The non-contradiction principle related to natural negations of fuzzy implication functions.

The translation of the classical Non-Contradiction (NC) principle, as well as its dual law: the Excluded-Middle (EM) principle, to the framework of fuzzy logic leads to a well-known functional equation involving appropriate aggregation and negation functions. When the involved negation is the natural negation of a fuzzy implication function, the NC principle becomes specially important because then it is a necessary condition for the fulfillment of the Modus Ponens inequality, as well as for the residuation property for residuated implications. In this paper, the functional equation corresponding to the NC principle is deeply studied in the case when both, the aggregation function and the fuzzy negation, are as general as possible. Moreover, the results are applied to the case when the fuzzy negation is the natural negation of a fuzzy implication function, extending this study to the most usual classes of fuzzy implication functions.