Exploring Logical Dynamics

by Van Benthem, Johan; Benthem, J. F. A. K. Van

• ISBN13: 9781575860589
• ISBN10: 1575860589
• Format: Paperback
• Copyright: 1996-06-01
• Publisher: Stanford Univ Center for the Study

Summary

This book is an exploration of current trends in logical theories of information flow across various fields, such as belief revision in computer science or dynamic semantics in linguistics. It provides one mathematical perspective encompassing all of these. This framework generates a new agenda of questions concerning dynamic inference and dynamic operators. The result is a mathematical theory of process models, simulations between these, and modal languages over them, which is developed in quite some detail. New results include theorems on expressive completeness, representation of styles of inference, and new kinds of decidable remodeling for standard logics. This theory is also confronted with practice in computer science, linguistics and philosophy.

Table of Contents

Introduction	ix
Part I The Dynamic Turn	1	(78)
1 Congnitive Actions	3	(14)
1.1 Recognizing Cognitive Processes	3	(2)
1.2 Mathematical Process Models	5	(2)
1.3 Standard Logics Dynamified	7	(2)
1.4 Propositions as Procedures	9	(4)
1.5 Notes	13	(4)
2 Dynamification	17	(30)
2.1 Updating Information in Propositional Logic	17	(5)
2.2 Changing Perspectives in Predicate Logic	22	(10)
2.3 Dynamic Modal Logic	32	(4)
2.4 Changing Perferences	36	(3)
2.5 General Conclusions	39	(1)
2.6 Notes	40	(7)
3 Technical Tools	47	(32)
3.1 First-Order Predicate Logic	47	(6)
3.2 Modal Logic	53	(11)
3.3 Relational Algebra	64	(7)
3.4 Dynamic Logic	71	(4)
3.5 Notes	75	(4)
Part II Logical Foundations	79	(124)
4 Process Simulation and Definability	81	(24)
4.1 Transition Systems and Process Equivalences	81	(3)
4.2 Languages over Transition Systems	84	(3)
4.3 Bisimulation Invariance and Modal Definability	87	(5)
4.4 Modal Fine-Structure of First-Order Logic	92	(9)
4.5 Notes	101	(2)
4.6 Questions	103	(2)
5 Relational Algebra of Process Operations	105	(18)
5.1 Spaces of Logical Operations	105	(4)
5.2 Boolean Structure	109	(3)
5.3 Safety for Bisimulation	112	(8)
5.4 Notes	120	(2)
5.5 Questions	122	(1)
6 Two-Level Static-Dynamic Architecture	123	(16)
6.1 Abstract Dynamic Logic	123	(7)
6.2 Dynamic Modal Logic	130	(6)
6.3 Notes	136	(2)
6.4 Questions	138	(1)
7 Dynamic Styles of inference	139	(20)
7.1 Styles and Structural Rules	139	(3)
7.2 Representation Theorems	142	(3)
7.3 A Modal Perspective	145	(5)
7.4 Inverse Logic	150	(2)
7.5 Switching Between Styles	152	(3)
7.6 Notes	155	(2)
7.7 Questions	157	(2)
8 Decidable Remodelling: Arrow Logic	159	(16)
8.1 Core Content Versus Formal Wrappings	159	(1)
8.2 Arrow Logic in a Nutshell	159	(5)
8.3 Dynamic Arrow Logic	164	(4)
8.4 Dynamic Logic with Arrows	168	(2)
8.5 Notes	170	(3)
8.6 Questions	173	(2)
9 Modal Foundations for Predicate Logic	175	(28)
9.1 The Modal Core of Tarski Semantics	175	(2)
9.2 Dependency Models	177	(1)
9.3 What Do First-Order Axioms Say	178	(3)
9.4 Quantifiers and Substitutions	181	(2)
9.5 Landscape of Deductive Strength	183	(2)
9.6 Rethinking the Language	185	(1)
9.7 Applications and Repercussions	186	(1)
9.8 Representation	187	(4)
9.9 Decidability	191	(3)
9.10 Translations	194	(2)
9.11 Notes	196	(4)
9.12 Questions	200	(3)
Part III Implications	203	(98)
10 Computational Process Theories	205	(24)
10.1 Definition of Process Equivalences	205	(3)
10.2 Axiomatization of Process Equivalences	208	(1)
10.3 Infinitary Theory of Transition Systems	209	(4)
10.4 Special Transition Systems	213	(2)
10.5 Process Algebra	215	(4)
10.6 Safety for Process Constructions	219	(4)
10.7 Appendix I: Modal Logic of Constructions	223	(2)
10.8 Appendix II: Concurrent Dynamic Logic	225	(1)
10.9 Notes	226	(1)
10.10 Questions	227	(2)
11 Imperative Aspects of Logic Programs	229	(22)
11.1 Declarative Versus Procedural?	229	(2)
11.2 Dynamics of Deductive Approximation	231	(7)
11.3 Proof Theory of Procedures	238	(6)
11.4 Calculus of Tasks	244	(2)
11.5 Notes	246	(3)
11.6 Questions	249	(2)
12 Understanding Natural Language	251	(34)
12.1 Syntax	251	(9)
12.2 Semantics	260	(6)
12.3 Pragmatics	266	(7)
12.4 Alternative Frameworks	273	(5)
12.5 Notes	278	(5)
12.6 Questions	283	(2)
13 Philosophical Repercussions	285	(16)
13.1 Meaning and Cognition as Activities	285	(2)
13.2 Three Models for Cognitive Dynamics	287	(4)
13.3 Cognitive Fine-Structure	291	(3)
13.4 Contents and Wrappings	294	(4)
13.5 Managing Diversity	298	(2)
13.6 Notes	300	(1)
Bibliography	301	(26)
Index	327