On Some Subclasses of the Fodor-Roubens Fuzzy Bi-implication

The paper deals with fuzzy versions of the classical bi-implication, that is, extensions of classical bi-implication to the canonical domain of mathematical fuzzy logics, the real-valued unit interval [0, 1]. Our approach to fuzzy bi-implication may be summarized as follows: first, we recall a well-known approach to bi-implications, by Fodor and Roubens, via the direct axiomatization of the properties of the corresponding class of operators; next, we investigate a particular defining standard of bi-implication in terms of t-norms and r-implications. We study four prospective classes of bi-implications based on such defining standard, by varying the properties of its composing operators, and show that these classes collapse into precisely two increasingly weaker subclasses of the Fodor-Roubens bi-implication.