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NON-NOMINAL QUANTIFICATION 35
interpretation of the analysis of truth by employing sentential quantification in the
satisfaction conditions as well:
A proposition α satisfies the open sentence ‘The Pythagorean Theorem = [P] & P’ if and
only if the Pythagorean Theorem is identical with the proposition that things are as α
says they are and things are as α says they are. 2 Alternatively: A proposition α
satisfies the open sentence ‘The Pythagorean Theorem = [P] & P’ if and only if, for
some S, α says that S, where the Pythagorean Theorem is the proposition that S, and S.
However, if one is willing to tolerate non-nominal quantification in the satisfaction conditions
then one might feel there is no need to appeal to a domain of objects in the first place. For one
can avoid mention of properties and propositions and account for any form of quantification
in the object language by using the same form of quantification in the metalanguage. Hugly
and Sayward (1996: 303–16) and Williamson (1999) have developed this approach.
Of course, one may think that truth and exemplification need not be explained in terms of
non-nominal generality: one might prefer to give an alternative account in terms of sets, for
example. A set-theoretic account, however, is not suitable for all purposes. Neo-Fregeans
employ non-nominal quantification in their attempted reduction of arithmetic to logic: if,
however, the semantics of non-nominal quantification is set-theoretic then the mathematics
is in the logic from the start. 3 The second problem with set-theoretic accounts would be that
one of the goals of a semantics for non-nominal quantification is to reveal the structure of our
language and thereby tell us something about our linguistic capacities. An account in which
the semantic values of predicates are given extensionally will not explain what speakers of a
language know when they understand a predicate. For speakers must understand that what
the objects in the extension of a predicate ‘F’ have in common is that that they are all F.
The difficulty of avoiding implicit non-nominal quantification in the satisfaction conditions
becomes more apparent when we consider intensional contexts. Note that in the satisfaction
conditions Künne gave for sentential quantification the appearance of a sentential variable in
the context of a that-clause was treated differently to its appearance outside of a that-clause: a
proposition satisfies ‘a = [P]’ if and only if it is identical with the referent of ‘a’; a proposition
satisfies ‘P’ if and only if it is true. Now, there is nothing wrong with this in itself, but it
illustrates the fact that while we might be able to explain truth in terms of non-nominal
quantification, we cannot explain non-nominal quantification in terms of truth. If both
appearances of the variable were treated the same way then ‘a = [P]’ would be satisfied by a
proposition α if and only if the referent of ‘a’ is identical with the proposition that α is true.
Clearly this is not what is intended: if the referent of ‘a’ is not a proposition about a
proposition then no proposition in the domain will satisfy the open sentence. It is for that
reason that the semantics must give the additional rule for appearances of a sentential
variable in the context of a that-clause: a proposition satisfies ‘a = [P]’ if and only if it is
identical with the referent of ‘a’.
However, this rule suffices only for the occurrence of a single sentential variable in a thatclause.
Were we to extend our language further, for instance to accommodate expressions
such as ‘the proposition that p and q’, we should need additional recourse to non-nominal
quantification in order to define new relations in the metalanguage. ‘[p & q]’ should refer to
the proposition expressed by ‘p & q’ but we must define what proposition this is in terms of
the semantic values of ‘p’ and ‘q’. One systematic way to do this would be to give designation
conditions to supplement the satisfaction conditions: the referent of ‘[φ & ϕ]’ is the
proposition which results from the conjunction of the proposition expressed by ‘φ’ with the
2
Like Künne I take the expressions here italicised to function as prosentences.
3
See Wright 2007 for a discussion of this point as well as an interesting suggestion as to how we should
understand interpretations of the quantifiers.