I'm just reading about this supposed paradox that apparently results from Jeffrey's logic of decision. It's a lot like the Newcomb's Paradox in that it raises the issue of whether causal back-tracking is metaphysically supported.
The problem is this. You are in the cinema and want some popcorn. What you prefer most is for you to go to the lobby, there to be popcorn, you buy some, return to your seat and enjoy it, even though you will miss some of the film. The next best option is if you decide not to get popcorn and there really wasn't any there, so at least you didn't miss the film. The next best alternative is if you go down to the lobby and there is no popcorn so you miss some of the film. The worst option is to miss out on popcorn by deciding not to leave the cinema, and find on the way out that there was popcorn there all along.
At the same time, you are fairly sure that there is no popcorn out there, because there very rarely is. Also, you're certain that if there was popcorn out there, the owners of the cinema would project onto the screen a subliminal message, reading "POPCORN!!!" every few seconds. You consider yourself to be suggestible, so if this were occurring you would definitely choose to go and look in the lobby for popcorn.
Sobel argues that of all the alternatives, the probability that popcorn being in the lobby given that you look for it, and the probability that there will be no popcorn given that you don't look for it, are both close to 1. As the first of these alternatives has the highest expected utility, using Jeffrey's conditional decision theory we ought to go and look for the popcorn, which would otherwise seem irrational given that we already think the probability of popcorn being there is suitably low.
This is similar to the Newcomb's problem because in both cases we suppose that our future actions could causally determine a past action to come to pass or be reversed. We know that the popcorn is either already out there or not. Our deciding to go and look for it should not affect it's being there. This is my conclusion and that of the 2-boxers, a group I am proud to share with the great David Lewis. I reject Sobel's argument for the following reason:
We are told simply that if they had popcorn then they would be displaying the subliminal message, and we would decide to look for popcorn. But presumably we could decide to look for popcorn independently of the subliminal message, which it seems is a lot like what's happening now to the 1-boxers. Can we be sure that we are in an identical mental state now to how we would have been had the subliminal message been aired, i.e. that it would cause us to invoke decision theory and make the reasoning we are currently using? Let's just say yes for the purpose of the problem. Now, if it were also established that these thoughts of decision theory would not have entered my head unless the subliminal message was displayed, then I would be certain (or at least highly suspect) that there is popcorn outside and go looking for it. If it isn't impossible, i.e. if I could independently think of that reasoning without the message, then I would need to know the frequency with which my thought is caused by the message and the frequency with which my thought is caused by my own curiosity or urge for popcorn. If I don't know the frequencies then I probably end up making a principle of indifference between them, assigning them equally 1/2. I therefore only go outside to get the popcorn if my utility for "leave and get popcorn" is more than double my utility for "stay and miss out on popcorn".
That is my current conclusion. I'll red what Sobel has to say about it and possibly edit this post later to account for belief revision.
