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Tools and Techniques in Modal Logic, Volume 142 [Hardback]

(Freie Universität Berlin, II. Mathematisches Institut, Berlin, Germany)
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Tools and Techniques in Modal Logic, Volume 142Palielināt
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Permanent link: https://www.kriso.lv/db/9780444500557.html
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This book treats modal logic as a theory, with several subtheories, such as completeness theory, correspondence theory, duality theory and transfer theory and is intended as a course in modal logic for students who have had prior contact with modal logic and who wish to study it more deeply. It presupposes training in mathematical or logic. Very little specific knowledge is presupposed, most results which are needed are proved in this book.


This book treats modal logic as a theory, with several subtheories, such as completeness theory, correspondence theory, duality theory and transfer theory and is intended as a course in modal logic for students who have had prior contact with modal logic and who wish to study it more deeply. It presupposes training in mathematical or logic. Very little specific knowledge is presupposed, most results which are needed are proved in this book.

Recenzijas

"....As a whole, the book deserves attention from specialists in the field....The author has succeeded in showing that the use of mathematical (algebraic and topological) tools provides an excellent means for deeper understanding of the theory." --Bulletin of the Symbolic Logic

About this Book v(2)
Overview vii
Part
1. The Fundamentals		 1(164)
Chapter
1. Algebra, Logic and Deduction
3(44)
1.1. Basic Facts and Structures
3(5)
1.2. Propositional Languages
8(6)
1.3. Algebraic Constructions
14(4)
1.4. General Logic
18(5)
1.5. Completeness of Matrix Semantics
23(3)
1.6. Properties of Logics
26(5)
1.7. Boolean Logic
31(7)
1.8. Some Notes on Computation and Complexity
38(9)
Chapter
2. Fundamentals of Modal Logic I
47(56)
2.1. Syntax of Modal Logics
47(8)
2.2. Modal Algebras
55(5)
2.3. Kripke-Frames and Frames
60(5)
2.4. Frame Constructions I
65(6)
2.5. Some Important Modal Logics
71(4)
2.6. Decidability and Finite Model Property
75(7)
2.7. Normal Forms
82(8)
2.8. The Lindenbaum-Tarski Construction
90(6)
2.9. The Lattices of Normal and Quasi-Normal Logics
96(7)
Chapter
3. Fundamentals of Modal Logic II
103(62)
3.1. Local and Global Consequence Relations
103(6)
3.2. Completeness, Correspondence and Persistence
109(7)
3.3. Frame Constructions II
116(6)
3.4. Weakly Transitive Logics I
122(2)
3.5. Subframe Logics
124(7)
3.6. Constructive Reduction
131(7)
3.7. Interpolation and Beth Theorems
138(8)
3.8. Tableau Calculi and Interpolation
146(10)
3.9. Modal Consequence Relations
156(9)
Part
2. The General Theory of Modal Logic		 165(230)
Chapter
4. Universal Algebra and Duality Theory
167(64)
4.1. More on Products
167(8)
4.2. Varieties, Logics and Equationally Definable Classes
175(6)
4.3. Weakly Transitive Logics II
181(9)
4.4. Stone Representation and Duality
190(8)
4.5. Adjoint Functors and Natural Transformations
198(7)
4.6. Generalized Frames and Modal Duality Theory
205(9)
4.7. Frame Constructions III
214(6)
4.8. Free Algebras, Canonical Frames and Descriptive Frames
220(5)
4.9. Algebraic Characterizations of Interpolation
225(6)
Chapter
5. Definability and Correspondence
231(44)
5.1. Motivation
231(1)
5.2. The Languages of Description
232(4)
5.3. Frame Correspondence -- An Example
236(3)
5.4. The Basic Calculus of Internal Descriptions
239(7)
5.5. Sahlqvist's Theorem
246(6)
5.6. Elementary Sahlqvist Conditions
252(6)
5.7. Preservation Classes
258(8)
5.8. Some Results from Model Theory
266(9)
Chapter
6. Reducing Polymodal Logic to Monomodal Logic
275(56)
6.1. Interpretations and Simulations
275(3)
6.2. Some Preliminary Results
278(3)
6.3. The Fundamental Construction
281(10)
6.4. A General Theorem for Consistency Reduction
291(5)
6.5. More Preservation Results
296(4)
6.6. Thomason Simulations
300(10)
6.7. Properties of the Simulation
310(12)
6.8. Simulation and Transfer -- Some Generalizations
322(9)
Chapter
7. Lattices of Modal Logics
331(64)
7.1. The Relevance of Studying Lattices of Logics
331(2)
7.2. Splittings and other Lattice Concepts
333(7)
7.3. Irreducible and Prime Logics
340(7)
7.4. Duality Theory for Upper Continuous Lattices
347(6)
7.5. Some Consequences of the Duality Theory
353(9)
7.6. Properties of Logical Calculi and Related Lattice Properties
362(6)
7.7. Splittings of the Lattices of Modal Logics and Completeness
368(10)
7.8. Blok's Alternative
378(6)
7.9. The Lattice of Tense Logics
384(11)
Part
3. Case Studies		 395(140)
Chapter
8. Extensions of K4
397(66)
8.1. The Global Structure of (Epsilon)K4
397(5)
8.2. The Structure of Finitely Generated K4-Frames
402(9)
8.3. The Selection Procedure
411(4)
8.4. Refutation Patterns
415(9)
8.5. Embeddability Patterns and the Elementarity of Logics
424(6)
8.6. Logics of Finite Width I
430(9)
8.7. Logics of Finite Width II
439(11)
8.8. Bounded Properties and Precomplete Logics above S4
450(6)
8.9. Logics of Finite Tightness
456(7)
Chapter
9. Logics of Bounded Alternativity
463(34)
9.1. The Logics Containing K.alt(1)
463(5)
9.2. Polymodal Logics with Quasi-Functional Operators
468(7)
9.3. Colourings and Decolourings
475(7)
9.4. Decidability of Logics
482(3)
9.5. Decidability of Properties of Logics I
485(5)
9.6. Decidability of Properties of Logics II
490(7)
Chapter
10. Dynamic Logic
497(38)
10.1. PDL -- A Calculus of Compound Modalities
497(2)
10.2. Axiomatizing PDL
499(6)
10.3. The Finite Model Property
505(3)
10.4. Regular Languages
508(6)
10.5. An Evaluation Procedure
514(9)
10.6. The Unanswered Question
523(6)
10.7. The Logic of Finite Computations
529(6)
Index 535(14)
Bibliography 549