Saturday, December 15, 2007

Absolute Generality 13: Skepticism about the quantifiers in particular

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.

4 comments:

Andrew Bacon said...

I wasn't quite convinced by the subtle skolemite.

Although there is, in some sense, a finite limit on what can be named, once you have demonstratives and indexicals in your language a lot of naming can be done with just one word. (e.g. "now" can pick out continuum many different times.)

Also there is the distinction between what constants you can feasibly add to your language all at once, and constants that could individually be added. It seems that all you need for open endedness to frustate the skolemite is that for each object, it is possible to extend the language (finitely) so that this object is named by some constant.

Peter Smith said...

Dear Andrew

>Although there is, in some sense, a finite limit on what can be named, once you have demonstratives and indexicals in your language a lot of naming can be done with just one word. (e.g. "now" can pick out continuum many different times.)

Really? How does "now" on our lips do that? We really can e.g. distinguish differences of time below the Planck scale? Surely not!

>Also there is the distinction between what constants you can feasibly add to your language all at once, and constants that could individually be added. It seems that all you need for open endedness to frustate the skolemite is that for each object, it is possible to extend the language (finitely) so that this object is named by some constant.

Again, names don't work by magic. If we are to genuinely name some object something in what we do has to distinguish that object. Given our finite capacities, are there really more than a finite number of objects which we can somehow or other single out to be named? So is it really the case that for each object (in some infinite domain), it is actually possible to extend the language (finitely) so that this object is named by some constant? I suggest not!

Cheers

Peter

Andrew Bacon said...

"If we are to genuinely name some object something in what we do has to distinguish that object."

I take it that this is the crucial premise. It doesn't sound quite right to me though. Take two indistinguishable twins, for example. If the first happens to be in front of me as I point and say "her", I have referred, even though I cannot distinguish the first from the second twin.

Imagine now that there are infinitely many twins. Then you have infinitely many people to whom it is possible to refer. For each one, it is possible that they be in front of you as you point and say "her" (and do what ever else you need to demostratively refer.)

Similarly for naming, except this time we do whatever is needed to dub the object in front of us with a name. As Kripke argued, the average man on the street can successfully refer to Gödel, even though he might not know any description that uniquely singles him out, and even though he couldn't pick him out of a lineup of several people.

Peter Smith said...

Sure, to refer to one twin rather than another I needn't be able to distinguish the twin in the sense of having a unique description to hand or possessing a suitably discriminating recognitional capacity. But I didn't say otherwise: I only claimed that "something in what we do has to distinguish that object" -- for example, saying "her" in the presence of one twin rather than the other will do.

Is there really, to return to your suggestion, something we can do in uttering "now" that makes it pick out one real-number precise time rather another (differing say by less than 1^-100000 seconds)?

(But I do agree that my way of putting things in the original post wasn't ideal. I guess we need to think more -- on both sides of the discussion -- about the nature of the modality in claims of the form "it is possible to extend the language so that ...".)