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Cut‐Rule Axiomatization of Product‐Free Lambek Calculus With the Empty String

Wojciech Zielonka

Mathematical Logic Quarterly 34 (2):135-142 (1988)

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Cut-rule axiomatization of the syntactic calculus NL.Wojciech Zielonka - 2000 - Journal of Logic, Language and Information 9 (3):339-352.details An axiomatics of the product-free syntactic calculus L ofLambek has been presented whose only rule is the cut rule. It was alsoproved that there is no finite axiomatics of that kind. The proofs weresubsequently simplified. Analogous results for the nonassociativevariant NL of L were obtained by Kandulski. InLambek's original version of the calculus, sequent antecedents arerequired to be nonempty. By removing this restriction, we obtain theextensions L 0 and NL 0 ofL and NL, respectively. Later, the finiteaxiomatization problem for L (...) Download Export citation Bookmark 2 citations
Cut-Rule Axiomatization of the Syntactic Calculus L0.Wojciech Zielonka - 2001 - Journal of Logic, Language and Information 10 (2):233-236.details 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 L0 of L. This modification, introduced in the early 1980s (see, e.g., Buszkowski, 1985; (...) Download Export citation Bookmark 3 citations
On reduction systems equivalent to the Lambek calculus with the empty string.Wojciech Zielonka - 2002 - Studia Logica 71 (1):31-46.details The paper continues a series of results on cut-rule axiomatizability of the Lambek calculus. It provides a complete solution of a problem which was solved partially in one of the author''s earlier papers. It is proved that the product-free Lambek Calculus with the empty string (L 0) is not finitely axiomatizable if the only rule of inference admitted is Lambek''s cut rule. The proof makes use of the (infinitely) cut-rule axiomatized calculus C designed by the author exactly for this purpose. Download Export citation Bookmark 2 citations
Linear axiomatics of commutative product-free Lambek calculus.Wojciech Zielonka - 1990 - Studia Logica 49 (4):515 - 522.details Axiomatics which do not employ rules of inference other than the cut rule are given for commutative product-free Lambek calculus in two variants: with and without the empty string. Unlike the former variant, the latter one turns out not to be finitely axiomatizable in that way. Download Export citation Bookmark 1 citation
On the directional Lambek calculus.Wojciech Zielonka - 2010 - Logic Journal of the IGPL 18 (3):403-421.details The article presents a calculus of syntactic types which differs from the calculi L and NL of J. Lambek in that, in its Gentzen-like form, sequent antecedents are neither strings nor phrase structures but functor-argument structures. The product-free part of the calculus is shown to be equivalent to the system AB due to Ajdukiewicz and Bar-Hillel. However, if the empty sequent antecedent is admitted, the resulting product-free calculus is not finitely cut-rule axiomatizable. Download Export citation Bookmark 1 citation

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