Why are Discrete Implications Necessary? An Analysis Through the Discretization Process

Discrete implications have been studied for almost two decades as those operators needed to perform inference processes when dealingwith qualitative information from a finite chain. However, it is known that, by means of some adequate transformations, fuzzy logic operators defined in [0,1] can generate the corresponding discrete operators. Thus, an immediate question arises: Do we need to study discrete implications or is it enough to study implications defined in [0,1], and then, to discretize them? The answer must rely on the preservation of the additional properties of fuzzy implication functions through these discretization methods. In this article, for two specific methods based on the ceiling and floor functions, it is proved that most of the additional properties are not preserved in general, showing that the preservation of the additional properties depends directly on the properties of the underlying operators considered in the discretization. Thus, sufficient, and for some properties necessary, conditions to guarantee the preservation are presented.