## 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.

### MSC:

 03B52 Fuzzy logic; logic of vagueness 03B48 Probability and inductive logic

### Keywords:

fuzzy logic; probabilistic theories
Full Text:

### References:

 [1] Bacchus, F., () [2] Biacino, L., Generated envelopes, J. math. anal. appl., 172, 179-190, (1993) · Zbl 0777.60004 [3] Biacino, L.; Gerla, G., Decidability, recursive enumerability and Kleene hierarchy for L-subsets, Z. math. logik grundl. math., 35, 49-62, (1989) · Zbl 0649.03031 [4] Chang, C.C.; Keisler, H.J., () [5] Dubois, D.; Lang, J.; Prade, H., Automated reasoning using possibilistic logic: semantics, belief revision and variable certainty weights, (), 81-87 [6] Gerla, G., Pavelka’s fuzzy logic and free L-subsemigroups, Z. math. logik grundl. math., 31, 123-129, (1985) · Zbl 0584.03015 [7] G. Gerla, Fuzzy refutations for probability and multivalued logics, Int. J. Approx. Reason. (to appear). · Zbl 0815.03016 [8] Halpern, J.Y., An analysis of first-order logics of probability, Artif. intell., 46, 311-350, (1991) · Zbl 0723.03007 [9] Horn, A.; Tarski, A., Measures in Boolean algebras, Trans. am. math. soc., 64, 467-497, (1948) · Zbl 0035.03001 [10] Nilsson, N.J., Probabilistic logic, Artif. intell., 28, 71-87, (1986) · Zbl 0589.03007 [11] Pavelka, J., On fuzzy logic I: many-valued rules of inference, Z. math. logik grundl. math., 25, 45-52, (1979) · Zbl 0435.03020 [12] Schrijver, A., () [13] Trillas, E.; Valverde, L., On inference in fuzzy logic, (), 294-297 [14] Vickers, J.M., () [15] Weichselberger, K.; Pöhlmann, S., () [16] Zadeh, L.A., Fuzzy sets, Inf. control, 12, 338-353, (1965) · Zbl 0139.24606 [17] Zadeh, L.A., Fuzzy logic and approximate reasoning, Synthese, 30, 407-428, (1975) · Zbl 0319.02016
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