Modal Logic An Introduction to its Syntax and Semantics

by Cocchiarella, Nino B.; Freund, Max A.

• ISBN13: 9780195366587
• ISBN10: 0195366581
• Format: Hardcover
• Copyright: 2008-08-04
• Publisher: Oxford University Press

Summary

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 Biography

Nino B. Cocchiarella, Professor Emeritus of Logic and Philosophy at Indiana University has published many new results on the logic of time, modality, logical necessity, philosophical and mathematical logic, formal ontology, theories of predication, reference, and nominalization. He is the author of several books, most recently Formal Ontology and Conceptual Realism.
Nino B. Cocchiarella, Professor Emeritus of Logic and Philosophy at Indiana University has published many new results on the logic of time, modality, logical necessity, philosophical and mathematical logic, formal ontology, theories of predication, reference, and nominalization. He is the author of several books, most recently Formal Ontology and Conceptual Realism. Max A. Freund is professor of logic and philosophy at the National University of Costa Rica, and has been a visiting professor at the most important universities in Latin America as well a visiting researcher at the University of Munich, Germany. He is the author of numerous articles on modal logic, sortal logic, epistemic logic, philosophy of language, philosophy of computability, second-order logic, judicial logic, and a forthcoming book entitled Judicial Logic.

Table of Contents

Introduction	p. 1
The Metalanguage	p. 1
Logical Syntax	p. 4
Symbols and Expressions	p. 4
Concatenation	p. 5
Formal Languages and Systems	p. 6
The Logistic Method	p. 8
Tautologous Implication	p. 13
The Syntax of Modal Sentential Calculi	p. 15
Sentential Modal Logic	p. 15
Modal CN-Formulas	p. 16
Modal-Free and Modally-Closed Formulas	p. 17
Modal CN-Calculi	p. 18
Classical Modal Calculi	p. 19
Regular and Normal Modal Calculi	p. 20
The MP Rule	p. 22
The Systems [Sigma subscript K]	p. 23
Some Standard Normal Modal CN-Calculi	p. 27
The Modal System Kr	p. 28
The Modal System M	p. 30
The Modal System Br	p. 31
The Modal System S4	p. 33
The Modal System S4.2	p. 34
The Modal System S4.3	p. 34
The Modal System S5	p. 36
The Systems S1, S2, and S3	p. 37
Modalities	p. 42
Matrix Semantics	p. 45
CN-Matrices	p. 46
The Standard Two-Valued CN-Matrix	p. 48
Modal CN-Matrices	p. 52
Henle Modal CN-Matrices	p. 55
Semantics for Logical Necessity	p. 61
The Problem of a Semantics for Logical Necessity	p. 62
Carnap's Adequacy Criterion	p. 64
Logical Atomism and Modal Logic	p. 66
Semantics for S5	p. 71
All Possible Worlds "Cut Down"	p. 71
Matrix Semantics for S5	p. 75
Decidability of L[subscript at] and S5	p. 78
Relational World Systems	p. 81
Relational World Systems Defined	p. 81
The Class of All Relational World Systems	p. 89
Reflexivity and Accessibility	p. 92
Transitive World Systems	p. 96
Quasi-Ordered World Systems	p. 98
Symmetric World Systems	p. 100
Reflexive and Symmetric World Systems	p. 101
Transitive and Symmetric World Systems	p. 102
Partitioned World Systems	p. 103
Connexity and Accessibility	p. 107
Connectable Accessibility	p. 113
Quantified Modal Logic	p. 119
Logical Syntax	p. 120
First-Order Languages	p. 122
Proper Substitution	p. 124
Quantified Modal CN-Calculi	p. 128
Quantified Extensions of Kr	p. 140
Omega-Completeness in Modal Logic	p. 147
The Semantics of Quantified Modal Logic	p. 153
Semantics of Standard Modal-Free Formulas	p. 154
The Semantics of Logical Necessity	p. 158
The Thesis of Anti-Essentialism	p. 159
Incompleteness of the Primary Semantics	p. 162
Secondary Semantics for Necessity	p. 164
Actualist-Possibilist Secondary Semantics	p. 169
Relational Model Structures	p. 177
Second-Order Modal Logic	p. 183
Second-Order Logical Syntax	p. 184
Second-Order Languages	p. 185
Proper Substitution	p. 188
Second-Order CN-Modal Calculi	p. 192
Second-Order Extensions of Kr	p. 202
Second-Order Omega-Completeness	p. 209
Semantics of Second-Order Modal Logic	p. 215
Semantics of Modal-Free Second-Order Formulas	p. 216
General Models	p. 221
Semantics of Standard Second-Order Modal Languages	p. 224
Actualist-Possibilist Second-Order Semantics	p. 231
Second-Order Relational World Systems	p. 243
Afterword	p. 253
Bibliography	p. 257
Index	p. 263
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