Abstract
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.
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Zielonka, W. (2009). Weak Implicational Logics Related to the Lambek Calculus—Gentzen versus Hilbert Formalisms. In: Makinson, D., Malinowski, J., Wansing, H. (eds) Towards Mathematical Philosophy. Trends in Logic, vol 28. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9084-4_10
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DOI: https://doi.org/10.1007/978-1-4020-9084-4_10
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