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82 LIEFKE
(15) {w s | [[John e]] exists in w & for all p , if @ ∈ p &
p ‘is about’ [[John]], then w ∈ p}
= {w s | [[John e]] exists in w & for all p , if @ ∈ p &
p ‘is about’ [[John]], then w ∈ p} ∩ {w s | w ∈ [[John whistles ]]}
= {w s | w ∈ [[John runs ]] & w ∈ [[John whistles ]]}
Since the property ‘whistles’ is true of John at w 1, the strong type- representations of
John and ‘John whistles’ at w 1 are the same semantic object.
In light of our previous considerations, the identity of rich local representations of entities
and their associated true propositions is arguably desirable. However, our strategy for the
rich representation of propositions in the type also allows the modeling of information
growth.
To see this, consider the representation of the proposition ‘John is bald’ at w 1 (in (16)). The
latter is obtained through the extension of the set of true propositions about John at w 1 by the
information encoded in the proposition ‘John is bald’. The latter corresponds to the
elimination of those indices from the set of indices from (15) at which the proposition ‘John is
bald’ is either false or undefined:
(16) {w s | [[John e]] exists in w & for all p , if @ ∈ p &
p ‘is about’ [[John]], then w ∈ p} ∩ {w s | w ∈ [[John is bald ]]}
= {w s | w ∈ [[John runs ]] & w ∈ [[John whistles]] & w ∈ [[John is bald]]}
The above suggests the representation of (type-) properties of entities at a given
index by type- functions from the local representation of entities in the
property’s domain (here, the w 1-specific representation of John from (15)) to the local
representation of the result of attributing the property to the relevant entity in its domain
(here, the w 1-specific representation of the result of attributing ‘is bald’ to John, cf. (16)). The
latter corresponds to the result of obtaining the type- representation of John at a betterdefined
index w 3, that distinguishes itself from w 1 at most with respect to the obtaining of the
proposition ‘John is bald’.
Our considerations from the preceding paragraphs show that the possibility of modelling
information growth in single-type semantics is conditional on the existence of a definedness
order on indices: If all indices in the domain of the type s are totally defined (such that, for all
indices w and propositions p, either w ∈ p or w ∈ ¬p), all single-type representations of
extensional properties are associated with improper extensions, that send single-type
representations of entities at some index to themselves (if the entity has the property at the
respective index) or to the empty set of indices (otherwise). But this makes it impossible to
capture the informativeness of propositions in single-type semantics.
To prevent the triviality of type- representations of propositions, we require the
existence of underdefined indices (so-called partial possible worlds, or possible situations
(Barwise and Perry, 1983; Kratzer, 1989)), that can be extended into totally defined possible
worlds. In virtue of the above, the type is only suitable as the semantic basis for
natural language if its objects are associated with functions from partial indices to functions
from partial indices to partial truth-values (or with functions from partial indices to
functions from total indices to total truth-values, see below).
In contrast to the above, the suitability of the single basic type is not conditional on the
adoption of partial indices or truth-values. This is due to the possibility 7 of representing
7
This holds modulo the absence of an algebraic structure on objects of this type (cf. Req. 2).