An Introduction to Non-Classical Logic: 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. All of the topics are explained clearly using devices such as tableau proofs, and their relation to current philosophical issues are explained. Title Copyright Dedication Contents Preface to the First Edition Preface to the Second Edition On Part I On Part II Book Website Mathematical Prolegomenon 0.1 Set-theoretic Notation 0.2 Proof by Induction 0.3 Equivalence Relations and Equivalence Classes Part I Propositional Logic 1 Classical Logic and the Material Conditional 1.1 Introduction 1.2 The Syntax of the Object Language 1.3 Semantic Validity 1.4 Tableaux 1.5 Counter-models 1.6 Conditionals 1.7 The Material Conditional 1.8 Subjunctive and Counterfactual Conditionals 1.9 More Counter-examples 1.10 Arguments for ... 1.11 Proofs of Theorems 1.12 History 1.13 Further Reading 1.14 Problems 2 Basic Modal Logic 2.1 Introduction 2.2 Necessity and Possibility 2.3 Modal Semantics 2.4 Modal Tableaux 2.5 Possible Worlds: Representation 2.6 Modal Realism 2.7 Modal Actualism 2.8 Meinongianism 2.9 Proofs of Theorems 2.10 History 2.11 Further Reading 2.12 Problems 3 Normal Modal Logics 3.1 Introduction 3.2 Semantics for Normal Modal Logics 3.3 Tableaux for Normal Modal Logics 3.4 Infinite Tableaux 3.5 S5 3.6 Which System Represents Necessity? 3.6a The Tense Logic Kt 3.6b Extensions of Kt 3.7 Proofs of Theorems 3.8 History 3.9 Further Reading 3.10 Problems 4 Non-normal Modal Logics; Strict Conditionals 4.1 Introduction 4.2 Non-normal Worlds 4.3 Tableaux for Non-normal Modal Logics 4.4 The Properties of Non-normal Logics 4.4a S0.5 4.5 Strict Conditionals 4.6 The Paradoxes of Strict Implication 4.7 ... and their Problems 4.8 The Explosion of Contradictions 4.9 Lewis' Argument for Explosion 4.10 Proofs of Theorems 4.11 History 4.12 Further Reading 4.13 Problems 5 Conditional Logics 5.1 Introduction 5.2 Some More Problematic Inferences 5.3 Conditional Semantics 5.4 Tableaux for C 5.5 Extensions of C 5.6 Similarity Spheres 5.7 C1 and C2 5.8 Further Philosophical Reflections 5.9 Proofs of Theorems 5.10 History 5.11 Further Reading 5.12 Problems 6 Intuitionist Logic 6.1 Introduction 6.2 Intuitionism: The Rationale 6.3 Possible-world Semantics for Intuitionism 6.4 Tableaux for Intuitionist Logic 6.5 The Foundations of Intuitionism 6.6 The Intuitionist Conditional 6.7 Proofs of Theorems 6.8 History 6.9 Further Reading 6.10 Problems 7 Many-valued Logics 7.1 Introduction 7.2 Many-valued Logic: The General Structure 7.3 The 3-valued Logics of Kleene and Lukasiewicz 7.4 LP and RM3 7.5 Many-valued Logics and Conditionals 7.6 Truth-value Gluts: Inconsistent Laws 7.7 Truth-value Gluts: Paradoxes of Self-reference 7.8 Truth-value Gaps: Denotation Failure 7.9 Truth-value Gaps: Future Contingents 7.10 Supervaluations, Modality and Many-valued Logic 7.11 Proofs of Theorems 7.12 History 7.13 Further Reading 7.14 Problems 8 First Degree Entailment 8.1 Introduction 8.2 The Semantics of FDE 8.3 Tableaux for FDE 8.4 FDE and Many-valued Logics 8.4a Relational Semantics and Tableaux for L3 and RM3 8.5 The Routley Star 8.6 Paraconsistency and the Disjunctive Syllogism 8.7 Proofs of Theorems 8.8 History 8.9 Further Reading 8.10 Problems 9 Logics with Gaps, Gluts and Worlds 9.1 Introduction 9.2 Adding $arrow $ 9.3 Tableaux for K4 9.4 Non-normal Worlds Again 9.5 Tableaux for N4 9.6 Star Again 9.7 Impossible Worlds and Relevant Logic 9.7a Logics of Constructible Negation 9.8 Proofs of Theorems 9.9 History 9.10 Further Reading 9.11 Problems 10 Relevant Logics 10.1 Introduction 10.2 The Logic B 10.3 Tableaux for B 10.4 Extensions of B 10.4a Content Inclusion 10.5 The System R 10.6 The Ternary Relation 10.7 Ceteris Paribus Enthymemes 10.8 Proofs of Theorems 10.9 History 10.10 Further Reading 10.11 Problems 11 Fuzzy Logics 11.1 Introduction 11.2 Sorites Paradoxes 11.3 ... and Responses to Them 11.4 The Continuum-valued Logic L 11.5 Axioms for LN 11.6 Conditionals in L 11.7 Fuzzy Relevant Logic 11.7a Appendix: t-norm Logics 11.8 History 11.9 Further Reading 11.10 Problems 11a Appendix: Many-valued Modal Logics 11a.1 Introduction 11a.2 General Structure 11a.3 Illustration: Modal Lukasiewicz Logic 11a.4 Modal FDE 11a.5 Tableaux 11a.6 Variations 11a.7 Future Contingents Revisited 11a.8 A Glimpse Beyond 11a.9 Proofs of Theorems Postcript: An Historical Perspective on Conditionals Part II Quantification and Identity 12 Classical First-order Logic 12.1 Introduction 12.2 Syntax 12.3 Semantics 12.4 Tableaux 12.5 Identity 12.6 Some Philosophical Issues 12.7 Some Final Technical Comments 12.8 Proofs of Theorems 1 12.9 Proofs of Theorems 2 12.10 Proofs of Theorems 3 12.11 History 12.12 Further Reading 12.13 Problems 13 Free Logics 13.1 Introduction 13.2 Syntax and Semantics 13.3 Tableaux 13.4 Free Logics: Positive, Negative and Neutral 13.5 Quantification and Existence 13.6 Identity in Free Logic 13.7 Proofs of Theorems 13.8 History 13.9 Further Reading 13.10 Problems 14 Constant Domain Modal Logics 14.1 Introduction 14.2 Constant Domain K 14.3 Tableaux for CK 14.4 Other Normal Modal Logics 14.5 Modality De Re and De Dicto 14.6 Tense Logic 14.7 Proofs of Theorems 14.8 History 14.9 Further Reading 14.10 Problems 15 Variable Domain Modal Logics 15.1 Introduction 15.2 Prolegomenon 15.3 Variable Domain K and its Normal Extensions 15.4 Tableaux for VK and its Normal Extensions 15.5 Variable Domain Tense Logic 15.6 Extensions 15.7 Existence Across Worlds 15.8 Existence and Wide-Scope Quantifiers 15.9 Proofs of Theorems 15.10 History 15.11 Further Reading 15.12 Problems 16 Necessary Identity in Modal Logic 16.1 Introduction 16.2 Necessary Identity 16.3 The Negativity Constraint 16.4 Rigid and Non-rigid Designators 16.5 Names and Descriptions 16.6 Proofs of Theorems 1 16.7 Proofs of Theorems 2 16.8 History 16.9 Further Reading 16.10 Problems 17 Contingent Identity in Modal Logic 17.1 Introduction 17.2 Contingent Identity 17.3 SI Again, and the Nature of Avatars 17.4 Proofs of Theorems 17.5 History 17.6 Further Reading 17.7 Problems 18 Non-normal Modal Logics 18.1 Introduction 18.2 Non-normal Modal Logics and Matrices 18.3 Constant Domain Quantified L 18.4 Tableaux for Constant Domain L 18.5 Ringing the Changes 18.6 Identity 18.7 Proofs of Theorems 18.8 History 18.9 Further Reading 18.10 Problems 19 Conditional Logics 19.1 Introduction 19.2 Constant and Variable Domain C 19.3 Extensions 19.4 Identity 19.5 Some Philosophical Issues 19.6 Proofs of Theorems 19.7 History 19.8 Further Reading 19.9 Problems 20 Intuitionist Logic 20.1 Introduction 20.2 Existence and Construction 20.3 Quantified Intuitionist Logic 20.4 Tableaux for Intuitionist Logic 1 20.5 Tableaux for Intuitionist Logic 2 20.6 Mental Constructions 20.7 Necessary Identity 20.8 Intuitionist Identity 20.9 Proofs of Theorems 1 20.10 Proofs of Theorems 2 20.11 History 20.12 Further Reading 20.13 Problems 21 Many-valued Logics 21.1 Introduction 21.2 Quantified Many-valued Logics 21.3 $forall$ and $exists$ 21.4 Some 3-valued Logics 21.5 Their Free Versions 21.6 Existence and Quantification 21.7 Neutral Free Logics 21.8 Identity 21.9 Non-classical Identity 21.10 Supervaluations and Subvaluations 21.11 Proofs of Theorems 21.12 History 21.13 Further Reading 21.14 Problems 22 First Degree Entailment 22.1 Introduction 22.2 Relational and Many-valued Semantics 22.3 Tableaux 22.4 Free Logics with Relational Semantics 22.5 Semantics with the Routley 22.6 Identity 22.7 Proofs of Theorems 1 22.8 Proofs of Theorems 2 22.9 Proofs of Theorems 3 22.10 History 22.11 Further Reading 22.12 Problems 23 Logics with Gaps, Gluts and Worlds 23.1 Introduction 23.2 Matrix Semantics Again 23.3 N4 23.4 N 23.5 K4 and K 23.6 Relevant Identity 23.7 Relevant Predication 23.8 Logics with Constructible Negation 23.9 Identity for Logics with Constructible Negation 23.10 Proofs of Theorems 1 23.11 Proofs of Theorems 2 23.12 Proofs of Theorems 3 23.13 History 23.14 Further Reading 23.15 Problems 24 Relevant Logics 24.1 Introduction 24.2 Quantified B 24.3 Extensions of B 24.4 Restricted Quantification 24.5 Semantics vs Proof Theory 24.6 Identity 24.7 Properties of Identity 24.8 Proofs of Theorems 1 24.9 Proofs of Theorems 2 24.10 History 24.11 Further Reading 24.12 Problems 25 Fuzzy Logics 25.1 Introduction 25.3 Validity in LN 25.4 Deductions 25.5 The Sorites Again 25.6 Fuzzy Identity 25.7 Vague Objects 25.8 Appendix: Quantification and Identity in t-norm Logics 25.9 History 25.10 Further Reading 25.11 Problems Postcript: A Methodological Coda References Index of Names Index of Subjects 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.