E-Book, Englisch, Band 153, 641 Seiten, Format (B × H): 150 mm x 225 mm
Gabbay / Skvortsov / Shehtman Quantification in Nonclassical Logic
1. Auflage 2009
ISBN: 978-0-08-093112-8
Verlag: Elsevier Science & Technology
Format: EPUB
Kopierschutz: 6 - ePub Watermark
E-Book, Englisch, Band 153, 641 Seiten, Format (B × H): 150 mm x 225 mm
Reihe: Studies in Logic and the Foundations of Mathematics
ISBN: 978-0-08-093112-8
Verlag: Elsevier Science & Technology
Format: EPUB
Kopierschutz: 6 - ePub Watermark
Quantification and modalities have always been topics of great interest for logicians. These two themes emerged from philosophy and
language in ancient times; they were studied by traditional informal
methods until the 20th century. In the last century the tools became
highly mathematical, and both modal logic and quantification found numerous applications in Computer Science. At the same time many other kinds of nonclassical logics were investigated and applied to Computer Science.
Although there exist several good books in propositional modal logics, this book is the first detailed monograph in nonclassical first-order quantification. It includes results obtained during the past thirty years. The field is very large, so we confine ourselves with only two kinds of logics: modal and superintuitionistic. The main emphasis of Volume 1 is model-theoretic, and it concentrates on descriptions of different sound semantics and completeness problem --- even for these seemingly simple questions we have our hands full. The major part of the presented material has never been published before. Some results are very recent, and for other results we either give new proofs or first proofs in full detail.
Dov M. Gabbay is Augustus De Morgan Professor Emeritus of Logic at the Group of Logic, Language and Computation, Department of Computer Science, King's College London. He has authored over four hundred and fifty research papers and over thirty research monographs. He is editor of several international Journals, and many reference works and Handbooks of Logic.
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Weitere Infos & Material
1;Front Cover;1
2;Quantification in Nonclassical Logic;4
3;Copyright Page;5
4;Contents;22
5;Preface;6
6;Introduction;10
7;Part I Preliminaries;26
7.1;Chapter 1 Basic propositional logic;28
7.1.1;1.1 Propositional syntax;28
7.1.2;1.2 Algebraic semantics;36
7.1.3;1.3 Relational semantics (the modal case);44
7.1.4;1.4 Relational semantics (the intuitionistic case);57
7.1.5;1.5 Modal counterparts;62
7.1.6;1.6 General Kripke frames;63
7.1.7;1.7 Canonical Kripke models;65
7.1.8;1.8 First-order translations and definability;69
7.1.9;1.9 Some general completeness theorems;72
7.1.10;1.10 Trees and unravelling;73
7.1.11;1.11 PTC-logics and Horn closures;77
7.1.12;1.12 Subframe and cofinal subframe logics;83
7.1.13;1.13 Splittings;90
7.1.14;1.14 Tabularity;93
7.1.15;1.15 Transitive logics of finite depth;95
7.1.16;1.16 ?-operation;97
7.1.17;1.17 Neighbourhood semantics;101
7.2;Chapter 2 Basic predicate logic;104
7.2.1;2.1 Introduction;104
7.2.2;2.2 Formulas;106
7.2.3;2.3 Variable substitutions;110
7.2.4;2.4 Formulas with constants;127
7.2.5;2.5 Formula substitutions;130
7.2.6;2.6 First-order logics;144
7.2.7;2.7 First-order theories;164
7.2.8;2.8 Deduction theorems;167
7.2.9;2.9 Perfection;171
7.2.10;2.10 Intersections;176
7.2.11;2.11 Cödel–Tarski translation;178
7.2.12;2.12 The Glivenko theorem;182
7.2.13;2.13 ?-operation;183
7.2.14;2.14 Adding equality;197
7.2.15;2.15 Propositional parts;205
7.2.16;2.16 Semantics from an abstract viewpoint;210
8;Part II Semantics;216
8.1;Introduction: What is semantics?;218
8.2;Chapter 3 Kripke semantics;224
8.2.1;3.1 Preliminary discussion;224
8.2.2;3.2 Predicate Kripke frames;230
8.2.3;3.3 Morphisms of Kripke frames;244
8.2.4;3.4 Constant domains;255
8.2.5;3.5 Kripke frames with equality;259
8.2.6;3.6 Kripke sheaves;268
8.2.7;3.7 Morphisms of Kripke sheaves;278
8.2.8;3.8 Transfer of completeness;284
8.2.9;3.9 Simulation of varying domains;291
8.2.10;3.10 Examples of Kripke semantics;293
8.2.11;3.11 On logics with closed or decidable equality;302
8.2.12;3.12 Translations into classical logic;306
8.3;Chapter 4 Algebraic semantics;318
8.3.1;4.1 Modal and Heyting valued structures;318
8.3.2;4.2 Algebraic models;326
8.3.3;4.3 Soundness;336
8.3.4;4.4 Morphisms of algebraic structures;344
8.3.5;4.5 Presheaves and ?-sets;353
8.3.6;4.6 Morphisms of presheaves;358
8.3.7;4.7 Sheaves;363
8.3.8;4.8 Fibrewise models;364
8.3.9;4.9 Examples of algebraic semantics;366
8.4;Chapter 5 Metaframe semantics;370
8.4.1;5.1 Preliminary discussion;370
8.4.2;5.2 Kripke bundles;376
8.4.3;5.3 More on forcing in Kripke bundles;381
8.4.4;5.4 Morphisms of Kripke bundles;384
8.4.5;5.5 Intuitionistic Kripke bundles;390
8.4.6;5.6 Functor semantics;399
8.4.7;5.7 Morphisms of presets;406
8.4.8;5.8 Bundles over precategories;411
8.4.9;5.9 Metaframes;413
8.4.10;5.10 Permutability and weak functoriality;422
8.4.11;5.11 Modal metaframes;429
8.4.12;5.12 Modal soundness;434
8.4.13;5.13 Representation theorem for modal metaframes;444
8.4.14;5.14 Intuitionistic forcing and monotonicity;447
8.4.15;5.15 Intuitionistic soundness;457
8.4.16;5.16 Maximality theorem;477
8.4.17;5.17 Kripke quasi-bundles;490
8.4.18;5.18 Some constructions on metaframes;492
8.4.19;5.19 On semantics of intuitionistic sound metaframes;494
8.4.20;5.20 Simplicial frames;498
9;Part III Completeness;506
9.1;Chapter 6 Kripke completeness for varying domains;508
9.1.1;6.1 Canonical models for modal logics;508
9.1.2;6.2 Canonical models for superintuitionistic logics;518
9.1.3;6.3 Intermediate logics of finite depth;526
9.1.4;6.4 Natural models;529
9.1.5;6.5 Refined completeness theorem for QH + K F;540
9.1.6;6.6 Directed frames;541
9.1.7;6.7 Logics of linear frames;549
9.1.8;6.8 Properties of ?-operation;553
9.1.9;6.9 ?-operation preserves completeness;557
9.1.10;6.10 Trees of bounded branching and depth;561
9.1.11;6.11 Logics of uniform trees;564
9.2;Chapter 7 Kripke completeness;578
9.2.1;7.1 Modal canonical models with constant domains;578
9.2.2;7.2 Intuitionistic canonical models with constant domains;580
9.2.3;7.3 Some examples of C-canonical logics;583
9.2.4;7.4 Predicate versions of subframe and tabular logics;588
9.2.5;7.5 Predicate versions of cofinal subframe logics;590
9.2.6;7.6 Natural models with constant domains;598
9.2.7;7.7 Remarks on Kripke bundles with constant domains;602
9.2.8;7.8 Kripke frames over the reals and the rationals;604
10;Bibliography;618
11;Index;628
