Sheaves, Games, and Model Completions: A Categorical Approach to Nonclassical Propositional Logics (Trends in Logic, 14) [Softcover reprint of hardcover 1st ed. 2002] 9048160367, 9789048160365
This book is an example of fruitful interaction between (non-classical) propo sitionallogics and (classical) model theo
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English Pages 254 Year 2010
Table of contents :
Title
Copyright
Dedication
Contents
1. Introduction
1 Motivating example
2 An overview of the book
3 How to read the book
4 Historical remarks
2. Preliminary notions
1 Basic algebraic structures
2 Model theoretic background
3 Subobjects and regular subobjects
4 Finitely presented algebras
5 Principal congruences
6 The amalgamation property
7 Exercises
3. Model completions
1 r-Heyting categories
2 Model completions and fp algebras
3 Exercises
4. Heyting algebras
1 Basic definitions
2 Finitely presented Heyting algebras
3 Duality for Heyting algebras
4 A combinatorial result
5 Properties of M_H and Ψ_H
6 Second order extensions of IpC
7 Quantifier-elimination
8 M_H is not exact
9 Some applications
10 Projective Heyting algebras
11 Exercises
5. Duality for modal algebras
1 Frames, evaluations and games
2 The category of finite S-frames
3 The category M_S
4 Finitely presented S-algebras
5 Duality
6 Regularity of monomorphisms
7 Combinatorial Conditions
8 Exercises
6. Model completions in modal logic
1 Negative results
2 Diagonalizable algebras
3 Varieties of interior algebras
4 Exercises
7. Algebraically closed models
1 FLEA's
2 Grothendieck Topologies on algebras
3 Sh(T^f, J^f) as a classifying topos
4 Models in Set
5 The case of Heyting algebras
6 Existentially closed algebras
7 Exercises
8. Open problems
9. Appendix
1 Glossary of basic categorical notions
2 Internal algebras in categories
3 Grothendieck topologies
4 Sheaves
5 Associated sheaf functor
6 Properties of Grothendieck toposes
7 Classifying toposes
References
Glossary of notation
Subject index
Title
Copyright
Dedication
Contents
1. Introduction
1 Motivating example
2 An overview of the book
3 How to read the book
4 Historical remarks
2. Preliminary notions
1 Basic algebraic structures
2 Model theoretic background
3 Subobjects and regular subobjects
4 Finitely presented algebras
5 Principal congruences
6 The amalgamation property
7 Exercises
3. Model completions
1 r-Heyting categories
2 Model completions and fp algebras
3 Exercises
4. Heyting algebras
1 Basic definitions
2 Finitely presented Heyting algebras
3 Duality for Heyting algebras
4 A combinatorial result
5 Properties of M_H and Ψ_H
6 Second order extensions of IpC
7 Quantifier-elimination
8 M_H is not exact
9 Some applications
10 Projective Heyting algebras
11 Exercises
5. Duality for modal algebras
1 Frames, evaluations and games
2 The category of finite S-frames
3 The category M_S
4 Finitely presented S-algebras
5 Duality
6 Regularity of monomorphisms
7 Combinatorial Conditions
8 Exercises
6. Model completions in modal logic
1 Negative results
2 Diagonalizable algebras
3 Varieties of interior algebras
4 Exercises
7. Algebraically closed models
1 FLEA's
2 Grothendieck Topologies on algebras
3 Sh(T^f, J^f) as a classifying topos
4 Models in Set
5 The case of Heyting algebras
6 Existentially closed algebras
7 Exercises
8. Open problems
9. Appendix
1 Glossary of basic categorical notions
2 Internal algebras in categories
3 Grothendieck topologies
4 Sheaves
5 Associated sheaf functor
6 Properties of Grothendieck toposes
7 Classifying toposes
References
Glossary of notation
Subject index
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