Abstract

The goal of this paper is to show how modal logic may be conceived as recording the derived rules of a logical system in the system itself. This conception of modal logic was propounded by Dana Scott in the early seventies. Here, similar ideas are pursued in a context less classical than Scott's.First a family of propositional logical systems is considered, which is obtained by gradually adding structural rules to a variant of the nonassociative Lambek calculus. In this family one finds systems that correspond to the associative Lambek calculus, linear logic, relevant logics, BCK logic and intuitionistic logic. Above these basic systems, sequent systems parallel to the basic systems are constructed, which formalize various notions of derived rules for the basic systems. The deduction theorem is provable for the basic systems if, and only if, they are at least as strong as systems corresponding to linear logic, or BCK logic, depending on the language, and their deductive metalogic is not stronger than they are.