Tools and Techniques in Modal Logic Marcus Kracht II ...

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xii Contents 3.7. Interpolation and Beth Theorems 132 3.8. Tableau Calculi and Interpolation 139 3.9. Modal Consequence Relations 149 Part 2. The General Theory of Modal Logic 157 Chapter 4. Universal Algebra and Duality Theory 159 4.1. More on Products 159 4.2. Varieties, Logics and Equationally Definable Classes 166 4.3. Weakly Transitive Logics II 172 4.4. Stone Representation and Duality 180 4.5. Adjoint Functors and Natural Transformations 188 4.6. Generalized Frames and Modal Duality Theory 194 4.7. Frame Constructions III 202 4.8. Free Algebras, Canonical Frames and Descriptive Frames 208 4.9. Algebraic Characterizations of Interpolation 213 Chapter 5. Definability and Correspondence 219 5.1. Motivation 219 5.2. The Languages of Description 220 5.3. Frame Correspondence — An Example 224 5.4. The Basic Calculus of Internal Descriptions 227 5.5. Sahlqvist’s Theorem 233 5.6. Elementary Sahlqvist Conditions 239 5.7. Preservation Classes 245 5.8. Some Results from Model Theory 252 Chapter 6. Reducing Polymodal Logic to Monomodal Logic 259 6.1. Interpretations and Simulations 259 6.2. Some Preliminary Results 261 6.3. The Fundamental Construction 265 6.4. A General Theorem for Consistency Reduction 274 6.5. More Preservation Results 279 6.6. Thomason Simulations 283 6.7. Properties of the Simulation 292 6.8. Simulation and Transfer — Some Generalizations 304 Chapter 7. Lattices of Modal Logics 313 7.1. The Relevance of Studying Lattices of Logics 313 7.2. Splittings and other Lattice Concepts 315 7.3. Irreducible and Prime Logics 321 7.4. Duality Theory for Upper Continuous Lattices 328 7.5. Some Consequences of the Duality Theory 334 7.6. Properties of Logical Calculi and Related Lattice Properties 342
Contents xiii 7.7. Splittings of the Lattices of Modal Logics and Completeness 348 7.8. Blok’s Alternative 357 7.9. The Lattice of Tense Logics 363 Part 3. Case Studies 373 Chapter 8. Extensions of K4 375 8.1. The Global Structure of EK4 375 8.2. The Structure of Finitely Generated K4–Frames 380 8.3. The Selection Procedure 388 8.4. Refutation Patterns 392 8.5. Embeddability Patterns and the Elementarity of Logics 400 8.6. Logics of Finite Width I 405 8.7. Logics of Finite Width II 414 8.8. Bounded Properties and Precomplete Logics above S4 424 8.9. Logics of Finite Tightness 430 Chapter 9. Logics of Bounded Alternativity 437 9.1. The Logics Containing K.alt1 437 9.2. Polymodal Logics with Quasi–Functional Operators 442 9.3. Colourings and Decolourings 449 9.4. Decidability of Logics 454 9.5. Decidability of Properties of Logics I 458 9.6. Decidability of Properties of Logics II 462 Chapter 10. Dynamic Logic 469 10.1. PDL — A Calculus of Compound Modalities 469 10.2. Axiomatizing PDL 471 10.3. The Finite Model Property 476 10.4. Regular Languages 480 10.5. An Evaluation Procedure 484 10.6. The Unanswered Question 493 10.7. The Logic of Finite Computations 499 Index 505 Bibliography 519
Page 1: Tools and Techniques in Modal Logic
Page 4 and 5: vi About this book the book [43] ha
Page 6 and 7: viii Overview proved in full genera
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Page 13: Part 1 The Fundamentals
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2.1. Syntax of Modal Logics 49 an o
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2.1. Syntax of Modal Logics 51 Howe
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2.2. Modal Algebras 53 for some non
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2.2. Modal Algebras 55 particular,
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2.3. Kripke-Frames and Frames 57 Fi
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2.3. Kripke-Frames and Frames 59 re
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2.4. Frame Constructions I 63 verif
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2.4. Frame Constructions I 67 Put [
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2.5. Some Important Modal Logics 69
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2.5. Some Important Modal Logics 71
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2.6. Decidability and Finite Model
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2.7. Normal Forms 79 Proposition 2.
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2.7. Normal Forms 81 formula of the
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2.7. Normal Forms 85 obtained as fo
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2.8. The Lindenbaum-Tarski Construc
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100 3. Fundamentals of Modal Logic
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Part 2 The General Theory of Modal
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260 6. Reducing Polymodal Logic to
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CHAPTER 7 Lattices of Modal Logics
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7.2. Splittings and other Lattice C
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7.8. Blok’s Alternative 357 as we
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7.8. Blok’s Alternative 359 x =
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CHAPTER 8 Extensions of K4 8.1. The
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8.1. The Global Structure of EK4 37
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8.2. The Structure of Finitely Gene
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8.3. The Selection Procedure 389 Le
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8.3. The Selection Procedure 391 a
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8.4. Refutation Patterns 393 succes
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8.4. Refutation Patterns 395 The no
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8.4. Refutation Patterns 397 Proof.
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8.4. Refutation Patterns 399 Figure
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8.5. Embeddability Patterns and the
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8.5. Embeddability Patterns and the
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8.6. Logics of Finite Width I 405 P
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CHAPTER 9 Logics of Bounded Alterna
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9.1. The Logics Containing K.alt1 P
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9.4. Decidability of Logics 455 �
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CHAPTER 10 Dynamic Logic 10.1. PDL
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10.2. Axiomatizing PDL 471 program
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10.2. Axiomatizing PDL 473 This log
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10.2. Axiomatizing PDL 475 if ϕ
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10.3. The Finite Model Property 477
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10.3. The Finite Model Property 479
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10.4. Regular Languages 481 A finit
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10.4. Regular Languages 483 Theorem
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10.5. An Evaluation Procedure 485 W
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10.5. An Evaluation Procedure 487 (
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10.5. An Evaluation Procedure 489 T
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10.5. An Evaluation Procedure 491 w
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10.6. The Unanswered Question 493
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10.6. The Unanswered Question 495 s
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10.6. The Unanswered Question 497
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10.7. The Logic of Finite Computati
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506 Index CanΛ(var), canΛ(var), 9
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508 Index 402, 410, 415, 422, 423,
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510 Index Parikh, Rohit, x, 505, 50
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512 Index computation trace, 515 co
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514 Index pointed, 62 refined, 206
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516 Index black, 311 negative, 250
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518 Index tack, 408 tautology, 20 t
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520 Bibliography [22] Wim Blok. The
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522 Bibliography [76] Zachary Gleit
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524 Bibliography [129] Marcus Krach
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526 Bibliography [182] Vladimir V.
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528 Bibliography [236] Frank Wolter
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