Analyzing the Core of Categorial Grammar
C Areces, R. Bernardi

Abstract:
Even though residuation is at the core of Categorial Grammar, it is
not always immediate to realize how standard logic systems like
Multi-modal Categorial Type Logics (MCTL) actually embody this
property. In this paper we focus on the basic system NL and its
extension with unary modalities NL(\Diamond), and we spell things out
by means of Display Calculi (DC).  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 \backslash, / and \bullet of binary operators, and
how the operators \Diamond and \Box^\downarrow introduced by Moortgat
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: 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.