Twenty years later: Remarks on a polemic

This paper just deals with the starting point in the twenty years old paper [1] by Charles Elkan, with which he tried to puzzle fuzzy logic: That the formula 1-min (a,1-b) = max (b, min (1-a, 1-a)) holds in the unit interval. As it is proven this formula only holds for the pairs a,b in [0,1] verifying the inequality min (b, 1-b) <; a. Such result follows from the theorem: In a De Morgan algebra (L; ., +, ´; 0, 1), it is 1) (a·b´)´ = b + a´·b´ q b·b´ <; a, and 2) It only holds in the Boolean sub-algebra of their Boolean elements. Hence, as the formula cannot be supposed, not only Elkan´s point is not sound, but opened the door to find the standard algebras of fuzzy sets A, B, C, ..., in which the formula (A·B´)´ = B + A´·B´ holds. Since among such many many algebras are all those that are isomorphic to that given in [0,1]^X, by . = W, + = prod*, and ´ = 1-id, it can be stated that Elkan´s argument is not puzzling fuzzy logic.