Quantifier variance, semantic collapse, and “genuine” quantifiers

Abstract

Quantifier variance holds that different languages can have unrestricted quantifier expressions that differ in meaning, where an expression is a “quantifier” just in case it plays the right inferential role. Several critics argued that J.H. Harris’s “collapse” argument refutes variance by showing that identity of inferential role is incompatible with meaning variance. This standard, syntactic collapse argument has generated several responses. More recently, Cian Dorr proved semantic collapse theorems to generate a semantic collapse argument against variance. The argument is significantly different from standard collapse, so it requires a new response. Here I clarify and analyze the semantic collapse argument, and explain how variantists can and should respond to it. The paper also includes an appendix showing the difficulties of positing identity variance without quantifier variance. The argument in the appendix has yet to appear in print, but is familiar to specialists.

Appendix: Mere identity variance

Suppose there is a language L such that L-speakers intuitively accept more objects than we English speakers do. More precisely, assume that English speakers reject and L-speakers accept a first-order counting sentence, “there are at least n things” for some natural number n. All finite counting sentences can be expressed in first-order logic with identity. For simplicity and with no loss of generality, assume that $n=3$:

$$\begin{aligned} (\mathrm {three})\qquad \exists x\exists y\exists z(x\ne y\wedge x\ne z\wedge y\ne z) \end{aligned}$$

According to the charity-based metasemantics which is key to quantifier variance, $(\mathrm {three})$ is false in English but true in L. And we can, in English, admit that $(\mathrm {three})$ is true in L. According to standard quantifier variance, we can admit this because the quantifiers (along with variables and identity predicate) differ in meaning between L and English.

Consider an alternative diagnosis: the quantifiers and variables (and conjunction and negation signs) mean the same thing in these two languages, only the identity symbol differs in meaning. This is supposed to capture all of the important claims made by quantifier variantists but without the quantifiers varying in meaning. The trouble is that this kind of identity variance is very difficult to maintain without also accepting some form of quantifier variance.

Assume mere identity variance for reductio. As English speakers, we admit that $(\mathrm {three})$ is true in L. This doesn’t commit us to thinking that $(\mathrm {three})$ is true in English, but it does commit us to thinking that the translation of $(\mathrm {three})$, from L into English, is true. To translate $(\mathrm {three})$ from L into English, we can introduce a two-place predicate “$=_{L}$”, stipulated to mean what “$=$” means in L. Minimally, this requires that “$=_{L}$” must be reflexive. This stipulation should work, since nothing about English conflicts with it. Since the L quantifiers and the English quantifiers mean the same thing, this stipulation merely introduces a new predicate over the same domain of quantification. Introducing a predicate like this can’t plausibly change the meanings of our English quantifiers.

This stipulation gives the following translation of $(\mathrm {three})$ from L into English$+$—English slightly extended by adding the predicate “$=_{L}$”:

$$\begin{aligned} (\mathrm {E\,three})\qquad \exists x\exists y\exists z(x\ne _{L}y\wedge x\ne _{L}z\wedge y\ne _{L}z) \end{aligned}$$

The assumption of charity—that identity variantists are committed to—commits us to $(\mathrm {E\,three})$ being true in English$+$. Since $(\mathrm {E\,three})$ is true, it must have a witness, a, and a true witnessing claim:

$$\begin{aligned} \exists y\exists z(a\ne _{L}y\wedge a\ne _{L}z\wedge y\ne _{L}z) \end{aligned}$$

And this claim must also have a witness, which can’t be a, by the reflexivity of “$=_{L}$”. So the witness is b, and the true witnessing claim is:

$$\begin{aligned} \exists z(a\ne _{L}b\wedge a\ne _{L}z\wedge b\ne _{L}z) \end{aligned}$$

This claim must also have a witness. So for some witnessing name “c”, we are committed to the truth of:

$$\begin{aligned} (a\ne _{L}b\wedge a\ne _{L}c\wedge b\ne _{L}c) \end{aligned}$$

The trouble is, by assumption, for English speakers, everything is either a or b, and not both. Yet the witness here violates this condition. It can’t be either a or b, and it can’t be something else without $(\mathrm {three})$ being true in English$+$ and so also in English. The assumption of mere identity variance have been reduced to absurdity.

The most plausible reaction is to fall back into quantifier variance. For instance, we English speakers might see the L-quantifiers as ranging over objects together with “guises” or the like. Since we are assuming that standard English quantifiers don’t work that way, translating L like this admits a form of quantifier variance. In general, non-homophonic translations of L's quantifiers seem to admit the central claims made by variantists.

Obviously, the reasoning here generalizes. The “smaller” ontological language will always have trouble accommodating the “bigger” ontological language without admitting quantifier variance. The above reasoning implicitly assumed that the names “a”, “b”, and “c” were not themselves objects in the range of our English quantifiers. This assumption can be avoided by considering disagreement over only non-syntactic objects.

How might mere identity variantists reply? They could try to claim that the introduction of “$=_{L}$”, and the corresponding move from English to English$+$, changed the meanings of the quantifiers. But as I already discussed, this isn’t very plausible. Relatedly, they could deny that the introduction of “$=_{L}$” is possible, but for the same reasons, this is implausible and unmotivated. And if the “smaller” language can’t translate the quantifiers of the “bigger” language homophonically, there is no motivation for saying, as the mere identity variantist does, that the quantifiers in both languages mean the same thing.

A follower of Dorr would likely deny this last claim and reply to the argument by denying the background assumption that quantifier expressions function the same way in both English and L. They would say that, yes, the quantifier expressions mean the same thing in both languages, but there is none-the-less a difference in function across languages, which is itself explained in some other way. There are a number of formal options available for making this out — perhaps “$\exists$”’s meaning is context sensitive, and L-uses and English-uses of “$\exists$” are always in different contexts. Or perhaps L and English have different semantic rules which activate different aspects of “$\exists$”’s shared meaning. These and other options are formally possible, but they divorce meaning from language use in a radical way. Put into these terms, what quantifier variantists care about is now “function within a language” rather than “formal meaning”. Once again we have a nice illustration of how the metasemantic focus of variantists differs from Dorr’s semantic focus. Other replies could be considered, but I think that all of the ways to coherently hold on to mere identity variance are unappealing.

This argument is related to early discussions by Hirsch, Sider, and others, but nothing exactly like it has been published. Jeffrey Russell gave a more compressed version of basically the same argument in a commentary on Aaron Segal’s paper “Identity Variance” at the 2010 meeting of the Eastern APA. I didn’t attend that meeting, but I discussed the issue with both authors in early 2011, having read both Segal’s paper and Russell’s commentary. Thanks to both of them, a decade later.