An Introduction to Non-Classical Logic

This book is an introduction to nonclassical propositional logics. It brings together for the first time in a textbook a range of topics in logic, many of them of relatively recent origin, including modal, conditional, intuitionist, many-valued, paraconsistent, relevant and fuzzy logics. Students with a basic understanding of classical logic will find this an invaluable introduction to an area that has become of central importance in both logic and philosophy. It will be of interest to readers studying logic and, more widely, to readers working in mathematics and computer science. Half-Title......Page 1 Title Page......Page 3 Copyright......Page 4 Dedication......Page 5 Contents......Page 7 Preface......Page 13 0.1 Set-theoretic notation......Page 17 0.2 Proof by induction......Page 19 1.1 Introduction......Page 23 1.2 The syntax of the object language......Page 24 1.3 Semantic validity......Page 25 1.4 Tableaux......Page 26 1.5 Counter-models......Page 30 1.6 Conditionals......Page 31 1.7 The material conditional......Page 32 1.8 Subjunctive and counterfactual conditionals......Page 33 1.9 More counter-examples......Page 35 1.10 Arguments for horseshoe......Page 36 1.11 *Proofs of theorems......Page 37 1.13 Further reading......Page 39 1.14 Problems......Page 40 2.2 Necessity and possibility......Page 42 2.3 Modal semantics......Page 43 2.4 Modal tableaux......Page 46 2.5 Possible worlds: representation......Page 50 2.6 Modal realism......Page 51 2.7 Modal actualism......Page 52 2.8 Meinongianism......Page 53 2.9 *Proofs of theorems......Page 55 2.10 History......Page 57 2.12 Problems......Page 58 3.2 Semantics for normal modal logics......Page 60 3.3 Tableaux for normal modal logics......Page 62 3.4 Infinite tableaux......Page 66 3.5 S5......Page 69 3.6 Which system represents necessity?......Page 70 3.7 *Proofs of theorems......Page 74 3.9 Further reading......Page 76 3.10 Problems......Page 77 4.2 Non-normal worlds......Page 80 4.3 Tableaux for non-normal modal logics......Page 82 4.4 The properties of non-normal logics......Page 84 4.5 Strict conditionals......Page 85 4.6 The paradoxes of strict implication......Page 86 4.7 ...and their problems......Page 87 4.8 The explosion of contradictions ,......Page 89 4.9 Lewis' argument for explosion......Page 90 4.10 *Proofs of theorems......Page 91 4.11 History......Page 93 4.13 Problems......Page 94 5.2 Some more problematic inferences......Page 96 5.3 Conditional semantics......Page 99 5.4 Tableaux for C......Page 100 5.5 Extensions of C......Page 102 5.6 Similarity spheres......Page 105 5.7 C1 and C2......Page 110 5.8 Further philosophical reflections......Page 113 5.9 *Proofs of theorems......Page 115 5.10 History......Page 117 5.12 Problems......Page 118 6.2 Intuitionism: the rationale......Page 121 6.3 Possible-world semantics for intuitionism......Page 123 6.4 Tableaux for intuitionist logic......Page 126 6.5 The foundations of intuitionism......Page 130 6.6 The intuitionist conditional......Page 132 6.7 *Proofs of theorems......Page 133 6.9 Further reading......Page 136 6.10 Problems......Page 137 7.2 Many-valued logic: the general structure......Page 139 7.3 The 3-valued logics of Kleene and Lukasiewicz......Page 141 7.4 LP and RM3......Page 144 7.5 Many-valued logics and conditionals......Page 145 7.6 Truth-value gluts: inconsistent laws......Page 147 7.7 Truth-value gluts: paradoxes of self-reference......Page 149 7.8 Truth-value gaps: denotation failure......Page 150 7.9 Truth-value gaps: future contingents......Page 152 7.10 Supervaluations, modality and many-valued logic......Page 153 7.11 *Proofs of theorems......Page 156 7.12 History......Page 158 7.14 Problems......Page 159 8.2 The semantics of FDE......Page 161 8.3 Tableaux for FDE......Page 163 8.4 FDE and many-valued logics......Page 166 8.5 The Routley star......Page 169 8.6 Paraconsistency and the disjunctive syllogism......Page 173 8.7 *Proofs of theorems......Page 174 8.9 Further reading......Page 181 8.10 Problems......Page 182 9.2 Adding ->......Page 184 9.3 Tableaux for K4......Page 185 9.4 Non-normal worlds again......Page 187 9.5 Tableaux for N4......Page 189 9.6 Star again......Page 190 9.7 Impossible worlds and relevant logic......Page 193 9.8 *Proofs of theorems......Page 196 9.11 Problems......Page 201 10.2 The logics......Page 204 10.3 Tableaux for B......Page 206 10.4 Extensions of B......Page 210 10.5 The system R......Page 215 10.6 The ternary relation......Page 219 10.7 Ceteris paribus enthymemes......Page 220 10.8 *Proofs of theorems......Page 224 10.9 History......Page 227 10.10 Further reading......Page 228 10.11 Problems......Page 229 11.2 Sorites paradoxes......Page 233 11.3 ...and responses to them......Page 234 11.4 The continuum-valued logic L......Page 236 11.5 Axioms for L-aleph......Page 240 11.6 Conditionals in L......Page 243 11.7 Fuzzy relevant logic......Page 244 11.8 History......Page 247 11.10 Problems......Page 248 12 Conclusion: an historical perspective......Page 251 References......Page 253 K......Page 259 Y......Page 260 E......Page 261 N......Page 262 T......Page 263 W......Page 264 Publisher Description (unedited publisher data) This book is an introduction to non-classical propositional logics. It brings together for the first time in a textbook a range of topics in logic, many of them of relatively recent origin, including modal, conditional, intuitionist, many-valued, paraconsistent, relevant and fuzzy logics. The material is unified by the underlying theme of world-semantics. All of the topics are explained clearly and accessibly, using devices such as tableaux proofs, and their relation to current philosophical issues and debates is discussed. Students with a basic understanding of classical logic will find this an invaluable introduction to an area that has become of central importance in both logic and philosophy, but which, until now, could be studied only through the research literature. It will interest those studying logic, those who need to know about non-classical logics because of their philosophical importance, and, more widely, readers working in mathematics and computer science. Library of Congress subject headings for this publication: Nonclassical mathematical logic 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. This volume is an introduction to non-classical propositional logics. It brings together a range of topics in logic, including modal, conditional, intuitionist, many-valued, paraconsistent, relevant and fuzzy logics. The material is unified by the underlying theme of world-semantics. Their relation to contemporary philosophical issues and debates is discussed