Introducing Interpolative Boolean Algebra for Interval-Valued Fuzzy Sets

Abstract

Interval-valued fuzzy sets (IVFSs) are an extension of traditional fuzzy sets by representing membership degrees as intervals rather than crisp numbers. It enables a more sophisticated modeling of imprecision. Despite these advantages, there remains substantial room for improvement in ensuring logical consistency and in adequately capturing dependencies between interval bounds. In this paper, we propose applying Interpolative Boolean algebra (IBA) as an algebraic framework for IVFS theory. Specifically, we define conjunction and disjunction operations for IVFS that are directly derived from IBA. These logical operations are based on the generalized product, an operator that can be realized by any t-norm whose result is greater than or equal to the minimum t-norm and less than or equal to the Lukasiewicz t-norm. Finally, we investigate the validity of the laws of contradiction and excluded middle within the proposed approach.