A simple and general method of solving the finite axiomatizability problems for Lambek's syntactic calculi

Studia Logica 48 (1):35 - 39 (1989)

Abstract
In [4], I proved that the product-free fragment L of Lambek's syntactic calculus (cf. Lambek [2]) is not finitely axiomatizable if the only rule of inference admitted is Lambek's cut-rule. The proof (which is rather complicated and roundabout) was subsequently adapted by Kandulski [1] to the non-associative variant NL of L (cf. Lambek [3]). It turns out, however, that there exists an extremely simple method of non-finite-axiomatizability proofs which works uniformly for different subsystems of L (in particular, for NL). We present it below to the use of those who refer to the results of [1] and [4].
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DOI 10.1007/BF00370632
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Axiomatizability of Ajdukiewicz-Lambek Calculus by Means of Cancellation Schemes.Wojciech Zielonka - 1981 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 27 (13-14):215-224.

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Two Weak Lambek-Style Calculi: DNL and DNL.Wojciech Zielonka - 2012 - Logic and Logical Philosophy 21 (1):53-64.

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