Analyzing the Core of Categorial Grammar

C. Areces and R. Bernardi. Analyzing the Core of Categorial Grammar. In Proceedings of ICoS-3, Siena, Italy, 2001.

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Abstract

Even though residuation is at the core of Categorial Grammar [11], it is not always immediate to realize how standard logic systems like Multi-modal Categorial Type Logics (MCTL) [17] actually embody this property. In this paper we focus on the basic system NL [12] and its extension with unary modalities NL(3) [16], and we spell things out by means of Display Calculi (DC) [3, 10]. The use of structural operators in DC permits a sharp distinction between the core properties we want to impose on the logic system and the way these properties are projected into the logic 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 [16] 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 [5]: Galois and dual Galois connections. Again, DC let us readily define logic calculi capturing them, and we will discuss different possibilities in which the logic operators so obtained can interact with those obtained from residuation. We also provide preliminary ideas on how to use them when modeling linguistic phenomena.

BibTeX

@InProceedings{Areces2001a,
  author =       "C. Areces and R. Bernardi",
  booktitle =    "Proceedings of ICoS-3",
  title =        "Analyzing the Core of Categorial Grammar",
  year =         "2001",
  address =      "Siena, Italy",
  abstract =     "Even though residuation is at the core of Categorial
                 Grammar [11], it is not always immediate to realize how
                 standard logic systems like Multi-modal Categorial Type
                 Logics (MCTL) [17] actually embody this property. In
                 this paper we focus on the basic system NL [12] and its
                 extension with unary modalities NL(3) [16], and we
                 spell things out by means of Display Calculi (DC) [3,
                 10]. The use of structural operators in DC permits a
                 sharp distinction between the core properties we want
                 to impose on the logic system and the way these
                 properties are projected into the logic 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 [16] 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 [5]: Galois and dual Galois connections.
                 Again, DC let us readily define logic calculi capturing
                 them, and we will discuss different possibilities in
                 which the logic operators so obtained can interact with
                 those obtained from residuation. We also provide
                 preliminary ideas on how to use them when modeling
                 linguistic phenomena.",
}