Abstract
An adaptation of string-semantic models to the Polymorphic Lambek Calculus is studied. We note that if quantifiers range over arbitrary sets of strings, the polymorphic calculus is incomplete. The semantics is refined so that quantifiers range over sets of strings that are the interpretation of categories, and we prove a completeness result. In addition some other models are considered and there are speculations on the relevance of model theory to linguistics.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
V. Michele Abrusci. Sequent calculus for intuitionistic linear propositional logic. In P.P.Petkov, editor, Proceedings of the Summer School and Conference on Mathematical Logic. Plenum Press, 1990.
Kazimierz Ajdukiewicz. Die syntaktische konnexität. Studia Philosophica, 1, 1935.
Yehoshua Bar-Hillel. A quasi arithmetic notation for syntactic description. Language, 29, 1953.
Wojczech Buszkowski. Compatibility of a categorial grammar with an associated category system. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 28, 1982.
Wojczech Buszkowski. Completeness results for lambek syntactic calculus. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 32, 1986.
Martin Emms. Logical ambiguity. PhD Thesis, Centre of Cognitive Science, Edinburgh, 1992.
Martin Emms. Extraction covering extensions of the lambek calculus are not context free. In P. Dekker and M. Stokhof, editors, Proceedings of Ninth Amsterdam Colloquium, pages 268–286, 1993.
Martin Emms. Parsing with polymorphism. In Sixth Annual Conference of the European Association for Computational Linguistics, 1993.
Martin Emms. Some applications of categorial polymorphism. In DYANA Deliverable R1.3.A. ILLC, University of Amsterdam, 1993.
Martin Emms. Movement in labelled and polymorphic calculi. In Proceedings of 1st Rome Workshop on Linear Logic and Categorial Grammar, 1994.
Martin Emms. Undecidability result for polymorphic lambek calculus. In Tenth Amsterdam Colloquium Proceedings, 1995.
Martin Emms and Hans Leiß. Cut elimination in the second order lambek calculus. In DYANA Deliverable Rl.l.A. ILLC, University of Amsterdam, 1993.
J.-Y. Girard. Linear logic. Theoretical Computer Science, 50(1–102), 1987.
J-Y. Girard. Linear logic: Syntax and semantics. In J.-Y.Girard, Y.Lafont, and L. Regnier, editors, Advances in Linear Logic, London Mathematical Society Lecture Note Series. Cambridge University Press, 1995. to appear.
Leon Henkin. Completeness in the theory of types. Journal of Symbolic Logic, 15:81–91, 1950.
Joachim Lambek. The mathematics of sentence structure. American Mathematical Monthly, 65:154–170, 1958.
Mitsuhiro Okada. Phase semantics for higher order completeness, cut-elimination and normalization proofs. In J.-Y. Girard, M. Okada, and A. Scedrov, editors, Linear'96, volume 3 of Electronic Notes in Computer Science. Elsevier, 1996. (available from http://pigeon.elsevier.nl/mcs/tcs/pc/menu.html).
Mati Pentus. Lambek grammar is l-complete. available by ftp, 1993.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Emms, M. (1997). Models for polymorphic Lambek Calculus. In: Retoré, C. (eds) Logical Aspects of Computational Linguistics. LACL 1996. Lecture Notes in Computer Science, vol 1328. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0052157
Download citation
DOI: https://doi.org/10.1007/BFb0052157
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63700-4
Online ISBN: 978-3-540-69631-5
eBook Packages: Springer Book Archive