According to Michael Glanzberg's "Context and Unrestricted Quantification", quantifiers always have to be understood as ranging over some contextually given domain; and paradoxes like Russell's show that, 'for any given context, there is a distinct context which provides a wider domain of quantification'. So he is defending 'a contextualist version of the view that there is no absolutely unrestricted quantification'.
The aim of this paper, however, isn't to directly defend the contextualist thesis as the best response to the paradoxes (Glanzberg has argued the case elsewhere), so much as to explore more closely how best to articulate the thesis, and in particular to explore how the idea that quantifiers always have to understood in terms of a background domain which is set contextually relates to more common-or-garden cases of quantifier domain restriction.
Consider, for example,
1. Every graduate student turned up to the party, and some undergraduates did too.
2. Everyone left before midnight.
In the first case there is are explicitly restricted quantifiers. But we of course don't mean every graduate student in the world turned up: there is also a contextually supplied restriction to e.g. students in the Cambridge philosophy department. In the second case, context does all the restricting -- e.g. to the people at the party.So far, so familiar. But what about
3. Absolutely everything that exists, without exception, is self-identical?
Here there is no explicit restriction to a subclass of what exists; nor need there be any common-or-garden-contextual restriction of the ordinary kind. Still, Glanzberg wants to say, in any given context there is a background domain ('the widest domain of quantification available' in that context). This is the domain over which quantifications as in an occurrence of (3) range, when there is no explicit restriction and no common-or-garden-contextual restriction. And, the argument goes, there is a kind of contextual relativity in fixing this background domain (so, in a sense, the likes of (3) involve contextually relative although unrestricted quantifiers):
Whereas the absolutist holds there is one fixed background domain, which is simply 'absolutely everything', the contextualist holds that different contexts can have different background domains.
But of course the contextualist needs to say more than that: it isn't just that different contexts might give different extensions to 'absolutely everything', it is also the case that there is no way of setting up a 'maximal' context in which our quantifiers do succeed in being maximally, unexpandably, all-embracing. For example, the contextualist must say that, even given a context of sophisticated philosophical reflection, when -- in fall awareness of the issues -- we essay a claim like
4. Absolutely everything that might fall under our quantifiers in any context whatsoever is self-identical,
we still somehow must fall short of our aim, because the context can be changed in a way that will expand what counts as everything. But how plausible is this? Well, we'll have to see how the explanations develop over the rest of the paper.