Counting the Number of Proofs in the Commutative Lambek Calculus

This paper is concerned with the study of the number of proofs of a sequent in the commutative Lambek calculus. We show that in order to count how many different proofs in βη-normal form a given sequent Γ ` α has, it suffices to enumerate all the ∆ ` β which are “minimal”, such that Γ ` α is a substitution instance of ∆ ` β. As a corollary we obtain van Benthem’s finiteness theorem for the Lambek calculus, which states that every sequent has finitely many different normal form proofs in the Lambek calculus.