A probabilistic approach to the modeling of the relationship between fuzzy sets

Instead of degrees, we model a partial membership of an object to a fuzzy set as a (crisp) subset of a special abstract universe of so-called membership quanta. That allows us to define the operations on fuzzy sets on the basis of classical set operations. As a result, there is not a freedom of choosing a t-norm for intersection or a t-conorm for union etc. Instead, the set operations are unique and satisfy all axioms of Boolean algebra. If a measure is defined on the universe of membership quanta, we can obtain membership degrees and show that similarly to the theory of probability also the membership degrees of intersections are governed by copulas.

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