Abstract
Nonassociative Lambek Calculus (NL) is a pure logic of residuation, involving one binary operation (product) and its two residual operations defined on a poset [26]. Generalized Lambek Calculus GL involves a finite number of basic operations (with an arbitrary number of arguments) and their residual operations [7]. In this paper we study a further generalization of GL which admits operations whose arguments and values can be of different sorts. This logic is called Multi-Sorted Lambek Calculus mL. We also consider its variants with lattice and boolean operations. We discuss some basic properties of these logics (completeness, decidability, complexity and others) and the corresponding algebras.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
van Benthem, J.: Language in Action. Categories, Lambdas and Dynamic Logic. North Holland, Amsterdam (1991)
Bimbó, K., Dunn, J.M.: Generalized Galois Logics. Relational Semantics of Nonclassical Logical Calculi. CSLI Lecture Notes, vol. 188 (2008)
Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001)
Blyth, T.S.: Lattices and Ordered Algebraic Structures. Springer, London (2010)
Buszkowski, W.: Generative Capacity of Nonassociative Lambek Calculus. Bull. Pol. Acad. Scie. Math. 34, 507–516 (1986)
Buszkowski, W.: Lambek Calculus with Nonlogical Axioms. In: Casadio, C., Scott, P.J., Seely, R.A.G. (eds.) Language and Grammar. Studies in Mathematical Linguistics and Natural Language. CSLI Lecture Notes, vol. 168, pp. 77–93 (2005)
Buszkowski, W.: Interpolation and FEP for logics of residuated algebras. Logic Journal of the IGPL 19(3), 437–454 (2011)
Buszkowski, W.: Many-sorted gaggles. A Talk at the Conference Algebra and Coalgebra Meet Proof Theory (ALCOP 2012). Czech Academy of Sciences, Prague (2012), http://www2.cs.cas.cz/~horcik/alcop2012/slides/buszkowski.pdf
Buszkowski, W., Farulewski, M.: Nonassociative Lambek Calculus with Additives and Context-Free Languages. In: Grumberg, O., Kaminski, M., Katz, S., Wintner, S. (eds.) Francez Festschrift. LNCS, vol. 5533, pp. 45–58. Springer, Heidelberg (2009)
Casadio, C.: Agreement and Cliticization in Italian: A Pregroup Analysis. In: Dediu, A.-H., Fernau, H., Martín-Vide, C. (eds.) LATA 2010. LNCS, vol. 6031, pp. 166–177. Springer, Heidelberg (2010)
Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Elsevier (2007)
de Groote, P., Lamarche, F.: Classical Nonassociative Lambek Calculus. Studia Logica 71(2), 355–388 (2002)
Hanikowá, Z., Horčik, R.: Finite Embeddability Property for Residuated Groupoids (submitted)
Horčik, R., Terui, K.: Disjunction property and complexity of substructural logics. Theoretical Computer Science 412, 3992–4006 (2011)
Jäger, G.: Residuation, structural rules and context-freeness. Journal of Logic, Language and Information 13, 47–59 (2004)
Jónsson, B., Tarski, A.: Boolean algebras with operators. Part I. American Journal of Mathematics 73, 891–939 (1952)
Jónsson, B., Tarski, A.: Boolean algebras with operators. Part II. American Journal of Mathematics 74, 127–162 (1952)
Kaminski, M., Francez, N.: Relational semantics of the Lambek calculus extended with classical propositional logic. Studia Logica (to appear)
Kanazawa, M.: The Lambek Calculus Enriched with Additional Connectives. Journal of Logic, Language and Information 1(2), 141–171 (2002)
Kandulski, M.: The equivalence of nonassociative Lambek categorial grammars and context-free grammars. Zeitschrift f. Math. Logik und Grundlagen der Mathematik 34, 41–52 (1988)
Kołowska-Gawiejnowicz, M.: On Canonical Embeddings of Residuated Groupoids. In: Casadio, C., et al. (eds.) Lambek Festschrift. LNCS, vol. 8222, pp. 253–267. Springer, Heidelberg (2014)
Kołowska-Gawiejnowicz, M.: Powerset Residuated Algebras. Logic and Logical Philosophy (to appear)
Kozak, M.: Distributive Full Lambek Calculus has the Finite Model Property. Studia Logica 91(2), 201–216 (2009)
Kusalik, T.: Product pregroups as an alternative to inflectors. In: Casadio, C., Lambek, J. (eds.) Computational Algebraic Approaches to Natural Language, p. 173. Polimetrica, Monza (2002)
Lambek, J.: The mathematics of sentence structure. American Mathematical Monthly 65, 154–170 (1958)
Lambek, J.: On the calculus of syntactic types. In: Jakobson, R. (ed.) Structure of Language and its Mathematical Aspects, pp. 166–178. AMS, Providence (1961)
Lambek, J.: From Categorial Grammar to Bilinear Logic. In: Schroeder-Heister, P., Došen, K. (eds.) Substructural Logics, pp. 207–237. Clarendon Press, Oxford (1993)
Lambek, J., Scott, P.J.: Introduction to higher order categorical logic. Cambridge University Press, Cambridge (1986)
Lin, Z.: Nonassociative Lambek Calculus with Modalities: Interpolation, Complexity and FEP (submitted)
Moortgat, M.: Categorial Type Logic. In: van Benthem, J., ter Meulen, A. (eds.) Handbook of Logic and Language, pp. 93–177. Elsevier, Amsterdam (1997)
Moortgat, M.: Symmetric Categorial Grammar. Journal of Philosophical Logic 38(6), 681–710 (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Buszkowski, W. (2014). Multi-Sorted Residuation. In: Casadio, C., Coecke, B., Moortgat, M., Scott, P. (eds) Categories and Types in Logic, Language, and Physics. Lecture Notes in Computer Science, vol 8222. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54789-8_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-54789-8_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-54788-1
Online ISBN: 978-3-642-54789-8
eBook Packages: Computer ScienceComputer Science (R0)