Algebraic Approach to Non-Classical Logics (eBook)

Lev D. Beklemishev (Autor)

eBook Download: PDF

2000 | 1. Auflage
400 Seiten
Elsevier Science (Verlag)
978-0-08-095488-2 (ISBN)

Infos

Provability, Computability and Reflection

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An Algebraic Approach to Non-Classical Logics

Inhalt

Front Cover 1
An Algebraic Approach to Non-Classical Logics 4
Copyright Page 5
Contents 12
PART ONE: IMPLICATIVE ALGEBRAS AND LATTICES 18
Chapter I. Preliminary set-theoretical, topological and algebraic notions 20
Introduction 20
1. Sets, mappings 20
2. Topological spaces 21
3. Ordered sets and quasi-ordered sets 24
4. Abstract algebras 26
5. Exercises 31
Chapter II. Implicative algebras 32
Introduction 32
1. Definition and elementary properties 33
2. Positive implication algebras 39
3. Implicative filters in positive implication algebras 43
4. Representation theorem for positive implication algebras 45
5. Implication algebras 47
6. Implicative filters in implication algebras 49
7. Representation theorem for implication algebras 51
8. Exercises 53
Chapter III. Distributive lattices and quasi-Boolean algebras 55
Introduction 55
1. Lattices 56
2. Distributive lattices 60
3. Quasi-Boolean algebras 61
4. Exercises 65
Chapter IV. Relatively pseudo-complemented lattices, contrapositionally complemented lattices, semi-complemented lattices and pseudo-Boolean algebras 68
Introduction 68
1. Relatively pseudo-complemented lattices 69
2. Filters in relatively pseudo-complemented lattices 73
3. Representation theorem for relatively pseudo-complemented lattices 74
4. Contrapositionally complemented lattices 75
5. Semi-complemented lattices 78
6. Pseudo-Boolean algebras 79
7. Exercises 82
Chapter V. Quasi-pseudo-Boolean algebras 84
Introduction 84
1. Definition and elementary properties 85
2. Equational definability of quasi-pseudo-Boolean algebras 92
3. Examples of quasi-pseudo-Boolean algebras 98
4. Filters in quasi-pseudo-Boolean algebras 107
5. Representation theorem for quasi-pseudo-Boolean algebras 117
6. Exercises 124
Chapter VI. Boolean algebras and topological Boolean algebras 127
Introduction 127
1. Definition and elementary properties of Boolean algebras 128
2. Subalgebras of Boolean algebras 129
3. Filters and implicative filters in Boolean algebras 130
4. Representation theorem for Boolean algebras 131
5. Topological Boolean algebras 132
6. I-filters in topological Boolean algebras 133
7. Representation theorem for topological Boolean algebras 137
8. Strongly compact topological spaces 138
9. A lemma on imbedding for topological Boolean algebras 139
10. Connections between topological Boolean algebras, pseudo-Boolean algebras, relatively pseudo-complemented lattices, contrapositionally complemented lattices and semi-complemented lattices 140
11. Lemmas on imbeddings for pseudo-Boolean algebras, relatively pseudo-complemented lattices, contrapositionally complemented lattices and semi-complemented lattices. 144
12. Exercises 146
Chapter VII. Post algebras 149
Introduction 149
1. Definition and elementary properties 150
2. Examples of Post algebras 159
3. Filters and D-filters in Post algebras 161
4. Post homomorphisms 167
5. Post fields of sets 173
6. Representation theorem for Post algebras 178
7. Exercises 180
PART TWO NON-CLASSICAL LOGICS 182
Chapter VIII. Implicative extensional propositional calculi 184
Introduction 184
1. Formalized languages of zero order 187
2. The algebra of formulas 189
3. Interpretation of formulas as mappings 191
4. Consequence operations in formalized languages of zero order 194
5. The class S of standard systems of implicative extensional propositional calculi 196
6. g-algebras 198
7. Completeness theorem 202
8. Logically equivalent systems 203
9. L-theories of zero order 206
10. Standard systems of implicative extensional propositional calculi with semi-negation. 209
11. Theorems on logically equivalent systems in S 210
12. Deductive filters 216
13. The connection between L-theories and deductive filters 220
14. Exercises 225
Chapter IX. Positive implicative logic and classical implicative logic 227
Introduction. 227
1. Propositional calculus I pl of positive implicative logic 229
2. gpl-algebras 230
3. Positive implicative logic Lpl 232
4. Lpl-theories of zero order 234
5. The connection between Lpl-theories and implicative filters 235
6. Propositional calculus Ixl of classical implicative logic 238
7. gxl-algebras 239
8. Classical implicative logic Lxl 240
9. Lxl-theories of zero order 243
10. The connection between Lxl-theories of zero order and implicative filters 245
11. Exercises 249
Chapter X. Positive logic 251
Introduction 251
1. Propositional calculus gx of positive logic 252
2. Ip-algebras 254
3. Positive logic Lp 255
4. On disjunctions derivable in the propositional calculi of positive logic 260
5. Lp-theories of zero order 261
6. The connection between Lp-theories and filters 262
7. Exercises 265
Chapter XI. Minimal logic, positive logic with semi-negation and intuitionistic logic 267
Introduction 267
1. Propositional calculus Ip of minimal logic 269
2. Minimal logic Lµ 270
3. Lµ-theories of zero order and their connection with filters 272
4. Propositional calculus Iv of positive logic with semi-negation 275
5. Positive logic with semi-negation Lv 276
6. Lv-thecries of zero order and their connection with filters 278
7. Propositional calculus Ix of intuitionistic logic 280
8. Intuitionistic logic Lx 282
9. Lx-theories of zero order and their connection with filters 284
10. Prime Lx-theories 288
11. Exercises 291
Chapter XII. Constructive logic with strong negation 293
Introduction 293
1. Propositional calculus IN of constructive logic with strong negation 296
2. IN-algebras 299
3. Constructive logic with strong negation LN 300
4. Connections between constructive logic with strong negation and intuitionistic logic 303
5. A topological characterization of formulas derivable in propositional calculi of constructive logic with strong negation 310
6. LN-theories of zero order 312
7. The connection between LN-theories of zero order and special filters of the first kind 315
8. Prime LN-theories 319
9. Exercises 326
Chapter XIll. Classical logic and modal logic 328
Introduction 328
1. Propositional calculus Ix of classical logic 330
2. Classical logic Lx 331
3. Lx-theories of zero order and their connection with filters 332
4. Propositional calculus I. of modal logic 335
5. Modal logic L. 336
6. L.-theories of zero order and their connection with I-filters 340
7. I-prime L.-theories 345
8. Exercises 348
Chapter XIV. Many-valued logics 350
Introduction 350
1. Propositional calculus Im of m-valued logic 352
2. Im-algebras 354
3. m-valued logic Lm 355
4. Lm-theories of zero order and their connection with D-filters 357
5. Exercises 363
SUPPLEMENT 364
First order predicate calculi of non-classical logics 364
Introduction 364
1. Formalized languages of first order 368
2. First order predicate calculi of a logic L 370
3. Elementary L-theories 374
4. The algebra of terms 375
5. Realizations of terms 376
6. Implicative algebras with generalized joins and meets 378
7. The algebra of an elementary L-theory 380
8. Realizations of first order formalized languages associated with a logic L 384
9. Canonical realizations for elementary L-theories 386
10. L-models 388
11. The completeness theorem for the first order predicate calculi of a logic L 391
12. The existence of L-models for consistent elementary L-theories 392
13. Exercises 394
Bibliography 397
List of symbols 408
Author index 410
Subject index 413

Produkt-Details

• Erscheint lt. Verlag: 1.4.2000
• Sprache: englisch
• Themenwelt: Informatik ► Software Entwicklung ► User Interfaces (HCI)
• Themenwelt: Informatik ► Theorie / Studium ► Algorithmen
• Themenwelt: Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
• Themenwelt: Naturwissenschaften
• Themenwelt: Technik
• ISBN-10: 0-08-095488-X / 008095488X
• ISBN-13: 978-0-08-095488-2 / 9780080954882