Characterizations of fuzzy implications satisfying the Boolean-like law y ≤ I(x, y)
Anderson Cruz, Benjam´ın Bedregal, and Regivan Santiago
Group of Theory and Intelligence of Computation - GoThIC Federal Rural University of Semi- ´Arido - UFERSA
59515-000, Angicos, RN, Brazil [email protected]
Group of Logic, Language, Information, Theory and Applications - LOLITA Federal University of Rio Grande do Norte - UFRN
59072-970, Natal, RN, Brazil {bedregal,regivan}@dimap.ufrn.br
Abstract. Properties valid on the classical theory (Boolean laws) have been extended to fuzzy set theory and so-called Boolean-like laws. The fact that they are not always valid in any standard fuzzy set theory induced a wide investigation. In this paper we show the sufficient and necessary conditions that the Boolean-like lawy≤I(x, y) holds in fuzzy logic. We focus the investigation on the following classes of fuzzy impli- cations: (S,N)-, R-, QL-, D-, (N,T)- andh-implications.
Keywords: Implications, Fuzzy Logics, Boolean-like Laws.
1 Introduction
Classical logic, well as its inference notion were the first formal elucidation of how would be a correct reasoning. Joined with those concepts is the classical implication definition (called material implication). Thus, such implication was the first one to be defined and disseminated. This fact induces us to believe that material implication is the correct notion (common sense) of what actually is a logical implication. However, other Boolean implications, such as intuitionistic, quantum or para-consistent implications, do give acceptable inference models and they must be regarded to understand the meaning of a logical implication.
In fuzzy theory, implication operator was originally used to define the rela- tion between antecedent and consequent of IF-THEN rules in Fuzzy Rule Based Systems (see [37]). However, the truth value of implication operators do not necessarily satisfy the classical implication truth table (see [21] for example).
In search of a classical implication generalization, Baldwin and Pilsworth in [7] and Blander and Kohout in [10], proposed a few basic properties that should be required by a fuzzy implication. Further, Trillas and Valverde in [32]
gave the first fuzzy implication axiomatic. Nowadays, the lack of a consensus on Boolean implication meaning entails non-equivalent acceptable fuzzy implication
