Interval valued fuzzy sets from continuous Archimedean triangular norms

Interval valued fuzzy sets are suggested in Turksen (1986) to model the situations where linguistic connectives as well as the variables are fuzzy. They are defined using the discrepancy of conjunctive and disjunctive Boolean normal forms in the fuzzy case. The discrepancy is due to relaxing some of the axioms of classical logic. The authors briefly investigate the basic operations in the unit interval. Specifically the literature on representing the negation functions and triangular norms is recalled. Archimedean triangular norms are investigated as possible candidates for the logical connective AND. De Morgan triples are constructed utilizing a general result for negations. Two broad families of De Morgan triples are identified; strict and strong, which are neither distributive nor idempotent. The authors introduce the concept of an interval valued fuzzy set and present the main results of the paper, namely for strict and strong De Morgan triples interval valued fuzzy sets are well defined. The paper is technical in nature and extends some results obtained in Turksen (1986) to more general settings using generator functions