Dempster–Shafer theory framed in modal logic

In this paper, the modal logic interpretation of plausibility and belief measures on an arbitrary universe of discourse, as proposed by Harmanec et al., is further developed by employing notions from set-valued analysis. In a model of modal logic, a multivalued mapping is constructed from the accessibility relation and a mapping determined by the value assignment function. This multivalued mapping induces a plausibility measure and a belief measure expressed in terms of conditional probabilities of inverse and superinverse images, or equivalently, in terms of conditional probabilities of truth sets of possibilitations and necessitations. Restricting to a finite universe of discourse, multivalued interpretations of basic probability assignments and of commonality functions are also obtained, in terms of conditional probabilities of pure inverse and subinverse images, or equivalently, in terms of conditional probabilities of truth sets of particular logical expressions involving possibilitations and necessitations.