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. (shrink)

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.