FUZZINESS VS. PROBABILITY

TLDR

Fuzziness is investigated as an alternative to randomness for describing uncertainty, with paradoxes of two‑valued logic corresponding to the midpoint of a fuzzy cube and prompting fundamental questions about the fuzziness of sets and their subset relationships. The authors develop a sets‑as‑points geometric view of fuzzy sets and use it to answer these fundamental questions geometrically via the Fuzzy Entropy, Subsethood, and Entropy‑Subsethood Theorems. This view represents a fuzzy set as a point in a unit hypercube and a nonfuzzy set as a vertex, provides a geometric proof that the relative frequency nA/N equals the deterministic subsethood S(X,A), and evaluates Bayesian polemics against fuzzy theory in this framework. Consequently, the frequency of successful trials is interpreted as the degree to which all trials are successful.

Abstract

Fuzziness is explored as an alternative to randomness for describing uncertainty. The new sets-as-points geometric view of fuzzy sets is developed. This view identifies a fuzzy set with a point in a unit hypercube and a nonfuzzy set with a vertex of the cube. Paradoxes of two-valued logic and set theory, such as Russell's paradox, correspond to the midpoint of the fuzzy cube. The fundamental questions of fuzzy theory—How fuzzy is a fuzzy set? How much is one fuzzy set a subset of another? —are answered geometrically with the Fuzzy Entropy Theorem, the Fuzzy Subsethood Theorem, and the Entropy-Subsethood Theorem. A new geometric proof of the Subsethood Theorem is given, a corollary of which is that the apparently probabilistic relative frequency nA /N turns out to be the deterministic subsethood S(X, A), the degree to which the sample space X is contained in its subset A. So the frequency of successful trials is viewed as the degree to which all trials are successful. Recent Bayesian polemics against fuzzy theory are examined in light of the new sets-as-points theorems.