.

0 (hodnocen0 x )

BK

Amsterdam : Elsevier, 2007

xx, 509 s. ; 24 cm

ISBN 978-0-444-52141-5 (váz.)

Studies in logic and the foundations of mathematics ; vol. 151

Obsahuje bibliografie na s. 479-495, rejstřík

000064495

Contents // Contents vii // List of Figures xix // List of Tables xxi // Introduction 1 // Chapter 1. Getting started 13 // 1.1. First-order languages and semantics 13 // 1.2. Concepts from universal algebra 24 // 1.3. Logic 38 // 1.4. Logic and algebra 51 // 1.5. Cut elimination in sequent calculi 58 // 1.6. Consequence relations and matrices 62 // Exercises 69 // Notes 72 // Chapter 2. Substruetural logics and residuated lattices 75 // 2.1. Sequent calculi and substruetur al logics 76 // 2.2. Residuated lattices and FL-algebras 91 // 2.3. Important subclasses of substruet ural logics 97 // 2.4. Parametrized local deduction theorem 119 // 2.5. Hilbert systems 124 // 2.6. Algebraization and deductive filters 130 // Exercises 135 // Notes 138 // Chapter 3. Residuation and structure theory 141 // 3.1. Residuation theory and Galois connections 142 // 3.2. Residuated structures 149 // 3.3. Involutive residuated structures 151 // 3.4. Further examples of residuated structures 156 // 3.5. Subvariety lattices 182 // 3.6. Structure theory 187 // Exercises 204 // vii // viii // CONTENTS // Notes 210 // Chapter 4. Decidability 211 // 4.1. Syntactic proof of cut elimination 211 // 4.2. Decidability as a consequence of cut elimination 217 // 4.3. Further results 226 // 4.4. Undecidability 230 // Exercises 240 // Notes 241 // Chapter 5. Logical and algebraic properties 245 // 5.1. Syntactic approach to logical properties 245 // 5.2. Maksimova’s variable separation property 254 // 5.3. Algebraic

characterizations 257 // 5.4. Maksimova’s property and well-connected pairs 265 // 5.5. Deductive interpolation properties 271 // 5.6. Craig interpolation property 279 // Exercises 287 // Notes 287 // Chapter 6. Completions and finite embeddability 289 // 6.1. Completions of posets 289 // 6.2. Canonical extensions of residuated groupoids 298 // 6.3. Nuclear completions of residuated groupoids 303 // 6.4. Negative results for completions 306 // 6.5. Finite embeddability property 310 // Exercises 319 // Notes 321 // Chapter 7. Algebraic aspects of cut elimination 323 // 7.1. Gentzen matrices for the sequent calculus FL 324 // 7.2. Quasi-completions and cut elimination 327 // 7.3. Cut elimination for other systems 332 // 7.4. Finite model property 339 // Exercises 342 // Notes 342 // Chapter 8. Glivenko theorems 345 // 8.1. Overview 345 // 8.2. Glivenko equivalence 348 // 8.3. Glivenko properties 352 // 8.4. More on the equational Glivenko property 360 // 8.5. Special cases 364 // 8.6. Generalized Kolmogorov translation 372 // Exercises 375 // CONTENTS ix // Notes 375 // Chapter 9. Lattices of logics and varieties 377 // 9.1. General facts about atoms 378 // 9.2. Minimal subvarieties of RL 380 // 9.3. Minimal subvarieties of FL 391 // 9.4. Almost minimal subvarieties of FLew 401 // 9.5. Almost minimal varieties of BL-algebras 415 // 9.6. Translations of subvariety lattices 417 // 9.7. Axiomatizations for joins of varieties and meets of logics 422 // 9.8. The subvariety lattices

of LG and LG“ 431 // Exercises 436 // Notes 437 // Chapter 10. Splittings 439 // 10.1. Splittings in general 439 // 10.2. Splittings in varieties of algebras 440 // 10.3. Algebras describing themselves 441 // 10.4. Construction that excludes splittings 446 // 10.5. Only one splitting 459 // Exercises 459 // Notes 460 // Chapter 11. Semisimplicity 463 // 11.1. Semisimplicity, discriminator, EDPC 463 // 11.2. Free FLeii;-algebras are semisimple: outline 465 // 11.3. A characterization of semisimple FLetl,-algebras 465 // 11.4. Sequent calculi for FLew 466 // 11.5. Semisimplicity of free FLeu;-algebras 470 // 11.6. Inside FLew semisimplicity implies discriminator: outline 471 // 11.7. A characterization of semisimple subvarieties of FLgw 472 // 11.8. Semisimplicity forces discriminator 474 // Exercises 477 // Notes 478 // Bibliography 479 // Index 497