Probability, Modal Logic and Fuzzy Logic

Fuzziness is often regarded as a special form of uncertainty, possibly related to probability but not necessarily reducible to the latter. Indeed, all the mathematical techniques that have been proposed to model fuzziness are substantially different from probabilistic methods, albeit sometimes loosely similar. As shown in this paper, a particular form of fuzzy logic can be derived from probability by means of standard logical techniques. It can be shown that, in a probability structure based on a first-order language, the definition of a particular class of fuzzy sets is straightforward. Consequently, a special form of fuzzy logic can be constructed by adopting a first-order, probabilistic modal language being interpreted in the above class of structures. The logic thus obtained shows several points of contact with fuzzy inferential techniques described in the literature and leads to further insights, with a fascinating informal interpretation.