Logics without the contraction rule and residuated lattices

In this paper, we will develop an algebraic study of substructural propositional logics over FL_, i.e. the logic which is obtained from the intuitionistic logics by eliminating the contraction rule. Our main technical tool is to use residuated lattices as the algebraic semantics for them. This enables us to study different kinds of nonclassical logics, including intermediate logics, BCK-logics, Łukasiewicz’s many-valued logics and fuzzy logics, within a uniform framework