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!