Abstract
There are two main methods to deal with belief in intelligent systems : a logical approach using modal logic and a numerical approach using probability, fuzzy measures, and so on. To study a theoretical relationship between the two approaches, an extended fuzzy-measure-based model as a family of minimal models for modal logic is defined where each minimal model corresponds to an intermediate value of a fuzzy measure. Then, graded modal operators can be defined in the model, which is an extension of our previous model which only deals with the value 1 of a fuzzy measure. Soundness and completeness results of several systems of modal logic are proved with respect to classes of these new models based on intermediate values of fuzzy, possibility, necessity, and Dirac measures. The inclusion relation between the classes of models as well as their corresponding systems is also shown.
