only if we use the quantifier itself in the description. For example,“some”is a relation
that obtains between F and G just in case there is something that is both F and G, and
“most”is a relation that obtains between F and G (in that order) just in case most Fs
are Gs. But many of these relations will be such that there are no natural language
expressions that have them as their semantic values. Perhaps some of these relations
are among those perfectly natural meanings for an existential quantifier.^50
It is acceptable for natural language semanticists to talk as if there are such
relations when providing models for the ways in which meaningful expressions in
natural language combine with each other to form larger meaningful units. We
shouldn’t begrudge these incursions into metaphysics any more than we should
complain when a physicist talks as though she were a Platonist when providing a
mathematical model of reality. But we are doing metaphysics now, and among our
metaphysical interlocutors are those who do not think that there are entities that
correspond to quantifier-expressions. So let’s consider a more neutral characteriza-
tion of what it is to be a quantifier in order to accommodate them.
Certain kinds of quantifiers, such as the quantifier“some,”can be characterized by
their inferential role. From the truth of“Ranger is a dog,”one may infer that
“something is a dog”is also a truth. And from the truth of“something is a dog”
one may infer that“not everything is not a dog”is also a truth. To be an existential
quantifier is to be a kind of expression that permits a certain range of inferences.
1.5.2 More on Quantification
Here we will discuss what to say if unrestricted quantification is impossible and if
either plural quantification or higher-order quantification is fundamental.
First, someone could believe that unrestricted quantification is impossible but also
hold that there are ways of being, perhaps for entirely independent reasons. There are
interesting worries about the coherence of quantifying over absolutely everything.^51
Suppose you believe in sets. Suppose you hold that whenever there are some things,
there is a set of those things, that is, if some things are in the domain of quantifica-
tion, there is a set of those things. You will be led to a contradiction if you assume
then that you can quantify over all the sets there are, for on these hypotheses one can
quantify over all of the sets that there are only if there is a set of all the sets that there
are; and there are proofs that there can be no such set on pain of contradiction.^52 You
might hold instead that, for every quantifier Q1, there is a more inclusive quantifier
Q2 that ranges over everything Q1 ranges over but not vice versa. On this view, every
quantifier is a restricted quantifier. And so any natural quantifier will be a restricted
(^50) I have been told that many linguists deny that“there is”is a quantifier. I am largely indifferent to
whether this is the case: although I will sometimes speak as if“there is”is a quantifier, I am happy to replace
sentences in which 51 “there is”is discussed with ones in which“some”is the object of focus.
52 See the papers in Rayo and Uzquiano (2007) for a discussion of the relevant issues.
Perhaps the thing to do is embrace a paraconsistent logic. See Priest (2006) for further motivation.
Paraconsistent logic will be discussed very briefly in chapter 2.