v`v p`p
v0 ' v0 v,P00 ' P00(v)
v, R, v0 ` R(v0)(v)
v,R ' λv0.R(v0)(v)
[\L]
[/L]
[/R] -η-;
p0 ` p0
v,R,Q ' Q(λv0.R(v0)(v))
R,Q ' λv.Q(λv0.R(v0)(v)) [\ ]
[\L]
p00 ` p00
P ' P Qf,R,Q ' Q0(λv.Q(λv0.R(v0)(v)))
P0 ' P0 D, P,R,Q ' D(P)(λv.Q(λv0.R(v0)(v)))
D, P, R, D0, P0 ' D(P)(λv.D0(P0)(λv0.R(v0)(v))) [/L]
[/L]
[/L]
Figure 2: Semantic tree for ‘a man adores a woman’
4 Boxes in Type Logic
In this section I will show that there is a natural way to emulate the DRT
language in the first-order part of type logic, provided that we adopt a few
axioms. This possibility to view DRT as being a fragment of ordinary type
logic will enable us to define our interface between Categorial Grammar and
DRT in the next section.
We shall have four types of primitive objects in our logic: apart from
the ordinary cabbages and kings sort of entities (type e) and the two truth
values (type t) we shall also allow for what I would like to call pigeon-holes or
registers (type π) and for states (type s). Pigeon-holes, which are the things
that are denoted by discourse referents, may be thought of as small chunks
of space that can contain exactly one object (whatever its size). States may
be thought of as a list of the current inhabitants of all pigeon-holes. States
are very much like the program states that theoretical computer scientists
talk about, which are lists of the current values of all variables in a given
program at some stage of its execution.
In order to be able to impose the necessary structure on our models, we
shall let V be some fixed non-logical constant of type π(se) and denote the
inhabitant of pigeon-hole u in state i with the type e term V(u)(i). We
define i [u1 . . . un]j to be short for
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