Compound Conditionals and Fuzzy Sets

FollowingCastronovo, L.Sanfilippo, G. the approach proposed by Coletti and Scozzafava, we propose a slightly different interpretation of fuzzy set theory in terms of conditional events and coherent conditional probabilities, examining it within the framework of the betting scheme. We propose a revised definition of fuzzy sets and introduce operations such as complement and intersection. To achieve this, we employ logical operations among conditional events within the framework of conditional random quantities. Specifically, we interpret the membership function of a fuzzy set as a coherent conditional prevision assessment on suitable compound conditionals. Furthermore, we provide examples and examine the class of Frank t-norms to describe coherent conditional previsions.