If we agree that we do need something to function as Kripke describes rigid designators' functioning, there seem to be only two candidates available within our present language: names or definite descriptions. This, of course, reflects the state of discussion of reference in the philosophy of language. Looking back to Frege we see that underlying these issues is the concern that language be able to put us into some sort of direct contact with the world in a way that allows us to by-pass all the issues of the state of our knowledge of that world, that after all is so often imperfect and transitory, to by-pass the way things just happen to be right now, again this is transitory, today my shirt is blue, tomorrow it's green and on either day it might have been purple.
Kripke begins by asking us to consider descriptions in this role of rigid designation. Taken at face-value, we've seen that these aren't suitable for the task. But, perhaps there's another way of seeing things. Descriptions don't function as rigid designators in the sense of ``defining'' what it is that we're talking about, but they can give us access to the object, their referent here, in this world, in these circumstances, and once we have this, we are able to go on to talk about that object without worrying about whether or not it has any of the features we initially used to identify it, the features we needed to ``get to it'' in the first place.
The distinction here is between fixing the referent of a term and defining it; between Donnellan's attributive and referential senses of descriptions. This seems promising, especially as a way of understanding names: the sense of a name according to Frege is some description, but we've seen that the problem with this is that it takes us only to an object as described, not to the OBJECT. The problem is not with sense or name, but with the DESCRIPTION as defining the object. If we could separate the object from the description, using the description only to locate the object, but then using only the object to make claims about its identity, &c. we might be able to preserve Leibniz's Law along with language's ability to allow us to talk about the world, convey information, and so on.
Kripke, however, argues against this. According to Kripke, descriptions cannot serve to fix the referent of a term, e.g., a name, without leaving a trace of the description as the name's meaning.
A proper understanding of [(1) Aristotle was fond of dogs] involves an understanding both of the (extensionally correct) conditions under which it is in fact true, and of the conditions under which a counterfactual course of history, ..., would be correctly (partially) described by (1) (6).
What does this have to do with descriptions as fixing the reference of a term? Let's suppose that the referent of ``Aristotle'' is picked out by some description: the teacher of Alexander the Great. Now we look at some sentence describing Aristotle. We not only have to use this description to locate Aristotle in this world to determine whether or not Aristotle is fond of dogs, say, but we now have to consider whether in some counterfactual situation, Aristotle is fond of dogs or not fond of dogs. How are we to locate the reference of Aristotle? If we use the description that fixed the reference, it might not take us to the same individual. If that individual were to turn out to like dogs it would not tell us whether being fond of dogs was a contingent or necessary feature of Aristotle, nor would it tell us whether or not we understood (1).
Kripke says of (1) in the actual world that it is true if and only if a certain man, ARISTOTLE, was fond of dogs. If we are to be able to assess it in another world, the same ``paradigm applies to the truth conditions of (1)''--that is to say, it's true if and only if a certain man, ARISTOTLE, is fond of dogs.
Russell's analysis, by contrast, is that
(3) There is one and only one last great philosopher of antiquity, and he was fond of dogs.
Kripke writes that Russel's analysis makes
that person's fondness for dogs the relevant issue for the correctness of (3) (7).
It's not entirely clear why Kripke thinks that on this account the referent will shift as it would on the account which defines ``Aristotle'' as ``the last great ...'' Well, why shouldn't it? If, according to Kripke, ``Aristotle'' reaches out and grabs ARISTOTLE in such a way that it can hold on to the MAN from counterfactual situation to counterfactual situation without any reliance upon any of that man's features, then why can't the reference fixing function work in the same way'' ``The last great ...'' points us, as it were, toward ARISTOTLE, once we have the man, that's what we look at in all the other possible worlds?
But there may be a stronger argument in favor of Kripke's view. It's not that this expression is used anew in every possible world, but that the referent which it picks out in the actual world must have a certain feature, in this case let us say that of being the teacher of Alexander the Great.'' Now suppose that the sentence is (A) ``Aristotle was the teacher of Alexander the Great''? Part of the advantage of the account in terms of reference fixing is that it is supposed to allow us to accommodate the contingency of such sentences as this, but if we use ``the teacher of Alexander the Great'' to fix the reference of ``Aristotle'' then it will still turn out that in every world, (A) will be true--i.e., it will not be contingent.