Elkan's theoretical argument, reconsidered (Q5932225)

In a price-winning, and therefore (in the fuzzy community) infamous paper [IEEE Expert 9, No. 4, 3-8 (1994; Zbl 1009.03517)] \textit{C. Elkan} had shown the almost obvious fact that elementary fuzzy logic (with min, max, and \(1-\ldots \) as truth degree functions for \(\land, \lor,\neg \)) together with the idea that logical equivalence for this fuzzy logic should be the same as for classical logic, gives nonsense. His reference point was the Boolean equation NEWLINE\[NEWLINE(*) \neg (a \land\neg b) = b \lor (\neg a \land\neg b). NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEIn the present paper the authors discuss this equation \((*)\) for fuzzy sets, with the logical connectives involved taken more generally as forming, e.g., a de Morgan triplet. NEWLINENEWLINENEWLINEThe relationship to the former Elkan paper is not essential, and a bit overstated by the authors. Elkan's answer (see below) states this fact and mentions a few more sloppy formulations in Trillas and Alsina's paper. Finally the authors answer (see below) to Elkan's response by explaining better some of their earlier, sloppier formulations. NEWLINENEWLINENEWLINEIt is only the present one of these papers that is of mathematical interest.