For example, if one were to ask the average person to express their credences regarding the outcome of a two horse race that they know absolutely nothing about, they will simply say "I don't know who will win" and refrain from assigning any odds, equal or otherwise. They will also tend to accept bets in which they have knowledge that the physical probabilities are 50/50 over bets that they are totally ignorant about. — sime

Two Envelopes seems to encourage abuse of the principle of indifference in exactly this way. Maybe it's just something like this: rationality requires treating "It's one or the other but I've literally no idea which" as an uninformative 50:50 prior only when there's the real possibility of acquiring new information upon which to update that prior. I'd rather just say, no, don't do that, "I don't know" doesn't mean "It's 50:50", but there are a great many usages in which the prior is quickly swamped by actual information, and the PoI is a harmless formality. --- In Two Envelopes, you know there will never be any new data, so that harmless prior metastasizes.

Y'all know the math much better than I, so maybe I'm speaking out of turn.

trying to move away from the idea that one's credence in the state H is entirely determined by the specification of the ways in which one can come to be in that state — Pierre-Normand

Even the word "state" feels too coarse for Sleeping Beauty, since it could denote the situation a robust well-defined subject finds themselves in, or it could denote the very identity of that subject. --- At least, that's how the two main camps look to me. One wonders, where am I? how did I get here? One wonders, what am I? what has made me into this?

As you say, it's all about individuation. Lacking a fixed point of individuation, you can push the lever however you like but you won't actually move anything.

The thirder's position is indeed a ratio of possible words, but there is scant evidence to support the idea that credences are accurately represented by taking ratios over possible worlds. — sime

Elsewhere Lewis is pretty careful about what he calls de se modality -- epistemic questions are not just about possible worlds but irreducibly about your epistemic counterpart's status in a given possible world.

That's in the neighborhood of what I've been musing about anyway.

Yeah there's some similarity to Bostrum's thing. In Stanford terms, you could say that thirders are identifying the self with the role they are playing at the moment, and it's a matter of chance that they are playing one role at a given moment rather than another.

Self and repetition are linked in Nietzsche's puzzling doctrine that truly to will something is to will its eternal recurrence. The Good Place is also oddly sleeping-beauty adjacent:

The Monty Hall "problem" has an answer: it's just an illustration of a tempting but fallacious way of reasoning about probability. Two Envelopes rises to the level of paradox because a clearly and provably wrong answer can be arrived at by reasoning, the flaw in which is so difficult to determine that there is no professional consensus on what the flaw is.

I'm not sure SB is the same sort of thing: which is the natural, obvious, tempting mistake and which the correction? If you haven't been around these sorts of puzzles, maybe "a fair coin is 50-50, period" is the obvious answer, and SB is one of the only scenarios bizarre enough to undermine that confidence; if you've been around these sorts of puzzles, carving the space into quarters and conditioning to get one third might be the obvious thing to do, but that's to be tricked into misreading the source of randomness here. Because the analysis of the 1200-sophomore study I posted is so straightforward, but getting standard SB to align with it is so difficult, I have even wondered if SB doesn't undermine the whole idea of "subjective probability". A more optimistic take would be that SB is unsolvable, and shows you what background assumptions are necessary for probabilistic reasoning to work -- memory, continuity of self, objective verification, something in here.

But as this debate has gone on long enough and I don't think I have the energy to continue it much more, I'm happy to just say that both 1/2 and 1/3 are correct answers to distinct but equally valid interpretations of the question. — Michael

I'm still mulling it over despite myself, but I think there's something to this.

Rather than getting back into the nitty-gritty, I'm thinking about the stuff I posted a while back, the possible self slices and all that.

It could be the two natural positions represent slightly different attitudes toward the self.

Halfers define participants as ending up having one interview, or ending up with two. It's a cumulative view of the self: I am my life story, and the story draws a connected arc passing through circumstances; what matters most is that storyline, and situations are only things I (robust, impervious) encounter along the way.

Thirders define participants by what they're experiencing at the moment, and seem less confident that what matters most is the unique historical self for whom this is merely an incident in their story; thirders seem to feel that the situation in part defines them, they are just the someone this is happening too -- anything could happen to anyone or not happen to them, and that's who you are at that moment.

It's hard to state each side clearly, but you could also say, roughly, that for a halfer the question is only: what happened? how did the coin land? For thirders, the question is: who am I? what's happening to me right now that defines who I am? (This is pretty clear in @JeffJo's thirder analysis, posted here on the forum and also here on a site dedicated to the sleeping beauty problem, in which the whole point is that participants don't know which sort of participant they are, and recognize that roles could be shuffled amongst them.)

The who am I? question doesn't really seem to be on the halfer radar at all, thus my suggesting that halfers view the self as more robust in the face of circumstances.

Maybe one more analogy: thirders might be more likely to buy the results Zimbardo claimed for the Stanford Prison Experiment. --- That's more than a bridge too far for me, but might be the clearest way to put the distinction.

It is obvious that we don't know with absolute certainty that objects persist when unobserved, but all the evidence of human experience, including observation of animal behavior, suggests that they do persist. — Janus

What evidence would that be? We don't observe what we don't observe, so ...

As far as I can tell this is not something we believe on evidence at all, but an assumption. Hume describes it so.

In a Bayesian frame of mind, you might say the furniture out in the living room is part of my model of the world, and when I'm not observing it, that part of the model is essentially frozen, not being updated because there is no relevant information upon which to base an update. No new observations. That's not much like believing, based on the evidence, that it's still there as I left it, and a lot more like just assuming that it is.

But I'm sure what you mean is that if I were now to go and check, everything would still be as my model says it is, and it's the experience over time we should trust. Thus:

Really all we mean by "persist" is that they are perceptually invariant over varying degrees of time, depending on the object — Janus

"Perceptually invariant" is a curious phrase, meaning something like "below our level of discrimination". We joke about watching paint dry or watching grass grow. You could, of course, do these things, and you would find that there is rather little in the world that never changes, even things that change too slowly for us to notice or care.

But of course invariance is, in some important ways, not a matter of observation exactly. I have an identity not just because I change slowly from day to day. So do many things. Or at least we're inclined to think so, so these identifications have at least the force of custom, and in some cases maybe that's all they have. (The identities of countries, firms, and so on.) If those identities are established by observation, it's by observation of the custom, not any object.

So if the couch has changed too little for me to notice or care since I last saw it an hour ago, I'm allowed to pretend it's the same and call it the same. Is that the metaphysics you had in mind?

they show perceptual commonality for almost all people and even some animals. — Janus

Of course this is not quite what you mean, but that we infer similar perceptions upon seeing similar behavior. Not saying that's a bad inference, but it's an inference, not an observation. We also know there is considerable variation in how things are perceived, even among humans. Beyond that, more variation: for instance, it turns out humans are relatively rare in not seeing the ultraviolet part of the spectrum, and a great many animals do. Flowers look quite different to most insects, for instance.

None of which is going to bother you because a flower, for you, is, in the final analysis just a whatever-it-is; all you need for the point is that sentient creatures all behave as if there's something there. And so it is with infants: there's research suggesting that infants develop an expectation of object permanence before the expectation of object identity. (When something goes behind a screen, the infant is satisfied if something comes out, even if it's not the same thing.)

But the theory you're defending is not that I hold, based on the evidence, that there's still a lot of somethings out in the living room, but the same chair, couch, and tables that were there when I last was. We need a lot more specificity than your fallback metaphysics of something-or-others.

Do you not think things exist when not being observed? — Janus

Is it a matter of opinion?

Hume agonizes over this; he can find no good reason to think objects persist, and yet he finds that he does believe so. It's a sort of prejudice; nature, he suggests, has taken the decision out of his hands, as a matter too important to leave to stumbling human reason.

that is the common, you might even say default, attitude to things. — Janus

Just so.

If this is all true, what are we to make of it? What do we do with this, as philosophers?

You could say belief in objects is a sort of quirk of human psychology, unsupported by reason, and that the only intellectually honest, and rational, position to hold is some sort of idealism.

That was an option for Hume, who had the example of Berkeley before him, and of course we have our choice of idealisms.

I think there may be an alternative, and thought we might begin to see the shape of it if we looked closely at the interplay of thought, object, existence, and absence in one of the things people typically say in these discussions, namely

I can imagine a rock existing without there being any conscious observer of it. — Janus

I admit, I was ignoring the chitchat about Kant you followed that with, because I find just this simple innocent claim terribly interesting.

But you're right, the discussion's gone nowhere.

I didn't mean to say that I can imagine, as in visualize — Janus

I understand that.

I can imagine that objects have attributes that cannot be observed, and that are not dependent on being observed. — Janus

Where we began was existence:

I can't imagine a particular rock without imagining it in terms of perceptible attributes, but I can imagine that a rock could exist without anyone perceiving it. — Janus

So your intention was to say that the existence of the rock is an attribute of it that is not dependent on being observed.

(Around here was where I mentioned Hume's suggestion that we seem only to think things as existing, which leaves open a question about whether existence is merely, as it were, an element of how we conceive things.)

Your idea then was never really to talk about unobserved objects, except incidentally, but to know which parts of our conception of an object we observe are down to us, and which aren't. And the existence of the object is not down to us, you say, so it's one of the properties we can still safely attribute to unobserved objects.

The most I would say is that whatever that existence is, it reliably gives rise to the spatiotemporal in-common perception of individuated objects. — Janus

But now here you have this free-floating attribute, existence, that isn't an attribute of anything, because the only sort of thing it can be an attribute of is apparently too contaminated by our conceptions.

Even a phrase like "whatever that existence is" doesn't work, because it's got a demonstrative in it. What existence are you referring to? You must be pointing at it, and you point at it just by saying it gives rise to all the conceptions you count as only for us. So we're right back where we started. You're still in a very roundabout way just saying "rock" while denying that you are.

My point is still that you're trying to bracket the "observedness" of the object, while depending on it completely to say anything at all, which means you haven't really bracketed it at all.

I'm talking about the unobserved aspect of rocks — Janus

Ah. That's rather different.

This all began with you defending the mind-independence of objects by saying that you could readily imagine an object that's unobserved. I questioned whether you could actually do that, and still do.

As a step toward an object being unobserved, we did at least pretend to pass through stripping an object of whatever observation "adds" to it; if that's even coherent, it ought to be part of the answer for what something not observed at all is like. It's just that even in "un-observing" something we've observed, all we can do is play with exactly the same categories as when we observe it, only we pretend not to be applying them -- or at least not some of them. We leave the spatial location of the something untouched, for instance, and it remains individuated just as it was when we observed it, and so on.

And the case is even worse with something not only not being observed at the moment, but never observed, perhaps impossible to observe.

There is no need for things, that's the point Descartes made. All that is required is that we have similar perceptions — Metaphysician Undercover

I think that just kicks the can down the road. I don't know why talk about "us" and the similarity of "our perceptions" should be countenanced when talk of other things is not.

Same for the treatment of convention you build on top of this:

And, if someone tried to argue that the earth was actually spinning instead, this person was wrong, or incorrect, as not obeying the convention. — Metaphysician Undercover

That makes conventions sound every bit as solid and consistent as any rock or table.

Except you seem to have forgotten that we were talking about unobserved rocks.

If you want to describe such a thing as having a "propensity" to produce rock conceptions, or an unrealized potential to, then you're still just saying it's a rock, using new words. And that means you still have to justify categorizing something unobserved. (Besides. if it's unobserved, you don't know anything about its propensities or potentials either.)

So why do you call this something-or-other you're conceiving "unobserved rocks"?
— Srap Tasmaner

It refers to whatever it is, apart from the human, that gives rise to observed rocks. — Janus

But why choose the word "rocks" if you're not attributing to it any rock properties? Why not "balloons" or "elegies"?

Is it like this: You start with rocks as observed and conceptualized by us, then peel off our conceptualizations leaving only a something that, on the occasion this something was observed, gave rise to our rock-conceiving, and then for the last step you just subtract the observation itself, leaving only the something that, were it observed, we would say was rocks.

The thing is, without observation, how do you know what conception it would give rise to in us and critters like us? How do you know it would be rocks? And if you don't know it would, why say there's a something we would call rocks if we observed it?

The whole procedure feels somehow disingenuous. (I don't mean this as a point about your character, you understand.) We're still talking about rocks, but we're embarrassed about it, so we kinda half-heartedly pretend we're not. "Something that when we observe it gives rise to the conception rock" -- we already have a word for that, and it's "rock". (Or "Stein", whatever.)

Roughly, I'm not convinced you've made any progress toward removing us from your conceptions.

So there is no real truth or falsity (in the sense of correspondence) with respect to distance, only conventional ways of acting and speaking, norms. — Metaphysician Undercover

Hmmm.

Given how we talk about distance, you're either using words the conventional way when you compare the distance from the earth to the sun and the distance from the earth to the moon, or you're not. Saying the former is "bigger", not the latter, is how we use the word "bigger". So there is a piece of a sort of "true by convention" account here.

Now you've granted that nature supports and enables our conceptualizations, and in this case using the word in the normal way is choosing that word instead of "smaller" only if the sun is further from here than the moon. The norm for usage of the word "bigger" requires something like this, else no one could understand and follow the norm.

(Quine tried to convince us many decades ago that trying to separate the empirical and the conventional elements of a statement was a fool's game.)

For "bigger" to be meaningful at all, there must be things (I'm speaking loosely and generally here) that are stably different sizes. If nature supports us coming up with a bigger / smaller pair of concepts, it's because they can be consistently applied to things that are what we choose to call "bigger" and "smaller" when compared to each other.

Do we disagree?

And this is how all concepts and ideas are — Metaphysician Undercover

How about a (I hope) non-mathematical example: stars and planets, for instance, are both celestial bodies, and they behave similarly as massive objects (gravitation and all that), but they are structurally quite different, have quite different life stories, and so on. Given how astronomers define these terms, their application to a given celestial object is correct or incorrect. (The "evening star" is in fact a planet, etc.)

We are not, under most circumstances, compelled by nature to distinguish stars from planets, but the distinction is there to be captured in our terminology, should we choose to. Nature supports making this distinction, enables it. For comparison, the "morning star" and the "evening star" turn out to be the same object. Nature supports both using two names, since the times of day Venus rises are distinguishable, and using just the one, picking out a unique body in our solar system.

The assumption that there is an existential distance which can be measured is the false and misleading assumption. The better assumption would be that the distance is produced, or created by the measurement. — Metaphysician Undercover

(1) Measurements that have not been done have not been done.
(2) Distances are created not discovered.

Certainly yes, if you start from (2), you can derive (1). But (1) is a tautology, so you can get it from anything.

The question is whether the truism (1) provides any support for (2).

The truth of this is demonstrated by the fact that different measuring techniques will produce a different measurement (as indicated by jorndoe's post), and each will be a valid measurement by the principles of the technique. — Metaphysician Undercover

This is an actual argument for your position, so you need to spell it out. How do various techniques for determining a distance differ, what principles are involved, and how are they valid with respect only to their own principles not each other?

Looking back, I see that you take this variation as evidence:

The distance between here and the moon is indeterminate until it's measured. This means that there is no fixed value. The variance in the numbers you [ i.e., @jorndoe ] gave are evidence of this. — Metaphysician Undercover

The other side would like various techniques to give the same answer, or, in the case of estimates, roughly the same answer -- which means: the same, but only to a certain degree.

(Funny, @Wayfarer used to make exactly the opposite argument, that because the content of a statement can be translated from one language to another -- as we might convert from imperial to metric, say -- this content must be somehow transcendent or whatever.)

the kind of existence they have is unimaginable to us, we can only imagine that they do not have the kind of existence they have as perceived phenomena, so it is an apophatic kind of imagining — Janus

Not obvious how you would even justify the "they" here...

So the upshot is that when you conceive of these unobserved rocks, you conceive of something unobserved which you can only say is not like what we usually think of as "rocks", not even in the sense of existing as we think rocks do.

So why do you call this something-or-other you're conceiving "unobserved rocks"?

in fact it is more difficult to imagine that they cease to exist when not being perceived — Janus

Something like this then: when I imagine a rock existing unobserved, I imagine a rock and then conceptually remove things like color and other perceptible attributes, until I can only say that right there, where we would observe rocks if we were observing, there is something about which we can say nothing, except that it's still there when we're not looking.

When I imagine, for example, a planet in a far distant galaxy I just have an image of a planet. However, in this case I am visualizing a planet, which means I am relying on perceptible attributes in order to do that. — Janus

I think it's stronger than that: I think you're imagining it as you or at least a creature a lot like you would see it, the attributes perceptible by us and those like us, and so on. When you visualize this, you visualize it from a certain vantage-point, yes? You can't visualize at all without picking the spot where the eye of the observer is situated.

Now if you do that but then add, "Only there's no observer at that spot," I think that just misrepresents the conception, which clearly has <person> as an element, only just off-screen.

And this is different to thinking that there could be a planet in a distant galaxy that has never been or ever will be seen by humans or any other percipient entities. — Janus

This is a thing we can say, but it remains to be seen what we mean by this. I think the conception here is still of the sort of thing we or critters like us might experience, it just happens that none of us do. And that means it's still all tangled up with us, and what is a possible experience for us. Which is fine, right? What else is there to talk about? We can pass by what isn't a possible experience for us, but we should at least be clear that we're still always in the picture in one sense or another. --- That's much vaguer than I'd like, but I'm just about done for the night. Saying what those senses are and aren't is exactly what we're about here.

I can visualize an empty room, for example — Janus

Good! If you hadn't said this, I was going to ask what your model for conceiving absence is, so you beat me to it. This bag used to have apples in it, now it doesn't, that sort of thing.

Think it'll work for conceiving unobserved rocks?

So, perhaps a better way of saying it would be 'I can, without contradiction or inconsistency, think that rocks exist when no one is looking at them'. — Janus

Yeah, but I'm not sure we can just switch from visualizing to something vague like "conceiving" and declare the problem solved. I think it was Hume who noted that to conceive of something is to conceive of it existing -- which cuts both ways: on the one hand, there's no "and existing" step, which either means existence is not a real predicate (which Hume says in almost so many words), or it means it's already baked in, i.e., it's at the very least part of how we think things. Maybe it even means that words like "existing" are names for our habits of thought more than anything else. (Hume, again, will end up saying almost exactly this, and he finds the idea horrifying.)

All that's about the rock's existence, I guess. I don't know if Hume's exactly right but I think he's on the trail of something, as he usually is. There's some connection between thinking and the object of thought's existence, and the various isms offer an account of what that connection is.

So you're thinking rocks, and thinking them existing, whether that's another step or not, and you're also thinking no one observing these rocks. Various ways to do this, I guess: you could conceive persons (or technological proxies for them, cameras and stuff, whatever), and they just happen not to observe some particular rocks, and the rocks continue peacefully existing undisturbed by not being noticed. Or you could strengthen this scenario: rather than just happening not to observe these rocks, though they could, you could make it impossible, make some particular rocks unobservable, even with technology. Simplest way to do that is conceive rocks in the distant past before what we take to be people had evolved. The only strengthening left would be to imagine there just aren't any persons, anywhere in the universe, and never will be. --- That's something more or less like the whole range.

Pointless though, right? I mean, to conceive the absence of persons to observe, you have to conceive them so you know what to keep out of your desired conception. You may claim to be able to conceive a universe without people, but you'll only get to what you call that conception by conceiving people and conceiving them absent. I don't see any way around that.

Does it matter? The idea is that the content of this conception is still person-free, even if your own mind isn't. But it's not blindingly obvious anymore what "conception" means; it's clear that even to define the pure person-free conception, you not only need the person conception as well, you need them to be constellated in a particular way.

I'll stop now noting that how these conceptions are related is particularly interesting because it runs through absence. Not perfectly obvious how dealing with that is going to work, but it makes a fitting third to go along with thought and existence.

Sure. I'm not really disagreeing with you. --- I'm just noting that our minds only work the way they work. But you're a better Kantian than I, so I'm not telling you anything.

I would give some thought, though, to exactly how this rock-imagining stuff works. Go slow. Is it like a regular picture of a rock but with a caption that says, "No on is looking at this"? There's something interesting here, don't rush past it.

The distance between here and the moon is indeterminate until it's measured. — Metaphysician Undercover

Is this another way of saying that it's not measured until it's measured? Or does "indeterminate" carry some meaning here unrelated to measurement?

But this does not imply that the value existed before the measurement. Prior to measurement there was just an assumption. — Metaphysician Undercover

With respect to the distance "itself", as it were, it is indeterminate before measurement; with respect to those who will measure, but haven't yet, there is an assumption that the distance is measurable, that it can be determined. Is this a way of saying that scientists, unless they are foolish indeed, ought to agree that values they have not determined are values they have not yet determined? Or is there more to this assumption?

if the measurement of the distance between here and the moon fixes the distance, this does not imply that the distance existed before the act of measurement. — Metaphysician Undercover

If by "distance" you mean a value, the result of a measurement, indeed it won't exist until it exists. Or do you mean that the spatial separation of the earth from the moon doesn't exist until someone thinks it does? Something must underwrite the assumption that "it" can be measured; its existence of that "it" to be measured would do nicely.

I can imagine a rock existing without there being any conscious observer of it — Janus

Really? You can imagine a rock without imagining yourself observing it?

I would like the halfer to explain why ruling out the Tuesday scenario doesn't affect their credence in the coin toss outcome at all. — Pierre-Normand

For Lewis, if I recall correctly, it raises her credence for heads from 1/2 to 2/3, which he finds curious, but that's it.

What should SB's credence be that this is her first (or second) interview?

Any thoughts?

But given the set {10,20}, E(z)=15=(5/4)12, and 12 isn't the value of the chosen envelope. — Michael

That's true. It's not in the set. Neither is 15. But 15 is, for all that, the expected value of both envelopes.

The total value of any pair of envelopes is 3x, where x is the smaller of the two. I choose one, and call it y; if I got the smaller then y is x, if the larger then y is 2x; so x is either y or y/2. The total value of the envelopes is then either 3y or 3y/2, so the average total value is 9y/4. Alternatively, you could just say that since x is y or it's y/2, the average value of x is 3y/4, again making a total of 9y/4, on average. By definition, my envelope is y, so the other envelope must be worth 5y/4, on average.

Where have I gone wrong?

Given a set {10, 20}, the expected value of a number selected from that set is 15. There's nothing wrong with your first set of equations, and it gives the right answer. You don't have to go through all that; you just need the average.

The second set of equations is different.

Are the situations described in the following questions the same? If not, what's the difference?

(1) What are the chances that y = x and the chances that y = 2x if y is chosen randomly from a set {x, 2x}? (You may, if you like, write it backwards as x = y and x = y/2.)

(2) What are the chances that a y chosen randomly from a set {x, 2x} was chosen from a set {y, 2y} and the chances it was chosen from a set {y/2, y}?

Consider what the halfer says next with my marble analogy: which marble-holding-self this is was selected by a coin toss. That's the same as saying whether the marble I have is white or black was determined by a coin toss.

But that's not good enough, because there's more than one black-marble-holding momentary self. There's just no way to select from more than two moments this might be with a single coin toss.

Here's a halfer analogy I found convincing before:

Two urns, one with a single white marble, one with many black marbles; you flip a coin to decide which urn to draw from; even though there are more black marbles than white, the chances of getting the white marble are equal to the chances of getting one of the many blacks.

The argument I've been making lately seems to be roughly this: if you close your eyes and someone selects a marble and places it in your hand, and if you know there are more black marbles than white, then you can figure it's more likely to be black. Fair enough.

But if you know they selected the marble in your hand by flipping a coin to select which urn to draw from, you should figure it's just as likely to be the one white as any of the blacks.

That looks like trouble for the thirder position, but it's missing the repetition, and missing SB's uncertainty about her own state. It's not just that she knows someone's put a marble in her hand, she knows they'll do it more than once for tails. So SB is justified in wondering, who is this person with a marble in her hand? Who is she more likely to be?

I think.

This relies on the intuition of repeating the experiment over and over. If so, then there are unconditionally more tail slices. But the coin is flipped exactly once. Therefore, even though there are more tail slices, they both exist only upon a tails flip. — hypericin

I think it's hard.

If you wake up and ask yourself "Which of the selves I could have become while I was asleep am I?" there are two ways to take that.

You could think of the self you're seeking as a continuous self (a sort of world-line, if I have the parlance right) accumulating experiences, in which case it's reasonable to describe that self from any point along the line: in the past you're the self who's going to end up having exactly one or exactly two interviews; in the future, you're the self that had exactly one or exactly two interviews.

But you could also think of your self as this momentary self, even if it is part of a larger stream of momentary selves. Your question then is, which stream is this self-moment likely to be part of? If any such moment is more likely to be part of one stream than another, then you are more likely to be the self living out that stream than the other.

What are we supposed to learn here?

I mainly take it as a logic puzzle: there's a bit of fallacious reasoning, you have to spot it, explain why it's fallacious, and why it's attractive, easy to fall into.

On the other hand there is a way of taking it as an interesting probability problem, where you have to model several different sorts of issues, and under some additional conditions there are interesting switching systems available. (I don't have the background for that stuff anyway, as you know.)

So there's kinda an old school and a new school response possible.

The old school take is just another case study in faulty reasoning, but might have some interest in how people elide different sorts of uncertainty and chance. And last time I recall speculating about some use for the new school approach. (Something about improving performance without feedback -- it was pretty half-baked.)

Besides an opportunity to play with our respective toys, do you get anything out of this?

I don't see any disagreement.

I see this has been gone over quite a bit — Mikie

Going back five years.

The "puzzle" is not figuring out what the right way to analyze this is -- although @Michael did argue, at length, in the old thread, for switching, and there are some interesting practical aspects to switching in the real world where the probability distributions need to be possible -- but why a seemingly natural way to analyze the game is wrong. There's not even agreement among analysts about whether this is a probability problem. (I don't think it is.)

As always, GOP cuts will worsen the deficit and benefit the wealthy while the GOP are bleating about 'cutting wasteful spending'. — Wayfarer

You may be right -- and surely this is true for many Republicans, but also Democrats -- that this is intended as a deliberate move to benefit the owner class, but don't underestimate how much the Freedom Caucus is motivated not by money but ideology. Some are against the IRS not because they are shills for donors far, far wealthier than they will ever be, but because "Muh liberty!" It's a, so to speak, principle.

I don't know case by case which explanation is best. Someone like Matt Gaetz is apparently just a piece of shit who plays a populist on TV but has always had his hand out to every lobbyist and donor he came anywhere near, and presumably has been more or less peddling influence since he entered politics. But some of these folks just aren't bright enough to be that corrupt. MTG probably only manages to be corrupt when someone explains to her in small words exactly what she needs to do for the money. In the meantime she's just a dangerous narcissist. But then there are those who are so dumb they really believe, you know, God hates the Fed and loves AR-15s. Not kidding. Not kidding at all. Most who make it to the national legislature have learned which of their true beliefs it's cool to say on a hot mic, but if you drop down to the Congressional minor leagues, state legislatures, it is shocking how absolutely stupid a lot of these tea-party-maga-evangelical-libertarian bozos are. Just plain stupid.

So gutting the IRS even though it will cost more, that's just a bit too subtle for some of these yahoos. The IRS are jackbooted thugs who hate our liberty, period. They want to take our money and use it to pay for abortions.

Crashing the world economy? That's just something some eggheads say will happen, I don't believe it. And even if the world economy crashes, what's that got to do with America? Fuck em. Let em take care of their economy and we'll take care of ours. -- No, seriously, this is the level of thought of the people who vote in Republican primaries and too many of those who end up getting elected.

Never attribute to malice what can be explained by greed or stupidity, the saying goes. You're right about the greed, but don't forget about the stupidity.

There are two reasonable ways to assign variables to the set from which you will select your envelope: {x, 2x} and {x, x/2}. Either works so long as you stick with it, but if you backtrack over your variable assignment, you have to completely switch to the other assignment scheme.

Thus you can say, if I have the smaller, I have x and the other is 2x, or you can say, I have x/2 and the other is x. The main thing is that you have to use the same scheme for the other case, where you have the larger.

If you choose to label the envelope you've chosen "A", it's true that the other envelope contains either 2A or A/2, but that's because there are two consistent ways to assign actual variables, and the "or" there is capturing the alternative variable schemes available, not the values that might be in the other envelope. "A" is an alias for one of the members of {x, 2x} or for one of the members of {x, x/2}, but you still have to choose which before you can think about expected values.

The key is that "or" up there ("2A or A/2") is not a matter of probability at all; once you've chosen a set, which member of the chosen set A is, is a matter of probability. But we won't be summing over the possible choices of variable scheme.

For an analogy: it's 11 am, and you're asleep in bed; how old are you? Odds are you're one of the ages where you were more often asleep in bed at 11 am, even if every year of your life has had at least one such day.

This is a quote from Descartes posted by @Fooloso4 next door in that thread:

For a life-span can be divided into countless parts, each completely independent of the others, so that from my existing at one time it doesn’t follow that I exist at later times, unless some cause keeps me in existence – one might say that it creates me afresh at each moment.

We can say of the sleeping beauty problem, just as a way of beginning, that there are three possible future SB slices which the SB being interviewed could be.

The halfer camp seems to take the view that two of those belong to one person, and one to another, and that the correct analysis is that the coin flip partitions SB's future slices into a heads set and a tails set, just two, equal chances of being in each set.

The thirder view is that only the current slice that you might be is relevant, and there are more being-interviewed slices in the tails partition, so you're more likely one of those.

It may not be true that I am more likely to be a person who is interviewed more than once, and yet be true that this slice of me being interviewed is more likely to be a somewhat common tails slice, considering the entire pool of possible slices of me, than to be a comparatively rarer heads slice.

What does the cut correspond to?
— Srap Tasmaner

Answering that gives you the origin of the paradox, right? — fdrake

In a sense yes. The cut is a "committed choice" thing, and you could take that as representing the dealer's not monkeying with the envelopes once he's offered them to you.

But we know how the game works objectively; the problem is why some ways of modeling your own epistemic situation as the player work and others, though quite natural, don't. It's clear that including the cut in your model works; but it's not as clear why you must include it.

It's not even perfectly clear to me what sort of epistemic move the cut is; what have I done when I've done that? I'm not considering certain options for backtracking; okay, but why shouldn't I consider those options? The dealer may not monkey with the envelopes after he presents them, but I still don't know which branch (perhaps of many) he went down, so why shouldn't I consider those?

It is, after all, simply true that the other envelope must contain half the value of mine or twice. (And the problem here doesn't seem to be spurious or-introduction, because {mine/2, mine*2} is the minimal set for which that claim is always true.)

But it's also just as clear that this not equivalent to the {x, 2x} framing. Both envelopes, mine included, are members of that set, but mine is of course never a member of {mine/2, mine*2}. Given a set that includes both, you get a proper disjunctive syllogism: whichever one I have, the other one can't be, so it's the only other one in the set. Reasoning from the "mine" framing isn't nearly so clean.

Feels like the explanation is right here, but I'm not quite seeing it.

Since we've all been throwing code around, here's a curiosity in Prolog:


Here's a swish link so you can try it out.

The idea is that if you run a query like this


you'll be able to backtrack all the way up the tree and back down, like this:


(There's a duplicate 10, but oh well. It shouldn't really be a tree but just a graph, since there's two routes from root to 10. Might redo it, but this was quick.)

This (nearly -- ignoring the dupe issue) represents @Michael's view of the problem construction.

BUT, the design of the actual problem is more like this:


If you insert a cut between the two predicates, backtracking all the way up to the selection of the envelope pair is blocked. When you ask for more solutions, you get just the one, not three.



Asking for another solution -- backtracking -- corresponds cleanly to swapping. (It's why the predicates are written a little odd, to be both randomized and backtrackable.)

What does the cut correspond to?

There is a difference between these two claims:

1. I have a 1 in 2 chance of picking the larger of two envelopes.

2. The envelope I have chosen has a 1 in 2 chance of being the larger of the two envelopes.

Suppose the pair is (5, 10). I have a 1 in 2 chance of picking 10. But 10 does not have a 1 in 2 chance of being greater than 5; it _is_ greater than 5.

just because I can plead, cajole, call in a favor, etc. and call them all “asking” doesn’t make the conditions allowing for a request to be any less specific nor the criteria for judging the line where it becomes pressuring any less clear. — Antony Nickles

Is that line particularly clear? Isn't this exactly the sort of thing people very often disagree about?

("Allowing"???)

And sure we can use language lazily if we like, but beating a nail in with a screwdriver doesn’t make it a hammer. — Antony Nickles

But this is odd. It takes considerable effort for Descartes to achieve the degree of abstraction he does in his reasoning, to extract himself from everyday ways of thinking. Doesn't look like laziness.

Now, f I don't understand the difference between a hammer and a screwdriver, I might select one or the other indifferently, much like a lazy person who does understand the difference but doesn't care. But that doesn't make me lazy. For that I would have to have deliberately shunned opportunities to learn the difference, and so on.

Besides, maybe you pound with the screwdriver because there's no hammer to hand. Recognizing that the screwdriver will do is not laziness, here, but insight, achieved by abstracting, and by flouting the rules about how tools ought to be used.

Affirming or doubting are acts with very specific criteria done in particular situations, just like asking, or thanking. — Antony Nickles

How specific?

Is there not more than one way of asking? Of thanking? Of affirming or doubting?

Are there not specific sorts of specificity?

How finely must we chop experience before the spectre of generality has been sufficiently warded off?

What you want is the odds that this interview is a heads-type interview. — Srap Tasmaner


Yes, that's the left hand side of the theorem: P(Heads|Questioned). — Michael

No, that's the probability that the coin was heads given that I'm being interviewed. The trouble is on the RHS, the probability that I am being interviewed given that my coin was heads.

Take a space and partition it evenly into a heads half and a tails half. Now, in the heads half put one point, an interview; in the tails half, put many interviews.

If you choose a half space, in each you will find at least one interview. You're interviewed either way.

But if you randomly select an interview among all the interviews, is it more likely to have come from the heads half, with its one interview, or the tails, where there are more?

That's what we want to capture with the base rate of heads interviews among all interviews, not the non-emptiness of the heads half of the space.

The probability that I will be questioned if the coin lands heads is 1. The probability that I will be questioned is 1. — Michael

But these are useless, uninformative, ambiguous categories. What you want is the odds that this interview is a heads-type interview. If Beauty could determine her current state in the world, what type of interview she is being given, she could answer with certainty. She cannot, but she can determine the likelihood of her being in each of the two possible states, states that she cannot otherwise distinguish. And she knows that it is more likely that she is being interviewed because it was tails.