An Introduction to Non-Classical Logic

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