Analyzing the core of categorial grammar

C. Areces and R. Bernardi. Analyzing the core of categorial grammar. Journal of Logic, Language and Information, 13(2):121–137, 2004. Extended version of ``Analyzing the Core of Categorial Grammar'' (Areces and Bernardi).

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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(3) (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 3 and 2↓ introduced by Moortgat in (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.

BibTeX

@Article{Areces2004,
  author =       "C. Areces and R. Bernardi",
  journal =      "Journal of Logic, Language and Information",
  title =        "Analyzing the core of categorial grammar",
  year =         "2004",
  note =         "Extended version of ``Analyzing the Core of Categorial
                 Grammar'' (Areces and Bernardi).",
  number =       "2",
  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(3) (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 3 and 2↓ introduced by Moortgat in
                 (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.",
  pages =        "121--137",
  volume =       "13",
}