This paper provides a logic framework for investigations of game theoretical problems. We adopt an infinitary extension of classical predicate logic as the base logic of the framework. The reason for an infinitary extension is to express the common knowledge concept explicitly. Depending upon the choice of axioms on the knowledge operators, there is a hierarchy of logics. The limit case is an infinitary predicate extension of modal propositional logic KD4, and is of special interest in applications. In Part I, (...) we develop the basic framework, and show some applications: an epistemic axiomatization of Nash equilibrium and formal undecidability on the playability of a game. To show the formal undecidability, we use a term existence theorem, which will be proved in Part II. (shrink)

Kripke completeness of some infinitary predicate modal logics is presented. More precisely, we prove that if a normal modal logic above is -persistent and universal, the infinitary and predicate extension of with BF and BF is Kripke complete, where BF and BF denote the formulas pi pi and x x, respectively. The results include the completeness of extensions of standard modal logics such as , and its extensions by the schemata T, B, 4, 5, D, and their combinations. The proof (...) is obtained by extending the correspondence between the representation of modal algebras and the completeness of propositional modal logic to infinite. (shrink)

In this paper we investigate first order common knowledge logics; i.e., modal epistemic logics based on first order logic with common knowledge operators. It is shown that even rather weak fragments of first order common knowledge logics are not recursively axiomatizable. This applies, for example, to fragments which allow to reason about names only; that is to say, fragments the first order part of which is based on constant symbols and the equality symbol only. Then formal properties of "quantifying into" (...) epistemic contexts are investigated. The results are illustrated by means of epistemic representations of Nash Equilibria for finite games with mixed strategies. (shrink)

This paper provides a Genzten style formulation of the game logic framework GLm (0 m ), and proves the cut-elimination theorem for GLm. As its application, we prove the term existence theorem for GL used in Part I.