Kripke's (Alleged) Argument for the Necessity of Identity Statements
I have often encountered in articles, talks and classes the following argument for the necessity of true identity statements, always attributed to Kripke:
1) a = b (assumption)
2) a = a
3) a = b (from 1, 2 by Leibniz' Law)
The argument is no good, and I think it is very doubtful that Kripke ever endorsed it.
A minor problem with the argument is that premise (2) is generally false because things don't exist necessarily. That is easily avoided by making [the identity claims in] (2) and (3) conditional on .
The main problem is the move from (1) and (2) to (3) via Leibniz' Law. This move isn't trivial: it is clearly invalid if the box is read as "it is well known that" or "Fred said that". The application of Leibniz' Law is valid iff " ... = ..." is an extensional context. That is just what "extensional context" means: a context wherein substitution of co-referential terms preserves truth.
So the question is whether " ... = ...", unlike "it is well known that ... = ...", is an extensional context, as the argument presupposes. That surely takes some argument, as sentences of the form " ..." are often presented as paradigm examples of non-extensional contexts. In fact, modulo existence, it is necessary that Hesperus = Hesperus, but not that Hesperus = the brightest star in the evening sky, even though "Hesperus" and "the brightest star in the evening sky" co-refer. Perhaps all such cases can be explained away by scope distinctions, but the extensionality of " ... = ..." at any rate remains a strong, and prima facie at least questionable, assumption.
The argument does not contain any defense of that assumption. Instead, the assumption is more or less just the conclusion which the argument is supposed to establish. Anybody who doubted that whenever a = b, then a = b (or rather, ) would certainly have doubted that " ... = ..." is extensional. So Kripke's alleged argument relies on a highly suspicious and question-begging assumption.
There are at least three ways how "" could fail to be extensional, two of which are often overlooked.
One way is for the context to be intensional: whether " Hesperus = the brightest star in the evening sky" is true depends not only on the actual denotation of the two terms involved, but also on what the terms denote at other possible worlds.
A second way is for the context to be hyper-intensional in the way in which "it is well known that ... = ..." are, where we can't even substitute expressions with the same (primary or secondary or 2D) intension (try "3 = 3" and "3 = the smallest prime whose reciprocal has period length 1/2(p-1)").
The division isn't clear. Consider Counterpart Theory, on which "the statue = the clay" may be true and " the statue = the clay" false because in normal contexts, different counterpart relations are invoked by the terms "the statue" and "the clay". Do these counterpart relations belong to the meaning of the terms? Arguably not, and certainly not as ordrinary primary or secondary intensions.
Anyway, here is the third, and most curious way. I don't know how to characterize it in general. Here is an example. Suppose it isn't perfectly determinate what "Hesperus" and "Phosphorus" denote: there are some atoms such that it is indeterminate whether the terms denote the planet together with those atoms or without them. Consider the operator "Det" defined as follows: "Det S" is true iff S is true on any resolution of such indeterminacies. Then "Det(Hesperus = Hesperus)" is true, but "Det(Hesperus = Phosophorus)" is false. So "Det(... = ...)" is not extensional. In this case, it doesn't matter whether "Hesperus" and "Phosphorus" have different intensions of whether there is any cognitive difference between them at all (which usually explains hyper-internsionality).
This third way is relevant here, because "" is arguably non-extensional in exactly this way. Suppose in a few million years, Venus will fission into two planets. Which of the two, if any, do we now denote by "Venus", "Hesperus" and "Phosphorus"? Suppose that's indeterminate. What then should we say about a sentence like "in 10 million years, Venus will still be part of our solar system"? No answer is entirely satisfactory. Perhaps the best answer is to say that such a sentence is true iff what it says is true of both planets between which "Venus" is indeterminate. Let's accept this proposal; that is, we take "in 10 million years, Phi(Venus)" to be true iff in 10 million years, both planets between which "Venus" is indeterminate satisfy Phi. Likewise for "Hesperus" and "Phosphorus". It follows that "in 10 million years, Hesperus = Phosphorus" is not true, even though "in 10 million years, Hesperus = Hesperus" is true. Hence "in 10 million years, ... = ..." is not extensional. And by exactly parallel reasoning, since Venus could just as well have already fissioned in the past, it is not true that at all worlds, Hesperus = Phosphorus, even though at all worlds, Hesperus = Hesperus (always add: "at all worlds where Hesperus exists").
Now I don't want to insist that this is the best way to handle fission cases. But anyone who claims that "" is extensional will have to come up with a better alternative.
Phew, on to the second point, that it is doubtful whether Kripke endorsed the bad argument.
The relevant place is Kripke's paper "Identity and Necessity", which begins with the following argument:
1)
2) (instance of Leibniz' Law)
3) (from (1), (2))
(Kripke forgot the pair of brackets in (2), but that's clearly what he meant.)
Kripke says that this argument "has been stated many times in recent philosophy" (p.136), and especially mentions David Wiggins. He then goes on to defend the conclusion and to argue at length that even instances of (3),
4) ,
with a and b proper names, are true. Interestingly, his argument for that is not that (4) follows from (3), which he has already defended, by universal instantiation. I'm not quite sure why. Maybe he (rightly!) thought that universal instantiation is very problematic in modal logic. (At every world, , but not at every world, , unless a necessarily exists. So either what logically follows from a necessary truth is sometimes not itself a necessary truth, or universal instantiation fails.) Or maybe he (again, rightly) thought that since the argument for (3) presupposes the extensionality of "", he'd need an independent argument to show that substitution of co-refering names cannot change the truth value in such a context.
Kripke's argument for (4) is on p.154:
First, recall the remark that I made that proper names seem to be rigid designators, as when we use the name 'Nixon' to talk about a certain man, even in counterfactual situations. [...] If names are rigid designators, then there can be no question about identities being necessary, because 'a' and 'b' will be rigid designators of a certain man or thing x. Then even in every possible world, 'a' and 'b' will both refer to this same object x [...].
More formally, Kripke's argument why identity statements between proper names are necessary goes like this:
1) Proper names are rigid designators;
2) rigid designators denote the same thing at every possible world;
3) if 'a' and 'b' denote the same thing at every possible world, then ;
4) for any proper names 'a', 'b', if a = b, then (from (1), (2), (3)).
Leibniz' Law doesn't figure in this argument at all. Unlike the original argument, this one is also not entirely suspicious and question-begging: Kripke has made a strong case that names are indeed in some sense rigid, and (2) and (3) are the most straightforward way to analyze rigidity. I prefer the counterpart analysis on which either (2) or (3) or both come out false (depending how they are interpreted). But at least this is something that can be properly called an argument.
I had a look back at Mark Sainsbury's lecture notes on this topic, and I think you're in agreement on much of this. But just a quick point, I take it you also want a = b in the antecedent of 3?