In this text, a variety of modal logics at the sentential, first-order, and second-order levels are developed with clarity, precision and philosophical insight. All of the S1-S5 modal logics of Lewis and Langford, among others, are constructed. A matrix, or many-valued semantics, for sentential modal logic is formalized, and an important result that no finite matrix can characterize any of the standard modal logics is proven. Exercises, some of which show independence results, help to develop logical skills. A separate sentential modal logic of logical necessity in logical atomism is also constructed and shown to be complete and decidable. On the first-order level of the logic of logical necessity, the modal thesis of anti-essentialism is valid and every de re sentence is provably equivalent to a de dicto sentence. An elegant extension of the standard sentential modal logics into several first-order modal logics is developed. Both a first-order modal logic for possibilism containing actualism as a proper part as well as a separate modal logic for actualism alone are constructed for a variety of modal systems. Exercises on this level show the connections between modal laws and quantifier logic regarding generalization into, or out of, modal contexts and the conditions required for the necessity of identity and non-identity. Two types of second-order modal logics, one possibilist and the other actualist, are developed based on a distinction between existence-entailing concepts and concepts in general. The result is a deeper second-order analysis of possibilism and actualism as ontological frameworks. Exercises regarding second-order predicate quantifiers clarify the distinction between existence-entailing concepts and concepts in general. Modal Logic is ideally suited as a core text for graduate and undergraduate courses in modal logic, and as supplementary reading in courses on mathematical logic, formal ontology, and artificial intelligence.
Author(s): Nino B. Cocchiarella, Max A. Freund
Publisher: OUP
Year: 2008
Language: English
Pages: 283
Cover......Page 1
Title Page......Page 5
Preface......Page 7
Contents......Page 11
1.1 The Metalanguage......Page 17
1.2.1 Symbols and Expressions......Page 20
1.2.2 Concatenation......Page 21
1.2.3 Formal Languages and Systems......Page 22
1.2.4 The Logistic Method......Page 24
1.3 Tautologous Implication......Page 29
2.1 Sentential Modal Logic......Page 31
2.1.1 Modal CN-Formulas......Page 32
2.1.2 Modal-Free and Modally-Closed Formulas......Page 33
2.2 Modal CN-Calculi......Page 34
2.2.1 Classical Modal Calculi......Page 35
2.2.2 Regular and Normal Modal Calculi......Page 36
2.2.3 The MP Rule......Page 38
2.2.4 The Systems Σ_K......Page 39
2.3 Some Standard Normal Modal CN-Calculi......Page 43
2.3.1 The Modal System Kr......Page 44
2.3.2 The Modal System M......Page 46
2.3.3 The Modal System Br......Page 47
2.3.4 The Modal System S4......Page 49
2.3.6 The Modal System S4.3......Page 50
2.3.7 The Modal System S5......Page 52
2.4 The Systems S1, S2, and S3......Page 53
2.5 Modalities......Page 58
3 Matrix Semantics......Page 61
3.1 CN-Matrices......Page 62
3.2 The Standard Two-Valued CN-Matrix......Page 64
3.3 Modal CN-Matrices......Page 68
3.4 Henle Modal CN-Matrices......Page 71
4 Semantics for Logical Necessity......Page 77
4.1 The Problem of a Semantics for Logical Necessity......Page 78
4.2 Carnap’s Adequacy Criterion......Page 80
4.3 Logical Atomism and Modal Logic......Page 82
5.1 All Possible Worlds “Cut Down”......Page 87
5.2 Matrix Semantics for S5......Page 91
5.3 Decidability of L_at and S5......Page 94
6.1 Relational World Systems Defined......Page 97
6.2 The Class of All Relational World Systems......Page 105
6.3 Reflexivity and Accessibility......Page 108
6.4 Transitive World Systems......Page 112
6.5 Quasi-Ordered World Systems......Page 114
6.6 Symmetric World Systems......Page 116
6.7 Reflexive and Symmetric World Systems......Page 117
6.8 Transitive and Symmetric World Systems......Page 118
6.9 Partitioned World Systems......Page 119
6.10 Connexity and Accessibility......Page 123
6.11 Connectable Accessibility......Page 129
7 Quantified Modal Logic......Page 135
7.1 Logical Syntax......Page 136
7.2 First-Order Languages......Page 138
7.3 Proper Substitution......Page 140
7.4 Quantified Modal CN-Calculi......Page 144
7.5 Quantified Extensions of Kr......Page 156
7.6 Omega-Completeness in Modal Logic......Page 163
8 The Semantics of Quantified Modal Logic......Page 169
8.1 Semantics of Standard Modal-Free Formulas......Page 170
8.2 The Semantics of Logical Necessity......Page 174
8.3 The Thesis of Anti-Essentialism......Page 175
8.4 Incompleteness of the Primary Semantics......Page 178
8.5 Secondary Semantics for Necessity......Page 180
8.6 Actualist-Possibilist Secondary Semantics......Page 185
8.7 Relational Model Structures......Page 193
9 Second-Order Modal Logic......Page 199
9.1 Second-Order Logical Syntax......Page 200
9.2 Second-Order Languages......Page 201
9.3 Proper Substitution......Page 204
9.4 Second-Order CN-Modal Calculi......Page 208
9.5 Second-Order Extensions of Kr......Page 218
9.6 Second-Order Omega-Completeness......Page 225
10 Semantics of Second-Order Modal Logic......Page 231
10.1 Semantics of Modal-Free Second-Order Formulas......Page 232
10.2 General Models......Page 237
10.3 Semantics of Standard Second-Order Modal Languages......Page 240
10.4 Actualist-Possibilist Second-Order Semantics......Page 247
10.5 Second-Order Relational World Systems......Page 259
Afterword......Page 269
Bibliography......Page 273
Index......Page 279
