Quantification in Nonclassical Logic (eBook)

Dov M. Gabbay, Valentin Shehtman, Dimitrij Skvortsov (Autoren)

eBook Download: EPUB

2009 | 1. Auflage
640 Seiten
Elsevier Science (Verlag)
978-0-08-093112-8 (ISBN)

Infos

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.

Inhalt

Front Cover 1
Quantification in Nonclassical Logic 4
Copyright Page 5
Contents 22
Preface 6
Introduction 10
Part I Preliminaries 26
Chapter 1 Basic propositional logic 28
1.1 Propositional syntax 28
1.2 Algebraic semantics 36
1.3 Relational semantics (the modal case) 44
1.4 Relational semantics (the intuitionistic case) 57
1.5 Modal counterparts 62
1.6 General Kripke frames 63
1.7 Canonical Kripke models 65
1.8 First-order translations and definability 69
1.9 Some general completeness theorems 72
1.10 Trees and unravelling 73
1.11 PTC-logics and Horn closures 77
1.12 Subframe and cofinal subframe logics 83
1.13 Splittings 90
1.14 Tabularity 93
1.15 Transitive logics of finite depth 95
1.16 ?-operation 97
1.17 Neighbourhood semantics 101
Chapter 2 Basic predicate logic 104
2.1 Introduction 104
2.2 Formulas 106
2.3 Variable substitutions 110
2.4 Formulas with constants 127
2.5 Formula substitutions 130
2.6 First-order logics 144
2.7 First-order theories 164
2.8 Deduction theorems 167
2.9 Perfection 171
2.10 Intersections 176
2.11 Cödel–Tarski translation 178
2.12 The Glivenko theorem 182
2.13 ?-operation 183
2.14 Adding equality 197
2.15 Propositional parts 205
2.16 Semantics from an abstract viewpoint 210
Part II Semantics 216
Introduction: What is semantics? 218
Chapter 3 Kripke semantics 224
3.1 Preliminary discussion 224
3.2 Predicate Kripke frames 230
3.3 Morphisms of Kripke frames 244
3.4 Constant domains 255
3.5 Kripke frames with equality 259
3.6 Kripke sheaves 268
3.7 Morphisms of Kripke sheaves 278
3.8 Transfer of completeness 284
3.9 Simulation of varying domains 291
3.10 Examples of Kripke semantics 293
3.11 On logics with closed or decidable equality 302
3.12 Translations into classical logic 306
Chapter 4 Algebraic semantics 318
4.1 Modal and Heyting valued structures 318
4.2 Algebraic models 326
4.3 Soundness 336
4.4 Morphisms of algebraic structures 344
4.5 Presheaves and ?-sets 353
4.6 Morphisms of presheaves 358
4.7 Sheaves 363
4.8 Fibrewise models 364
4.9 Examples of algebraic semantics 366
Chapter 5 Metaframe semantics 370
5.1 Preliminary discussion 370
5.2 Kripke bundles 376
5.3 More on forcing in Kripke bundles 381
5.4 Morphisms of Kripke bundles 384
5.5 Intuitionistic Kripke bundles 390
5.6 Functor semantics 399
5.7 Morphisms of presets 406
5.8 Bundles over precategories 411
5.9 Metaframes 413
5.10 Permutability and weak functoriality 422
5.11 Modal metaframes 429
5.12 Modal soundness 434
5.13 Representation theorem for modal metaframes 444
5.14 Intuitionistic forcing and monotonicity 447
5.15 Intuitionistic soundness 457
5.16 Maximality theorem 477
5.17 Kripke quasi-bundles 490
5.18 Some constructions on metaframes 492
5.19 On semantics of intuitionistic sound metaframes 494
5.20 Simplicial frames 498
Part III Completeness 506
Chapter 6 Kripke completeness for varying domains 508
6.1 Canonical models for modal logics 508
6.2 Canonical models for superintuitionistic logics 518
6.3 Intermediate logics of finite depth 526
6.4 Natural models 529
6.5 Refined completeness theorem for QH + K F 540
6.6 Directed frames 541
6.7 Logics of linear frames 549
6.8 Properties of ?-operation 553
6.9 ?-operation preserves completeness 557
6.10 Trees of bounded branching and depth 561
6.11 Logics of uniform trees 564
Chapter 7 Kripke completeness 578
7.1 Modal canonical models with constant domains 578
7.2 Intuitionistic canonical models with constant domains 580
7.3 Some examples of C-canonical logics 583
7.4 Predicate versions of subframe and tabular logics 588
7.5 Predicate versions of cofinal subframe logics 590
7.6 Natural models with constant domains 598
7.7 Remarks on Kripke bundles with constant domains 602
7.8 Kripke frames over the reals and the rationals 604
Bibliography 618
Index 628

Produkt-Details

• Erscheint lt. Verlag: 20.6.2009
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
• Themenwelt: Technik
• ISBN-10: 0-08-093112-X / 008093112X
• ISBN-13: 978-0-08-093112-8 / 9780080931128