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Definability and Interpolation: Model-theoretic investigations

Hoogland, E.

Publication date 2001

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Citation for published version (APA):

Hoogland, E. (2001). Definability and Interpolation: Model-theoretic investigations. Institute for Logic, Language and Computation.

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Download date:24 Sep 2022

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Contents s

A c k n o w l e d g m e n t ss xi

11 I n t r o d u c t i o n 1 1.11 W h a t to expect from this dissertation? 2

1.22 Organization of the dissertation 4 1.33 About the origins of the various chapters 5

22 A general p i c t u r e 7 2.11 Preface 8 2.22 Definability 9

2.2.11 Predicate logics 9 2.2.22 Prepositional logics 14 2.2.33 Related definability properties 17

2.33 Interpolation 20 2.3.11 Craig interpolation property 20

2.3.22 Applications: definability and preservation 26

2.3.33 A global interpolation property 30 2.3.44 Robinson's consistency property 33

2.3.55 Discussion 35 2.44 Table of results 39 2.55 Algebraic equivalents 41

2.5.11 Amalgamation properties 41 2.5.22 Surjectiveness of epimorphisms 46 2.5.33 On the relations between ES. AP and SAP 50

2.66 Logic versus algebra 53 2.6.11 Interpolation and amalgamation 53

2.6.22 Definability and epimorphisms 56

2.6.33 A schematic overview 58 vii i

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33 A l g e b r a i c e q u i v a l e n t s of definability p r o p e r t i e s 6 1

3.11 Introduction to abstract algebraic logic 62 3.22 Logic as a consequence relation 65

3.2.11 A'-deductive systems 65 3.2.22 Basics on matrices 67 3.2.33 Matrix semantics 68 3.2.44 Protoalgebraic and equivalential deductive systems . . . . 70

3.2.55 Translations and equivalent deductive systems 73 3.33 An algebraic characterization of the Beth property 76

3.3.11 T h e Beth property in abstract algebraic logic 76 3.3.22 Characterization of Beth property for equivalential logics . 79

3.3.33 Characterization of Beth property for algebraizable logics . 84

3.44 Characterizing the projective Beth property 86

3.55 Examples and applications 89 3.5.11 Sufficient conditions for failure of ES 89

3.5.22 Applications of the characterization theorem 92

3.66 The weak Beth property in algebraic logic 101 3.6.11 The model-theoretic framework 101 3.6.22 A characterization of the weak Beth property 108

44 G u a r d e d f r a g m e n t s 117 4.11 Introducing some guarded fragments 118

4.22 Preliminaries 121 4.33 Failure of interpolation in G F and P F 125

4.44 Restoring interpolation 130 4.55 An excursion into definability 139 4.66 Guarded finite variable fragments with interpolation 143

4.77 A Lyndon theorem for GF 147

4.88 Appendix 152 55 I n t e r p r e t a b i l i t y logics 155

5.11 Introduction to interpretability logics 156

5.22 Preliminaries 157 5.33 The interpolation theorem for IL 162

5.44 Corollary: interpolation for I LP 170 5.55 Failure of interpolation in ILW 170 5.66 Beth definability and fixed points 173

5.6.11 From Beth definability to fixed points 175 5.6.22 From fixed points to Beth definability 177

AA A p p e n d i x 181 A.0.33 Normal modal logics 182

A.0.44 Some other logics 185 viii i

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Bibliographyy 187

Indexx 201 Listt of Symbols 205

Samenvattingg 207

IX X

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