Contents
Introduction page x
1 What Is This Book About? xii
2 What Is Not in This Book? xvii
1 Why Justification Logic? 1
1.1 EpistemicTradition 1
1.2 MathematicalLogicTradition 4
1.3 Hyperintensionality 8
1.4 Awareness 9
1.5 Paraconsistency 10
2 The Basics of Justification Logic 11
2.1 Modal Logics 11
2.2 Beginning Justification Logics 12
2.3 J0, the Simplest Justification Logic 14
2.4 JustificationLogicsinGeneral 15
2.5 Fundamental Properties of Justification Logics 20
2.6 The First Justification Logics 23
2.7 A Handful of Less Common Justification Logics 27
3 The Ontology of Justifications 31
3.1 Generic Logical Semantics of Justifications 31
3.2 Models for J0 and J 36
3.3 BasicModelsforPositiveandNegativeIntrospection 38
3.4 Adding Factivity: Mkrtychev Models 39
3.5 Basic and Mkrtychev Models for the Logic of Proofs LP 42
3.6 TheInevitability of Possible Worlds: ModularModels 42
3.7 Connecting Justifications, Belief, and Knowledge 45
3.8 History and Commentary 46
4 Fitting Models 48
4.1 ModaipossibleWorldSemantics 48
4.2 Fitting Models 49
4.3 Soundness Examples 52
4.4 Canonical Models and Completeness 60
4.5 Completeness Examples 65
4.6 Formulating Justification Logics 72
5 Sequents and Tableaus 75
5.1 Background 75
5.2 Classical Sequents 76
5.3 Sequents for S4 79
5.4 Sequent Soundness, Completeness, and More 81
5.5 Classical Semantic Tableaus 84
5.6 Modal Tableaus for K 90
5.7 Other Modal Tableau Systems 91
5.8 Tableaus and Annotated Formulas 93
5.9 Changing the Tableau Representation 95
6 Realization - How It Began 100
6.1 TheLogic LP 100
6.2 Realization for LP 103
6.3 Comments 108
7 Realization - Generalized 110
7.1 WhatWeDoHere 110
7.2 Counterparts 112
7.3 Realizations 113
7.4 Quasi-Realizations 116
7.5 Substitution 118
7.6 Quasi-Realizations to Realizations 120
7.7 proving Realization Constructively 126
7.8 Tableau to Quasi-Realization Algorithm 128
7.9 Tableau to Quasi-Realization Algorithm Correctness 131
7.10 AnIllustrativeExample 133
7.11 Realizations, Nonconstructively 135
7.12 PuttingThingsTogether 138
7.13 A Brief Realization History 139
8 TheRangeofRealization 141
8.1 Some Examples We Already Discussed 141
8.2 Geach Logics 142
8.3 Technical Results 144
8.4 GeachjustificationLogicsAxiomatically 147
8.5 GeachJustificationLogicsSemantically 149
8.6 Soundness, Completeness, and Realization 150
8.7 AConcrete S4.2/JT4.2 Example 152
8.8 Why Cut-Free Is Needed 155
9 Arithmetical Completeness and BHK Semantics 158
9.1 Arithmetical Semantics of the Logic of Proofs 158
9.2 A Constructive Canonical Model for the Logic of Proofs 161
9.3 Arithmetical Completeness of the Logic of Proofs 165
9.4 BHK Semantics 174
9.5 Self-Referentiality of Justifications 179
10 QuantifiersinJustificationLogic 181
10.1 Free Variables in Proofs 182
10.2 Realization of FOS4 in FOLP 186
10.3 PossibleWorldSemanticsfor FOLP 191
10.4 Arithmetical Semantics for FOLP 212
11 Going Past Modal Logic 222
11.1 Modeling Awareness 223
11.2 Precise Models 225
11.3 Justification Awareness Models 226
11.4 The Russell Scenario as a JAM 228
11.5 KripkeModelsandMasterJustification 231
11.6 Conclusion 233
References 234
Index 244