# Sleeping Beauty and Prisoner X

Here’s what I think is a new argument against the ‘halfer’ solution to the Sleeping Beauty puzzle. (I assume here a lot of familiarity with the puzzle!) I’m particularly interested in responding to this kind of argument for the halfer view, versions of which you can find in papers by David Lewis, Carrie Jenkins and Joe Halpern.

1. On Sunday, the rational credence in Heads is ½.
2. Nothing surprising happens between Sunday and Monday, so Beauty doesn’t learn anything new.
3. If nothing surprising happens between Sunday and Monday, Beauty’s credence in Heads shouldn’t change.
4. So on Monday, the rational credence in Heads is ½.

I think premise 2 in this argument fails, for reasons set out by Bob Stalnaker in Our Knowledge of the Internal World. But Stalnaker’s positive theory is controversial; it would be good to have independent arguments against premise 2 here.

Start by modifying the story. Add in a new character, who I’ll call Prisoner X (or X for short). X will wake once, on either Monday or Tuesday. When he wakes, he’ll be kept awake for exactly as long as Beauty is awake. He’ll be told, truthfully, that Beauty is awake, then returned to sleep. There won’t be any messing around with his memories in any way; he’ll wake on Wednesday with memory intact.

Which day X wakes on will be determined by the following algorithm:

• If Heads, then X will wake on Monday.
• If Tails, X will wake on Monday with probability ½, and Tuesday with probability ½.

X knows all these facts before he goes to sleep on Sunday, as does Beauty. Now for the argument against the halfer position.

1. When X wakes up, his rational credence in heads is ½.
2. When X wakes up, X’s evidence is just Beauty’s evidence plus the fact that X is awake.
3. If 1 and 2, then when she wakes up, Beauty’s credence in Heads conditional on X being awake should be ½.
4. When Beauty wakes up, her credence in Heads conditional on X being asleep is 0.
5. When Beauty wakes up, her credence in X being asleep should be greater than 0.
6. So, when Beauty wakes up, her credence in Heads should be less than ½.

I hope the validity of the argument is obvious. In general, if Pr(A | B) = x, and Pr(A | ¬ B) = 0, and Pr(¬ B) > 0, then Pr(A) < x. That’s all we’re appealing to in the argument.

Premise 1 seems intuitively plausible and hard for the halfer to deny. If Beauty isn’t surprised by anything, then neither is Prisoner X. X knows on Sunday that he’ll wake up once, and knows that Beauty will be awake when he does. So on Sunday he can think about the one and only waking he’ll have during the experiment. (That’s in sharp contrast to Beauty, who can’t think that.) And on Sunday he can say that conditional on Beauty being awake when he is, the probability of Heads is ½. So when he conditionalises on what he is told on waking, his credence in Heads is ½, as suggested.

The motivation for premise 2 comes from thinking about what it would take to get Beauty and X into the same evidential situation. It seems that simply telling Beauty that X is awake would be enough. X knows that he and Beauty are both awake. Beauty knows that she is awake. So that X is awake is enough evidence to put Beauty in the same position.

Premise 3 follows from a form of evidentialism. Rational credence supervenes on evidence. X’s evidence is Beauty’s evidence plus the fact that X is awake. So conditional on X being awake, Beauty’s credences should be just like X’s. And the first two premises (which are, note, the antecedents of premise 3) imply that X should have credence ½ in Heads. So, if everything is working so far, should Beauty.

Premise 4 follows immediately from the setup of the problem. The only way Beauty can be awake while X is asleep is if Tails.

Premise 5 is an instance of a regularity principle. Regularity principles say we should always give positive probability to epistemic possibilities. They are in general a little strong. In cases where there are infinitely many epistemic possibilities, we should give probability 0 to many of them. But in this case, where in some good sense there are only 3 possibilities of the same kind as the one being considered (the possibilities being Heads, Tails and X wakes on today, Tails and X wakes on the other day Beauty is awake), having 0 credence in any of them that is a live possibility seems a little close-minded.

So I think this argument is sound. Of course, one could argue that the presence of X changes things. Perhaps Beauty should have credence ½ in Heads in the original example, but not in this one. But I don’t see how that could be motivated. Beauty doesn’t find out anything about what happens to X on Monday or Tuesday. Her experiences are just like they would be if X didn’t exist, and she knows this at the start. So this looks like a good argument against having credence ½ in Heads in the original puzzle.

## 9 Replies to “Sleeping Beauty and Prisoner X”

1. wo says:

Doesn’t premise 2 rest on the assumption that all evidence is uncentered? Otherwise it would seem that SB has a lot of evidence that X doesn’t have, e.g. that if the coin lands tails, then she* will undergo memory erasure. But if you discount centered evidence, then what you need for premise 3 isn’t just evidentialism, but the assumption that rational belief about uncentered matters supervenes on uncentered evidence. In other words, the assumption would be that centered information is evidentially irrelevant to uncentered propositions.

But that seems false. For example, suppose you are certain that the universe contains infinitely many Earth-like planets that are perfect duplicates of one another until the time when someone first says “cabbage” 583 times in a row. You are torn between two hypotheses about what happens then. On hypothesis A, it starts to rain cabbage on all the Earth-like planets except for one. On hypothesis B, it doesn’t rain cabbage on any of the planets. Now you say “cabbage” 583 times and observe that it doesn’t rain cabbage. It seems to me that this should strongly increase your credence in hypothesis B and that this increase is not due to any uncentered information you receive.

2. “When X wakes up, his rational credence in heads is 1/2.”

I’m not sure. X, by the setup, will only be woken when Beauty is also woken, and X knows this.

So on waking, X learns that both he and Beauty are awake together. So, among other things, he learns that this waking of Beauty is one at which he is also being awoken.

He also knows that if Heads, a randomly selected waking of Beauty is guaranteed to be one where he is also awake.

And he knows that if Tails, a randomly selected waking of Beauty is /not/ guaranteed to be one where he is also awake.

So he should think Heads is more likely on his evidence, which is that this particular waking of Beauty is one where he is also awake.

So maybe this is another case of apparently null information which really isn’t (like Beauty being told it’s Monday).

3. Wo,

In that case, you get some very important uncentered information, namely that it doesn’t rain cabbage on this planet. That seems to account for the difference between A and B.

Carrie,

The proposition that Beauty is awake just is, in informational terms, the proposition that either (It’s Monday and Heads) or (It’s Monday or Tuesday and Tails). So change the case a little – let X be told when he wakes that either (It’s Monday and Heads) or (It’s Monday or Tuesday and Tails). That seems clearly null information – he knew exactly what he would be told before he went to sleep, and he knew it would be true. (Or do you disagree?)

There is perhaps a complication here if you think that when X wakes he can get de re information that he couldn’t get before. But I think in X’s case (crucially unlike Beauty’s) X can think about his one waking during the experiment, as such. And I think (perhaps wrongly) that means the new de re information he gets doesn’t matter. (If, that is, X even gets de re information. On the kind of Hawthorne/Manley picture I find attractive, he doesn’t even really get that. Being able to think about his one waking as such suffices for singular thought about that waking, even in the absence of causal contact with the waking.)

4. patrick says:

Does the support for premise 3 require that
(if Tails) Beauty has the same evidence on
Monday and Tuesday?

Suppose the coin lands tails, and also that X
wakes on Tuesday, not Monday. Consider
Beauty’s situation when awake on Monday.

You say “… conditional on X being awake, Beauty’s credences should be just like X’s”.

X’s credences when? On Tuesday? X is asleep on Monday.
Does Beauty on Monday have the same evidence as
X on Tuesday? Or Beauty on Tuesday?

5. Patrick,

I think you’re right – premise 3 is too strong. What I should have said in premise 3 is that when X and Beauty both wake, then Beauty’s credence conditional on X being awake is just like X’s credence.

That’s enough to get an anti ‘halfer’ conclusion if the halfer position is that Beauty’s credence in heads should be ½ whenever she wakes up.

I don’t really think Beauty’s evidence is the same both days, so I shouldn’t be assuming that in my arguments. My only excuse here is that I was thinking of this argument as a kind of reductio, so I was helping myself to assumptions that others make. But that’s sloppy – you’re right that we should distinguish her Monday and Tuesday evidence.

6. wo says:

In that case, you get some very important uncentered information, namely that it doesn’t rain cabbage on this planet. That seems to account for the difference between A and B.

That sounds like centered information to me. But anyway, let X be the supposedly uncentered information that it never rains cabbage on this planet, and suppose you already learned X yesterday. Then you were put into a spaceship that brought you to a randomly selected Earth-like planet. By chance, you ended up on the same planet you started from, but you don’t know this. Your travel experience doesn’t cause you to forget any uncentered information about the universe, so you still know X. Now the story continues as above: you say “cabbage” a lot, observe that it doesn’t rain cabbage, and increase your credence in hypothesis B. This time, the increase can’t be due to your learning of X, because you knew X all along.

Setting aside the example, I take it from your response that you actually do believe that centered information is never evidentially relevant to uncentered propositions?

7. patrick says:

“That’s enough to get an anti ‘halfer’ conclusion if the halfer position is that Beauty’s credence in heads should be ½ whenever she wakes up.”

OK… but the halfer can comfortably say that the
argument for 1/2 is about Beauty’s credences on
Monday, not Tuesday. (You put the argument in terms of
surprisingness, but I think the argument is better
stated as relying on the claim that Beauty neither
gains nor loses information relevant to the coin
flip between Sunday and Monday, as I put it in
a forthcoming paper.)
It seems to me that the halfer can reasonably
say that the halfer argument does not also immediately
show 1/2 on Tuesday because the cognitive
disturbance (memory erasure) Monday evening
blocks the 1/2 conclusion for Tuesday.

As far as I can tell, the best motivation for the halferish
suggestion of 1/2 also on Tuesday involves
the internalistic view that (on Tails) Beauty has the same
evidence on Monday on Tuesday, and the
evidentialist thought that same evidence means
same credences. So perhaps what you’ve shown is that
the halfer should not accept the same evidence view.

8. Wo,

I do want to say that rational credence supervenes on uncentered evidence. So I have to deal with some of these kinds of cases.

I think the spaceship example means we need to individuate propositions more finely than some direct reference theorists might. The move I prefer right now is to have something like Fregean senses be constituents of the propositions believed. So when you say “cabbage” a lot the first time, your evidence includes a singular proposition that includes that very planet under one mode of presentation, and after the spaceship comes back, your evidence includes a similar proposition that involves a different mode of presentation. (The picture I’m sketching here is very heavily influenced by Stalnaker’s Our Knowledge of the Internal World, though I doubt he’d put everything exactly as I am doing here.)

The crucial thing I want to insist on is that modes of presentation, unlike centered worlds propositions, can be shared both between speakers and between time-slices of a speaker. The sharing between speakers explains how testimony can provide evidence reasonably directly; how, as we’d commonly put it, people can share their evidence. The sharing across time is how we can consider the same evidence, and how it bears on the same propositions, as we move to different centres/times. These seem like big advantages over the centered world approach, for reasons Stalnaker sets out.

9. Patrick,

I think that would be a bit quick for the halfer to just make this move.

For one thing, once we drop internalism about evidence, I’m not sure we can motivate the halfer position. After all, I think there’s a good case that the object-dependent proposition she learns on Monday is relevant to how the coin lands.

For another, X might wake up on Monday. If X does, it seems the argument goes through. So I think it’s plausible that at least Beauty might be mistaken in having credence ½ in Heads. And that’s a problem for philosophical halfers.