Recall that in (2) and (2') which you refer to above, there is no interuption of consciousness. In (2) her mind is read twice during her hour awake when tails, once during her hour awake when heads. In (2') she is asked her credence twice during her hour awake when tails, once during her hour awake when heads.

So in quoting the above, are you saying she should change her credence in (2') because she is asked twice rather than just knowing her mind will be read twice? If so, why?



I'm not sure what argument you think you're refuting. In (2) and (2') she is always awakened once for an hour. In (2) her mind is read twice when tails and once when heads. In (2') she is asked her credence twice when tails and once when heads (no short term memory). When you say "what if she is awakened twice if it is tails and ZERO times if it is heads?", what scenario are you talking about. (2) and (2') or the original Sleeping Beauty problem? What argument are you claiming to refute? If the original Sleeping Beauty problem, there is no mind reading involved so why do you mention mind reading? If you're referring to (2) and (2') you seem to misunderstand that she is always awakened only once for an hour in both those scenarios.

If you are trying to refute the 1/2 argument for the original Sleeping Beauty by appealing to an altered version where she is not awakened at all for heads then that argument has been addressed, probably best by jason1990 in his post #374, shown below. Pay special attention to the Theorem he states.


In the Sleeping Beauty problem, SB knows she will be awakened and asked her credence at least once. So when she experiences an awakening she learns nothing new. She already knew she would with certainty she would experience an awakening. But in your scenario above she doesn't know beforehand that she will certainly experience an awakening. There is a non zero chance she will not experience an awakening. So if (not "when" as before) she experiences an awakening she now does indeed learn something new and if she conditions on that information she will get new probabilities.

Notice in the Theorem, E is the new information she gets if she experiences an awakening in your altered version (no awakening on Heads). Notice the P(not E) > 0 thus satisfying the condition of the Theorem. Whereas, in the original Sleeping Beauty, when she experiences an awakening, what she learns, E, is that at least one awakening occurs. But in that case, P(not E) = 0 so the condition of the Theorem is not met. She knows with certainty beforehand that she will experience at least one awakening.


jason1990's post #374

PairTheBoard

It was already addressed somewhere in the beginning of the threat.
If you run original SB experiment you have two possible outcomes:
a)SB was awakened on MH (Monday and heads)
b)SB was awakened on both MT and TT

Probability of either occurring is 50%. If SB finds herself in the experiment she knows she is in one of those two worlds with equal probability, in your example she knows exactly she is in tails world, because a) is never recorded during long run (in original experiment it's recorded half the time).

So now think about this experiment:
We flip a coin and put a ball in a bucket and then we again put a ball in the bucket every time it's tails and no balls it's heads (so 2 balls for tail flip overall).
1/3'ers now reason: "Well, there were 1000 trials of the experiment and 1000 balls in a bucket, so probability of putting a ball in a bucket is 100%". Mistake in this is obvious, putting two balls in a bucket is not two separate events, it's one thing, it just happen we do something twice during it. The same goes for original SB problem.

Similar example is this: we put 1 green ball in a bucket on heads, 2 blue balls on tails. We do our thing and ask you to draw random ball from the bucket without looking at it. You hold a ball in your hand and somehow are teleported inside the ball, you can't see it's color from inside. What is now your credence the coin was heads ?
If you are now told that in case there are 2 balls to choose from you could have been given amnesia pills before, does it change anything ?

It's very difficult to address 1/3'ers argument adequately as they talk about events, outcomes and experiment but refuse to say how an outcome of one run of experiment is determined. I see you are back in this thread and you advocated "50/150" argument before. What about you address this ?
You claim one awakening is one "event". What is experiment which is used to determine if that event is satisfied ? Remember that every experiment produces exactly one outcome and events are sets of outcomes (given event is satisfied if one of the outcomes which belongs to it is the result of the experiment).

This is just silly.

OP basically says, "Now if I give you reason to believe heads is less likely, would you change your answer?"

Yes, of course.

She could very easily give an answer greater than 1/2 if the converse of this additional information were revealed: waking twice on heads, once on tails.



Edit: To clarify rephrase OP as such

SB doesn't know anything in advance. She wakes up and is asked, "A coin was flipped. What do you think it landed?" She of course assigns a 50/50 chance for each side.

Then she is told, "What if I were to tell you *insert all of OP here*" what would your credence be now? Obv she says 1/3.

Now it's easy to illustrate that information could be given to adjust the probability in the opposite direction. The only thing I've changed is that she's not told the experiments details until after. Clearly that won't change the odds, but it refutes Jason's main problem with the 1/3 side.



Edit #2: If you still don't understand how information can be added to this effect, you basically just need to realize, "the turn card has already been dealt" (using Jason's own metaphor).

It's even more spectacular when you know RLK is also convinced he has proven, "Theism is in fact the only logical philosophy for guiding life choices if one considers the implications of the alternatives rationally."

It's ridiculous how theists have a way of using 32409x as many words as necessary. This post needed only two words: "Pascal's Wager". Perhaps a GG at the end for flair.

Thanks for the link

You might consider reading jason1990's posts more carefully. His application of the theorem only defeats arguments that claim post-awakening information allows Beauty to update her credence to 1/3.

Indeed, at the end of that same post, he writes: "There are, however, arguments for 1/3 which do not rely on frequencies, and instead rely on seemingly reasonable probability assignments based on the indifference principle. In accordance with the above theorem, they produce the surprising result that the probability of heads is 1/3 before the experiment even begins."

(Emphasis mine.)

Yes, I've already addressed this. The short version is, "She receives new information that she knew in advance she was going to receive."

I mean, clearly you've spent time thinking about this, so I'm ready to believe that this notion of a "postdated" prior probability makes sense within your framework.

Nonetheless, as things stand I'm very far from convinced that it can be made rigorous in a way that evades the theorem while preserving your frequentist intuition...but I'll keep reading!

Not really sure what you're talking about but as I read that it is the complete opposite of what he says. His theorem says that updating her credence to 1/3 REQUIRES post-awakening information.

Jason's point is this:
You believe something to be 50/50, you now receive information that makes you believe the odds are now 2/3. This can only be true if it is possible to receive the reverse information which changes the odds in the other direction.

If something (A) happens that makes some event (X) more likely
P(X |A) > P(X)

some probability of X remains which has to go in the NOT-A category

P(X) = P(X |A)*P(A) + P(X |~A)*P(~A)

where P(X |~A)*P(~A)>0 and P(X |~A) < P(X)

therefore, there must be some chance that NOT-A occurs.

which IMO is the case in SBP because she can wake up on tuesday outside of the experiment, which would lead her to conclude that P(heads)=1

Well, the assumption is that "experiment ends" on after either MH or TT so it would be more accurate to say that she can wake up next Sunday (or w/e) after all the mess. If we ask her then what is her credence about the coin being head she can just recall what happened during the experiment (one awakening or two). If she can't it's still 50%.
She can't wake up on Tuesday after the experiment and conclude that otherwise the experiment would be still going on, it's specifically given in SBP desctription.

Read Kenross' post 445

It's not what SB experiment is.
If the experiment was something like: "we flip 2 coins to determine one time we wake SB up: MH, TH, MT, TT then wake SB up and tell her that it's not TH then her credence for heads should be 1/3. This is completely different scenario though as she would be awakened once per experiment so we can see well defined 4 outcomes of the experiment.

Also, it's not practical to analyse it any further after seeing this:
three probability spaces ? wtf
We need one space to describe experiment. Probability space is set of outcomes of experiment with events defined as subsets of set of all outcomes. There is nothing magical about it. Unfortunately it seems that many 1/3'ers try to sound formal while ignoring definitions entirely thus confusing themselves in the process.
This is nothing more than trying to justify intuitive answer "1/3" which comes from flawed frequency argument ("50/150 = 1/3" one).

PairTheBoard -
"Notice that "Guess" is singular"

Ok, how about the guess of heads she makes when the coin is heads?

I think what you're really saying is, "let's consider an arbitrary SB guess." That would be fine in a logical or mathematical argument where you go on and prove certain properties such an arbitrary guess must have. But you're doing something special with this arbitrary guess. You are assigning probabilities for it. How do I know this? Because you conclude that

(1) P( [SB's Guess of Heads] is True) = 1/3

Beauty decides on her guess of heads Sunday night. So the guess of heads is fixed. There is no randomness involved in what she guesses. So for your 1/3 probability in (1) to hold there must be randomness involved for the various Conditions within the experiment under which she makes her arbitrary guess. What are these Conditions?

For SB to make an arbitrary guess she must be in an arbitrary awakening. That is, she is in an awakening where the coin outcome {h,t} and the day {M,T} are unknown. So there are three possibilities for the Conditions under which she makes her arbitrary guess. (h,M), (t,M), (t,T). You are rolling up the guesses of heads she makes under all those three conditions into 1 "random awakening guess of heads". But (h,M) is the only one of the three Conditions where her Guess of heads is true. So when you claim in (1) that the probability is 1/3 her heads-guess is True you are claiming the probability is 1/3 she makes this unspecified, arbitrary, random-Conditions heads-guess when having an unspecified, arbitrary, "random" awakening in (h,M). You are claiming probability 1/3 for a "random" awakening in (h,M).

I think this is what you have in mind. SB is in a "random" awakening where the coin's outcome {h,t} and the day {M,T} are unknown. In this "random" awakening she faces the prospect of making a guess of heads. You claim she should consider the probability of her heads-guess being True as 1/3. But that can only be true if the probability is 1/3 that the "random" awakening in which she makes her heads-guess is the awakening for (h,M).

So you have not escaped the problem of explaining how your perspective of this experiment can produce a "random" awakening with 1/3 probability for (h,M) via a probability space consisting of distinct, mutually exclusive possible atomic outcomes for the "random" awakening.

As far as your frequentist argument, in light of the above it's the same one that's been made before. You might consider this comment by jason1990 in post #37

Notice that the Truth values of her heads-guesses are not independent.


PairTheBoard

This is exactly what I just addressed. If she were told she would be awoken twice if heads and once if tails, that pushes the odds in the opposite direction.

He means without changing the original setup. Which is in the SB-scenario impossible.

No, you're changing the experiment instead of changing the information.

Prior to the flip she says the odds of heads (P(X)) is 1/2, after she wakes (A) up she says it's P(X |A)=1/3, clearly some information has been added. Halfers argue that P(A)=1, if this is true then P(X)=P(X |A)=1/2, which you disagree with. So P(A)<1 and P(X |~A)>1/2, there must be some situation where she does not wake up (inside the experiment ~A), and she must conclude that P(X |~A)>1/2

Something (some information) has to change (BUT NOT THE RULES OF THE GAME) between her evaluation before the coinflip and when she wakes up. For this to make any sense the opposite change must be possible aswell.

If you know an opponents range to be nuts(X) or air(~X) and he always bluffs (bets (A)) his air and always valuebets (bets (A)) his nuts then the fact that he bets does not give you additional information.
P(X) = P(X |A)P(A) + P(X |~A)P(~A). In this example he always bets, so P(A)=1 and P(~A)=0, so P(X)=P(X |A).

Now if he bluffs only 50% of the time when he has air en still vbets all the time then when he bets it's more likely to be nuts than before, because he doesn't bet all the time anymore
P(~A)>0 and P(X |A)>P(X |~A)

The original setup is a coin is flipped. The additional information is how awakenings/interviews occur.


Just because it's told to her in advance that she's going to receive information, doesn't mean it doesn't count as additional information. It's just like the Montey Hall problem. You are just confused by the order things happen in, but if we change the order, we have the exact same problem, and now this information clearly appears to be new additional information that was not in the original setup.

Just for the sake of argument, let's assume she was also awoken on tuesday&heads and was told, "It's tuesday and heads, and we're not asking your credence today", do you believe this should change her credence in the awakenings where she IS asked her credence? And if so, why?

First, it is not my view that A, B, and C are the same propositions. They have the same truth-value (when A is true, B and C are also true, when A is false, B and C are also false), but they are different in meaning (interestingly enough, and possibly relevant, this means that they can have a different truth value when placed within a propositional attitude report. For instance, it is true that I believe that Newt Gingrich is fat. But it is not true that I believe that jason1990 is fat. This would be so even if jason1990 is Newt Gingrich.). Rather, you should see my claim as being that A is a complete proposition when placed within a context of utterance, as then the referent of "my father" is fixed, so that we know that it refers to the father of the speaker. In other words, we can think of "my father" as operating like a kind of temporary name that can be used in some contexts to refer to the speaker's father.

Now, you said that in your view, "a proposition is a declarative sentence with a definite, stand-alone truth value, that does not rely on the context in which it is uttered." This is slightly ambiguous. If you mean to say that the truth-value of the proposition is not context dependent, then there is no problem with statements that include indexicals (as it is the meaning, not the truth-value, that is context dependent on the theory I'm putting forward). If, however, you mean to say that the meaning of the proposition is not context-dependent, then there is a problem. If the latter is your view, I would challenge why you have such a narrow view of propositions and point out that you would have to then develop a theory of propositions* for declarative sentences with a definite, stand-alone truth value, where that truth-value does not rely on the context in which the sentence is uttered, but which propositions we pick out by that sentence does.

Finally, you ask thirders to present an indexical-free analysis. My response would be that we cannot. We can only capture the difficulty of this problem by using indexicals. Here's what I mean. Let's say that we name the three possible outcomes HM, TM, TT. Now, if Sleeping Beauty knew when she woke which outcome she was in, then she could know with certainty whether or not the coin was heads or tails. But of course, she cannot. The interviewer knows when he interviews Sleeping Beauty which outcome he is in. He can then ask Sleeping Beauty to guess which outcome she is in. What he is asking her to do is pick out which outcome (HM, TM, or TT) is identical with the outcome she is in. However, as a matter of language, Sleeping Beauty can only refer to the outcome she is in by using an indexical (as she doesn't know its name). Thus, Sleeping Beauty would have to say something like, "The outcome I am in right now is HM."

Your view seems to be that the phrase, "the outcome I am in right now" doesn't pick out an single outcome and so Sleeping Beauty's identification doesn't have a truth-value. But this just seems wrong to me. Sleeping Beauty is picking out a single outcome--the outcome in which she is being asked that question to which she is providing the answer. After all, the interviewer knows which outcome he is in when interviewing Sleeping Beauty. Let's say that he writes at the top of the his notes where it says "date and outcome" HM. Thus, all he has to do is look at his notes to see if Sleeping Beauty's identification is correct. It is not as if he then looked at his notes on the next day that he would think that the truth-value of Sleeping Beauty's answer had changed.

Say that on heads, she wakes up and does 1 interview in a red room.
And on tails, she wakes up and does 1 interview in a red room, then wakes up and does N-1 interviews in a blue room.

----

Given she is awake, thirders for the same reasons as before will claim:

P(heads|red,awake) = 1/2
P(heads|awake) = 1/(N+1)
----

So Alice reasons that when she awakens, but before she opens her eyes, that her new credence for heads must be:

P(heads|awake) = P(red|awake)P(heads|red,awake) + P(blue|awake)P(heads|blue,awake)

implying
1/(N+1) = (1/2)P(red|awake)

implying
P(blue|awake) = (N-1)/(N+1).


In particular, for arbitrarily large N, this probability approaches 1. But she knew that at the beginning of the experiment, there was a 50% chance she would never see a blue room upon awakening; that is:

P(blue|awake) <= 1/2.

Your experiment leaves out a crucial feature of the original scenario. If you could identify the second time you were interviewed as the second time (which you can't in the original SB scenario because of the amnesia drug), then clearly the rational thing to do is guess heads 1/2 in your first interview and 0 heads in your second interview.

Is there an argument about whether new information is added when SB wakes up? It seems to clear to me that there is. And that new information is "I am in the experiment".

New information added means it is possible for the probability of head to be something other than 1/2.

An example which is similar, but makes it clearer (I think it's been discussed previously in this thread):
1. SB put to sleep
2. Coin flipped
3a. If heads SB is brutally murdered.
3b. If tails SB is woken.
4. SB is asked whether coin was heads or tails

Would the '1/2ers' say SB should indifferent to guessing heads or tails here?

Going back to my original point. The full set of outcomes from {h,t} and {M,T} are hM,hT,tM,tT, but the new information of "being in the experiment" removes hT, which leads to 1/3 heads.

I strongly believe that 1/3 is correct and cannot comprehend the 1/2 argument, but I do not understand much of the advanced 'probability language' in this thread, so perhaps I'm missing something.

Wrong. The theorem implies that if post-awakening information E leads Beauty to update P(h) ⇒ 1/3, then ~E must have a nonzero probability, so that sometimes Beauty updates P(h) ⇒ >1/2.

I was trying to remain somewhat comprehensible, but if I needed to be more precise (for example, if I intended to pursue this as a line of research), then I would begin with the informal notion that a proposition is an equivalence class of declarative sentences under the relation of logical equivalence, where the sentences are formed in some previously agreed upon universe of discourse. I would then begin my literature search by investigating the work of Alfred Tarski from the 1930s, since I believe this was the approach he took in order to establish the connection between propositional logic and Boolean algebras. It is necessary to make this connection if we wish to use probability on propositions. (This is because, by definition, a probability measure is a function whose domain is a Boolean algebra.)

The point about indexicals is that they are not, in and of themselves, objects in the universe of discourse. I am fine with the word "today" serving as a kind of temporary name for some object in the universe of discourse. But if Sleeping Beauty does not know which object that is, then she does not know which element of the Boolean algebra is signified by the sentence, "Today is Monday," and she is therefore unable to evaluate the probability measure.