Fuzzy logics are many-valued logics that are well suited to reasoning in the context of vagueness. They provide the basis for the wider field of Fuzzy Logic, encompassing diverse areas such as fuzzy control, fuzzy databases, and fuzzy mathematics. This book provides an accessible and up-to-date introduction to this fast-growing and increasingly popular area. It focuses in particular on the development and applications of "proof-theoretic" presentations of fuzzy logics; the result of more than ten years of intensive work by researchers in the area, including the authors. In addition to providing alternative elegant presentations of fuzzy logics, proof-theoretic methods are useful for addressing theoretical problems (including key standard completeness results) and developing efficient deduction and decision algorithms. Proof-theoretic presentations also place fuzzy logics in the broader landscape of non-classical logics, revealing deep relations with other logics studied in Computer Science, Mathematics, and Philosophy. The book builds methodically from the semantic origins of fuzzy logics to proof-theoretic presentations such as Hilbert and Gentzen systems, introducing both theoretical and practical applications of these presentations.
Author(s): George Metcalfe, Nicola Olivetti, Dov M. Gabbay
Series: Applied Logic Series
Edition: 1
Publisher: Springer
Year: 2008
Language: English
Pages: 279
Contents......Page 7
1. Introduction......Page 9
1. Truth values......Page 15
2. Ands and ors......Page 18
1. Basic properties......Page 19
2. t-norms......Page 20
3. t-conorms......Page 25
4. Uninorms......Page 26
3. Nots and ifs......Page 29
1. Basic notions......Page 32
2. Commutative residuated lattices......Page 35
3. The Dedekind-McNeille completion......Page 37
5. Languages and logics......Page 38
6. Historical remarks......Page 41
1. Structures and systems......Page 44
2. Core axioms and rules......Page 47
3. Axiomatic extensions......Page 49
1. Truth, falsity, negation......Page 51
2. Distributivity and prelinearity......Page 53
3. Weakening......Page 54
4. Contraction and mingle......Page 55
5. Divisibility......Page 56
6. Excluded middle and non-contradiction......Page 57
4. A local deduction theorem......Page 58
5. Soundness and completeness......Page 62
6. The density rule......Page 66
7. Historical remarks......Page 72
1. Sequents and hypersequents......Page 74
1. Sequents......Page 76
2. Hypersequents......Page 80
2. Core systems......Page 83
3. Adding structural rules......Page 87
1. External weakening and external contraction......Page 88
2. Communication and split......Page 90
3. Weakening......Page 92
4. Contraction......Page 94
5. Cancellation......Page 97
4. Non-standard logical rules......Page 98
5. Density again......Page 99
6. Soundness and completeness......Page 100
7. Historical remarks......Page 105
5. Syntactic Eliminations......Page 108
1. Cut elimination......Page 109
1. Regular calculi......Page 111
2. The main theorem......Page 115
3. Conservative extensions......Page 117
4. Decidability......Page 119
2. Cancellation elimination......Page 121
1. The proof......Page 122
2. Abelian l-groups......Page 127
3. Density elimination......Page 129
1. Calculi with weakening......Page 130
2. Calculi without weakening......Page 136
3. Standard completeness......Page 140
4. Historical remarks......Page 141
1. Goedel logic......Page 143
1. The hypersequent calculus GG......Page 144
2. A sequent calculus......Page 146
3. Another hypersequent calculus......Page 150
4. A sequent of relations calculus......Page 151
2. Lukasiewicz logic......Page 152
1. A hypersequent calculus......Page 153
2. A sequent calculus......Page 158
3. An embedding into Abelian logic......Page 164
4. McNaughton functions......Page 165
5. Giles' game......Page 167
3. Product logic......Page 170
4. Related logics......Page 177
5. Historical remarks......Page 180
1. Uniform systems......Page 182
1. Uniform logical rules......Page 183
2. Revised logical rules......Page 185
3. Structural rules......Page 187
1. The goal-directed methodology......Page 190
2. Uniform rules......Page 192
3. Goal-directed systems......Page 195
3. Complexity......Page 197
1. Co-NP-hardness......Page 198
2. Bi-coloured graphs......Page 199
3. Linear programming......Page 200
4. Historical remarks......Page 202
1. Syntax and semantics......Page 205
2. Hilbert systems......Page 208
3. Gentzen systems......Page 214
4. Herbrand's theorem and Skolemization......Page 218
1. An approximate Herbrand theorem......Page 224
2. Gentzen systems......Page 228
6. Historical remarks......Page 230
1. Modalities and truth stressers......Page 233
1. Axioms and algebras......Page 234
2. Gentzen systems......Page 237
3. Embeddings......Page 239
2. Propositional quantifiers......Page 241
3. Non-commutative logics......Page 244
1. Residuated logics......Page 245
2. Hilbert systems......Page 246
3. Gentzen systems......Page 247
4. Finite-valued logics......Page 250
1. Logical matrices......Page 251
2. n-sequents......Page 252
3. Hypersequents......Page 255
5. Comparative logics......Page 257
6. Basic logic and other open problems......Page 260
References......Page 263
Index......Page 272
