Similarity of States in Modal Logic

Abstract

The similarity between states is an important issue in modal logic. There are many formalisms that have been introduced to express different kinds of similarities in conditional logic, probabilistic logic, and dynamic logic and so on. However, in these cases similarities between states are directly given by the models. In this paper, we introduce two kinds of models and provide two different judgment standards of the similarity between states, while the accessibility of relations on our models are generated by similarities instead of giving this in the definitions of frames or models. In our first type of models, we compare the propositional assignments at states to measure the degree of similarity among states. We give the axiomatizations of the formula sets of this class of models and prove the completeness and soundness. The second type of model is about conditionals, in which we assign to each proposition a group of sets of propositions called the context of the proposition and extend this assignment to all the formulas. Finally, we assign two formulas $\phi$ and a world w, a collection of sets of worlds by the contexts of $\phi$.