Nugatory said:
That's certainly a blemish (arguably, THE blemish) in QM, but I don't see it as a contradiction. It's ugly and aesthetically displeasing that we have to stick in this extra postulate about projection happening, but it doesn't lead to a contradiction in the formal sense of having valid derivations of both A and not-A.
I am sincerely grateful to you for admitting this is at least a matter of concern.
First, a technical note (or you may call it "nit-picking":-) ). This is not about derivations, as postulates are not derived, they are postulated. However, they are typically postulated to achieve agreement with experiments. On the other hand, as far as I know, there is no experimental basis for the projection postulate. I offered the following quote several times here:
"(i) the universal validity of unitary dynamics and the superposition principle has been confirmed far into the mesoscopic and macroscopic realm in all experiments conducted thus far;
(ii) all observed ‘‘restrictions’’ can be correctly and completely accounted for by taking into account environmental decoherence effects;
(iii) no positive experimental evidence exists for physical state-vector collapse;
(iv) the perception of single ‘‘outcomes’’ is likely to be explainable through decoherence effects in the neuronal apparatus."
(M. Schlosshauer, Annals of Physics, 321 (2006) 112-149).
I may have an issue with (iv), but this quote does give a modern summary of experimental results. No positive evidence of collapse so far.
So you don't see the measurement problem as "a contradiction in the formal sense". Perhaps one can defend such a point of view, for example, saying (following von Neumann) that unitary evolution only takes place between measurements, but this is indeed "ugly and aesthetically displeasing". What exactly do we call "measurement"? How and why is it different from "normal" physical processes?
Let me also note that in my recent work (
http://link.springer.com/content/pdf/10.1140/epjc/s10052-013-2371-4.pdf , Eur. Phys. J. C (2013) 73:2371, open access) I gave examples of local realistic models reproducing unitary evolution of quantum field theory (but not the projection postulate). This is another indication that you need the projection postulate (or something similar) to prove that violations of the Bell inequalities are possible in standard quantum theory (this proof is an important part of the Bell theorem). On the other hand, there is no loophole-free experimental evidence of the violations.