I’ve been thinking a bit about Sleeping Beauty, and I’ve found it a little easier to think about this variation on the original case. I was wondering whether anyone thinks this changes the case substantially.
On Sunday, Sleeping Beauty is told about the game setup, and a coin is tossed, but Beauty isn’t told the results.
On Monday, Beauty is woken iff the coin lands tails, is put back to sleep, and has her memory erased.
On Tuesday, Beauty is woken, stays awake for as long as she would have stayed awake for if she had been woken on Monday, then is put back to sleep.
On Wednesday, Beauty is woken, is told that it is Wednesday, and goes on with her life.
And the interesting question is, on each day, what should be her credence that the coin landed heads?
The primary change from the standard form of the story is to make the only day of waking if heads to be Tuesday, not Monday. This makes it somewhat easier to think about what Beauty should think on Wednesday. I also made it explicit that she doesn’t have her memory erased on Tuesday, and that she’s told on Wednesday that it is Wednesday.
The reason I’ve been thinking about this version of the case is that it makes it easier to see what Beauty should say on Wednesday. And the reason I was thinking about that is that I think it pulls apart two ways of thinking about the problem. (This is all inspired by Robert Stalnaker’s book Our Knowledge of the Internal World. But I don’t say that Stalnaker would endorse any of this.)
One way of thinking about the problem is in terms of centered worlds propositions. Beauty’s knowledge, at any moment, consists of the centered worlds that for all she knows are her centered world at that time. So on Tuesday, she can think the thought “This is Tuesday”, and this means that the centre is on Tuesday. And that’s a thought that is true on Tuesday and false on Wednesday.
Another way of thinking about the problem uses regular possible worlds propositions, but makes free use of demonstrative reference to times to ‘latch on’ to propositions about the time. So on Tuesday Beauty can think the thought “This1 is Tuesday.” She might not know whether that is true, but she can consider the proposition. And she can think that thought on Wednesday, if she has sufficient memory to track that demonstrated time. And that proposition doesn’t change its truth value over time.
Now here’s one nice consequence of the latter way of thinking about the puzzle. Let’s say we want to say that on Tuesday, her credence in heads should be 1/3. (I’m not endorsing this, but a lot of people do. And I’m silent here about (a) what we should say about Monday, or for that matter Sunday, or (b) what the dynamic explanation is of how we get from Sunday to Monday to Tuesday.) And let’s also say, as I think we really must, that when she learns it is Wednesday, her credence in heads should be 1/2. What should be the explanation of the change from Tuesday to Wednesday?
On the regular propositions approach, all that happens is that Beauty conditionalises on her new information. On Tuesday she thinks the thought “This1 is Tuesday.” She gives it credence 2/3, since she gives equal credence to each of the following possibilities.
- This1 is Tuesday and the coin landed heads.
- This1 is Tuesday and the coin landed tails.
- This1 is Monday and the coin landed tails.
Then on Wednesday she learns that that1 was Tuesday, so she can drop the third possibility. Conditionalising on the falsehood of that possibility gives her a new credence in heads of 1/2. That seems like an elegant solution to one part of the Sleeping Beauty problem.