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SINGLE-TYPE ONTOLOGY 73
(b) The type α is related† to some derived type , o> or ( # o), where β 1, . . . , β n are single-type types.
Given the existence of a unique single-type representation for every basic Montagovian object
(clause (a)), the rule ST ensures the existence of a unique single-type representation for every
Montagovian object of a complex type. As a result, it suffices for a demonstration of the
satisfaction of the Representability requirement to show that (2a) obtains.
We will give concrete examples of the success and failure of the representability of single-type
objects in Sections 3.2 and 3.3, respectively. The next subsection presents the requirement of
their intensionality (Property 3).
2.3. Intensionality
The intensionality requirement on single basic types is a response to the granularity problem
for linguistic meanings from (Frege 1892). The latter concerns the fact that interpretations of
natural language expressions do not individuate semantic objects as finely as speakers’
intuitions about strict synonymy, to the effect that there are too few intensions to enable
correct predictions about linguistic entailment (Muskens 1995).
Most logics for natural language adopt a version of the axiom scheme of Extensionality from
(Ext), where the variables X and Y range over objects of some type and
where the variables x 1, . . . , x n have the Montague types α 1, . . . , α n:
Extensionality (Ext)
∀X ∀Y. ∀ x 1 . . . x n (X (x 1) . . . (x n) ↔ Y (x 1) . . . (x n)) → X = Y)
As a result, models of such logics identify all objects of some type (or
) that are logically equivalent. For the case of (type-) propositions,
these models identify all propositions which have the same truth-values across all indices.
Thus, the proposition ‘John seeks a unicorn’ will be treated as identical to the propositions
‘John seeks a unicorn and 1 3 + 12 3 = 9 3 + 10 3 ’ and ‘John seeks a unicorn and Bill walks or does
not walk’. 2
In an extensional setting (where we are concerned with a description of the actual physical
world, and assume the availability of all relevant facts about this world), the identification of
the above propositions is unproblematic. Thus, the inference from (1a) to (1b) or (1c) is intuittively
valid, where @ denotes planet Earth on Wednesday 12th December, 2012 at 10:52am.
(1) a. At @, John seeks a unicorn.
b. At @, John seeks a unicorn and 1 3 + 12 3 = 9 3 + 10 3 .
c. At @, John seeks a unicorn and Bill walks or does not walk.
However, in epistemic contexts, the extension of our commitment from a single proposition
to the set of its semantic equivalents is much less-warranted. This is due to the fact that a
cognitive agent may possess only partial information about the physical world, such that
(s)he may assume the truth of one, but not of another proposition. The substitution salva non
veritate of the proposition ‘John seeks a unicorn’ in the complement of the verb believes in
(2a) by any of the propositions from (1b) or (1c) (in (2b), resp. (2c)) illustrates the special
status of such contexts:
(2) a. Mary believes that John seeks a unicorn.
b. Mary believes that John seeks a unicorn and 1 3 + 12 3 = 9 3 + 10 3 .
2
We will justify the equivalence of these three propositions in Section 2.4.