Leibniz’ Law says that if X has a property that Y doesn’t, then X is distinct from Y.  That’s the principle that Descartes’ argument seems to rely on.  But Leibniz’ Law fails for what I shall call representation-dependent properties, properties that involve someone’s particular way of representing or thinking of the individual in question.  Oedipus wanted to marry Jocasta.  But he did not want to marry his mother.<1>   That he wanted to marry Jocasta but did not want to marry his mother does not show that Jocasta was not his mother; the poor woman was his marriage-object under one description or mode of presentation but not under the other.  So being wanted as a wife by Oedipus is a representation-dependent property: Whether Oedipus wants to marry the woman depends on whether he is thinking of her as “Jocasta” or as “Mom.”
Someone might reason as follows:

        1.  Jocasta was wanted as a wife by Oedipus.
        2.  Oedipus’ mother was not wanted as a wife by him.
        __________________________________________________________________
        3.  Jocasta is distinct from Oedipus’ mother.

Once again, true premises but false conclusion.  The diagnosis is the same:  Being wanted as a wife by Oedipus is a representation-dependent, description-relative property, not a property that someone either has or doesn’t.  Jocasta had it only qua “Jocasta”; she did not have it under the description “Oedipus’ mother.”  Leibniz’ Law does not apply.

    Let me give two more examples of “representation-dependent” properties and show in more detail why Leibniz’ Law must be restricted so as not to apply to them.  A classic from Quine:  Ralph sees a man in a dark brown hat and sunglasses  behaving suspiciously, and forms the belief that the man is a foreign spy.  The next day Ralph is at the beach and sees a rather distinguished looking man who is recognized as a pillar of the community.  Unbeknownst to Ralph, the “two” men he has seen are one and the same; the man’s name (Quine tells us) is Bernard J. Ortcutt.

    Ralph believes that the man in the brown hat is a spy.  Ralph also believes that the man at the beach is no spy.  (This is no reflection on Ralph’s rationality or logical powers, because of course he has no way of knowing that the man in the brown hat is also the man at the beach.)  So by Descartes-style reasoning:

    1.  The man in the hat is believed by Ralph to be a spy.
    2.  The man at the beach is not believed by Ralph to be a spy.
    ____________________________________________________________________________
    3.  The man in the hat is distinct from the man at the beach.

True premises, false conclusion.  Yet the inference seems to be licensed by Leibniz’ Law:  The man in the hat has a property that the man at the beach does not, namely, the property of being believed by Ralph to be a spy.  So how can the two men be one and the same?

    Answer:  Being believed by Ralph to be a spy is a representation-dependent property.  To see this, ask yourself, “Does the man Ortcutt have that property?”  The answer is, at best, “Yes and no”; it depends on how Ralph is representing Ortcutt to himself, how he is thinking of him.  If he mentally refers to Ortcutt as “the man in the hat,” then he does believe him to be a spy, but if he refers to Ortcutt as “the man at the beach,” then  under that description  he does not believe him to be a spy.  So “being believed by Ralph to be a spy” is not a property that someone simply either has or doesn’t have.  It’s a relative property, that one and the same thing can have under one description or mode of presentation and not have under a different description or mode of presentation.  Ortcutt has that property qua man in the hat, but does not have it qua man at the beach.  There is no property that the man in the hat has and the man at the beach does not have, period; “both” men are, after all, just Ortcutt.

    Second example, from Russell’s “On Denoting” (slightly simplified):  King George IV wondered whether the author of Waverley was Sir Walter Scott.  But of course George IV did not wonder whether Scott was Scott.  (As Russell put it, “an interest in the law of identity can hardly be attributed to the first gentleman of Europe”).  So, as before, it looks as though the author of Waverley had a property that Scott did not have, namely, that of being wondered about (in regard to being Scott) by George IV.  If Leibniz’ Law remains unrestricted, that would show right there that the author of Waverley was not Scott.<2>   But of course the author of Waverley was, in fact, Scott.

    Diagnosis just as usual:  Being wondered about is a representation-dependent property.  Imagine that it is 1805 A.D. in Edinburgh, and you and I see a man in a kilt, riding a horse,<3> and we recognize him as Sir Walter.  I ask, “Is that man there wondered about by George IV (in regard to being Scott)?”  What should you answer?  Either “Yes” or “No” would be at best misleading.  For the answer is again relative: it depends on how we represent or refer to the man.  Qua author of Waverley, he was indeed wondered about by George IV, qua “Scott” he was not.  There is no property that the author of Waverley just plain had but Scott just plain did not have.

    On the present understanding, the reason Descartes’ argument fails is that dubitability is a representation-dependent property.  I can doubt something under one description or mode of presentation but not doubt the same thing under another.  That’s what’s happening in the Cross-Dressing Contest example.

    (Actually we can talk either of two ways:  As in the first paragraph above, we can talk of restricting Leibniz’ Law to exclude representation-dependent properties, or we can say that representation-dependent “properties” aren’t genuinely properties of a thing.  It doesn’t matter much.)
 

1  We know that because of his historic reaction when, after having married Jocasta, he learned that she was his mother:  In the words of Tom Lehrer, “When he found what he had done, he tore his eyes out, one by one.”)

2  Notice that George IV could have worked that out himself.  If we and he were allowed to invoke Leibniz’ Law in Descartes’ way, he would himself just conclude that the author of Waverley was not Scott, and that would stop his wondering.  The wonderment would be self-terminating; George IV would stop it as soon as he realized he was doing it.

3  I actually have no idea whether a Scotsman of the early 19th century would have worn a kilt while on horseback.  Cut me some slack.