Logic and Meaning: The Philosophical Significance of the Sequent Calculus

Abstract

In his doctoral dissertation of I935,1 Gentzen introduced two new approaches to the theory of proofs: natural deduction and the sequent calculus. He formulated both classical and intuitionist predicate logics using these approaches. For the resulting sequent calculi, he proved his Hauptsatz, or main theorem, which states that the rule of cut may be dispensed with in all proofs. The significance of this 'cut-elimination theorem' has been much debated over the past fifty years, but no clear consensus has arisen. In this paper, I will examine some of the ways in which this theorem has been interpreted. In particular, I will focus on the idea that the introduction rules for a logical constant in a sequent calculus may be taken as defining the constant or 'giving its meaning'; and I will consider the relation of cut-elimination to this thesis. My discussion will be illustrated by a specific example, drawing on recent work on truth and the semantic paradoxes.