A man in a wizard clothing appears to Blaise Pascal.
Wizard: Pascal, shuffle this deck of cards, please.
Pascal obliges. They flip the deck of cards up; it seems to be in a fairly random order—5 of hearts, 3 of diamonds, jack of diamonds, king of spades, 10 of spades, ace of diamonds, 5 of clubs, 10 of spades, 10 of clubs, 9 of diamonds, etc.
Wizard: Pascal, I will bet you 5 dollars that I can convincingly argue that this deck, which you think you shuffled, was more likely to have been rigged by a pixie than have been randomly shuffled by you. Do you accept the deal?
Pascal: Well, given how well things have gone in the past when wizards appear to me asking for money, I accept your terms. Why do you think it was rigged by a pixie?
Wizard: It’s simple! If a pixie rigged the deck to be as it is, 5 of hearts, 3 of diamonds, jack of diamonds, king of spades, 10 of spades, ace of diamonds, 5 of clubs, 10 of spades, 10 of clubs, 9 of diamonds, etc, the odds are 100% that the deck would have been arranged as it was. In contrast, if you randomly shuffled it, because there are 52! (52x51x50x49…) which is equal to 8×10^67 possible arrangements, each roughly equally probable, the odds of getting the arrangement you did is 1/52!, which is roughly zero. So we have evidence that is 8×10^67 more strongly predicted on my hypothesis than on yours.
Pascal: Woah there, slow down. It’s true that the hypothesis that you rigged the deck to get the exact arrangement you got gets a huge probabilistic boost from the exact arrangement. But this doesn’t provide evidence for the hypothesis that it’s rigged, because conditional on it having been rigged, the odds that it would be rigged exactly as it is are 1/8×10^67. Given that there are 52! ways to rig it, and this one isn’t special, the odds it would be rigged in this exact way are probably less than 1/52!. Therefore, the prior of it having been rigged in exactly the way it was rigged must be less than 1/52!, meaning that even after that hypothesis gets a big boost, it’s still super unlikely that it was rigged. To illustrate, we could compare the hypothesis that it was rigged to the hypothesis that it was shuffled randomly but would happen to turn up exactly the arrangement that it did turn up. So I can play your game too.
Wizard: Ah, I see the confusion. Yes, I agree that if our credence was equally distributed across the possible arrangements of decks then we should guess that it wasn’t rigged. But it’s not—conditional on it being rigged by pixies, there’s a very high probability—at least 10%—that it would be rigged in exactly the way that it is, to get 5 of hearts, 3 of diamonds, jack of diamonds, king of spades, 10 of spades, ace of diamonds, 5 of clubs, 10 of spades, 10 of clubs, 9 of diamonds.
Pascal: This seems super implausible. Why would pixies be especially likely to rig a deck that way?
Wizard: That’s just a specially privileged arrangement. It’s sort of like asking why simple physical laws are more intrinsically probable than more complicated physical laws or why continuous physical parameters are more probable than disjointed ones. Our quest for explanation must end somewhere, and this seems like a fine place for it to stop. It seems very intuitive to me that this would be especially privileged. I’m also a wizard, and consequently very smart! So even if intuitively this seems wrong to you, you should defer to me on such questions.
Pascal: Well, I don’t know if you are a wizard! You’re just wearing a cloak—but anyone could do that. And it just seems like you’re trying to swindle me out of my money—I don’t believe that you actually find that sequence to be an especially intrinsically probable sequence.
Wizard: How confident are you in that?
Pascal: What?
Wizard: How confident are you that I’m a bullshit artist trying to swindle you out of money by buying a cheap cloak from the target across the street rather than an interdimensional wizard with ideal priors?
Pascal: I’m 99.9999% confident. After all, there is a target sicker still on the cloak.
Wizard: Okay, perfect. Your prior in me being a wizard who is telling the truth is 0.0001. Conditional on me being right, the odds that the deck would come up as it did would be .1. Conditional on you being right, it would be 1/8×10^67. So we updated on the evidence and get posterior odds of me being right of 1.25x10^62 to 1. So I’m almost guaranteed to be right!
Pascal: Okay, something seems quite fishy. Why can’t I just say that the prior in you being a wizard accurately reporting your priors is less than 1/52!. After all, conditional on you being a wizard, from my perspective, each random arrangement you could describe as being entailed by the pixies is equally likely. So the odds that you’d be a wizard who has a high prior in the particular arrangement that is actual would be almost zero—low enough to get a huge boost.
Wizard: Maybe there’s a .000000001% prior that I’d be some kind of truthful wizard who would assert that some deck of cards is especially likely to be picked by pixies. But conditional on me being a genuine wizard, I’d be accurate on what one’s priors rationally should be. So updating on the fact that I tell you that the deck of cards you actually got is especially likely to be picked, you conclude that there’s maybe a .000000001% chance, given that I informed you of that, that that is the rational prior. Thus, you update your prior in pixies being especially likely to rig it to get the exact hand you got to be at least .000000001x.1. Then, conditional on that hypothesis, it swamps the other hypotheses.
The basic idea is simple: there’s some probability that I’m telling the truth—not high but some. But the truth I’m telling is that the rational prior to have in you getting the exact deck of cards you got is very high. If this is right, then there’s some probability that the correct prior in that hypothesis is reasonably high. But if there’s some sizeable probability that the reasonable prior in some event is pretty high then your prior in it should be really high. If you think there’s a 10% chance that an ideal thinker would have a 10% credence in X, then you should have at least a 1% credence in X. So updating on the evidence, you are informed of what the rational priors might be, then you update and compare it to the vast improbability of the chance hypothesis, and it swamps it probabilistically.
Analogously, suppose that you find a random hand. Someone tells you that it’s evidence for cheating because it’s a really good hand in poker. Even if you’re not sure if they’re right, you update in favor of that hypothesis because it best explains why they got that hand. The hypothesis that a straight is especially likely to be chosen in poker best explains why a particular potential cheater got a straight rather than some other hand.
Pascal: I feel like you can make this argument for a lot of things. Start with some event where every possible outcome is insanely improbable. Then have some person assert that the actual outcome is especially likely. Then as long as you give any credence to them being right, you change your assessment of the rational priors, so you abandon the hypothesis of chance. For instance, when arguing for the resurrection, you might say that the odds that there would be a resurrection appearance at all are extremely low. However, if Swinburne is right, then the odds of an apparent resurrection conditional on theism is high. Therefore you give some deference to him, have a reasonable prior in an apparent resurrection conditional on theism, and then update based on an apparent resurrection towards Swinburne’s view, even if you’re almost certain that it’s wrong.
Wizard: Yes, I agree that this is the argumentative schema being employed. But what’s wrong with it? This seems like a perfectly kosher way to argue.
Pascal: I don’t know.
Wizard: Then pay up! If you can’t figure out where this goes wrong, then you should pay up.
Pascal: Okay, but I should only pay up if doing so is morally right, correct?
Wizard: Right.
Pascal: Okay but here’s the thing. I received divine revelation, as I often do, that you would appear to me saying exactly the words you did and that paying up would be morally wrong. Now you might think I’m lying. But let’s say the odds of you saying the exact sequence of words you did is 1/googol. Well, my hypothesis predicts you’d say it with certainty. So my hypothesis gets a massive probabilistic boost over chance.
Wizard: Using my own tricks against me. But I have another hypothesis that you would say exactly that and you should pay up. And mine’s more likely to be right because of the whole maybe a wizard thing I have going for me.
Pascal: Okay, but I have another hypothesis that you’d say that and that some arbitrarily large number of other actually true facts would be true. My theory gets a big boost, etc, and so I win.
Wizard: This can clearly keep going for ever. Draw?
Pascal: Deal!