Some mathematical tools for linguistic probabilities

We propose and develop, in this paper, some concepts and techniques useful for the theory of linguistic probabilisies introduced by L.A. Zadeh. These probabilities are expressed in linguistic rather than numerical terms. The mathematical framework for this study is based upon the possibility theory.We formulate first the problem of optimization under elastic constraints which is not only important for mathematical programming but will be served to justify the extension of possibility measure to linguistic variables. Next, in connection with translation rules in natural languages we study some transformations of fuzzy sets using a relation between random sets and fuzzy sets. Finally, we point out some differences between random variables and fuzzy variables, and present the mathematical notion of possibility, in the setting of set-functions, as a special case of Choquet capacities.