Cut-Rule Axiomatization of the Syntactic Calculus L0

In Zielonka (1981a, 1989), I found an axiomatics for the product-free calculus L of Lambek whose only rule is the cut rule. Following Buszkowski (1987), we shall call such an axiomatics linear. It was proved that there is no finite axiomatics of that kind. In Lambek's original version of the calculus (cf. Lambek, 1958), sequent antecedents are non empty. By dropping this restriction, we obtain the variant L ₀ of L. This modification, introduced in the early 1980s (see, e.g., Buszkowski, 1985; Zielonka, 1981b), did not gain much popularity initially; a more common use of L ₀ has only occurred within the last few years (cf. Roorda, 1991: 29). In Zielonka (1988), I established analogous results for the restriction of L ₀ to sequents without left (or, equivalently, right) division. Here, I present a similar (cut-rule) axiomatics for the whole of L ₀. This paper is an extended, corrected, and completed version of Zielonka (1997). Unlike in Zielonka (1997), the notion of rank of an axiom is introduced which, although inessential for the results given below, may be useful for the expected non-finite-axiomatizability proof. The paper follows the same way of subject exposition as Zielonka (2000) but it is technically much less complicated. I restrict myself to giving bare results; all the ideological background is exactly the same as in case of the non-associative calculusNL ₀ and those who are interested in it are requested to consult the introductory section of Zielonka (2000). This revised version was published online in August 2006 with corrections to the Cover Date.