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74 LIEFKE
c. Mary believes that John seeks a unicorn and Bill walks or does not walk.
To block invalid inferences of the above form, formal semanticists have, in the last 30 years,
sought for more strongly intensional notions of proposition, whose objects exhibit more finegrained
identity criteria. In particular, such notions have been developed in the frameworks
of ‘structured meanings’ theories (Cresswell 1985), Situation Semantics (Barwise and Perry
1983), Data Semantics (Landman 1986), Property Theory (Chierchia and Turner 1988),
impossible world semantics (Rantala 1982), and partial type theory (Muskens 1995).
Our stipulation of the intensionality requirement on single-type candidates is motivated by
the characterization of single-type semantics as a theory of meaning for natural language,
that therewith also models epistemic statements of the form from (2a) to (2c).
We finish our discussion of single-type requirements with a presentation of the requirement
of partiality (Property 4). The latter concerns the possibility of leaving some single-type
objects underdefined, such that they can be extended into better-defined, total objects.
2.4. Partiality
In single-type semantics, partiality serves double duty as a strategy for the obtaining of finegrained
linguistic meanings, and for the modelling of information growth. On its former use,
the adoption of partial single-type objects constitutes a means of satisfying the intensionality
requirement from Section 2.3, that follows the approach of partial type theory. On its latter
use, the adoption of partial single-type objects constitutes a means of accommodating the
dynamics of linguistic meanings (cf. van Benthem 1991).
We discuss the two rationales for the adoption of partial single-type objects below. To prepare
their presentation, we first provide a brief characterization of partial objects (or functions).
We then show that the properties of dynamicity and intensionality arise naturally from the
property of partiality.
The characterization of single-type objects as partial objects relates to their algebraic
structure – in particular to the identity of the type-t domain. Thus, since the ‘ingredient’-type
t of the type for propositions is classically associated with the two truth-values true (T)
and false (F), objects of some type are taken to be total (or
Boolean) functions, that obey the law of Excluded Middle. As a result, one can directly obtain
a function’s complement from the function itself.
Our description of single-type objects as partial objects involves a generalization of the set of
truth-values to the set {T, F, N}, where N is the truth-valuationally undefined element
(neither-true-nor-false). As a result of its introduction, some functions of the type will send certain arguments to the truth-value N and do, thus, fail to satisfy
the law of Excluded Middle (Fact 1). The partiality of the set of truth-values further induces a
definedness ordering on all partial single-type domains (Fact 2). We will see below that our
use of partial single-type objects for the obtaining of fine-grained linguistic meanings uses
Fact 1. Our use of partial single-type objects for the modelling of information growth employs
Fact 2.
We will demonstrate the dynamicity of the partial single-type objects in Section 3.4. Their
intensionality (cf. Sect. 2.3) is illustrated below:
The fine-grainedness of partial single-type objects is a consequence of our adoption of the
partial set of truth-values {T, F, N}. Our consideration of the propositions ‘John seeks a unicorn’
and ‘John seeks a unicorn and Bill walks or does not walk’ from Section 2.3 illustrates
this point: The logical equivalence of these propositions is conditional on the adoption of the
law of Excluded Middle in the underlying logic, and the attendant possibility of identifying
propositions of the form (p∨¬p) with universally true propositions. Thus, the proposition
‘John seeks a unicorn’ is only equivalent to the proposition ‘John seeks a unicorn and Bill