Justification Logic: Reasoning with Reasons (Cambridge Tracts in Mathematics)

Description

Classical logic is concerned, loosely, with the behaviour of truths. Epistemic logic similarly is about the behaviour of known or believed truths. Justification logic is a theory of reasoning that enables the tracking of evidence for statements and therefore provides a logical framework for the reliability of assertions. This book, the first in the area, is a systematic account of the subject, progressing from modal logic through to the establishment of an arithmetic interpretation of intuitionistic logic. The presentation is mathematically rigorous but in a style that will appeal to readers from a wide variety of areas to which the theory applies. These include mathematical logic, artificial intelligence, computer science, philosophical logic and epistemology, linguistics, and game theory. Justification Logic: Reasoning with Reasons Contents Introduction 1 What Is This Book About? 2 What Is Not in This Book? 1 Why Justification Logic? 1.1 Epistemic Tradition 1.2 Mathematical Logic Tradition 1.3 Hyperintensionality 1.4 Awareness 1.5 Paraconsistency 2 The Basics of Justification Logic 2.1 Modal Logics 2.2 Beginning Justification Logics 2.3 J0, the Simplest Justification Logic 2.4 Justification Logics in General 2.5 Fundamental Properties of Justification Logics 2.6 The First Justification Logics 2.7 A Handful of Less Common Justification Logics 2.7.1 K4³ and J4³ 2.7.2 S5 and JT45 2.7.3 Sahlqvist Examples 2.7.4 S4:2 and JT4:2 2.7.5 KX4 and JX4 3 The Ontology of Justifications 3.1 Generic Logical Semantics of Justifications 3.2 Models for J0 and J 3.3 Basic Models for Positive and Negative Introspection 3.4 Adding Factivity: Mkrtychev Models 3.5 Basic and Mkrtychev Models for the Logic of Proofs LP 3.6 The Inevitability of Possible Worlds: Modular Models 3.7 Connecting Justifications, Belief, and Knowledge 3.8 History and Commentary 4 Fitting Models 4.1 Modal Possible World Semantics 4.2 Fitting Models 4.3 Soundness Examples 4.3.1 J(CS) 4.3.2 LP 4.3.3 K4³ and J4³ 4.3.4 S5 and JT45 4.3.5 Sahlqvist Examples 4.3.6 S4:2 and JT4:2 4.3.7 KX4 and JX4 4.3.8 A Remark about Strong Evidence Functions 4.4 Canonical Models and Completeness 4.4.1 Canonical Modal Logics 4.4.2 Canonical Justification Models 4.4.3 Strong Evidence and Fully Explanatory 4.5 Completeness Examples 4.5.1 LP and Sublogics 4.5.2 J4³ 4.5.3 JT45 4.5.4 Sahlqvist Examples 4.5.5 S4:2 and JT4:2 4.5.6 KX4 and JX4 4.6 Formulating Justification Logics 5 Sequents and Tableaus 5.1 Background 5.2 Classical Sequents 5.3 Sequents for S4 5.4 Sequent Soundness, Completeness, and More 5.5 Classical Semantic Tableaus 5.6 Modal Tableaus for K 5.7 Other Modal Tableau Systems 5.8 Tableaus and Annotated Formulas 5.9 Changing the Tableau Representation 6 Realization – How It Began 6.1 The Logic LP 6.2 Realization for LP 6.3 Comments 7 Realization – Generalized 7.1 What We Do Here 7.2 Counterparts 7.3 Realizations 7.4 Quasi-Realizations 7.5 Substitution 7.6 Quasi-Realizations to Realizations 7.7 Proving Realization Constructively 7.8 Tableau to Quasi-Realization Algorithm 7.9 Tableau to Quasi-Realization Algorithm Correctness 7.10 An Illustrative Example 7.11 Realizations, Nonconstructively 7.12 Putting Things Together 7.13 A Brief Realization History 8 The Range of Realization 8.1 Some Examples We Already Discussed 8.2 Geach Logics 8.3 Technical Results 8.4 Geach Justification Logics Axiomatically 8.5 Geach Justification Logics Semantically 8.6 Soundness, Completeness, and Realization 8.7 A Concrete S4:2/JT4:2 Example 8.8 Why Cut-Free Is Needed 9 Arithmetical Completeness and BHK Semantics 9.1 Arithmetical Semantics of the Logic of Proofs 9.2 A Constructive Canonical Model for the Logic of Proofs 9.3 Arithmetical Completeness of the Logic of Proofs 9.4 BHK Semantics 9.5 Self-Referentiality of Justifications 10 Quantifiers in Justification Logic 10.1 Free Variables in Proofs 10.2 Realization of FOS4 in FOLP 10.2.1 Implications for First-Order BHK Semantics 10.3 Possible World Semantics for FOLP 10.3.1 The Ideas Informally 10.3.2 FOLP Fitting Models 10.3.3 Nonvalidity Examples 10.3.4 Soundness 10.3.5 Language Extensions 10.3.6 The Canonical Fitting Model 10.3.7 Completeness 10.3.8 Fully Explanatory Models 10.3.9 Mkrtychev Models 10.4 Arithmetical Semantics for FOLP 10.4.1 Free Variables in First-Order Derivations 10.4.2 Basic Semantics: Derivations from Hypotheses in PA 10.4.3 Completeness Is Not Attainable 10.4.4 Limitations of Provability Semantics 11 Going Past Modal Logic 11.1 Modeling Awareness 11.2 Precise Models 11.3 Justification Awareness Models 11.4 The Russell Scenario as a JAM 11.4.1 Closure Conditions for Acceptance and Knowledge-Producing Predicates 11.4.2 Can Russell’s Scenario Be Made Modal? 11.5 Kripke Models and Master Justification 11.6 Conclusion References Index