Lattice-valued possibility measures on the basis of multimodal logic

Although the consideration of L-fuzzy sets shows that the membership values may not be limited to the unit interval, such an extension of the possibility measure has not been studied. The aim of the present paper is to show that an extension to lattice-valued possibility measures is possible, and moreover the extension is done by applying multimodal logic. An axiomatic system of multimodal logic is defined and completeness between a Kripke model with an indexed relation and the axiomatic system is shown. The set of indices is assumed to be a lattice, and the possibility measure of a proposition is defined by the supremum of the subset of indices by which the proposition possibly holds, provided that the lattice is complete. Duality between the possibility measure and the necessity measure is discussed. Relationship between the present measure and the standard possibility measure is shown when the lattice is the unit interval