Admissibility of Logical Inference Rules (eBook)

V.V. Rybakov (Autor)

eBook Download: PDF

1997 | 1. Auflage
616 Seiten
Elsevier Science (Verlag)
978-0-08-052599-0 (ISBN)

Infos

The aim of this book is to present the fundamental theoretical results concerning inference rules in deductive formal systems. Primary attention is focused on:

&bull, admissible or permissible inference rules

&bull, the derivability of the admissible inference rules

&bull, the structural completeness of logics

&bull, the bases for admissible and valid inference rules.

There is particular emphasis on propositional non-standard logics (primary, superintuitionistic and modal logics) but general logical consequence relations and classical first-order theories are also considered.

The book is basically self-contained and special attention has been made to present the material in a convenient manner for the reader. Proofs of results, many of which are not readily available elsewhere, are also included.

The book is written at a level appropriate for first-year graduate students in mathematics or computer science. Although some knowledge of elementary logic and universal algebra are necessary, the first chapter includes all the results from universal algebra and logic that the reader needs. For graduate students in mathematics and computer science the book is an excellent textbook.

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The aim of this book is to present the fundamental theoretical results concerning inference rules in deductive formal systems. Primary attention is focused on:* admissible or permissible inference rules* the derivability of the admissible inference rules* the structural completeness of logics* the bases for admissible and valid inference rules.There is particular emphasis on propositional non-standard logics (primary, superintuitionistic and modal logics) but general logical consequence relations and classical first-order theories are also considered.The book is basically self-contained and special attention has been made to present the material in a convenient manner for the reader. Proofs of results, many of which are not readily available elsewhere, are also included.The book is written at a level appropriate for first-year graduate students in mathematics or computer science. Although some knowledge of elementary logic and universal algebra are necessary, the first chapter includes all the results from universal algebra and logic that the reader needs. For graduate students in mathematics and computer science the book is an excellent textbook.

Inhalt

Cover 1
Copyright Page 5
Contents 6
0.1 Preface and Acknowledgments 8
0.2 Introduction 13
Chapter 1. Syntaxes and Semantics 22
1.1 Syntax of Formal Logic Systems 23
1.2 First-Order Semantics and Universal Algebra 34
1.3 Algebraic Semantics for Propositional Logics 53
1.4 Admissible Rules in Algebraic Logics 65
1.5 Logical Consequence Relations 74
1.6 Algebraizable Consequence Relations 80
1.7 Admissibility for Consequence Relations 93
1.8 Lattices of Logical Consequences 102
Chapter 2. Semantics for Non-Standard Logics 108
2.1 Algebraic Semantics for Intuitionistic Logic 109
2.2 Algebraic Semantics, Modal and Tense Logics 129
2.3 Kripke Semantics for Modal and Temporal Logic 151
2.4 Kripke Semantics for Intuitionistic Logic 181
2.5 Stone's Theory and Kripke Semantics 193
2.6 The Finite Model Property 223
2.7 Relation of Intuitionistic and Modal Logics 237
2.8 Advanced Tools for the Finite Model Property 250
2.9 Kripke Incomplete Logics 265
2.10 Advanced Tools for Kripke Completeness 268
Chapter 3. Criteria for Admissibility 284
3.1 Reduced Forms 285
3.2 T-translation of inference rules 296
3.3 Semantic Criteria for Admissibility 300
3.4 Some Technical Lemmas 313
3.5 Criteria for Determining Admissibility 330
3.6 Elementary Theories of Free Algebras 350
3.7 Scheme-Logics of First-Order Theories 366
3.8 Some Counterexamples 380
3.9 Admissibility through Reduced Forms 388
Chapter 4. Bases for Inference Rules 398
4.1 Initial Auxiliary Results 399
4.2 The Absence of Finite Bases 404
4.3 Bases for Some Strong Logics 427
4.4 Bases for Valid Rules of Tabular Logics 453
4.5 Tabular Logics without Independent Bases 465
Chapter 5. Structural Completeness 480
5.1 General Properties, Descriptions 481
5.2 Quasi-characteristic Inference Rules 489
5.3 Some Preliminary Technical Results 500
5.4 Hereditary Structural Completeness 514
5.5 Structurally Complete Fragments 523
Chapter 6. Related Questions 530
6.1 Rules with Meta-Variables 531
6.2 The Preservation of Admissible for S4 Rules 555
6.3 The Preservation of Admissibility for H 566
6.4 Non-Compact Modal Logics 576
6.5 Decidable Kripke Non-compact Logics 587
6.6 Non-Compact Superintuitionistic Logics 594
Index 600
Bibliography 608

Produkt-Details

• Erscheint lt. Verlag: 14.3.1997
• Sprache: englisch
• Themenwelt: Informatik ► Theorie / Studium ► Künstliche Intelligenz / Robotik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
• Themenwelt: Technik
• ISBN-10: 0-08-052599-7 / 0080525997
• ISBN-13: 978-0-08-052599-0 / 9780080525990