Preface to the First Edition. xxvii 0.1 Set-theoretic Notation xxvii 0.2 Proof by Induction xxix 0.3 Equivalence Relations and Equivalence Classes xxx

Transcription

1 Table of Preface to the First Edition Preface to the Second Edition page xvii xxi Mathematical Prolegomenon xxvii 0.1 Set-theoretic Notation xxvii 0.2 Proof by Induction xxix 0.3 Equivalence Relations and Equivalence Classes xxx Part I Propositional Logic 1 1 Classical Logic and the Material Conditional Introduction The Syntax of the Object Language Semantic Validity Tableaux Counter-models Conditionals The Material Conditional Subjunctive and Counterfactual Conditionals More Counter-examples Arguments for Proofs of Theorems History Further Reading Problems 18 2 Basic Modal Logic Introduction Necessity and Possibility 20 vii

2 Table of viii 2.3 Modal Semantics Modal Tableaux Possible Worlds: Representation Modal Realism Modal Actualism Meinongianism *Proofs of Theorems History Further Reading Problems 34 3 Normal Modal Logics Introduction Semantics for Normal Modal Logics Tableaux for Normal Modal Logics Infinite Tableaux S Which System Represents Necessity? a The Tense Logic K t b Extensions of K t *Proofs of Theorems History Further Reading Problems 60 4 Non-normal Modal Logics; Strict Conditionals Introduction Non-normal Worlds Tableaux for Non-normal Modal Logics The Properties of Non-normal Logics a S Strict Conditionals The Paradoxes of Strict Implication and their Problems The Explosion of Contradictions Lewis Argument for Explosion *Proofs of Theorems History 79

3 Table of ix 4.12 Further Reading Problems 80 5 Conditional Logics Introduction Some More Problematic Inferences Conditional Semantics Tableaux for C Extensions of C Similarity Spheres C 1 and C Further Philosophical Reflections *Proofs of Theorems History Further Reading Problems Intuitionist Logic Introduction Intuitionism: The Rationale Possible-world Semantics for Intuitionism Tableaux for Intuitionist Logic The Foundations of Intuitionism The Intuitionist Conditional *Proofs of Theorems History Further Reading Problems Many-valued Logics Introduction Many-valued Logic: The General Structure The 3-valued Logics of Kleene and Lukasiewicz LP and RM Many-valued Logics and Conditionals Truth-value Gluts: Inconsistent Laws Truth-value Gluts: Paradoxes of Self-reference Truth-value Gaps: Denotation Failure Truth-value Gaps: Future Contingents Supervaluations, Modality and Many-valued Logic 133

4 Table of x 7.11 *Proofs of Theorems History Further Reading Problems First Degree Entailment Introduction The Semantics of FDE Tableaux for FDE FDE and Many-valued Logics a Relational Semantics and Tableaux for L 3 and RM The Routley Star Paraconsistency and the Disjunctive Syllogism *Proofs of Theorems History Further Reading Problems Logics with Gaps, Gluts and Worlds Introduction Adding Tableaux for K Non-normal Worlds Again Tableaux for N Star Again Impossible Worlds and Relevant Logic a Logics of Constructible Negation *Proofs of Theorems History Further Reading Problems Relevant Logics Introduction The Logic B Tableaux for B Extensions of B a Content Inclusion The System R The Ternary Relation 206

5 Table of xi 10.7 Ceteris Paribus Enthymemes *Proofs of Theorems History Further Reading Problems Fuzzy Logics Introduction Sorites Paradoxes and Responses to Them The Continuum-valued Logic L Axioms for L ℵ Conditionals in L Fuzzy Relevant Logic a *Appendix: t-norm Logics History Further Reading Problems a Appendix: Many-valued Modal Logics a.1 Introduction a.2 General Structure a.3 Illustration: Modal Lukasiewicz Logic a.4 Modal FDE a.5 Tableaux a.6 Variations a.7 Future Contingents Revisited a.8 A Glimpse Beyond a.9 *Proofs of Theorems 255 Postscript: An Historical Perspective on Conditionals 259 Part II Quantification and Identity Classical First-order Logic Introduction Syntax Semantics Tableaux Identity 272

6 Table of xii 12.6 Some Philosophical Issues Some Final Technical Comments *Proofs of Theorems *Proofs of Theorems *Proofs of Theorems History Further Reading Problems Free Logics Introduction Syntax and Semantics Tableaux Free Logics: Positive, Negative and Neutral Quantification and Existence Identity in Free Logic *Proofs of Theorems History Further Reading Problems Constant Domain Modal Logics Introduction Constant Domain K Tableaux for CK Other Normal Modal Logics Modality De Re and De Dicto Tense Logic *Proofs of Theorems History Further Reading Problems Variable Domain Modal Logics Introduction Prolegomenon Variable Domain K and its Normal Extensions Tableaux for VK and its Normal Extensions Variable Domain Tense Logic Extensions 336

7 Table of xiii 15.7 Existence Across Worlds Existence and Wide-Scope Quantifiers *Proofs of Theorems History Further Reading Problems Necessary Identity in Modal Logic Introduction Necessary Identity The Negativity Constraint Rigid and Non-rigid Designators Names and Descriptions *Proofs of Theorems *Proofs of Theorems History Further Reading Problems Contingent Identity in Modal Logic Introduction Contingent Identity SI Again, and the Nature of Avatars *Proofs of Theorems History Further Reading Problems Non-normal Modal Logics Introduction Non-normal Modal Logics and Matrices Constant Domain Quantified L Tableaux for Constant Domain L Ringing the Changes Identity *Proofs of Theorems History Further Reading Problems 397

8 Table of xiv 19 Conditional Logics Introduction Constant and Variable Domain C Extensions Identity Some Philosophical Issues *Proofs of Theorems History Further Reading Problems Intuitionist Logic Introduction Existence and Construction Quantified Intuitionist Logic Tableaux for Intuitionist Logic Tableaux for Intuitionist Logic Mental Constructions Necessary Identity Intuitionist Identity *Proofs of Theorems *Proofs of Theorems History Further Reading Problems Many-valued Logics Introduction Quantified Many-valued Logics and Some 3-valued Logics Their Free Versions Existence and Quantification Neutral Free Logics Identity Non-classical Identity Supervaluations and Subvaluations 469

9 Table of xv *Proofs of Theorems History Further Reading Problems First Degree Entailment Introduction Relational and Many-valued Semantics Tableaux Free Logics with Relational Semantics Semantics with the Routley Identity *Proofs of Theorems *Proofs of Theorems *Proofs of Theorems History Further Reading Problems Logics with Gaps, Gluts and Worlds Introduction Matrix Semantics Again N N K 4 and K Relevant Identity Relevant Predication Logics with Constructible Negation Identity for Logics with Constructible Negation *Proofs of Theorems *Proofs of Theorems *Proofs of Theorems History Further Reading Problems Relevant Logics Introduction 535

10 Table of xvi 24.2 Quantified B Extensions of B Restricted Quantification Semantics vs Proof Theory Identity Properties of Identity *Proofs of Theorems *Proofs of Theorems History Further Reading Problems Fuzzy Logics Introduction Quantified Lukasiewicz Logic Validity in L ℵ Deductions The Sorites Again Fuzzy Identity Vague Objects *Appendix: Quantification and Identity in t-norm Logics History Further Reading Problems 582 Postscript: A Methodological Coda 584 References 587 Index of Names 603 Index of Subjects 607