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Preface 1 Modal Operators 1.1 A new language 1.2 Flavours of modality 1.3 The turnstile 1.4 Duality 1.5 A system of modal logic 2 Possible Worlds 2.1 The possible-worlds analysis of possibility and necessity 2.2 Models 2.3 Basic entailment and validity 2.4 Explorations in S5 2.5 Trees 3 Accessibility 3.1 Variable modality 3.2 The systems K and S5 3.3 Some other normal systems 3.4 Frames 3.5 More trees 4 Models and Proofs 4.1 Soundness and completeness 4.2 Soundness for trees 4.3 Completeness for trees 4.4 Soundness and completeness for axiomatic calculi 4.5 Loose ends 5 Epistemic Logic 5.1 Epistemic accessibility 5.2 The logic of knowledge 5.3 Multiple Agents 5.4 Knowledge, belief, and other modalities 6 Deontic Logic 6.1 Permission and obligation 6.2 Standard deontic logic 6.3 Norms and circumstances 6.4 Further challenges 6.5 Neighbourhood semantics 7 Temporal Logic 7.1 Reasoning about time 7.2 Temporal models 7.3 Logics of time 7.4 Branching time 7.5 Extending the language 8 Conditionals 8.1 Material conditionals 8.2 Strict conditionals 8.3 Variably strict conditionals 8.4 Restrictors 9 Towards Modal Predicate Logic 9.1 Predicate logic recap 9.2 Modal fragments of predicate logic 9.3 Predicate logic proofs 9.4 Modality de dicto and de re 9.5 Identity and descriptions 10 Semantics for Modal Predicate Logic 10.1 Constant domain semantics 10.2 Quantification and existence 10.3 Variable-domain semantics 10.4 Trans-world identity 11 Answers to the Exercises
Logic 2: Modal Logic

Preface

These notes are aimed at philosophy students who have taken an introductory course in formal logic. They provide an introduction to modal logic, with may philosophical applications. Along the way, they introduce general ideas that might be taught in an intermediate logic course: different methods of proof, the concept of a model, soundness and completeness, compactness, three-valued logics, free logics, supervaluation, properties of relations and orders, etc.

Chapters 1–3 introduce the standard toolkit of modal propositional logic: Kripke models, frame correspondence, some popular systems, the tableau method and axiomatic calculi. Chapter 4 goes through soundness and completeness. Chapters 5–8 turn to philosophical applications. Each of these chapters also extends the toolkit from chapter 3. Chapter 5 introduces multi-modal logics, chapter 6 ordering models and neighbourhood semantics, chapter 6 two-dimensional semantics and supervaluationism, chapter 7 conditional logics and Lewis-Stalnaker models. Chapters 9 and 10 look at some of the complexities that arise in first-order modal logic.

Apart from chapter 9, which sets the stage for chapter 10, every chapter after chapter 3 can be skipped or skimmed without affecting the accessibility of later chapters.

The best way to learn logic is by solving problems. That’s why the text is frequently interrupted by exercises. As a student, you should try to do the exercises as soon as you reach them, before continuing with the text.

Next chapter: 1 Modal Operators