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.