Unfortunately, many philosophers have recently doubted that genuinely unrestricted quantification is even coherent, much less attainable. Note that these philosophers face at least two preliminary challenges, both of which are forcefully pressed by Williamson (2003). First, they face the question of what to make of the prospects of ontological inquiry without unrestricted generality. How should we formulate substantive ontological positions such as nominalism, if we cannot hope to quantify over all objects at once? The second challenge for the skeptics is to state their own position. To the extent to which the thesis that we cannot quantify over everything appears to entail that there is something over which cannot quantify, skeptics seem to find themselves in a bind by inadvertently quantifying over what, by their own lights, lies beyond a legitimate domain of quantification.
I haven't read the Williamson article referred to here, and I might just be thrown off by the concision, but I find both of these suggested problems, as stated, simply baffling. When we're talking about 'objects' here, we aren't talking just about things but whatever you feel like talking or thinking about. It includes not just things like dogs and cats but also days, colors, styles, what theoretical mathematicians talk about, and even (contrary to what some would like) fictional characters and impossible objects. Why in the world would we need unrestricted quantification to formulate nominalism? If we take 'nominalism' in the sense in which we usually take it, it is either the denial of abstract objects or of universals. Let's just take denial of abstract objects as our example. In order to do this, we don't need to consider an infinite universe of objects; we just need to know enough about what 'abstract object' purports in order to what less specific categories it purports to say things exist in; and nominalists would simply have to deny the abstract objects that are purported to be there. This is the way rational inquiry works. If you are considering whether you should deny that there are any unicorns, you don't look at socks, squares, body parts, printers, blades of grass, or (unless you're in a Madeleine L'Engle story) subatomic particles; you look at things a reasonable person might under some conceivable circumstances count as a unicorn. Denial of abstract objects is certainly of greater scope than denial of unicorns, but there's no reason to think that reasonable inquiry magically goes out the window when talking about whether nominalism is true or not. I suppose the idea is that nominalists would, for some reason, have to insist that even negative logical statements can only be made about nonabstract objects; there's no particular reason to believe that this is a real commitment of nominalism, but even if true would have nothing to do with unrestricted quantification -- if you insist on saying that X doesn't exist while simultaneously insisting that you can't possibly say anything about X, you're already in trouble, whether your quantification is restricted or unrestricted.
But the second attack strikes me as not just puzzling but so absurd that what is actually said cannot seriously be what is meant. Denial of unrestricted quantification does not entail that there is something over which we cannot quantify; it only entails that there is something over which we do not quantify for any particular instance of quantification. This is similar to the mistake someone would make if they thought that insisting that "any given dictionary leaves words out" were the same as saying that "there are words that are left out of every possible dictionary". It's like claiming that 'every integer has an integer greater than it' implies 'there are integers greater than any integer'. Uzquiano himself must recognize this, because he earlier insisted on the fact that unrestricted quantification was quantifying over all objects at once, i.e., in a single instance of quantification, so I don't know what's going on here.
Perhaps I am missing something. I suspect, however, that this is of a piece with the tendency in some analytic circles to want to pull ontology out of this or that logical system, as if there weren't a jillion different logical systems already on the table, and as if it weren't the case that every single one of them admits of a jillion different ontological interpretations.
