The Tarskian Turn: Deflationism and Axiomatic Truth

Contents

Contents Preface xi 1 About This Book 1 1.1 What Is It About? 1 1.2 What Is It Not About? 5 1.3 For Whom Is It Intended? 6 1.4 How Is It Structured? 7 1.5 Note on Technicality and Notation 8 2 Axiomatic Theories of Truth 11 2.1 Pilate’s Question 11 2.2 Essence and Function 14 2.3 Tarski’s Distress 19 2.4 The Framework 22 2.5 Soundness and Strength 24 3 On the Shoulders of Giants 27 3.1 Introduction 27 3.2 Coding in the Language of Peano Arithmetic 28 3.3 The Diagonal Lemma 29 3.4 Metatheorems 31 3.4.1 The First Incompleteness Theorem 31 3.4.2 The Second Incompleteness Theorem 32 3.4.3 The Completeness Theorem 34 3.4.4 The Undefinability Theorem 35 3.5 More Strengthenings 36 3.5.1 The Extended Diagonal Lemma 36 3.5.2 The Naive Theory of Truth 37 3.5.3 The Paradox of the Knower 39 3.6 Bounded Truth Predicates 41 3.6.1 Complexity Classes of Arithmetical Formulae 41 3.6.2 True Equations 42 3.6.3 Defining Bounded Truth Predicates 43 3.7 Simplify, Simplify 44 viii Contents 4 The Disquotational Theory 47 4.1 Tarski on Defining Truth 47 4.2 The Disquotational Theory of Truth 49 4.3 The Soundness of the Disquotational Theory 51 4.4 Climbing Tarski’s Ladder 53 4.5 The Uniformity of the Concept of Truth 55 4.6 Contextual Theories of Truth 56 5 Deflationism 59 5.1 The Unbearable Lightness of Truth 59 5.2 Commitments of Deflationist Theories 60 5.2.1 The Meaning of the Concept of Truth 61 5.2.2 The Function of the Concept of Truth 63 5.2.3 Truth as a Logico-Linguistic Notion 65 5.3 Foreign Tongues 66 6 The Compositional Theory 69 6.1 Clouds on the Horizon 69 6.2 The Compositional Theory of Truth 71 6.3 Truth and Satisfaction 73 6.4 The Power of Truth 74 7 Conservativeness and Deflationism 79 7.1 Defining Conservativeness 79 7.2 Conservative Over What? 81 7.2.1 Conservativeness Over Logic 81 7.2.2 Conservativeness Over Arithmetic 82 7.2.3 Conservativeness Over Empirical Science and Metaphysics 85 7.3 Truth and Epistemology 86 7.4 Truth and Meaning 91 7.5 Deflating Arithmetical Nonconservativeness 92 7.6 Substantiality and Irreducibility 94 7.6.1 Reducibility and Interpretability 94 7.6.2 A Conservative but Noninterpretable Truth Theory 95 7.6.3 Disquotationalism Revisited 97 7.7 A Serious Game 99 8 Maximizing Classical Compositionality 103 8.1 Typed and Untyped Theories 103 8.2 The Friedman–Sheard Theory 104 8.3 The Revision Theory of Truth 105 8.4 Probing the Friedman–Sheard Theory 107 8.5 Is the Friedman–Sheard Theory Sound? 111 8.6 A Somewhat Frivolous Game 113 8.6.1 Subsystems of Second-Order Arithmetic 113 8.6.2 The Strength of FS 114 8.6.3 The Weakness of FS 115 Contents ix 9 Kripke’s Theory Axiomatized 117 9.1 Kripke’s Semantical Theory of Truth 117 9.1.1 Constructing Models for Self-Referential Truth 117 9.1.2 Properties and Variations 120 9.2 Kripke–Feferman 124 9.3 The Inner Logic of KF 126 9.4 A Conservative Type-Free Theory 130 9.5 Partial Kripke–Feferman 132 9.5.1 Restricted Conditionalization 132 9.5.2 Partial Arithmetic and Determinate Truths 133 9.5.3 The Truth Rules 134 9.6 Analysis and Evaluation 135 9.6.1 Naturalness 135 9.6.2 Methodology 136 9.6.3 Soundness 137 9.6.4 Strength 138 10 Truth and Philosophy 141 10.1 Strong Theories of Truth 141 10.2 Inferential Deflationism 143 10.2.1 Truth and Logical Notions 143 10.2.2 Silence 144 10.2.3 A Concept Without an Essence 146 10.3 How Truth Rules Are Used 148 10.3.1 Truth in the Foundations of Mathematics 148 10.3.2 Truth in Philosophy 150 10.4 A Last Look at Deflationism 150 Bibliography 153 Glossary 159 Author Index 161 Subject Index 163 ...