Sleeping Beauty Variations and Explanations

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.

6 Replies to “Sleeping Beauty Variations and Explanations”

  1. First off, I don’t think your variation changes the case in any significant way. However, I am a bit worried about your “demonstrative reference” approach. You suggest that Beauty can think the thought “This_1 is Tuesday” on Wednesday as well. But how does she pick out that thought, especially given the possibility that she had a thought on Monday that looked exactly the same (so to speak) to her then? On Wednesday she can consider the thought “This (as I use it on Tuesday) is Tuesday”, but that’s a tautology! Moreover, that’s a thought she can consider on Tuesday, and is distinct then from the thought “This_1 is Tuesday.”

    Curiously, I have less of a problem if Beauty doesn’t sleep between Tuesday and Wednesday. Somehow it feels intuitively possible to me that she might think the demonstrative thought “This is Tuesday” on Tuesday, keep it fixed before her mind, and then keep keeping it fixed as the clock ticks over at midnight from Tuesday to Wednesday. But if she loses her grasp on the thought (as I suppose happens if she goes to sleep), I’m not sure how she can reclaim it without trivializing it.

  2. I think Brian’s problem and solution are both interesting.

    First, he raises the problem of what credence SB should assign to the coin’s landing heads (hereafter: H) when she is told “that1 (day) was Tuesday” on Wednesday. As far as I know, nobody discussed to what degree SB should believe H on Wednesday. All attention has been paid to what happens on Monday and Tuesday.

    Second, Brian’s solution provides a way to the following problem: On Tuesday, SB refers to [the day in which she is then located] in an essentially indexical way. On waking up on Tuesday, she does not know whether it is Monday or Tuesday but not which. Hence, “today” is not a word that can be used by SB with the same referent on a later day. When SB is told that the day previously denoted by “today” was Monday, it seems natural to think that she should conditionalize on “___ was Tuesday,” but what should we (or she) fill into the blank?

    Brian’s approach uses “this1 (day)” instead of “today,” and “this1 (day)” can be used later by SB without the problem of essential indexicality. Hence, SB can conditionalize on “that1 (day) was Tuesday,” leading to the result that Cw(H)=Ct(H/this1 day is Tuesday), where Ct is her cred. function on Tuesday and Cw is her cred. function on Wednesday.

    That said, I am not sure whether this is a big progress. In David Lewis’s version of the SB problem (in which Monday is the only day when SB wakes up in the case of H), an analogous process occurs from [the moment of wakeup] to [that of being told “Monday”]: Some time after SB wakes up on Monday, she is told “today is Monday.” And, “today” refers to the same day as it did when she woke up before. Hence, SB can conditionalize on “today is Monday.” As a result, Cm+(H)=Cm(H/today is Monday), where Cm is her cred. function on waking up and Cm+ is her cred. function on being told “Monday.” Both Thirders and Halfers agree to this result, and so both groups think that SB’s credence in H decreases as a result. Thus, Brian’s result seems to say only something analogous to what both parties agree to.

    If Brian could apply his view to SB’s credence updating from Sunday to Monday, it would be more interesting. However, the result is unclear.

    First, let me describe the problem faced by the traditional approach: When SB wakes up on Monday, she receives the following evidence: “I am waking up today (with a memory of Sunday as the last memory).” This evidence includes “today,” which, on Sunday, did not denote Monday. Thus, it seems irrational that Cm(H)=Cs(H/I am waking up today), where Cm is her cred. function on Monday and Cs her cred. function on Sunday.

    Second, does Brian’s solution do any better if we apply it to SB’s cred. updating from Sunday to Monday? When SB wakes up on Monday, she perhaps receives this evidence: “I am waking up on this1 day (with a memory of…).” My problem is that it is unclear to me what credence SB on Sunday was supposed to assign to H on the condition of “I will wake up on this1 day.” To me, “this1” is a new word. What is supposed to be the value of Cs(H/I am waking up on this1 day), where “this1 day” refers to some day in the future? I have no idea, and I doubt that anybody has any idea about it.

    P.S. If you are interested, you can check my paper about the SB problem. Find “Sleeping Beauty and Shifted Jeffrey Conditonalization” in the Online First page of Synthese’s website. In that paper, I present and defend an updating principle applicable to SB’s credal transition from Sunday to Monday.

  3. This is an interesting variation, and Brian’s analysis seems right to me. I agree that the reversal of Monday/Tuesday should make no difference to the original puzzle, and that it simplifies the analysis of the situation of Wednesday. But the complications (for the Wednesday situation) that are avoided by the reversal are also interesting, and I think it is worth considering how to understand SB’s beliefs on the Wednesday that follows the original scenario, assuming as Brian does, that SB knows, on Wednesday, that it is Wednesday, and she remembers exactly one waking from the previous two days – the most recent one, in case there were two.

    Mike Titelbaum is worried about the demonstrative reference on Wednesday, with the identification of Tuesday’s thought, “this1 is Tuesday” with Wednesday’s thought “that1 was Tuesday. He asks, “how does she pick out that thought, especially given the possibility that she had a thought on Monday that looked exactly the same (so to speak) to her then?” But if we are to make sense of updating from a prior belief state, we need to assume that the agent remembers that state, even if she does not necessarily know when it was. We need demonstrative ‘then’s that pick up the ‘now’s from the time remembered. Even if SB had a similar – even indiscernible – thought (“Is this day Tuesday?”) at another earlier time, that will be irrelevant if that thought is not remembered, and so cannot be the reference of the memory demonstrative, ‘then’.

    In Brian’s variation, SB didn’t know, on Tuesday, whether her ‘now’ (or ‘this1’) referred to Monday or Tuesday, but on Wednesday she does know that her ‘then’ (or ‘that1’) refers to Tuesday, so that is why she can rule out one of the three possibilities that were still open on Tuesday. But if we add the Wednesday scenario onto the original version of the story, the situation is slightly more complicated, even if the result is the same. In this case, she still doesn’t know, on Wednesday, whether it was Tuesday then (whether that1 was Tuesday). Nevertheless, she is (on Wednesday) able to rule out one of the scenarios that she was unable to rule out when she was in the epistemic situation she is remembering, namely the situation in which the coin landed tails, and it was then Monday. The Wednesday ‘then’ can refer to Monday only when the coin landed heads, since only in that case will she remember the Monday waking.

    Mike Titelbaum has elsewhere considered a version of the original puzzle in which SB receives some irrelevant information on just one of the two days. (There is either a piece of red paper visible in the room on Monday, but not Tuesday, or the other way around, depending on the role of a fair die). He argues persuasively that this should make no difference to SB’s beliefs on Monday/Tuesday. The information will still be irrelevant on Wednesday, but it will be perhaps more obvious that she will be able to exclude the possibility that she does not in fact remember.

  4. Unsurprisingly, I agree with Bob. I don’t see why we should think there’s any problem with SB being able to have a singular thought about her last awakening. She can remember that very wakening, and that memory can ground a thought about it, even if the memory is no part of the thought about it.

    I agree that the story I’ve told here doesn’t explain the Sunday/Monday, or the Sunday/Tuesday dynamic. Bob’s book has a fair bit to say about that, and I’m trying to work out what I think about that question. But newitx is correct that what I say would be more useful if I answered that question as well.

  5. Thanks for explaining this, Bob. I now think you (and Brian) are right that Beauty can remember the thought she had on Tuesday and think about it as such. But of course, this takes advantage of the special feature of Brian’s version of the problem that Beauty knows on Wednesday that she’s remembering Tuesday. We can imagine other versions of the problem (even with the days inverted as Brian wants) on which either 1) Beauty’s memory is erased after the Tuesday awakening as well as the Monday awakening; or 2) if Beauty is awake on both days, the experimenters do something random to determine which day she gets to remember when she awakens on Wednesday. Either way, Beauty would have trouble picking out “This_1” thoughts on Wednesday.

    So what we have now, Brian, are some close variants of your version of the problem that your technique may have trouble with, along with newitx’s point that your technique also has trouble with the Sunday/Monday transition within your own variant. Still, even with all that said, your argument seems to work quite well for the places where it can profitably be applied….

  6. Brian,
    I’ve been thinking about this more, and while I still agree that your argument works, I’ve started to wonder whether the use of demonstrative reference makes any difference. Call Monday and Tuesday “experiment days.” On Tuesday Beauty can assign credences to things like “The only experiment day I currently remember is Tuesday and the coin landed heads.” Using these sorts of locutions, we can generate analogues of your three regular propositions. Since “the only experiment day I currently remember” doesn’t change for Beauty between Tuesday and Wednesday (and she knows that on Wednesday), she can update her credences between Tuesday and Wednesday by conditionalizing, and we’ll get your result that if she assigns 1/3 to each possibility on Tuesday she’ll assign 1/2 to heads on Wednesday.
    But talking in terms of “the only experiment day I currently remember” doesn’t involve any demonstrative reference. It can be done perfectly well with sentences involving ordinary indexicals. And it can probably be done using centered worlds propositions as well—-after all, there have to be sets of centered worlds in which the center is on someone who can currently remember only one experiment day and that day is a Tuesday (or whatever). So while reversing the order of the days and thinking about Wednesday credences provides support for the thirder solution to the Sleeping Beauty Problem, I’m not sure that support depends on getting demonstrative reference involved.

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