Study of multi-modal logics and its application to game theory

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Among other multi-modal logics, we dealt mainly with multi-agent epistemic logics. We can describe inter-personal epistemic inference by making use of multi-agent epistemic logics, and hence we can apply them to game theory. As application, we tried mainly to analyze the game theoretical decision-making process. The idea of applying multi-modal epistemic logics enables us to get new perspectives on relations between epistemic logic and game theory. Many suggestions on future research were obtained.We found that the restriction of inter-personal epistemic inference to "shallow depths" is an important aspect of bounded rationality.In this system IGEF, we can bescribe such restriction by making use of thought sequents. Our system enables us to 'isolate' inter-personal epistemic inference from other ability, and to consider bounded rationality as restrictions on the ability of inter-personal epistemic inference. Moreover, we succeeded to construct the extended Kripke-type semantics for such multi-modal epistemic logics.The main and ralated results are the following.1. Systems IGEF of multi-agent epistemic logics with the feature of the restriction of inter-personal epistemic inference to shallow depths are constructed. These systems enable us to describe contentwise cost of inter-personal epistemic inference in game theoretical situations.2. The proof theory and Kripke-type model theory for such multi-modal epistemic logics are established.3. Algebraic study of substructural logics.4. Results on Martin's axiom on fragmnets of aleph-one dense sets.