Wolfgang Schwarz

Two Kinds of Rigidity

I want to write something about rigidity in the philosophy of mind. But first I have to say more about rigidity. (Apologies in advance: this is all going to be rather basic. But I'll need it, and I found that many people disagree with it.)

Recently I argued that the assumption that ordinary proper names are rigid designators leads to an implausibly excessive form of essentialism. But I don't want to deny the useful distinction between rigid and non-rigid designators. That is, in a sense I do believe in rigid designators. But they are not quite what rigid designators are usually supposed to be.

Rigidity concerns the denotation of our terms at other possible worlds. In itself this is a somewhat strange question. Why believe that our terms denote anything at all at other worlds? And how are we supposed to find out what they denote? The question becomes clearer if we say what we want to use the other-worldly denotations for, namely as intensions. Still this doesn't settle the answer because there are lots of different uses of intensions, and I don't believe that the same intensions will do in all these cases. In particular, if we want to use the intensions to do semantics of statements about epistemic possibility or propositional attitudes, we must chose them differently from if we want to analyse statements about metaphysical possibility. The distinction between rigid and non-rigid designators applies to the latter use of intensions.

So let T be a singular term. How do we assign denotations to T at other worlds in such a way that we get the right truth-conditions for modal statements? The general idea is that "possibly, T is F" is true iff at some world w, the denotation of T at w satsfies F (or satisfies what F expresses at w -- I'll ignore that for simplicity). Now "possibly, the biggest planet is not the biggest planet" is false. (At any rate, there is one reading on which it is false. Focus on that one.) It follows that at no world, "the biggest planet" denotes something that is not the biggest planet at that world. That is, at every world, "the biggest planet" denotes the biggest planet at that world.

Let "Qwertz" denote the biggest planet of our universe. Qwertz is big, but there could well have been a bigger planet. So "possibly, Qwertz is not the biggest planet" is true. Thus "Qwertz" does not denote at every world the biggest planet of that world. What then does it denote at other worlds? That is, how are the denotations of "Qwertz" at different worlds related, what do they have in common, what qualifies something at another world as a member of the intension of "Qwertz"?

One answer, Kripke's, is that what the members of the intension of "Qwertz" have in common is that they are all identical. They are related by identity. Another answer, Lewis', is that they all resemble the this-worldly Qwertz in certain salient respects. Who is right?

It is sometimes said that Kripke's answer is more intuitive, but I'm not so sure what to make of that. Firstly, I don't think questions about details in the possible-world semantics of modal statements can be settled by common sense. I'd say that intuition is only relevant in deciding which truth-conditions delivered by the accounts are correct. Secondly, Lewis' answer is strictly weaker than Kripke's, hence if Kripke's answer is true then Lewis' is true as well, and it is hard to see how intuition could favour the stronger over the weaker position. One gets Kripke's position out of Lewis' by explicating "salient resemblance" as identity. Thirdly, there might be something about Kripke's account of possible worlds and their inhabitants that is more plausible than Lewis' modal realism, but I doubt that this is relevant here. Counterpart theory (Lewis' theory of rigidity) is certainly just as compatible with non-realism as Kripke's theory. (It's true that modal realism is probably incompatible with Kripke's theory. But that is also irrelevant because the advantages of Lewis' theory are independent of modal realism.) Finally, most of the alleged intuitions in favour of Kripke's theory turn out to be misunderstandings of Lewis' theory. For example, it is agreed by all sides that Humphrey himself could have won the presidency. If Lewis denied that, he would get the intuitive truth conditions of "Humphrey could have won the presidency" wrong, and that would indeed be a fatal flaw. But he doesn't deny it. On Lewis' account, "Humphrey could have won the presidency" is true iff at some other world there is somebody saliently resembling Humphrey who wins the presidency.

Taken by themselves, Lewis' and Kripke's theories imply very little about the actual truth-conditions and truth-values of de re modal predications: Is there a world where somebody saliently resembling Humphrey wins the presidency? Is there a world where somebody identical with Humphrey wins the presidency? Both Lewis and Kripke will say that exactly those worlds exist that make the intuitively true modal statements true. (Lewis will also add that "salient resemblance" is to be understood in such a way that it makes the intuitively true modal statements true.) So how are we to decide between the two accounts?

Fortunately, Kripke's account has some consequences that Lewis' account is not committed to. These consequences have to do with the fact that identity is a precise and context-independent relation. It doesn't make sense to suppose that some (determinate) thing at some (determinate) world might be vaguely or indeterminately identical to a (determinate) thing at another world, or that whether "they" are identical depends on various features of context. Hence Kripke's account implies that de re modal predications (with precise predicates) can never be vague or indeterminate or context-dependent. Lewis' account does not imply any of these because "resembling in salient respects" clearly allows for vagueness and indeterminacy and context-dependence. So the question is: Are some de re modal predications (intuitively) vague or indeterminate or context-dependent? And the answer obviously is: Sure.

What, now, is a rigid designator? If it is a term that behaves more like "Qwertz" than like "the biggest planet", then of course there are rigid designator. It is somewhat difficult to formally define this difference without taking a stance on the Lewis versus Kripke question of how the denotations of rigid designators are related to each other. That's why one usually defines a rigid designator as a term whose referents at different worlds are related by identity. Formally, T rigidly denotes x iff for all things y and worlds w, T denotes y at w iff y exists at w and y=x. And in this sense I doubt that there are many rigid designators. Let's call terms that are rigid in that sense "Kripke-rigid". On my view, mathematical terms are plausible candidates for Kripke-rigid terms, but ordinary proper names and rigidified (or rigidly interpreted) descriptions are not.

By contrast, a Lewis-rigid designator is a term whose referents at different worlds all saliently resemble some thing at our world. Formally, T Lewis-rigidly denotes x iff for all things y and worlds w, T denotes y at w iff y exists at w and saliently resembles x.

A problem with this definition is that it might be too tolerant. For if we are sufficiently relaxed about the counterpart relation (salient resemblance), one can construe every designator as Lewis-rigid: "the biggest planet" rigidly denotes Qwertz, where the contextually salient counterpart relation selects all and only biggest planets as Qwertz-counterparts. Following Lewis' "things qua truthmakers", we can say that "the biggest planet" rigidly denotes Qwertz-qua-biggest-planet.

The problem could be easily solved by not being so relaxed about the counterpart relation: Things that resemble each other only in being the biggest planet of their universe don't count as counterparts. But in fact I do believe that we should be so relaxed. So what to do?

Nothing. The distinction between rigid and non-rigid designators ends up being somewhat blurry, but that's fine. We only need to account for clear cases, and we can do that, for instance by noticing that the resemblance relation that unites other-worldly denotations of paradigmatic rigid designators is in some sense more natural than the resemblance relation that unites all biggest planets. Compare the temporal analogy: We might call a term "temporally rigid" if it denotes the same thing at all times. "George W. Bush" is temporally rigid, "the president of the US" is not. (Let's suppose that at every time t, "the president of the US" denotes the president at t.) But now let Asdf be the fusion of all stages of persons that are presidents of the US. Then in a sense "Asdf" is temporally rigid -- it denotes that same fusion at every time --, but its temporal intension (its denotation at different times) coincides with that of "the president of the US", which we claimed was not temporally rigid. So do we have to give up the distinction? No. The thing to do is to say that what distinguishes the temporal intension of temporally rigid designators from that of temporally non-rigid designators is that the former are in some sense more natural than the latter.

[Update 2003-12-18: I just realized that virtually all the points I wanted to make in this entry have already been made by Brian Weatherson last year. He calls Kripke-rigidity strong rgidity and Lewis-rigidity weak rigidity, which are obviously much nicer terms.

The repetition is particulary embarrassing because I remembered that Brian had written about this, but only now cared to reread his posting. I should really link more and write less (even more so as I still have RSI).]

[Update 2003-12-18, later: Well, in the meantime I reread the final section of On the Plurality of Worlds, where I remembered Lewis talks about related issues. Which he does. He calls strong rgidity rigidity and weak rigidity quasi-rigidity.]