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Games, Logic, and Constructive Sets cover

Games, Logic, and Constructive Sets

Edited by Grigori Mints and Reinhard Muskens

Mathematical game theory has been embraced by a variety of scholars: social scientists, biologists, linguists, and now, increasingly, logicians. This volume illustrates the recent advances of game theory in the field. Logicians benefit from things like game theory's ability to explain informational independence between connectives; meanwhile, game theorists have even begun to benefit from logical epistemic analyses of game states. In concert with such pioneering work, this volume also present surprising developments in classical fields, including first-order logic and set theory.

Grigori Mints is Professor of Philosophy and Professor (by Courtesy) of Mathematics and Computer Science at Stanford University. Reinhard Muskens is Associate Professor of Logic and Linguistics in the Department of Linguistics at Tilburg University.

Contents

    I Logic and Games

  • Logic and Game Theory: Close Encounters of the Third Kind
    Johan van Benthem
    • 1 Encounters of three kinds
    • 2 Games from a logical viewpoint
      • 2.1 Pictures
      • 2.2 Games as models for logical languages
      • 2.3 Intentional logic as usual
      • 2.4 Major issues in process theories
    • 3 Levels of representation
      • 3.1 Games: from actions to outcomes
      • 3.2 Actions in modal and dynamic logic
      • 3.3 Powers and a forcing language
      • 3.4 Intermediate game levels
    • 4 Game operations and game algebra
      • 4.1 Game operations
      • 4.2 Game algebra
      • 4.3 Languages
    • 5 From finite to infinite games
    • 6 Coping with imperfect information
      • 6.1 From perfect to imperfect information
      • 6.2 Uniform outcome equivalence and a forcing modality
      • 6.3 Actions and information: dynamic-epistemic logic
    • 7 Preferences and rational behavior
      • 7.1 Game values, equilibrium and backward induction
      • 7.2 Logic issues revisited
    • 8 Conclusion

  • Informationally Independent Connectives Gabriel Sandu and Ahti Pietarinen
    • 1 Sentential logic and games
      • 1.1 Extensive games of perfect information
      • 1.2 Extensive games of imperfect information
      • 1.3 Extensive forms as Kripke frames
    • 2 Informationally independent connectives
      • 2.1 Games of perfect information
      • 2.2 Semantical games of imperfect information
      • 2.3 Informationally independent

  • Descriptions of Game States
    Hans van Ditmarsch, Wiebe van der Hoek, Barteld Kooi
    • 1 Introduction
    • 2 Epistemic logic
    • 3 Description of Hexa
      • 3.1 Derived characteristics of Hexa
    • 4 Description of initial game states
      • 4.1 Derived characteristics: factual knowledge
      • 4.2 Derived characteristics: private knowledge
      • 4.3 Derived characteristics: private ignorance
      • 4.4 Derived characteristics: seedontknow
    • 5 Description of the pre-initial state
    • 6 Further observations
    • 7 Conclusion

    II Classical Logic

  • Resource Consciousness in Classical Logic
    Andreas Blass
    • 1 Introduction
    • 2 An Example
    • 3 Herbrand's Theorem
    • 4 Simple Herbrand Validity
    • 5 Universal Simple Herbrand Validity
    • 6 Modus Ponens
    • 7 Connection with Affine Logic

  • Quick Cut-Elimination for Monotone Cuts
    Grigori Mints
    • 1 Introduction
    • 2 System LK
    • 3 Cut-Elimination

    III Constructive Set Theory

  • The Anti-Foundation Axiom in Constructive Set Theories
    Michael Rathjen
    • 1 Introduction
    • 2 The anti-foundation axiom
    • 3 AFA in constructive set theory
      • 3.1 The theory of CZFA
      • 3.2 Interpreting AFA in Martin-Löf type theory
      • 3.3 Upper bounds
      • 3.4 Lower Bounds
    • 4 Anti-foundation with inaccessible sets

  • On Non-wellfounded Constructive Set Theory: Construction of Non-wellfounded Sets in Explicit Mathematics Sergei Tupailo
    • 1 Constructive Set Theory with Natural Numbers
    • 2 Explicit Mathematics: a reminder
    • 3 Realization of NCF‾ into EETJ

  • Index

July 2003

ISBN (Paperback): 9781575864501
ISBN (Cloth): 9781575864495

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