Cut-rule axiomatization of the syntactic calculus NL

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 0 andNL 0 was partially solved, viz., for formulas withoutleft (or, equivalently, right) division and an (infinite) cut-ruleaxiomatics for the whole of L 0 has been given. Thepresent paper yields an analogous axiomatics forNL 0. Like in the author's previous work, the notionof rank of an axiom is introduced which, although inessentialfor the results given below, may be useful for the expectednonfinite-axiomatizability proof.
Keywords axiomatizability  cut rule  Lambek calculus
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DOI 10.1023/A:1008309805070
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