Interval Valued Fuzzy Sets and Fuzzy Connectives

The generalization of two-valued (crisp) sets to fuzzy sets gave us a rich and powerful representation scheme within the domain of semantic uncertainty. A parallel generalization of conjunctions and disjunction is needed from tnorms and t-conorms to what we might appropriately call linguistic (fuzzy) connectives, “AND," “OR," etc. In fact a closer look at the canonical forms of the combination of concepts, known as the Disjunctive and Conjunctive Normal Forms, DNF and CNF, respectively, suggest that linguistic (fuzzy) connectives “AND," “OR," etc., should be represented with interval-valued fuzzy sets as particular realizations of type II fuzziness that could represent the semantic uncertainty contained in linguistic (fuzzy) connectives which exist in everyday natural language expressions of human use. In fact, the experimentally discovered “Compensatory ‘AND’ " can be grounded within the boundaries of the interval-valued fuzzy sets so suggested and proposed. This leads to the possibility of defining other fuzzy relations such as linguistic (fuzzy) IMPLICATION to enrich its representation by a new definition which we may call “Compensatory ‘IMPLICATION’ ," etc.