Imagine an only slightly different problem: You volunteer for an experiment where you may be wakened once, or twice, on Monday and/or Tuesday. The administrators of the experiment will flip a coin to decide whether it will be once, or twice; but they do not tell you what coin result determines that the number of wakings. And they do not tell you whether the day you will be left asleep, if only one waking is to occur, will be Monday or Tuesday. Oh, and you will be given the amnesia drug on Monday, if you are wakened on that day.
Whenever you are awake, you will be asked to assess, from your perspective based on these procedures, the probability that you will be wakened only once during the experiment. Surely (call this assertion A) the answer to this question must be the same as in the original Sleeping Beauty Problem.
But when you are awakened, you are told that you are one of four volunteers undergoing the exact same procedures based on the same coin flip; with one exception. The choices for the coin results, and the days, are different for each of the four volunteers. (Since there are four possible combinations, each is used.) You are not told which choices were made for you, or introduced to the others. Surely (Assertion B) this gives you no information about your own situation, so it cannot affect your answer.
But it does allow you to evaluate your perspective with a bit more clarity. Of the four volunteers, you know that one must still be asleep, and it isn’t you. That volunteer will be wakened only once. Of the other three, including you, two will be wakened on both days, and one will be wakened only once. Surely (Assertion C) this means that, from your perspective, the probability that you will be wakened once is 1⁄3.
Unless you can find a flaw with one of my assertions, this means that the answer to the original problem is also 1⁄3. There are valid, if unorthodox, mathematical proofs of this. But those who trust their intuition over mathematics want the answer to be 1⁄2. The so-called “double halfer” approach is just a rationalization for ignoring valid mathematics.
The rationalization: an indexical is an identifier for a value relative to some index. If no way to associate the indexical with actual values is provided, as in “the probability it will rain tomorrow,” then the indexical cannot be used to assess your perspective. But if such a means is provided, EVEN AS A PROBABILITY DISTRIBUTION, then it can be assessed. For example, say a different volunteer is told she will be wakened once or twice based on the roll of two standard dice; 1=Monday, 2=Tuesday, etc. If both dice results the same, she will be wakened only once. Upon being awake, she can determine that the probability that it is Tuesday is 1⁄6. “Tuesday” may ne an indexical, but context is provided to index it.
Hello Jeff. Good seeing you again. Happy new year.
A typical thirder argument follows the Self-Indication Assumption. That this awakening should be regarded as being randomly selected from one of the three possible awakenings. One awakening for Monday-Heads, one for Monday-Tails and another for Tuesday-Tails. The way I understand what your experiment does is, very cleverly, using actually existing participants to represent these possibilities. I.e. today, out of the three awaking participants two of witch would be awake on both days and one would be awake on one day only. So my argument against you is the same as my argument against SIA (or SSA as a matter of fact): in the same logical framework it is wrong to use terms such as [today] or [myself] as self-explanatory concepts while also treat them as in the same reference class of all days and all people. Because [now] or [I] is only meaningful if reasoning from a first-person perspective for they are inherently unique to oneself. If they are treated as unique then it is logically inconsistent to treat them as ordinary time or people. Even though they might be ordinary by objective measures, i.e. in the same reference class as other time and people from a third-person perspective/uncentered reasoning. More specifically applying to your argument, I disagree with assertion C. Just because there are three people awake today does not mean I should regard myself as randomly selected among those three or as if today is randomly selected among Monday or Tuesday.
In this question Monday and Tuesday are not indexical. They are defined by a calendar which could be interpreted as defined by the relative positions of planetary bodies. In this sense the dates are defined by objective events. They can be treated as ordinary compare to one another from a third-person perspective. What is indexical is rather the concept of [today]. Which require a perspective center (first-person) to define. My argument is that first-person and third-person reasoning (or centered and uncentered reasoning) should not be mixed together in the same logic.
Imagine an only slightly different problem: You volunteer for an experiment where you may be wakened once, or twice, on Monday and/or Tuesday. The administrators of the experiment will flip a coin to decide whether it will be once, or twice; but they do not tell you what coin result determines that the number of wakings. And they do not tell you whether the day you will be left asleep, if only one waking is to occur, will be Monday or Tuesday. Oh, and you will be given the amnesia drug on Monday, if you are wakened on that day.
Whenever you are awake, you will be asked to assess, from your perspective based on these procedures, the probability that you will be wakened only once during the experiment. Surely (call this assertion A) the answer to this question must be the same as in the original Sleeping Beauty Problem.
But when you are awakened, you are told that you are one of four volunteers undergoing the exact same procedures based on the same coin flip; with one exception. The choices for the coin results, and the days, are different for each of the four volunteers. (Since there are four possible combinations, each is used.) You are not told which choices were made for you, or introduced to the others. Surely (Assertion B) this gives you no information about your own situation, so it cannot affect your answer.
But it does allow you to evaluate your perspective with a bit more clarity. Of the four volunteers, you know that one must still be asleep, and it isn’t you. That volunteer will be wakened only once. Of the other three, including you, two will be wakened on both days, and one will be wakened only once. Surely (Assertion C) this means that, from your perspective, the probability that you will be wakened once is 1⁄3.
Unless you can find a flaw with one of my assertions, this means that the answer to the original problem is also 1⁄3. There are valid, if unorthodox, mathematical proofs of this. But those who trust their intuition over mathematics want the answer to be 1⁄2. The so-called “double halfer” approach is just a rationalization for ignoring valid mathematics.
The rationalization: an indexical is an identifier for a value relative to some index. If no way to associate the indexical with actual values is provided, as in “the probability it will rain tomorrow,” then the indexical cannot be used to assess your perspective. But if such a means is provided, EVEN AS A PROBABILITY DISTRIBUTION, then it can be assessed. For example, say a different volunteer is told she will be wakened once or twice based on the roll of two standard dice; 1=Monday, 2=Tuesday, etc. If both dice results the same, she will be wakened only once. Upon being awake, she can determine that the probability that it is Tuesday is 1⁄6. “Tuesday” may ne an indexical, but context is provided to index it.
Hello Jeff. Good seeing you again. Happy new year.
A typical thirder argument follows the Self-Indication Assumption. That this awakening should be regarded as being randomly selected from one of the three possible awakenings. One awakening for Monday-Heads, one for Monday-Tails and another for Tuesday-Tails. The way I understand what your experiment does is, very cleverly, using actually existing participants to represent these possibilities. I.e. today, out of the three awaking participants two of witch would be awake on both days and one would be awake on one day only. So my argument against you is the same as my argument against SIA (or SSA as a matter of fact): in the same logical framework it is wrong to use terms such as [today] or [myself] as self-explanatory concepts while also treat them as in the same reference class of all days and all people. Because [now] or [I] is only meaningful if reasoning from a first-person perspective for they are inherently unique to oneself. If they are treated as unique then it is logically inconsistent to treat them as ordinary time or people. Even though they might be ordinary by objective measures, i.e. in the same reference class as other time and people from a third-person perspective/uncentered reasoning. More specifically applying to your argument, I disagree with assertion C. Just because there are three people awake today does not mean I should regard myself as randomly selected among those three or as if today is randomly selected among Monday or Tuesday.
In this question Monday and Tuesday are not indexical. They are defined by a calendar which could be interpreted as defined by the relative positions of planetary bodies. In this sense the dates are defined by objective events. They can be treated as ordinary compare to one another from a third-person perspective. What is indexical is rather the concept of [today]. Which require a perspective center (first-person) to define. My argument is that first-person and third-person reasoning (or centered and uncentered reasoning) should not be mixed together in the same logic.