Nonassociative Substructural Logics and their semilinear Extensions: Axiomatization and Completeness Properties

Substructural logics extending the full Lambek calculus FL have largely benefited from a systematical algebraic approach based on the study of their algebraic counterparts: residuated lattices. Recently, a non-associative generalization of FL (which we call SL) has been studied by Galatos and Ono as the logics of lattice-ordered residuated unital groupoids. This paper is based on an alternative Hilbert-style presentation for SL which is almost (MP)-based. This presentation is then used to obtain, in a uniform way applicable to most (both associative and non-associative) substructural logics, a form of local deduction theorem, description of filter generation, and proper forms of generalized disjunctions. A special stress is put on semilinear substructural logics (i.e. logics complete w.r.t. linearly ordered algebras). Axiomatizations of the weakest semilinear logic over SL and other prominent substructural logics are provided and their completeness with respect to chains defined over the real unit interval is proved.