construal of concepts: only those locked up in variables are allowed to count. This is not just unsatisfactory: it also eliminates conceptivism as a plausible way of formalising the no-new-concept position on belief. For the position rejects such implications as x believes p x believes ~~~~~~p on the ground that x may not have or (like Griss) not acknowledge the concept of negation. The arbitrariness of the conceptivist construal becomes embarrassingly obvious when the question is raised as to what conceptivist quantification logic looks like. Is universal instantiation, (x)A(x) -»• A(t), to be rejected or qualified because t is a new term not in the antecedent, or is it to be admitted because the antecedent is really an infinite "conjunction" and so contains A(t) as a conjunct or because universal quantification says something about everything and so about t? If we say the latter we are on the way to the claim (contemplated by Prior and others) that every statement, and certainly every inconsistent statement, says something about everything, in which case A -»• A v B, should be readmitted, certainly when A is inconsistent. Qualifying instantiation is perhaps the more appealing alternative, and a qualification is suggested by the way Addition and other principles get patched up in Parry's system,2 namely by tacking on newly appearing variables in antecedents, Instantiation would correspondingly be dressed up as (x)A(x) & B(t) A(t>, or given a predicate T (read, say 'is a term') in free logic style as (x)A(x) & T(t) -* A(t). The resemblance to free logic is however superficial: for free logic correctly admits Contraposition and also incorrectly Antilogism, and thereby arrives at a qualified form of existential generalisation, of the form A(t) & T(t) •*• (3x)A(x) . But nothing stops conceptivism from accepting unqualified generalisation - for A(t) -*• (3x)A(x) is like A — A isn't it, both just introducing new logical operators - unless of course (3x)A(x) is construed as an implicit disjunction, in which case existential generalisation will po the way of Addition. But if generalisation is at fault, can't it be (footnote 3 from previous page) Since A ->- B is true iff c(B) £ c(A), A -»-. A v B is true. The content analysis provides one direct defence of Addition. Other intuitive modellings for entailment can likewise be put to work to vindicate Addition. 1

Parry is aware of this, and is tempted by a stronger containment of concepts requirement according to which 'entailment should be treated like empirical concepts, which may not appear in the consequent unless they (or empirical concepts from which they may be derived) are in the antecedent'. The principle of counterexample, A & ~B ~(A B) illustrates the point. 'From my point of view it is perhaps objectionable as introducing the concept of entailment without antecedent, and perhaps I should have this rather in such a form as p & q & +(r s) -*• -(p ->- q) ' (Parry 76, p.25).

2

Parry's Proscriptive Principle, that there is no free variable in the consequent of a (correct) entailment that is not also in the antecedent, requires qualification of Instantiation, and rejection of the picture of a universal quantifier as an infinite conjunction.

99

2.5 THE RELEVANCE

QUALIFIED

PARADOXES

Of PARRY

SYSTEMS

repaired as follows1:A(t) & (3x)B(x) -*• (3x)A(x)? Similar arbitrariness appears also in such questions as to whether change of bound variable is to be admitted or qualified, as to the connections of free variables with bound variables, and so on. In application to formal logic concept preservation is reduced - none too satisfactorily as we have seen - to variable inclusion. According to Parry's so-called "Proscriptive Principle" 'no formula with analytic implication as main relation holds universally if it has a free variable occurring in the consequent but not the antecedent'. The Proscriptive Principle is an attempt to reduce to syntactical form an essentially semantical matter, interrelation of concepts, and suffers from most of the difficulties of such attempts (e.g. that concepts are presented in various linguistic guises, that for interrelations of concepts such as inclusion a notion of entailment is presupposed, so that there is an endemic circularity in the proposal that what entails what should depend on what concepts are included). Systems which conform to the Proscriptive Principle we call Parry systems. The sentential conceptivist, or analytic implicational, systems that have actually been presented - those of Parry, Dunn 72, Urquhart, and Fine 74b - all conform to the principle, as a matrix argument will show (see below)> but they all leave a great deal to be desired. Parry's system, which is included in all these other systems has as an axiom the near paradox A v (B & ~B) -*• A, which permits the suppression of impossible disjuncts. It is only because the addition principle B & ~B A v (B & ~B) is excluded that the systems escape outright paradox. Parry's system also has as an axiom a generalised form of Mingle, namely the scheme .. .A.. .-*•. A -*• A, i.e. any wff containing A entails the law of identity in A. This is but another relevance qualified paradox. This feature is even more blatant in the next weakest of these systems, that of Fine (which Parry now seems inclined to accept), which has a theorem scheme (A & ~A) & ~B -*• B, i.e. the negation of B guarantees the connection. More generally, paradoxes are in order provided variable containment is ensured, e.g. where V(A) represents the variables of A, if V(A) £ V(B) then A - o B -*• A and ~B—frB -*• A. Fine's system also contains a range of other implausible theorems not directly tied to the Lewis paradoxes> e.g. DA & (A -*• B) ~A •*• ~B, in effect a fallacy of conversion. For a counterexample, let A represent 'Some objects have a definite shape' and B 'Some objects have size' (and in case the necessity of A is in doubt construe 'object' in Meinong's fashion). Since definite shape logically guarantees size the.antecedent is true, but that no objects have a definite shape does not entail that no objects have size, since, for example, objects without a definite shape can have a large size. In Dunn's demodalisation of the Fine system (obtained effectively by adding the scheme A -»• OA) , the patently false conversion principle A & (A -»• B) ~A -»• ~B of course results, and a great many other oddities also follow. The semantical analysis of Fine's and Dunn's systems reveals that they are effectively strict and material systems, with a variable or contents inclusion requirement thrown on top. The oddities emerging help to show that the trouble with strict and material systems, is not merely, but only incidentally, their variable-sharing failure. The real troubles go deeper and are not repaired simply by throwing on a variable-inclusion filter - any more than by just ruling out paradoxical cases. These fallacies, near paradoxes, and oddities are not, however, inevitable features of conceptivist systems. It is not too difficult to design other Parry logics which escape these features by amending the axiomatisation of Parry's system (see Part II). But some peculiarities remain, e.g. prefixing principles are in order but suffixing ones are not. For example, l And correspondingly A

— A as:

A & ~C -»• ~~A.

100

2.5 PARRV LOGICS ALSO FAIL PARACONSISTENCY

REQUIREMENTS

while C -*- A & B ->•. C -»• B is derivable in Parry's system its suffixing mate, B -»• C -»•. A & B + C is not. Yet these principles should stand or fall together. (What we have is a further asymmetry induced through the unwarranted failure of Contraposition.) Moreover the general objections made to conceptivism apply equally to "relevancised" Parry systems. So too does the telling objection made by Kielkopf 75 to Parry's system, that it has as a derived rule the spread principle A, ~A-frB. The argument, a variation of the hard Lewis argument, is as follows:A & ~A -*(B v ~B) & (A & ~A),

adjoining the theorem B v ~B*

—p(A & ~A) & ~B v (A & ~A) & B, -*A v B

distributing

, by Simplification and v-Composition.

That is, rulewise the effect of Addition can be obtained indirectly. as A & ~A -»• ~A, by Rule Composition, A & ~A

Hence,

~A & (A v B) -»B;

by Disjunctive Syllogism.

Thus Parry logics fail to meet the paraconsistency condition of adequacy (cf.1.7). The matter is not rectified by abandoning - what is at fault Disjunctive Syllogism. For it is a central tenet of conceptivism that it is Addition that is the faulty move in the hard Lewis argument, and that Disjunctive Syllogism, which meets the variable inclusion requirement, is perfectly in order. Were Disjunctive Syllogism rejected, then - as Antilogism is rejected by conceptivism because it, like Contraposition, fails on variable inclusion - conceptivism would be able to resolve the paradoxes in the relevance fashion and there would be no need to reject Addition and Contraposition. To sum up. The conceptivist objections to Addition - a principle which is validated by underlying informal accounts of entailment - are, when carefully examined, unimpressive, and if they succeeded they would similarly condemn other quite acceptable principles. Moreover the conceptivist objections do not rest on a solid base, but on a narrow and arbitrary assumption as to what counts as a concept or term. Finally, conceptivist accounts fail on important requirements of adequacy, in particular on the paraconsistency test. For all these reasons we reject conceptivism as offering a viable account of entailment.2 But before we turn back to logics not yet out of the running in the entailment and implication stakes, let us glance at other proposals which, like Dunn's demodalisation of Parry's system, essentially characterise entailment by the addition of variable sharing requirements to classical logic (or, equivalently at the first degree stage, modal logic). The proposals take the form: A |- B (or A B) iff A 3 B is a classical tautology and A and B satisfy the proposed variable sharing requirement. Where A and B are zero-degree wff, i.e. contain no occurrences of intensional connectives but only of connectives &, v and there are several possibilities for the interrelations of their variables that 1

The corresponding, and fairly harmless positive .paradoxes - A B v ~B and A C where C is a thesis - are immediate from the classical character of the rule theory. Since Contraposition is not correct for rule theory the positive rule paradoxes do not directly yield the damaging negative paradoxes. The design of non-classical rule theories will be taken up again in chapter 15. 2 Subsequently we shall present however, semantical analysis of certain conceptivist systems. But such analyses do not of course, without much further ado, show the satisfactoriness of an account for a given philosophical purpose.

101

2.5 VARIABLE

INCLUSION,

ZINOV'EV'S

SYSTEM,

A W PARRY PROBLEMS

REPEATEV

have been thought to merit investigation, as follows on the classical provability of A => B, namely, where V(A) represents the class of variables of.A; i) V(B) } C V(A), ii) V(B) = V(A), and iii) V(A) n V(B) 4 { }. The system which emerges from the first proposal, which we call ZV after Zinov'ev, is axiomatised in Zinov'ev 73 (in systems Sj and S 1 : the adequacy of the axiomatisation is proved, in the same text, by Smirnov1). A B is a theorem of ZV iff A B is a classical tautology and V(B) c V(A). ZV is of course decidable, variable relevant, and almost as simple-minded as classical logic. It is a first degree implicational subsystem of all the Parry systems mentioned, since it is a sublogic of the weakest of these, Parry's original system, as derivation of ZV (as formulated in 73, p.79) in Parry's system shows. Indeed ZV is the first degree implicational part of certain of the Parry systems, e.g. of Dunn's system (as the semantics of Dunn 72 will show). Furthermore the following Parry matrices, used to show that the Proscriptive Property holds for Parry Systems, are characteristic for ZV:

*1 *2 3 4

1

2

3

4

1 2 1 2

4 2 4 2

3 4 1 2

4 4 4 2

3 4 1 2

&

1

2

3

4

1 2 3 4

1 2 3 4

2 2 4 4

3 4 3 4

4 4 4 4

That is to say, A (- B is a theorem of ZV iff A j- B takes either value 1 or value 2 for each assignment of values from {1, 2, 3, 4} to its initial obs. (This will be demonstrated, and the Parry matrices analysed subsequently, when normal negation has been introduced, and matrices for normal systems semantically analysed.) The above Parry matrices also satisfy the other Parry systems introduced, that is to say each axiom takes designated values and the rules preserve designated values. Accordingly the Parry matrices yield a simple proof that these systems are indeed Parry systems. For the following assignments show that any wff A (- B (or A -»• B) in which B contains some obs (or variables) not in A takes an undesignated value:- assign to each ob in B but not in A value 4 and to all other obs value 3. Then, by induction, A takes the value 1 or 3 and B takes the value 2 or 4, and hence A + B has value 4; so A ->• B is not provable. The first degree implication system ZV is unsatisfactory in much the way that other Parry logics are: it contains relevancised paradoxes e.g. (A & ~A) &}6(B)

B and C&}S(B) B v ^B, where 6(B) is any extensional funcv v tion of B, and A & ~A &}6(p i ,. . . ,p n ) -»• B and C&}6(p 1 ,.. . ,p^) B v ~B, where 6(p,,...,p ) is any extentional function of p,,...,p and all the variables l n i n of B are included in p ,...,p ; it excludes, without - so far as we can see substantial basis, the principles of Rule Contraposition and Additon which are correct for a sufficiency implication and for such associated notions as inclusion of logical content; and it fails requirements of adequacy, such as the paraconsistency requirement. How is it then that Zinov'ev manages to *It should be a straightforward matter to find less cumbrous axiomatisations than those of Zinov'ev. ^Since this chapter was drafted the system satisfying the third proposal has achieved some celebrity - though it is none the better for that - as the leading example of a relatedness logic (see especially Philosophical Studies 36, 1979). Other relatedness logics vary the implication A ^ B to which the relatedness (or relevance) requirement R(A,B) - generalising iii) - is applied, the connectives to which it is applied, and the conditions on R in addition to reflexivity and usually symmetry.

102

2.5 THE INADEQUACY

OF ZINOV'EV'S

ACCOUNTS

OF LOGICAL ENTAILMENT

AND OF LOGIC

arrive at the conclusion that with system ZV 'the problem of logical entailment can be considered solved in principle' (pp.85-61)? In a thoroughly unsatisfactory and question-begging way is the short answer. The reason Zinov'ev offers is this (p.86): 'there are no other apriori criteria of meaningful sentential relations, which would be as developed and defined as relations of sets of variables occurring in premisses and conclusions'. This already assumes that the correct account of entailment is to be obtained by imposing a (relevance) sieve on classical logic - a seriously mistaken assumption (as we have seen). Even the sieve Zinov'ev mentions is far from uniquely determined, since it would be required that the variable sets concerned are related in other ways, e.g. they are identical or they merely overlap. And many other criteria, both syntactical and semantical, can be just as well developed and just as sharply defined. That is, such criteria do not delineate ZV. Nor does Zinov'ev's other procedure, that of imposing - in a quite unsupported way - conditions of adequacy, uniquely determine ZV. According to Zinov'ev an axiomatisation of the general theory of logical entailment ought to satisfy the conditions: 1) in axiomatic construction all the above assertions should be provable; the construction ought to correspond to the intuitive understanding of logical entailment; 2) in axiomatic construction not just any assertions should be provable (outside of those introduced above) but only those which satisfy the requirement I). For one thing intuitive understandings differ. On ordinary understanding at least, truth functions of entailment statements make sense; so a correct theory should be at least first degree. But on Zinov'ev's theory even (A |- B) v ~(A |- B) is ill-formed, and in fact meaningless. Furthermore if higher degree statements are admitted - as they should be, again on intuitive grounds - then even given variable inclusion there is a wide variety of Parry logics to choose among, and many proposed theses about which intuitions differ. Actually Zinov'ev has the audacity to write down, without discussion, as intuitively acceptable, principles such as Disjunctive Syllogism and Conjunctive Simplification about which there has been considerable discussion and which are far from intuitively acceptable to some thinkers. On the other side implicational paradoxes are now intuitively acceptable to many, so the conditions of adequacy do not strictly rule out material and strict implication, or at least their common first degree implicational part, as providing axiomatisations. The idea of a unique solution, such as that offered by ZV, to the problem of entailment is, moreover, incompatible with Zinov'ev's general relativism about logic (see especially 73, chapter 21). At the outset of his discussion of the problem of logical entailment (p.69 ff) he rejects the 'prejudice that there is some single, unchanging, "natural", "basic", etc. ["intuitive"?], logical entailment and all logic has to do is to find the most accurate and complete (adequate) description of it'. ... there is no single, perfect, "natural", etc. logical entailment which simply has not been adequately described up to now. ... Logic has in fact come to recognise different forms of logical entailment. No one of them can be considered any more "basic" than another. In a certain sense they are 1

See also (on p.86) such overstatements as 'the solution of the problem is basically achieved'. It is amusing to note that 'the systems of Ackermann, Anderson, Belnap and others' are said to be 'in a sense, "improper" ("deformed" etc.) systems', perhaps even "revisionist". 103

2.5 SIMILAR OBJECTIONS

APPLY AGAINST HINTIKKA'S

ACCOUNT ANV RELATIONAL

LOGICS

all equal. The problem of the most adequate description of logical entailment is no longer a matter of finding some logical system as the final and unique theory of logical enailment but of constructing different types of logical systems ...1 Ergo, it is not a matter of finding ZV and its variants. simply incoherent.

Zinov'ev's theory is

Among the alternatives to variable inclusion that have been looked at in the accumulating literature on implication are variable overlap and variable identity. Kielkopf has studied and discarded the proposal that implication requires as well as classical provability variable sharing, i.e. A -+ B iff A ^ B is a tautology and V(A) n v(B) ^ { }. The "implication" defined fails transitivity, which is enough to discredit it, and also replacement of coimplications; and it lets through such relevance qualified paradoxes as (A & ~A) & C -»•. B & ~C.2 But, unlike both Zinov'ev's and Hintikka's accounts, it does admit Contraposition and Addition. Hintikka has defined an implication which, at the sentential level, is as follows: A B iff A = B is a classical tautology and V(A) « V(B); and A -> B iff A •** A & B. Thus Hintikka's theory is but an earlier formulation of Zinov'ev's. For A -*• B is a Hintikka thesis iff A = B & B is a tautology and V(A) = V(A & B), i.e. iff A => B is a tautology and V(B) c V(A), i.e. iff A B is a theorem of ZV. Thus the objections already lodged against ZV apply to Hintikka's account. So concludes our detour into Parry logics: we return at last to logics not yet out of the running in the implication stakes. DLL (Distributive Lattice Logic) is set within a simplified ML: f- in prefix form is omitted (since no formal objects of DLL are theses) and F , A , etc. are (inessentially) restricted to finite sets of formal objects. DLL is built up from the following components (vocabulary from one point of view): Initial formal objects (there is no need to specify what these are). Operations and auxiliaries: &, v, (,); Relation: f- . Formal objects (obs) are defined as follows: 1. Initial formal objects are formal objects. 2. If A and B are formal objects so are (A & B) and (A v B). Statements of DLL are defined thus: If r is a finite (non-null) set of formal objects and C is a formal object then r b B is a statement (wff). The postulates of DLL are these: Axioms: A A Rules: A A r r, J

A (- A & B f- A A & B (• B (- A v B B (- A v B (- B; B, F |- C -h> A , r | - C (- B; A c r r(-B I- B; r I- C r|-B&C A |- C; r , B|- C - » r , A V B

(• C

(Identity) (Simplification) (Addition) (Transitivity, Cut) (Augmentation) (&-Composition) (v-Composition)

The question of the extent to which entailment can be uniquely characterised will concers us much in what follows. The important issue of logical relativism, versus substitutionism, is taken up in detail in chapter 15.

Relational logics - which continue the nontransitivity tradition (discussed above, p.74ff.) - not only wrongly validate basic substitutivity principles, but also validate several principles already or subsequently rejected, e.g. A -»-. A ->• B -»• B and A -*• (A •*• B) -*•. A B (and also A -*• ~A -»• ~A) , and hence fail further important conditions of adequacy for implication in its central roles. Some of these points, especially paraconsistency failure, are elaborated in the disucssion of relational logics in PL.

104

2.5 FEATURES OF SYSTEM DLL OF VISTRIBUTIVE

LATTICE

LOGIC

Logical versions of all the results of distributive lattice theory now follow, but a few sample results suffice here since simple semantical arguments can be used once the adequacy theorem is proved to verify theorems and derived rules. T1 (Adjunction). Pf. B, C |- B

A, B [• A & B

B, C |- C

B, C |- B & C

, by Identity and Augmentation. , by &-Composition.

Unlike the axioms this thesis, which could replace the rule of &-Composition, has multiple antecedents. DR1.

A | - C, B |- C - » A v B I- C

Pf.

A f- C, B

C-frA v B, A |- C, A v B, B (• C,

by Augmentation (invoking an obvious enlargement of the rule notation) —P A v B, A v B (- C , by v-Composition — • A v B |- C

, by set properties.

T2 (&-Commutation) A & B f- B & A Pf. By Simplification and fit-Composition, T3

(v-Commutation). A v B f- B v A

DR2. A, B | - C - * A & B | - C Pf.

A, B |- C, A & B |- A - * A & B, B A & B, B

C, A & B

C,

by Transitivity

B - * A & B, A & B (- C, by Transitivity.

Hence the result using set properties and rule features. T4

(Distribution).

Pf.

A & (B v C) f- (A & B) v C.

A, B |- (A & B) v C, by Tl, Addition, Transitivity.

Since C (- (A & B) v C, by Addition, T3, Transitivity, A, C (- (A & B) v C, by Augmentation. Hence A, B v C (A & B) v c , by v-Composition. Distribution then follows by DR2. As is well-known from lattice theory, multiple antecedents in v-Composition are essential for this proof. It is tempting to suggest that it is only defective formulations of earlier entailment logics that have made distribution an independent and apparently questionable axiom. Orthogonal lattice logic, OLL, discussed below results simply by limiting v-Composition to: A |- C, B

C-»A V B

|- C.

But sufficiency and its offshoot, relevance, demand no such restriction. Weak Relevance Lemma 2.2. If F j- B, then some ob is a component both of B and of some element of T. Proof cannot quite be by induction over proofs in DLL. For though it is a matter of inspection that each axiom satisfies the requirement, and is readily shown that each rule, except Cut, preserves the property, the inductive argument breaks down (in a very characteristic way) on Cut unless Cut is reformulated in a way (such as putting the desired premiss B into the premisses of the conclusion) that no longer cuts out a connecting ob (namely B). But such a reformulation would appear to weaken the scope of the logic:1 unrestricted 1

The argument set out below shows that the restriction would not in fact reduce the class of theorems. Accordingly a more satisfactory, relevant, (footnote continued on the next page)

105

2.5 RELEVANCE

SV THE WAJSBERG MATRICES,

WHICH ARE CHARACTERISTIC

FOR VLL

transitivity is, to repeat this important point, essential to a good entailment and for applications of the logic. The failed argument suggests reformulating DLL in a Gentzen fashion and proving an appropriate Cut-elimination theorem (cf. Kleene 52, p.422 ff.). Such a strategy will indeed succeed (as we shortly indicate). However a simple matrix argument will show that Cut does indeed preserve relevance; and since the argument does all the other cases of our proposed induction for us, we can forget the induction and begin again. We show, by a different induction, that every theorem of DLL takes only designated values for every assignment to initial obs under the following Wajsberg matrices (these matrices are semantically derived, and thereby explained, in 2.6):

h *1

1

4

1 | 1

2

1

1

3 |

3

1

4

1 J ~~3

3

4

1

1

2 I

4

1 1

2

3

The matrices are just the Lewis and Langford Group III matrices (32, p.493), but 1 is taken as sole designated value. To evaluate wff of DLL we construe commas on the left of f- as conjunctions (and commas on the right of }- in multiple formulation as disjunctions), i.e. Ai,...,An |- B is evaluated as Aj&.-.&A^ |- B.

Then, by induction over proofs of DLL,

(1) Every theorem of DLL is satisfied by the matrices. (2) Where T and B share no ob, T |- B takes an undesignated value. To establish (2), assign every initial ob occurring in B one of the values 3 or 4 and every initial ob occurring in an element of T one of the values 1 or 2. Such an assignment is possible because T and B share no ob. A simple inductive argument (confirming inspection of matrix blocks) shows that B takes one of the values 3 and 4 and that T takes one of the values 1 and 2, whence it follows that T B takes value 4. Finally, the theorem follows from (1) and (2). The Wajsberg matrices not merely satisfy DLL; they are characteristic for DLL, i.e. a wff of DLL is a theorem iff it takes a designated value (i.e. value 1) for every assignment of values to initial obs. This will follow from the semantics for DLL, to which we turn after one further preliminary. DR3.

Bi & A bCi, B 2 h c 2 v A

?f.

Bi & A f-Ci, B 2 h C 2 v A

•Bi & B 2 h c i

v C

J

B 2 ) & A |-

B

R 1- A,

B

e

U© r (- e

1

>

r

h

6

Because sets are used in the formulation rather than sequences, structural rules other than Augmentation can be dispensed with. Also, as with Gentzen's systems, it can be shown that Cut is eliminable, i.e. a Cut elimination theorem is provable by the usual double induction. (Note that elimination does not follow directly from the semantics, since Cut is needed in proving DR3.) With Cut eliminated weak relevance of DLLM does follow by a simple inductive argument (the formalisation of inspection), since DLLM and GDLLM are equivalent systems in the sense of having the same theorems. §6'. Negation: De Morgan lattice logic and the parting of classical and relevant ways. Thus far classical, modal and relevant positions on deducibility ride along side by side. Other positions such as conceptivism and standard ccmnexivism have departed on their separate ways, but not intuitionist logic and not relevant logic. For DLL i^s a relevant logic. Not so Boolean logic that classicists would have us buy - and that includes all the motley band of modal logicians, for their differences from the classical position and among themselves only emerge when the first degree, involving truth-functions of deducibility, and the higher degree, involving iterations of deducibility, are reached,1 if they are. BL (Boolean Logic) results from DLL by the addition of a one place operation, subject to the following postulates:Axioms: A |- — A ; — A j- A (Double Negation) Rule; r , A f - B - * T , ~B (Rule Antilogism) It is easy to see that BL violates relevance: A, B {- A by, e.g. Tl, T2, Identity; A, ~A (- ~B by Rule Antilogism; and it is easy to trace the source of this violation: Rule Antilogism. The negation axioms do not interfere with the extension of the relevance lemma to encompass negation, but Antilogism does. For suppose, to extend the pseudo-inductive argument for DLL, T, A (- B has the sharing property. It does not follow that T, ~B |- ~A preserves it, for the sharing may be done by B and T with A irrelevant to B and T. Then ~A will be irrelevant to T u {~B}. The argument also reveals that it is not difficult to restore relevance.2 Delete parameter T; that is, restrict Rule Antilogism to: A |- B ~B (- ~A (Rule Contraposition) . Then if A and B share an ob, so do ~B and ~A, so relevance is preserved. DML (De Morgan Lattice Logic) results simply by replacing the classical rule of Antilogism by the relevance-preserving rule of Contraposition, i.e. Truth functional combinations of deducibility statements, such as the erroneous principle (A (| B) v (B [| A), are enough to separate modal (and more generally intensional) positions from the classical one. But the separation of the main modal positions only occurs with the introduction of higher degree statements. Relevance may be restored in more than one way. A different way from that adopted in the text - a way with some appeal - is the following: T, A |- B —* T, ~B |- ~A, provided A and B share a component.

110

2.6 SEMANTICS

FOR VE MORGAN LATTICE LOGIC VML: THE STAR

OPERATION

DML adds to DLL just Double Negation and Rule Contraposition. For ease in logical investigation of DLM, it pays to expose a little of its De Morgan character. T5. ~(A & B) ~A v ~B Pf.

~(~A v ~B)

—A,

by Addition, Rule Contraposition

~(~A v ~B)

A,

using Double Negation, Transitivity.

Similarly ~(~A v ~B) (- B Hence

~(~A v ~B) j- A & B,

by & - Composition.

The result then follows by Contraposition, Double Negation and Transitivity. T6.

~A & ~B (- ~(A v B).

Double Negation

Pf.

A j- ~(~A & ~B),

by Simplification, Contraposition, Double Negation.

Similarly B |- ~(~A & ~B) Hence A v B (- ~(~A & ~B), by v - Composition. The result then follows by Contraposition, etc. The converses of T5 and T6 of course hold also, as semantical adequacy will quickly show. Semantical analysis of DML, unlike that of BL, requires enrichment of ML models, in order that a correct negation rule can be semantically supplied. A DML model M is a structure M • where K and I are as before, i.e. respectively a set of worlds and an interpretation function, and * is an involutary operation on K, i.e. a* e K where and only where a e K, and a** • a. Though it is quite unnecessary to furnish further interpretational details for *, it can in fact be modelled in various ways, e.g. as a reversal operation, topologically as an inside-out operation, and so forth..1 To the rules extending I from initial obs to all obs, the following normal negation rule is added: I(~A, a) = 1 iff I(A, a*) * 1. A f- A is DML valid iff A |- A is true in every DML model, i.e. for every DML model M = and for every a e K, if I(B, a) = 1 for every B e A then I(A, a) = 1. Adequacy Theorem for DML 2.5. A |- A is a theorem of DML iff A |- A is valid. Proof extends the adequacy theorem for DDL. That the Double Negation axioms do not upset soundness follows from the modelling condition a** = a. ad Rule Contraposition. Suppose, for arbitrary DML model M, A |- B is true in M, i.e., for every a e K, if I(A, a) = 1 then I(B, a) = 1. Then, as * is an operation, for every a e K, if I(B, a*) ^ 1 then I(A, a*) ^ 1, that is, for every a e K, if I(~B, a) = 1 then I(~A, a) = 1, which is to say that ~B f- ~A is true in M. Completeness is also a matter of adding the occasional detail to the proof for DLL. In the canonical DML model M » , I is as before for elements of K , K is given a practical adjustment, and for a e K^, a* = {A: ~A i a}. The practical adjustment to the class K c of prime DML theories is that closure under |- is redefined to allow for the non-parametric form of Contraposition (as opposed to Antilogism). A theory c is closed under j- , in the new sense, iff whenever A e c and A (- B then B e e . (In the presence of Adjunction and Simplification the new sense is equivalent to the old, since compactness holds.) Then a* e K iff a e K c , since a* is a prime theory when and only when a is. For (i) a* is closed under |- . Suppose A e a* and A |- B. Then ~A I a and x

The interpretation and derivation of the star operation are much discussed in subsequent sections: see also NC. 111

2.6 AVEOUACV

FOR 8L; M L AS CONSERVATIVELY

EXTENDING

TAUTOLOGICAL

ENTAILMENT

~B |- ~A. Hence since a is closed under |- , ~B I a, i.e. B e a* as required. (ii) a* is closed under Adjunction, Suppose otherwise A, B e a* but (A & B) i a*. Then ~A I a, ~B i a but ~(A & B) e a. Since a is closed under f- and ~(A & B) (- ~A v ~B, by T5, ~A v ~B e a. But a is prime, so either ~A e a or ~B e b, which is impossible. (iii) a* is prime. Suppose otherwise A v B e a*, A e a*, B e a*. Then ~A e a, ~B e a, but ~(A v B) / a . Since a is closed under Adjunction, ~A & ~B e a, whence as ~A & ~B |—(AvB), by T6, ~(AVB) e a, which is impossible. Now suppose r B. There is, as before, by the Extension Lemma, a prime DML theory a extending T such that B { a. Hence since I(D,a) = 1 iff D e a for every a, I(D,a) = 1 for every D e T but I(B,a) i 1, that is f B is not valid. One new inductive step is required in the reduction of holding at a to membership, namely that for negation: I(~C,a) = 1

iff

I(C,a*) ^ 1, by the rule for ~

iff

C i a*, by induction hypothesis

iff ~C e a, by definition of a* Corollary 1. (Adequacy for Boolean Logic). A j- A is a theorem of BL iff A (- A is true in every DML model M = for which a* = a for a e K; hence the operation * can be eliminated, and the normal negation collapses into the classical one, I(~A,a) « 1 iff I(A,a) t 1 for a e K. Proof. Given a* = a, Antilogism is verified. For completeness observe that the proof of adequacy for DML works just as well if only non-degenerate (n.d.) prime theories are considered in the canonical model, that is prime theories that are neither null nor universal (i.e. contain every ob). Use of n.d. theories together with the following paradoxical BL theorems enables proof that a* = a, and so completes the corollary: T7.

A £. ~A j- B

T8.

B (• A v ~A

Pf.

B j- ~(A & ~A) B

~A v — A

~A v — A

|- A v ~A

,

from Rule Antilogism

,

by T7, Antilogism.

,

by T5, Transitivity.

,

from Double Negation,

v-Composition.

ad a £ a*. Suppose otherwise A e a and A d a * . Then ~A e a, so A & ~A whence, by T7, B e a for every ob B, contradicting the non-universality ad a* £ a. Since a is non-null some ob D e a. Hence, by T8, A v ~A e Thus, by primeness, A e a or ~A e a, i.e. A t a*. Since this holds for ob A, a* c a.

e a, of a. a. every

De Morgan Lattice Logic is nothing but the logic of tautological, i.e. first degree, entailment (of ABE, chapter 3) in another guise. Rewrite B, T |- C as B & G C where G is the conjunction of elements of T, and the logic of tautological entailment is forthcoming. Formation rules for system FDE of first degree entailment are obtained from those for DML by deleting the rule for |- , and substituting the following rule for -»•: If A and B are formal objects then A -»• B is a statement (wff). Alternatively FDE has as wff those wff of the full sentential logic of implication (of chapter 3) which are of the form A -»• B where A and B are zerodegree wff, i.e. contain no occurrences of connective i.e. have as their only connectives the extensional connectives &, v and The postulates of FDE can be chosen as follows: Axioms: A A A&B-+A A & B

+ B& A

A +

A V B

A V B -*• B v A

—A

A 112

2.6 THE INTERCONNECTIONS

Rules:

A

B, B

A

B, A -*• C -f A

D 6. A

C

BETWEEN WORLV MODELLINGS

A

AND MATRICES

ANV

LATTICES

C B & C

C, D & B - » - C - * D & (A v B)

C

A -*• ~ B - * B -*• ~A Equivalence Theorem 2.6. A -> B is a theorem of FDE iff A j- B is a theorem of DML. Hence too the semantical analysis of DML may be transferred to FDE'. In fact the semantical analysis given readily furnishes many of the main results concerning tautological entailments. To illustrate the power of the analysis decidability of DML and the Smiley matrices, which are characteristic for DML, are derived. Corollary 2. A f- A is a theorem of DML iff A f- A is true in every two-world DML model M • , i.e. K - {d,d*} in every case (and worlds are accordingly dispensible). Proof. Soundness is automatic since truth in every model implies truth in every two-world model. For the converse observe that the extension lemma is only applied once in the completeness argument, thereby delivering d which then, together with its image d*, carries the remainder of the argument; that is, a canonical model M with K = {d,d*} suffices for the argument. Corollary 3. DML is decidable.C For corollary 2 has delivered an effectively specified finite model in which each non-theorem can be refuted. The class of interpretations on a two-world model structure , or {d,d*} for short, can be reexpressed as a class of assignments in terms of four-valued matrices. To avoid repetition in later sections, we first set out, in a more general and systematic way than is required merely for system DML, the interconnections between world modellings on the one hand, and matrices and lattices on the other. A model M can be resolved into a pair <S,I> (or <S,v>) where S which comprises every component of M except I is a model structure (m.s.) and I (or v) is an interpretation (or valuation) on m.s. S. In the case of B models (of 4.4) S = with R a 3-place relation, provides the model structure; in the case of I models (of 7) S = , with R a 2 place relation and N (like 0) a subset of K, is the m.s.; in the case of DML models or, as we sometimes prefer to say since the introduction of base T makes no important difference, is the relevant model structure. The characteristic m.s. for DML, and DLL, , supplied by corollary 2 is called the standard DML m.s. Consider now the class CM of models obtained by taking all two-valued valuations I on model structure S on set K. Following Kripke (63a,p.92), define a matrix value as a mapping whose domain is K and whose range is the set 2 = {1,0}, i.e. as a function in 2^. Accordingly a matrix-value is tantamount to a subset of K; for each matrix value determines that unique subset of K which it maps into 1 and conversely each subset of K specifies a mapping. Furthermore valuation I maps sentential variables into matrix values; for, given variable p, I assigns as matrix-value {a:I(p,a) = l}. Thus the class CM of models provides a matrix. Where p and O are matrix values, define matrix operations 'v' and as follows: (~p)(a) = 1

iff

p(a*) + 1;

(p&a)(a) = 1

iff

p(a) = 1 = a(a);

(pva)

iff

p(a) = 1 or a(a) = 1;

(a)

= 1

and given a B model, (p-KJ) (a) • 1 iff for every b and c in K, if Rabc and p(b) = 1 then a(c) = 1, while given an I model, (p-*0) (a) = 1 iff a e N and,

113

2.6 THE SMILEV MATRICES ARE CHARACTERISTIC FOR DML AND FPE

for every a', if aRa* and p(a') = 1 then a(a') = 1. For DML and first degree models this latter condition reduces, for requisite a, to (p |- 0)(a) = 1 iff, for every a 1 , p(a') | 1 or a(a') = 1. It follows: ~p = {a:p(a*) = 0} and p&O = p 1 ^, and pvct = pUcr, where n and U are set or lattice operations. Finally, in models where truth is evaluated in T, p is a designated value iff p(T) = 1. For standard DML models p is designated iff p(d) = 1 = p(d*). Where S is an m.s. the matrix (or algebra) S + = <M,D,~,&,v,-*> on S is defined as follows: M is the class of matrix values, D is the class of designated matrix values, and &, v, are the class-closing matrix operations defined above. Then S + is a matrix in the standard sense. Lemma 2.4. Where S + is the matrix on m.s. S, and where truth is evaluated at T. S + satisfies A iff, for every I on S, I(A,T) = 1, for every wff A. Where S + is defined on standard DML model S, S + satisfies P iff, for every I on S, I(P,d) = 1 = l(P,d*), for every statement P, i.e. P is of the form r f- A, (Here P, i.e. T (- A,is validated by I at d, written I(P,d) = 1, iff, whenever each element of T holds on I at d, A also holds on I at d.) Proof. Let A + (p + ,... ,p + ) be the matrix function of p"!",...,p+ corresponding to i "n i n wff A(p j,.. p ) under the replacement of connectives by corresponding matrix operations, where I maps p

to p

Let A + be the matrix value of A. tives (or operations),

...,pn to p*. Thus p* = {a: I (p.,,a) « l}. n 'i " 'i] Then, it follows by induction over connec-

(*) a e A + iff I(A,a) = 1. Applying (*), I(A,T) = 1 iff T € A + , i.e. iff A + is designated, i.e. iff for the given assignment S + satisfies A. Generalising, S + satisfies A iff for every assignment I on S,I(A,T) = 1. Similarly I(P,d) = 1 = I(P,d*) iff d e P + & d* e P + , i.e. iff P + is designated; and hence S + satisfies P for DML statements iff for every I, I(P,d) = 1 = I(P,d*). Theorem 2.7. T (- A is a theorem of DML iff the Smiley matrices,

*1

1

2

*1

2

2

2

2

3

3 4

4 4

3 4

4

4 4 1

4 4 4

1

1

satisfy T j- A; i.e. these matrices are characteristic for DML and for first degree entailments. Proof. Let P abbreviate r (- A. P is provable iff P is DML valid. Define matrix values as follows: 1 = {d,d*} = K; 2 - {d}; 3 = {d*}; 4 = A. Then, with matrix operations defined as before for DML models, the Smiley matrices result. The argument is by cases for each operation, ad As matrix value 1 maps both d and d* to 1, ~1 maps neither to 1, i.e. ~1 = 4 ; similarly as 4 maps neither to 1, ~ 4 maps both d* and d = d** to 1, i.e. ~ 4 = 1 . As 2 maps just d to 1, ~2 maps just d* to 0 , i.e. ~2 maps just d to 1, so ~2 = 2; similarly ~3 = 3. ad &. As p&0 = fAJ, the matrix values in the & matrix are given by the intersection of the values of p and a . Hence 4 & 0 = a&4 • 4 , since the intersection of A with any set is A . Hence also lHa = ani = a. Finally 3^2 = 2H3 = {d}n{d*} = A , so 2&3 = 3&2 = 4 ; and ana = a. so 2&2 = 2 and 3&3 = 3. ad |- . By the evaluation rule, (pj-cr)(d) = 1 iff both p(d) = 0 or 0(d) « 1 and p ( d * ) = 0 or a(d) = 1; similarly for (p|-