Vague and fuzzy t-norms and t-conorms

Abstract

t-norms and t-conorms are the natural connectives “and” and “or” in fuzzy logic. The unit interval with a t-norm or a t-conorm is a special monoid and some submonoids like discrete t-norms and t-conorms have been proved useful in many cases. In the first part of this article these submonoids will be fuzzified to fuzzy t-subnorms and fuzzy t-subconorms in order to deal with imprecision. As particular examples we will provide fuzzifications of the classical and the Łukasiewicz three-valued conjunctions. The second part of the article will define and study vague t-norms and t-conorms as fuzzy operations ∘˜:[0,1]3→[0,1] where ∘˜(x,y,z) is the degree in which z is T(x,y) (S(x,y) respectively) where T (S) is a t-norm (t-conorm).