Tuesdays and Heads is clearly and obviously not in the sample space. It is not a possible outcome, she will be never be interviewed on Heads and Tuesday. — Jeremiah

It is a possible outcome; she just won't be asked about it.

What's more, Wednesday & Heads and Wednesday & Tails are both part of the sample space. In those cases, she'll be awakened but not asked.

We can only ask her about her credence if she's awake; and we can't ask her if we're not asking her. But she can have a credence whether she's awake and whether we ask.

Think about this -- the variation we haven't discussed -- why does Beauty's credence in HEADS increase by 1/6 when she's told it's Monday? For Elga, it's an increase from 1/3 to 1/2 (and he's right), but for Lewis it's from 1/2 to 2/3 (and he's wrong). Where does this magic 1/6 come from?

It comes from the sample space, which has 6 equiprobable possibilities.

We can ask what Beauty's credence that a fair coin will land HEADS:

• whether we awaken her or not: 3/6
• given we've awakened her: 2/5
• given that we're asking: 1/3
• given that we told her it's Monday: 1/2
• given that it's Tuesday: 1/2
• given that it's Tuesday and she's awake: 0
• given that it's Tuesday and we're asking: 0.

Looked at this way, it's a whole lot of fuss over nothing. The whole point of all the epistemic this-and-that, the centered and uncentered worlds business, all that, is just to get us to pretend we don't know things we know, on the theory that Beauty could not possibly know them, that she is in some astonishing epistemic quandary.

Beauty just needs to know how to count. Her credences are all consistent given the various conditions we might impose. It's pure semantics really to say that her credence changes at all.

No that's wrong.

We can imagine that on HEADS, we wake up Beauty Tuesday and send her on her way. That's no different from putting her back to sleep after a Tuesday interview and waking her up Wednesday. She will not know the difference between sleeping two days and only one. So that would be a sample space of 5 instead of 6 -- the experiment can conclude on two different days, depending on the coin, so that's not the explanation for the magic 1/6.

It's also the same as not counting Tuesday & Heads.

It is a possible outcome; she just won't be asked about it. — Srap Tasmaner

And Tuesday first week of April in the year 3030 is also in the sample space, I suppose.


The sample space is either head and tails, or awakened states.

No worries. The only reason to throw in awake-but-unasked is to show there's yet another way to carve up Beauty's credence. (And we cannot leave her asleep forever -- part of the setup is awakening her and sending her on her way. She will know when this has happened.)

1/6 is just 1/2 - 1/3, of no interest in itself, and we don't really need to throw in the Wednesday states or worry about whether the experiment concludes only on Wednesday or on either of Tuesday or Wednesday. Some denominators change. None of it matters. It's still just counting.

There's no conflict between believing P(HEADS) = 1/2 unconditionally and believing P(HEADS | you're asking) = 1/3. That's all there is to it. If asked, she should say "1/3". I don't think there's anything like her credence changing going on it at all. It only looks that way because she's answering the question "What is your credence?" If you don't ask, it's one thing; if you do, it's another. Just the usual silliness.

Notice how you've decided (correctly) not to consider Heads + Tuesday and argue that in my example there's a 1/3 chance of it being heads. Why the inconsistency? — Michael

Because unlike in the Sleeping Beauty problem, in your example if there's x probability of going through any day you can draw the conclusion there's x probability of it being that day. The heads + Tuesday option is still there, but because of the rules of your variation the odds of going though any outcomes with a specific coin flip need to add up to 50%.

so that's not the explanation for the magic 1/6. — Srap Tasmaner

1/2+1/6 = 2/3

P+(Heads) = P(Heads)+1/6

1/6 is the differences between P+(Heads) and P(Heads)

I think it is just a statement noting the difference, to show they are not the same thing.

I think several of you are way overthinking this, but to be honest, I think philosophers and the like tend to do that. This whole problem was summed up pages ago.

1/2+1/6 = 2/3

P+(Heads) = P(Heads)+1/6 — Jeremiah

That is P(HEADS | you told me it's Monday) = P(HEADS | you asked me) + 1/6

For Elga, that's 1/2 = 1/3 + 1/6

Here's how I got the idea to include Wednesday, and it's still a good argument, I just dropped it because it all comes down to counting anyway.

Lewis accepts that being told it's Monday, and therefore not Tuesday, is relevant new information. (It tells you you're not in the future!) He does not accept that being asked for your credence is relevant new information, but it is: it tells you it's not Wednesday. If he accepts one, he ought to accept the other. Game over.

I had this whole analysis worked out about Beauty's credence upon being awakened Monday or Tuesday or Wednesday before being interviewed -- awakened on Wednesday, she won't know the experiment's over until they tell her. You can redo the weighted average credence I did before (and which is the same argument Elga makes) including Wednesday:

$\small \cfrac{1}{3}(50\%)+\cfrac{1}{3}(0\%)+\cfrac{1}{3}(50\%)\approx 33\%$

I still find that convincing. Maybe it's even more convincing than

$\small \cfrac{2}{3}(50\%)+\cfrac{1}{3}(0\%)\approx 33\%$

because it bakes in why being asked the question is informative, and why Lewis ought to be willing to conditionalize on H1 ∨ T1 ∨ T2.

That is P(HEADS | you told me it's Monday) = P(HEADS | you asked me) + 1/6 — Srap Tasmaner

I already know this. It is a statement showing P+(Heads) does not equal P(Heads). Elga's argument is that her credence in H, because she knew she would be awaken on Monday, should shift when she is awakened on Monday. Lewis is merely showing that the two are not the same thing.

Let H be the proposition that the outcome of the coin toss is Heads. Before being put
to sleep, your credence in H was 1/2. I’ve just argued that when you are awakened
on Monday, that credence ought to change to 1/3. This belief change is unusual. It is
not the result of your receiving new information — you were already certain that you
would be awakened on Monday.3 (We may even suppose that you knew at the start of the
experiment exactly what sensory experiences you would have upon being awakened on
Monday.) Neither is this belief change the result of your suffering any cognitive mishaps
during the intervening time — recall that the forgetting drug isn’t administered until well
after you are first awakened. So what justifies it?
The answer is that you have gone from a situation in which you count your own
temporal location as irrelevant to the truth of H, to one in which you count your own
temporal location as relevant to the truth of H. - Elga

https://www.princeton.edu/~adame/papers/sleeping/sleeping.pdf

The problem with Elga's argument is that you can't count your temporal location relevant if you don't know what it is.

Never-mind misread.

:smile:

The problem with Elga's argument is that can't count your temporal location relevant if you don't know what it is. — Jeremiah

When you are first awakened, you are here:

$\small P(HEADS) = \cfrac{1}{3}(50\%)+\cfrac{1}{3}(0\%)+\cfrac{1}{3}(50\%)\approx 33\%$

The $\small \cfrac{1}{3}$'s there are the days -- you don't know if it's Monday or Tuesday or Wednesday, but you know you've been awakened.

(Do we really have to say you already knew this was going to happen? That you were certain? What if you're awakened by a bloody research assistant surrounded by rubble? Do you assume the experiment is still running? What's become of this certainty about the future Elga attributes to you?)

As soon as you're asked for your credence, you switch to

$\small P(HEADS) = \cfrac{2}{3}(50\%)+\cfrac{1}{3}(0\%)+\cfrac{0}{3}(50\%)\approx 33\%$

and now you're weighting by possible interviews per day.

Wednesday is not from the same population of awakening where she is interviewed. — Jeremiah

Well, yeah. She's awakened but not interviewed. According to the setup this does happen.

you don't know if it's Monday or Tuesday or Wednesday, but you know you've been awakened. — Srap Tasmaner

Sorry, I actually misread a few words in your post, however. . .

Wednesday is irrelevant. The interview is what triggers the response, before then you are merely speculating. Also the probability if it being Wednesday would drop to 0% as soon as the interview happened, and her response to the interview is what we are concerned with. And one more point, Wednesday is not new information. Everything we know, she knows.

When you are first awakened, you are here: — Srap Tasmaner

We are already there and no one has been put to sleep yet.

Speculative claims about what Beauty might be thinking before the interview is speculative, and you can't update your probability model on that. The nature of the problem is to give Beauty's response to the question itself. So she already knows it is not Wednesday when trying to determine the credence and even if you do see that as "new" information, that still does not justify additional reallocation of credibility, you need to justify them all.

One might argue that all else being equal Beauty should go with the 50% credence, since 50 is greater than 33, which gives her a higher chance of being right. If both arguments are equal that is.

I think one of the best ways to envision probability is to imagine what would happen if you did the event 10,00 times, so that is what I have done


Here is the output:

[1] 5037 - Number of Tails
[1] 4963 - Number of Heads

[1] 4963 - Number of Monday and Heads
[1] 5037 - Number of Monday and Tails
[1] 5037 - Number of Tuesday and Tails

You can see that the 1/3 argument does in fact lead to a 33% split and the 1/2 argument does lead to a 50% split; however, you should go with the 1/2 argument, as 4963/10000 is better odds than 4963/15037, therefore you have a better chance of being correct.

The simplest way to block the Wednesday argument is to change the experiment: they send you on your way immediately after your last interview. It's cleaner. The wake you up on Monday, flip a coin, and ask for your credence. If it was heads, they send you on your way. If it was tails, you get the amnesia sleep, then they wake you up on Tuesday, ask for your credence and send you on your way. No letting you sleep through Tuesday, on heads, no waking up Wednesday not knowing if you'll be interviewed.

Now when you wake up, all you have is the original weighted average:

$\small P(HEADS) = \cfrac{2}{3}(50\%)+\cfrac{1}{3}(0\%)\approx 33\%$

I don't know if anything here counts as "updating".

One might argue that all things being equal Beauty should go with the 50% credence, since 50 is greater than 33, which gives her a higher chance of being right. If the both argument are equal that is. — Jeremiah

The problem is that wagering confirms the odds are 2-1, which, duh, there are 2 tails interviews for every heads interview. If it's all about getting to give the right answer most often, there's no way to go but tails.

Geez, I've been staring at this far too long. The magic 1/6 is right there. We already have $\small \cfrac{2}{3}$ credence that it's Monday, when our credence would be 50%. Tell me it's Monday and I get to add the last $\small \cfrac{1}{3}(50\%)$, or 1/6.

The problem is that wagering confirms the odds are 2-1, which, duh, there are 2 tails interviews for every heads interview. If it's all about getting to give the right answer most often, there's no way to go but tails. — Srap Tasmaner

She is asked about if the coin landed on heads, we can't go tails. The short and dry of it is all, is that a coin flip is still a coin flip. I posted my simulation up above, if you are considering the 1/2 argument then the odds are not 2:1 in favor of tails, they are 1:1 and those are better odds.

Oh right. Give the 50% answer you'd give on Monday, because there are more Monday interviews. I remember thinking about that a while ago -- you get to be right 2/3 of the time.

The simulation proves that both views are valid ways to look that the possible outcomes. All this nonsense about to reallocate or not to reallocate is just that nonsense, both proportions can be used to represent the problem accurately. There is no mistake in the math on either side, there is no mistake in the theory on either side and people are grasping at straws over this idea of new information just to argue their point of view. The reasons being given for why it should be 1/3 over 1/2 would never be a practical reason to give for application.

Oh right. Give the 50% answer you'd give on Monday, because there are more Monday interviews. I remember thinking about that a while ago -- you get to be right 2/3 of the time. — Srap Tasmaner

This no good though. If you know you're wrong 1/3 of the time, your credence is really the 33%.

The simulation proves that both views are valid ways to look that the possible outcomes. — Jeremiah

I understand Elga's argument. I understand the wagering argument. Do you understand Lewis's argument? I don't. He tries to get you to accept P+(HEADS)=2/3 because of something about prophecy and backwards-in-time causation and accepting credences you wouldn't normally. I've read it several times but I don't know what he's on about.

He is trying to show Elga's argument leads to a contradiction, the same contradiction you posted here: https://thephilosophyforum.com/discussion/comment/184884

But I have already said several times, I think they are both wrong.

Is there any argument for not reporting both the 1/2 and 1/3? Seems like a perfectly valid solution to me and if I ever had a similar dilemma in the real world that is exactly what I would do.

Huh. Didn't realize my first argument might be a contradiction.

I'll slog through the Lewis some more. He also notes that you can't jump straight to indifference about which of the three interviews is happening -- although you can argue for them being equiprobable -- and I didn't quite get that either.

If your position is that it's just a prior and you can pick whatever you want (thus Bayesianism is intellectually bankrupt), then the section at the end of Lewis's article is relevant right? It was all this stuff about believing now a credence you know you should believe in the future.

Is there any argument for not to reporting both the 1/2 and 1/3? Seems like a perfectly valid solution to me and if I ever has a like dilemma in the real world that is exactly what I would do. — Jeremiah

I think "What do you mean?" is a good answer. "If you weren't asking me in this bizarre manipulative way, I'd say 50%; since you are I'll say 33%."

If your position is that it's just a prior and you can pick whatever you want — Srap Tasmaner

That was never my point. My point is that you don't update priors with priors. Priors can be subjective but the "new information" needs to be objective. Priors on the other hand need solid reasoning to be used, and I have said many time both are valid positions; as such, you could justify either one as a prior.

You are too fast for my ninja edits, so you end catching my typos. I don't want to review any more Lewis tonight, maybe tomorrow I'll take another look at the end.

However, she will be able, and she will be taught how, to distinguish her brief awakenings during the experiment from her Wednesday awakening after the experiment is over, and indeed from all other actual awakenings there have ever been, or ever will be. — Lewis

Hey look at that. He saw the Wednesday argument and slipped in a defeater!