The thing is, if the theory is true, how do we know we are not in such a universe? We can say it is extremely unlikely because they are very rare - but we say it is “unlikely” and “rare” because we assume the global (multiverse-wide) probability distribution is similar to the local (this universe) one - but isn’t that assumption effectively equivalent to the assumption that we are not in such a universe? An argument which begins by assuming its conclusion is not much of an argument.
However, if we can’t rely on that assumption, it seems in principle impossible for us to know what the global probability distribution is - how is that not a lethal blow to the entire theory?
I don't think MWI is assuming the global (multiverse-wide) probability distribution is similar to the local (this universe) one, but rather than local probabilities directly arise from the global probability distribution, they are the same. If you do an experiment (e.g. wavefunction collapse) the outcomes we observe in a given experiment are each a single sample from the global distribution. Some outcomes can be highly unlikely and give a skewed view of the global distribution, but a larger number of experiments will always converge to the global distribution.
But don't there exist universes in which that fails to happen? Consider a binary quantum experiment for which the global distribution is 0.5 (we might call it a "quantum coin flip"). If I repeat the experiment often enough, will it always converge to the global distribution? Well, suppose I have an ordinary (non-quantum) fair coin, and flip it one million times – what is the odds of it coming up heads every time? If I've got my maths right, 2^(-10^6) – so beyond astronomically unlikely, its probability is for all practical purposes indistinguishable from zero.
And yet, if MWI is right, then if I flip a "fair quantum coin" one million times, there are universes (just as "real" as ours) in which it comes up heads every single time. 2^(-10^6) is unbelievably small, but it isn't zero. Indeed, no matter how many observations occur, the probability of getting them all wrong just by chance remains non-zero – and, according to the MWI, everything with a non-zero probability in the global distribution actually exists. If MWI is true, there is no limit to how misled some actually existent observers will be.
Hence, by MWI, there are universes, just as real as ours, containing observers who (purely by chance) are consistently misled by their experiments, and therefore conclude that the global distribution is very different from what it actually is. But, if such observers exist, how do we know we are not them? We can say that, by the global distribution, they must be exceedingly rare, so it is exceedingly unlikely we are among them – but that argument relies on the assumption that our locally observed distribution is a reliable guide to the global distribution, which is the very thing it is setting out to prove – and hence must fail as a circular argument. With that argument dismissed, we are left with this conclusion: if MWI is true, we cannot know what the global distribution actually is. That contradicts one of the foundational claims of MWI; therefore, reductio ad absurdum, MWI is false.
This is different from classical sceptical arguments "what if our senses mislead us?", because it argues (if MWI is true) that such misled observers will exist, and the only question is how do we know we are not among them. Classical sceptical arguments are a lot weaker because they are not arguing from the (assumed) actual existence of such deceived observers, only from the (even remote) abstract possibility of their existence. But, if MWI gives sceptical arguments a huge boost - isn't that in itself a good argument against MWI? It renders MWI a self-undermining theory, and theories which undermine themselves ultimately refute themselves.
One might save MWI from this argument by assuming there is some "minimum probability", such that only universes whose probability rises to that minimum actually exist – if all the "misleading" universes are beneath that probability cutoff, no misleading universes exist, so we who exist could not possibly belong to any of them. However, this solution seems rather reminiscent of Ptolemy's epicycles.
