Two extensions of the structurally free logic LC

Abstract

The paper considers certain extensions of the system LC introduced in Dunn & Meyer 1997. LC is a structurally free system (it has no structural rules), but it has combinators as formulas in the place of structural rules. We consider two ways to extend LC with conjunction and disjunction depending on whether they distribute over each other or not. We prove the elimination theorem for the systems. At the end of the paper we give a Routley-Meyer style semantics for the distributive extension, including some new definitions and an algorithm which are used in proving a representation theorem in general, i.e., for arbitrary combinators. Keywords:Gentzen system, substructural logic, combinators, cut elimination, algebraic semantics