Saul Kripke's Paderewski puzzle put me in mind of a rather similar puzzle -- call it the Ortcutt puzzle -- from W.V. Quine's seminal 1956 J. Phil. paper, "Quantifiers and Propositional Attitudes" (in The Ways of Paradox, Harvard UP, 1976, pp. 185-196). Back to Ortcutt!
The ordinary language 'Ralph believes that someone is a spy' is ambiguous as between the de dicto
a. Ralph believes that (∃x)(x is a spy)
and the de re
b. (∃x)(Ralph believes that x is a spy).
To believe that someone is a spy is very different from believing, of a particular person, that he is a spy. Most of us believe the former, but few of us believe the latter.
Despite Quine's queasiness about quantifying into belief contexts, and intensional contexts generally, (b) is intelligible. Suppose (b) is true: someone is believed by Ralph to be a spy. This existentially general sentence cannot be true unless some particular person is believed by Ralph to be a spy. Let that person be Bernard J. Ortcutt.
Now suppose Ralph has several times seen a man in a brown hat hanging around dubious venues, a man Ralph takes to be a spy. There is also a man that Ralph has seen once on the beach, an elderly gray-haired gent who Ralph takes to be a pillar of the community. (Assume that, in Ralph's mind at least, no pillar of a community is a spy.) Unbeknownst to Ralph, the 'two' men are one and the same man, Ortcutt.
Does Ralph believe, of Ortcutt, that he is a spy or not?
Suppose de re belief is irreducible to de dicto belief. What we then have is a relation (possibly triadic) that connects Ralph to the concrete individual Ortcutt himself and not to a name or description or a Fregean sense or any doxastic intermediary in the mind of Ralph such as a concept or idea, or to any incomplete object that is an ontological constituent of Ralph such as one of Hector-Neri Castaneda's ontological guises, or to anything else other than Ortcutt himself, that completely determinate chunk of extramental and extralinguistic reality.
It would seem to follow on the above supposition that Ralph believes, of Ortcutt, that he is both a spy and not a spy. It seems to follow that Ralph has contradictory beliefs. How so? Well, if there is de re belief, and it is irreducible to de dicto belief, then there is a genuine relation, not merely an intentional 'relation' or a notional 'relation' that connects Ralph to Ortcutt himself who exists. (A relation is genuine just in case its holding between or among its relata entails that each relatum exists.) Under the description 'the man in the brown hat,' Ralph believes, of Ortcutt, that he is a spy. But under the description 'the man on the beach,' he believes, of Ortcutt, that he is not a spy. So Ralph believes, of one and the same man, that he is a spy and not a spy. Of course, Ralph does not know or suspect that the 'two' men are the same man. But he doesn't need to know or suspect that for the de re belief relation to hold.
The above seems to amount to a reductio ad absurdum of the notion of irreducible de re belief. For if we accept it, then it seems we must accept the possibility of a rational person's having contradictory beliefs about one and the same item. Why not then try to reduce de re belief to de dicto belief? Roderick Chisholm, following Quine, attempts a reduction in Appendix C of Person and Object (Open Court, 1976, pp. 168-172)
A Reductio ad Absurdum Argument Against a Millian Theory of Proper Names
c. If a normal English speaker S, on reflection, sincerely assents to a sentence 'a is F,' then S believes that a is F. (Kripke's disquotational principle)
d. If a Millian theory of proper names is correct, then the linguistic function of a name is exhausted by the fact that it names its bearer.
e. Peter sincerely assents to both 'Paderewski is musical' and 'Paderewski is not musical.' (Kripke's Paderewski example)
f. Peter believes both that Paderewsi is musical and that Paderewski is not musical. (From c)
g. Peter believes, of one and the same man, Paderewski, that he is both musical and not musical. (From f, d)
h. Peter believes a contradiction. (From g)
i. Peter is rational, and no rational person believes a contradiction.
j. Peter is rational and Peter is not rational. (From h,i)
k. (d) is false: Millianism about proper names is incorrect.
Interim Tentative Conclusion
Millianism about proper names entails that there are cases of de re belief that are irreducible to cases of de dicto belief. This is turn entails contradictions, as in Paderewski-type cases. Therefore, Millianism about proper names entails contradictions. So we have here a powerful argument against Millianism. But there are also poweful arguments against the alternatives to Millianism. So I conjecture that we are in the presence of a genuine aporia, an insoluble problem (insoluble by us), that is yet genuine, i.e., not a pseudo-problem.