A first course in logic

250 Pages·2019·1.97 MB·English

by  Lawson, Mark V

A First Course in Logic

A First Course in Logic

Mark V. Lawson
Heriot-Watt University, Edinburgh
CRC Press
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Library of Congress Cataloging‑in‑Publication Data
Names: Lawson, Mark V., author.
Title: A first course in logic / Mark V. Lawson.
Description: Boca Raton, Florida : CRC Press, [2018] | Includes
bibliographical references and index.
Identifiers: LCCN 2018024672| ISBN 9780815386643 (hardback : alk. paper) |
ISBN 9780815386650 (pbk. : alk. paper) | ISBN 9781351175388 (ebook).
Subjects: LCSH: Logic, Symbolic and mathematical--Problems, exercises, etc. | Logic.
Classification: LCC QA9 .L37256 2018 | DDC 511.3--dc23
LC record available at https://lccn.loc.gov/2018024672
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This book is dedicated to the memory of my mother
Shirley Lawson (1935–2017)
“...for the growing good of the world is partly dependent on un-
historicacts;andthatthingsarenotsoillwithyouandmeasthey
might have been, is half owing to the number who lived faithfully
a hidden life ...” – George Eliot.

Contents
Preface ix
Introduction xi
1 Propositional logic 1
1.1 Informal propositional logic . . . . . . . . . . . . . . . . . . . 1
1.2 Syntax of propositional logic . . . . . . . . . . . . . . . . . . 9
1.3 Semantics of propositional logic . . . . . . . . . . . . . . . . 16
1.4 Logical equivalence . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.2 Logical patterns . . . . . . . . . . . . . . . . . . . . . 25
1.4.3 Applications . . . . . . . . . . . . . . . . . . . . . . . 27
1.5 PL in action . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.5.1 PL as a programming language . . . . . . . . . . . . . 32
1.5.2 PL can be used to model some computers . . . . . . . 36
1.6 Adequate sets of connectives . . . . . . . . . . . . . . . . . . 42
1.7 Truth functions . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.8 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . 48
1.8.1 Negation normal form (NNF) . . . . . . . . . . . . . . 48
1.8.2 Disjunctive normal form (DNF) . . . . . . . . . . . . . 49
1.8.3 Conjunctive normal form (CNF) . . . . . . . . . . . . 50
1.8.4 Prologue to PROLOG . . . . . . . . . . . . . . . . . . 51
1.9 P =NP? or How to win a million dollars . . . . . . . . . . . 56
1.10 Valid arguments . . . . . . . . . . . . . . . . . . . . . . . . . 61
1.10.1 Definitions and examples . . . . . . . . . . . . . . . . 62
1.10.2 Proof in mathematics (I) . . . . . . . . . . . . . . . . 66
1.11 Truth trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
1.11.1 The truth tree algorithm . . . . . . . . . . . . . . . . 72
1.11.2 The theory of truth trees . . . . . . . . . . . . . . . . 83
1.12 Sequent calculus . . . . . . . . . . . . . . . . . . . . . . . . . 87
1.12.1 Deduction trees . . . . . . . . . . . . . . . . . . . . . . 89
1.12.2 Truth trees revisited . . . . . . . . . . . . . . . . . . . 94
1.12.3 Gentzen’s system LK . . . . . . . . . . . . . . . . . . 99
vii
viii Contents
2 Boolean algebras 103
2.1 More set theory . . . . . . . . . . . . . . . . . . . . . . . . . 103
2.2 Boolean algebras . . . . . . . . . . . . . . . . . . . . . . . . . 110
2.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . 110
2.2.2 Definition and examples . . . . . . . . . . . . . . . . . 112
2.2.3 Algebra in a Boolean algebra . . . . . . . . . . . . . . 114
2.3 Combinational circuits . . . . . . . . . . . . . . . . . . . . . 120
2.3.1 How gates build circuits . . . . . . . . . . . . . . . . . 120
2.3.2 A simple calculator . . . . . . . . . . . . . . . . . . . . 127
2.4 Sequential circuits . . . . . . . . . . . . . . . . . . . . . . . . 134
3 First-order logic 145
3.1 First steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
3.1.1 Names and predicates . . . . . . . . . . . . . . . . . . 146
3.1.2 Relations . . . . . . . . . . . . . . . . . . . . . . . . . 148
3.1.3 Structures . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.1.4 Quantification . . . . . . . . . . . . . . . . . . . . . . 152
3.1.5 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
3.1.6 Variables: their care and maintenance . . . . . . . . . 156
3.1.7 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . 160
3.1.8 De Morgan’s laws for quantifiers . . . . . . . . . . . . 167
3.1.9 Quantifier examples . . . . . . . . . . . . . . . . . . . 168
3.2 Gödel’s completeness theorem . . . . . . . . . . . . . . . . . 173
3.2.1 An example . . . . . . . . . . . . . . . . . . . . . . . . 173
3.2.2 Truth trees for FOL . . . . . . . . . . . . . . . . . . . 174
3.2.3 The soundness theorem . . . . . . . . . . . . . . . . . 180
3.2.4 The completeness theorem. . . . . . . . . . . . . . . . 182
3.3 Proof in mathematics (II) . . . . . . . . . . . . . . . . . . . . 188
3.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 190
Solutions to all exercises 193
Bibliography 225
Index 231
Preface
When you come to a fork in the road, take it! — Yogi Berra.
This book grew out of notes written to accompany my Heriot-Watt Uni-
versitymoduleF17LP Logic and proofwhichIhavebeenteachingsince2011.
ThismodulewasinfactinstigatedbymycolleaguesinComputerScienceand
was therefore intendedoriginally forfirst-year computersciencestudents, but
the module was subsequently also offered as an option to second-year mathe-
matics students (in the Scottish university system). There are three chapters.
Chapters 1 and 2 focus on propositional logic and Boolean algebras, respec-
tively, and Chapter 3 is a short introduction to the basic ideas of first-order
logic.Ihavetreatedpropositionallogicasasubjectinitsownrightratherthan
as just a stopover to a more interesting destination. In particular, I outline
the intuitive background to the question of whether P = NP, describe how
thisquestionisrelatedtothesatisfiabilityproblemofpropositionallogic,and
why all this matters. The theory of Boolean algebras is the algebraic face of
propositionallogicandisusedtodesigncircuits:boththecombinationalones
thathavenomemoryandthesequentialonesthatdo.Eachsectionconcludes
withexercises,allofwhosesolutionscanbefoundattheendofthebook.Let
me stress that this is very much a first introduction to logic. I have therefore
assumed as few prerequisites as possible; specifically, I have tried to keep any
mathematical background to the absolute minimum. The real mathematical
prerequisiteisanabilitytomanipulatesymbols:inotherwords,basicalgebra.
Anyone who can write programs should have this ability already. In addition,
Idonottrytofolloweverylogicalbyway,rathermygoalistoinspireinterest
and curiosity about this subject and lay the foundations for further study. In
writingthisbook,Ihavebeenparticularlyinfluenced,likemanyothers,bythe
workofSmullyan1,particularly[67].Chapters1and3coverroughlythesame
material as the first 65 pages of his [67] together with part of his Chapter XI.
For a complete list of all the references I consulted, see the bibliography, but
letmehighlightafewparticularexampleshere.InbothChapter1andChap-
ter 3, I have used truth trees. These are vastly superior to the Hilbert-style
systems often containedin manyintroductions to logic. Though such systems
have their place, they are a stumbling block in an introduction such as this. I
first learnt about them from [65]. I have, however, included a short introduc-
1RaymondSmullyan(1919–2017)wasthedoyenoflogic,andhisworkdidmuchtobring
ittoawideraudience.
ix