Logics for belief functions on MV-algebras

In this paper we present a generalization of belief functions over fuzzy events. In particular we focus on belief functions defined in the algebraic framework of finite MV-algebras of fuzzy sets. We introduce a fuzzy modal logic to formalize reasoning with belief functions on many-valued events. We prove, among other results, that several different notions of belief functions can be characterized in a quite uniform way, just by slightly modifying the complete axiomatization of one of the modal logics involved in the definition of our formalism.

► We introduce a notion of belief function on MV-algebras of fuzzy sets whose focal elements are arbitrary fuzzy sets. ► We generalize Dempster spaces in such a way that their induced lower measures coincide with belief functions in our sense. ► Our logics for belief functions on MV-algebras are probabilistic logics over modal expansions of Łukasiewicz logic Łk. ► They allow for a uniform treatment of several classes of belief functions characterized by their particular focal elements. ► Our logics are sound and complete w.r.t. both Kripke-style probabilistic semantics and belief function based semantics.