GAMES IN LOGIC, LOGIC IN GAMES, AND META GAMES

RANDOLPH RUBENS GOLDMAN¹*
¹Department of Mathematics, Hawaii Pacific University, Honolulu, HI
* Corresponding Author : rgoldman@hpu.edu

Received : 09-05-2011     Accepted : 19-06-2011     Published : 01-09-2011
Volume : 2     Issue : 1       Pages : 7 - 14
J Stat Math 2.1 (2011):7-14

This is a survey on the relationship between logic and games. What do games have to save about logic, and conversely what does logic have to say about games? Johan Van Benthem in his lengthy manuscript Logic In Games sets forth an axiomatization of game equivalence and asks two main questions. The first is whether the axioms are complete for the semantic notion of game equivalence. Van Benthem also cautions that it is important to distinguish that games are “dynamic” activities, and that the meaning of a game is not fully captured by the assertion player has a winning strategy in it, and hence the second question is what constitutes this “dynamic aspect”. In this survey project, I will briefly discuss the difference between using games to determine results about logic and using logic to determine results about games. I then will discuss two responses in the affirmative to the first question by Van Benthem about the axiomatization of game logic with regard to logic in games. One is by Goranko which employs translations into modal logic to obtain the completeness result; the second is by Venema which uses a more general approach to show that game algebras and board algebras are isomorphic. I will also offer what seems to be a novel approach in responding to Van Benthem’s second question by suggesting that games are not fully captured by understanding whether a player has a winning strategy or not because games involve a dynamic action between intelligent agents who are trying to out think each other. In order to represent this dynamic process mathematically I propose that one must classify strategies themselves, and I will suggest ways of classifying strategies in the context of modal logic.

[1] Van Benthem J. (2000) Logic in Games, Lecture notes, ILLC, University of Amsterdam  
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[2] Phokion Kolaitis (2002) Combinatorial Games in Finite Model Theory, Lecture Notes,University of California at Santa Cruz  
» CrossRef   » Google Scholar   » PubMed   » DOAJ   » CAS   » Scopus  

[3] William Hodges (1993) Cambridge University Press  
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[4] Goranko V.F. (2003) Studia Logica, 75: 221-238.  
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[5] Yde Venema (2003) Studia Logica 75:239- 256  
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[6] Rohit Parikh (1985) Annals of Discrete Mathematics 24: 111-140  
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[7] Marc Pauly and Rohit Parikh (2003) Studia Logica 75:165-182  
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[8] Randolph Rubens Goldman (2000) Godel’s Ontological Argument, Dissertation, University of California at Berkeley  
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