Here is yet another variation of the problem that I think perfectly identifies the source of the controversy. The experiment's methodology is the same as the original, except in these four details:
(1) Two coins, a Nickel and a Quarter, are flipped on Sunday Night.
(2) On either day of the experiment, Beauty is wakened if either of the two coins is showing Tails.
(3) On Monday Night, while Beauty is asleep, the Nickel is flipped over to show its opposite face.
(4) Beauty is asked the same question, but about the Quarter.
The only functional difference is that there is a 50% chance that the "optional" waking occurs on Monday instead of Tuesday. Since Beauty does not know the day in either version, this cannot affect the result; she is still wakened once if the Quarter landed on Heads, and twice if it landed on Tails.
The controversy boils down to whether the current state of the Nickel can be called a random variable. Or, much like Schrodenger's Cat while its box is unopened, the Nickel has to be considered to be in both states simultaneously for the purposes of the experiment.
Halfers treat it as both. The Nickel shows both Heads and Tails during the experiment, so Beauty cannot use it as a random variable. This is the crux of Radford Neal's argument, in the original experiment. That "Today" is an indexical becasue it has both the value "Monday" and "Tuesday" during the experiment, so it can't be used as evidence.
The thirder's argument is that what the Nickel is currently showing is not an indexical at all. While Beauty is awake, it has only one value. That value is unknown, and can have either value with probability 50%. So there are four states for {Nickel, Quarter} that, at any time during the experiment, are equiprobable in the prior. And that the evidence Beauty has, based on the fact that she is awake, eliminates {Heads, Heads} as a possibility. This makes the probability that the Quarter landed on Heads 1/3.
Well, I never checked back to see replies, and just tripped back across this.
The error made by halfers is in thinking "the entire analysis" spans four days. Beauty is asked for her assessment, based on her current state of knowledge, that the coin landed Heads. In this state of knowledge, the truth value of the proposition "it is Monday" does not change.
But there is another easy way to find the answer, that satisfies your criterion. Use four Beauties to create an isomorphic problem. Each will be told all of the details on Sunday; that each will be wakened at least once, and maybe twice, over the next two days based on the same coin flip and the day. But only three will be wakened on each day. Each is assigned a different combination of a coin face, and a day, for the circumstances where she will not be wakened. That is, {H,Mon}, {T,Mon}, {H,Tue}, and {T,Tue}.
On each of the two days during the experiment, each awake Beauty is asked for the probability that she will be wakened only once. Note that the truth value of this proposition is the same throughout the experiment. It is only the information a Beauty has that changes. On Sunday or Wednesday, there is no additional information and the answer is 1/2. On Monday or Tuesday, an awake Beauty knows that there are three awake Beauties, that the proposition is true for exactly one of them, and that there is no reason for any individual Beauty to be more, or less, likely than the others to be that one. The answer with this knowledge is 1/3.
You said: "The standard textbook definition of a proposition is a sentence that has a truth value of either true or false.
This is correct. And when a well-defined truth value is not known to an observer, the standard textbook definition of a probability (or confidence) for the proposition, is that there is a probability P that it is "true" and a probability 1-P that it is "false."
For example, if I flip a coin but keep it hidden from you, the statement "The coin shows Heads on the face-up side" fits your definition of a proposition. But since you do not know whether it is true or false, you can assign a 50% probability to the result where "It shows Heads" is true, and a 50% probability the event where "it shows Heads" is false. This entire debate can be reduced to you confusing a truth value, with the probability of that truth value.
During this part of the experiment, the statement "today is Monday" has the truth value "true", and does not have the truth value "false." So by your definition, it is a valid proposition. But Beauty does not know that it is "true."
During this part of the experiment, the statement "today is Monday" has the truth value "false", and does not have the truth value "true." So by your definition, it is a valid proposition. But Beauty dos not know that it is "false."
In either case, the statement "today is Monday" is a valid proposition by the standard definition you use. What you refuse to acknowledge, is that it is also a proposition that Beauty can treat as "true" or "false" with probabilities P and 1-P.
And the purpose of a thought experiment, is to define how ideal concepts work when you can't run them in principle. And strawman arguments do not change that.
She is allowed any reasoning she wants to use. The condition explicitly stated in the thought problem (see https://en.wikipedia.org/wiki/Thought_experiment, for why we shouldn't care about realism) is that experiences during the day will not help her to deduce what day it is, not that she can't use it to determine her initial belief about the day or the coin.
What this means, is that if Xi represents her ordered experiences, with X0 representing only the experience of waking up as defined by the experiment, that Pr(Today=Monday|Xi+1) = Pr(Today=Monday|Xi) for all i>=0. Not that she can't define Pr(Today=Monday|X0).
But you are right, there is no point in continuing if you insist on violating the problem statement.
"You're failing to distinguish between though experiments that are only mildly-fantastic, like ones assuming perfectly fair coins, when real ones have (say) a 50.01% chance of landing heads, versus highly-fantastic thought experiments, such as ones assuming that on Sunday you know exactly, in complete detail, what all your experiences will be on Monday."
I'm not failing to distinguish anything. I'm intentionally not bothering to distinguish what the problem statement says we should treat as indistinguishable. "While awake she obtains no information that would help her infer the day of the week." Whether or not you think it is more realistic, the problem you are solving is not the Sleeping Beauty Problem.
And I'm not saying that Beauty's experiences are the same. I'm just following the instructions in the problems statement, that says any information contained in the experiences of one day cannot be used to infer anything about the other.
And this is exactly what makes thought experiments interesting. Isolating one factor, and determining what its effect is when treated alone.
"Your Argument 1 doesn't seem persuasive to me, because I don't see how Beauty can be said to have prior beliefs when she is unconscious." And she similarly can have different beliefs, then she can project during the experiment, than she had on Sunday. If she can project back a state in the past, why does it matter if she was awake. (If you want a comparison, you seem to be saying that the Sailor's Child can't hold a belief about the coin that was flipped before he was born.)
If you want real experiences in Argument #2, go back and read the Tim and Tom version. I just get tired of people saying "you changed the problem" when all I did was introduce an element that instantiates the day with out proving information, which is valid an necessary.
In #3, I have no problem with experiences that an external observers sees as differentiating the day, as long as Beauty can't. The differences can exist, but provide Beauty with no information that identifies the day to her.
"[Elga and Lewis] don't realize that that is relevant information. They're mistaken." They are not. The very premise of the problem is that it cannot be relevant. The same reasoning suggests we don't need to accept that the coin is fair, or that Beauty might wake on Tuesday after Heads.
"Surely you would agree that a thought experiment is uninteresting if the conditions for it are actually impossible?" I absolutely would not. There is no coin, or a methodology for flipping one, that produces exactly 50%. In fact, if we could achieve the level of detail you try to with Beauty, a coin flip is deterministic.
And the conditions that we assume for nearly all of mathematics are "actually impossible." There are no dimensionless points, and no two lines have the exact same length. You can even debate whether the numbers "i", "-7", "pi,", or even "23" actually exist. See https://www.quora.com/Does-infinity-exist-If-it-exists-then-what-is-it .
The point of mathematics is to postulate an ideal circumstance, and deduce what happens in that circumstance regardless of whether it is "actually possible." Even philosophers know this: "In pure mathematics, actual objects in the world of existence will never be in question, but only hypothetical objects having those general properties upon which depends whatever deduction is being considered." Bertrand Russell, from the preface to Principles of Mathematics, page XLV.
The reason it is interesting, even if you restrict yourself to "actually possible" conditions, is that the "actual" answer is derived from the ideal one.
"I myself think that there is no need to use words like 'today' for this problem." I don't think it is possible to address it without it, or some substitute that performs the same indexing. And that saying there is no need, is affirming the consequent: If the solution you want to be true does not distinguish the days, then distinguishing the days is unnecessary. Regardless, if they are distinguishable, then we cannot go wrong by distinguishing them.
"I don't understand your argument for '1/3'."
A capsule of Argument 1: Beauty's prior is not the state on Sunday, since that state does not describe a measure that can vary over the course of the experiment. It is the state just before she is wakened, which includes a variable for the current day. There are four equiprobable combinations of this variable, and the variable "coin." One is eliminated because she is awake.
A capsule of a different version of Argument 2 (with Tim and Tom): On Sunday, Beauty places an imaginary, invisible coin that only she can find under her pillow. Since it is invisible, she doesn't know if it is Heads or Tails. But when she wakes, she can find it, and flip it over. Now there are three random variables: the real coin, Sunday's value for the imaginary coin, and the current value for the imaginary coin (These last two can be combined into one if you want). The eight possible combinations are again equiprobable, and two are eliminated.
A capsule of Argument 3: Four Beauties are used, based on the same coin flip (it must be flipped before Monday morning). Each will sleep through a different combination of the day and coin. Each is asked for her belief that the coin result is the one where she will sleep through a day.
One of these is in the exact experiment we are debating. The others' experiments are functionally equivalent. Each has the same answer. If I am one of them, I know (A) that three are currently awake, (B) that one will sleep through a day and two will wake both days, and (C) I have no information about which of the three I am. My belief that I am the one who will sleep can only be 1/3.
It is consensus on how one uses experiences as evidence, not the usage itself, that is only possible if the method is uncontroversial. Controversy just means that two people see its applicability differently. Not that it is impossible to use, or that either is correct or incorrect.
But neither Lewis, nor Elga, say anything about using Beauty's experiences during the day of an awakening as evidence. Elga's footnote is defining what he means by "new information," which we are calling "evidence." He never relates it to experiences during the day, only to experiences inherited from Sunday at the start of a day. So while Beauty's nose may itch, she has no idea why that itch should be more, or less, likely on Monday than on Tuesday. And how fantastic, or mundane, the experiment is, is completely irrelevant. We are told to assume that nothing affects Beauty's ability to distinguish Monday from Tuesday.
Removing hyperbole and the double negative from your compound sentence, it seems you said "[How] the conditions of a thought experiment can be achieved is relevant to its possible solution." I canceled the double negative by changing "is not at all irrelevant" to "is relevant." If I misinterpreted that, or you misstated your thought, I'm sorry - please correct me. But I disagree completely with that sentiment. A thought experiment is used when its conditions are not easily achievable in the real world, and applies only to the ideal conditions it describes.
I am not discussing your analysis, I am discussing KSVANHORN's. He says that the use of the word "today" is "problematic." He argues that evidence that identifies a day with the use of "today" is needed in order for Beauty to use it. This is incorrect. Her "today" refers to the moment when she uses the word, and the fact that she does not know the current moment may not prevent it from having a valid, logical meaning as a random variable with a probability distribution.
The issue in the Sleeping Beauty Problem, is when that prior is evaluated. Halfer's want it to be evaluated on Sunday Night, where KSVANHORN is correct: "today is Monday" and "today is Tuesday" are not valid propositions, let alone mutually exclusive propositions. My point is that her prior is just before she was awakened, where they are. This leads easily to the answer "1/3."
But I understand why that may not be easy to accept. That's why I have presented two alternative, but equivalent, formulations of the problem. They both lead to the inescapable answer that her belief should be 1/3.
Lewis says that the evidence that it is Monday or Tuesday is identical, not the totality of her thoughts and experiences is identical. A window and rain on only one of the days constitutes different experiences, but requires knowledge of the weather forecast to extrapolate that difference into evidence.
The context you omitted from the Elga quote was comparing Sunday's knowledge to Monday's, with no mention of Tuesday. He even added a footnote: "To say that an agent receives new information (as I shall use that expression) is to say that the agent receives evidence that rules out possible worlds not already ruled out by her previous evidence." His point was that she does not receive new information when she is woken on Monday.
But I had two points. First, the problem statement specifically says that Beauty has no evidence. Lewis' statement is describing this result; how it might be achieved is irrelevant to the thought experiment. Second, any analysis that uses such evidence is treating the propositions "Today is Monday" and "Today as Tuesday" as logically valid statements, even before evidence from her different thoughts and experiences accumulates. So you can't also claim that they aren't logically valid.
Sleeping Beauty (SB) volunteers for this experiment, and is told all these details by a Lab Assistant (LA):
Questions:
1) Is the question "What side is the Dime currently showing?" functionally different, in any way and on either day, than the question "How did the Dime land on Sunday Night?"
2) Is LA's probability distribution wrong in any way?
I think these answers are both "no." So LA can answer the probability question (s)he asks. Since case HH is eliminated, in LA's world the probability that the Dime is showing Heads is 1/3.
3) Is SB's prior probability distribution for the two coins the same as LA's?
4) Is SB's information the same as LAX's?
I think these answers are both "yes." SB's answer is the same as LAX's. The probabiltiy is 1/3.
As far as I can tell, a halfer will say that the answer to #4 is a definite "no." #3 is unclear to me, but how they address it seems to be how they justify saying her information is different.
I think Halfers use a Shcrodinger's-Cat-like argument where HH and HT both true at the same time. HH cannot be eliminated because HT can't. Literally, they seem to say that SB can't consider the current state of the Nickel (which corresponds to the day in the original experiment) to be a random variable, since it shows both faces during the experiment. That's an invalid argument here, since the sensor is based on what it is currently showing.
5) How is this question, in SB's world, any different than the original SB problem?
I'd really like to hear a halfer's answer. Because it isn't.