The following are distilled from emails I received from Guido Bacciagaluppi at the University of Sydney.
Objections, like Stapp's (i.e,
http://sts.lbl.gov/~stapp/bp.PDF or his book
Mindful Universe) or Rosenblum's and Kuttner's (
Quantum Enigma), that state decoherence does not give rise to a basis in the strict sense of a complete set of orthogonal states is correct but not important. There is no need then for consciousness to play the role
some people think it needs to in QM. Decoherence and a proper interpretation of the wave function is all that's required.
[e.g, Each of the current interpretations of quantum mechanics has its pros and cons, but if you are prepared to accept either many-worlds or the Bohm theory as good candidates for the solution of the measurement problem, then decoherence plays an important role in both these proposed solutions. In the case of many-worlds, it appears that it provides the key to which decomposition or decompositions of the universal wave function define the many worlds of the interpretation. In the case of the Bohm theory it guarantees that the result of a measurement, once it has been obtained, is not undone (or that Schroedinger's cat, once it is dead, does not resuscitate).]
There are two reasons, one mathematical and one philosophical. The mathematical reason is that one does not need a basis (i.e. a complete set of orthogonal vectors) in order to define probabilities, one can work also with more general objects, so-called POV measures. See e.g. in the book by A. Peres, Quantum Theory: Concepts and Methods (Dordrecht: Kluwer, 1993), on pp. 282-289. Also, see links below to papers by Wallace where Wallace also remarks on why one should expect worlds to be merely vaguely defined.
The philosophical reason is that the best proposal so far for understanding probabilities in many-worlds is in terms of subjective probabilities (see again for instance David Wallace's papers). If that is the correct way to do it, then the fact that at the objective level of the wave function there might (possibly) be only an inexact correlate to the probabilities is not very worrying.
It is interesting to note that Daniel Dennett's writing on
functionalism play a significant role in Wallace's work. Either of these two papers are about worlds and about decoherence in many-worlds (one of them is more accessible than the other):
PhilSci Archive - Everett and Structure PhilSci Archive - Worlds in the Everett Interpretation
And probably this one for the issue of probabilities in many-worlds. This, however, is mainly a technical paper. Guido was not sure what to recommend as a review of these very recent developments:
PhilSci Archive - Quantum Probability from Subjective Likelihood: improving on Deutsch's proof of the probability rule
I found this recent paper by Wallace on the state of play on the measurement problem in QM.
The Quantum Measurement Problem: State of Play
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