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.
“neither Lewis, nor Elga, say anything about using Beauty’s experiences during the day of an awakening as evidence”
Yes, they don’t realize that that is relevant information. They’re mistaken. That’s not particularly unusual for philosophers.
“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”
Surely you would agree that a thought experiment is uninteresting if the conditions for it are actually impossible? And don’t you think that it becomes less interesting if the conditions may be impossible (but we’re not sure)? Finally, if the conditions are possible only when in some highly unusual (but let us suppose possible) situation, you have to be very careful to not argue while assuming things that are normally true but are not true in such unusual situations. Situations in which people can anticipate the previous day what all their experiences will be the next day are so unusual that I have no confidence in my ability to reason about what such “people” should believe. I doubt that many people would find the Sleeping Beauty problem interesting if it were made clear that Beauty has to be a “person” of this sort.
I’m not trying to explicate ksvanhorn’s argument. I myself think that there is no need to use words like “today” for this problem. Perhaps it’s OK to do so, but since it isn’t necessary, I think people should just rephrase what they want to say while avoiding the word.
I don’t understand your argument for “1/3”. Of course, I agree that 1⁄3 is the correct answer, in the normal non-very-fantastical version of the problem, but that doesn’t mean that all arguments for 1⁄3 are correct.
“[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.
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 detai, what all your experiences will be on Monday. If such prevision is impossible, then thought experiments assuming it are uninteresting. More to the point, such a fantastic thought experiment is NOT THE SAME as a somewhat simillar thought experiment that makes no such fantastic assumption. Sleeping Beauty purports to be an only-mildly-fantatic thought experiment. If you want to talk about a fantastic Sleeping Beaurty in which the mental properties of Beauty are such that she is a completely different entity than any human could possibly be, then go right ahead. But don’t try to maintain that this is the same as the original Sleeping Beauty problem. Whether Elga realized that making fantastic assumptions matters is irrelevant—it does matter.
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.
The only way your Argument 2 makes any sense to me is as a way of introducing new experiences for Beauty—albeit ones involving an imaginary (and invisible!) coin. If you just accepted that the problem involves an actual person, this would be unnecessary, since real people have new, unique experiences all the time. If we’re talking about a version of S.B. in which Beauty is required to be an AI whose experiences are completely controlled by the experimenter, then I don’t think imaginary invisible coins are going to settle the issue.
Your Argument 3 seems persuasive to me, but I’m not sure why you find it persuasive. If you are unwilling to admit that Beauty has any experiences other than the single fact that she is awake, then it seems dubious that there are actually three awake people rather than one on Monday. If an AI is simulated redundantly by three computers operating synchonously, which all function properly, and so produce exactly the same thoughts of the AI, are there three AIs or only one?
“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.
I think there’s no point in continuing this discussion. As far as I can tell, you are willing to admit that Beauty will have a variety of experiences on Monday, which will very probably be different from those she will have on Tuesday (if awake), although (by the conditons of the problem) she doesn’t know ahead of time how the two days will differ. But you are saying that she is NOT ALLOWED to use the fact that she has such experiences when reasoning about whether the coin landed heads. Debates in which one party to the discussion just says by fiat that the other party isn’t allowed to use certain forms of reasoning are pointless.
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.
As you’re defined it, the thought experiment is simply impossible to run in principle, since if you apply standard principles of probability, Pr(Today=Monday|Xi+1) is NOT equal to Pr(Today=Monday|Xi). It’s like saying that as a condition of your thought experiment, arguments about the solution must assume that pi is exactly equal to three.
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.
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.
“neither Lewis, nor Elga, say anything about using Beauty’s experiences during the day of an awakening as evidence”
Yes, they don’t realize that that is relevant information. They’re mistaken. That’s not particularly unusual for philosophers.
“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”
Surely you would agree that a thought experiment is uninteresting if the conditions for it are actually impossible? And don’t you think that it becomes less interesting if the conditions may be impossible (but we’re not sure)? Finally, if the conditions are possible only when in some highly unusual (but let us suppose possible) situation, you have to be very careful to not argue while assuming things that are normally true but are not true in such unusual situations. Situations in which people can anticipate the previous day what all their experiences will be the next day are so unusual that I have no confidence in my ability to reason about what such “people” should believe. I doubt that many people would find the Sleeping Beauty problem interesting if it were made clear that Beauty has to be a “person” of this sort.
I’m not trying to explicate ksvanhorn’s argument. I myself think that there is no need to use words like “today” for this problem. Perhaps it’s OK to do so, but since it isn’t necessary, I think people should just rephrase what they want to say while avoiding the word.
I don’t understand your argument for “1/3”. Of course, I agree that 1⁄3 is the correct answer, in the normal non-very-fantastical version of the problem, but that doesn’t mean that all arguments for 1⁄3 are correct.
“[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.
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 detai, what all your experiences will be on Monday. If such prevision is impossible, then thought experiments assuming it are uninteresting. More to the point, such a fantastic thought experiment is NOT THE SAME as a somewhat simillar thought experiment that makes no such fantastic assumption. Sleeping Beauty purports to be an only-mildly-fantatic thought experiment. If you want to talk about a fantastic Sleeping Beaurty in which the mental properties of Beauty are such that she is a completely different entity than any human could possibly be, then go right ahead. But don’t try to maintain that this is the same as the original Sleeping Beauty problem. Whether Elga realized that making fantastic assumptions matters is irrelevant—it does matter.
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.
The only way your Argument 2 makes any sense to me is as a way of introducing new experiences for Beauty—albeit ones involving an imaginary (and invisible!) coin. If you just accepted that the problem involves an actual person, this would be unnecessary, since real people have new, unique experiences all the time. If we’re talking about a version of S.B. in which Beauty is required to be an AI whose experiences are completely controlled by the experimenter, then I don’t think imaginary invisible coins are going to settle the issue.
Your Argument 3 seems persuasive to me, but I’m not sure why you find it persuasive. If you are unwilling to admit that Beauty has any experiences other than the single fact that she is awake, then it seems dubious that there are actually three awake people rather than one on Monday. If an AI is simulated redundantly by three computers operating synchonously, which all function properly, and so produce exactly the same thoughts of the AI, are there three AIs or only one?
“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.
I think there’s no point in continuing this discussion. As far as I can tell, you are willing to admit that Beauty will have a variety of experiences on Monday, which will very probably be different from those she will have on Tuesday (if awake), although (by the conditons of the problem) she doesn’t know ahead of time how the two days will differ. But you are saying that she is NOT ALLOWED to use the fact that she has such experiences when reasoning about whether the coin landed heads. Debates in which one party to the discussion just says by fiat that the other party isn’t allowed to use certain forms of reasoning are pointless.
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.
As you’re defined it, the thought experiment is simply impossible to run in principle, since if you apply standard principles of probability, Pr(Today=Monday|Xi+1) is NOT equal to Pr(Today=Monday|Xi). It’s like saying that as a condition of your thought experiment, arguments about the solution must assume that pi is exactly equal to three.
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.