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In this paper we present the logic F P (Ł n , 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 F P (Ł n , Ł) w.r.t. both weak and strong models. We also present two extensions of F P (Ł n , Ł): the first one is the logic F P (Ł n , RP L), obtained by expanding the F P (Ł n , Ł)-language with truth-constants for the rationals in [0, 1], while the second extension is the logic F CP (Ł n , Ł 1 2 ) allowing to reason about conditional states.

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Paraconsistent logics are specially tailored to deal with inconsistency, while fuzzy logics primarily deal with graded truth and vagueness. Aiming to find logics that can handle inconsistency and graded truth at once, in this paper we explore the notion of paraconsistent fuzzy logic. We show that degree-preserving fuzzy logics have paraconsistency features and study them as logics of formal inconsistency. We also consider their expansions with additional negation connectives and first-order formalisms and study their paraconsistency properties. Finally, we compare our approach to other paraconsistent logics in the literature.Keywords Mathematical fuzzy logic, degreepreserving fuzzy logics, paraconsistent logics, logics of formal inconsistency. This is an extended and revised version of the conference communication "Exploring paraconsistency in degree-preserving fuzzy logics",