Indexed Rough Approximations, A Polymodal System, and Generalized Possibility Measures

Indexed rough approximations that generalize fuzzy rough sets are proposed. A family of indexed relations between objects with the set of indices being a lattice is considered. Relations in the family are ordered by the inclusion, and moreover the ordering is assumed to be consistent with the ordering of the lattice. Thus, a collection of rough approximations, each of which is induced from a relation in the family, is obtained. A polymodal system in which the modal operators with the indices are defined; the completeness between the axiomatic system and the Kripke model which is the above collection of rough approximations is proved. A possibility and necessity measures for sentences that takes the values of the lattice are derived from the polymodal system. These measures are proved to be equivalent to the ordinary possibility and necessity measures when the lattice is the unit interval.