A logic for reasoning about the probability of fuzzy events

In this paper we present the logic FP (! Ln, ! L) which allows to reason about the probability of fuzzy events formalized by means of the notion of state in a MValgebra. This logic is defined starting from a basic idea exposed by Hájek in [13]. 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 (! Ln, ! L) w.r.t. both weak and strong models. We also present two extensions of FP (! Ln, ! L): the first one is the logic FP (! Ln, RPL), obtained by expanding the FP (! Ln, ! L)-language with truth constants for the rationals in [0, 1], while the second extension is the logic FCP (! Ln, ! LΠ2) allowing to reason about conditional states.