On the definition of modal operators in fuzzy logic

The classical modal logic with Kripke semantics is generalized in two ways. First, the set {0,1} consisting of the logical values 0 (false) and 1 (true) is replaced by the set [0,1] of all real numbers x with 0⩽x⩽1. The concept of Kripke semantics is generalized to this logic, but the accessibility relation R is still assumed as crisp, i.e., RWS×WS, where WS is the set of worlds considered. The accessibility relation is generalized by taking it as a soft relation S:WS×WS→[0,1]. Several problems of interpreting two of the modal operators on the basis of this concept are discussed