Intermediate Logic This page intentionally left blank Intermediate Logic DAVID BOSTOCK CLARENDON PRESS • OXFORD This book has been printed digitally and produced in a standard specification in order to ensure its continuing availability OXFORD UNIVERSITY PRESS Great Clarendon Street, Oxford 0X2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Sao Paulo Shanghai Singapore Taipei Tokyo Toronto with an associated company in Berlin Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © David Bostock 1997 The moral rights of the author have been asserted Database right Oxford University Press (maker) Reprinted 2002 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer ISBN 0-19-875141-9 ISBN 0-19-875142-7 (pbk) Preface This book is intended for those who have studied a first book in logic, and wish to know more. It is concerned to develop logical theory, but not to apply that theory to the analysis and criticism of ordinary reasoning. For one who has no concern with such applications, it would be possible to read this book as a first book in the subject, since I do in fact introduce each logical concept that I use, even those that I expect to be already familiar (e.g. the truth-functors and the quantifiers). But it would be tough going. For in such cases my explanations proceed on a fairly abstract level, with virtually no discussion of how the logical vocabulary relates to its counterpart in everyday language. This will be difficult to grasp, if the concept is not in fact familiar. The book is confined to elementary logic, i.e. to what is called first-order predicate logic, but it aims to treat this subject in very much more detail than a standard introductory text. In particular, whereas an introductory text will pursue just one style of semantics, just one method of proof, and so on, this book aims to create a wider and a deeper understanding by showing how several alternative approaches are possible, and by introducing com- parisons between them. For the most part, it is orthodox classical logic that is studied, together with its various subsystems. (This, of course, includes the subsystem known as intuitionist logic, but I make no special study of it.) The orthodox logic, however, presumes that neither names nor domains can be empty, and in my final chapter I argue that this is a mistake, and go on to develop a 'free' logic that allows for empty names and empty domains. It is only in this part of the book that what I have to say is in any way unortho- dox. Elsewhere almost all of the material that I present has been familiar to logicians for some time, but it has not been brought together in a suitably accessible way. The title of the book shows where I think it belongs in the teaching of the subject. Institutions which allow a reasonable time for their first course in logic could certainly use some parts of this book in the later stages of that course. Institutions which do not already try to get too much into their advanced courses could equally use some parts of it in the earlier stages of those courses. But it belongs in the middle. It should provide a very suitable background for those who wish to go on to advanced treatments of model V PREFACE theory, proof theory, and other such topics; but it should also prove to be an entirely satisfying resting-place for those who are aware that a first course in logic leaves many things unexplored, but who have no ambition to master the mathematical techniques of the advanced courses. Moreover, I do not believe that the book needs to be accompanied by a simultaneous course of instruction; it should be both comprehensible and enjoyable entirely on its own. While I have been interested in logic ever since I can remember, I do not think that I would ever have contemplated writing a book on the topic, if it had not been for my involvement fifteen years ago in the booklet Notes on the Formalization of Logic. This was compiled under the guidance of Professor Dana Scott, for use as a study-aid in Oxford University. Several themes in the present work descend from that booklet, and I should like to acknowledge my indebtedness not only to Dana Scott himself, but also to the others who helped with the compilation of that work, namely Dan Isaacson, Graeme Forbes, and Gören Sundholm. But, of course, there are also many other works, more widely known, which I have used with profit, but with only occasional acknowledgement in what follows. David Bostock Merton College, Oxford vi Contents Parti. SEMANTICS 1 1. Introduction 3 1.1. Truth 3 1.2. Validity 5 1.3. The Turnstile 8 2. Truth-Functors 14 2.1. Truth-Functions 14 2.2. Truth-Functors 17 2.3. Languages for Truth-Functors 21 2.4. Semantics for these Languages 24 2.5. Some Principles of Entailment 30 2.6. Normal Forms (DNF, CNF) 37 2.7. Expressive Adequacy I 45 2.8. Argument by Induction 48 2.9. Expressive Adequacy II 56 2.10. Duality 62 2.11. Truth-value Analysis 65 3. Quantifiers 70 3.1. Names and Extensionality 70 3.2. Predicates, Variables, Quantifiers 74 3.3. Languages for Quantifiers 77 3.4. Semantics for these Languages 81 3.5. Some Lemmas on these Semantics 91 3.6. Some Principles of Entailment 96 3.7. Normal Forms (PNF) 109 3.8. Decision Procedures I: One-Place Predicates 115 3.9. Decision Procedures II: V3-Formulae 126 3.10. The General Situation: Proofs and Counter-examples 131 vii CONTENTS Part II. PROOFS 139 4. Semantic Tableaux 141 4.1. The Idea 141 4.2. The Tableau Rules 147 4.3. A Simplified Notation 152 4.4. Constructing Proofs 157 4.5. Soundness 165 4.6. Completeness I: Truth-Functors 168 4.7. Completeness II: Quantifiers 174 4.8. Further Remarks on Completeness, Compactness, and Decidability 182 4.9. Appendix: A Direct Proof of the Cut Principle 187 5. Axiomatic Proofs 190 5.1. The Idea 190 5.2. Axioms for the Truth-Functors 193 5.3. The Deduction Theorem 200 5.4. Some Laws of Negation 208 5.5. A Completeness Proof 217 5.6. Axioms for the Quantifiers 220 5.7. Definitions of Other Logical Symbols 227 5.8. Appendix: Some Alternative Axiomatizations 232 6. Natural Deduction 239 6.1. The Idea 239 6.2. Rules of Proof I: Truth-Functors 242 6.3. Rules of Proof II: Quantifiers 254 6.4. Alternative Styles of Proof 262 6.5. Interim Review 269 7. Sequent Calculi 273 7.1. The Idea 273 7.2. Natural Deduction as a Sequent Calculus 277 viii CONTENTS 7.3. Semantic Tableaux as a Sequent Calculus 283 7.4. Gentzen Sequents; Semantic Tableaux Again 291 7.5. Comparison of Systems 299 7.6. Reasoning with Gentzen Sequents 307 Part III. FURTHER TOPICS 321 8. Existence and Identity 323 8.1. Identity 323 8.2. Functions 333 8.3. Descriptions 341 8.4. Empty Names and Empty Domains 348 8.5. Extensionality Reconsidered 355 8.6. Towards a Universally Free Logic 360 8.7. A Formal Presentation 366 8.8. Appendix: A Note on Names, Descriptions, and Scopes 375 REFERENCES 379 LIST OF SYMBOLS 383 LIST OF AXIOMS AND RULES OF INFERENCE 384 INDEX 387 ix This page intentionally left blank Part I SEMANTICS This page intentionally left blank 1 Introduction 1.1. Truth 3 1.2. Validity 5 1.3. The Turnstile 8 1.1. Truth The most fundamental notion in classical logic is that of truth. Philo- sophers, of course, have long debated the question 'what is truth?', but that is a debate which, for the purposes of the present book, we must leave to one side. Let us assume that we know what truth is. We are concerned with truth because we are concerned with the things that are true, and I shall call these things 'propositions'. Philosophers, again, hold differing views on what is to count as a proposition. A simple view is that a proposition is just a (declarative) sentence, but when one thinks about it for a moment, there are obvious difficulties for this suggestion. For the same sentence may be used, by different speakers or in different con- texts, to say different things, some of them true and others false. So one may prefer to hold that it is not the sentences themselves that are true or false, but particular utterings of them, i.e. utterings by particular people, at particular times and places, in this or that particular situation.