ControlsFreak
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User ID: 1422
ControlsFreak
8d ago
Which number in the above examples do you think that is? The one Alice bets, or the one she tells Bob, for him to use to make his bet? Or maybe the one she tells Bob to bet in Variant 2? Which bet? Which version of "came up heads"? The one that you observe some variable number of times? Or, like, "the one true one"?
EDIT: Or even just in your comment. You gave two possibilities. Which one do you think that is?
ControlsFreak
8d ago
Can't tell if really good joke because that's what we actually see the culture warriors roll with... or if actually missed the point.
ControlsFreak
8d ago
I think if Alice was specifically directed to input her "credence that the coin flip came up heads" then it's not really ambiguous if everyone is on the same page, as it were.
This is actually kind of the core of the problem! The original problem statement, long ago, used this phrase like "credence that the coin flip came up heads". But what does that mean? Obviously, if they all get on the same page and say, "It specifically means this and not that," then there's no ambiguity. But the "ambiguous question" position is saying, "Actually, maybe you need to specify, because maybe there are just multiple different things?"
You're perfectly correct. I semi-strategically left this possibility open for Variant 1. That you are able to realize that either can be done means that you adeptly realize that all of these probabilities can be 'things' at the same time. The only thing that matters is that Alice and Bob both know whether Alice is going to put p_tails or P(O_A(H|wake)) (probability of Alice's observation function) into the computer.
...I waited until Variant 3 to add the constraint that Bob doesn't really have a clue what's going on with Alice's observation function, just so that by that point, it became really really clear that we can do whatever it takes to force Alice to give a 'true' (or whatever you want to call it) estimate of p_tails apart from her estimate of what she's going to observe.
EDIT: This is extra important for actually driving home the Wiki description of it being an "ambiguous question". In Variant 1, it's ambiguous which one they're gonna communicate, right!? They have to specify in order to be able to communicate properly!
ControlsFreak
8d ago
I definitely agree that Groisman did it. I think that Groisman's very slight issue with the pre-filling of the box has apparently left a lot of people unconvinced. They're still publishing papers about it!
So, what I think is useful about my framing is that 1) It doesn't have this issue. Everything is very cleanly just in line with the original Sleeping Beauty setup. Alice is still even making her same bets! 2) I think more important than assigning them to Alice and Bob, my setup with the computer communication is demonstrating that Alice is, herself, retaining knowledge of the different probability spaces. You know this, because you can get her to tell you this (through the computer and her own bets). Even if you just had Alice and Bob doing independent experiments, one could very plausibly still go off the deep end of weird anthropics. By forcing all the conceptual distinctions to be contained within one hypothetical brain, I think you're pretty forced to realize that one brain can, indeed, hold different probabilities for different purposes, rather than "updating" your worldview because they sound similar at first glance or whatever weird timeline causality argument you want to twist your brain into.
That Damn Sleeping Beauty Problem
This is apparently Culture War, so whatever, I'll put it in the Culture War Thread. We discussed it a couple weeks ago. In the between time, I seriously considered writing up something to actually submit for publication, but I've decided against it after determining that it would be the absolute worst literature review I've ever had to do. There's just so much incoherence out there; I can't even bring myself to try to write brief sentences describing what it is these various papers are trying to say with their silly jargon.
So buckle up, you're about to get a clarifying contribution that, frankly IMHO, puts it to bed. I mean, I said in the linked comment that I thought Groisman's paper put it to bed (it's mentioned in the "Ambiguous-question position" section of the Wiki article), but I did acknowledge that I could see some people complaining. I referred to it in terms of moving sums around, but that was kind of opaque. So while I think that Lewis has come around to a more Groisman-like position (shrouded in jargon), folks like Piva are unconvinced, citing the N=1 funniness of the problem.1
I make a modification to the Sleeping Beauty problem. Suppose there are two people who are subject to this experimentation, in parallel. Alice goes through the canonical version, woken up either once or twice, with only one extremely minor relaxation to enable the rest of the thought experiment - the coin is possibly weighted, coming up tails with probability p. Alice is told what p is at all times; it can be, like, written on the wall in her room or something, or the magic memory drugs can just magically not erase that part of her memory.2 Bob is in a separate room, but his experiment is controlled by the same coin used for Alice. Bob goes through one of the following variants:
Variant 1) Bob is woken up with the opposite pattern. That is, if it's heads, Bob is woken up on both Monday and Tuesday, but if it's tails, Bob is only woken up on Monday. But Bob is never informed about what p is. Bob is scheduled to be woken up strictly later than Alice on any given day (i.e., Alice is woken up and put back to sleep between noon and 1pm and Bob is woken up and put back to sleep between 1-2pm). Alice has a computer terminal in her room, and the only thing she can do with this computer terminal3 is input into it a single number, her "credence that the coin flip came up heads". Alice knows that Bob will get to see that number when he is woken4. Of course, because of the set-up, she cannot put different numbers into this computer on different awakenings, for she has no way of distinguishing which awakening she is in. Alice knows that Bob will be computing how to make his bet based on the number she puts into the computer. Alice and Bob do not know each other, will never meet again, there is no way for them to come to some agreement to arbitrage their bets or anything, but in deciding what number to put into the computer, Alice is altruistic and wants Bob to be able to maximize his own payout.
Variant 2) Bob doesn't even know what his pattern of awakenings will be, but Alice does. This time, they both know that Alice is not putting in a probability "for the coin flip", but is putting in a probability that reflects how Bob should bet. Bob is still, in actuality, awoken according to this "opposite" pattern.
Variant 3) Bob is going to be awoken some number of days n, if the coin is flipped heads, but only once if the coin is flipped tails.5 Bob knows n, but not p. Alice knows p, but not Bob's n. For its and giggles, we could even say that Bob doesn't know Alice's pattern of awakenings (it shouldn't matter).
For all of these variants, assume that once a number is input into Alice's computer, it will forevermore be displayed in Bob's room. Alice's own computer will reset, so she can't tell that she put a number in it before, and again, since she can't know which awakening she is in, she'll always put the same number in. Even if Alice is only woken on Monday, if she puts a number in the computer, Bob will still see it on Tuesday (and possibly Wednesday, Thursday, etc.).
I contend that it is obvious that in Variant 1, Alice should still tell Bob that the probability of the coin flip is p, even though she is going to personally bet on heads with probability (1-p)/(p+1). That is, if p=1/2, Alice should bet heads with probability 1/3, but tell Bob that the probability of the coin flip is 1/2. She knows that Bob will be taking this number and doing math with it. In fact, she knows that Bob will see p=1/2 and choose to bet on tails with probability 1/3! Opposite of her own bet! Alice absolutely knows that there is a difference between the probability of the coin flip, itself, and the probability that one observes a particular result, given their relative experimental setups.
Variant 2 shows us that Alice is fully aware of this difference. She should make exactly the same computation that Bob would have done, had he known his own experimental setup. And so, she should, herself, bet on heads with probability 1/3... but tell Bob (by putting it in the computer) that he should bet on tails with probability 1/3. They're just different probabilities!
Finally, Variant 3 really drives home that there should be no doubt that Alice is still capable of simultaneously holding the knowledge that "the coin flip" has a different probability than her observation of the coin flip. This time, she can't compute Bob's best betting strategy. He knows his n; she doesn't. Bob just needs to know "the probability of the coin flip", so that he can compute his betting strategy.6 Alice does not "update" her estimate of "the coin flip"; she doesn't tell Bob that she actually thinks that the probability of the coin flip was 1/3 likely to be heads. She happily tells Bob that the probability of the coin flip was 1/2 (what other number would she put in?! what other number could she possibly compute that could be useful to Bob?), lets him compute his own optimal betting strategy appropriately, and proceeds to, herself, bet that she's 1/3 likely to observe heads.
If Alice tells Bob anything different in any of these variants, than Bob will lose money in his wagers. Since Alice is altruistic towards Bob's wagering, Alice would be wrong to "update" rather that simply remain cognizant that there is a difference between the probability of the coin flip and the probability that a particular person, in a particular experimental setup, will observe an outcome.
This should put to bed the idea that Alice "gains information" upon awakening that actually "updates" her estimation of the probability of the coin flip, itself. She had all the information she needed, from the beginning, to make all of the above bets and put all of the above numbers into the computer. Every single time, she's fully aware that there is just a difference between "the coin flip", itself, and the observation function defined by the various experimental setups. I think Lewis has mostly come around to this with his "centered/uncentered" language, but I think these variants make it as clear as can possibly be.
1 - This sort of thing is what ultimately led me to talk about it in vague terms of "moving sums around", because so many of the betting-based arguments still inherently rely on some sort of, "Assume you run this Sleeping Beauty experiment a bunch of times; in the long run, if you bet poorly, you lose money..." and so, really, the question is whether the pre-filled sums are essentially equivalent to the post-filled sums. I'm pretty sure my main argument kills this concern dead.
2 - This is consistent with the original version, as there is no sense in the original that SB does not always know the 'original' properties of the coin flip.
3 - Nothing about this computer business can affect Alice's own payout. Alice still wants to maximize her own payout. AFAICT, it doesn't matter whether you have her bet first, then use the computer or vice-versa. It shouldn't matter if it's structured such that she's woken up twice on each day, once to bet and another time to put a number into the computer, with no memory of the other awakening.
4 - Alice will always have put a number in before Bob is woken up, since Alice is always woken up on Monday.
5 - This is still the "opposite" sort; Bob is awoken more often on heads, whereas Alice is awoken more often on tails, just generalized to a larger possible n.
6 - np/((n-1)p+1) or (1-p)/((n-1)p+1) for heads/tails, as computed in the linked comment.
ControlsFreak
15d ago
they want to maximize the number of requested marriages implemented
I don't know that I agree. This is sort of a weird and arbitrary thing to try to maximize. I think plenty of effort has gone into messaging that marriage is a big, serious thing, shouldn't be entered into lightly, and really annoying for the State to unwind if it goes poorly. Plenty of States have processes that take some time and effort, in part so that they're not just maximally implementing all marriage requests, when they could be really rash and hastily/carelessly requested.
both because that's what the citizens want
I don't buy this one, because I don't think many citizens want to care about some cousins getting married. It's a tiny portion of the population. I think plenty of citizens are perfectly fine just not letting them get married. That's a perfectly fine default. Most citizens think they probably shouldn't even be having sex in the first place! There's basically no point in even thinking about them getting married. There's almost certainly not a ton of folks clamoring to create some special process for this for apparently no reason other than some vague quantity maximization. In fact, I think most citizens don't even know that this sort of case exists! On first impression, I imagine plenty would be perfectly happy with just reverting to the default of 'you're cousins, so you don't get married'.
and because it will make administration easier later down the line
I don't see how that's the case, either. It doesn't make administration much easier to have such a tiny percentage of people having sex marginally getting married, especially not for some weird special case that most people disapprove of anyway. This would be a tiny tiny change in the numbers and almost certainly not worth the effort.
Therefore any one couple failing to get legal recognition of their union is lost value
Yeah, I just don't see how there's "value" in them just getting married. Even if there was, then there seems to be little reason for the rigmarole of proving infertility. The biggest issue with your account is that there's just no reason for the rigmarole if they're just maximizing requested marriages implemented.
Instead, what I think is far more parsimonious is that the State is using marriage as an incentive. They know that there will be some cousins out there who want to be having sex and such. They can't just ban this. But they certainly don't want irresponsible, inbred procreation. So hey, Bob and Alice; you'd like to get married, right? Ya know what, Bob, if you just cut off your balls (or take some less drastic measure to ensure infertility), we'll let you get married. I think this is much more parsimonious than some vague quantity maximization, especially if they're going to go to the trouble to set up a whole process for this, with what are likely to be some necessarily complex rules (how exactly do you verify infertility, what is sufficient verification, etc.).
Would you disagree?
ControlsFreak
15d ago
Why would the State care if Bob got his balls cut off years ago? Why would they make some special process to 'allow' this? It's extra work; it seems to serve little purpose on your account. They have a perfectly good default to revert to - you're cousins, so you don't get married. Why would they do this other mess?
ControlsFreak
15d ago
"By default any man/woman pair who ask for it can be legally married, but we will deny it to couples that could produce inbred children with defects in the hope that that'll make them give upon fucking one another at all"
What about the bit about letting them marry if they show that they're infertile?
ControlsFreak
15d ago
Specifically concerning the example of some people only being able to marry if they show that they are infertile. I thought I was speaking plainly about this, but apparently, it didn't come across. What do you think they were trying to do?
I am not treating this as a fight, but it's clear that you are. You call it such and your demeanor is indicative that you may have something like cortisol levels going on which correspond to you perceiving it as a fight. I just want you to think about a brute fact in the world and give some impression as to what you think is going on. If I was being a jerk, I'd say that your immediate reaction to lash out at your interlocutor rather than have a respectable conversation about the topic is, yes, why the wokies won so many political fights. Bullying and anger won a lot of political victories, but left a lot of people privately unconvinced and resentful that such tactics managed to ram through major societal changes, rather than reasoned discourse.
ControlsFreak
15d ago
Possibly. Possibly not. I'm not really viewing it as a "debate". I'm just encouraging you to think about things. It would be nice to get your perspective on how you think about it. Perhaps it's something you've never thought about before; it would then be useful to get your fresh perspective on the matter rather than simply treating it as a "debate" to be "won", because that often leads to people simply trying to shove things into a pre-canned bin where they think they can just draw from their pre-canned set of talking points. So far, I think it's apparent that you don't have a simple pre-canned talking point for this, specifically, so it's useful to get your first impressions concerning the brute fact of such laws.
ControlsFreak
16d ago
if the line between who should be allowed to marry
Again, the perspective change needs to be pretty deep. It is not about who is "allowed" to marry. It's about what the State is trying to encourage/discourage. Think about the example I gave; see if you can come up with an idea for what it is that they're trying to do.
ControlsFreak
16d ago
If you are saying the line between who can marry and who cannot, which puts gay couples on the "cannot" side, is drawn on the grounds of who can produce children and who cannot
This "if" is precisely what my example points out is not true. The entire premise of the argument is simply false. The entire frame of reference simply does not make sense. Basically the entire remainder of this comment is sort of pointless from the get-go because of this flaw.
ControlsFreak
16d ago
This sounds like needless complexity, and it would invite a whole host of additional complex questions. Is there an expiration on a provisional marriage? Suppose you want to get married early, but delay having children a bit, is that allowed? Why or why not? The outrageous news stories will kill you, too. "This couple has had two miscarriages and is now about to hit the deadline on their provisional marriage!" This kinda thing will never fly with the public.
ControlsFreak
16d ago
then where's the law banning infertile people from marrying? Because on the axis of "family formation," there's no difference between them and the gays, is there?
As mentioned below, there are actually laws saying that some people couldn't marry unless they could show that they were infertile. Your entire frame of reference simply does not make sense, and you need a pretty significant perspective change.
Further, rather than there being "no difference", there is actually quite a huge difference, particularly in terms of intrusiveness to privacy. The government can very very simply look at the government documents which state that they're the same sex. What kind of standards, and what kind of intrusive nightmare would it be to require something like proof of fertility? @WandererintheWilderness would call it "Chinese-style authoritarian social engineering". These examples are worlds apart rather than being "no different".
ControlsFreak
16d ago
What do you think the purpose of such laws is?
ControlsFreak
16d ago
entirely material, but do not support discrimination between same-gender couples and opposite-gender couples one or both members of which is entirely infertile
Interestingly, many states had laws on the books that some people couldn't marry unless they showed that they were infertile. Namely, close relatives.
This has been trod over time and time again, but people still draw on this silly argument.
Do not impose your religious beliefs on people who do not share them.
Do not impose your atheistic beliefs on people who do not share them.
ControlsFreak
16d ago
In general, how sure should we be that the stock market today is doing well because of Donald Trump and not in spite of/unrelated to him?
In general, I've found that the answer to whether the president causes the market to go up/down is an XNOR function with inputs "Is the market up?" and "Is the current president on my team?"
ControlsFreak
22d ago
For example, we know there was at least one decent Pharisee, Nicodemus. And yet, Jesus doesn’t say “Beware of the leaven of the Pharisees and Sadducees! Except Nicodemus, he’s one of the good ones.”
He just says, “Beware of the leaven of the Pharisees and Sadducees!”
I think plenty of people see daylight between treating people like a class and being able to speak with labels. Even going back to the Scholastics, this could probably be viewed as a component (lol) of mereology.
ControlsFreak
22d ago
lollipop example
Sure, there are ways to add actual gain of information that is relevant for X. I'd have to work through different precise formulations.
Anyway, this morning, after having written my last comment (and before reading yours, as it happens), I was feeling very confident in it. I figured (as I should figure) that I should actually check out the literature in the area a bit, and see what's there. Of course, I was also looking for whether anyone in the literature had proposed a similar solution... and if so, whether there was any responding literature saying that it was insufficient in some way.
I proceeded with a mix of Wikipedia cites and Google Scholar, but it turns out that Wikipedia actually sums up what I now think is a great representation of my view pretty well, with reference to Groisman's 2008 paper. It's in the Wiki section on Ambiguous-question position:
Imagine tossing a coin, if the coin comes up heads, a green ball is placed into a box; if, instead, the coin comes up tails, two red balls are placed into a box. We repeat this procedure a large number of times until the box is full of balls of both colours. A single ball is then drawn from the box. In this setting, the question from the original problem resolves to one of two different questions: "what is the probability that a green ball was placed in the box" and "what is the probability a green ball was drawn from the box". These questions ask for the probability of two different events, and thus can have different answers, even though both events are causally dependent on the coin landing heads.
I hadn't quite hit on the right language, but I was getting there with random variables X and Y. I was pretty sure, and I'm still pretty sure, that if one actually spells out, in detail, formal definitions of X and Y, one can see that they do not relate in the form of a simple conditional probability that can be used to 'update' X. What I hadn't yet specified was that the main way that they differ is that they're describing different sample spaces.
One can make this analogy even more explicit by saying that if heads is flipped, a green ball that has written on it, "Monday, heads" is placed in the box. If tails is flipped, one red ball labeled "Monday, tails" and one red ball labeled "Tuesday, tails" are placed in the box.
I think very clearly here, one can say that when you pull a ball from the box, there is a 1/3 chance that you see a green ball. That is exactly the same as saying that there's a 1/3 chance that you see "heads" written on the ball. Similarly, there is a 2/3 chance that you see "tails" written on a red ball. "Seeing heads/tails on a ball" is random variable Y.
...but you cannot say that there was a 2/3 chance of tails having been flipped (random variable X). That's just a different sample set. You don't "gain information" about what was flipped by knowing that a ball has been drawn from the box (waking up). You had all the information you needed at the first moment, because you knew the experimental setup and how the depositing/withdrawing mechanism worked.
Balls are deposited according to the sample set {One Green, Two Red}, where they have some stuff written on them, but they're withdrawn according to the sample set {Green/Monday, Red/Monday, Red/Tuesday}.
This is also, I believe, the key intellectual step that justifies the naive thirder position against the naive halfer position in the first place - that because you have no information about which situation you're waking up in, you have to realize that the set of possibilities has three elements (over which, you take a typical uniform distribution), only one which has you seeing heads (green) and two which have you seeing tails (red).
To reiterate, yes, the correct betting strategy for what you observe will be 2/3 red/tails, but I don't believe that any property of conditional probability implies that your estimate should be that tails was flipped with probability 2/3. I think it would actually directly violate the laws of probability, for if you apply the laws of probability to the actual mechanics of the experiment and say that tails is flipped with probability 2/3, then you should observe tails with probability 4/5. This is actually a pretty straightforward calculation.
p = probability of flipping tails/depositing 2red
Un-normalized probability of observing tails/observing red: 2p
Un-normalized probability of observing heads/observing green: (1-p)
2p + (1-p) = p+1 is our normalization constant over the three possible balls
Normalized probability of observing tails/observing red: 2p/(p+1)
Normalized probability of observing heads/observing green: (1-p)/(p+1)
This is just the mechanics of the game. For p=1/2, we get 2/3 and 1/3. For p=2/3, we get 4/5 and 1/5.
This math should work for the extended versions of the game, too. If you wake up once (have one green ball) on heads, but wake up n times (have n green balls) on tails, then
p = probability of flipping tails/depositing n red
Un-normalized probability of observing tails/observing red: np
Un-normalized probability of observing heads/observing green: (1-p)
np + (1-p) = (n-1)p+1 is our normalization constant over the n+1 possible balls
Normalized probability of observing tails/observing red: np/((n-1)p+1)
Normalized probability of observing heads/observing green: (1-p)/((n-1)p+1)
I also took a little time on Google Scholar to check some of the papers that cited this Groisman paper. Many of them did the typical thing of just citing a paper because it came up in their own GS search, clearly not having read it (this has happened to my own work plenty, much to my chagrin). I already can't remember whether there was one or two papers that actually said something about what Groisman did and complained about it, but their complaint wasn't really comprehensible to me (maybe if I spent more than a morning on it, I could figure out whether I think it's valid or not). Perhaps you'll still disagree along some lines like that and be able to explain it better.
Maybe I could still imagine a critique, perhaps in terms of moving sums around (I.e., there are cases of multiple summation where you can/can't moving an inner sum out to an outer sum), but sums are often not that hard to move around. I'd definitely need to see a pretty detailed formal argument of where exactly a problem occurs. Otherwise, I'm pretty doubtful that any more informal argument is going to move me much.
It also comports with my casino game example. A static, non-feedback policy is just queried a different number of times, so it observes tails more often. I know that it'll observe tails more often (2p/(p+1) of the time), so that's how I should bet on what it observes. Perhaps to reach your preference for saying that whether you bet right matters the most, let's say this casino has you play two games simultaneously. In the first game, you're just betting on the outcome of a coin flip with probability p (maybe we even remove p=1/2 to remove possible degeneracies). In the second game, at the same time, you're betting on this modified game where your policy is queried twice if it's tails. They use the same coin and then evaluate both games, with your separate bets. If someone is not betting according to p and 2p/(p+1) in the two respective games, then I think you would declare that they are wrong. The difference between these two bets is simply that these two static policies have different observation/evaluation functions. The second policy doesn't somehow update mid-game and think that the properties of the coin flip have changed. If it did, your two policies would have weird and conflicting estimates for the properties of the coin flip. How would you even make your second set of bets?
...I guess finally, since I can't shut up, go back to computing policies for parallel Sleeping Beauty games. One is betting on a normal coin flip, while the other has this weird observation function. They use the same coin. Should those policies (people) have different estimates for the coin flip when they wake up... or just different estimates for what they will observe in their appropriately-blinded state?
ControlsFreak
22d ago
Let me be clear: nothing in my comment implies that you have ever said or implied that you are God. It is purely a matter of a tool for biblical interpretation. AFAICT, the Bible says that there is a difference between you and God. (Nothing to do with anything you have or haven't said.) Ergo, presumably, the Bible may think that there are things that God does which may not necessarily be things that you should do. One possible thing that might be in that category could be "treating people as a class". But of course, it could be complicated; maybe it's not in that category! But I don't think one can generally reason from, "Here is an example of God doing X," to, "Therefore, I should do X."
ControlsFreak
22d ago
SMBC does the philosophy of mathematics joke. As a bonus, throwing shade on the "unreasonable effectiveness of mathematics" line.
There are tons of culture war topics where this could be applicable, and I'm sure I'll link it many times in the future in those conversations, but I won't bring up any specific topics for a Friday thread. Just enjoying the funny today. It nails the sort of Internet Brashness that you get from various folks on a whole variety of topics when mathematics/philosophy of mathematics may be relevant.
ControlsFreak
22d ago
Sorry to belabor this, because I think we've made progress and are maybe not on the same page, perhaps somewhere in the same chapter... but...
My only issue is that I really, honestly cannot wrap my mind around a mindset that doesn't treat Y as the obvious thing the question's about.
I think it's because people... sorry to say, like yourself... say things like...
You can learn things about past events that change your probability estimates!
and present it as though someone told you that they rolled an even number, which would be a case in which you are genuinely gaining information about the past event.
And I think that's probably the core of the philosophical debate and why people try to connect this problem to anthropics. Many people genuinely think that there is something here that "updates" (or "changes" or something) their belief about a past event. This is a genuinely tricky question, and I'm not completely confident of my own perspective. I clearly lean toward just saying that they're separate mathematical objects, and you're not saying anything about changing your estimate of X when you make an estimate of Y. But tons of people want it to say something about changing their estimate of X and they present it with language that clearly indicates that they're trying to say something about changing their estimate of X.
I think that if you mostly agree with my presentation that you can simply cleave them apart and say something separate about X and Y, and that your estimate for Y doesn't necessarily have some temporally-bound back-implications for beliefs about X, then you're actually taking a particular philosophical position... one that I think a lot of thirders would disagree with. One that many of them (like yourself, frankly) would start off vehemently denying and claiming that it's just obvious mathematics that you're saying something about X.
ControlsFreak
22d ago
There are multiple examples of God, in the Bible, treating people as a class.
You are not God. God is not you.
But in any event, the biblical account of God also has multiple examples of God engaging differently with some individuals out of a class. These things are not trivial to just take one way or another.
Verily, in the Monty Hall problem. There, you actually do have a very very clear moment where information is gained and there is no ambiguity about which question you are being asked. But in this problem, if Alice tells Bob what you seem to want to have her tell him, we would say that she is wrong. We'd even say that she's extra wrong if she said she "updated".
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