Basic Fuzzy Logic is the logic of continuous t-norms and their residua

Abstract

 In this paper we prove that Basic Logic (BL) is complete w.r.t. the continuous t-norms on [0, 1], solving the open problem posed by Hájek in [4]. In fact, Hájek proved that such completeness theorem can be obtained provided two new axioms, B1 and B2, were added to the original axioms of BL. The main result of the paper is to show that B1 and B2 axioms are indeed redundant. We also obtain an improvement of the decomposition theorem for saturated BL-chains as ordinal sums whose components are either MV, product or Gödel chains, in an analogous way as for continuous t-norms. Finally we provide equational characterizations of the variety of BL-algebras generated by the three basic BL subvarieties, as well as of the varieties generated by each pair of them, together with completeness results of the calculi corresponding to all these subvarieties.