On reduction systems equivalent to the Lambek calculus with the empty string (Q1611249)

This is one of the author's papers concerning the axiomatizability of the Lambek Calculus. The main result of the paper is the theorem saying that the product-free Lambek calculus with the empty string is not axiomatizable if Lambek's cut rule is the only admitted rule of inference. The proof of the theorem is technically rather complicated. It uses the cut-rule axiomatized calculus \(C\) and its special subsystems \(C_{n}\), \(C_{n}^{R}\), \(C_{n}^{r}\) introduced in order to simplify the proof. This reduction calculus, being the union of the chain \((C_{n})\), is equivalent to the Lambek Calculus with the empty string. The difficulties appearing in the solution of the problem described in the paper are connected with the type raising elimination.