An Introduction to Formal Logic(2020) [pdf]

A friendly request to the reader! As the Preface explains, this is a very slightly corrected reprint of the second edition of my Introduction to Formal Logic, originally published in 2020 by CUP. I am now able to make it freely available to download as a PDF. You can also buy a print-on-demand physical copy at a price as low as I can make it from Amazon. In these troubled times, giving students easy access to what they need seems the least we can do. No introductory logic text will suit all readers. This one will be too longwinded for some, and too brisk for others. Some readers will want more ‘informal logic’ at the outset; others with complain that I take over fifty pages before we start formal logic proper. The tone will be too ‘mathematical’ for some, too relaxed for others. And so it goes. I can’t please everyone. I can only hope that it works for you! But if you do read the book – because it is set as a course text, or just because you want to teach yourself some basic logic – do let me know how you find it. Are there any particular sections that you find obscure? Are there parts that go too fast/too slowly? Would you e.g. like more historical asides? More philosophical asides? Oh, and did you spot any outright mistakes? Even quick notes saying ‘It all went pretty well for me!’ are appreciated. I’m keen to know because I have it mind to write a third edition sooner or later. Because the book is now self-published, it is a lot easier to keep improving it, even if it means adding extra pages here or there. And I’d like to make the book as readable and as useful as I can. So do please send comments to me, peter smith at logicmatters.net. Thanks! Peter Smith, before he retired, was Senior Lecturer in Philosophy at the Uni- versity of Cambridge. His other books include Explaining Chaos (1998) and An Introduction to G¨odel’s Theorems (2007; 2013). The page left intentionally blank An Introduction to Formal Logic Second edition Peter Smith Logic Matters c© Peter Smith 2003, 2020 All rights reserved.

Permission is granted to distribute this PDF as a complete whole, including this copyright page, for educational purposes such as classroom use. Otherwise, no part of this publication may be reproduced, distributed, or transmitted in any form or by any means, including photocopying or other electronic or mechanical methods, without the prior written permission of the author, except in the case of brief quotations embodied in critical reviews and certain other noncommercial uses permitted by copyright law. For permission requests, write to peter smith@logicmatters.net. First published by Cambridge University Press, 2003 Second edition, Cambridge University Press, 2020 Reprinted with corrections, Logic Matters, August 2020 A low-cost paperback of this book is available by print on demand from Amazon ISBN 979-8-675-80394-1 Paperback Additional resources for this publication at www.logicmatters.net, where the latest version of this PDF will always be found. Since this revised edition is not produced by a publisher with a marketing depart- ment, your university librarian will not get to know about it in the usual way. You will therefore need to give them the details and ask them to order a printed copy for the library. Contents Preface vii 1 What is deductive logic? 1 2 Validity and soundness 9 3 Forms of inference 20 4 Proofs 28 5 The counterexample method 39 6 Logical validity 44 7 Propositions and forms 52 Interlude: From informal to formal logic 59 8 Three connectives 61 9 PL syntax 72 10 PL semantics 81 11 ‘P’s, ‘Q’s, ‘α’s, ‘β’s – and form again 94 12 Truth functions 104 13 Expressive adequacy 113 14 Tautologies 120 15 Tautological entailment 127 16 More about tautological entailment 137 17 Explosion and absurdity 145 18 The truth-functional conditional 148 19 ‘If’s and ‘→’s 162 Interlude: Why natural deduction?

172 20 PL proofs: conjunction and negation 174 21 PL proofs: disjunction 191 22 PL proofs: conditionals 203 23 PL proofs: theorems 211 24 PL proofs: metatheory 216 Interlude: Formalizing general propositions 228 25 Names and predicates 230 26 Quantifiers in ordinary language 240 v vi Contents 27 Quantifier-variable notation 249 28 QL languages 258 29 Simple translations 271 30 More on translations 282 Interlude: Arguing in QL 290 31 Informal quantifier rules 293 32 QL proofs 299 33 More QL proofs 315 34 Empty domains? 329 35 Q-valuations 333 36 Q-validity 346 37 QL proofs: metatheory 354 Interlude: Extending QL 359 38 Identity 361 39 QL= languages 367 40 Definite descriptions 375 41 QL= proofs 382 42 Functions 391 Appendix : Soundness and completeness 402 The Greek alphabet 412 Further reading 413 Index 415 vi Preface The first and second printed editions The world is not short of introductions to logic aimed at philosophy students. They differ widely in pace, style, coverage, and the ratio of formal work to philosophical commentary. Like other authors, my initial excuse for writing yet another text was that I did not find one that offered quite the mix that I wanted for my own students (first-year undergraduates doing a compulsory logic course). Logicians are an argumentative lot and get quite heated when discussing the best route into our subject for beginners. But, standing back from our differences, we can all agree on this: in one form or another, the logical theory eventually arrived at in this book – classical first-order quantification theory, to give it its trade name – is a wonderful intellectual achievement by formally-minded philosophers and philosophically-minded mathematicians.

It is a beautiful theory of permanent value, a starting point for all other modern work in logic. So we care greatly about passing on this body of logical knowledge. And we write our logic texts – like this one – in the hope that you too will come to appreciate some of the elements of our subject, and even want to go on to learn more. This book starts from scratch, and initially goes pretty slowly; I make no apology for working hard at the outset to nail down some basic ideas. The pace gradually picks up as we proceed and as the idea of a formal approach to logic becomes more and more familiar. But almost everything should remain quite accessible even to those who start off as symbol-phobic – especially if you make the effort to read slowly and carefully. You should check your understanding by tackling at least some of the routine exercises at the ends of chapters. There are worked answers available at the book’s online home, www.logicmatters.net. These are often quite detailed. For example, while the book’s chapters on formal proofs aim to make the basic principles really clear, the online answers provide extended ‘examples classes’ exploring proof-strategies, noting mistakes to avoid, etc. So do make good use of these further resources. As well as exercises which test understanding, there are also a few starred exercises which introduce additional ideas you really ought to know about, or which otherwise ask you to go just a little beyond what is in the main text. The first edition of this book concentrated on logic by trees. Many looking for a course text complained about this. This second edition, as well as significantly vii viii Preface revising all the other chapters, replaces the chapters on trees with chapters on a natural deduction proof system, done Fitch-style. Which again won’t please everyone! So the chapters on trees are still available, in a revised form. But to keep the length of the printed book under control, you will find these chapters – together with a lot of other relevant material including additional exercises – at the book’s website. This PDF version The second edition is now made freely available as a PDF download (note, by the way, that internal cross-references are live links). Apart from altering this Preface to note the new publication arrangements, and tak- ing the opportunity to correct a very small number of misprints, the book is unchanged. However, . . . A third edition?