Probable equivalence, superpower sets, and superconditionals

Abstract

A natural measure of probabilistic equality between sets leads to two measures of probabilistic conditioning that form the endpoints of a conditioning interval. The interval's lower bound is the standard conditional probability or "subconditional" that describes the probability of a subset relation. The upper bound is a new "superconditional" that describes the probability of the corresponding superset relation. These dual conditioning operators correspond to dual set collections and enjoy optimality relations with respect to these set collections. Fuzzy cubes illustrate these set-collection relations in the two-dimensional case. The subconditional operator corresponds to the usual "power set" of a given set. The dual superconditional operator corresponds to what we call the "superpower set" or the set of all supersets of the given set. The two dual conditioning operators can eliminate each other through simple equalities. They obey dual Bayes theorems but differ in how they respond to statistical independence.

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