A subjective approach to fuzzy events and their coherent probability assessment for decision making under uncertainty

This research starts from the concepts of conditional event and conditional probability introduced by de Finetti, Dubins and Scozzafava and from the concepts on fuzzy probability introduced by Zadeh, Klir, Yuan, and Yager. We propose a formal adjustment (but logically equivalent) to the definition of conditional event and related concepts, which allows us to give a definition of fuzzy event as an extension of a conditional event. Then, we suggest a formal modification (but logically equivalent) of the concept of conditional probability in the sense of de Finetti-Dubins, and we introduce some possible definitions of fuzzy event's probability as an extension of the conditional probability. Each of these definitions is an extension of the operational de Finetti definition of subjective conditional probability based on a coherent bet. Moreover, it contains a subjective evaluation, which is represented by a score function, that, in appropriate contexts, can be interpreted as a utility function.