Probability Logic as a Fuzzy Logic

The basic principles of fuzzy logic have been formulated by Zadeh (1975) and successively examined by several other authors (as an example, see Pavelka, 1979). Now, in spite of the fact that fuzzy logic is usually considered rather far from probability logic, the purpose of some recent researches of mine is to show that fuzzy logic is a useful tool to manage information that is probabilistic in nature. Namely, we propose a fuzzy logic in which the complete theories (that is, the models) are finitely additive probabilities defined on the set F of formulas of a given language. In such a logic, given a fuzzy subset v of axioms, the fuzzy subset C(υ) of logical consequences of υ is defined as the intersection (in fuzzy set theory sense) of the class of complete theories containing υ, i.e. the lower envelope generated by υ. For the syntactical part two directions are possible, a refutation approach (see Gerla, 1994a) and Hilbert-style approach (see Gerla, 1994b), both enabling to give an effective way to to compute the operator C. Not differently from the classic logic, this enable to show that if υ is decidable then C(υ) is recursively enumerable, and if v is decidable and complete then C(υ) is decidable (see Biacino and Gerla, 1987; 1989; 1992). Notice that the approach we propose for probability logic is rather different from the usual ones. Indeed, while from my point of view the elements in [0,1] play a semantic role, in the existing litterature these elements play a linguistic role. Namely, a probability logic is usually obtained by extending the first order language in a two-sorted language (see, for example Halpern, 1991; Bacchus, 1990 and others). A sort is for the domain we want to reason about, the other sort is devoted to represent numbers in [0, 1]. As a consequence, the deductive systems have to contain axioms for the structure of [0,1]. The price paid for this is either the necessity of allowing as truth values nonstandard elements in arbitrary ordered fields (Bacchus, 1990) or the impossibility to have a complete axiomatization (Halpern, 1991).