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On the coincidence of conditional functions

This paper deals with a problem in fuzzy logic that can be traced back to classical logic: the reduction of one conditional to another. In particular, the reduction of R-implications to S-implications and to Q-operators is considered, although the more complex case is the reduction of a Q-operator to an S-implication, equivalent to find the complete solution of a functional equation involving strong negations, continuous t-conorms, and continuous t-norms. Necessary and sufficient conditions for the validity of such equation are presented in several cases, but other cases remain open and deserve to be studied.