Further results on fuzzy negations and implications induced by fuzzy conjunctions and disjunctions
Xin-Tong Zhang, Xue-ping Wang
TL;DR
This work develops a unified theoretical framework connecting fuzzy negations induced by fuzzy conjunctions and disjunctions to $(D,N)$-implications. It defines natural negations $N_C$ and $N_D$, analyzes their continuity and strength properties, and represents $(D,N)$-implications as $I_{D,N}(x,y)=D(N(x),y)$. The authors derive necessary and sufficient conditions for implication properties (IP) and (OP) in terms of a law of excluded middle $(LEM1)$ and explore continuous/unique representations via derived negations $N_I$, enriching the theory of fuzzy connectives. These results advance a cohesive treatment of fuzzy negations and implications, with potential implications for robust fuzzy reasoning systems.
Abstract
In this article, we deeply investigate some properties of fuzzy negations induced from fuzzy conjunctions (resp. disjunctions), which are then applied to characterizing the fuzzy negations. We further use the obtained characterization of fuzzy negations to explore some properties of $(D,N)$-implications generated from fuzzy disjunctions and negations. We finally describe $(D,N)$-implications (resp. continuous $(D,N)$-implications) generated from fuzzy disjunctions and negations.