Weak implicational logics related to the Lambek calculus

Gentzen versus Hilbert formalisms

Wojciech Zielonka

pp. 201-212

in: David Makinson, Jacek Malinowski, Heinrich Wansing (eds), Towards mathematical philosophy, Berlin, Springer, 2009

Abstract

It has been proved by the author that the product-free Lambek calculus with the empty string in its associative (L ₀) and non-associative (NL ₀) variant is not finitely Gentzen-style axiomatizable if the only rule of inference is the cut rule. We give here rather detailed outlines of the proofs for both L ₀ and NL ₀. In the last section, Hilbert-style axiomatics for the corresponding weak implicational calculi are given.

Cited authors

Hilbert David

Gentzen Gerhard