Abstract
Even though residuation is at the core of Categorial Grammar (Lambek, 1958), it is not always immediate to realize how standard logical systems like Multi-modal Categorial Type Logics (MCTL) (Moortgat, 1997) actually embody this property. In this paper, we focus on the basic system NL (Lambek, 1961) and its extension with unary modalities NL(♦) (Moortgat, 1996), and we spell things out by means of Display Calculi (DC) (Belnap, 1982; Goré, 1998). The use of structural operators in DC permits a sharp distinction between the core properties we want to impose on the logical system and the way these properties are projected into the logical operators. We will show how we can obtain Lambek residuated triple \, / and • of binary operators, and how the operators ♦and □↓ introduced by Moortgat (1996) are indeed their unary counterpart.
In the second part of the paper we turn to other important algebraic properties which are usually investigated in conjunction with residuation (Birkhoff, 1967): Galois and dual Galois connections. Again, DC let us readily define logical calculi capturing them. We also provide preliminary ideas on how to use these new operators when modeling linguistic phenomena.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abrusci, M., 1991, “Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic,” The Journal of Symbolic Logic 56, 1403-1451.
Ajdukiewicz, K., 1935, “Die Syntaktische Konnexität,” Studia Philosophica 1, 1-27. (English translation in Storrs McCall, ed., Polish Logic, 1920-1939, Oxford, 1996, pp. 207-231).
Areces, C., Bernardi, R., and Moortgat, M., 2001, “Galois connections in categorial type logic,” in Proceedings of FGMOL'01, R. Oherle and L. Moss, eds., Special Issue of the Electronic Notes in Theoretical Computer Science, Vol. 53.
Bar-Hillel, Y., 1953, “A quasi-arithmetical notation for syntactic description,” Language 29, 47-58.
Belnap, N., 1982, “Display logic,” Journal of Philosophical Logic 11, 375-417.
Bernardi, R., 2002, “Reasoning with polarity in categorial type logic,” Ph.D. Thesis, UiL OTS, University of Utrecht.
Bernardi, R. and Moot, R., 2002, “Scope ambiguities from a proof-theoretical perspective,” pp. 9-23 in Proceedings of ICoS-2, J. Bos and M. Kohlhase, eds., Special Issue of the Journal of Language and Computation, to appear.
Birkhoff, G., 1940, 1948, 1967, Lattice Theory, Providence, RI: American Mathematical Society.
Blyth, T. and Janowitz, F., 1872, Residuation Theory, New York: Pergamon Press.
Dunn, J., 1991, “Gaggle theory: An abstraction of Galois connections and residuation with applications to negation and various logical operations,” pp. 31-51 in JELIA 1990: Proceedings of the European Workshop on Logics in Artificial Intelligence, J. van Eijck, ed., Lecture Notes in Computer Science, Vol. 478, Berlin: Springer-Verlag.
Fuchs, L., 1963, Partially-Ordered Algebraic Systems, New York: Pergamon Press.
Goré, R., 1998, “Gaggles, Gentzen and Galois: How to display your favourite substructural logic,” Logic Journal of the IGPL 6, 669-694.
Goré, R., 1998, “Substructural logics on display,” Logic Journal of the IGPL 6, 451-504.
Lambek, J., 1958, “The mathematics of sentence structure,” American Mathematical Monthly 65, 154-170.
Lambek, J., 1961, “On the calculus of syntactic types,” pp. 166-178 in Structure of Languages and Its Mathematical Aspects, R. Jakobson, ed., Providence, RI: American Mathematical Society.
Lambek, J., 1993, “From categorial to bilinear logic,” pp. 207-237 in Substructural Logics, K.D.P. Schröder-Heister, ed., Oxford: Oxford University Press.
Lambek, J., 2001, “Type grammars as pregroups,” Grammars 4, 21-39.
Montague, R., 1974, Formal Philosophy: Selected Papers of Richard Montague, New Haven: Yale University Press.
Moortgat, M., 1996, “Multimodal linguistic inference,” Journal of Logic, Language and Information 5, 349-385.
Moortgat, M., 1997, “Categorial type logics,” pp. 93-178 in Handbook of Logic and Language, J. van Benthem and A. ter Meulen, eds., Cambridge, MA: The MIT Press.
Partee, B. and Rooth, M., 1983, “Generalized conjunction and type ambiguity,” pp. 361-383 in Meaning, Use, and Interpretation of Language, R. Bäuerle, C. Schwarze, and A. von Stechow, eds., Berlin, New York: De Gruyter.
Restall, G., 2000, An Introduction to Substructural Logics, London: Routledge.
van der Wouden, T., 1994, “Negative Contexts,” Ph.D. Thesis, University of Groningen.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Areces, C., Bernardi, R. Analyzing the Core of Categorial Grammar. Journal of Logic, Language and Information 13, 121–137 (2004). https://doi.org/10.1023/B:JLLI.0000024730.34743.fa
Issue Date:
DOI: https://doi.org/10.1023/B:JLLI.0000024730.34743.fa