Presuppositions and Pronouns - Nijmegen Centre for Semantics

44 Presuppositions and Pronouns
The second of these definitions is straightforward. It says that a set of DRSs
K is consistent iff a model and an embedding function can be found for ~K. ®K.
The definition of entailment in (18) says that K entails 9 q> iff, for every model
M and every function / that embeds ~K ®K into M, we can find a function g
which extends/and / embeds the merge of K and 9 q> into M. This definition (or
something like it) is required in view of the DRS semantics in (16). The point
is that a function can never embed a DRS into any model if the function is
too big. For example, if/is a function which maps x onto a tomato and y onto
a potato, then/can / never be an embedding for [x: tomato x], because this
DRS doesn't say anything about y. However, although no function which
embeds [x, y: tomato x, potato y] into a given model can ever be extended to
a function which embeds [x: tomato x], we do want the set {[x: tomato x],
[y: potato y]} to entail [x: tomato x], and it is for this reason that we take the
merge of the premise set and [x: tomato x], which in this case is [x, y: tomato
x, potato y].
What I have presented in the foregoing is essentially the standard version
of DRT. In the remainder of this section I will first reformulate the semantics
given in (16), and then extend the DRS language with a possibility operator.
The reason for doing this is that I want to make preparations for the
extensions to DRT that will be proposed in Chapters 5 and 6.
In (16) the semantics of the DRS language is given by stating what it means
for a function to embed a DRS in a model. That is, we have defined M 1=^-9, F= fq>,
for any M, /, and q>. 9. Given this relation, we might say that, relative to an
embedding function/, each DRS denotes a set of functions g which extend/
and which embed q> 9 in M. For example, [[x: banana x]]M,f(the M /. denotation of
the DRS [x: banana x] in the model M relative to the functionj) function/) would be the
set of functions which extend/, whose domain is {x}, and which map x onto a
banana in M (this set may be empty, of course). In Chapter 5,1 I will define an
extended version of DRT whose semantics will be given directly in terms of
[.], I.], and therefore I want to show here how the same can be done for the
standard DRS language:
(20) DRS semantics (second version)
Let M = (D, I) be a model, where D is some domain and I is an
interpretation function. Let / be a partial function from the set of
reference markers to D. [q>]M,fis M ,is either lor 1 0 if 9 q> is a condition, and
a (possibly empty) set of embedding functions if q> 9 is a DRS.
a. [q>]M,f= [cp] M f = {g I / c ~ g and dom(g) = Acc(9) Acc(q» and
V'I' Vxj/e E Con(9):[\|/] q»: ['I']M, M;g g = =l} I}
b. [PU Ml 1
... Un]M,f uJ Mt/ = 1 iff (j(u > ,/(u n ) E e I(P)
c. [u[ M = v]M,f M/ =liff/(u)=/(v)
= 1 = lev)