CONTENTS

1 What is Logic? I 1.1 Logical consequence and logical truth 1.2 Formalization 2 1.3 Metalogic 4 Exercises 1.1-1.2 6 1.4 Application 6 1.5 The nature of logical consequence 7 Exercise 1.3 9 1.6 Logical constants 9 1.7 Extensions, deviations, variations II 1.8 Settheory 12 Exercises 1.4-1. 5 24 2 Propositional Logic 25 2.1 Grammar of PL 25 2.2 The semantic approach to logic 28 2.3 Semantics of PL 29 Exercise 2.1 35 2.4 Establishing validity and invalidity in PL 35 Exercise 2.2 36 2.4.1 Schemas, validity, and invalidity 36 2.5 Sequent proofs in PL 37 2.5.1 Sequents 38 2.5.2 Rules 4° 2.5.3 Sequent proofs 42 2.5.4 Example sequent proofs 43 Exercise 2. 3 46 2.6 Axiomatic proofs in PL 46 Exercise 2.4 5° 2.7 Soundness of PL and proof by induction 5° Exercises 2.5-2.10 55 2.8 PL-proofs and the deduction theorem 56 Exercises 2.11-2.12 62 2.9 Completeness of PL 62 2.9.1 Maximal consistent sets of wffs 62 2.9.2 Maximal consistent extensions 63 2.9.3 F eatures of maximal consistent sets 65 2.9.4 The proof 65 x Contents

3 Beyond Standard Propositional Logic 67 3.1 Alternate connectives 67 3.1.1 Symbolizing truth functions in PL 67 3.1.2 Sheffer stroke 7° 3.1.3 Inadequate connective sets 7° Exercises 3.1-3.3 71 3.2 Polish notation 71 Exercise 3.4 72 3.3 Nonclassical propositionallogics 72 3.4 Three-valued logic 73 3.4.1 Lukasiewicz's system 75 Exercises 3.5-3.6 77 3.4.2 Kleene's tables 77 Exercises 3.7-3.9 79 3.4.3 Determinacy 79 3.4.4 Priesťs logic of paradox 80 Exercises 3.10-3.11 82 3.4.5 Supervaluationism 82 Exercises 3.12-3.16 86 3.5 Intuitionistic propositionallogic: proof theory 86 Exercise 3.17 89 4 Predicate Logic 9° 4.1 Grammar of PC 9° 4.2 Semantics of PC 91 Exercise 4.1 96 4.3 Establishing validity and invalidity in PC 96 Exercises 4.2-4.3 98 4.4 Axiomatic proofs in PC 99 Exercise 4.4 104 4.5 Metalogic of PC 104 Exercise 4.5 106 5 Beyond Standard Predicate Logic 107 5.1 Identity 107 5.1.1 Grammar for the identity sign 107 5.1.2 Semantics for the identity sign 108 5.1.3 Symbolizations with the identity sign 108 Exercises 5.1-5.2 109 5.2 Function symbols 110 Exercise 5.3 III 5.2.1 Grammar for function symbols III 5.2.2 Semantics for function symbols 112 Exercise 5.4 113 5.3 Definite descriptions 113 Contents xi

5.3.1 Grammar for I 114 5.3.2 Semantics for I 114 Exercises 5.5-5.6 117 5.3.3 Elimination of function symbols and descriptions 117 Exercises 5.7-5.8 119 5.4 Further quantifiers 119 5.4.1 Generalized monadic quantifiers 120 Exercise 5.9 121 5.4.2 Generalized binary quantifiers 122 Exercise 5.10 12 3 5.4.3 Second-order logic 12 3 Exercise 5.11 126 5.5 Complex predicates 126 Exercises 5.12-5.13 129 5.6 Free logic 129 5.6.1 Semantics for free logic 129 Exercises 5.14-5.15 13 2 5.6.2 Proof theory for free logic 13 2

6 Modal Propositional Logic 133 6.1 Grammar of MPL 135 6.2 Symbolizations in MPL 135 6.3 Semantics for MPL 137 6.3.1 Kripke models 13 8 Exercise 6.1 143 6.3.2 Establishing validity in MPL 143 Exercise 6.2 145 6.3.3 Establishing invalidity in MPL 145 Exercise 6.3 158 6.4 Axiomatic systems of MPL 158 6.4.1 System K 159 Exercises 6.4-6. 5 166 6.4.2 System D 166 Exercise 6.6 167 6.4.3 System T 167 Exercise 6. 7 167 6.4.4 System B 168 Exercise 6.8 168 6.4.5 System S4 168 Exercise 6.9 169 6.4.6 System Ss 169 Exercise 6.10 170 6.4.7 Substitution of equivalents and modal reduction 170 Exercise 6.11 172 XlI Contents

6.5 Soundness in MPL 173 Exercises 6.12-6.13 174 6.5.1 Soundness of K 174 6.5.2 Soundness of T 175 6.5.3 Soundness of B 175 Exercises 6.14-6.1 5 175 6.6 Completeness in MPL 175 6.6.1 Definition of canonical models 176 6.6.2 Facts about maximal consistent sets 177 Exercise 6.16 178 6.6.3 "Mesh" 179 Exercise 6.17 180 6.6.4 Truth and membership in canonical models 180 6.6.5 Completen es s of systems of MPL 181 Exercises 6.18-6.20 182 7 Beyond Standard Modal Propositional Logic 18 3 7.1 Deontic logic 18 3 Exercises 7.1-7.2 185 7.2 Epistemic logic 185 Exercise 7.3 186 7.3 Propositional tense logic 186 7.3.1 The metaphysics of time 186 7.3.2 Tense opera tors 188 7.3.3 Possible-worlds semantics for tense logic 189 Exercises 7.4-7.5 190 7.3.4 Formal constraints on ::s 190 Exercise 7.6 192 7.4 lntuitionistic propositionallogic: semantics 192 7.4.1 Proof stages 192 Exercises 7.7-7.8 195 7.4.2 Validity and semantic consequence 195 Exercises 7.9-7.10 196 7.4.3 Soundness 196 Exercises 7.11-7.13 198 8 Counterfactuals 199 8.1 Natural-language counterfactuals 199 8.1.1 Antecedents and consequents 199 8.1.2 Can be contingent 200 8.1.3 No augmentation 200 8.1.4 No contraposition 201 8.1.5 Some implications 201 8.1.6 Context dependence 202 8.2 The Lewis-Stalnaker theory 20 3 Contents xiii

8.3 Stalnaker's system 2°4 8.3.1 Syntax ofSC 204 8.3.2 Semantics of SC 2°5 Exercise 8.1 2°7 8.4 Establishing validity in SC 2°7 Exercise 8.2 208 8.5 Establishing invalidity in SC 208 Exercises 8.3-8.4 216 8.6 Logical features of SC 216 8.6.1 No exportation 21 7 8.6.2 No importation 21 7 8.6.3 No permutation 218 8.6.4 No transitivity 218 8.7 Lewis's criticisms of Stalnaker's theory 21 9 8.8 Lewis's system 221 Exercises 8.5-8.6 223 8.9 The problem of disjunctive antecedents 223 8.10 Counterfactuals as strict conditionals 224 9 Quantified Modal Logic 227 9.1 Grammar ofQML 227 9.2 De re and de dicto 227 9.3 A simple semantics for QML 23° 9.4 Establishing validity and invalidity in SQML 232 Exercise 9.1 23 6 9.5 Philosophical questions about SQML 23 6 9.5.1 The necessity of identity 23 6 9.5.2 The necessity of existence 23 8 Exercise 9.2 241 9.5.3 N ecessary existence defended? 241 9.6 Variable domains 244 9.6.1 Contingent existence vindicated 245 Exercises 9.3-9.4 246 9.6.2 lncreasing, decreasing domains 246 Exercise 9.5 247 9.6.3 Strong and weak necessity 247 9.6.4 Actualist and possibilist quantification 249 9.7 Axiomatic proofs in SQML 249 Exercise 9.6 25 2 10 Two-dimensional Modal Logic 253 10.1 Actuality 253 10.1.1 Kripke models with designated worlds 253 Exercise 10.1 254 10.1.2 Semantics for @ 254 XlV Contents

10.1.3 Establishing validity and invalidity 255 10.2 x 255 10.2.1 Two-dimensional semantics for x 256 Exercise 10.2 25 8 10.3 Fixedly 25 8 Exercises 10.3-lO. 5 260 10.4 Necessity and apriority 260 Exercises lO.6-lO.9 265 Appendix A Answers and Hints to Selected Exercises 266 References 281 Index 28 5