Towards Mathematical Philosophy (eBook)

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)"

Contents 6
List of Contributors 11
From Logic to Mathematical Philosophy 14
Commutativity of Quantifiers in Varying-Domain Kripke Models 21
Introduction and Overview 21
1. Model Structures 24
2. Premodels and Models 26
3. Soundness and M-Equivalence 29
4. Validating CQ 32
5. A Countermodel to CQ 35
6. Completeness and the Barcan Formulas 40
The Method of Tree-hypersequents for Modal Propositional Logic 43
1. Introduction 43
2. The Calculi CSK* 46
3. Admissibility of the Structural Rules 49
4. The adequateness of the calculi 55
5. Cut-elimination Theorem for CSK* 57
6. Conclusions and Further Work 61
All Splitting Logics in the Lattice NExt(KTB) 64
1. Introduction 64
2. Preliminaries 65
3. Splitting 67
4. Connected KTB-frames 70
5. Few splittings theorem 72
6. Some questions and conjectures 76
A Temporal Logic of Normative Systems 79
1. Introduction 79
2. Normative Temporal Logic 80
3. Symbolic Representations 90
4. Model Checking 96
5. Case Study: Traffic Control 103
6. Discussion 110
Reasoning with Justifications 117
1. Introduction 117
2. Hintikka’s Logics of Knowledge 117
3. Awareness Logic 120
4. Explicit Justifications 120
5. Internalization 123
6. Information Hiding and Recovery 124
7. Original Intent 125
8. Realizations As First-Class Objects 126
9. Generalizations 130
10. The Goal 131
Monotone Relations, Fixed Points and Recursive Definitions 134
1. Partially Ordered Sets 136
2. Monotone relations 143
3. Arithmetic Recursion and Fixed-Points 155
4. The downward Loewenheim-Skolem-Tarski Theorem 170
Processing Information from a Set of Sources 174
1. Introduction 174
2. The framework 175
3. Existential strategy for standard structures 182
4. The universal strategy 188
5. Proof systems for the existential strategy 188
6. Future research 193
The Classical Model Existence Theorem in Subclassical Predicate Logics I 196
1. Introduction 196
2. Classical model existence theorem in propositional logics 198
3. A Herbrand-Henkin style proof of the classical model existence theorem for prenex normal form sentences 200
4. Prenex normal form theorem holds in logics weaker than first order logic 204
5. Concluding remarks 206
Weak Implicational Logics Related to the Lambek Calculus — Gentzen versus Hilbert Formalisms 209
1. Introduction 209
2. Preliminaries 211
3. The associative case 213
4. The non-associative case 215
5. Hilbert-style formalism 217
Faithful and Invariant Conditional Probability in Lukasiewicz Logic 221
Introduction: Conditionals and de Finetti coherence criterion 221
1. The i-dimensional volume of a formula 223
2. Conditionals in Lukasiewicz propositional logic L8 228
3. A faithful invariant conditional for L8 230
4. Proof: construction of a faithful conditional P 232
5. Conclusion of the proof: P is invariant 235
A Fuzzy Logic Approach to Non-scalar Hedges 241
1. Introduction 241
2. Lakoff’s proposal 242
3. Some new machinery 245
4. The generic fuzzy logic for non-scalar hedges FLh 248
5. Conclusion 255
The Procedures for Belief Revision 257
1. Introduction 258
2. Nonmonotonicity on classical base 264
3. Nonmonotonicity on intuitionistic base 271
4. Generalization 274
Shifting Priorities: Simple Representations for Twenty-seven Iterated Theory Change Operators 277
1. Introduction 277
2. Representing doxastic states: Prioritized belief bases, entrench-ment, systems of spheres 278
3. Variants of expansion 283
4. Radical revision 284
5. Conservative revision 285
6. Moderate revision 286
7. Restrained revision 287
8. Variants of contraction 288
9. Refinement: Neither revision nor contraction 289
10. Two-dimensional operators: Revision by comparison 290
11. Two-dimensional operators: Cantwell’s lowering 291
12. Gentle raising and lowering 293
13. Two-dimensional operators: Raising and lowering by strict comparisons 293
14. Two-dimensional operators: Bounded revision 294
15. Conclusion 296
The Coherence of Theories — Dependencies and Weights 305
1. Introduction 305
2. Internalist Coherence 307
3. Application to Game Theory 319
4. Summary and Discussion 325
On Meta-knowledge and Truth 327
Introduction 327
1. Ideas 328
2. Main Assumptions of the Theory of Syntax and Semantics 330
3. Three notions of truthfulness 342
4. Final remarks 347