Implication operators in fuzzy logic

is obvious, and then what always hold in the Mamdani's case is the equivalence (p 2 q ! r) = (p ! r) ^ (q ! r): (14) Consequently, if I is a Mamdani–Larsen's implication (8) never holds, but (14) is always true. IV. CONCLUSION After remembering the four main types of implications used in Fuzzy Logic, the equality [p ^ q ! r] = [(p ! r) _ (q ! r)] (*) is analyzed bot in complemented lattices and in standard theories of fuzzy sets. In the case of complemented lattices, it is shown that in most of them (*) do not hold but that only the inequality [p ^ q ! r] [(p ! r) _ (q ! r)] is a law. An example, inspired in a paper by the late K. Menger, helps to see how (*) depends as a law on the actual relationship between p ^ q ! r and (p ! r) _ (q ! r), once decided which operation ! modelizes the current rules " If, then " in a given situation. Regardless of the distributive, character of the lattice, (*) is a law if the complemented lattice verifies De Morgan's Laws and ! is taken as the so-called material implication. Then, a fortiori (*) is a law in Boolean Algebras. and it is shown that its validity implies S = Max and that, if J belongs to the four main types of implication functions, it should also be T = Min. Hence, only in the theories ([0; 1] X ; Min; Max; N) (**) can give the logical law (*). Then • in the cases of S-implications and R-implications, (**) is always a law; • in the case of Q-implications J (r; s) = S 1 (N 1 (r); T 1 (r; s)), it should be S 1 in the family of the t-conorm W 3 , and (**) has uncountable many different possibilities of being a law; • in the case of Mamdani–Larsen's implications, (**) is never a law. " When QM-operators are implication functions and conditional fuzzy relations?, " Int. Combinatorial rule explosion eliminated by a fuzzy rule configuration, " IEEE Trans. Comments on " Combinatorial rule explosion eliminated by a fuzzy rule configuration " , " IEEE Trans. Abstract—The choice of fuzzy implication as well as other connectives is an important problem in the theoretical development of fuzzy logic, …