a) Regardless of how sleeping beauty makes their decision, we can model it as an algorithm decided ahead of time. If part of the decision is random, we can program that in too. So we can assume they make the same meta-level decision, so have the same expected pay-off for both interviews.
b) I don’t follow the argument here? You seem to just be assuming that I am wrong?
c) We can’t just say, “Radford Neal” for sailors child without defining who will have that name.Is it one particular mother who will call their child Radford Neal if they have one? Or is a random child assigned that name?
a) But Beauty is actually a human being. If your argument depends on replacing Beauty by a computer program, then it does not apply to the usual Sleeping Beauty problem. Why are you so reluctant to actually address the usual, only-mildly-fantastic Sleeping Beauty problem?
In any case, why is it relevant that she has the same expected payoff for both interviews (which will indeed likely be the case, since she is likely to make the same decision)? Lots of people make various decisions at various times that happen to have the same expected payoff. That doesn’t magically make these several decisions be actually one decision.
b) If I understand your setup, if the coin lands Heads, Beauty gets one dollar if she correctly guesses on Monday, which is the only day she is woken. If the coin lands Tails, a ball is drawn from a bag with equal numbers of balls labeled “M” and “T”, and she gets a dollar if she makes a correct guesses on the day corresponding to the ball drawn, with her guess the other day being ignored. For simplicity, suppose that the ball is drawn (and then ignored) even if the coin lands Heads. There are then six possible situations of coin/ball/day when Beauty is considering her decision:
1) H M Monday
2) H T Monday
3) T M Monday
4) T T Monday
5) T M Tuesday
6) T T Tuesday
If Beauty is a Thirder, she considers all of these to be equally likely (probability 1⁄6 for each). In situations 4 and 5, her action has no effect, so we can ignore these in deciding on the best action. In situations 1 and 2, guessing Heads results in a dollar reward. In situations 3 and 6, guessing Tails results in a dollar reward. So she is indifferent to guessing Heads or Tails.
c) Really, can you actually not suppose that in the Sailor’s Child problem, which is explicitly designed to be a problem that could actually occur in real life, the child has not been given a name? And if so, do you also think that if the child gets cancer, as in the previous discussion, that they should refuse chemotherapy on the grounds that since their mother did not give them a name, they are unable to escape the inapplicability of probability theory to statements with indexicals? I’m starting to find it hard to believe that you are actually trying to understand this problem.
a) Even if sleeping beauty is a human, they are still a deterministic (or probabilistically deterministic) machine, so their responses in any scenario can be represented by an algorithm.
b) The halfer gets the same solution (indifference) too as 1), 2), 5) and 6) are all assigned a probability of 1⁄4; whilst 3) and 4) are ignored.
c) My point isn’t that the child might not have a name. My point is that in order to evaluate the statement: “Radford Neal has a half-sibling” we have to define the scheme in which someone comes to be called Radford Neal.
So, suppose the two potential mothers are Amy and Barbara. The first possibility is that Amy calls their child, if they have one, “Radford Neal”. However, if this is the case, it may come to pass that Amy doesn’t have a child so no-one is called Radford Neal and the reference fails. Alternatively, we might want to ensure that there is someone always called Radford Neal. If they only have one child, this is trivial, if there’s two, we could pick randomly. My point is that there isn’t a unique way of assigning the name, so I don’t know what scheme you want to use to replace the indexical.
a) You know, it has not actually been demonstrated that human consciousness can be mimicked by Turing-equivalent computer. In any case, the only role of mentioning this in your argument seems to be to push your thinking away from Beauty as a human towards a more abstract notion of what the problem is in which you can more easily engage in reasoning that would be obviously fallacious if your thoughts were anchored in reality.
b) Halfer reasoning is invalid, so it’s difficult to say how this invalid reasoning would be applied in the context of this decision problem. But if one takes the view that probabilities do not depend on what decision problem they will be used for, it isn’t possible for possibilities 5) and 6) to have probability 1⁄4 while possibilities 3) and 4) have probability zero. One can imagine, for example, that Beauty is told about the balls from the beginning, but is told about the reward for guessing correctly, and how the balls play a role in determining that reward, only later. Should she change her probabilities for the six possibilities simply because she has been told about this reward scheme? I suspect your answer will be yes, but that is simply absurd. It is totally contrary to normal reasoning, and if applied to practical problems would be disastrous. Remember! Beauty is human, not a computer program.
c) You are still refusing to approach the Sallor’s Child problem as one about real people, despite the fact that the problem has been deliberately designed so that it has no fantastic aspects and could indeed be about real people, as I have emphasized again and again. Suppose the child is considering searching for their possible sibling, but wants to know the probability that the sibling exist before deciding to spend lots of money on this search. The child consults you regarding what the probability of their having a sibling is. Do you really start by asking, “what process did your mother use in deciding what name to give you”? The question is obviously of no relevance whatsoever. It is also obvious that any philosophical debates about indexicals in probability statements are irrelevant—one way or another, people solve probability problems every day without being hamstrung by this issue. There is a real person standing in front of you asking “what is the probability that I have a sibling”. The answer to this question is 2⁄3. There is no doubt about this answer. It is correct. Really. That is the answer.
Thanks for taking the time to write all of these responses, but I suspect that we’ve become stuck. At some point I’ll write up some posts aimed at trying to argue for my position, rather than primarily aimed at addressing rebuttal and perhaps it will clear up some of these issues.
a) Regardless of how sleeping beauty makes their decision, we can model it as an algorithm decided ahead of time. If part of the decision is random, we can program that in too. So we can assume they make the same meta-level decision, so have the same expected pay-off for both interviews.
b) I don’t follow the argument here? You seem to just be assuming that I am wrong?
c) We can’t just say, “Radford Neal” for sailors child without defining who will have that name.Is it one particular mother who will call their child Radford Neal if they have one? Or is a random child assigned that name?
a) But Beauty is actually a human being. If your argument depends on replacing Beauty by a computer program, then it does not apply to the usual Sleeping Beauty problem. Why are you so reluctant to actually address the usual, only-mildly-fantastic Sleeping Beauty problem?
In any case, why is it relevant that she has the same expected payoff for both interviews (which will indeed likely be the case, since she is likely to make the same decision)? Lots of people make various decisions at various times that happen to have the same expected payoff. That doesn’t magically make these several decisions be actually one decision.
b) If I understand your setup, if the coin lands Heads, Beauty gets one dollar if she correctly guesses on Monday, which is the only day she is woken. If the coin lands Tails, a ball is drawn from a bag with equal numbers of balls labeled “M” and “T”, and she gets a dollar if she makes a correct guesses on the day corresponding to the ball drawn, with her guess the other day being ignored. For simplicity, suppose that the ball is drawn (and then ignored) even if the coin lands Heads. There are then six possible situations of coin/ball/day when Beauty is considering her decision:
1) H M Monday
2) H T Monday
3) T M Monday
4) T T Monday
5) T M Tuesday
6) T T Tuesday
If Beauty is a Thirder, she considers all of these to be equally likely (probability 1⁄6 for each). In situations 4 and 5, her action has no effect, so we can ignore these in deciding on the best action. In situations 1 and 2, guessing Heads results in a dollar reward. In situations 3 and 6, guessing Tails results in a dollar reward. So she is indifferent to guessing Heads or Tails.
c) Really, can you actually not suppose that in the Sailor’s Child problem, which is explicitly designed to be a problem that could actually occur in real life, the child has not been given a name? And if so, do you also think that if the child gets cancer, as in the previous discussion, that they should refuse chemotherapy on the grounds that since their mother did not give them a name, they are unable to escape the inapplicability of probability theory to statements with indexicals? I’m starting to find it hard to believe that you are actually trying to understand this problem.
a) Even if sleeping beauty is a human, they are still a deterministic (or probabilistically deterministic) machine, so their responses in any scenario can be represented by an algorithm.
b) The halfer gets the same solution (indifference) too as 1), 2), 5) and 6) are all assigned a probability of 1⁄4; whilst 3) and 4) are ignored.
c) My point isn’t that the child might not have a name. My point is that in order to evaluate the statement: “Radford Neal has a half-sibling” we have to define the scheme in which someone comes to be called Radford Neal.
So, suppose the two potential mothers are Amy and Barbara. The first possibility is that Amy calls their child, if they have one, “Radford Neal”. However, if this is the case, it may come to pass that Amy doesn’t have a child so no-one is called Radford Neal and the reference fails. Alternatively, we might want to ensure that there is someone always called Radford Neal. If they only have one child, this is trivial, if there’s two, we could pick randomly. My point is that there isn’t a unique way of assigning the name, so I don’t know what scheme you want to use to replace the indexical.
a) You know, it has not actually been demonstrated that human consciousness can be mimicked by Turing-equivalent computer. In any case, the only role of mentioning this in your argument seems to be to push your thinking away from Beauty as a human towards a more abstract notion of what the problem is in which you can more easily engage in reasoning that would be obviously fallacious if your thoughts were anchored in reality.
b) Halfer reasoning is invalid, so it’s difficult to say how this invalid reasoning would be applied in the context of this decision problem. But if one takes the view that probabilities do not depend on what decision problem they will be used for, it isn’t possible for possibilities 5) and 6) to have probability 1⁄4 while possibilities 3) and 4) have probability zero. One can imagine, for example, that Beauty is told about the balls from the beginning, but is told about the reward for guessing correctly, and how the balls play a role in determining that reward, only later. Should she change her probabilities for the six possibilities simply because she has been told about this reward scheme? I suspect your answer will be yes, but that is simply absurd. It is totally contrary to normal reasoning, and if applied to practical problems would be disastrous. Remember! Beauty is human, not a computer program.
c) You are still refusing to approach the Sallor’s Child problem as one about real people, despite the fact that the problem has been deliberately designed so that it has no fantastic aspects and could indeed be about real people, as I have emphasized again and again. Suppose the child is considering searching for their possible sibling, but wants to know the probability that the sibling exist before deciding to spend lots of money on this search. The child consults you regarding what the probability of their having a sibling is. Do you really start by asking, “what process did your mother use in deciding what name to give you”? The question is obviously of no relevance whatsoever. It is also obvious that any philosophical debates about indexicals in probability statements are irrelevant—one way or another, people solve probability problems every day without being hamstrung by this issue. There is a real person standing in front of you asking “what is the probability that I have a sibling”. The answer to this question is 2⁄3. There is no doubt about this answer. It is correct. Really. That is the answer.
Thanks for taking the time to write all of these responses, but I suspect that we’ve become stuck. At some point I’ll write up some posts aimed at trying to argue for my position, rather than primarily aimed at addressing rebuttal and perhaps it will clear up some of these issues.