I think your criticism that the usual thirder arguments fail to properly apply probability theory misses the mark. The usual thirder arguments correctly avoid using the standard formalization of probability, which was not designed with anthropic reasoning in mind. It is usually taken for granted that the number of copies of you that will be around in the future to observe the results of experiments is fixed at exactly 1, and that there is thus no need to explicitly include observation selection effects in the formalism. Situations in which the number of future copies of you around could be 0, and in which this is correlated with hypotheses you might want to test, do occur in real life, but they are rare enough that they did not influence the development of probability theory. The fact that you can fit anthropic reasoning into a formalism that was not designed to accommodate it by identifying your observation with the existence of at least one observer making that observation, but that there is some difficulty fitting anthropic reasoning into the formalism in a way that weights observations by the number of observers making that observation, is not conclusive evidence that the former is an appropriate thing to do.
Imagine a world where people getting split into multiple copies and copies getting deleted is a daily occurrence, in which most events will correlate with the number of copies of you around in the near future, so that these people would not think it makes sense to assume for simplicity that the hypotheses they are interested in are independent of the number of copies of them that will be around. How would people in this world think about probability. I suspect they would be thirders. Or perhaps they would be halfers, or this would still be a controversial issue for them, or they’d come up with something else entirely. But one thing I am confident they would not do is think about what sources of randomness are available to them that might make different copies have slightly different experiences. This behavior is not useful. Do you disagree that people in this world would be very unlikely to handle anthropic reasoning in the way you suggest? Do you disagree that people in this world are better positioned to think about anthropic reasoning than we are?
I don’t find your rebuttal to travisrm89 convincing. Your response amounts to reiterating that you are identifying observations with the existence of at least one observer making that observation. But each particular observer only sees one bit or the other, and which bit it sees is not correlated with anything interesting, so all the observers finding a random bit shouldn’t do any of them any good.
And I have a related objection. The way of handling anthropic reasoning you suggest is discontinuous, in that it treats two identical copies of an observer much differently from two almost identical copies of an observer, no matter how close the latter two copies get to being identical. In an analog world, this makes no sense. If Sleeping Beauty knows that on Monday, she will see a dot in the exact center of her field of vision, but on Tuesday, if she wakes at all, she will see a dot just slightly to the right of the center of her field of vision, this shouldn’t make any difference if the dot she sees on Tuesday is far too close to the center for her to have any chance of noticing that it is off-center. But there also shouldn’t be some precise nonzero error tolerance that is the cutoff between “the same experience” and “not the same experience” (if there were, “the same” wouldn’t even be transitive). A sharp distinction between identical experiences and similar experiences should not play any role in anthropic reasoning.
...the standard formalization of probability… was not designed with anthropic reasoning in mind. It is usually taken for granted that the number of copies of you that will be around in the future to observe the results of experiments is fixed at exactly 1, and that there is thus no need to explicitly include observation selection effects in the formalism.
1. Logic, including probability theory, is not observer-dependent. Just as the conclusions one can obtain with classical propositional logic depend only on the information (propositional axioms) available, and not on any characteristic or circumstance of the reasoner, epistemic probabilities also depend only on the information available. Logic—including probability theory—was designed to be fully general. If you want to argue that probability theory is not, in its standard formulation, suitable for anthropic reasoning, you need to point out the specific points in its rationale that are incompatible with anthropic effects. As I have shown (preprint), all you have to assume to get probability theory from classical propositional logic is that certain properties of propositional logic are retained in the extended logic.
2. No, neither classical logic nor probability theory as the extension of classical propositional logic assumes anything about observers, or their numbers, or experiments, or what may happen in the future.
3. Selection effects are routinely handled within the framework of standard probability theory. You don’t need to go beyond standard probability theory for this.
2. No, neither classical logic nor probability theory as the extension of classical propositional logic assumes anything about observers, or their numbers, or experiments, or what may happen in the future.
Right, probability theory itself makes no mention of observers at all. But the development of probability theory and the way that it is applied in practice were guided by implicit assumptions about observers.
3. Selection effects are routinely handled within the framework of standard probability theory. You don’t need to go beyond standard probability theory for this.
You seemed to argue in your first post that selection effects were not routinely handled within standard probability theory. Unless perhaps you see a significant difference between the selection effect that suggests that the coin has a 1⁄3 chance of having landed heads in the Sleeping Beauty problem and other selection effects? I was attempting to concede for the sake of argument that accounting for selection effects as typically practiced depart from standard probability theory, not advance it as an argument of my own.
1. …
Certainly agreed as to logic (which does not include probability theory). As for probability theory, it should not be a priori surprising if a formalism that we had strong intuitive reasons for being very general, in which we made certain implicit assumptions about observers (which do not appear explicitly in the formalism) in these intuitive justifications, turned out not to be so generally applicable in situations in which those implicit assumptions were violated. As for whether probability theory does actually lack generality in this way, I’m going to wait to address that until you clarify what you mean by applying standard probability theory, since you offered a fairly narrow view of what this means in your original post, and seemed to contradict it in your point 3 in the comment. My position is that “the information available” should not be interpreted as simply the existence of at least one agent making the same observations you are, while declining to make any inferences at all about the number of such agents (beyond that it is at least 1). I take no position on whether this position violates “standard probability theory”.
But the development of probability theory and the way that it is applied in practice were guided by implicit assumptions about observers.
I don’t think that’s true, but even if it is an accurate description of the history, that’s irrelevant—we have justifications for probability theory that make no assumptions whatsoever about observers.
You seemed to argue in your first post that selection effects were not routinely handled within standard probability theory.
No, I argued that this isn’t a case of selection effects.
Certainly agreed as to logic (which does not include probability theory).
Why are you ignoring what I wrote about proofs that probability theory is either a or the uniquely determined extension of classical propositional logic to handle degrees of certainty? That places probability theory squarely in the logical camp. It is a logic.
in which we made certain implicit assumptions about observers
No, we made no such implicit assumptions. There are no assumptions, implicit or otherwise, about observers at all. If you think otherwise, show me where they occur in Cox’s Theorem or in my theorem.
I’m going to wait to address that until you clarify what you mean by applying standard probability theory, since you offered a fairly narrow view of what this means in your original post, and seemed to contradict it in your point 3 in the comment.
I have no idea what you’re talking about here.
My position is that “the information available” should not be interpreted as simply the existence of at least one agent making the same observations you are, while declining to make any inferences at all about the number of such agents (beyond that it is at least 1).
Um, there’s only one agent here, but if by “agent” you mean the pair (person, day), then the above is just wrong—it’s very clearly part of the model that if the coin comes up Heads, there is exactly one day on which the remembered observations could be made, and if the coin comes up Tails, there are exactly two days on which the remembered observations could be made. I even worked out the probabilities that the observations occurred on just Monday, just Tuesday, or both Monday and Tuesday.
Listen, if you want to argue against my analysis, you need to
1. Propose a different model of what Beauty knows on Sunday, and/or
2. Propose a different proposition that expresses the additional information Beauty has on Monday/Tuesday and that accounts for her altered probabilities. This proposition should be possible to sensibly state and talk about on Sunday, Monday, Tuesday, or Wednesday, by either Beauty or one of the experimenters, and mean the same thing in all these cases.
I’m not sure how people would reason if people were often duplicated. Lots of issues would need to be addressed once that is common. Are two duplicates that have had identical experiences actually different people (and count twice in moral calculations, for instance)? It seems just as reasonable to count duplicates separately only if they have had different experiences. And in most scenarios with ems or AIs, duplication capability would go with the ability to completely control the duplicate’s experiences, making it hard to see how the duplicate can have any justified knowledge of the external world.
But Sleeping Beauty is not a problem with such characteristics. As described, it is only mildly-fantastical, with perfect memory erasure (which needn’t actually be perfect, just very good) the only unusual feature. So it should be possible to solve it using the tools used for reasoning about ordinary situations, and one would expect the answer obtained that way to still be correct according to some hypothetical more general theory of inference that might be devised in the future, just as the answers to problems solved 200 years ago using Newtonian mechanics are still regarded as correct today, despite the subsequent developments of relativity and quantum mechanics.
The questions of whether two duplicates are actually different people and of whether they count twice in moral calculations are different questions, and would likely be answered differently. People often answer these questions differently in the real world: people are usually said to remain the same person over time, but I think if you ask whether it is better to improve the daily quality of life of someone who’s about to die tomorrow, or improve the daily quality of life of someone who will go on to live a long life by the same amount, I think most people would agree that the second one is better because the beneficiary will get more use out of it, despite each time-slice of the beneficiary benefiting just as much in either case. Anyway, I was specifically talking about the extent to which experiences being differentiated would influence the subjective probability beliefs of such people. If they find it useful to assign probabilities in ways that depend on differentiating copies of themselves, this is probably because the extent to which they care about future copies of themselves depends on how those copies are differentiated from each other, and I can’t see why they might decide that the existence of a future copy of them decreases the marginal value of an additional identical copy down to 0 while having no effect on the marginal value of an additional almost identical copy.
Sleeping Beauty may be less fantastical, but it is still fantastical enough that such problems did not influence the development of probability theory. As I said, even testing hypotheses that correlate with how likely you are to survive to see the result of the test are too fantastical to influence the development of probability theory, despite such things actually occurring in real life. My point was that people who see Sleeping Beauty-like problems as a normal part of everyday life would likely have a better perspective on the problem than we do, so it might be worth trying to think from their perspective. The fact that Sleeping Beauty-type problems being normal is more fantastical than a Sleeping Beauty-type problem happening once doesn’t change this.
“My point was that people who see Sleeping Beauty-like problems as a normal part of everyday life would likely have a better perspective on the problem than we do”
Yes, I agree.
“so it might be worth trying to think from their perspective.”
Yes, it might. But I think we shouldn’t expect to be very successful in this attempt. So if trying to do that gives a result that contradicts ordinary reasoning, which really ought to suffice for Sleeping Beauty, then we’re probably not thinking from their perspective very well.
I agree that it is difficult to see things from the perspective of people in such a world, but we should at least be able to think about whether certain hypotheses about how they’d think are plausible. That may still be difficult, but ordinary reasoning is not easy to do reliably in these cases either; if it was, then presumably there would be a consensus on how to address the Sleeping Beauty problem.
I think your criticism that the usual thirder arguments fail to properly apply probability theory misses the mark. The usual thirder arguments correctly avoid using the standard formalization of probability, which was not designed with anthropic reasoning in mind. It is usually taken for granted that the number of copies of you that will be around in the future to observe the results of experiments is fixed at exactly 1, and that there is thus no need to explicitly include observation selection effects in the formalism. Situations in which the number of future copies of you around could be 0, and in which this is correlated with hypotheses you might want to test, do occur in real life, but they are rare enough that they did not influence the development of probability theory. The fact that you can fit anthropic reasoning into a formalism that was not designed to accommodate it by identifying your observation with the existence of at least one observer making that observation, but that there is some difficulty fitting anthropic reasoning into the formalism in a way that weights observations by the number of observers making that observation, is not conclusive evidence that the former is an appropriate thing to do.
Imagine a world where people getting split into multiple copies and copies getting deleted is a daily occurrence, in which most events will correlate with the number of copies of you around in the near future, so that these people would not think it makes sense to assume for simplicity that the hypotheses they are interested in are independent of the number of copies of them that will be around. How would people in this world think about probability. I suspect they would be thirders. Or perhaps they would be halfers, or this would still be a controversial issue for them, or they’d come up with something else entirely. But one thing I am confident they would not do is think about what sources of randomness are available to them that might make different copies have slightly different experiences. This behavior is not useful. Do you disagree that people in this world would be very unlikely to handle anthropic reasoning in the way you suggest? Do you disagree that people in this world are better positioned to think about anthropic reasoning than we are?
I don’t find your rebuttal to travisrm89 convincing. Your response amounts to reiterating that you are identifying observations with the existence of at least one observer making that observation. But each particular observer only sees one bit or the other, and which bit it sees is not correlated with anything interesting, so all the observers finding a random bit shouldn’t do any of them any good.
And I have a related objection. The way of handling anthropic reasoning you suggest is discontinuous, in that it treats two identical copies of an observer much differently from two almost identical copies of an observer, no matter how close the latter two copies get to being identical. In an analog world, this makes no sense. If Sleeping Beauty knows that on Monday, she will see a dot in the exact center of her field of vision, but on Tuesday, if she wakes at all, she will see a dot just slightly to the right of the center of her field of vision, this shouldn’t make any difference if the dot she sees on Tuesday is far too close to the center for her to have any chance of noticing that it is off-center. But there also shouldn’t be some precise nonzero error tolerance that is the cutoff between “the same experience” and “not the same experience” (if there were, “the same” wouldn’t even be transitive). A sharp distinction between identical experiences and similar experiences should not play any role in anthropic reasoning.
1. Logic, including probability theory, is not observer-dependent. Just as the conclusions one can obtain with classical propositional logic depend only on the information (propositional axioms) available, and not on any characteristic or circumstance of the reasoner, epistemic probabilities also depend only on the information available. Logic—including probability theory—was designed to be fully general. If you want to argue that probability theory is not, in its standard formulation, suitable for anthropic reasoning, you need to point out the specific points in its rationale that are incompatible with anthropic effects. As I have shown (preprint), all you have to assume to get probability theory from classical propositional logic is that certain properties of propositional logic are retained in the extended logic.
2. No, neither classical logic nor probability theory as the extension of classical propositional logic assumes anything about observers, or their numbers, or experiments, or what may happen in the future.
3. Selection effects are routinely handled within the framework of standard probability theory. You don’t need to go beyond standard probability theory for this.
Right, probability theory itself makes no mention of observers at all. But the development of probability theory and the way that it is applied in practice were guided by implicit assumptions about observers.
You seemed to argue in your first post that selection effects were not routinely handled within standard probability theory. Unless perhaps you see a significant difference between the selection effect that suggests that the coin has a 1⁄3 chance of having landed heads in the Sleeping Beauty problem and other selection effects? I was attempting to concede for the sake of argument that accounting for selection effects as typically practiced depart from standard probability theory, not advance it as an argument of my own.
Certainly agreed as to logic (which does not include probability theory). As for probability theory, it should not be a priori surprising if a formalism that we had strong intuitive reasons for being very general, in which we made certain implicit assumptions about observers (which do not appear explicitly in the formalism) in these intuitive justifications, turned out not to be so generally applicable in situations in which those implicit assumptions were violated. As for whether probability theory does actually lack generality in this way, I’m going to wait to address that until you clarify what you mean by applying standard probability theory, since you offered a fairly narrow view of what this means in your original post, and seemed to contradict it in your point 3 in the comment. My position is that “the information available” should not be interpreted as simply the existence of at least one agent making the same observations you are, while declining to make any inferences at all about the number of such agents (beyond that it is at least 1). I take no position on whether this position violates “standard probability theory”.
I don’t think that’s true, but even if it is an accurate description of the history, that’s irrelevant—we have justifications for probability theory that make no assumptions whatsoever about observers.
No, I argued that this isn’t a case of selection effects.
Why are you ignoring what I wrote about proofs that probability theory is either a or the uniquely determined extension of classical propositional logic to handle degrees of certainty? That places probability theory squarely in the logical camp. It is a logic.
No, we made no such implicit assumptions. There are no assumptions, implicit or otherwise, about observers at all. If you think otherwise, show me where they occur in Cox’s Theorem or in my theorem.
I have no idea what you’re talking about here.
Um, there’s only one agent here, but if by “agent” you mean the pair (person, day), then the above is just wrong—it’s very clearly part of the model that if the coin comes up Heads, there is exactly one day on which the remembered observations could be made, and if the coin comes up Tails, there are exactly two days on which the remembered observations could be made. I even worked out the probabilities that the observations occurred on just Monday, just Tuesday, or both Monday and Tuesday.
Listen, if you want to argue against my analysis, you need to
1. Propose a different model of what Beauty knows on Sunday, and/or
2. Propose a different proposition that expresses the additional information Beauty has on Monday/Tuesday and that accounts for her altered probabilities. This proposition should be possible to sensibly state and talk about on Sunday, Monday, Tuesday, or Wednesday, by either Beauty or one of the experimenters, and mean the same thing in all these cases.
I’m not sure how people would reason if people were often duplicated. Lots of issues would need to be addressed once that is common. Are two duplicates that have had identical experiences actually different people (and count twice in moral calculations, for instance)? It seems just as reasonable to count duplicates separately only if they have had different experiences. And in most scenarios with ems or AIs, duplication capability would go with the ability to completely control the duplicate’s experiences, making it hard to see how the duplicate can have any justified knowledge of the external world.
But Sleeping Beauty is not a problem with such characteristics. As described, it is only mildly-fantastical, with perfect memory erasure (which needn’t actually be perfect, just very good) the only unusual feature. So it should be possible to solve it using the tools used for reasoning about ordinary situations, and one would expect the answer obtained that way to still be correct according to some hypothetical more general theory of inference that might be devised in the future, just as the answers to problems solved 200 years ago using Newtonian mechanics are still regarded as correct today, despite the subsequent developments of relativity and quantum mechanics.
The questions of whether two duplicates are actually different people and of whether they count twice in moral calculations are different questions, and would likely be answered differently. People often answer these questions differently in the real world: people are usually said to remain the same person over time, but I think if you ask whether it is better to improve the daily quality of life of someone who’s about to die tomorrow, or improve the daily quality of life of someone who will go on to live a long life by the same amount, I think most people would agree that the second one is better because the beneficiary will get more use out of it, despite each time-slice of the beneficiary benefiting just as much in either case. Anyway, I was specifically talking about the extent to which experiences being differentiated would influence the subjective probability beliefs of such people. If they find it useful to assign probabilities in ways that depend on differentiating copies of themselves, this is probably because the extent to which they care about future copies of themselves depends on how those copies are differentiated from each other, and I can’t see why they might decide that the existence of a future copy of them decreases the marginal value of an additional identical copy down to 0 while having no effect on the marginal value of an additional almost identical copy.
Sleeping Beauty may be less fantastical, but it is still fantastical enough that such problems did not influence the development of probability theory. As I said, even testing hypotheses that correlate with how likely you are to survive to see the result of the test are too fantastical to influence the development of probability theory, despite such things actually occurring in real life. My point was that people who see Sleeping Beauty-like problems as a normal part of everyday life would likely have a better perspective on the problem than we do, so it might be worth trying to think from their perspective. The fact that Sleeping Beauty-type problems being normal is more fantastical than a Sleeping Beauty-type problem happening once doesn’t change this.
“My point was that people who see Sleeping Beauty-like problems as a normal part of everyday life would likely have a better perspective on the problem than we do”
Yes, I agree.
“so it might be worth trying to think from their perspective.”
Yes, it might. But I think we shouldn’t expect to be very successful in this attempt. So if trying to do that gives a result that contradicts ordinary reasoning, which really ought to suffice for Sleeping Beauty, then we’re probably not thinking from their perspective very well.
I agree that it is difficult to see things from the perspective of people in such a world, but we should at least be able to think about whether certain hypotheses about how they’d think are plausible. That may still be difficult, but ordinary reasoning is not easy to do reliably in these cases either; if it was, then presumably there would be a consensus on how to address the Sleeping Beauty problem.