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

Probabilistic Fuzzy Reasoning in a Coherent Setting

“…In fact, we can claim both "X is ϕ" and "X is ¬ϕ", or only one of them or finally neither of them (similarly for ϕ and "very ϕ"). As proved in [8], for any given x in the range of X, the assessment P (E ϕ ∧ E ψ |x) is coherent if and only if it holds…”

“…Here we study this problem by adopting the interpretation of fuzzy sets in terms of coherent conditional probabilities introduced in [7,8,9] (and developed in [40,6,14,15]) whose main concepts are recalled in Appendix A. Under this interpretation the membership function µ ϕ (·) of a fuzzy set related to the property ϕ is represented by a conditional probability P (E ϕ |·) regarded as function of the conditioning event.…”

“…Then P (E ϕ |·) comes out to be a natural interpretation of the membership function µ ϕ (·), according to [8] (see also [7,9] …”

“…By referring to [8] we recall the operations between fuzzy subsets: the binary operations of union and intersection and that of complementation can be obtained directly by using the rules of coherent conditional probability and the logical independence between E ϕ and E ψ w.r.t. the partition {(X = x)} x∈C X (or w.r.t.…”

“…One of the main objectives is to jointly handle probability and vagueness [3,4,6,8,9,11,14,15,16,21,22,23,31,32,39,40,41,46,43,44,47,48,49]. For this task it is suitable to have a general framework in which it is possible to handle uncertainty encoding the different aspect by maintaining consistency with the model of reference: this is crucial to ensure a sound inference.…”