Inferences in probability logic. (English) Zbl 0821.03014

This somewhat misleadingly titled paper is about the incorporation of probabilistic ideas in the framework of fuzzy logic. The general idea is this: probabilistic theories are fuzzy sets of formulas, i.e., functions from formulas to \([0,1]\) indicating degree of membership. The models of a given fuzzy set of formulas \(v\) are “the probabilities greater than or equal to \(v\)” (p. 34). It seems that this should be “the set of probability functions \(p\), such that for every formula \(\alpha\), \(p(\alpha)> v(\alpha)\)”. Such difficulties abound in the text, but the general idea comes through. There are difficulties also of another kind: the author offers a proof of the proposition that a fuzzy set of formulas is satisfiable if and only if every finite subset of it is satisfiable. He claims that a proof in the general case can be found in F. Bacchus’s book: Representing and reasoning with probabilistic knowledge (MIT Press, Cambridge, MA, 1990). This is not so, since Bacchus does not deal with the fuzzy notion of satisfiability at all. On the other hand, the proof offered for the denumerable case seems perfectly all right.
The upshot is that the logic of envelopes, which is the fuzzy logic developed by the author, leads to the conclusion that probabilities are, in this framework, the analogs of complete theories in classical logic (p. 48). Other results are also of interest, for example that every axiomatizable complete probabilistic theory in this framework is decidable.


03B52 Fuzzy logic; logic of vagueness
03B48 Probability and inductive logic
Full Text: DOI


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