A logic for reasoning about the probability of fuzzy events

In this paper we present the logic FP(Łn,Ł) which allows to reason about the probability of fuzzy events formalized by means of the notion of state in a MV-algebra. This logic is defined starting from a basic idea exposed by Hájek [Metamathematics of Fuzzy Logic, Kluwer, Dordrecht, 1998]. Two kinds of semantics have been introduced, namely the class of weak and strong probabilistic models. The main result of this paper is a completeness theorem for the logic FP(Łn,Ł) w.r.t. both weak and strong models. We also present two extensions of FP(Łn,Ł): the first one is the logic FP(Łn,RPL), obtained by expanding the FP(Łn,Ł)-language with truth-constants for the rationals in [0,1], while the second extension is the logic allowing to reason about conditional states.