If I am a determined £10-taker, then the probability that the envelope contains the £1000000 is zero. If I am a determined £10-refuser, then the probability that the envelope contains the £1000000 is one.
But I am neither of those things. I am some more complicated decision-making system which is capable of either taking or refusing the £10, depending on which act is to my advantage. And I can see that if I refuse the £10, then there must be £1000000 in the envelope, which I get to keep. So, I refuse the £10.
No. If you knowably refuse the £10 in this situation that makes you a determined £10-refuser. The fact that you personally did not know that you are a determined £10-refuser even though Omega did does not have any magical consequences.
Basically you can't simultaneously take the fact that you have a choice and the fact that Omega is actually standing before you as given.
Apparently someone thinks there is something wrong with this. Could they please explain?
This is a variant built on Gary Drescher's xor problem for timeless decision theory.
You get an envelope from your good friend Alpha, and are about to open it, when Omega appears in a puff of logic.
Being completely trustworthy as usual (don't you just hate that?), he explains that Alpha flipped a coin (or looked at the parity of a sufficiently high digit of pi), to decide whether to put £1000 000 in your envelope, or put nothing.
He, Omega, knows what Alpha decided, has also predicted your own actions, and you know these facts. He hands you a £10 note and says:
"(I predicted that you will refuse this £10) if and only if (there is £1000 000 in Alpha's envelope)."
What to do?
EDIT: to clarify, Alpha will send you the envelope anyway, and Omega may choose to appear or not appear as he and his logic deem fit. Nor is Omega stating a mathematical theorem: that one can deduce from the first premise the truth of the second. He is using XNOR, but using 'if and only if' seems a more understandable formulation. You get to keep the envelope whatever happens, in case that wasn't clear.