Kripke-Style Semantics for Non-Commutative Monoidal t-Norm Logic

A t-norm is a binary operation on [0, 1] that is associative, commutative, with identity 1 and non-decreasing in both argument. The notion of t-norm is very important in the context of fuzzy logic, for it is used in modeling a general form of the propositional conjunction. The basic fuzzy logic (BL logic) was introduced by Hájek in [9,10] and it was inspired by a continuous t-norm and its residual. If the continuity hypothesis is weakened, by replacing it with left-continuity (a necessary and sufficient condition for a t-norm to have a residuum), then we obtain a different fuzzy logic, namely monoidal t-norm based logic (MTL logic) introduced by Esteva and Godo in [2]. In the algebraic framework, the algebras corresponding to BL logic are called BL algebras [10] and they are residuated lattices with two new properties: the divisibility condition and the prelinearity condition. MTL algebras [2], the algebraic structures for MTL logic, are a generalization of BL algebras by dropping down the divisibility condition. MTL algebras are also known under the name of weak BL algebras [4].