# Category Archives: Indeterminacy

## Rigidity and inscrutability

In response to something Dan asks in the comments in the previous post, I thought it might be worth laying out one reason why I’m thinking about “rich” forms of rigidity at the moment.

Vann McGee published a paper on inscrutability of reference recently. The part of it I’m particularly interested in deals with the permutation argument for radical inscrutability. The idea of the permutation arguments, in brief, is: twist the assignments of reference to terms as much as you like. By making compensating twists to the assignments of extensions to predicates, you’ll can make sure the twists “cancel out” so that the distribution of truth values among whole sentences matches exactly the “intended interpretation”. So (big gap) there’s no fact of the matter whether the twisted-interpretation or rather the intended-interpretation is the correct description of the semantic facts. (For details (ad nauseum) see e.g. this stuff)

Anyway, Vann McGee is interested in extending this argument to the intensional case. V interesting to me, since I’d be thinking about that too. I started to get worried when I saw that McGee argued that permutation arguments go wrong when you extend them to the intensional case. That seemed bad, coz I thought I’d proved a theorem that they go over smoothly to really rich intensional settings (ch.5, in the above). And, y’know, he’s Vann McGee, and I’m not, so default assumption was that he wins!

But actually, I think what he was saying doesn’t call into question the technical stuff I was working on. What it does is show that the permuted interpretations that I construct do strange things with rigidity. Hence my now wanting to think about rigidity a little more.

McGee’s nice point is this: if you permute the reference scheme wrt each world in turn, you end up disrupting facts about rigidity. To illustrate suppose that A is the actual world, and W a non-actual one. Choose a permutation for A that sends Billy to the Taj Mahal, and a permutation for W that sends Billy to the Great Wall of China. Then the permuted interpretation of the language will assign to “Billy” an intension that maps A to the Taj Mahal, and W to the Great Wall of China”. In the familiar way, we make compensating twists to the extension of each predicate wrt each world, and the intensions of sentences turn out invariant. But of course, “Billy” is no longer a rigid designator.

(McGee offers this as one horn of a dilemma concerning how you extend the permutation argument to the intensional case. The other horn concerns permuting the reference scheme for all worlds at once, with the result that you end up assigning objects as the reference of e in w, when that object doesn’t exist in w. I’ve also got thoughts about that horn, but that’s another story).

McGee’s dead right, and when I looked at (one form of) my recipe for extending the permutation argument to waht I called the “Carnapian” intensional case, I saw that this is exactly what I got. However, the substantial question is whether or not the non-rigidity of “Billy” on the permuted interpretation gives you any reason to rule out that interpretation as “unintended”. And this question obviously turns on the status of rigidity in the first place.

Now, if the motivation for thinking names were rigid, were just that assigning names rigid extensions allows us to assign the right truth conditions to “Billy is wise”, then it looks like the McGee point has little force against the permutation argument. Because, the permuted interpretation does just as well at generating the right truth conditions! So what we should conclude is that it becomes inscrutable whether or not names are rigid: the argument that names are rigid is undermined.

However, maybe there’s something deeper and spookier about rigidity, above and beyond getting-the-truth-conditions-right. Maybe, I thought, that’s what people are onto with the de jure rigidity stuff. And anyway, it’d be nice to get clear on all the motivations for rigidity that are in the air, to see whether we could get some (perhaps conditional) McGee-style argument against permutation inscrutability going.

p.s. one thing that I certainly hadn’t realized before reading McGee, was that the permuted interpretations I was offering as part of an inscrutability argument had non-rigid variables! As McGee points out, unless this were the case, you’d get the wrong results when looking at sentences involving quantification over a modal operator. I hadn’t clicked this, since I was working with Lewis’s general-semantics system, where variables are handled via an extra intensional index: it had quite passed me by that I was doing something so kooky to them. You live and learn!

## 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.)

## 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.