People grumble that it isn’t what it was, but the output on BBC Radio 3 remains pretty amazing, and these days there is the great bonus of being able to “listen again” via the web for up to a week. While driving earlier today, I caught a few minutes of what seemed a strikingly good performance of the Beethoven op. 132 quartet. It turned out to by the Skampa Quartet. I’ve just “listened again” to the whole performance: I’d say it is extremely well worth catching here (click on “Afternoon on 3” for Tuesday, the Beethoven quartet starts about 19 minutes in). Terrific stuff.
Shaughan Lavine’s is one of two fifty-page papers in Absolute Generality (I’m not sure that the editors’ relaxed attitude to overlong papers does either the authors or the readers a great service, but there it is). In fact, the paper divides into two parts. The first six sections review four anti-absolutist arguments, and criticizes McGee’s response on behalf of the absolutist to (in particular) the Skolemite argument. The last five sections are much more interesting, arguing that we can in fact do without absolute unrestricted generality — rather, “full schematic generality will suffice”, where Lavine is going to explain at some length what such schematic generality comes to.
But first things first. What are the four anti-absolutist arguments that Lavine considers? (1) First, there’s the familiar argument from the paradoxes that suggests that certain concepts (set, ordinal) are indefinitely extensible and that it is not possible for a quantifier to have all sets or all ordinals in its domain. Now, unlike McGee, Lavine recognizes that that the “objection from paradox” raises serious issues. However, he evidently thinks that any direct engagement with the objection just leads to a stand-off between the absolutist and anti-absolutist sides, “each finds the other paradoxical”, so he initially sets the argument aside.
(2) The second argument is the “framework objection” that Hellman also discusses in his contribution. “Different metaphysical frameworks differ on what there is … If the answers to [questions like are there any mathematical entities? is space composed of points?] are not matters of facts, but of choice of framework, … [then there is only] quantification over everything there is according to the framework [and not] absolutely unrestricted quantification.” Well, as I noted before, if two frameworks differ just in what they take to be ontologically basic and what they take to be (in some broad sense) constructions out of the basics, then that is beside the present point. We can still quantify over all the same things in the different frameworks — for quantifying over everything isn’t to be thought of as restricted quantification over whatever is putatively basic. So to make trouble here, the idea would have to be that there can be equally good rival frameworks, with only a conventional choice to be made between them, where Xs exist according to one framework, and cannot even be constructed according to the other. If there are such cases, then there may be an argument to be had: but that is a pretty big “if”, and Lavine doesn’t give us any reason to suppose that the condition can be made good, so let’s pass on.
[To be continued.]
I’ve added a link alongside to Tim Gower’s blog (thanks to Carrie Jenkins for recommending it), and I’ve removed links to a couple of seemingly dormant blogs.
I’m a bit staggered to find that the number visitors to this blog have doubled over recent weeks, to over a thousand this week. A sudden enthusiasm out there for musings about absolute generality or recommendations for logicky reading? Somehow I doubt it! Perhaps unsurprisingly, the tracker shows that this geeky post still gets a number of hits a day, even though it just links to someone else’s neat solution to a Leopard irritation. Rather more surprisingly, there is a steady if small stream of people who’ve ended up here because they are searching for photos of Monica Vitti (so as not to disappoint, here is another screen-capture from L’Eclisse). But I wonder why they are searching … Other one-time film buffs, perhaps moved by Antonioni’s death earlier in the year to revisit the haunting films first seen so long ago? Or is it that the films, and Vitti’s iconic presence, are being rediscovered? I find it so very difficult to imagine how they would now seem, if being seen for the first time, against such an utterly different cinematic background to that of the early 60s.
The final section of McGee’s paper is called “A rule for “everything”‘. He argues that “the semantic values of the quantifiers are fixed by the rules of inference”. The claim rests on noting that (i) two universal quantifiers governed by e.g. the same UE and UI rules will be interderivable [McGee credits “a remarkable theorem of J.H. Harris”, but it is an easy result, which is surely a familiar observation in the Gentzen/Prawitz/Dummett tradition]. McGee then claims that, assuming the quantifier rules don’t misfire completely [like the tonk rules?], this implies that (ii) they determine a uniquely optimal candidate for their semantic value. And further, (iii) “the Harris theorem … gives us reason to anticipate that, when we develop a semantic theory, it will favor unambiguously unrestricted quantification.”
The step from (i) to (ii) needs some heavy-duty assumptions — after all, the intuitionist, for example, doesn’t differ from the classical logician about the correct quantifier rules, but does have different things to say about semantic values. But McGee seems to be assuming a two-valued classical background; so let that pass. More seriously in the present context, the step on to (iii) is just question-begging, if it is supposed to be a defence of an absolutist reading of unrestricted quantification. Consider a non-absolutist like Glanzberg. He could cheerfully accept that the rules governing the use of unrestricted universal quantifiers fix that they run over the whole background domain, whatever that is (and that a pair of quantifiers governed by the same rules would both run over that same domain): but that leaves it entirely open whether the background domain available to us at any point is itself contextually fixed and can be subject to indefinite expansion.
Of course, says the anti-absolutist, there is no God’s eye viewpoint from which we can squint sideways at our current practice and comment that right now “(absolutely) everything” on our lips doesn’t really run over all the exists. “Everything” always means everything. What else? But that isn’t what the anti-absolutist denies, and so it seems that McGee fails to really engage with the position.
A few weeks back, I was asked to put together a twenty minute logicky multiple choice test for undergraduates applying to Cambridge to read philosophy. I had better not say much here at all, but I think I managed to produce a fairly tricky paper to do in the time. I’m told that scores spread out from worse-than-chance to almost-perfect, so I guess it worked to filter out the not-so-hot reasoners. I did, though, find it surprisingly difficult to put together a template that I was happy with and which we could weave variations around. I think next year we should instead use this splendid alternative.
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.
I was intending to look at the papers in Absolute Generality in the order in which they are printed. But Glanzberg’s piece is followed by a long one by Shaughan Lavine which is in significant part a discussion of Vann McGee’s views, including those expressed in the latter’s contribution to this book. So it seems sensible to discuss McGee’s paper first.
The first section of this paper addresses “semantic skepticism in general”. McGee writes
The prevalent skeptical view, which is sometimes called deflationism or minimalism, allows that a speaker can say things that are true, but denies that her ability to do so depends on the linguistic practices of herself and her community. … [D]isquotationalism doesn’t connect truth-conditions with patterns of usage. … The (T)-sentences for own language are, for the deflationist, an inexplicable brute fact.
And there is more in the same vein. Well, I thought I was a kind of deflationist, but certainly I don’t take myself to be wedded to the idea that truth-conditions aren’t connected to patterns of usage. Au contraire. I’d say that it is precisely because of facts about the way I use “snow is white” that I am interpretable as using it to say that snow is white. And because “snow is white” is used by me to say that snow is white, then indeed “snow is white” on my lips is true just in case snow is white. So, I’d certainly say that the truth of such a (T)-sentence isn’t inexplicable, it isn’t an ungrounded brute fact. But the core deflationist thought — that there is in the end, bells and whistles apart, no more to the content of the truth predicate than is given in such (T)-sentences (the notion, so to speak, lacks metaphysical weight) — is surely quite consistent with that.
Still, let’s not fuss about who gets to choose which position counts as properly “deflationist”. My point is merely that the sort of extreme position which McGee seems to talking about (though he is far from ideally clear) is remote from plausible versions of deflationism, is therefore to my mind not especially interesting, and in any case — the key point here — hasn’t anything particularly to do with issues about absolute generality. So exactly why does he think it is going to be illuminating to come at the topic this way? I’m rather stumped. So I propose just to pass over his first section with a rather puzzled shrug.
I mentioned that Glanzberg’s paper focuses on Williamson’s version of Russell’s paradox for interpretations. I can’t say that I find that version very illuminating, but there it is. But it does shape Glanzberg’s discussion, and he tells the story about background domain expansion in terms of someone’s reflecting directly about the interpretation of their own language. But I don’t think that this is of the essence, nor the clearest way to present a discussion about what Dummett would call indefinite extensibility.
What is central to the discussion is Glanzberg’s reflections on how far we should iterate the expansion of our domain of “absolutely everything”, once we grasp the (supposed) Dummettian imperative to start on the process. Dummett’s talk about indefinite extensibility suggests that he thinks that there is no determinate limit point (which, I take it, isn’t to say that the expansion definitely goes on for ever, but that there is no point where we have a clear reason to stop). Now, Glanzberg by constrast, things here might be reason just to embrace iteration up to the first non-recursive ordinal, or up to the first α + 1 ordinal, or up to the first α+ ordinal. He then says
In considering multiple options, I do not want to suggest that there is nothing to distinguish among them … Rather, I think the moral to be drawn is that we do not yet know enough to be certain just how far iteration really does go.
But, once we play the Dummettian game, I just don’t see why we should think that there will be a determinate answer here, and certainly Glanzberg gives us no clear reason to suppose otherwise.
Another day, another new book to mention. My current and recent colleagues Mary Leng, Alexander Paseau, and Michael Potter have edited revised versions of some of the papers from the 2004 conference held here in Cambridge on Mathematical Knowledge. It is quite a slim, expensive, hardback; but it should certainly at least be in your university library, if only for the three papers by the editors.
I wonder why they thought that Durer’s ‘Melancholia’ was an apt illustration for the cover …?