In Sec. 2 of his paper, McGee reviews a number of grounds that might be offered for skepticism about absolutely unrestricted quantification. But he doesn’t take the classic indefinite extensibility argument very seriously — indeed he doesn’t even mention Dummett, but rather offers a paragraph commenting on the disparity between what Russell and Whitehead say they are doing in PM to avoid vicious circles, and what they end up doing with the Axiom of Reducibility. Given the actual state of play in the debates, just ignoring the Dummettian version of the argument seems pretty odd to me.
But be that as it may, “the bothersome worry,” according to McGee, “is not that our domain of quantification is always assuredly restricted [because of indefinite extensibility] but that the domain is never assuredly unrestricted [because of Skolemite arguments]”. Here I am, trying to quantify all-inclusively in some canonical first-order formulation of my story of the world, and by the LS theorem there is a countable elementary model of story. So what can make it the case that I’m not talking about that instead?
OK, it is a good question how we should best respond to the Skolemite argument, and McGee offers some thoughts. He suggests two main responses. The first appeals very briefly to considerations about learnability. I just don’t follow the argument (but I note that Lavine is going to discuss it, so let’s hang fire on this argument for the moment). The second is that “[t]he recognition that the rules of logical inference need to be open-ended … frustrates Skolemite skepticism.” Why?
The LS construction requires that every individual that’s named in the language be an element of the countable subdomain S. If the individual constant c named something outside the domain S, then if ‘(∀x)’ is taken to mean ‘for every member of S‘, the principle of universal instantiation [when c is added to the language] would not be truth-preserving. Following Skolem’s recipe gets us a countable set S with the property that interpreting the quantifiers as ranging over S makes the classical modes of inference truth-preserving, but when we expand the language by adding new constants, truth preservation is not maintained. The hypothesis that the quantified variables range over S cannot explain the inferential practices of people whose acceptance of universal instantiation is open-ended.
But this line of response by itself surely won’t faze the subtle Skolemite. After all, there is a finite limit to the constants that a finite being like me can add to his language with any comprehension of what he is doing. So start with my actual language L. Construct the ideal language L+ by expanding L with all those constants I could add (and add to my theory of the world such sentences involving the new constants that I would then accept). Now Skolemize on that, and we are back with trouble that McGee’s response, by construction, doesn’t touch.
Actually, it seems to me that issues about the Skolemite argument are orthogonal to the distinctive issues, the special problems, about absolute generality. Suppose we do have a satisfactory response to Skolemite worries when applied e.g. to talk about “all real numbers” (supposing here that “real number” doesn’t indefinitely extend): that still leaves the Dummettian worries about “all sets”, “all ordinals” and the like in place just as they were. Suppose on the other hand we struggle to find a response to the Skolemite skeptic. Then it isn’t just quantifications that aim to be absolutely general that are in trouble, but even some seemingly tame highly restricted ones, like generalizations about all the reals. Given this, I’m all for trying to separate out the distinctive issues about absolute generality and focussing on those, and then treating quite separately the entirely general Skolemite arguments which apply to (some) restricted and unrestricted quantifications alike.