A Uniform Algebraic Approach to Cut Elimination via Semi-completeness

Abstract

This is an attempt to present a uniform algebraic framework for semantical approach to cut elimination. The basic idea came from a paper by S. Maehara of 1991. Using the notion of quasi-homomorphisms essentially due to Maehara, an algebraic condition called semi-completeness of a given sequent system is introduced. It is shown that for a given sequent system S, semi-completeness for the system \(\mathbf{S}^-\) implies cut elimination for S, where \(\mathbf S^-\) is obtained from S by deleting cut rule. In the present paper it is confirmed that many of existing semantical proofs of cut elimination using either Kripke semantics or algebraic one will fall into our algebraic framework. In fact, semi-completeness is considered to be an intelligible algebraic criterion of cut elimination which is applicable to both single- and multiple-succedent sequent systems for wide variety of nonclassical logics, including modal logics and substructural logics. For modal logics and intuitionistic logic, connections of quasi-homomorphisms with downward saturations, the conditions which are used in the constructions of canonical Kripke models, will be clarified. On the other hand, for substructural logics, quasi-homomorphisms will be discussed in relation to quasi-embeddings which are crucial in algebraic approaches to cut elimination. In the last three sections, semi-completeness arguments are extended so as to cover semantical proofs of cut elimination for nonclassical predicate logics. This can be carried out by generalizing quasi-homomorphisms on expanded algebraic structures. In this way, semi-completeness will provide a unified view of comprehending various semantical approaches to cut elimination.