This comprehensive text demonstrates how various notions of logic can be viewed as notions of universal algebra. It is aimed primarily at logisticians in mathematics, philosophy, computer science and linguistics with an interest in algebraic logic, but is also accessible to those from a non-logistics background. The premise of the text is that standard algebraic results (representations) translate into standard logical results (completeness) and it identifies classes of algebras appropriate for classical and non-classical logic studies, including: gaggles, distributoids, partial- gaggles, and tonoids. Also discused is the idea that logic is fundamentally information based, with its main elements being propositions, that can be understood as sets of information states. Logics are considered in various senses such as systems of theorems, consequence relations and, symmetric consequence relations.
Author(s): J. Michael Dunn, Gary M. Hardegree
Series: Oxford Logic Guides 41
Publisher: Oxford University Press
Year: 2001
Language: English
Pages: 487
Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Dedication......Page 6
Preface......Page 8
Contents......Page 12
1 Introduction......Page 18
2.2 Relational and Operational Structures (Algebras)......Page 27
2.3 Subrelational Structures and Subalgebras......Page 28
2.4 Intersection, Generators, and Induction from Generators......Page 30
2.5 Homomorphisms and Isomorphisms......Page 32
2.6 Congruence Relations and Quotient Algebras......Page 36
2.7 Direct Products......Page 42
2.8 Subdirect products and the Fundamental Theorem of Universal Algebra......Page 45
2.9 Word Algebras and Interpretations......Page 50
2.10 Varieties and Equational Definability......Page 53
2.11 Equational Theories......Page 54
2.12 Examples of Free Algebras......Page 56
2.13 Freedom and Typicality......Page 58
2.14 The Existence of Free Algebras; Freedom in Varieties and Subdirect classes......Page 61
2.15 Birkhoff's Varieties Theorem......Page 64
2.16 Quasi-varieties......Page 66
2.17 Logic and Algebra: Algebraic Statements of Soundness and Completeness......Page 68
3.2 Partially Ordered Sets......Page 72
3.3 Strict Orderings......Page 75
3.4 Covering and Hasse Diagrams......Page 77
3.5 Infima and Suprema......Page 80
3.6 Lattices......Page 84
3.7 The Lattice of Congruences......Page 87
3.8 Lattices as Algebras......Page 88
3.9 Ordered Algebras......Page 91
3.10 Tonoids......Page 94
3.11 Tonoid Varieties......Page 99
3.12 Classical Complementation......Page 102
3.13 Non-Classical Complementation......Page 105
3.14 Classical Distribution......Page 109
3.15 Non-Classical Distribution......Page 115
3.16 Classical Implication......Page 122
3.17 Non-Classical Implication......Page 126
3.18 Filters and Ideals......Page 132
4.2 The Algebra of Strings......Page 142
4.3 The Algebra of Sentences......Page 147
4.4 Languages as Abstract Structures: Categorial Grammar......Page 150
4.5 Substitution Viewed Algebraically (Endomorphisms)......Page 153
4.6 Effectivity......Page 154
4.7 Enumerating Strings and Sentences......Page 155
5.1 Introduction......Page 158
5.2 Categorial Semantics......Page 159
5.3 Algebraic Semantics for Sentential Languages......Page 161
5.4 Truth-Value Semantics......Page 163
5.5 Possible Worlds Semantics......Page 165
5.6 Logical Matrices and Logical Atlases......Page 169
5.7 Interpretations and Valuations......Page 172
5.8 Interpreted and Evaluationally Constrained Languages......Page 175
5.9 Substitutions, Interpretations, and Valuations......Page 179
5.10 Valuation Spaces......Page 183
5.11 Valuations and Logic......Page 186
5.12 Equivalence......Page 189
5.13 Compactness......Page 193
5.14 The Three-Fold Way......Page 198
6.1 Motivational Background......Page 201
6.2 The Varieties of Logical Experience......Page 202
6.3 What Is (a) Logic?......Page 204
6.4 Logics and Valuations......Page 206
6.5 Binary Consequence in the Context of Pre-ordered Sets......Page 208
6.6 Asymmetric Consequence and Valuations (Completeness)......Page 211
6.7 Asymmetric Consequence in the Context of Pre-ordered Groupoids......Page 213
6.8 Symmetric Consequence and Valuations (Completeness and Absoluteness)......Page 216
6.9 Symmetric Consequence in the Context of Hemi-distributoids......Page 219
6.10 Structural (Formal) Consequence......Page 225
6.11 Lindenbaum Matrices and Compositional Semantics for Assertional Formal Logics......Page 226
6.12 Lindenbaum Atlas and Compositional Semantics for Formal Asymmetric Consequence Logics......Page 228
6.13 Scott Atlas and Compositional Semantics for Formal Symmetric Consequence Logics......Page 230
6.14 Co-consequence as a Congruence......Page 231
6.15 Formal Presentations of Logics (Axiomatizations)......Page 233
6.16 Effectiveness and Logic......Page 241
7.1.1 Background......Page 243
7.1.2 Lukasiewicz matrices/submatrices, isomorphisms......Page 244
7.1.4 Sugihara matrices/homomorphisms......Page 247
7.1.6 Tautology preservation......Page 249
7.1.7 Infinite matrices......Page 250
7.1.8 Interpretation......Page 251
7.2 Relations Among Matrices: Submatrices, Homomorphic Images, and Direct Products "......Page 254
7.3 Proto-preservation Theorems......Page 256
7.4 Preservation Theorems......Page 260
7.5.1 Unary assertional logics......Page 263
7.5.2 Asymmetric consequence logics......Page 264
7.6 Congruences and Quotient Matrices......Page 266
7.7 The Structure of Congruences......Page 271
7.8 The Cancellation Property......Page 274
7.9 Normal Matrices......Page 279
7.10 Normal Atlases......Page 283
7.11 Normal Characteristic Matrices for Consequence Logics......Page 287
7.12 Matrices and Algebras......Page 288
7.13 When is a Logic "Algebraizable"?......Page 290
8.1.1 Partially ordered sets......Page 294
8.1.2 Implication structures......Page 295
8.2 Semi-lattices......Page 304
8.3 Lattices......Page 305
8.4 Finite Distributive Lattices......Page 310
8.5 The Problem of a General Representation for Distributive Lattices......Page 312
8.6 Stone's Representation Theorem for Distributive Lattices......Page 314
8.7 Boolean Algebras......Page 317
8.9 Maximal Filters and Prime Filters......Page 319
8.10 Stone's Representation Theorem for Boolean Algebras......Page 320
8.11 Maximal Filters and Two-Valued Homomorphisms......Page 322
8.12 Distributive Lattices with Operators......Page 330
8.13 Lattices with Operators......Page 334
9.1 Preliminary Notions......Page 338
9.2 The Equivalence of (Unital) Boolean Logic and Frege Logic......Page 339
9.3 Symmetrical Entailment......Page 341
9.4 Compactness Theorems for Classical Propositional Logic......Page 343
9.5 A Third Logic......Page 350
9.6 Axiomatic Calculi for Classical Propositional Logic......Page 351
9.7 Primitive Vocabulary and Definitional Completeness......Page 352
9.8 The Calculus BC......Page 354
9.9 The Calculus D(BC)......Page 358
9.10 Asymmetrical Sequent Calculus for Classical Propositional Logic......Page 363
9.11 Fragments of Classical Propositional Logic......Page 365
9.12 The Implicative Fragment of Classical Propositional Logic: Semi-Boolean Algebras......Page 366
9.13 Axiomatizing the Implicative Fragment of Classical Propositional Logic......Page 367
9.14 The Positive Fragment of Classical Propositional Logic......Page 369
10.1 Modal Logics......Page 373
10.2 Boolean Algebras with a Normal Unitary Operator......Page 375
10.4 The Kripke Semantics for Modal Logic......Page 378
10.5 Completeness......Page 380
10.6 Topological Representation of Closure Algebras......Page 381
10.8 Henle Matrices......Page 384
10.9 Alternation Property for S4 and Compactness......Page 386
10.10 Algebraic Decision Procedures for Modal Logic......Page 387
10.11 S5 and Pretabularity......Page 392
11.1 Intuitionistic Logic......Page 397
11.2 Implicative Lattices......Page 398
11.4 Representation of Heyting Algebras using Quasi-ordered Sets......Page 400
11.5 Topological Representation of Heyting Algebras......Page 401
11.7 Translation of H into S4......Page 403
11.8 Alternation Property for H......Page 404
11.9 Algebraic Decision Procedures for Intuitionistic Logic......Page 405
11.10 LC and Pretabularity......Page 407
12.1 Introduction......Page 411
12.2 Residuation and Galois Connections......Page 412
12.3 Definitions of Distributoid and Tonoid......Page 415
12.4 Representation of Distributoids......Page 417
12.5 Partially Ordered Residuated Groupoids......Page 423
12.6 Definition of a Gaggle......Page 425
12.7 Representation of Gaggles......Page 426
12.8 Modifications for Distributoids and Gaggles with Identities and Constants......Page 429
12.9 Applications......Page 431
12.10 Monadic Modal Operators......Page 432
12.11 Dyadic Modal Operators......Page 434
12.12 Identity Elements......Page 437
12.13 Representation of Positive Binary Gaggles......Page 438
12.14 Implication......Page 439
12.14.1 Implication in relevance logic......Page 440
12.14.3 Modal logic......Page 441
12.15.1 The gaggle treatment of negation......Page 442
12.15.2 Negation in intuitionistic logic......Page 443
12.15.3 Negation in relevance logic......Page 444
12.15.4 Negation in classical logic......Page 446
12.16 Future Directions......Page 447
13.1 Representations and Duality......Page 448
13.2 Some Topology......Page 450
13.3 Duality for Boolean Algebras......Page 452
13.4 Duality for Distributive Lattices......Page 455
13.5 Extensions of Stone's and Priestley's Results......Page 458
References......Page 462
Index......Page 472
