Table Of ContentLogic and Language
Neville Dean
© C.Neville Dean 2003
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First published 2003 by
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Contents
Listof Tables vii
Listof Figures ix
Preface xi
Acknowledgements xiii
1 Language, Logic andSymbols 1
1.1 Information andtechnology 1
1.2 Characteristics of natural language 2
1.3 Connectives 3
1.4 Semantics 4
1.5 Truth values 4
1.6 Conjunction, Disjunction andNegation 5
1.7 Reasoningin naturallanguage 8
1.8 Symbols 10
2 Compound Propositions 15
2.1 Symbolicconnectives 15
2.2 Operators 16
2.3 Conjunction asanoperator 17
2.4 Schematicletters 19
2.5 Negation asan operator 21
2.6 Disjunction asanoperator 22
2.7 Use of schematicletters 23
2.8 More complexcompound propositions 24
2.9 Parse trees 25
2.10 Compound propositions from parsetrees 28
2.11 Connective priorities 29
2.12 Removingparentheses 33
2.13 Truth valuesof compound propositions 34
iii
iv Contents
3 Propositional Forms 37
3.1 Compoundpropositions from propositional forms 37
3.2 Propositional formsfor compound propositions 38
3.3 Truth tablesforpropositional forms 42
3.4 Languageandmetalanguage 46
3.5 Propertiesof propositional forms 49
3.6 Equivalentpropositional forms 54
3.7 Somelawsof equivalence 57
3.8 Semanticentailment 63
3.9 Uniformreplacement 69
4 Natural Deduction 73
4.1 Arguments andvalidity 73
4.2 Natural deduction 82
4.3 New inference forms 89
4.4 Deduction trees 95
4.5 Other methodsof deduction 103
4.6 Theoremsof natural deduction 113
4.7 Syntacticequivalence 115
5 Conditional Connective 117
5.1 Symbolicrepresentation of information 117
5.2 Causality,conditional statementsandimplication 119
5.3 Anew connective 121
5.4 Propertiesof the conditional connective 125
5.5 Priority of the conditional connective 127
5.6 Equationallogic 130
5.7 Natural deduction with the conditional connective 133
5.8 Derivedrules 137
5.9 Thebiconditional connective 139
6 Predicate Logic 143
6.1 Propositions andpredicates 143
6.2 Predicateswithmore thanone gap 145
6.3 Free variables 145
6.4 Compoundpredicates 148
6.5 Constantsandfunctions 149
6.6 Predicateforms 154
6.7 Quantifiers 156
6.8 Semantics 157
6.9 Deduction with quantifiedpredicates 164
6.10 Methodsof deduction 168
7 FirstOrder Theories 175
7.1 First orderlogic with identity 175
7.2 Theories 177
Contents v
7.3 Digital circuits 178
7.4 Equationaltheories 184
7.5 Boolean algebras 188
7.6 Equationaltheory of logic 189
7.7 First order logic 196
8 An Introduction toLogic Programming 199
8.1 Limitationsof naturaldeduction 199
8.2 Consistency andrefutation 201
8.3 Clauses 203
8.4 Refutation inclausallogic 211
8.5 Horn clauses 213
A Solutions toExercises 217
B Summary of notation 277
B.1 Letters 277
B.2 Connectives 277
B.3 Quantifiers 278
B.4 Propositional formsandtruth values 278
B.5 Arguments andnatural deduction 278
C Glossary 279
D Summary of deduction rules 287
D.1 Inference forms 287
D.2 Methods of deduction 287
D.3 Identity 287
D.4 Derived rules 288
E Summary of equivalences 289
E.1 Propositional forms 289
E.2 Quantifiers 291
F Bibliography 293
Index 295
List of Tables
3.1 EquivalencesinvolvingT,F, ¬,∧ and∨ 58
3.2 Properties of = 61
T
3.3 A GoldenRule forDoing Logic 62
5.1 Specialpropertiesof ⇒ 131
vii
List of Figures
4.1 Deduction tree for (P∧Q)∨(P∧R)(cid:5)P∧(Q∨R) 112
6.1 Solution toExample6.38 172
7.1 Solution toExample7.4 180
8.1 Truth tablesfor Example8.1 200
A.1 Solution toExercise 15, Question 18 227
A.2 Solution toExercise 30, Question 5 238
A.3 Solution toExercise 30, Question 6 239
A.4 Solution toExercise 32,Question 4 241
A.5 Solution toExercise 34, Question 2(j) 244
A.6 Solution toExercise 34, Question 2(l) 245
A.7 Solution toExercise 34, Question 4(g) 248
A.8 Solution toExercise 37, Question 9 253
A.9 Solution toExercise 50, Question 3 262
A.10 Solution toExercise 51, Question 2 264
ix
Preface
The range of intended readership for logic books is wide, and includes com-
puter scientists, philosophers, mathematicians and the lay reader. The aims
of thesebooks canvarywidelytoo: somearemeanttobereadforgeneralin-
terest;othersareintendedtodeveloplogicalthinking;somearemeanttogive
the reader an understanding of logic sufficient to support their professional
activities; others are intended to develop the deeper understanding required
bytheprofessionallogician. Therearemanydifferentsystemsoflogic,includ-
ingclassicalsystemsbaseduponnaturallanguageandawiderangeofsystems
of symbolic logic. Finally, there are the pedagogical issues to be considered.
Howshouldthe logicalsystemsbepresentedandexplainedin awaythatwill
bestachievethepurposesofthebook fortheintendedreadership?
Where does this book fit into this scheme? The complete answer to this
questioncanonlybeascertainedbyreadingthebookinitsentirety;neverthe-
less it is possible to give an overview. The book arose out of the need for a
text which would be suitable for graduates from a wide range of disciplines
studyingonaconversionM.Sc. incomputing. Littlepriormathematicalability
isassumed;furthermorethereisalittlemoreemphasisontherelationshipto
languagethanmanyotherbooksofthislevel,whilststillretainingtheimport-
anceofformalism. Thusthebookissuitablenotonlyforcomputerscientists
beginning a study of logic, but also to those studying logic in philosophy or
mathematics. Its purpose is to develop an understanding of the nature and
application of symbolic logic, and also of its relationship to language. It sets
out to develop the skills of reasoning and an ability to work with abstract
formalism; as well as being important in their own right, these will help im-
prove skills of program design and development in the computer scientist.
The main logical system developed in the book is that of natural deduction,
though necessarilytruthsemanticsare alsodeveloped. In addition, thereis a
briefintroductiontoautomatedreasoningandlogicprogramming.
There are a number of distinctive pedagogical features to the book. The
material is presented in a carefully explained step-by-step approach with co-
pious worked examples and exercises. The solutions to these exercises are
considered to be an integral part of the exposition and so all the solutions
xi
xii Preface
have been included in an appendix. One difficulty of teaching any symbolic
subject, whether mathematics or formal logic, is that many students feel un-
comfortableunlesstheycanattachanameandameaningtoallsymbolsthey
encounter; this problemis particularlyacute for studentswhoare just begin-
ning their studies. In logic, this problem manifests itself most notably with
theconditionalconnective ⇒,andisaggravatedbythealmostuniversalhabit
of introducing this connective at the very beginning. In this book, the condi-
tional is not introduced until after many of the important concepts of logic
have already been introduced. At this stage it is now possible to present a
deeperexplorationof theconceptof theconditional connective.
Another difficulty that can arise for many students is understanding the
notionof‘term’. Partofthedifficultyofthismaylieintheconfusionsurround-
ing the different uses of the word ‘variable’. This book attempts to alleviate
thisdifficultybyintroducingtheconceptofarbitraryconstant,asdistinctfrom
that of proper constant; and by defining the concept of term using only con-
stants and functions. Although this may seem a little unconventional, there
is precedent for it in the books by Lemmon (1965) and Galton (1990). The
advantage of this approach, as pointed out by Lemmon, is pedagogical. The
disadvantage is that if any readers move on to more advanced work in logic,
theywillencounteradifferentdefinition(andonethatisarguablybetterfroma
technicalpointofview). However,anysuchreadershouldbeabletocopewith
a variety of formalisms and to see the relationships between them, without
beingthrownbyalternativedefinitionsanduseofsymbols.
The material is presented in eight chapters. The first chapter considers
the need for rules of reasoning and the concept of symbol. Chapter 2 intro-
duces symbolic connectives for conjunction, disjunction and negation, and
considershowthesecanbeusedinthesymbolicrepresentationofcompound
propositions. Acentralconceptofthisbookisthatofpropositionalforms(or
schemas); this concept is introduced in Chapter 3, together with truth tables
and properties of propositional forms. Chapter 4 considers the notion of ar-
gument andwhat constitutes avalid argument;this in turn leadstoa consid-
erationofasystemofnaturaldeductioninvolvingthethreeconnectives ¬, ∧
and ∨. TheconditionalconnectiveisintroducedinChapter5,whereitisseen
tobeanessentiallyabstractconceptforwhichanymeaningcanbegivenonly
in terms of logic itself. Chapters6 and 7 develop first order logic, as distinct
fromthelogicofpropositionalforms,andhowthismaybeusedtobuildthe-
ories. AbriefintroductiontologicprogrammingispresentedinChapter8;the
intention hereis not togive a basicunderstandingof logic programming,but
simply to illustrate that there are alternative systems of logic, and that some
of thesemaybemoresuitableforautomationthannaturaldeduction.
Throughoutthebook,copiousworkedexamplesandexercisesareusedto
aidunderstandingoftheconceptsandtodeveloplogicalskills. Sincetheexer-
cisesareanintegralpartofthebook,itwasfeltimportanttoincludesolutions
toalltheexercises(AppendixA).AppendicesB–Eprovideusefulsummariesof
thecontentofthebook. FinallyashortbibliographyisincludedinAppendixF;
thisshouldprovehelpfultoanyonewishingtostudylogicfurther.
