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Sleeping Beauty (SB) volunteers for this experiment, and is told all these details by a Lab Assistant (LA):
- I will put you to sleep tonight (Sunday) with a drug that lasts 12 hours. After you are asleep, I will flip two coins - a Dime and a Nickel. I will lock them in an opaque box that has a sensor which can tell if at least one coin inside is showing Tails.
- I will then administer a drug to myself, that erases my memory of the last 12 hours, and go to sleep in the next room.
- Until I am stopped (which will happen on Wednesday morning), when I wake up in the morning I will perform the following procedure:
- If the box's sensor says neither coin is showing Tails, I will administer a drug to you (in your sleep) that extends your sleep another 24 hours.
- If the box's sensor says that at least one coin is showing Tails, I will let you wake up. I will sa to you: "Before I looked at the box this morning, the probabilities the coins were showing HH, HT, TH, or TT were all 1/4. Now that we've proceeded to the awake portion of this procedure, what probability should each of us give that the Dime is currently showing Heads?" After receiving an answer, I will administer the amnesia drug to you, and then the 12-hour sleep drug.
- In either case, after you are asleep I will open the box, turn the Nickel over to show its other face, administer the amnesia drug to myself, and go to sleep in the next room.
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.
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.
- On Monday Beauty is awakened. While awake she obtains no information that would help her infer the day of the week. Later in the day she is put to sleep again.
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."
- On Tuesday the experimenters flip a fair coin. If it lands Tails, Beauty is administered a drug that erases her memory of the Monday awakening, and step 2 is repeated.
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.
You said "Discussion of Sleeping Beauty often slides [into a] problem in which Beauty's thoughts and experiences on Tuesday (if she is woken then) are ABSOLUTELY IDENTICAL to her thoughts and experiences on Monday." No, they don't. They assume that nothing in the set of experiences can change the assessment of what day it currently is. But that, of course, requires one to recognize that "today" is a valid random variable.
Carl works Monday through Friday in the European Rain Recording Society in Berlin. He records daily data from two field agents: Colin in London, and Carlos in Madrid.
On Sunday Night, the temporary janitor in his office is careless with his cigarette, and accidentally sets fire to some papers on Carl's desk. Most are totally destroyed; just the bottom half of one piece of paper remains. It says "It rained here today." Knowing nothing of the work that is performed in the office, he thinks this is a very odd message. "Today" and "here" are indexicals, and cannot have meaning at face value alone. So the note seems to convey no information. Still, he leaves it on Carl's desk.
When Carl arrives at work on Monday, and the accident is explained to him, he knows some details that can provide context. Each report he receives from his agents is a single piece of paper with a header at the top that states the date and the name of the reporting agent.
Case A: His agents only report on days when it rained. So he knows that it rained at least once, in at least one of the two cities. This isn't very useful to Carl, but it isn't "nothing."
Case B: His agents report every day, rain or shine. So he knows that four reports were one his desk when the fire started. The burned papers included three reports, and the half-burned one is the fourth. In addition to the information in case A, he can now place a 25% probability on each of the propositions that the report referred to Saturday in London, Saturday in Madrid, Sunday in London, and Sunday in Madrid, respectively.
In other words, when the words "today" and "here" can only be used in a context that resolves the issues linguistics has with indexicals, there is no such issue present. This is different from knowing that context, which is where probability is used.
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"Indexical" is an adjective meaning "of or pertaining to an index." As used in linguistics, and discussed in the article, it also means "without sufficient context to determine the value that is indexed." Much like the pronouns "he, she, it, that, ..." when no antecedent is evident. With the added meaning, it refers to the usage of the word, not to the word itself.
If an index's context is not supplied, as in the janitor's reading of the half-burned paper, it conveys no meaning. But if the context is explicit, as in "Colin Cumberbund; London, England; Saturday June 2, 2018: It rained here today," then the meaning is explicit despite the fact that by itself the word "today" conveys no information.
So it is a non sequitur to say that such words cannot be used in a logic problem. They can, if context is supplied for the index. And in a probability problem, they can be used if the context narrows the range to a set of values - a sample space - to which you can attach a probability distribution.
Example: On Sunday, Beauty is put to sleep. A six-sided die is rolled. Beauty is wakened on whichever day, over the next six days, that is indexed by the die roll (1=Monday, 2=Tuesday, etc.). She is asked "What is the probability that today is Wednesday?"
The word "today" does not refer to the entire range of days Monday thru Saturday. It is used on only one day, and it has one unchanging value over the period when Beauty is awake. The fact that she does not know the value makes it a random variable, not an unfathomable reference. The answer to the question is 1/6.
The answer is the same if she is woken twice (with the amnesia drug), based on rolling two dice until the result is not doubles. Or if she is woken once or twice by accepting doubles. It can be used because, even if she is awake on another day, its usage refers to a fixed index into the range. The answer is 1/6 because no day is preferred over the others, even when the number of awakenings is uncertain.
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I agree with the above analysis, about betting arguments. But not about the rest.
The error in the above argument, is that the details of the experiment do provide context for "today." But as a random variable, not an explicit value. This is complicated by the coin toss, but using the non sequitur "'today' is an indexical so we can't evaluate it" is a placebo used to avoid analyzing the context. "Raising technical challenges" does not mean "challenges that can't be met," it actually means they can.
Still, there are ways to avoid using an indexical in a solution. I suggested one in a comment to part 1: use four Beauties, where each is left asleep under a different combination of {Coin,Day}. Three are wakened each day of the experiment. One of those three will be awakened only once during the experiment. They are asked, essentially, "what is the probability that you will be awake only once?" It was agreed that this question is equivalent the original problem. Since there is no information that makes any of the three more, or less, likely to be the one, the answer is 1/3.
Another is to use identical twins Tim and Tom as interviewers. They play "Rock, Paper, Scissors" on Sunday night to see who will interview Beauty first. Beauty can describe the possible outcomes of the experiment based on the two propositions C="Heads" and I="Tim first." Each outcome in the sample space {TT, TF, FT, FF} has a prior (that is, on Sunday) probability of 1/4.
Case A: They wear nametags to the interview. If Beauty sees that she is being interviewed by Tim, she knows that the outcome is not TF. That is, that it is not possible that the coin was/will be Heads and Tom interviews first. From this, she can update her probability for the proposition C=True from 1/2 to 1/3. She can do the same if she is interviewed by Tom.
Case B: They conceal their names. She can assign a probability Q to the proposition that her interviewer is TIm, and 1-Q to the proposition that it is Tom. Regardless of what value Q has, the Law of Total Probability says the answer is 1/3.
"At any point in the history that Beauty remembers in step 2 of step 3, the proposition has a simple, single truth value."
No, it doesn't. This boils down to a question of identity. Absent any means of uniquely identifying the day -- such as, "the day in which a black marble is on the dresser" -- there is a fundamental ambiguity.
At any point in the history that Beauty remembers when she is in one of those steps, the proposition M, "Today is Monday," has a simple, single truth value. All day. Either day. If she is in step 2, it is "true." If she is in step 3, it is "false."
The properties of "indexicals" that you are misusing apply when, within her current memory state, the value of "today" could change. Not within the context of the overarching experiment.
This has nothing to do with whether she knows what that truth value is. In fact, probability is how we represent the "fundamental ambiguity" that the simple, single truth value belonging to a proposition is unknown to us. If you want to argue this point, I suggest that you try looking for the forest through the trees.
If Beauty's remembered experiences and mental state are identical at a point in time on Monday and another point in time on Tuesday, then "today" becomes ill-defined for her.
I tell you that I will flip a coin, ask a question, and then repeat the process.
If the question is "What is the probability that the coin is showing Heads?", and I require an answer before I repeat the flip, then coin's state has a simple, single truth value that you can represent with a probability.
If the question is "What is the probability that the coin is showing Heads?", and I require an answer only at after the second flip, the question only applies to the second since it asks about a current state.But it has a simple, single truth value that you can represent with a probability.
If the question is "What is the probability of showing Heads?" then the we have the logical conundrum you describe.
"Showing" is an indexical. It can change over time. But it is only an issue if we refer to it in the context of a range of time where it does change. That's why indexicals are a problem in general, but maybe not in a specific case.
"Today" is never ill-defined for Beauty.
"To an awake Beauty, the "experiment" she sees consists of Sunday and a single day after it."
No, it doesn't. She knows quite well that if the coin lands Tails, she will awaken on two separate days. It doesn't matter that she can only remember one of them.
The entirety of the experiment includes Sunday, Wednesday, and two other days. She knows that. The portion that exists in her memory state at the time she is asked to provide an answer consists of Sunday (when she learned it all), which cannot be "Today," and Today, which has a simple, single value.
I do not understand why you are so insistent on using "propositions" that include indexicals
Because the property that defines an indexical is that it can change over the domain where it is evaluated. Beauty is asked for her answer within a domain where "Today" does not change.
You didn't answer my questions, about the variable Sleeping Beauty Problem.
They're irrelevant.
I've learned from experience that I need halfers to answer them while they seem irrelevant. Otherwise, they argue that there is a difference, but can't say what that difference is. Yes, this has happened more than once.
Each of the four card outcomes leads to a problem equivalent to the first. But randomly choosing one of four problems equivalent to the first problem doesn't tell you what the solution to the first problem is.
Not yet, but it does tell you that the same answer applies to the original problem, and to the random-card problem.
So use four Beauties. Deal one card to each, but don't show them. And flip the coin on Sunday (necessary since we need the result on Monday).
In your step 2, bring the three awake volunteers together to discuss their answers. Tell them, truthfully, what they already know: "One of you was dealt card where the coin value matches the flip we performed on Sunday. Two were dealt a card with the opposite coin result. What probability should you assign the propositions that each of you is the one whose card matches?"
There are three possibilities. Each must have the same probability, since they have no information that distinguishes any one from the other. The probabilities must add up to 1.
They are all 1/3.
(Not in order)
The problem is this: it seems propositions, being the objects of belief, cannot in general be spatially and temporally unqualified.
Note the clause "in general." Any assertion that applies "in general" can have exceptions in specific contexts.
We similarly cannot deduce, in general, that a coin toss which influences the path(s) of an experiment, is a 50:50 proposition when evaluated in the context of only one path.
"In the philosophy of language, an indexical is any expression whose content varies from one context of use to another."
An awake Beauty is asked about her current assessment of the proposition "The coin will/has landed Heads." Presumably, she is supposed to answer on the same day. So, while the content of the expression "today" may change with the changing context of the overarching experiment, that context does not change between asking and answering. So this passage is irrelevant.
The problem with indexicals is that they have meanings that may change over the course of the problem being discussed.
And the problem with using this argument on the proposition "Today is Monday," is that neither the context, nor the meaning, changes within the problem Beauty addresses.
The above is telling us that a "proposition" involving an indexical is not a single proposition, but a set of propositions that you get by specifying a particular time/location.
No, it analyzed two specific usages of an indexical, and showed that they represented different propositions. And concluded that, in general, indexicals can represent different propositions. It never said that multiple usages of a time/location word cannot represent the same proposition, or that we can't define a situation where we know they represent the same proposition.
If we accept this agreement, we must avoid words such as 'I', 'my', 'now', or 'this', whose meaning or reference depends on the circumstances of their use.
So my corner bar can post a sign saying "Free Beer Tomorrow," without ever having to pour free suds. But if it says "Free Beer Today," they will, because the context of the sign is the same as the context when somebody asks for it. Both are indexicals, but the conditions that would make it ambiguous are removed.
"words will continue to be used in the same way" They do not change meaning within the discussion.
And over the duration of when Beauty considers the meaning of "today," it does not change.
the same word used at different points in the argument must have the same meaning.
"Today" means the same thing every time Beauty uses it. This is different than saying the truth value of the statement is the same at different points in Beauty's argument; but it is. She is making a different (but identical) argument on the two days.
"we must avoid words such as... 'now', ... whose meaning or reference depends on the circumstances of their use."
Only if those circumstances might change within the scope of their use.
requires that [words] be replaced by words that we can treat as uniform in meaning or reference throughout a discussion.
And throughout Beauty's discussion of the probability she was asked for, the meaning of "Today" does not change.
[The proposition "today is Monday" is] not a simple, single truth value; that's a structure built out of truth values.
At any point in the history that Beauty remembers in step 2 of step 3, the proposition has a simple, single truth value. But she cannot determine what it that value is. This is basis for being able to describe its truth value with probabilities.
"The proposition 'coin lands heads' is sometimes true, and sometimes false, as well."
No, it is not. It has the same truth value throughout the entire scenario, Sunday through Wednesday.
In some instances of the experiment, it is true. In others, it is false.
Just like "today is Monday" has the same truth value at any point in the history that Beauty remembers in step 2 of step 3. Your error is in falling to understand that, to an awake Beauty, the "experiment" she sees consists of Sunday and a single day after it. She just doesn't know which. In her experiment, the proposition "today is Monday" has a simple, single truth value. The truth of "it is Monday" never changes in any point of the scenario she sees after being wakened.
The point you are missing is that Day changes throughout the problem you are analyzing.
And the point I am trying to get across to you is that it cannot change at any point of the problem Beauty is asked to analyze.
The problem that I am analyzing is the problem that Beauty was asked to analyze. Not what an outside observer sees. She was told some details on Sunday, put to sleep, and is now awake on an indeterminate day.
She is asked about a coin that may have been flipped, or has already been flipped, but to her that difference is irrelevant. "Today is Monday" is either true, or false (which means "Today is Tuesday"). She doesn't know which, but she does know that this truth value cannot change within the scope of the problem as she sees it now.
Things like "today" and "now" are known as indexicals, and there is an entire philosophical literature on them because they are problematic for classical logic.
No, "time" is an indexical. That means that the value of time can change the context of the problem when you consider different values to be part of the same problem. Not that a problem that deals with only one specific value, and so an unchanging context, has that property.
While Beauty is awake, the day does not change. While Beauty is awake, the context of the problem does not change. While Beauty is awake, the other day of the experiment does not exist in her context. So for our problem, this resolves the issue that classical logic has with the word "today."
The problem with indexicals is that they have meanings that may change over the course of the problem being discussed.
But the meaning of "Today" does not change of the course of the problem Beauty is asked to address. This is different than her not know what that value is.
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And you didn't answer my questions, about the variable Sleeping Beauty Problem. They really are simple.
Are you really claiming that the statement "today is Monday" is not a sentence that is either true or false?
Yes. It does not have a simple true/false truth value.
It most certainly does. It is true on Monday when Beauty is awake, and false on Sunday Night, on Tuesday whether or not Beauty is awake, and on Wednesday.
A better random variable might be D, which takes values in {0,1,2,3} for these four days. What you refuse to deal with, is that its uninformed distribution depends on the stage of the experiment: {1,0,0,0} when she knows it is Sunday, {0,1/2,1/2,0} when she is awakened but not told the experiment is over, and {0,0,0,1} when she is told it is over.
Or you could just recognize that the probability space when she awakes is not derived by removing outcomes from Sunday's. Which is how conventional problems in conditional probability work. That a new element of randomness is introduced by the procedures you use in steps 2 and 3.
To illustrate this without obfuscation, ignore the amnesia part. Wake Beauty just once. It can happen any day during the rest of the week, as determined by a roll of a six-sided die. When she is awake, "Die lands 3" is just as valid a proposition - in fact, the same proposition - as "today is Wednesday." It has probability 1/6.
If you add in the amnesia drug, and roll two dice (re-rolling if you get doubles so that you wake her on two random days), the probability for "a die lands 3" is 1/3, but for "today is Wednesday" it is 1/6.
Since it is sometimes true and sometimes false, its truth value is a function from time to {true, false}. That makes it a predicate, not a proposition.
The proposition "coin lands heads" is sometimes true, and sometimes false, as well. In fact, you have difficulty expressing the tense of the statement for that very reason.
But, it is a function of the parameters that define how you flip a coin: start position, force applied, etc. What you refuse to deal with, is that in this odd experiment, the time parameter Day is also one of the independent parameters that defines the randomness of Beauty's situation, and not one that makes Monday's state predicated on Sunday's.
It is not a fixed moment in time; if it were, the SB problem would be trivial and nobody would write papers about it.
By being asked about the proposition H, Beauty knows that she is in either step 2 or step 3 of your experiment. This establishes a fixed value of the time parameter Day. And the problem is trivial - people write papers about it because they don't understand how Day is an independent parameter that defines the randomness of the situation, and not one that predicates one state on another.
The sentence "the sensor detects white" is not a valid proposition.
Then "Coin lands heads" is similarly a predicate, and so not a valid proposition.
But your argument about being a predicate, and not a valid proposition, does apply to the statement "It is the 9 o'clock hour." Because "hour" it is not a parameter you use to define the randomness of the situation.
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Here's some simple questions for you, to illustrate how randomness is being defined. Write the four labels {"Heads,Monday", "Tails,Monday","Heads,Tuesday", "Tails,Tuesday"} on four cards. Deal one at random to Beauty. Change step 3 of your experiment so that the day and coin result it mentions, are those on the dealt card. Change step 2 so that it mentions the other day. Change the proposition Beauty is asked to evaluate a probability for to "Coin lands on the face written on the dealt card."
If, on Sunday, she is shown that she was dealt, "Heads,Tuesday," this is identically your problem.
If, on Sunday, she is shown a different label, does this represent an equivalent problem with the same answer?
If she is not shown the label, does have the same answer? And is it still an equivalent problem?
(Sorry about the typo - I waffled between several isomorphic versions. The one I ultimately chose should have "both showed Heads.")
In the OP, you said:
Another serious error in many discussions of this problem is the use of supposedly mutually exclusive “propositions” that are neither mutually exclusive nor actually legitimate propositions. HM, TM, and TT can be written as
HM=H and (it is Monday)TM=(not H) and (it is Monday)TT=(not H) and (it is Tuesday).
These are not truly mutually exclusive because, if not H, then Beauty will awaken on both Monday and Tuesday.
Now you say:
A proposition is a sentence that is either true or false.
Are you really claiming that the statement "today is Monday" is not a sentence that is either true or false? That it is not "mutually exclusive" with "today is Tuesday"? Or are you simply ignoring the fact that the frame of reference, within which Beauty is asked to assess the proposition "The coin lands Heads," is a fixed moment in time? That she is asked to evaluate it at the current moment, and not over the entire time frame of the experiment?
Let me insert an example here, to illustrate the problem with your assertion about functions. One half of a hidden, spinning disk is white; the other, black. It spins at a constant rate V, but you don't know its position at any previous time. There is a sensor aligned along its rim that can detect the color at the point in time when you press a button. You are asked to assess the probability of the proposition W, that the sensor will detect "white" when you first press the button.
This is a valid proposition, even though it varies with time. It is valid because it doesn't ask you to evaluate the proposition at every time, but at a fixed point in time.
The problem with a statement whose truth varies with time is that it does not have a simple true/false truth value; instead, its truth value is a function from time to the set {true,false}.
It does have a simple true/false truth value if you are asked to evaluate it at fixed point in time. Your assertion applies to functions where every value of the dependent variable are considered to be "true" simultaneously.
I did give you the math, but I'll repeat it in a slightly different form. Consider the point in time just before (A) in my version, when Beauty is awake and could be interviewed, or (B) in yours, when Beauty could be awakened. At this point in time, there are two valid-by-your-definition propositions: H, the proposition that "the coin lands Heads" and M, the proposition that "today is Monday." Each is asking about a specific moment in time, so your unsupported assertion that we need to consider all possible values of the time parameter is wrong. The two propositions are independent, because at the moment in time where I asked you to evaluate it, H does not influence M.
The sample space (the set set of possible outcomes described by {H,M}) is {{t,t},{t,f},{f,t},{f,f}}. The probability distribution for this sample space is {1/4,1/4,1/4,1/4}. If Beauty is (A) interviewed or (B) awakened, she knows that the outcome that applies to the current moment in time is not {t,f}. So the probability distribution can be updated to {1/3,0,1/3/,1/3}.
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The error that halfer's make, is considering all values of the time parameter to applicable when Beauty is asked to make an assessment at a single, unknown time.
You point out that Elga's analysis is based on an unproven assertion; that "it is Monday” and “it is Tuesday” are legitimate propositions. As far as I know, there is no definition of what can, or cannot, be used as a proposition. In other words, your analysis is based on the equally unproven assertion that they are not valid. Can remove the need to decide?
- On Sunday, the steps of the following experiment are explained to Beauty, and she is put to sleep with a drug that somehow records her memory state. After she is put to sleep, two coins are flipped; a quarter and a nickel.
- On Monday, Beauty is awakened. While awake she obtains no information that would help her infer the day, or if she has been previously awakened. If the two coins are not both showing Heads, she is interviewed. An hour after being awakened, she is put back to sleep with a drug that resets her memory to the recorded state.
- The nickel is turned over.
- On Tuesday, Beauty is awakened. While awake she obtains no information that would help her infer the day, or if she has been previously awakened. If the two coins are not both showing Heads, she is interviewed. An hour after being awakened, she is put back to sleep with a drug that resets her memory to the recorded state.
- On Wednesday, Beauty is awakened once more and told that the experiment is over.
In each interview, Beauty is asked for her epistemic probability that the quarter is showing Heads.
Non-consequential option: replace "Beauty is awakened ... If the two coins show the same face, she is interviewed" with "If the two coins show the same face, Beauty is awakened and interviewed".
I hope we can all agree that the order of the potential awakenings changes nothing. She is not asked whether, and her answer cannot depend on if, it is Monday or Tuesday. Or which day she might sleep through. So this Beauty no longer cares about the propositions "it is Monday" and "it is Tuesday."
Beauty can assess the probability space that describes the two coins before the decision to interview her (or to wake her) was made. Most significantly, it doesn't change when the nickel has been manually turned over. Yes, the day changes the path it takes to get there, but the states look identical. In that state, there are four possible outcomes: {HH,HT,TH,TT}. The probability distribution is {1/4,1/4,1/4,1/4}.
But when she is interviewed, she knows that {HH} has been ruled out. The probability distribution can be updated to {0,1/3,1/3,1/3}.
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The point of contention in the Sleeping Beauty Problem is whether the probability state at the end of your step 1 is the same state as at the beginning of your steps 2 and 3. The mere fact that the two steps can be distinguished, and can result in different paths, demonstrates that they are not. If your definition of what constitutes a "valid proposition" cannot model this difference, then I suggest that it is that definition that is faulty.
And yes, there are other ways that I can demonstrate that the answer must be 1/3.
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Note: The worst red herring in this thread is about how Beauty might be able to tell how she has aged. This is a thought problem in probability, not an exercise in human physiology. We must assume that the mechanisms described in the problem function ideally as described. That includes "While awake she obtains no information that would help her infer the day of the week." Considering how these mechanisms might not be achievable is not productive.
You mis-characterize what Elga does. He never directly formulates the state M1, where Beauty is awake. Instead, he formulates two states that are derived from information being added to M1. I'll call them M2A (Beauty learns the outcome is Tails) and M2B (Beauty learns that it is Monday). While he may not do it as formally as you want, he works backwards to show that three of the four components of a proper description of state M1 must have the same probability. What he skips over, is identifying the fourth component (whose probability is now zero).
What it seems Elga was trying to avoid - as everybody does - is that Beauty still "exists" on Tuesday, after Heads. She just can't observe it. But it is a component you need to consider in your more formal modeling. To illustrate, here's a simple re-structuring of your steps that changes nothing relevant to the question she is asked:
- On Sunday the steps of the experiment are explained to Beauty, and she is put to sleep.
- On Monday Beauty is awakened. She has no information that would help her infer the day of the week. Later in the day she is interviewed. Afterwards, she is administered a drug that resets her memory to its state when she was put to sleep on Sunday, and puts her to sleep again.
- On Tuesday Beauty is awakened. She has no information that would help her infer the day of the week. The experimenters flip a fair coin. If it lands Tails, Beauty is interviewed again; if it lands Heads, she is not. In either case, she is then administered a drug that resets her memory to its state when she was put to sleep on Sunday, and puts her to sleep again.
- On Wednesday Beauty is awakened once more and told that the experiment is over.
In the interview(s), Beauty is asked to give a probability for her belief that the coin in step 3 lands Heads.
I'm sure you can make this more formal, so I'll be brief: State M, on Sunday, requires only proposition C describing what Beauty thinks the coin result is (*not* for what it actually is, which becomes deterministic at different times in different versions of the problem). There is no information in state M that favors either result, so the Principle of Indifference applies and the probability for each is 1/2.
State M1, when Beauty is first awakened, requires another proposition: D, for what day Beauty thinks it is (*not* for what day it actually is, which is deterministic). Due to the memory-reset drug, the same state M1 applies on both Monday and Tuesday. Since there is no information in state M1 that favors either result, the Principle of Indifference applies and the probability for each is 1/2. And (what seems to be overlooked by denying the existence of Tuesday when Beauty sleeps through it) D and H are independent. So M1 comprises four possible combinations of D and H that all have a probability of 1/4.
State M2 applies when Beauty is interviewed. The information that takes Beauty from M1 to M2 is that one of the four combinations is ruled out. The remaining three now have probability 1/3.
State M1 applies to your version of the problem at the point in time just before Beauty could be wakened, in either step 2 or step 3. It applies, and can be determined later when Beauty is awake, whether or not Beauty is awake at that time. Elga's solution is essentially the same as mine, except he does it in two parts by adding more information to each. It just avoids identifying the component of the state that Beauty sleeps through.