Elementary Logic

192 Pages·2014·1.489 MB·English

by  Brian Garrett

ELEMENTARY LOGIC
ACU-ElemLogic-PROOF4.indd i 27/07/2012 12:25:44
This(cid:2)page(cid:2)intentionally(cid:2)left(cid:2)blank
ELEMENTARY LOGIC
BRIAN GARRETT
RO Routledge
U
T
L
ED Taylor & Francis Group
G
E
LONDON AND NEW YORK
ACU-ElemLogic-PROOF4.indd iii 27/07/2012 12:25:49
First published 2012 by Acumen
Published 2014 by Routledge
2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN
711 Third Avenue, New York, NY 10017, USA

Routledge is an imprint of the Taylor & Francis Group,
an informa business
© Brian Garrett, 2012
Th is book is copyright under the Berne Convention.
No reproduction without permission.
All rights reserved. No part of this book may be reprinted or
reproduced or utilised in any form or by any electronic, mechanical,
or other means, now known or hereafter invented, including
photocopying and recording, or in any information storage or
retrieval system, without permission in writing from the publishers.

Notices
Practitioners and researchers must always rely on their own
experience and knowledge in evaluating and using any information,
methods, compounds, or experiments described herein. In using such
information or methods they should be mindful of their own safety
and the safety of others, including parties for whom they have a
professional responsibility.

To the fullest extent of the law, neither the Publisher nor the
authors, contributors, or editors, assume any liability for any injury
and/or damage to persons or property as a matter of products
liability, negligence or otherwise, or from any use or operation of
any methods, products, instructions, or ideas contained in the
material herein.

isbn: 978-1-84465-517-5 (hardcover)
isbn: 978-1-84465-518-2 (paperback)
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Designed and typeset in Minion Pro.
ACU-ElemLogic-PROOF4.indd iv 27/07/2012 12:25:49
CONTENTS
Preface vii
1 Overview 1
2 Logical connectives and truth-tables 9
3 Conditional 22
4 Conjunction 35
5 Conditional proof 47
6 Solutions to selected exercises, I 57
7 Negation 67
8 Disjunction 80
9 Biconditional 95
10 Solutions to selected exercises, II 100
11 Derived rules 120
12 Truth-trees 128
13 Logical refl ections 139
14 Logic and paradoxes 155
Glossary 169
Further reading 173
References 177
Index 179
CONTENTS v
ACU-ElemLogic-PROOF4.indd v 27/07/2012 12:25:49
This(cid:2)page(cid:2)intentionally(cid:2)left(cid:2)blank
PREFACE
Th e book off ers a clear and concise introduction to propositional logic. It can be
used in a three-month course, but also could be expanded for use in a longer
course. Th e last two chapters contain philosophical material, which should be
accessible even to beginning students.
My aim has been to produce a logic text that off ers a concise, ground-up introduc-
tion to elementary logic, with an emphasis on the ideas underlying logical principles
and rules of inference. I am writing for the student who wants to understand the
basic ideas of elementary logic, but may have no intention of doing more advanced
work in logic. My concern is to get students to see why a given proof is valid or what
the rationale is for a particular rule of inference. Such insights are of more value than
being able to zip through proofs in record time. To that end, Chapters 6 and 10 are
given up to answering some of the questions at the end of preceding chapters with,
quite deliberately, much in the way of authorial intervention.
Since the target audience is philosophy students, I thought it might be useful to
look at various meta-logical or philosophical issues that arise from propositional
logic, and outline more advanced logics that some students might like to investi-
gate further. It is well for students to know, for example, that the classical logic that
underpins elementary logic has counter-intuitive aspects, and that there are rival
systems of logic. Such refl ections make it apparent that logic and the philosophy of
logic cannot be cleanly separated. Th ese issues are discussed in Chapter 13.
In Chapter 14 I try to show that, and why, logic is important, and not a mere
game played with symbols. In this chapter we examine four (logic-relevant) para-
doxes: the Liar paradox, Curry’s paradox, one of Lewis Carroll’s paradoxes, and the
sorites paradox. Some of these paradoxes have been thought to require revisions
to classical logic for their solution, thus connecting with a theme from Chapter 13.
Since paradoxes matter, logic matters too.
PREFACE vii
ACU-ElemLogic-PROOF4.indd vii 27/07/2012 12:25:49
Th ree further features of the book are worth noting: the concept boxes, the
Glossary and the Further Reading section. In many of the chapters I have inter-
spersed the text with concept boxes. Th ese boxes either summarize some concept
or technique central to a given chapter or else further elaborate on some claim
made therein. In the Glossary I give clear and explicit defi nitions of many of the
key words and phrases used in the book. Key words or phrases in the text are indi-
cated using bold toning on fi rst appearance. A fi rm grasp of the terms that appear
in the Glossary is crucial to understanding the nature of logic and logical truth. As
well as the articles and books referred to in the text and footnotes, I have selected a
number of useful and interesting books and articles for the Further Reading. Th ese
readings are themselves divided into various subsections, making it easy for the
student to follow up any particular topic.
Some are sceptical of the value of teaching students elementary logic. I am not.
Many arguments in philosophy and elsewhere are fallacious, and we need logic to
expose them. Even if many philosophical debates turn on the truth or falsity of
premises, rather than the validity of arguments, it is still important to be able to
identify the form of argument in question. In addition, it is undeniable that com-
petency in logic, like any intellectual skill, helps to sharpen and perfect the mind.
A student trained in logic will be better placed to assess and evaluate any piece of
reasoning they come across, in philosophy or in real life.
ACKNOWLEDGEMENTS
Th is book has grown out of a short Introduction to Logic course that I teach at
the Australian National University (ANU). Th anks to various students in my
logic classes over the years for their helpful comments. I am also grateful to three
anonymous Acumen referees, and to Peter Eldridge-Smith, Katrina Hutchison, J.
J. Joaquin, Th omas Mautner, Peter Roeper and Ryan Young for useful feedback.
viii PREFACE
ACU-ElemLogic-PROOF4.indd viii 27/07/2012 12:25:49
1 OVERVIEW
Th e aim of this book is to introduce students to the ideas and techniques of symbolic
logic. Logic is the study of arguments. Aft er working through this book the reader
should be in a position to identify and evaluate a wide range of arguments.
Once an argument has been identifi ed, we need to determine whether it is a
good argument or a bad one. By ‘good argument’ we mean a valid argument; by
‘bad argument’ we mean an invalid argument. Our primary method for determin-
ing validity will be natural deduction proofs, but we also use the (simpler but
more cumbersome) method of truth-trees. In addition, we briefl y show how truth-
tables can also be used to test for validity.
Elementary logic studies arguments, and, in doing so, it studies the logical or
inferential properties of the so-called logical connectives: ‘and’, ‘if … then …’, ‘or’,
‘not’ and ‘if and only if’. We use these logical words much of the time, even if we
might fi nd it hard to say what they mean. In logic, however, these key words have
a clear and explicit meaning.
SOME KEY TERMS AND IDEAS
Premises and conclusion
In elementary logic, the premises and conclusion of an argument are all declarative
sentences; that is, they are sentences that are either true or false. Th ere are only two
truth-values and each declarative sentence has one and only one of them. ‘Th e cat
is on the mat’, ‘no one loves Raymond’ and ‘all bachelors are bald’ are examples of
declarative sentences.
OVERVIEW 1
ACU-ElemLogic-PROOF4.indd 1 27/07/2012 12:25:49