Conditional probability, fuzzy sets, and possibility: a unifying view

We propose an interpretation of fuzzy set theory (both from a semantic and a syntactic point of view) in terms of conditional events and coherent conditional probabilities. During past years, a large number of papers has been devoted to support either the thesis that probability theory is all that is required for reasoning about uncertainty, or the negative view maintaining that probability is inadequate to capture what is usually treated by fuzzy theory. In this paper we emphasize the role of conditioning (in a proper framework, i.e. de Finettiʹs coherence) to get rid of many controversial aspects. Moreover, we introduce suitable operations between fuzzy subsets, looked on as corresponding operations between conditional events endowed with the relevant conditional probability. Finally, we show how the concept of possibility function naturally arises as a coherent conditional probability.