A logical foundation of graded modal operators defined by fuzzy measures

To give rigid semantics to graded modal operators, an extended fuzzy-measure-based model is defined as a family of minimal models for modal logic, each of which corresponds to an intermediate value of a fuzzy measure. Soundness and completeness results of several systems of modal logic are proved with respect to classes of newly introduced models based on intermediate values of fuzzy, possibility, necessity, and Dirac measures, respectively. It is emphasized that a fuzzy measure inherently contains a multimodal logical structure.<>