Logic : the laws of truth

545 Pages·2012·1.943 MB·English

by  Nicholas J J Smith

LOGIC
TheLawsofTruth
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LOGIC
The Laws of Truth
NICHOLAS J. J. SMITH
PRINCETONUNIVERSITYPRESS PRINCETONANDOXFORD
.
Copyright©2012byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,41WilliamStreet,Princeton,NewJersey08540
IntheUnitedKingdom:PrincetonUniversityPress,6OxfordStreet,Woodstock,
OxfordshireOX201TW
press.princeton.edu
AllRightsReserved
LibraryofCongressCataloging-in-PublicationData
Smith,NicholasJ.J.(NicholasJeremyJosef),1972–
Logic:thelawsoftruth/NicholasJ.J.Smith.
pages cm
Includesbibliographicalreferencesandindex.
ISBN978-0-691-15163-2(hardcover:alk.paper)
1. Logic. I. Title.
BC71.S616 2012
160—dc23 2011048269
BritishLibraryCataloging-in-PublicationDataisavailable
ThisbookhasbeencomposedinMinionandMyriadProusingZzTEX
byPrincetonEditorialAssociates,Inc.,Scottsdale,Arizona.
Printedonacid-freepaper.
PrintedintheUnitedStatesofAmerica
1 3 5 7 9 10 8 6 4 2
ForCath,JeremyandOliver
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CONTENTS
Preface xi
Acknowledgments xv
PARTI PropositionalLogic 1
1 PropositionsandArguments 3
1.1 WhatIsLogic? 3
1.2 Propositions 5
1.3 Arguments 11
1.4 LogicalConsequence 14
1.5 Soundness 21
1.6 Connectives 23
2 TheLanguageofPropositionalLogic 32
2.1 Motivation 32
2.2 BasicPropositionsofPL 32
2.3 ConnectivesofPL 36
2.4 WffVariables 39
2.5 SyntaxofPL 40
3 SemanticsofPropositionalLogic 49
3.1 TruthTablesfortheConnectives 49
3.2 TruthValuesofComplexPropositions 51
3.3 TruthTablesforComplexPropositions 54
3.4 TruthTablesforMultiplePropositions 58
3.5 ConnectivesandTruthFunctions 59
4 UsesofTruthTables 63
4.1 Arguments 63
4.2 SinglePropositions 67
4.3 TwoPropositions 69
4.4 SetsofPropositions 74
4.5 MoreonValidity 75
5 LogicalForm 79
5.1 AbstractingfromContent:FromPropositionstoForms 81
5.2 Instances:FromFormstoPropositions 82
5.3 ArgumentForms 84
5.4 ValidityandForm 87
5.5 InvalidityandForm 91
5.6 NotableArgumentForms 94
5.7 OtherLogicalProperties 95
6 Connectives:TranslationandAdequacy 97
6.1 AssertibilityandImplicature 97
6.2 Conjunction 103
6.3 ConditionalandBiconditional 110
6.4 Disjunction 117
6.5 Negation 122
6.6 FunctionalCompleteness 124
7 TreesforPropositionalLogic 134
7.1 TreeRules 136
7.2 ApplyingtheRules 140
7.3 UsesofTrees 146
7.4 Abbreviations 156
PARTII PredicateLogic 161
8 TheLanguageofMonadicPredicateLogic 163
8.1 TheLimitationsofPropositionalLogic 164
8.2 MPL,PartI:NamesandPredicates 167
8.3 MPL,PartII:VariablesandQuantifiers 172
8.4 SyntaxofMPL 182
9 SemanticsofMonadicPredicateLogic 189
9.1 Models;TruthandFalsityofUncomplicatedPropositions 191
9.2 Connectives 196
9.3 QuantifiedPropositions:TheGeneralCase 197
9.4 SemanticsofMPL:Summary 204
9.5 AnalysesandMethods 206
10 TreesforMonadicPredicateLogic 211
10.1TreeRules 212
10.2UsingTrees 223
10.3InfiniteTrees 228
11 Models,Propositions,andWaystheWorldCouldBe 242
11.1Translation 243
11.2Valuation 247
11.3Axiomatization 251
viii Contents
11.4Propositions 253
11.5LogicalConsequenceandNTP 257
11.6Postulates 261
12 GeneralPredicateLogic 264
12.1TheLanguageofGeneralPredicateLogic 264
12.2SemanticsofGPL 276
12.3TreesforGeneralPredicateLogic 282
12.4Postulates 286
12.5MovingQuantifiers 293
13 Identity 298
13.1TheIdentityRelation 299
13.2TheIdentityPredicate 303
13.3SemanticsofIdentity 306
13.4TreesforGeneralPredicateLogicwithIdentity 311
13.5NumericalQuantifiers 321
13.6DefiniteDescriptions 326
13.7FunctionSymbols 343
PARTIII FoundationsandVariations 355
14 Metatheory 357
14.1SoundnessandCompleteness 358
14.2DecidabilityandUndecidability 368
14.3OtherLogicalProperties 374
14.4ExpressivePower 382
15 OtherMethodsofProof 385
15.1AxiomaticSystems 386
15.2NaturalDeduction 407
15.3SequentCalculus 421
16 SetTheory 438
16.1Sets 438
16.2OrderedPairsandOrderedn-tuples 449
16.3Relations 453
16.4Functions 454
16.5Sequences 458
16.6Multisets 460
16.7Syntax 462
Notes 467
References 509
Index 515
Contents ix