Paradoxes of fuzzy logic, revisited

This paper is a reply to the article entitled Elkan’s Theoretical Argument, Reconsidered by Prof. Enric Trillas and Prof. Claudi Alsina. I would like to express my thanks to Dr. Piero Bonissone for inviting me to write this paper and for showing me the article by Trillas and Alsina in advance of its publication. Ever since mathematical studies of fuzziness began with a 1965 paper entitled Fuzzy Sets by Lotfi Zadeh [Zad65], it has been important to distinguish between between fuzzy logic and fuzzy set theory. The theorem in my paper The Paradoxical Success of Fuzzy Logic, as published in 1993 [Elk93] and also as revised in 1994 [Elk94], is a result about standard fuzzy logic. The paper of Trillas and Alsina concerns fuzzy sets, and therefore erroneously states that the theorem is false. At the start of their Section 2, Trillas and Alsina write: