Cut Elimination for Sequent Systems

Abstract

Cut elimination for a given sequent system \(\mathbf L\) means that if a sequent is provable in \(\mathbf L\) then it is also provable in \(\mathbf L\) without using cut rule. Any proof P of \(\mathbf L\) is said to be cut-free when P contains any application of cut rule in it. Intuitively, any cut-free proof is a kind of a proof without detours, or a direct proof, though it may be longer than a proof with cut. On the other hand, such a direct proof has many ‘good’ properties. Because of this, it is one of most important goals of syntactic study of logic to formalize a given logic in a sequent system and to show cut elimination for this system, although cut elimination holds for rather a limited number of sequent systems.