Weak Implicational Logics Related to the Lambek Calculus—Gentzen versus Hilbert Formalisms
It has been proved by the author that the product-free Lambek calculus with the empty string in its associative (L 0) and non-associative (NL 0) variant is not finitely Gentzen-style axiomatizable if the only rule of inference is the cut rule. We give here rather detailed outlines of the proofs for both L 0 and NL 0. In the last section, Hilbert-style axiomatics for the corresponding weak implicational calculi are given.
KeywordsLambek calculus implicational logics finite axiomatizability
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