Extended Kripke Semantics and its Application to Epistemic Logics and Game Theory

我们主要讨论了能够描述人际认知推理的多通道认知逻辑。将多模式认知逻辑应用于博弈论所描述的博弈论决策过程的分析，使我们对认知逻辑与博弈论的关系有了新的认识。对未来的研究提出了许多建议。我们发现，人与人之间的认识论推理被限制在“浅层”是有限理性的一个重要方面。有限理性是近期博弈论文献中比较感兴趣的一个概念。我们成功地为这种多通道认知逻辑构造了扩展的Klipke型语义。主要研究结果如下：1.人与人之间的认知推理局限于浅层是有限理性的一个重要方面。我们证明了多通道认知逻辑为这一限制提供了理论框架。建立了这类多通道认知逻辑的证明论和Kriske型模型理论。2.将层次理论中的自然变换和范畴理论中的函子解释为扩展的Klipke语义。利用这些技巧，研究了非经典谓词逻辑中的Hellden-完备性，并与命题逻辑中的情况进行了比较。3.度量空间逻辑的公理化和可判定性。4.代数的模糊子代数上的所有格恒等式的集合与普通子代数上的所有格恒等式的集合重合。5.给出了一些模糊逻辑的标准完备性证明。

We dealt mainly with multi-modal epistemic logics which can describe interpersonal epistemic inference. The idea of applying multi-modal epistemic logics to analysis of the "game theoretical decision-making process" described by game theory enables us to get new perspectives on relations between epistemic logic and game theory. Many suggestions on future research were obtained. We find that the restriction of inter-personal epistemic inference to "shallow depths" is an important facet of the bounded rationality. The bounded rationality is a concept interested in the recent literature of game theory. We succeeded to construct the extended Kripke-type semantics for such multi-modal epistemic logics. The main and ralated results are the following.1.The restriction of inter-personal epistemic inference to shallow depths is found to be an important facet of the bounded rationality. We showed that multi-modal epistenic logics provide a theoretical framework to this restriction. The proof theory and Kripke-type model theory for such multi-modal epistemic logics are established.2.Natural transformations in sheaf theory and functors in category theory are interpreted into extended Kripke semantics. By means of these techniques, Hellden-completeness in non-classical predicate logics is investigated and compared with the case in propositional logics.3.(in computer science) axiomatization and decidability of the logic of metric spaces.4.The set of all lattice-identities hold on the fuzzy subalgebra of an algebra coincides with the set of all lattice-identities hold on the ordinary subalgebra.5.The standard completeness proofs of some fuzzy logics are given.