Categorial Grammar and Discourse

The same semantical recipe can be used to obtain a translation for sentence
(6), we find it in (7). But (1) and (6) have alternative derivations in the
Lambek calculus too. Some of these lead to semantical recipes equivalent to
(2), but others lead to recipes that are equivalent to (8) (for more explana-
tion consult Hendriks [1993]). If we apply this recipe to the translations of
the words in (6), we obtain (9), the interpretation of the sentence in which
a² woman has a wide scope specific reading and is available for anaphoric
reference from positions later in the text.

I leave it to the reader to verify that the little text in (10) translates
as (11) by the same method (note that the stop separating the first and
second sentences is lexicalised as an item of category s\(txt/s)), and that
(12) translates as (13). A reader who has worked himself through one or two
of these examples will be happy to learn from Moortgat [1988] that there are
relatively fast Prolog programs that automatically find all semantic recipes
for a given sentence.

6 From Boxes to Truth Conditions

We now have a way to provide the expressions of our fragment automatically
with Discourse Representation Structures which denote relations between
states, but of course we are also interested in the truth conditions of a given
text. These we equate with the domain of the relation that is denoted by its
box translation (as is done in Groenendijk & Stokhof [1991]).

Theoretically, if we are in the possession of a box Φ, we also have its truth
conditions, since these are denoted by the first-order term λi∃j (Φ(i)(j)), but
in practice, reducing the last term to some manageable first-order term may
be a less than trivial task. Therefore we define an algorithmic function that
can do the job for us. The function given will in fact be a slight extension of
a similar function defined in Kamp & Reyle [1993].

First some technicalities. Define adr (Φ), the set of active discourse ref-
erents of a box Φ, by adr ([~u | ~γ]) = {~u} and adr (Φ ; Ψ) = adr (Φ) ∪ adr (Ψ).
Let us define [t/u]Γ, the substitution of the type e term t for the discourse
referent u in the construct of the box language Γ, by letting [t/u]u = t and
[t/u]u ⁰ = u⁰ if u⁰ 6= u; for type e terms t⁰ we let [t/u]t ⁰ = t⁰. For complex
constructs [t/u]Γ is defined as follows.

11

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