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Analyzing the Core of Categorial Grammar

Carlos Areces 1 Raffaella Bernardi 2
1 LANGUE ET DIALOGUE - Human-machine dialogue with a significant language component
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
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.
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Submitted on : Friday, September 23, 2005 - 4:36:17 PM
Last modification on : Friday, February 4, 2022 - 3:24:55 AM

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Carlos Areces, Raffaella Bernardi. Analyzing the Core of Categorial Grammar. Journal of Logic, Language and Information, Springer Verlag, 2001, Journal of Logic, Language and Information, 13 (2), pp.121--137. ⟨10.1023/B:JLLI.0000024730.34743.fa⟩. ⟨inria-00000292⟩



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