Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-22T20:38:49.965Z Has data issue: false hasContentIssue false
  • English
  • Français

DE MORGAN INTERPRETATION OF THE LAMBEK–GRISHIN CALCULUS

Published online by Cambridge University Press:  26 February 2019

MICHAEL KAMINSKI  and
NISSIM FRANCEZ
MICHAEL KAMINSKI*
Affiliation:
Department of Computer Science, Technion – Israel Institute of Technology
NISSIM FRANCEZ*
Affiliation:
Department of Computer Science, Technion – Israel Institute of Technology
*
*DEPARTMENT OF COMPUTER SCIENCE TECHNION – ISRAEL INSTITUTE OF TECHNOLOGY HAIFA 32000, ISRAEL E-mail: [email protected]E-mail: [email protected]
*DEPARTMENT OF COMPUTER SCIENCE TECHNION – ISRAEL INSTITUTE OF TECHNOLOGY HAIFA 32000, ISRAEL E-mail: [email protected]E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We present an embedding of the Lambek–Grishin calculus into an extension of the nonassociative Lambek calculus with negation. The embedding is based on the De Morgan interpretation of the dual Grishin connectives.

Keywords

03B4703B47Lambek–Grishin calculusdualitynegationDe Morgan’s lawssequent calculuscut elimination
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 4 , December 2020 , pp. 845 - 856
Check for updates
Copyright
Copyright © Association for Symbolic Logic 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Buszkowski, W. (1996). Categorial grammars with negative information. In Wansing, H., editor. Negation. A Notion in Focus. Berlin: de Gruyter, pp. 107–126.Google Scholar
Buszkowski, W. (2005). Lambek calculus with nonlogical axioms. In Casadio, C., Scott, P., and Seely, R., editors. Language and Grammar: Studies in Mathematical Linguistics and Natural Language. CSLI Lectures Notes, Vol. 168. Stanford, CA: Center for the Study of Language and Information, pp. 77–93.Google Scholar
Buszkowski, W. (2011). Interpolation and FEP for logics of residuated algebras. Logic Journal of the IGPL, 19, 437–454.CrossRefGoogle Scholar
de Groote, P. (2015). Proof-theoretic aspects of the Lambek–Grishin calculus. In de Paiva, V., de Queiroz, R. J. G. B., Moss, L. S., Leivant, D., and de Oliveira, A. G., editors. Logic, Language, Information, and Computation - 22nd International Workshop, WoLLIC 2015. Lecture Notes in Computer Science, Vol. 9160. Heidelberg: Springer, pp. 109–123.Google Scholar
de Groote, P. & Lamarche, F. (2002). Classical nonassociative Lambek calculus. Studia Logica, 71, 355–388.CrossRefGoogle Scholar
Grishin, V. N. (1983). On a generalization of the Ajdukiewicz-Lambek system. In Mikhailov, A. I., editor. Studies in Nonclassical Logics and Formal Systems. Moscow: Nauka, pp. 315–334. (English translation in: Abrusci, V. M. & Casadio, C., editors. New Perspectives in Logic and Formal Linguistics. Proceedings of the Vth Roma Workshop, Bulzoni, Rome, 2002.)Google Scholar
Kaminski, M. & Francez, N. (2016). The Lambek calculus extended with intuitionistic propositional logic. Studia Logica, 104, 1051–1082.CrossRefGoogle Scholar
Lambek, J. (1958). The mathematics of sentence structure. American Mathematical Monthly, 65, 154–170. (Also in Categorial Grammars, Buszkowski, W., Marciszewski, W., and van Benthem, J., editors. John Benjamins, Amsterdam, 1988.)Google Scholar
Troelstra, A. S. & Schwichtenberg, H. (2000). Basic Proof Theory. Cambridge: Cambridge University Press.CrossRefGoogle Scholar