General Characterization of Implication's Distributivity Properties: The Preference Implication

Abstract

It is widely accepted that distributivity properties play a key role in fuzzy research, especially in fuzzy control. Making use of the solution of the autodistributivity functional equations, we give a characterization of all the four types of distributivity of fuzzy implication. The necessary and sufficient condition for all the four distributive equation is that the operator belongs to the pliant operator class. This theorem leads to a new implication called preference implication. It is well known that there is no implication in (continuous-valued) fuzzy logic that satisfies all the properties that are valid in (classical) two-valued logic. We show that preference implication fulfills: 1) the law of contraposition, 2) the $T$ -conditionality, 3) the ordering property, 4) the exchange principle, 5) the law of importation, 6) the identity principle, and 7) the general hypothetical reasoning. Preference implication has a $\nu$ parameter, i.e., the fixed point of the negation. This $\nu$ value serves as a threshold and with this value we can go back by projection to the two-valued logic case. At the end of the article, we indicate that the preference implication is closely related to the preference relation used in multicriteria decision making. We point out that if the preference implication is multiplicative, transitive, and reciprocal, then the pliant system is reduced to some particular generator function of Dombi.