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!

4 responses to “Rigidity and inscrutability

  1. I haven’t read the McGee paper, but isn’t the second horn the more straightforward one? I always thought this was the standard way of extending the permutation arguments to intensions. Though I don’t know where I have that idea from. (Maybe from the Hale and Wright paper about Putnams permutation argument in the Companion to the Philosophy of Language? I recall that this contains something about intensions.)
    If things can satisfy predicates at worlds where they don’t exist (like people can satisfy “famous” at times when they no longer exist), it seems to me that worlds where only one of the candidates exists shouldn’t be a problem.

    If you permute from world to world, don’t you get semantically primitive expressions whose 2-intension is neither rigid nor equal to their 1-intension? It seems to me that there are no such expressions in English or German, and I suspect there is a good reason why not. (Though I’m not sure what that reason is.) If so, there would be a good reason to rule out those permutations.


  2. I had a bit of a look through the literature when I came across this stuff, and really couldn’t find much said about the issue. I’d very much like references to places where people suggest how to extend the argument. There are some tricky issues that arise, particularly over exactly how to deal with variables (it’s all ok in Lewis’s general semantic framework, but Lewis himself says in the appendices that to get all the (higher order) quantification linguists need, he’ll have to move to a Cresswell-style lambda-categorial language. I had to think hard to see how exactly to extend it to that case (effectively, you’ve got to tweak the axiom governing the lambda-operator, as well as twisting all the semantic expressions of categoramic expressions).

    In their presentation, Hale and Wright work with world-by-world permutations (following Putnam’s appendix). They don’t discuss the rigidity issue, and for them it doesn’t surface when they get to the quantified case because they use Mates-style treatment of quantifiers, rather than Tarski-style ones, where each variable is assigned a definite name on the model. So e.g. “ExFx” is true on M iff there is some M’ differing from M only in what intension is assigned to x, such that “Fx” is true on M’. Since variables are in effect names, they get non-rigid intensions on the permuted interpretation, just as other names do.

    I’m totally with you on the sort of things to say to the second horn of McGee’s dilemma: I don’t see why we should insist that things have to refer (at w) to things that exist in w. And the temporal case makes a nice analogy. There’s also some stuff by Salmon on referring to non-existents, which might be wheeled in at this point.

    The way I like to think of the McGee point is as providing an inscrutability argument for the following claim: that there is no fact of the matter about whether any of the things that our terms refer to exist. At that point, the usual dialectic starts: incredulous stares from some, followed by challenges to say what goes wrong with accepting the result.

    FWIW, I think that it’s important to bear in mind both ways of generalizing permutation arguments to the intensional case. The world-by-world (more generally, index-by-index) method gives you a more powerful result: no fact of the matter concerning what intension our predicates have. That’s significant, in my view, since I think that the upshot of all this is: (a) there’s no way of getting inscrutability of reference without inscrutability of intensions; (b) there are reasons to think that inscrutability of intensions will be a BAD THING. So it plays a crucial role in my best shot at explaining why we can’t just accept the arguments for inscrutability. But that’s another story.

  3. Wo: on the point about 2-intensions. I think the most general result in the vicinity is that if you take the 2D matrix for a name (first axis: contexts; second axis: n-tuples of other indices) and then fill in the blanks with objects absolutely any way you choose, then there’ll be a way of choosing suitable matrices for other expressions so as to get the standard truth conditions for sentences. All else equal the inscrutabilist will urge that there’s no fact of the matter whether this matrix, or a more standard one, is the “real” 2D intension of the term.

    As a special case, you can write down 2D matrices that have the feature you mention: the diagonal doesn’t have the same object in it wherever it goes, and the diagonal and horizontal aren’t equal.

    (Disclaimer: this only holds in general if you the intensions are “Carnapian”, e.g. if the semantic values for derived categories of expression take the form of functions from indices to extensions. On Lewis’s general semantics, this isn’t the case: semantic value for a predicate, for example, will be a function from name-intensions to sentence-intensions. This is because of the stuff about (putative) “intensional predicates” like “is rising”.)

    I’d be very interested to know if and why English/German etc can’t have terms of the kind just described. But just like the McGee argument that (e.g.) variables should be rigid, but aren’t on the permuted interpretation, I’d want to know how this got to constrain the selection of semantic theory, for the interpretationist. My interest in the de jure rigid stuff (or whatever we should call it) is that it might provide a rationale for requiring rigidity in a semantic theory. I’d want to look for some similiar rationale for the kind of thing you suggest.

  4. sorry, that should have been:

    “As a special case, you can write down 2D matrices that have the feature you mention: the *horizontal* doesn’t have the same object in it wherever it goes, and the diagonal and horizontal aren’t equal.”

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