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Journal of Logic and Computation Advance Access published online on December 21, 2007
Journal of Logic and Computation, doi:10.1093/logcom/exm084
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© The Author 2007. Published by Oxford University Press on behalf of the Association of Physicians. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original papers

Lambek Calculus in Natural Deduction

Ernst Zimmermann

Boeblingen, Germany

E-mail: ernstzimmermann{at}de.ibm.com

Received 21 June 2007.


  Abstract

A formulation of Lambek calculus in natural deduction is given. New rules for Lambek's multiplicative, non-commutative conjunction are proposed, rules for Lambek's two implications are standard. Rules for Lambek's conjunction are variants of general elimination rules: a symmetric elimination rule and its specializations, left elimination rule and right elimination rule. Conversions hold for all these rules, but only the symmetric elimination rule is fully permutable. Due to a natural transformation for left and right elimination rules to the symmetric elimination rule with partial empty sequences of assumptions and vice versa, there hold two normalization theorems, one with a minimal set and one with a maximal set of permutations.

Keywords: Natural deduction; Lambek calculus


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