A logic for reasoning about the probability of fuzzy events

Abstract

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.

Weighted logics for artificial intelligence – an introductory discussion

“…[59,58,60] for some logical formalisms that are able to deal with uncertainty over non-Boolean (fuzzy) events).…”

An Extension of Gödel Logic for Reasoning under Both Vagueness and Possibilistic Uncertainty

“…Starting from the basic ideas exposed by Hájek in [13], various kinds of uncertainties can be studied by using various kinds of modal-fuzzy logics (see, e.g., [10,11]). The very basic idea allowing a treatment of the certainty of classical (crisp) events inside a fuzzy-logical setting consists of interpreting the certainty degree of a (classical) proposition φ as the truth value of a modal proposition φ which reads "φ is certain".…”

Strong non-standard completeness for fuzzy logics

“…Then, in those last years, many improvements of this approach have been proposed (see for instance Esteva et al 2000;Flaminio 2005Flaminio , 2007Godo 2006, Flaminio andMontagna 2005;Hájek 1998;Marchioni and Godo 2004). In particular in Flaminio and Godo (2006), we introduced a modal fuzzy logic to reason about the probability of fuzzy events.…”

“…(4) In Sect. 4, we will apply the results of the previous section to provide a partial solution for a problem left in Flaminio and Godo (2006), Hájek (1998) about the (non-)standard completeness for some modal fuzzy logics allowing a treatment of probability of infinite-valued events.…”