New results of fuzzy implications satisfying I(x,I(y,z))=I(I(x,y),I(x,z))

Abstract

Cruz et al. (2018) [10] investigated the fuzzy generalization of Frege's Law: x→(y→z)≡(x→y)→(x→z), i.e., I(x,I(y,z))=I(I(x,y),I(x,z)), which is called generalized Frege's Law. They showed conditions such that the generalized Frege's Law holds for (S,N)-implications (R-, QL-, D-, (T,N)-, H-, respectively).In this paper, firstly, a new necessary condition such that the generalized Frege's Law holds is given: NI, the natural negation of I, is not continuous or has no fixed point. Based on this result, some propositions in [10] with contradictory assumptions are pointed out, and a correction is given. Secondly, new solutions of the equation I(x,I(y,z))=I(I(x,y),I(x,z)) in (S,N)-implications are given. Finally, the necessary and sufficient conditions under which the generalized Frege's Law holds for the (U,N)-implications (f-, g-, T-Power based implications, respectively) are studied.