“Quantifier variance” is a term generally credited to Eli Hirsch, who in turn claimed inspiration from Hilary Putnam. It’s a version of ontological pluralism which poses questions about whether terms like “exists”, “there are”, “object”, and “Ǝx” necessarily have privileged meanings or designations.

Here’s a good description from Hale and Wright (2009):

Quantifier-Variance is the doctrine that there are alternative, equally legitimate meanings one can attach to the quantifiers – so that in one perfectly good meaning of ‛there exists’, I may say something true when I assert ‛there exists something which is a compound of this pencil and your left ear’, and in another, you may say something true when you assert ‛there is nothing which is composed of that pencil and my left ear’. And on one view – perhaps not the only possible one – the general significance of this variation in quantifier meanings lies in its deflationary impact on ostensibly head-on disagreements about what kind of objects the world contains: [it may be] a matter of their protagonists choosing to use their quantifiers (and other associated vocabulary, such as ‛object’) to mean different things – so that in a sense they simply go past each other. — Bob Hale and Crispin Wright

So this is a way to clarify or dissolve problems in mereology and, perhaps even more basically, to help resolve ancient quarrels between nominalists and Platonists about abstracta. Do numbers exist? It depends on how you restrict/define your quantifiers, and there is no correct or privileged way to do this, says the proponent of quantifier variance. This is the “deflationary impact” that Hale and Wright talk about: It reduces ontology to something much less than it appears. As Matti Eklund puts it here: “Both Hirsch and Putnam take their ontological pluralism to entail that ontological questions are shallow. Hirsch says for instance that the proponent of quantifier variance ‛will address a typical question of ontology either by shrugging it off with Carnapian tolerance for many different answers, or by insisting with Austinian glee that the answer is laughably trivial.’” In other words, there are either many, equally good ontologies that will allow the scientist and/or philosopher to accomplish what she wants to accomplish, or else the whole question is verbal, and silly -- “it depends what you mean,” and you can mean whatever is consistent, or at least acceptable to your fellow language-users.

This is a super-rough sketch, but I hope good enough to pose the meta-problem I want to raise. To set it up, let me bring in Theodore Sider, not a fan of slapdash quantifier variance. He says, in "Ontological Realism": “Every serious theory of the world that anyone has ever considered employs a quantificational apparatus, from physics to mathematics to the social sciences to folk theorists. Quantification is as indispensable as it gets.” No one, Sider says, can avoid “choosing fundamental notions with which to describe the world.” If your fundamental notion is that “reality” and “the world” and “fundamental” are non-objective, that is still a fundamental notion. However, you could claim to “accept the idea of structure as applied to logic but deny that there is distinguished quantificational structure in particular.” Sider calls this “in essence, quantifier variance,” and he doesn’t think it works.

Can I accept logical structure (aka the rules of logic or rationality or inferential validity) while leaving empty – or at least unspecified – the “existence stuff” we want to plug into our ontology? Here’s another way of asking the question: Does logical structure entail ontological commitments about things like grounding, “simples,” existence, Ǝx, and other tools of the trade? Is there any way that such commitments can be rescued from the charge that they’re either pragmatic or merely verbal? And yet a third, very simple way to pose the question: Can quantifiers really “mean different things,” as long as inferential rules and other logical apparatus are respected?

I think this question, in whatever version, is a hard one. Just to give a taste of the kind of complexity involved: Borrowing an example from Sider, let’s say I am a non-native English speaker who has recently learned the language. I mistakenly believe that the word for “number” is “fish”. You and I have a conversation in which we discover we’re both nominalists. You say, correctly from our shared point of view, “numbers do not exist”. I agree with you, saying “fish do not exist”. Sider claims, I think rightly, that this is not a “verbal dispute” in the classic sense of two people talking past each other because they use words differently. You and I both mean the same thing – we are each thinking the same thing about numbers – but I have made a verbal mistake. Presumably, genuine disagreements between languages can’t be analyzed and resolved in this way. And what about disagreements about quantifiers? (This is me now, not Sider.) If I say “mereological composites exist” and you say “there is no such thing as a mereological composite”, which kind of dispute is going on? Are we disagreeing about concepts, while using the same words? Or are we holding the concept of “existence” steady, while (someone is) making a mistake in terminology? How could we know which of us is making the mistake?!

I don’t know nearly enough about quantifier variance to have solid positions on any of this. I know just enough to realize that the question is of critical importance for ontology. So I’d welcome any responses, especially from philosophers who have been wrestling with this for a while. Can we have quantifiers without “distinguished metaphysical status”?