# Varities of Rigidity

This post over on metaphysical values by Ross Cameron has got me thinking about reference and rigidity.

There’s a familiar distinction between singular terms that are “de facto” rigid and those that are “de jure” rigid. Paradigm example of the former: “the smallest prime”; paradigm example of the latter: “Socrates” (or, variables).

I’m not sure exactly how “de jure” rigidity is typically characterized. I’ve seen it done through slogans such as: what the name contributes to the truth conditions expressed by sentences in which it figures is just the object it stands for. I’ve seen it done like this: a name is de jure rigid if its rigidity is “due to” the semantics of language, and not to metaphysical facts about the world.
Those two definitions seem to come apart: “the actual inventer of the zip” is plausibly de jure rigid in the second, but not the first, sense.

Let’s concentrate on the first sense of de jure rigidity (so a constraint on getting this right is that actualized descriptions won’t count as de jure rigid in this sense). How could we tighten it up? Well, the task is pretty easy if your semantic theory takes the right shape. For example, suppose you have a semantic theory which in the first instance assigns structured propositions to sentences, and then says what truth conditions these propositions (and thus sentences) have. Then you can say precisely what it is for “name to contribute an object” to the truth conditions of sentences in which it figures: it’s for you to shove an object into the structured prop associated with the sentence.

Notice two things:
(1) this is a semantic characterization: you can read off from the semantics of the language whether or not a given term is de jure rigid. (In this sense, it’s like the characterization of “rigidity” as “referring to the same thing wrt every world”).
(2) this is a local characterization: it only works if you’re working within the right semantic framework (the structured-props one). You can’t use it if you’re working e.g. with Davidsonian truth theories, or possible world semantics.

This raises a natural question: how can we capture de jure rigidity in this, that and the next semantic framework? What interests me is what we can do to this end, working with a general semantics in the sense of Lewis (1970). I can’t see any way to read off de jure rigidity from semantic theory.

But if we appeal to metasemantics (i.e. the theory of how semantic facts get fixed) it looks like we have some options. Suppose, for example you’re one of the word-first guys: that is, like early Field, Fodor, Stalnaker et al, you think that the metasemantic story operates first at the level of lexical items (names, predicates), and then we can offer a reduction of the semantic properties of complex expressions (e.g. definite descriptions, sentences) to the semantic properties of their parts. The de jure rigid terms will be those whose semantic properties are fixed in the following way:

(1) term T refers (simpliciter) to an object X.
(2) term T has the as intension that function from worlds to objects, which, at each world w, will pick out the entity that is identical to what T refers to (simpliciter).

So here’s my puzzle: this looks like a characterization that’s turns essentially on the word-first metasemantic theory. Fair do’s, if you like that kind of thing. But I’m more sympathetic to metasemantic theories like Lewis’s, where the semantic properties of language get determined holistically. If you’re an “interpretationist” (and if you haven’t got the semantic characterizations to help you out, because you’re working with a trad possible world semantics), is there any content in the notion of de jure rigidity? More on this to follow.

### 21 responses to “Varities of Rigidity”

1. Dan López de Sa

Hi Robbie!

I have some worries about the first kind of characterization you focus on: “what the name contributes to the truth conditions expressed by sentences in which it figures is just the object it stands for.”

I would have thought that contributing an object to the truth-conditions of sentences in which it figures makes an expression a singular term, regardless of whether it is rigid or flexible. If a name was flexible, it would still contribute just an object—but a different one with respect to the different worlds. (Compare ‘the great philosopher of antiquity, on the assumption that singular definite descriptions are singular terms.)

One possible amendment would then be along the lines of: an expression contributing, to the truth conditions of sentences in which if occurs, the same object with respect to all (relevant) possible world.

But then it seems rigidity that is being characterized, not specifically de jure rigidity.

Sorry if I am missing something!

2. Robbie

Right, I’m sure we can’t just leave the phrase “contributes an object…” unexplained. The way I could think to give it substance is to explicate it within a structured propositions framework. So if e.g. “Socrates is wise” expresses the struc proposition (Socrates, wisdom), then there’s a good sense in which what the word “Socrates” has contributed is just the object.

The other part of the story here is a definition of truth conditions for structured propositions. I guess the standard way to do that gives you that the struc prop (a,Fness) relative to world w, is true iff a instantiates Fness. (Notice we’re no longer mentioning words at all here). And the net effect is rigidity.

Of course, that way of doing things won’t work for other settings. Which is why I wanted to look at alternative (metasemantic) characterizations.

Thoughts?

3. Robbie

NB: I should have written:

(a,Fness) is true wrt world w iff a instantiates Fness in world w.

(If you’re lewis, you can leave off the final relativization. But I’m not.)

4. Dan López de Sa

Oh, I think I see. You are assuming that sentences have (associated structured propositions which are identical to or at least determine) truth-conditions simpliciter, whereas I was following Kripke (N&N, 6-7 for instance) in having them just relative to worlds. Is this right?

In any case, I was curious about whether there is reason to belief that there will be an interesting variety of rigidity telling proper names and “rigidified” singular definite descriptions apart (at least, assuming them to be singular terms). I seem to remeber being told that Kripke acknowledged them to be de jure rigid in his intended sense. What do you think?

5. Robbie

Right: I see the passage you refer to

“the actual truth conditions of [“exactly one person was last among the great philosophers of antiquity, and any such person was fond of dogs”] agree extensionally with those metnioned above for [“Aristotle was fond of dogs”]… But counterfactually, Russell’s conditions can can vary wildly from those supposed by the rigidity thesis”.

I don’t know what to make of that! I can only think that Kripke’s not using “truth conditions” in the way I was taught to. I hope it’s clear what I was meaning though: truth conditions of a sentence in that sense determines the truth value of a sentence wrt each world.

On whether actualized definite descriptions are de jure rigid. As I mentioned in the post, I’m just not sure how the terminology goes here. But it does seem to me plausible that there’s some important distinction between rigidified descriptions and simple names. Whether that should be labelled “de jure rigidity” is another issue (one I’d be glad to know the accepted answer to, by the way).

Do you know a neat way of characterizing de jure rigidity in the other sense? Wasn’t quite clear to me how to flesh it out.

6. Anonymous

Hi Robbie, how about this: a term is rigid if its 2-intension is a constant function; it is de jure rigid if it is rigid and its 1-intension is not a constant function?
Of course, this again isn’t neutral about the underlying semantic framework.

I’m also very dissatisfied with the usual characterisations of rigidity, including the one in terms of “denoting the same thing at every world”. I think the only correct (and neutral) way to introduce the concept is in terms of modal and temporal embeddings.

7. Robbie

I’m not so worried by lack-of-neutrality at this stage: just getting some workable proposals on the table for the pw-semantics framework would be a good start! (Maybe we could then abstract from these to get a more “transcendent” definition).

What you suggest is interesting. So “I” would be de jure rigid, because in my context it (rigidly) picks out me, and in your context it (rigidly) picks out you.

“the smallest prime” wouldn’t be de jure rigid, because in any context it’d pick out 2.

I guess “the actual inventor of the zip” would be de jure rigid too (no?).

Let me introduce the indexical adjective “blurg”, which’ll pick out smallness when uttered on an even year, and loveliness when uttered on an odd year. “the blurgest pirime” thus rigidly picks out 2 this year; though as uttered next year, it’d (rigidly) pick out the loveliest prime (which is 3, by the way).

Anyway, “the blurgest prime” is rigid. And it’s got an inconstant 1-intension. So it’d count as de jure rigid by the criterion you suggest. Seems wrong to me: whaddya think?

8. Dan López de Sa

Hi again!

Re Kripkean “truth-conditions,” they also satisfy your characterization: that they determine a truth-value wrt each world. (Actually, they might be modelled by structured porpositions, set of worlds, or what have you). It is only that, in general, when the language might include flexible expressions, they are assigned only relative to worlds. Hence that that is true wrt w iff Aristotle was fond of dogs in w is assigned to ‘The last great philosopher of antiquity was fond of dogs’ wrt actuality; whereas that that is to true wrt w iff Plato was fond of dogs in w is assigned to ‘The last great philosopher of antiquity was fond of dogs’ wrt a relevant counterfactual situation.

Re proper names vs “rigidified” singular definite description, I see that one might hold they are crucially different. But why think that this will be a difference in the kind of rigidity they exhibit? (I haven’t read yet your newer post, maybe the response is there.)

Re the characterization of de jure rigidity, what would be wrong with one along the lines you mention: a rigid expression is de jure rigid if its rigidity is due to (can be read off/explained in terms of/holds in virtue of) its semantics?

9. Robbie

Hi Dan,

I’m wondering what work it does to relativize truth conditions?

I may have misunderstood, but it looks like what’s being suggested is the following: that the semantic properties of sentences take the form:

S expresses p wrt w

Where (as you say) p could be a set of worlds. Let’s define “S expresses* q” to hold iff w is a member of q iff S expresses p wrt w and w is a member of p. Isn’t “q” what deserves the name “truth conditions”.

I mean, basically this is like the 2D framework (with p as the secondary proposition, and q as the primary proposition). To motivate that you need to tell a whole big story about the why it’s useful to double-index things in this way. I guess I’m just asking for the motivation to do the kind of extra relativization Kripke seems to be talking about (as you read him). (BTW, it’s definitely not the 2D framework itself, right? Otherwise what “rigidity” in the Kripke quote would amount to is constancy of the 1-intension of the name.).

On the “reading off the semantics” characterization of de jure rigidity. Perhaps that is the way to go (if you don’t mind rigidified descriptions counting as such). But I do want to see it written out. It’d be interesting if we had to appeal to “in virtue of” to define it, for example. I’m thinking that something like that will be needed in order to construct a metasemantic characterization of de jure rigidity for the Lewisian. But that’s another story.

(Truth be told, I was putting off thinking about these kind of characterizations because I want to get hold of Kit Fine’s new book on metaphysics of language to see what he has to say on the matter (I remember he was saying interesting stuff on this at the Arche Modality conference). It’s listed as coming out March 2006 on Amazon, but doesn’t seem to have appeared. Anybody know anything about it?)

10. Anonymous

yes, “the actual inventor of the zip” would be de jure rigid. I don’t have any clear grasp of that notion (“de jure rigid”), but it seems to me that rigidified descriptions should definitely count.

Re “the blurgest prime”, I don’t know. I think there’s a reading on which this isn’t rigid at all: “people don’t care much about the blurgest prime this year, unlike last year, when the blurgest prime was the number 3, which everybody adored.” On the reading on which it is rigid — “the *actually/presently* blurgest prime” –, I don’t think it’s wrong to count it as de jure rigid. But as I said, I don’t have any strong intuitions about the de re/de jure distinction, and I’ve never used it. So I certainly couldn’t offer you any deep and spooky account of rigidity.

–wo.

11. Anonymous

That’s a funny passage from Kripke. It definitely doesn’t look like 2D semantics. And maybe one shouldn’t read too much into it: Kripke says that the two truth conditions “agree *extensionally*” for the actual world but not for counterfactual worlds. The extension of a truth condition is a truth value, so one could read that as saying simply that the truth conditions assign different values to different worlds.

–wo.

12. Robbie

Hi Wo (thought it was you!),

Given the way I introduced “blurg” (explicitly as an indexical) I thought that the sort of gloss you offered on which it turns out non-rigid wouldn’t be available. E.g. since the tense operator “At 2005” won’t shift the context, “At 2005, the blurgest prime was the the number 2” will be true. So I’d be inclined to say that there’s only a misreading of the definite description on which it’s non-rigid!

I have to say that my intuitions on technical terms like “de jure rigid” aren’t all that hot either! Likewise for “directly referential” and other stuff around here. I’m quite open to the sceptical answer to these questions: there just isn’t a categorization that does the work I’m describing. But I would be kinda disappointed.

(By the way, I do think that the metasemantic characterization, really does capture what intuitions I have in the vicinity of de jure rigidity: it’s just that I want to see if something similar can be done for those of us who aren’t words-first kinda people.)

13. Daniel Elstein

Hi Robbie!

How about this: a term is de jure rigid if its rigidity doesn’t depend on which worlds are possible/accessible? ‘The smallest prime’ is only rigid because worlds with smaller primes than 2 aren’t possible. But ‘the actual inventor of the zip’ is rigid regardless of of which worlds are possible, as are names (on the standard view). This seems to deal correctly with the ‘the blurgest prime’ example too. I guess I’m an enthusiast for de jure rigidity in the sense that it seems strange to me to say that ‘the smallest prime’ is rigid (but that just reflects my prejudices I suppose).

On the issue of how to fit this into Lewisian metasemantics, I want to check I understand the worry. I’m guessing it’s something like this: according to Lewis, the reference of our terms depends holistically on how the structure of our theories match up with the structure of the world. The eliteness of kinds interacts with the theoretical role played by terms within the theory to determine which terms refer to which kinds. So is ‘water’ de jure rigid or only de facto rigid? Its being rigid depends on how the actual world is: if there were no sufficiently elite kind whose causal role corresponded to the theoretical role of ‘water’ then ‘water’ would not be rigid. And that seems to be a persuasive reason to hold that ‘water’ is only de facto rigid. I’d resist that line. The rigidity of ‘water’ depends on which world is actual, but not on which other worlds are possible. So my way of drawing the distinction still makes ‘water’ de jure rigid (except that I have to modify the criterion to make an exception for the actual world). I think the Lewisian should say that just as we don’t get to choose what our words refer to, we also don’t get to choose which terms are de jure rigid; the world helps to make the rules too. Is that taking anti-voluntarism about reference too far?

Another question: will the Lewisian think that the world helps to determine the contribution which ‘actual’ makes to singular terms? If not, then maybe ‘the actual F’ is rigid in a stronger de jure way. But maybe a strong enough holism will involve anti-voluntarism all the way down.

14. Robbie

Hi Daniel,

Very interesting idea. De Jure rigidity=rigidity even at impossible worlds! Will have to ponder that one.

My concern with the Lewis semantics/metasemantics pairing was that I didn’t know even how to begin to make out a notion of de jure rigidity there (in contrast with the Fieldian metasemantic setting, where there’s an obvious candidate notion). I didn’t have an argument in mind of the kind you suggest against de jure rigidity.

FWIW, I was playing around with characterizing de jure rigidity for the Lewisian in terms of what role the eliteness of an intension played in determining that that term had that intension. Intuitively, “the smallest prime” has the intension it does because “smallest” and “prime” have the intensions they do; and ultimately, then, it’s the eliteness of “small” and “prime” that do the work. Whereas “2” has the intension it does because of the eliteness of that very intension. I’ll post something more on this soon, if I can convince myself it’s got a chance of being a goer.

15. Aidan

Hi Robbie,

The US Amazon site is much more depressing – it lists the publications date as June ’07:

http://www.amazon.com/Semantic-Relationism/dp/1405108436/sr=1-3/qid=1156998630/ref=sr_1_3/104-9249686-6652718?ie=UTF8&s=books

16. Daniel Elstein

Robbie, I think the idea you mention might be on the right track, but a couple of worries:

With rigidified descriptions (‘the actual F’) it isn’t the eliteness of the intension which makes it rigid. So either you end up denying that rigidified descriptions are de jure rigid, or else you get an ugly disjunctive definition of de jure rigidity.

And I don’t think it’s quite right to say that it’s the eliteness of ‘small’ and ‘prime’ which make ‘the smallest prime’ rigid. Pedantic point (and possibly not even correct?): it’s smallness and primeness which are elite, whereas the thing to say about ‘small’ and ‘prime’ is that our theory assigns to their referents the eliteness which smallness and primeness in fact have. More substantive point: it matters what role ‘small’ has in our theory; if it had the role of ‘famous’ or ‘large’ then ‘the smallest prime’ would not be rigid. So it’s not just the eliteness of the properties (or the eliteness ascribed to the referents of theoretical terms) that matters. What’s special about de facto rigidity is that the details matter: even though largeness is just as elite as smallness it has to be smallness that’s involved here to generate the rigidity.

17. Dan López de Sa

Hi Robbie,

Well, one work that relativized truth-conditions of sentences à la Kripke directly do seems to be precisely to allow for a nice characterization of rigidity/flexibility of consitutent expressions ;-).

Anyway, I agree that “trivialization” concerns like the one you voice can certainly exist—also for the (popular) view that singular terms signify (refer, designate) entities only relative to worlds, or for the (less popular) view that predicates do. I have discussed the case of predicates here.

18. Robbie

Daniel:

On “the actual F”. I was happy to think that these are non-de jure rigid. I’m also happy to drop the term if the convention is that “the actual F” should fall under that category.

You’re right that I should have said that smallness and primeness are the elite things, not the words!

And eliteness of course isn’t the whole story. The idea was that just looking at the eliteness component of the reference-fixing story we’d be able to make the distinction. It’s actually quite difficult to state the thought exactly, which is one reason I was delaying in putting it up. E.g. in drafting the comment above I first tried:

2 has the intension it does “partially in virtue of the eligibility of that the intension that takes any world to the number 2”. But consider merely de fact rigid definite descriptions with that intension, where the decriptive clause features “2”. Then the definite description has the intension it does partially in virtue of the eliteness of that very intension (+ the eliteness of the intensions of the other components of the descriptive clause).

We need to disinfect this kind of problem, without falling into problems due to the holistic nature of the Lewisian project.

So, on a number of counts, it’s not entirely clear how to formulate what I had in mind (as I say, that was one reason why I didn’t put it in the post).

One last thing: I’m not quite following your last sentence?

cheers
Robbie

19. Daniel Elstein

I just meant that ‘the largest prime’ isn’t rigid, and this difference with ‘the smallest prime’ isn’t explained by smallness being more elite than largeness. But since you accept that eliteness isn’t the whole story, I guess you already agree.

20. Robbie

Still not getting it! Sorry to be a muppet.

A (probably irrelevant) aside. What do we say about terms that suffer reference failure? (I’m taking it we’re not taking definite descriptions as quantifiers in this context). “The largest prime” doesn’t change reference to world to world: to that extent, they’re rigid. Presumably empty terms don’t have to be rigid (“the present king of france”). And some may be de jure rigid (“Pegasus”) while others aren’t (“the largest prime”).

R

21. Daniel Elstein

Sorry, I was kind of assuming that empty terms don’t count as rigid. If they can, then my example is no good. And I suspect that then I can’t really make the point when talking about numbers, because all the contingent properties of numbers are less elite than the necessary ones (at least I can’t think of any counterexamples). But then I’m not sure that properties of numbers are well-described as more or less elite. And I guess that all the candidates for de facto rigidity (leaving aside the rigidified descriptions) are things like definite descriptions of numbers, where there aren’t non-empty, non-rigid descriptions which involve equally elite properties. As I said, ‘the most famous prime’ isn’t rigid, but then fame isn’t a very elite property. So I think I was basically wrong on that final point.