It is a pleasure, as always, to turn to a paper by Charles Parsons (a long time ago, his "Frege's Theory of Numbers" was one of the papers that grabbed me when I very first started philosophy and it helped get me really enthused by Frege's project). Nice too, in a volume of overlong papers that 'The Problem of Absolute Universality' sticks to a reasonable length.
In his first section, Parsons quickly reviews a few reasons for supposing that sometimes, at any rate, we aim to make claims that do involve absolutely general quantification -- offering the usual sort of cases. For example, there are logical examples like Everything is self identical (which surely is indeed intended to be about everything.) And there are more humdrum examples like There are no unicorns. (If we suppose 'there are' ranges only over some domain D, then the statement could be true even if there are unicorns outside D. Then, asks Parsons, "can we exclude this outcome short of admitting a D that is absolutely everything?").
So what are the problems about taking such cases at face value? Parsons takes the main problems to be logical in character, and more about them in due course. But first, in his second section, he discusses "a problem of a more metaphysical character".
Our problem is that for statements of an absolutely general kind to have a definite truth-value, it appears that there has to be a final answer to the question what objects there are, ... That is metaphysical realism.
If we are suspicious about metaphysical realism, so understood, we should therefore be suspicious about quantifiers which purporting to really capture everything, once and for all.
But why be suspicious about realism, so understood? Parsons mentions the possibility of going for a trope ontology rather than object/property ontology and ending up with a different catalogue of the fundamental constituents of the universe. But it isn't at all clear why that possibility counts against quantifying over everything, as I said before in talking about a similar discussion in Hellman's paper. For a start, note that Parsons says the problem is that there has to be a final answer to the question of what objects there are. But the tropist and the traditionalist (if we can call her that) needn't disagree at all about the objects that there are. In particular, the tropist needn't deny any of the objects that the traditionalist posits; it's just that he has his own special story about what objects are (how they are constructed from tropes).
Perhaps Parsons mis-spoke and meant to say that absolutely general quantification involves fixing not the objects but the entities in some all-embracing cross-category sense. But I suppose that well brought up Fregeans might start getting unhappy about the coherence of that idea. And in any case, since the traditionalist can treat tropes as logical constructions, the tropist and the traditionalist don't have different stories about what entities (broad sense) there are either, but rather a different story about the entities are interrelated by metaphysical dependence relations (whatever they exactly are).
Now in fact Parsons himself raises the same concerns about the trope example. But he comments
The mere fact of the possibility of construction is not sufficient, since constructions that may be offered will not necessarily satisfy the metaphysical intuitions that drive the alternative framework. That's a reason for thinking that even if this possibility gives us a way of talking about everything in the world that does not commit us to metaphysical realism, even making sense of it gets us into heavy-duty metaphysics.
But I'm not getting the force of that. For remember the dialectical situation. Someone purports to quantify over everything. The objector says "Ahah! Do you realize you are committed to metaphysical realism in a bad way?". The proponent of unrestricted quantification says "Why so?". The objector responds "You are committed to thinking the world carves up into entities in a unique way: and what about e.g. the choice between a traditionalist and trope ontology". We've imagined a come-back: "You've not shown that that's a substantive choice about what there is, rather than a choice about how we organize the world into basic entities and constructions out of them". And now Parsons is offering the opponent the retort: "Hmmmm, even making sense of that gets us into heavy-duty metaphysics". To which the original proponent might reasonably protest that it was the objector who started playing the heavy-duty metaphysics game, so he can hardly complain about that. Rather it is the objector who needs to say more about e.g. the trope example and why (i) on the one hand it is supposed to be a contentful and a genuinely different story about what there is (not just a different story about "dependence"), yet (ii) on the other hand there is some kind of free choice about whether to adopt it rather than the traditional story, i.e. there isn't an objective fact of the matter about which is the right story, which explains why we shouldn't be metaphysical realists.
So for the moment, until he hears rather more, the proponent of absolutely general quantification can reasonably suppose himself to live to fight another day!