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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Rerefences
M. Kandulski, The non-associative Lambek calculus, In: Categorial Grammar, Benjamins, Amsterdam (to appear).
J. Lambek, The mathematics of sentence structure, American Mathematical Monthly 65 (1958), pp. 154–170.
J. Lambek, On the calculus of syntactic types, In: R. Jakobson (ed.), Structure of Language and Its Mathematical Aspects, AMS, Providence 1961, pp. 166–178.
W. Zielonka, Axiomatizability of Ajdukiewicz-Lambek calculus by means of cancellation schemes, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 27 (1981), pp. 215–224.
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Zielonka, W. A simple and general method of solving the finite axiomatizability problems for Lambek's syntactic calculi. Stud Logica 48, 35–39 (1989). https://doi.org/10.1007/BF00370632
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DOI: https://doi.org/10.1007/BF00370632