A fuzzy modal logic for belief functions

In this paper we introduce a new logical approach to reason explicitly about Dempster-Shafer belief functions. We adopt the following view: one just starts with Boolean formulas phi and a belief function on them; the belief of phi is taken to be the truth degree of the (fuzzy) proposition Bphi standing for "phi is believed". For our complete axiomatization (Hylbert-style) we use one of the possible definitions of belief, namely as probability of (modal) necessity. This enables us to define a logical system combining the modal logic S5 with an already proposed fuzzy logic approach to reason about probabilities. In particular, our fuzzy logic is the logic LPi(1)/(2) which puts Lukasiewicz and Product fuzzy logics together.