Towards Mathematical Philosophy

by Makinson, David; Malinowski, Jacek; Wansing, Heinrich

• ISBN13: 9781402090837
• ISBN10: 1402090838
• Format: Hardcover
• Copyright: 2008-12-01
• Publisher: Springer Verlag

Summary

This volume contains a collection of articles applying methods of logic or, more generally, of mathematics to solve problems, some of which come from logic itself, others from other sciences. Its range of subjects is far from complete, but broadly representative.The first group of papers in this volume consists of contributions to pure and applied modal logic. The problems discussed here range from the structure of lattices of normal and other modal propositional logics to modal proof theory and to the semantics of quantified modal logic. The second group of papers deals with Many-valued logics - an extensive domain of strictly logical investigations rooting in philosophical questions concerning the nature of logical values. Logical investigations in cognitive science have successfully utilized methods and systems of belief revision, non-monotonic logic and dynamic epistemic logic. Towards Mathematical Philosophy deals with focal issues of belief revision. The volume concludes with contributions which may be seen to belong to the field of formal epistemology, the area applying logical, probabilistic, game-theoretic and other formal methods to problems and issues in epistemology and philosophy of science, such as those concerning anti-realism, skepticism, theory comparison and theory choice, justification, sources of knowledge and learning theories.

Author Biography

David Makinson, Visiting Professor in Department of Philosophy, London School of Economics, author of "Bridges from Classical to Nonmonotonic Logic" (College Publications, 2005) and "Sets Logic and Maths for Computing" (Springer 2008) Jacek Malinowski, Professor of Logic at Institute of Philosophy, Polish Academy of Sciences and at Department of Logic, Nicolaus Copernicus University. Editor-in-Chief of Studia Logica Heinrich Wansing, Professor of Philosophy of Science and Logic, Dresden University of Technology; managing editor of Studia Logica; author of "The Logic of Information Structures" (1993) and "Displaying Modal Logic (1998)"

Table of Contents

From Logic to Mathematical Philosophy	p. 1
Introduction	p. 1
Modal Logic	p. 3
Non-Classical and Many-Valued Logics	p. 4
Belief Management	p. 6
Commutativity of Quantifiers in Varying-Domain Kripke Models	p. 9
Introduction and Overview	p. 9
Model Structures	p. 12
Premodels and Models	p. 14
Soundness and <$>{\cal M}<$>-Equivalence	p. 17
Validating CQ	p. 20
A Countermodel to CQ	p. 23
Completeness and the Barcan Formulas	p. 28
References	p. 30
The Method of Tree-Hypersequents for Modal Propositional Logic	p. 31
Introduction	p. 31
The Calculi CSK*	p. 34
Admissibility of the Structural Rules	p. 37
The Adequateness of the Calculi	p. 43
Cut-Elimination Theorem for CSK*	p. 45
Conclusions and Further Work	p. 49
References	p. 50
All Splitting Logics in the Lattice NExt (KTB)	p. 53
Introduction	p. 53
Preliminaries	p. 54
Splitting	p. 56
Connected KTB-Frames	p. 59
Few Splittings Theorem	p. 61
Some Questions and Conjectures	p. 65
References	p. 66
A Temporal Logic of Normative Systems	p. 69
Introduction	p. 69
Normative Temporal Logic	p. 70
Symbolic Representations	p. 80
Model Checking	p. 86
Case Study: Traffic Control	p. 93
Discussion	p. 100
References	p. 104
Reasoning with Justifications	p. 107
Introduction	p. 107
Hintikka's Logics of Knowledge	p. 107
Awareness Logic	p. 110
Explicit Justifications	p. 110
Internalization	p. 113
Information Hiding and Recovery	p. 114
Original Intent	p. 115
Realizations As First-Class Objects	p. 116
Generalizations	p. 120
The Goal	p. 121
References	p. 122
Monotone Relations, Fixed Points and Recursive Definitions	p. 125
Partially Ordered Sets	p. 127
Monotone Relations	p. 134
Arithmetic Recursion and Fixed-Points	p. 146
The Downward Löwenheim-Skolem-Tarski Theorem	p. 161
References	p. 163
Processing Information from a Set of Sources	p. 165
Introduction	p. 165
The Framework	p. 166
Existential Strategy for Standard Structures	p. 173
The Universal Strategy	p. 179
Proof Systems for the Existential Strategy	p. 179
Future Research	p. 184
References	p. 185
The Classical Model Existence Theorem in Subclassical Predicate Logics I	p. 187
Introduction	p. 187
Classical Model Existence Theorem in Propositional Logics	p. 189
A Herbrand-Henkin Style Proof of the Classical Model Existence Theorem for Prenex Normal Form Sentences	p. 191
Prenex Normal Form Theorem Holds in Logics Weaker than First Order Logic	p. 195
Concluding Remarks	p. 197
References	p. 198
Weak Implicational Logics Related to the Lambek Calculus-Gentzen versus Hilbert Formalisms	p. 201
Introduction	p. 201
Preliminaries	p. 203
The Associative Case	p. 205
The Non-Associative Case	p. 207
Hilbert-Style Formalism	p. 209
References	p. 211
Faithful and Invariant Conditional Probability in Łukasiewicz Logic	p. 213
Introduction: Conditionals and de Finetti Coherence Criterion	p. 213
The i-Dimensional Volume of a Formula	p. 215
Conditionals in Łukasiewicz Propositional Logic Ł∞	p. 220
A Faithful Invariant Conditional for Ł∞	p. 222
Proof: Construction of a Faithful Conditional <$>{\cal P}<$>	p. 224
Conclusion of the Proof: <$>{\cal P}<$> is Invariant	p. 227
References	p. 231
A Fuzzy Logic Approach to Non-Scalar Hedges	p. 233
Introduction	p. 233
Lakoff's Proposal	p. 234
Some New Machinery	p. 237
The Generic Fuzzy Logic for Non-Scalar Hedges FLh	p. 240
Conclusion	p. 247
References	p. 247
The Procedures for Belief Revision	p. 249
Introduction	p. 250
Nonmonotonicity on Classical Base	p. 256
Nonmonotonicity on Intuitionistic Base	p. 263
Generalization	p. 266
References	p. 267
Shifting Priorities: Simple Representations for Twenty-Seven Iterated Theory Change Operators	p. 269
Introduction	p. 269
Representing Doxastic States: Prioritized Belief Bases, Entrenchment, Systems of Spheres	p. 270
Variants of Expansion	p. 275
Radical revision	p. 276
Conservative Revision	p. 277
Moderate Revision	p. 278
Restrained Revision	p. 279
Variants of Contraction	p. 280
Refinement: Neither Revision nor Contraction	p. 281
Two-Dimensional Operators: Revision by Comparison	p. 282
Two-Dimensional Operators: Cantwell's Lowering	p. 283
Gentle Raising and Lowering	p. 285
Two-Dimensional Operators: Raising and Lowering by Strict Comparisons	p. 285
Two-Dimensional Operators: Bounded Revision	p. 286
Conclusion	p. 288
References	p. 290
The Coherence of Theories-Dependencies and Weights	p. 297
Introduction	p. 297
Internalist Coherence	p. 299
Application to Game Theory	p. 311
Summary and Discussion	p. 317
References	p. 317
On Meta-Knowledge and Truth	p. 319
Introduction	p. 319
Ideas	p. 320
Main Assumptions of the Theory of Syntax and Semantics	p. 322
Three Notions of Truthfulness	p. 334
Final Remarks	p. 339
References	p. 340
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