Towards mathematical philosophy : papers from the Studia Logica Conference Trends in Logic IV

358 Pages·2014·2.643 MB·English

by  David Makinson, Jacek Malinowski, Heinrich Wansing

Towards Mathematical Philosophy
TRENDS IN LOGIC
Studia Logica Library
VOLUME 28
Managing Editor
Ryszard W´ojcicki, Institute of Philosophy and Sociology,
Polish Academy of Sciences, Warsaw, Poland
Editors
Vincent F. Hendricks, Department of Philosophy and Science Studies,
Roskilde University, Denmark
Daniele Mundici, Department of Mathematics “Ulisse Dini”,
University of Florence, Italy
Ewa Or(cid:2)lowska, National Institute of Telecommunications,
Warsaw, Poland
Krister Segerberg, Department of Philosophy, Uppsala University,
Sweden
Heinrich Wansing, Institute of Philosophy, Dresden University of Technology,
Germany
SCOPE OF THE SERIES
Trends in Logic is a bookseries covering essentially the same area as the
journal Studia Logica – that is, contemporary formal logic and its applica-
tions and relations to other disciplines. These include artificial intelligence,
informatics, cognitive science, philosophy of science, and the philosophy of
language. However, this list is not exhaustive, moreover, the range of appli-
cations,comparisonsandsourcesofinspirationisopenandevolvesovertime.
Volume Editor
Ryszard W´ojcicki
For other titles published in this series, go to www.springer.com/series/6645
David Makinson • Jacek Malinowski • Heinrich Wansing
Editors
Towards
Mathematical Philosophy
Papers from the Studia Logica conference
Trends in Logic IV
David Makinson Jacek Malinowski
London School of Economics Polish Academy of Sciences
Department of Philosophy Institute of Philosophy and Sociology
Houghton Street Nowy Swiat 72
London WC2A 2AE 00-330 Warszawa
United Kingdom Poland
e-mail: [email protected] e-mail: [email protected]
Heinrich Wansing
Dresden University of Technology
Institute of Philosophy
01062 Dresden
Germany
e-mail: [email protected]
ISBN 978-1-4020-9083-7 e-ISBN 978-1-4020-6867-6
LibraryofCongressControlNumber: 2008934403
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Contents
1 From Logic to Mathematical Philosophy, David Makinson,
Jacek Malinowski and Heinrich Wansing . . . . . . . . . . . 1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Modal Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Non-Classical and Many-Valued Logics . . . . . . . . . . . . 4
Belief Management . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Commutativity of Quantifiers in Varying-Domain Kripke Models,
Robert Goldblatt and Ian Hodkinson. . . . . . . . . . . . . . 9
Introduction and Overview. . . . . . . . . . . . . . . . . . . . . 9
1 Model Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Premodels and Models . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Soundness and M-Equivalence . . . . . . . . . . . . . . . . . . 17
4 Validating CQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5 A Countermodel to CQ . . . . . . . . . . . . . . . . . . . . . . . 23
6 Completeness and the Barcan Formulas. . . . . . . . . . . . 28
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 The Method of Tree-Hypersequents for Modal Propositional
Logic, Francesca Poggiolesi . . . . . . . . . . . . . . . . . . . . . . 31
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 The Calculi CSK* . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Admissibility of the Structural Rules. . . . . . . . . . . . . . 37
4 The Adequateness of the Calculi . . . . . . . . . . . . . . . . . 43
5 Cut-Elimination Theorem for CSK* . . . . . . . . . . . . . . 45
6 Conclusions and Further Work . . . . . . . . . . . . . . . . . . 49
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 All Splitting Logics in the Lattice NExt(KTB), Tomasz
Kowalski and Yutaka Miyazaki . . . . . . . . . . . . . . . . . . . 53
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3 Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 Connected KTB-Frames. . . . . . . . . . . . . . . . . . . . . . . 59
5 Few Splittings Theorem . . . . . . . . . . . . . . . . . . . . . . . 61
6 Some Questions and Conjectures. . . . . . . . . . . . . . . . . 65
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
vi
5 A Temporal Logic of Normative Systems, Thomas ˚Agotnes,
Wiebe van der Hoek, Juan A. Rodr´ıguez-Aguilar, Carles
Sierra and Michael Wooldridge . . . . . . . . . . . . . . . . . . 69
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2 Normative Temporal Logic . . . . . . . . . . . . . . . . . . . . . 70
3 Symbolic Representations. . . . . . . . . . . . . . . . . . . . . . 80
4 Model Checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5 Case Study: Traffic Control . . . . . . . . . . . . . . . . . . . . 93
6 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6 Reasoning with Justifications, Melvin Fitting . . . . . . . . . . . 107
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
2 Hintikka’s Logics of Knowledge. . . . . . . . . . . . . . . . . . 107
3 Awareness Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4 Explicit Justifications . . . . . . . . . . . . . . . . . . . . . . . . 110
5 Internalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6 Information Hiding and Recovery . . . . . . . . . . . . . . . . 114
7 Original Intent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8 Realizations As First-Class Objects. . . . . . . . . . . . . . . 116
9 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
10 The Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7 Monotone Relations, Fixed Points and Recursive Definitions,
Janusz Czelakowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
1 Partially Ordered Sets . . . . . . . . . . . . . . . . . . . . . . . . 127
2 Monotone Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3 Arithmetic Recursion and Fixed-Points . . . . . . . . . . . . 146
4 The Downward L¨owenheim-Skolem-Tarski Theorem . . . 161
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
8 Processing Information from a Set of Sources, Arnon Avron,
Jonathan Ben-Naim and Beata Konikowska . . . . . . . . . 165
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
2 The Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
3 Existential Strategy for Standard Structures . . . . . . . . 173
4 The Universal Strategy . . . . . . . . . . . . . . . . . . . . . . . 179
5 Proof Systems for the Existential Strategy. . . . . . . . . . 179
6 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
vii
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
9 The Classical Model Existence Theorem in Subclassical Predicate
Logics I, Jui-Lin Lee. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
2 Classical Model Existence Theorem in Propositional
Logics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
3 A Herbrand-Henkin Style Proof of the Classical Model
Existence Theorem for Prenex Normal Form Sentences . 191
4 Prenex Normal Form Theorem Holds in Logics Weaker
than First Order Logic. . . . . . . . . . . . . . . . . . . . . . . . 195
5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 197
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
10 Weak Implicational Logics Related to the Lambek
Calculus—Gentzen versus Hilbert Formalisms, Wojciech
Zielonka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
3 The Associative Case . . . . . . . . . . . . . . . . . . . . . . . . . 205
4 The Non-Associative Case . . . . . . . . . . . . . . . . . . . . . 207
5 Hilbert-Style Formalism . . . . . . . . . . . . . . . . . . . . . . . 209
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
11 Faithful and Invariant Conditional Probability in L(cid:2)ukasiewicz
Logic, Daniele Mundici . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Introduction: Conditionals and de Finetti Coherence
Criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
1 The i-Dimensional Volume of a Formula . . . . . . . . . . . 215
2 Conditionals in L(cid:2) ukasiewicz Propositional Logic L(cid:2) ∞ . . . 220
3 A Faithful Invariant Conditional for L(cid:2) ∞ . . . . . . . . . . . 222
4 Proof: Construction of a Faithful Conditional P. . . . . . 224
5 Conclusion of the Proof: P is Invariant . . . . . . . . . . . . 227
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
12 A Fuzzy Logic Approach to Non-Scalar Hedges, Stephan van
der Waart van Gulik. . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
2 Lakoff’s Proposal. . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
3 Some New Machinery . . . . . . . . . . . . . . . . . . . . . . . . 237
viii
4 The Generic Fuzzy Logic for Non-Scalar Hedges FL . . 240
h
5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
13 The Procedures for Belief Revision, Piotr L(cid:2) ukowski . . . . . . 249
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
2 Nonmonotonicity on Classical Base . . . . . . . . . . . . . . . 256
3 Nonmonotonicity on Intuitionistic Base . . . . . . . . . . . . 263
4 Generalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
14 Shifting Priorities: Simple Representations for Twenty-Seven
Iterated Theory Change Operators, Hans Rott . . . . . . . . . . 269
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
2 Representing Doxastic States: Prioritized Belief Bases,
Entrenchment, Systems of Spheres . . . . . . . . . . . . . . . 270
3 Variants of Expansion . . . . . . . . . . . . . . . . . . . . . . . . 275
4 Radical revision. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
5 Conservative Revision . . . . . . . . . . . . . . . . . . . . . . . . 277
6 Moderate Revision. . . . . . . . . . . . . . . . . . . . . . . . . . . 278
7 Restrained Revision . . . . . . . . . . . . . . . . . . . . . . . . . . 279
8 Variants of Contraction . . . . . . . . . . . . . . . . . . . . . . . 280
9 Refinement: Neither Revision nor Contraction . . . . . . . 281
10 Two-Dimensional Operators: Revision by Comparison . 282
11 Two-Dimensional Operators: Cantwell’s Lowering . . . . 283
12 Gentle Raising and Lowering . . . . . . . . . . . . . . . . . . . 285
13 Two-Dimensional Operators: Raising and Lowering by
Strict Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . 285
14 Two-Dimensional Operators: Bounded Revision . . . . . . 286
15 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
15 The Coherence of Theories—Dependencies and Weights, Jason
Jingshi Li, Rex Bing Hung Kwok and Norman Y. Foo . 297
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
2 Internalist Coherence . . . . . . . . . . . . . . . . . . . . . . . . . 299
3 Application to Game Theory . . . . . . . . . . . . . . . . . . . 311
4 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . 317
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
ix
16 On Meta-Knowledge and Truth, Urszula
Wybraniec-Skardowska . . . . . . . . . . . . . . . . . . . . . . . . . 319
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
1 Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
2 Main Assumptions of the Theory of Syntax and
Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
3 Three Notions of Truthfulness . . . . . . . . . . . . . . . . . . 334
4 Final Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340