On functional equations and inequalities related to some reasoning schemes that involve fuzzy implications

In this paper, we investigate some functional equations and inequalities which are associated with recognized reasoning schemes like hypothetical syllogism, modus ponens, modus tollens and reduction to absurdity. First, we recall facts and theorems that might be found in the literature. Next, we show some properties and relationships that take place between these equations and inequalities. Moreover, we present some of their solutions among well-known families of fuzzy implications – R-implications, (S,N)-implications and Yager's implications.