Tuesday, February 07, 2006

Pro globalization

Writing the last post reminded me of something that came up when I was last up in St Andrews visiting the lovely people at Arche (doubly lovely that time since they gave me a phD the same week). While thinking about stuff presented by (among others) Achille Varzi, Greg Restall and Dominic Hyde, I suddenly realized something disturbing about super and sub-valuationists notions of "local validity". (Local validity, by the way, is important because everyone accepts that *its* not revisionary. The substantial question is whether *global* validity is revisionary. Lots of people think it is, and I'm inclined to think not). Below the fold, I explain why....

It's easiest to appreciate the worry in the dual "subvaluationist" setting. Take a standard sorites argument, taking you from Fa, through loads of conditional premises, to the repugnant conclusion Fz. Now the standard subvaluationist line is that though every premise is (sub-)true, the reasoning is invalid (*global* subvaluational consequence departs from classical consequence on multi-premise reasoning of just this sort.). But local validity matches classical validity even on multi-premise reasoning (details are e.g. in the paper Achille Varzi presented to Arche).

Problem! We've got a valid argument with true premises, whose conclusion is absurd (and in particular, it's not true: even a dialethist can't accept it). It really doesn't come much worse than that.

You can reconstruct the same problem for a supervaluationist using local validity, if you take multi-conclusion logic seriously. And you should. It addresses this question: if you've established that a load of propositions fail to be true, what can you conclude? If the conclusions C follow from the premises A, then if each of the conclusions are "rejectable" (fails to be true) one of the premises is rejectable (fails to be true).

Take a sorites series a, b, c,....,z and consider the following set of formulae: {Fa&~Fb; Fb&~Fc; ....;Fy&~Fz}. In a classical multi-conclusion setting, the premises {Fa, ~Fz} entail this set of conclusions. The result therefore carries over to a supervaluationist setting under local validity (but - crucially - not with global validity).

Now, each of the conclusions is really bad (only an epistemicist could buy into one of them). For the supervaluationist, they're each rejectable. So one of the premises must be rejectable too. But of course, neither is.

Either way, this seems to me pretty devastating for "local validity" fans. (NB: I chatted about this to Achille Varzi, and he's put forward a response in the footnotes of the paper cited above. I don't think it works, but it raises some really nice questions about what we want a notion of consequence for.)

Monday, February 06, 2006

Illusions of validity

I seem to spend loads of time thinking how to defend supervaluationism these days. That's reasonably peculiar, since I don't defend its application in many areas: not to vagueness, especially not as a cure-all to the problem of the many (I'm a many-man myself: there are *billions* of mountains around Kilimanjaro). I'm not particularly chuffed with it as a way of handling the inscrutability of reference, either. So basically we're down to a few bits and pieces: perhaps partially defined predicates, perhaps theoretical terms (though even there I have my doubts).

I do like the spirit of the thing, though, and some relatives of supervaluationism appeal to me as a way of thinking about vagueness (e.g. Edgington-style degree theory).I also like something isomorphic to supervaluationism as a way of thinking about ontic indeterminacy and the like. So I've got some investment in it. (continued below the fold)

I've recently had a go at defending supervaluationism from the charge that it's logically revisionary. My line, in affect, is that the arguments that it's revisionary (most famously pushed by Tim Williamson in the marvelous "Vagueness" book) work only if you think "definitely" is a logical operator. And I can't see any reason to believe that. (A draft is available here).

Because of this, I was intrigued to find an argument that supervaluationists are (and should be!) logically revisionary in a recent paper by Delia Graff (it's in the JC Beall "Liars and Heaps" volume). The idea is the following. Suppose that we have a sorites series on the predicate F, and R is an "adjacency" relation along the series. Then from Fa and ~Fb, it should follow for the supervaluationist that ~Rab. For the whole supervaluationist thing is that if there's a gap between the last F's and the first ~F's. But the contrapositive principle (simplifying) is that from Rab you can get ~(Fa v~Fb). That gives you all you need for a negated-conjunction "long sorites" argument.

I think that defender of non-revisionary supervaluation should say that *in no sense* does ~Rab follow from Fa and ~Fb. Yet *intuitively* it does follow (just repeat it to yourself!). But we've come against this sort of situation before: the answer is going to be that we *confuse* the inference from Fa and ~Fb to ~Rab with the inference from Def[Fa] and Def[~Fb] to Def[~Rab]. That inference may well be in goodstanding in some sense (it's obviously not logically valid, but still...) but we won't get in trouble if we take the contrapositive to be in equal goodstanding. (My moves here are independently motivated because I'm basically replaying the Fine/Keefe "confusion hypothesis" moves that the supervaluationist (and others) need in order to account for the seductiveness of the sorites (there's a brief presentation of this here).)

So *I think* the Graff thing doesn't force us to be revisionists any more than the Williamson arguments. But there's lots of rich stuff around here: plenty more things to think about.

Friday, February 03, 2006