Willard Van Orman Quine

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just mentioned, then quantification works whatever kind of
objects one quantifies over.
As we noted earlier, Church argued that one can quantify
into modal contexts provided the quantifier has an intensional
range – a range, for instance, composed of attributes
rather than classes. However, what saves his “logic of sense
and denotation” from collapse is not this feature but the
Frege-inspired reference shift that takes place within modal
contexts: What object a variable takes as value depends
on the modal operators (or, in Church’s case, modal predicates)
within whose scope it occurs. Thanks to this feature,
Church’s system does not have any opaque contexts. All its
contexts are referentially and extensionally transparent, and
there is no need to have a two-sorted semantics in order to
prevent a collapse of modal distinctions. Although it uses
symbols like ‘⊓’, Church’s system is not a system of modal
logic; instead, it is a purely extensional system.
Carnap proposes in Meaning and Necessity (1947) a system
of modal logic, S2, where, in effect, he interprets the
quantifiers as ranging over intensions. (This is not clear from
his presentation, since he operates with two “identity” relations,
identity of extension ‘≡’ and identity of intension
‘=’, where ‘a ≡ b’ is defined as ‘N(a ≡ b)’. However, only
identity of intension has the properties characteristic of an
identity relation.) Carnap states that “in order to avoid certain
complications, which cannot be explained here, it seems
advisable to admit in S, only descriptions which do not contain
‘N’.” 6 Carnap never mentioned what the complications
are. He may not have discovered that one of them was the
collapse of modal distinctions. If so, he might have seen, as
Quine saw later, that the root of the trouble is singular terms
that contain descriptive elements.
The first systems of quantified modal logic that were proposed
had no singular terms other than variables. Since variables
keep their reference from one possible world to another,
the collapse discussed by Quine was not brought to the fore
until one got systems of quantified logic that included singular
terms other than variables.
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