An Introduction to Non-Classical Logic, Second Edition: From If to Is (Cambridge Introductions to Philosophy)

This revised and considerably expanded 2nd edition brings together a wide range of topics, including modal, tense, conditional, intuitionist, many-valued, paraconsistent, relevant, and fuzzy logics. Part 1, on propositional logic, is the old Introduction, but contains much new material. Part 2 is entirely new, and covers quantification and identity for all the logics in Part 1. The material is unified by the underlying theme of world semantics. All of the topics are explained clearly using devices such as tableau proofs, and their relation to current philosophical issues and debates are discussed. Students with a basic understanding of classical logic will find this book an invaluable introduction to an area that has become of central importance in both logic and philosophy. It will also interest people working in mathematics and computer science who wish to know about the area. Cover......Page 1 Half-title......Page 3 Title......Page 5 Copyright......Page 6 Dedication......Page 7 Contents......Page 9 Preface to the First Edition......Page 19 On Part I......Page 23 On Part II......Page 25 Book Website......Page 28 0.1 Set-theoretic Notation......Page 29 0.2 Proof by Induction......Page 31 0.3 Equivalence Relations and Equivalence Classes......Page 32 Part I Propositional Logic......Page 35 1.1 Introduction......Page 37 1.2 The Syntax of the Object Language......Page 38 1.3 Semantic Validity......Page 39 1.4 Tableaux......Page 40 1.5 Counter-models......Page 44 1.6 Conditionals......Page 45 1.7 The Material Conditional......Page 46 1.8 Subjunctive and Counterfactual Conditionals......Page 47 1.9 More Counter-examples......Page 48 1.10 Arguments for .........Page 49 1.11 Proofs of Theorems......Page 50 1.14 Problems......Page 52 2.2 Necessity and Possibility......Page 54 2.3 Modal Semantics......Page 55 2.4 Modal Tableaux......Page 58 2.6 Modal Realism......Page 62 2.7 Modal Actualism......Page 63 2.8 Meinongianism......Page 64 2.9 Proofs of Theorems......Page 65 2.10 History......Page 67 2.12 Problems......Page 68 3.2 Semantics for Normal Modal Logics......Page 70 3.3 Tableaux for Normal Modal Logics......Page 72 3.4 Infinite Tableaux......Page 76 3.5 S5......Page 79 3.6 Which System Represents Necessity?......Page 80 3.6a The Tense Logic Kt......Page 83 3.6b Extensions of Kt......Page 85 3.7 Proofs of Theorems......Page 90 3.10 Problems......Page 94 4.2 Non-normal Worlds......Page 98 4.3 Tableaux for Non-normal Modal Logics......Page 99 4.4 The Properties of Non-normal Logics......Page 101 4.4a S0.5......Page 103 4.6 The Paradoxes of Strict Implication......Page 106 4.7 ... and their Problems......Page 107 4.8 The Explosion of Contradictions......Page 108 4.9 Lewis' Argument for Explosion......Page 110 4.10 Proofs of Theorems......Page 111 4.11 History......Page 113 4.13 Problems......Page 114 5.2 Some More Problematic Inferences......Page 116 5.3 Conditional Semantics......Page 118 5.4 Tableaux for C......Page 120 5.5 Extensions of C......Page 121 5.6 Similarity Spheres......Page 124 5.7 C1 and C2......Page 128 5.8 Further Philosophical Reflections......Page 131 5.9 Proofs of Theorems......Page 132 5.10 History......Page 134 5.12 Problems......Page 135 6.2 Intuitionism: The Rationale......Page 137 6.3 Possible-world Semantics for Intuitionism......Page 139 6.4 Tableaux for Intuitionist Logic......Page 141 6.5 The Foundations of Intuitionism......Page 146 6.6 The Intuitionist Conditional......Page 147 6.7 Proofs of Theorems......Page 148 6.8 History......Page 150 6.10 Problems......Page 151 7.2 Many-valued Logic: The General Structure......Page 154 7.3 The 3-valued Logics of Kleene and Lukasiewicz......Page 156 7.4 LP and RM3......Page 158 7.5 Many-valued Logics and Conditionals......Page 159 7.6 Truth-value Gluts: Inconsistent Laws......Page 161 7.7 Truth-value Gluts: Paradoxes of Self-reference......Page 163 7.8 Truth-value Gaps: Denotation Failure......Page 164 7.9 Truth-value Gaps: Future Contingents......Page 166 7.10 Supervaluations, Modality and Many-valued Logic......Page 167 7.11 Proofs of Theorems......Page 171 7.12 History......Page 173 7.14 Problems......Page 174 8.2 The Semantics of FDE......Page 176 8.3 Tableaux for FDE......Page 178 8.4 FDE and Many-valued Logics......Page 180 8.4a Relational Semantics and Tableaux for L3 and RM3......Page 183 8.5 The Routley Star......Page 185 8.6 Paraconsistency and the Disjunctive Syllogism......Page 188 8.7 Proofs of Theorems......Page 189 8.10 Problems......Page 195 9.2 Adding $arrow $......Page 197 9.3 Tableaux for K4......Page 198 9.4 Non-normal Worlds Again......Page 200 9.5 Tableaux for N4......Page 202 9.6 Star Again......Page 203 9.7 Impossible Worlds and Relevant Logic......Page 205 9.7a Logics of Constructible Negation......Page 209 9.8 Proofs of Theorems......Page 213 9.9 History......Page 218 9.11 Problems......Page 219 10.2 The Logic B......Page 222 10.3 Tableaux for B......Page 224 10.4 Extensions of B......Page 228 10.4a Content Inclusion......Page 231 10.5 The System R......Page 237 10.6 The Ternary Relation......Page 240 10.7 Ceteris Paribus Enthymemes......Page 242 10.8 Proofs of Theorems......Page 245 10.9 History......Page 250 10.10 Further Reading......Page 251 10.11 Problems......Page 252 11.2 Sorites Paradoxes......Page 255 11.3 ... and Responses to Them......Page 256 11.4 The Continuum-valued Logic L......Page 258 11.5 Axioms for LN......Page 261 11.6 Conditionals in L......Page 264 11.7 Fuzzy Relevant Logic......Page 265 11.7a Appendix: t-norm Logics......Page 268 11.8 History......Page 271 11.9 Further Reading......Page 272 11.10 Problems......Page 273 11a.2 General Structure......Page 275 11a.3 Illustration: Modal Lukasiewicz Logic......Page 277 11a.4 Modal FDE......Page 278 11a.5 Tableaux......Page 281 11a.6 Variations......Page 284 11a.7 Future Contingents Revisited......Page 285 11a.8 A Glimpse Beyond......Page 288 11a.9 Proofs of Theorems......Page 289 Postcript: An Historical Perspective on Conditionals......Page 293 Part II Quantification and Identity......Page 295 12.2 Syntax......Page 297 12.3 Semantics......Page 298 12.4 Tableaux......Page 300 12.5 Identity......Page 306 12.6 Some Philosophical Issues......Page 309 12.7 Some Final Technical Comments......Page 311 12.8 Proofs of Theorems 1......Page 312 12.9 Proofs of Theorems 2......Page 317 12.10 Proofs of Theorems 3......Page 319 12.12 Further Reading......Page 321 12.13 Problems......Page 322 13.2 Syntax and Semantics......Page 324 13.3 Tableaux......Page 325 13.4 Free Logics: Positive, Negative and Neutral......Page 327 13.5 Quantification and Existence......Page 329 13.6 Identity in Free Logic......Page 331 13.7 Proofs of Theorems......Page 334 13.8 History......Page 338 13.10 Problems......Page 339 14.2 Constant Domain K......Page 342 14.3 Tableaux for CK......Page 343 14.4 Other Normal Modal Logics......Page 348 14.5 Modality De Re and De Dicto......Page 349 14.6 Tense Logic......Page 352 14.7 Proofs of Theorems......Page 354 14.8 History......Page 359 14.9 Further Reading......Page 360 14.10 Problems......Page 361 15.2 Prolegomenon......Page 363 15.3 Variable Domain K and its Normal Extensions......Page 364 15.4 Tableaux for VK and its Normal Extensions......Page 365 15.5 Variable Domain Tense Logic......Page 369 15.6 Extensions......Page 370 15.7 Existence Across Worlds......Page 373 15.8 Existence and Wide-Scope Quantifiers......Page 375 15.9 Proofs of Theorems......Page 376 15.11 Further Reading......Page 380 15.12 Problems......Page 381 16.1 Introduction......Page 383 16.2 Necessary Identity......Page 384 16.3 The Negativity Constraint......Page 386 16.4 Rigid and Non-rigid Designators......Page 388 16.5 Names and Descriptions......Page 391 16.6 Proofs of Theorems 1......Page 392 16.7 Proofs of Theorems 2......Page 396 16.9 Further Reading......Page 398 16.10 Problems......Page 399 17.2 Contingent Identity......Page 401 17.3 SI Again, and the Nature of Avatars......Page 407 17.4 Proofs of Theorems......Page 410 17.7 Problems......Page 416 18.2 Non-normal Modal Logics and Matrices......Page 418 18.3 Constant Domain Quantified L......Page 419 18.4 Tableaux for Constant Domain L......Page 420 18.5 Ringing the Changes......Page 421 18.6 Identity......Page 425 18.7 Proofs of Theorems......Page 427 18.10 Problems......Page 431 19.2 Constant and Variable Domain C......Page 433 19.3 Extensions......Page 437 19.4 Identity......Page 442 19.5 Some Philosophical Issues......Page 447 19.6 Proofs of Theorems......Page 449 19.9 Problems......Page 453 20.2 Existence and Construction......Page 455 20.3 Quantified Intuitionist Logic......Page 456 20.4 Tableaux for Intuitionist Logic 1......Page 458 20.5 Tableaux for Intuitionist Logic 2......Page 461 20.6 Mental Constructions......Page 465 20.7 Necessary Identity......Page 466 20.8 Intuitionist Identity......Page 468 20.9 Proofs of Theorems 1......Page 471 20.10 Proofs of Theorems 2......Page 482 20.13 Problems......Page 487 21.2 Quantified Many-valued Logics......Page 490 21.3 $forall$ and $exists$......Page 491 21.4 Some 3-valued Logics......Page 493 21.5 Their Free Versions......Page 495 21.6 Existence and Quantification......Page 496 21.7 Neutral Free Logics......Page 499 21.8 Identity......Page 501 21.9 Non-classical Identity......Page 502 21.10 Supervaluations and Subvaluations......Page 503 21.11 Proofs of Theorems......Page 505 21.12 History......Page 507 21.14 Problems......Page 508 22.2 Relational and Many-valued Semantics......Page 510 22.3 Tableaux......Page 513 22.4 Free Logics with Relational Semantics......Page 515 22.5 Semantics with the Routley......Page 517 22.6 Identity......Page 520 22.7 Proofs of Theorems 1......Page 523 22.8 Proofs of Theorems 2......Page 527 22.9 Proofs of Theorems 3......Page 533 22.12 Problems......Page 536 23.1 Introduction......Page 538 23.3 N4......Page 539 23.4 N......Page 542 23.5 K4 and K......Page 544 23.6 Relevant Identity......Page 546 23.7 Relevant Predication......Page 549 23.8 Logics with Constructible Negation......Page 551 23.9 Identity for Logics with Constructible Negation......Page 555 23.10 Proofs of Theorems 1......Page 557 23.11 Proofs of Theorems 2......Page 561 23.12 Proofs of Theorems 3......Page 564 23.14 Further Reading......Page 566 23.15 Problems......Page 567 24.2 Quantified B......Page 569 24.3 Extensions of B......Page 571 24.4 Restricted Quantification......Page 575 24.5 Semantics vs Proof Theory......Page 577 24.6 Identity......Page 582 24.7 Properties of Identity......Page 587 24.8 Proofs of Theorems 1......Page 589 24.9 Proofs of Theorems 2......Page 593 24.11 Further Reading......Page 595 24.12 Problems......Page 596 25.1 Introduction......Page 598 25.3 Validity in LN......Page 599 25.4 Deductions......Page 604 25.5 The Sorites Again......Page 606 25.6 Fuzzy Identity......Page 607 25.7 Vague Objects......Page 610 25.8 Appendix: Quantification and Identity in t-norm Logics......Page 612 25.9 History......Page 615 25.11 Problems......Page 616 Postcript: A Methodological Coda......Page 618 References......Page 621 Index of Names......Page 637 Index of Subjects......Page 641 This revised and considerably expanded edition of An introduction to Non-Classical Logic brings together a wide range of topics, including modal, tense, conditional, intuitionist, many-valued, paraconsistent, relevant and fuzzy logics. Part I, on prepositional logic, is the old introduction, but contains much new material. Part II is entirely novel, and covers quantification and identity for all the logics in Part I. The material is unified by the underlying theme of world semantics. All of the topics are explained clearly and accessibly, using devices such as tableau proofs, and their relation to current philosophical issues and debates is discussed students with a basic understanding of classical logic will find this book an invaluable introduction to an area that has become of central importance in both logic and philosophy. It will also interest people working in mathematics and computer science who wish to know about the area.