Games, Logic, and Constructive Sets

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.

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

Games, Logic, and Constructive Sets

Contents

I Logic and Games

II Classical Logic

III Constructive Set Theory