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Relevant Logic a Philosophical Examination of Inference

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Relevant Logic A Philosophical Examination of Inference Stephen Read February 21, 2012 i c Stephen Read First published by Basil Blackwell 1988 Corrected edition (incorporating all errata noted since 1988) c Stephen Read 2010 ii To my wife, Gill, without whose encouragement and support it could not have been completed Contents Acknowledgements vi Introduction1 1 Scepticism and Logic6 1.1 Scepticism.............................6 1.2 Fallacy...............................9 1.3 Descartes' Test.......................... 11 1.4 Defeasibility............................ 12 1.5 Demonstrativism......................... 13 1.6 Fallibilism............................. 15 2 Classical Logic 19 2.1 The Classical Account of Validity................ 19 2.2 The Deduction Equivalence................... 20 2.3 Material Implication....................... 22 2.4 Truth and Assertibility...................... 24 2.5 Three Morals........................... 28 2.6 The Lewis Argument....................... 31 2.7 Three Senses of Disjunction................... 32 2.8 Conclusion............................ 34 3 An Intensional Conjunction 36 3.1 Conjunction............................ 36 3.2 The Deduction Equivalence................... 38 3.3 Bunches.............................. 40 3.4 Consequence........................... 41 3.5 Relevant Logic.......................... 44 3.6 Distribution............................ 46 3.7 Antilogism............................. 48 iii CONTENTS iv 3.8 Conclusion............................ 50 4 Proof-Theory for Relevant Logic 51 4.1 The Bunch Formulation..................... 52 4.2 A Deductive System for DW .................. 55 4.3 Extensions of DW ........................ 58 4.4 Derivations in DW ........................ 61 4.5 A Deductive System for R and R ............... 70 4.6 Derivations in R ........................ 73 5 Worlds Semantics for Relevant Logics 78 5.1 Worlds Semantics......................... 78 5.2 Relevant Semantics........................ 80 5.3 Formal Semantics for DW .................... 82 5.4 Formal Results for DW-Semantics............... 84 5.5 Extending the Semantics..................... 90 5.6 Semantics for R and R ..................... 94 Appendix 100 5A.1 Completeness for DW ...................... 101 5A.2 Extending DW .......................... 107 5A.3 Completeness for R ....................... 109 6 Relevance 114 6.1 The Pre-History of Relevance.................. 114 6.2 The Later History of Relevance................. 118 6.3 Variable-Sharing......................... 121 6.4 Derivational Utility........................ 124 6.5 Logical Relevance......................... 126 6.6 Fusion............................... 128 6.7 Conclusion............................ 130 7 Logic on the Scottish Plan 131 7.1 Geach's Challenge........................ 131 7.2 The Dialetheic Response..................... 132 7.3 The Australian Plan....................... 134 7.4 Truth-functionality........................ 135 7.5 The American Objection..................... 137 7.6 Classical Relevant Logic..................... 139 7.7 Local Consistency........................ 140 CONTENTS v 7.8 The Scottish Plan........................ 141 8 Deviant Logic and Sense of Connectives 144 8.1 The Deviant's Predicament................... 144 8.2 Rigid Designation......................... 146 8.3 Rigid Connectives........................ 148 8.4 The Homophonic Principle................... 150 8.5 A Metalanguage for R ...................... 152 8.6 Logical Consequence....................... 154 8.7 Conclusion............................ 157 9 Semantics and the Theory of Meaning 159 9.1 Semantics............................. 159 9.2 Tonk................................ 161 9.3 Conservative Extension..................... 163 9.4 Assertion-conditions....................... 165 9.5 Proof-conditions......................... 166 9.6 Cut................................ 168 9.7 Negation.............................. 171 9.8 Normalisation........................... 175 9.9 Tonk Revisited.......................... 178 10 Conclusion 180 Index 184 Acknowledgements In working on the matters discussed in this book, I have received help, advice, criticism and encouragement from many students, friends and col- leagues. I would particularly like to thank my colleagues in the Department of Logic and Metaphysics at St Andrews for allowing me to take study leave in 1987, during which time this book was completed. I was also fortunate to be invited to the Australian National University to a three-month Visiting Fellowship during this time. I am especially grateful to Michael MacRobbie for arranging for the invitation, and for providing me with such a stimulating research environment. I am indebted to the Australian National University, the British Academy, the British Council and the University of St Andrews for financial assistance which made the visit possible. For their helpful and searching comments on earlier drafts of the present material I would like to thank by name: Ross Brady, Jack Copeland, Max and Mary Cresswell, Roy Dyckhoff, John Fox, Andr Fuhrmann, Allen Hazen, Reinhard Hlsen, Frank Jackson, Peter Lavers, Errol Martin, Bob Meyer, Joan Northrop, Charles Pigden, Graham Priest, Peter Schroeder-Heister, John Slaney, Richard Sylvan and Neil Tennant. I am especially indebted to John Slaney, who first introduced me to the idea of the bunch formulation of a natural deduction calculus. John and I ran joint seminars in Edinburgh and St Andrews from 1984 to 1987, from which I learned a great deal. I am also grateful to Stephan Chambers of Basil Blackwell Ltd. for his help and encouragement in arranging publication. Stephen Read St Andrews December 1987 vi Introduction The logician's concern is with validity, with the relation of consequence be- tween premises and conclusion. In order to justify an assertion, we may adduce other statements, from which we claim the assertion follows. But what is the criterion by which to decide if the conclusion really does fol- low? The question has two aspects: concretely, to decide in particular cases whether the conclusion follows from the premises|in technical language, whether a consequence relation holds; and abstractly, to understand in gen- eral what the relation between premises and conclusion in a valid argument is. The purpose of this book is to explore and defend a particular answer to this abstract question, an answer which will be characterised in chapter 6 as the Relevant Account of Validity. Relevant logic evolved as a general framework of logical investigation in the early 1960s, out of work by Alan Anderson and Nuel Belnap, themselves extending ideas of Wilhelm Acker- mann, Alonzo Church and others (see chapter 6 below). In a valid argument, the premises must be relevant to the conclusion. My aim in this book is to show why this must be so, and what the content of this claim is. Interest in relevant logic has grown enormously in the last decade, follow- ing publication of the first volume of Anderson and Belnap's encyclopedic work Entailment: the logic of relevance and necessity.1 Nonetheless, in- terested readers will experience some difficulty in finding out what relevant logic involves. Entailment itself is a large work containing a wealth of detail. Yet, for example, it includes nothing on the possible worlds semantics (see chapter 6 below) for relevant logic developed in the early 1970s, that topic being reserved for the second volume, whose publication has been badly, and sadly, delayed. Details of the semantics can certainly be unearthed from Relevant Logics and their rivals by Richard Routley and others,2 but 1A.Anderson and N.Belnap, Entailment, vol.I (Princeton, 1975). 2F.R. Routley et al., Relevant Logics and their rivals (Atascadero, Calif., 1982). 1 INTRODUCTION 2 many readers, particularly those not already well-versed in the subject, will find that work an uncompromising and difficult one. By far the best in- troduction so far to the subject is Michael Dunn's survey article, `Relevant Logic and Entailment'.3 The present book does not attempt to compete with Dunn's survey. I have set myself two aims here. The first is to lay forth what I believe are the correct philosophical reasons for rejecting classical logic and adopting a rel- evant logic as a correct description of the basis of inference. These are not, in general, the reasons which led historically to the development of the sub- ject, and are not those emphasised in the writings of Anderson, Belnap and Dunn. In many ways relevance as a core notion is displaced in the present framework by direct consideration of inference itself. Secondly, I have tried to keep the work intelligible to anyone who has some training in logic and in philosophical analysis, but no more than the average second or third year undergraduate with an introductory course on formal and philosophical logic behind them. Anyone who has read and understood Lemmon's Beginning Logic or Guttenplan's The Languages of Logic should have no difficulty with the technical material in chapters 4 and 5. The most difficult material, on the completeness proof, I have relegated to an appendix to chapter 5. Later chapters do not presuppose that readers have mastered the material in this appendix. I hope, however, that chapters 4 and 5 themselves do not con- tain any gratuitous technicality. One cannot reason clearly and correctly about logic without proper acquaintance with the technical development of the subject. Equally, neither can one develop the formal material usefully
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