New development of Kripke semantics by combining with graph theory

本研究基于“Kripke frame is a graph”的观察，旨在通过引入图论中的知识和技术，重构框架理论对模态逻辑语义研究的方法。经过三年的调查，我发现了以下事实。(1)对于无限图类，我们必须利用图论，而不是普通的图论；(2)图小定理中的“小关系”可能破坏由两个Kripke框架确定的两个模态逻辑的顺序。因此，这种关系不适用于分析由某一类模态逻辑构成的格结构，并且(3)至少有两种方法可以扩展通常的模态语言来表征由所有无向图类所决定的逻辑。

I started this research on basis of the observation "Kripke frame is a graph", and I aimed at restructuring the frame-theoretic approch to semantical study of modal logics by introducing the knowledge and techniques in graph theory. As the results after three-year investigation, I found the following facts. (1) We have to utilize graph theory for the class of infinite graphs, not ordinary graph theory, (2) The "minor relation" in Graph Minor Theorem may destroy the order of two modal logics which are determined by two Kripke frames. Therefore this relation is not suitable for analizing the lattice structure which is made of some class of modal logics, and (3) There are at least two methods for extending the usual modal language to characterize the logic determined by the class of all undirected graphs.