## Rigid Designation

October 23, 2009

Imagine the following set up. There are two tribes, A and B, who up until now have never met. It turns out that tribe A speaks English as we speak it now. However, tribe B speaks English* – a language much like English except it doesn’t contain the names “Aristotle” or “Plato”, and contains two new names, “Fred” and “Ned”.

Suppose now that these two tribes eventually meet and learn each others language. In particular tribe A and B come to agree that the following holds in the new expanded language: (1) necessarily, if Socrates was a philosopher, Fred was Aristotle and Ned was Plato, and (2) necessarily, if Socrates was never a philosopher, Fred was Plato and Ned was Aristotle.

Now we introduce to both tribes some philosophical vocabulary: we tell them what a possible world is, what it means for a name to designate something at a possible world. Both tribes think they understand the new vocabulary. We tell them a rigid designator is a term that designates the some object at every possible world.

Before meeting tribe B, tribe A will presumably agree with Kripke in saying that “Aristotle” and “Plato” are rigid designators, and after learning tribe B’s language will say that “Fred” and “Ned” are non-rigid (accidental) designators.

However tribe B will, presumably, say exactly the opposite. They’ll say that “Aristotle” is a weird and gruesome name that designates Fred in some worlds and Ned in others. Indeed whether “Aristotle” denotes Fred or Ned depends on whether Socrates is a philosopher or not, and, hence, tribe A are speaking a strange and unnatural language.

Who is speaking the most natural language is not the important question. My question is rather, how do we make sense of the notion of ‘rigid designation’ without having to assume English is privileged in some way over English*. And I’m beginning to think we can’t.

The reason, I think, is that the notion of rigid designation (and, incidentally, lots of other things philosophers of modality talk about) cannot be made sense of in the simple modal language of necessity and possibility – the language we start off with before we introduce possible worlds talk. However the answer to whether or not a name is a rigid designator makes no difference to our original language. For any set of true sentences in the simple modal language involving the name “Aristotle” I can produce you two possible worlds models that makes those sentences true: one that makes “Aristotle” denote the same individual in every world and the other which doesn’t.* If this is the case, how is the question of whether a name is a rigid designator ever substantive? Why do we need this distinction? (Note: Kripke’s arguments against descriptivism do not require the distinction. They can be formulated in pure necessity possibility talk.)

To put it another way, by extending our language to possible world/Kripke model talk we are able to postulate nonsense questions: Questions that didn’t exist in our original language but do in the extended language with the new technical vocabulary. An extreme example of such a question: is the denotation function a set of Kuratowski or Hausdorff ordered pairs? These are two different, but formally irrelevant, ways of constructing functions from sets. The question has a definite answer, depending on how we construct the model, but it is clearly an artifact of our model and corresponds to nothing in reality.

Another question which is well formed and has a definite answer in Kripke model talk: does the name ‘a’ denote the same object in w as in w’. There seems to be no way to ask this question in the original modal language. We can talk about ‘Fred’ necessarily denoting Fred, but we can’t make the interworld identity comparison. And as we’ve seen, it doesn’t make any difference to the basic modal language how we answer this question in the extended language.

[* These models will interpret names from a set of functions, S, from worlds to individuals at that world and quantification will also be cashed out in terms of the members of S. We may place the following constraint on S to get something equivalent to a Kripke model: for $f, g \in S$, if f(w) = g(w) for some w then f=g.

One might want to remove this constraint to model the language A and B speak once they’ve learned each others language. They will say things like: Fred is Aristotle, but they might have been different. (And if they accept existential generalization they’ll also say there are things which are identical but might not have been!)]

1. 1) necessarily, if Socrates was a philosopher, Fred was Aristotle and Ned was Plato, and (2) necessarily, if Socrates was never a philosopher, Fred was Plato and Ned was Aristotle.

I wonder if there’s something important here about the fact that you’re using indicative conditionals rather than subjunctive ones. As indicative conditionals, they don’t contradict the thesis of rigid designation. (Compare, if the brightest star seen in the morning on date X and the brightest star seen on the evening on date Y are the same object, then Hesperus is Phosphorus, but otherwise not. That’s not incompatible with rigid designation.)

Is there something that can make the two tribes agree to your claims as counterfactuals, rather than just as indicatives?

2. Thanks Kenny. I guess I wasn’t really paying enough attention to that subtlety.

The claims I wanted them to agree on were the material conditional versions: i.e. necessarily either Socrates was a philosopher or Fred was Plato and Ned was Aristotle, and necessarily either Socrates was never a philosopher or Fred was Aristotle and Ned was Plato.

So the question becomes: is there something that tribe A could learn about tribe B’s use of “Fred” and “Ned” that could cause them to accept these claims (and vice versa for tribe B and “Aristotle” /”Plato”.)

On the one hand, if Aristotle and Plato differ over their essential properties then one could distinguish Fred and Aristotle by their modal properties. But I think that we can then make a good case that tribe B are speaking a very unnatural language.

If we chose to reformulate the argument with objects that have no non-haecceitistic modal differences, like two twins in a symmetric universe, then the two languages seem more on a par in terms of naturalness. But then I also can’t see what fact they could ever come to learn that would make them accept (1) and (2).

That said, in the latter case I can’t think what would make them come to agree that Fred was necessarily twin1 and Ned was necessarily twin2 either. My inclination is to say there is no matter of fact in such cases. (Compare the case with two mathematician tribes who independently come up with names for the roots of -1. One tribe calls them i and -i, the other / and \ and it seems there’s no matter of fact whether to translate i as / or \.) I need to spend some more time sorting this out in my head though.

3. I imagine there’s an obvious answer to this, but it wasn’t obvious to me from the post. Why does $\exists x \Box Fred' denotes x$ not suffice to express at least something like the rigid designation claim. This is different from the case you consider where we have: Fred’ necessarily denotes Fred, where `Fred'(and “Fred”) is within the scope of the modal.

4. hmmm not sure how to make logical symbols appear here. I thought it was just the Tex commands but obviously not.

5. Hi Mike,

Sorry for not getting back to you sooner.

Part of the problem with that was that you can construct models where $\exists x \Box \mbox{'Fred' denotes } x$ is true but actually Fred denotes different individuals in different worlds (as does the variable x on the assignment that makes this sentence true.)

Another point is that we have the following two claims:

1) Necessarily “Aristotle” denotes Aristotle

2) It’s contingently the case that “Aristotle” denotes Fred.

If we followed the suggestion at the end of the post we could existentially generalize to get the result that “Aristotle” both contingently denotes some object and necessarily denotes another. So if we were following you’re suggestion it woulb both be a rigid and an accidental designator.