Kripke chapter II,
Here I want to get a little clearer about why this rigid designation stuff is so important.
(1) What rigid designation is
Kripke states that a proper name rigidly designates if it names the same object at every possible world. This has the consequence that if the names "a" and "b" rigidly designate, and "a=b" is true, then "a=b" is necessarily true ([ ]a=b).
To see why this is the case, suppose that it is false. That is, suppose that "a" and "b" rigidly designate, that "a=b" is true, and it is not necessarily the case that "a=b" (~[ ]a=b). But by the possible worlds analysis, to say that it is not necessarily the case that a is equal to b is to say that it is not the case that at every possible world a equals b. But if it is not the case that at every possible world a equals b, then there exists a possible world where it is not the case that a equals b. But if "a" and "b" rigidly designate, then they name the same things at every possible world, so if there exists a possible world where it is not the case that a equals b, then it follows that in the actual world a is not equal to b. This contradicts our initial assumption. Thus it can't be the case that "a" and "b" rigidly designate, that "a=b" is true, and it is not necessarily the case that "a=b" (~[ ]a=b). But this is the same as saying that if the names "a" and "b" rigidly designate, and "a=b" is true, then "a=b" is necessarily true ([ ]a=b).
A more direct proof would be something like this. Assume a is equal to b. If "a" and "b" rigidly designate then they name the same things at every possible world. But then given that "a" and "b" name the same thing in this world (call that thing "x"), they will have to name x in every possible world. Thus it follows that a is equal to b at every possible world, and hence "a=b" is necessarily true.
(2) Why Kripke needs proper names to rigidly designate for his argument to go through.
O.K. So now we know what rigid designation is. Why is it important to say that proper names rigidly designate? If names do not rigidly designate, then Searle or Russell will have a very good way to defuse all of Kripke's thought experiments.
First, consider the following argument, which sets the two views in bold relief.
The description theorist could say to this "big deal," if she believed that names do not rigidly designate. However, if names rigidly designate, then Kripke can say that if Hesperus is equal to Phosphorous is true in the actual world, then it has to be true at all possible worlds, so we know that the conclusion of the above argument must be false. Kripke will then conclude that what got us into trouble were premisses 1 and 2.
So it seems that Kripke needs the thesis that names rigidly designate for his argument to go through, and Kripke's opponent needs the view that they don't to be able to hold onto the description theory.
We really haven't established that Kripke needs rigid designation for his arguments to go through though, because he offers different arguments than the one above. However if we see how his standard argument can be blocked if names don't rigidly designate, then we'll quickly appreciate how important the notion is in the context of these debates. Here's the standard argument.
If names do not rigidly designate then line 3 might not follow. To see this assume that names don't rigidly designate. Then the x denoted by "Aristotle" in this world may not be the same object denoted by "Aristotle" in another world. So lets say there is another world where the x in question (the one picked out by the world "Aristotle" in this world) was not born in Stagira. However, since "Aristotle" does not rigidly designate, the fact that the x in question was born in Crete, does not make it the case that there is a possible world where ``Aristotle is the most famous philosopher born in Stagira'' is false. For as long as names don't rigidly designate, the description theorist can just say that "Aristotle" only names "x" in worlds where "x" is the most famous philosopher born in Stagira, without denying that there are possible worlds where that very same "x" was not born in Stagira. Since (on this view) "Aristotle" does not name that "x," the "x" in question does not make false any claims about Aristotle in that world.
So, again, Kripke needs proper names to rigidly designate, and Kripke's opponent needs to deny that they do. Thus, Kripke better block arguments that they don't.
(3) Conversely, how he's going to block arguments that they don't
Kripke has a general strategy for blocking arguments to the conclusion that rigid designation fails.
Again, remember any true identity of two rigid designators entails the necessary truth of that identity (where "a" and "b" rigidly designate a=b entails [ ]a=b). One who opposes Kripke then need only argue that for some terms ("a" and "b") that Kripke takes to be rigid designators, that a=b and it is not the case that necessarily a=b (i.e. a=b & ~[ ]a=b). Then it will follow that the "a" and "b" in question are not rigid designators. Consider the following argument.
Kripke points out here that, "For all we know P" does not entail "It is possible that P." Consider Goldbach's Conjecture that every even number is the sum of two primes. Since the claim has not been proven, it is true to say both for all we know Goldbach's conjecture is false and for all we know Goldbach's conjecture is true. Now if "For all we know P" entailed "It is possible that P," we would have "it is possible that Goldbach's conjecture is false and it is possible that Goldbach's conjecture is true." But this can't be the case, since Mathematical claims like Goldbach's conjecture are either true or false at all possible worlds (2+2=4 is true at all possible worlds, and 2+2=5 is false at all possible worlds).
So, at least for math, when we say "For all we know P" we are merely asserting that we do not know that P is false. We are not asserting that it is possible that P is true. We can say "for all we know P" when it is impossible for P to be true.
Kripke wants to say the same thing for claims such as "Hesperus = Phosphorous." However, you might doubt this, as mathematical claims are a priori. Kripke claims that "Hesperus = Phosphorous" is both necessarily the case and a posteriori. Why should reflection about how we use the phrase "for all we know" in dealing with a priori claims that are clearly necessary say anything at all about whether or not an a posteriori claim is necessary or contingent? This is worth puzzling over. Kripke seems to be aware of it at one point (I don't have the citation with me).
I think at best Kripke has shown that he has a response to the previous argument. However, I don't think he's shown that it's the best response.
At this point maybe all Kripke's got going for him is that his theory is much more elegant than his opponents. I'm with Plato (and most mathemeticians as well, I think) in thinking that beauty and elegance are guides to truth. So this is some evidence that Kripke is right.
(4) Arguments that they do rigidly designate.
Philosophers like David Lewis think of the set of possible worlds (really we should say universes) as if it something that exists, like the set of galaxies in our universe. Kripke showed that thinking this way makes it extremely difficult to make sense of our modal intuitions. The Lewisian needs to make sense of which objects in all of the possible worlds count as counterparts to objects in this world, and this is extremely difficult to do. For the Lewisian the sentence "Nixon was necessarily a meanie" is true if and only if in every posssible world where a Nixon counterpart exists, that counterpart is a meanie. Then you have to ask how similar to the actual Nixon an object must be to be a Nixon counterpart, and this is egregiously difficult. Even if you could see all of the possible worlds through your possible worlds telescope you would still be in bad shape. Because you wouldn't know if the Nixon-like objects in those worlds are counterparts to Nixon.
On the other hand, if you take possible worlds to just be a model that we use to make sense of modal judgements about objects in the actual world (as Kripke wants us to), the task becomes much easier. Here "Nixon was necessarily a meanie" is true if and only if in every possible world where Nixon exists he is a meanie." We assume that we are talking about the actual Nixon, and contemplating whether or not he would be a meanie in hypothetical situations.
So Kripke claims that the counterpart/telescope theory of possible worlds has horrible problems and the possible-worlds-are-only-a-model theory does not. His explanation of this other theory involves rigid designation.
So it seems he has presented some evidence for rigid designation, though, of course, there are places one could quibble.