An Introduction to Formal Methods for Philosophy Students Thomas Forster February 20, 2021 2 Contents 1 Introduction 13 1.1 What is Logic? . 13 1.1.1 Exercises for the first week: “Sheet 0” . 13 2 Introduction to Logic 17 2.1 Statements, Commands, Questions, Performatives . 18 2.1.1 Truth-functional connectives . 20 2.1.2 Truth Tables . 21 2.2 The Language of Propositional Logic . 23 2.2.1 Truth-tables for compound expressions . 24 2.2.2 Logical equivalence . 26 2.2.3 Non truth functional connectives . 27 2.3 Intension and Extension . 28 2.3.1 If–then . 31 2.3.2 Logical Form and Valid Argument . 33 2.3.3 The Type-Token Distinction . 33 2.3.4 Copies . 35 2.4 Tautology and Validity . 36 2.4.1 Valid Argument . 36 2.4.2 V and W versus ^ and _ .................... 40 2.4.3 Conjunctive and Disjunctive Normal Form . 41 2.5 Further Useful Logical Gadgetry . 46 2.5.1 The Analytic-Synthetic Distinction . 46 2.5.2 Necessary and Sufficient Conditions . 47 2.5.3 The Use-Mention Distinction . 48 2.5.4 Language-metalanguage distinction . 51 2.5.5 Semantic Optimisation and the Principle of Charity . 52 2.5.6 Inferring A-or-B from A . 54 2.5.7 Fault-tolerant pattern-matching . 54 2.5.8 Overinterpretation . 54 2.5.9 Affirming the consequent . 55 3 4 CONTENTS 3 Proof Systems for Propositional Logic 57 3.1 Arguments by LEGO . 57 3.2 The Rules of Natural Deduction . 57 3.2.1 Worries about reductio and hypothetical reasoning . 60 3.2.2 What do the rules mean?? ................... 62 3.2.3 Goals and Assumptions . 63 3.2.4 The Small Print . 64 3.2.5 Some Exercises . 66 3.3 Soundness of the Natural Deduction Rules . 69 3.4 Harmony and Conservativeness . 72 3.4.1 Conservativeness . 72 3.4.2 Harmony . 72 3.4.3 Maximal Formulæ . 74 3.5 Sequent Calculus . 74 3.5.1 Soundness of the Sequent Rules . 78 3.5.2 The rule of cut . 79 3.5.3 Two tips . 81 3.5.4 Exercises . 82 3.6 Hilbert-style Proofs . 83 3.6.1 The Deduction Theorem . 84 3.7 Interpolation . 86 3.8 Completeness of Propositional Logic . 91 3.8.1 Completeness . 91 3.8.2 Completeness using Sequents . 93 3.9 What is a Completeness Theorem? . 93 3.10 Compactness . 95 3.10.1 Why “compactness”? . 96 3.11 Why do we need proof systems for propositional Logic. ?? . 96 3.12 Some advanced exercises for enthusiasts . 97 3.13 Formal Semantics for Propositional Logic . 97 3.14 Eager and Lazy Evaluation . 99 4 Predicate (first-order) Logic 103 4.1 Towards First-Order Logic . 103 4.1.1 The Syntax of First-order Logic . 106 4.1.2 Warning: Scope ambiguities . 108 4.1.3 First-person and third-person . 109 4.2 Some exercises to get you started . 109 4.3 Comparatives and Superlatives . 114 4.4 Russell’s Theory of Descriptions . 114 4.5 First-order and second-order . 115 4.5.1 Higher-order vs Many-Sorted . 115 4.6 Validity . 116 4.7 Natural Deduction Rules for First-Order Logic . 120 4.8 Sequent Rules for First-Order Logic . 121 4.8.1 Repeat a warning . 122 CONTENTS 5 4.8.2 Some more exercises . 122 4.9 Equality and Substitution . 123 4.9.1 Substitution . 124 4.9.2 Leibniz’s law . 125 4.10 Prenex Normal Form . 125 4.11 Soundness again . 126 4.12 Hilbert-style Systems for First-order Logic . 126 4.13 Semantics for First-order Logic . 126 4.13.1 stuff to fit in . 126 4.13.2 Syntactic types for the various pieces of syntax . 127 4.14 Truth and Satisfaction . 128 4.14.1 Definition of “truth-in-a-structure” . 130 4.14.2 An illustration . 133 4.14.3 Completeness . 135 4.15 Interpolation . 138 4.16 Compactness . 138 4.17 Skolemisation . 138 4.18 What is a Proof? . 138 4.19 Relational Algebra . 139 5 Constructive and Classical truth 143 5.1 A Radical Translation Problem . 149 5.2 Classical Reasoning from a Constructive Point of View . 150 5.2.1 Interpretations, specifically the Negative Interpretation . 153 5.3 Prophecy . 159 6 Possible World Semantics 161 6.0.1 Quantifiers . 163 6.1 Language and Metalanguage again . 163 6.1.1 A Possibly Helpful Illustration . 165 6.2 Some Useful Short Cuts . 165 6.2.1 Double negation . 165 6.2.2 If there is only one world then the logic is classical . 166 6.3 Persistence . 167 6.4 Independence Proofs . 170 6.4.1 Some Worked Examples . 170 6.4.2 Exercises . 173 6.5 Modal Logic . 175 7 Curry-Howard 177 7.1 Decorating Formulæ . 178 7.1.1 The rule of !-elimination . 178 7.1.2 Rules for ^ ........................... 178 7.1.3 Rules for _ ........................... 179 7.2 Propagating Decorations . 180 7.2.1 Rules for ^ ........................... 181 6 CONTENTS 7.2.2 Rules for ! ........................... 181 7.2.3 Rules for _ ........................... 183 7.2.4 Remaining Rules . 183 7.3 Exercises . 184 7.4 Combinators and Hilbert Proofs . 187 8 How Not to Use Logic 189 8.1 Many-Valued Logic is an Error . 192 8.1.1 Many states means many truth-values . ? . 192 8.2 Beware of the concept of logically possible .............. 195 9 Other Logics 197 9.1 Relevance Logic . 198 9.2 Resource Logics . 199 10 Some Applications of Logic 201 10.1 Berkeley’s Master Argument for Idealism . 201 10.1.1 Priest on Berkeley . 202 10.2 Fitch’s Knowability Paradox . 205 10.3 Curry-Howard Unifies Two Riddles . 206 10.3.1 What the Tortoise Said to Achilles . 207 10.3.2 Bradley’s regress . 207 10.4 The Paradoxes . 208 11 Appendices 211 11.1 Notes to Chapter one . 211 11.1.1 The Material Conditional . 211 11.1.2 The Definition of Valid Argument . 213 11.1.3 Valuation and Evaluation . 213 11.2 Notes to Chapter 3 . 215 11.2.1 Hypothetical Reasoning: an Illustration . 215 11.2.2 _-elimination and the ex falso ................. 216 11.2.3 Negation in Sequent Calculus . 217 11.2.4 What is the right way to conceptualise sequents? . 217 11.3 Notes to Chapter 4 . 218 11.3.1 Subtleties in the definition of first-order language . 218 11.3.2 Failure of Completeness of Second-order Logic . 219.