• The language of thought.
    ↪fdrake

    Pfft. I was only using Gary Larson to support (!) to @unenlightened's point.

    ((Need clones! Would love to be here in this thread more, but I'm still trying to deal with Sleeping Beauty,
    and I've promised to participate in the Tractatus thread, which is now starting. Oh, and I have a job.))
  • Why be rational?
    2. is unacceptable because P is possible through mere coincidence. — TheMadFool

    True. It's also true that "If its predictions are accurate, then logic is justified" sounds more like a definition than anything else, that it says this is what we mean by "justified". Your syllogism, seen that way, is just figuring out whether the word "justified", so defined, applies to logic. And it does, despite the possibility that every test of logic has succeeded by the merest coincidence rather than because This Is How The World Is. Justification is for us, isn't it? Or were you thinking of some other meaning for "justification"?
  • Why be rational?
    1. If logic is justified then predictions it makes must come true
    2. Predictions it makes are true
    Therefore
    3. Logic is justified

    The fallacy the argument commits, per logic itself, is that of affirming the consequent.     — TheMadFool

    Are you quite clear what you want to argue here and what you don't?

    How would you feel about this same argument with the first premise replaced by its converse? Or replaced by a biconditional? The one would be valid, but maybe not what you want to say. The other only "partially" valid.
  • The language of thought.
    It looks to me that the dog knows when the cat is angry sans a common language. — unenlightened

    48d3c7d5a69a2871f787dd9ddd0318cc.jpg
  • Mathematical Conundrum or Not? Number Five
    Probability is about expectations. Success should be measured not by what proportion of your predictions were accurate, but by what proportion of outcomes you successfully predicted. That won't usually make much of a difference, but SB is skewed so that you get twice the credit when you're right about tails but are only singly penalized for being wrong about heads. If you guess heads all the time, in our guessing version of SB, you get exactly the same proportion of outcomes right as guessing tails, but you get doubly penalized for being wrong so that only 1/3 of your predictions were right.

    Because the SB scenario doubles tails outcomes, it is difficult, or at least unnatural, to express your confidence about the outcome through wagering. For instance, suppose the coin is biased 2:1 in favor of heads. You have inside information and are thrilled to be given even money odds. Then we get these results:
    100 tosses, 67 heads, 33 tails, 133 interviews.
    Betting heads consistently on a biased coin at even money, you break even. WTF?
    The sucker who was betting tails, who didn't know the coin was biased? He breaks even too.

    ADDED:
    Here's the spontaneous version of guessing-SB:
    Suppose I'm going to teach Andy & Michael a little about probability. I'm going to flip a coin a bunch of times, but before each flip, they each guess. When they're right they get an M&M, and when we're done we'll count the M&M's and stuff. Now suppose before one toss, Michael guesses "Heads! Heads heads heads heads heads!!!!" If the coin lands heads, do I give him 1 M&M or 6?
  • Mathematical Conundrum or Not? Number Five
    What happens on Tuesday&HEADS is a part of the HEADS protocol, so you excluded part of it. — JeffJo

    (a) No it isn't. From the OP:

    A fair coin will be tossed to determine which experimental procedure to undertake: if the coin comes up heads, Beauty will be awakened and interviewed on Monday only. If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday. In either case, she will be awakened on Wednesday without interview and the experiment ends.

    I personally think it's slightly cleaner to describe the experiment as immediately sending her home at the conclusion of her final interview, whether that's Monday or Tuesday -- it reduces the temptation to argue about Tuesday and Wednesday -- but this is a standard presentation and it works just fine.

    There's also little difference if you modify the experiment as @Michael suggested and push the tails interviews back a day to Tuesday & Wednesday, and give Monday entirely to heads. Or do them all at different times of day on Monday, as some internet poster (on one of the LW threads, I think) has suggested.

    The only thing that matters is one for heads and two for tails.

    (b) If it were part of the heads protocol, by eliminating it, you would be eliminating heads as an outcome. Simply being interviewed would tell you the coin landed tails.

    If that seems like a tendentious interpretation, consider what happens as you increase the number of tails interviews: whatever the ratio, that's your odds it was tails. Do a thousand tails interviews, and it's a near certainty -- according to thirders -- that a fair coin lands tails.

    The "help" I am trying to offer — JeffJo

    I was speaking as Beauty there. I appreciate your input very much -- my point was that Beauty reasoning in this way makes no progress.

    you won't address my "four volunteers" proof that the answer is 1/3. — JeffJo

    I'll look at it. After 18 days now in this thread, I had grown weary of alternative presentations that require analysis to figure out if they're even equivalent to SB. But I'll look at it.
  • The language of thought.
    it is a sophistication to de-animate the world, rather than a struggle to animate others — unenlightened

    This is straight-up brilliant.

    ((Hey @frank -- remember that thread about whether we do in some sense "talk" to the world, ask it questions and listen for an answer, etc.))
  • Mathematical Conundrum or Not? Number Five
    ↪Andrew M

    Suppose we really were asking Beauty to guess the result of the coin toss, rather than give her credence. We'll do this with 100 Beauties and tally the results.

    100 tosses, presumed result of 50 heads and 50 tails, 150 interviews.
    If the Beauties all guess tails all the time, they will get 100 right out of their 150 answers.

    That looks like a 2/3 success rate, right? But is it?

    Out of the 100 tosses, they got 50 of them wrong. Looked at this way, that's a 50% success rate.

    There's an element to committed tails-guessing of over-performing in districts you're sure to win, if you see what I mean. That extra interview, the one that tips that the result was tails, it happens in the tails track. If you're guessing tails, you've already guessed it. You're not "extra right" about tails just because you get to be right twice about the same event.

    Some of the math puzzles me. Figuring out how to formalize it puzzles me. It's not a situation there's an off-the-shelf model for. Getting the math to work in a satisfying way is a chance to learn. None of that makes me at all uncertain about 1/2 being the right answer though.
  • Mathematical Conundrum or Not? Number Five
    ↪Andrew M

    Yeah that one was too easy. If I could have come up with a way for you not to know the difference between getting one box and getting two, I would have done that. Every now and then I think if I can find just the right analogy, I'll convince you!

    I'm disappointed that I'm still struggling with this, but it's a chance to learn.
  • Mathematical Conundrum or Not? Number Five
    when she is in an interview she knows that SOTAI is not happening — JeffJo

    If I condition on ~(Tuesday & HEADS), I exclude neither the heads protocol nor the tails protocol, as neither included it. This helps me not at all.
  • Mathematical Conundrum or Not? Number Five
    ↪Andrew M

    Here are the rules:
    • I will flip a coin.
    • On heads I will give you a box with 1 red marble in it.
    • On tails I'll give you a box with either 1 blue marble in it or 2 blue marbles in it.
    • You cannot tell the difference between a box with 1 marble in it and a box with 2.
    • You don't see the outcome of the toss.
    • You don't get to look in the box.

    I have tossed the coin and given you a box.
    What are the chances there's a red marble in the box?
  • Mathematical Conundrum or Not? Number Five
    what an agent knows about the outcome of a particular fair coin toss. — Andrew M

    Which in Beauty's case is zilch, isn't it?

    I agree about your double-header example, but don't see the similarity to SB at all. Interviewing here clearly gives you information. In SB it does not.
  • The language of thought.
    Of course an individual will not understand the sense of the words unless they have had private experiences of ecstasy or suffering that they can associate with the public expressions of these private states — Janus

    Is that true? It really might be -- I'm not disagreeing -- but there's also a little waystation we get to stop at sometimes of knowing at least what kind of word we don't understand. Oddly, stopping here seems to mean that even though you don't know how to use the word properly yourself, and could not judge whether someone else is, you get to have a partial sense of what someone using the word is saying. ("He's referring to a color I don't know by name." ”She's referring to a feeling I may never have felt.")

    ETA: Learning a new word usually means getting to this waystation first, I think.
  • The language of thought.
    ↪Sam26

    I had hoped I was suggesting an alternative approach and providing some motivation for it, why one might invest some time in pursuing such an alternative.

    Just tinkering, as usual.
  • The language of thought.
    It's not just catastrophic misunderstandings, but also subtle misunderstandings, so subtle that much of the time they're missed — Sam26

    Or this whole approach is wrong, this is the sort of thing language takes in stride, and the only question is how well or poorly it's done. (All models are wrong. All analogies suck.)

    I'm deeply suspicious of this Wittgensteinian "You think you're making sense but you're not, and it's so subtle you don't even know it, but I do." People ripping bits off one machine and shoving them into another, and making it work , making a machine do something it couldn't do before, or making a new machine -- this sort of bricolage might be perfectly commonplace. It's not a terrible description of uttering a sentence no one's ever uttered before and being understood.
  • Mathematical Conundrum or Not? Number Five
    ↪Andrew M

    Here's a story where Lewis's table seems to make sense:

    Beauty wonders to herself whether she's already been interviewed, and whether she'll be interviewed again. She reasons that there's a 1/2 chance of tails, and then a 1/2 chance that this is the first of two interviews, for 1/4; and there's also a 1/2 chance that this is the second of two of interviews, for another 1/4. The chance of either one of those being the case is 1/4 + 1/4 = 1/2, so she concludes that, since the chances this is not her only interview are the chances of tails, then the chances of tails are 1/2. Huzzah!

    But this is pretend reasoning. She starts out knowing the chances of tails are 1/2.
  • The language of thought.
    The contrast-space of 'use' here would simply be something like 'not-a-use in a langauge-game, rather than 'incorrect use'. — StreetlightX

    Yes, I think this is certainly the right approach, and the proviso is only that a word that does not belong, that has no use in a language-game, is a word that has not yet been given a role, a use in the language-game.

    What I think Wittgenstein is interested in blocking, as a sort of catastrophic misunderstanding, is taking a word as it used in one language-game, and bringing it into a another language-game where it is expected to play that same role, to have the same usage. The bare sign can readily be re-used. But if a word's functionality in a language-game is its interface, the way it connects to other words and how they're used, etc., there's no reason to expect that such a piece of machinery will even slot into another machine properly, that everything will connect so that it can function at all much less function here as it functioned there.

    I have some qualms about this view. Machines are of course quite rigid and specific in their requirements. Sometimes you can take something quite generic from one machine and use it in another, but most of the time hardly any part from any machine will fit into another machine. What's more, machines are largely designed not to change.

    Language use doesn't appear to be nearly this rigid. I'm not trying to start a war about LW here, but I think this way of reading him, even if a bit of a caricature, is natural, widespread, and a genuine tendency within his thinking, even if it's not the whole story.
  • The language of thought.
    If for e.g., I'm learning English words and I confuse the use of the word pain with being happy, then it's clearly incorrect. — Sam26

    In @StreetlightX's sense thread, it occurred to me that we might look at the rules of language as permissives, or as enabling communication, rather than as constraints.

    One way is this: "The word for that is 'thimble'", suggesting a natural predicative/model-theoretic approach -- you sort things into what "thimble" is true of and what not, which sentences including "thimble" are true and which not, etc. All the machinery of Frege-Tarski style semantics.

    Another is this: "You can/may call that 'thimble'", that is, using "thimble" to refer to that is a recognized usage in our speech community, using it that way you're likely to be understood, etc. There is "correctness" here as conformity to convention, but conventions are not carved in stone, and you participate in their revision.

    The permissive view deliberately leaves open a pair of possibilities: (1) finding new uses for "thimble" (as in Peter Pan, for instance); (2) finding other ways to refer to thimbles. (When Homer can't remember the word "spoon" he asks for a "thing you dig food with".)
  • Mathematical Conundrum or Not? Number Five
    ↪Andrew M

    No. I'm not sure how to formalize this (@fdrake help!), but I think if we want to do this as a table, it will be n+1-dimensional, where n is the number of tails interviews. We're going to multiply at each step, but that's just multiplying by 1 since each interview is a certainty. At the front, we're multiplying by 1/2 for the outcome of the toss.

    (We usually write about chains of independent events "combinatorially" -- HHHH, HHHT, HHTH, etc. We could do that here: "H" and "T" toss outcomes, "h" and "t" interviews, and then we're choosing between Hh and Ttt. Each has a chance of 1/2. Other permutations are eliminated by the rules: there is no Ht, no Tht, and so on.)

    There is no uncertainty in the interview pools themselves. This I have tried to express by having a single interview available to be selected. In the case of tails, that selection is with replacement, so you can repeat that same certain selection indefinitely. The only uncertainty here is in the outcome of the coin toss.

    Consider that from the experimenter's point of view, there is never any doubt about which interview comes next. There is uncertainty for Beauty, of course, but again if she can figure out the objective chances, that's what she sets her credences to. Knowing which of, say, 1000 tails interviews she's in would be useless to her -- either she knows it's a tails interview or not, and she'll never care which one it is. Knowing that it's not Monday would be useful, but by stipulation she can't.
  • History of a Lie: The Stanford Prison Experiment
    But saying that "people did what they chose to, period" is not even an attempt at an explanation, this is just giving up. We don't have to give up trying to find explanations, we just have to be honest and patient and never trust stereotypes and preconceptions. — SophistiCat

    Of course, and it goes without saying that I have oversimplified. ("One might say that oversimplification is the occupational hazard of philosophy, if it were not the occupation." -Austin)

    The book is well worth reading. There are explanations offered for macro phenomena (crucially, why were German Jews more likely to survive the Holocaust than, say, Polish Jews?), and there's interesting stuff about the political resources available to resist and so on. The main argument of the book has to do with institutions and state structures. I think he does his job as an historian, but stops cold at psychology and does not explain why individuals do what they did. Again, I'm simplifying.

    I was surprised to find that by the end of the paragraph I was writing about Snyder I was once again addressing the issue supposedly raised by Zimbardo, the responsibility of individuals in situations. Snyder is not a psychologist, but he works as an anti-Zimbardo.
  • The language of thought.
    I agree with this absolutely — fdrake

    That's a relief!

    My posts tried to achieve this by situating the public/private distinction within the use of language — fdrake

    Yes, I think it's an interesting move, and wanted to say that something similar happens with semantic distinctions that formally would require a meta-language. You might also look at Lewis's scorekeeping this way: the domain of discourse should be fixed before we start predicating, assigning truth-values, and so on, but instead the domain of discourse is (implicitly) negotiated as we go.
  • Donald Trump (All General Trump Conversations Here)
    Please take 5 minutes to watch this video and try to understand the impact on our nation.
    And I ask you, at what point does the necessity of self preservation come in?     — ArguingWAristotleTiff

    Did anyone actually watch the link I provided? — ArguingWAristotleTiff

    Yes. Here's the link again for anyone who missed it. I also recommend watching it. (It has nearly 5 million views, so don't feel guilty about giving him a few more.)

    There's some question about his numbers, but even if he's off by an order of magnitude, the point would stand that we cannot solve world poverty by taking in a million people a year, and inviting all the poor people in the world to move to America is probably Not A Great Plan™.

    I suspect BS King (linked above) slightly missed the boat on why this video went viral: it's because people see that little tiny brandy glass on the left and imagine pouring ALL of the world's poor gumballs into it. I suspect Roy Beck knows that's what people will imagine while he's talking. Evidently you did, because you remember this as a video about the threat to America, which is not what it's supposed to be about at all. He blathers on about this and that, demolishing a position no one holds, spewing out numbers (which again are probably wrong) but the important thing is the visual people will remember: a massive gumball horde that must be held at bay.
  • Mathematical Conundrum or Not? Number Five
    ↪Andrew M

    I'll take one more stab at this. I did it with marbles above, but here's the application.

    Which protocol to use is determined by the toss of a fair coin. Each protocol is an interview pool; when they awaken Beauty before Wednesday, they select an interview from the pool. Each pool has a single member.

    This is where Lewis goes wrong, and where I went wrong when I first came around to the halfer position. It's natural to imagine the tails interview pool as a collection of 2 interviews or 1000 interviews or whatever, in which case you end up with each interview being "discounted", as I put it. If there are 2 tails interviews, they each have a 1/2 chance of being selected; since the pool as a whole has a 1/2 chance, they're each 1/4.

    There are two problems with this view: (a) there are absurd consequences, like the 2/3 heads advantage on Monday, the likelihood of a second interview being 1/4 instead of 1/2, etc; (b) it does not represent how the experiment is conducted. Remember me frustratedly asking, back when I was a thirder, when anyone ever randomly selects between the two tails interviews? No one ever does, not even Beauty.

    The key for me was to recognize that there is only a single interview in each pool, but under the heads protocol you select that interview from the pool (100% chance on heads) without replacement, but under the tails protocol you select that interview from the pool (100% chance on tails) with replacement. Thus the chance of that tails interview is 1/2, just as it was for the heads interview, and the next time you do a tails interview, its chance is once again 1/2, and it's always 1/2, as often as you go back to the tails pool and select that interview again.

    (As a thirder I argued for conditioning this 50:50:50 to 33:33:33, but that's also wrong. The heads and tails interviews are never part of the same pool; it's one or the other. The wagering payoffs make it clear that this conditioning does not happen: every 1/2 stays a 1/2. The chance of a second interview is clearly 1/2, not 1/3. Now I understand how all this is possible.)

    The question Beauty needs to answer is, which protocol is in force? Each has a 1/2 chance. That's it. Either she's being interviewed once about a heads or repeatedly about a tails, but the chances of heads and tails remain 1/2 regardless.
  • Mathematical Conundrum or Not? Number Five
    That is of no use to her. When awakened, she doesn't know whether she is in an awake state that she should assign a probability of 1/2 to or 1/4 to. — Andrew M

    I see the problem.

    Yes, I argued recently for "discounting", basically the model that Lewis presents.

    Now I think that's wrong. There is no discounting. None of the 1/2's should be reduced to 1/4's. Monday is not 1/2:1/4 either.

    When Beauty is asked, "What is your credence that the coin landed heads?" she knows there's a chance the experiment is using the heads protocol, in which case this is her one and only interview, and a chance that it is using the tails protocol, in which case this may be her first interview, last, or one of many, depending. By stipulation, there is no evidence she can use to distinguish one interview from another; all she has to go on is her knowledge of the experiment's design. So the right answer is that there is a 1/2 chance of heads protocol -- which is the chance of heads -- and a 1/2 chance of tails protocol. It's 1/2 everywhere, all the time. When she is told is Monday, this makes no difference. The number of interviews conducted in the tails protocol also makes no difference. They are all interviews about the same solitary outcome.

    And the question of "updating" never arises because she never does.
  • Mathematical Conundrum or Not? Number Five
    I think you are both looking at the experiment from an independent observer's perspective (or Beauty's Sunday perspective) and not from Beauty's perspective when she is awakened and interviewed in the experiment. — Andrew M

    I'm just following the principal principle. If I can figure out what the objective chances are, so can Beauty, and she can set her credences accordingly.
  • History of a Lie: The Stanford Prison Experiment
    Was there something that history wasn't telling Milgram, Simbardo, et al about manipulation, brutality, dehumanization, submission, studied ignorance, and so forth that wasn't available in the histories? — Bitter Crank

    As far as that goes, reading Timothy Snyder's Black Earth can be a pretty strange experience. There's chapter after chapter of atrocities and no explanation offered. Some explanations are brought up only to be rejected. (For instance, he denies that Lithuanian people were more anti-Semitic than other Europeans and this explains something.) He makes a point of bringing up instances like a police unit being redeployed from Belgium and within weeks (maybe it was days) is shooting Jews lined up in front of ditches. You get the idea that he's making a point, that anyone might do this, but that's not quite right, or not the whole story. He ends with stories about people who rescue Jews and again refuses to explain. Makes a point of how those you might expect to rescue don't, and those you might not, do. Finally there is a glimpse of what he's about: people did what they chose to, period. You can explain a lot about the circumstances in which those choices are made, and he does, but people are not pawns of circumstance.
  • The language of thought.
    @fdrake, @Sam26

    Perhaps I missed it, but one point that doesn't seem to come out clearly in your exchange is that language use includes non-linguistic elements, which include private, personal, phenomenal experiences -- however you'd like to put that. People's actual pain is part of the "talking about pain" language-game -- even if only by its absence, as in shamming, lying, exaggerating, etc., and its absence would be important. (The blocks too are part of the builders game.)

    And then we note again how all of this is underwritten by the broadly similar biology of speaker and audience, blah blah blah. Ultraviolet is part of the spectrum just like green, but we talk about it differently.

    Really, all I have is a suspicion that as uses of language are dynamic, languages evolve, uses are introduced for novel phenomena, and the box the beetle is in shrinks. — fdrake

    Temporarily anyway. Experience isn't finite. (With all this talk of novelty, creativity and sense-making, why isn't @StreetlightX here?)

    Two short thoughts on this:
    (1) Imagining an instance of this sort of thing, of trying to get an idea (or an experience or sensation, novel or not) into words for someone else to understand, one interesting indicator of success is when the listener, trying to figure out what you're getting at, says something right that you haven't said yet -- on the right track! -- and even better when they say something right that you hadn't even thought of yet. In the best cases, you seem to pick up on this immediately, can recognize whether they've gotten it, even when their thought is now, in turn, new to you, though in tune, or in line, with the thought you were trying to express to them. (Again, all of SX's stuff about sense making is swirling around this for me.)

    (2) Back during the discussion of Russell's paradox, it occurred to me that while it may be true that natural language has no meta-language we can kick paradoxes up to, we seem to be able to temporarily switch levels as needed, and do so all the time. Even the simplest cases, like saying, "That's not what 'literally' means," betray an ability to temporarily, on-the-fly, conjure a meta-language we'll dispose of when we're done, which might be at the end of a single sentence. This is not so far from where Davidson lands, that people only ever speak local, temporary, negotiated shared languages, not some capital-L Language. And oddly, this starts to look like the scenario in (1) there.

    It is interesting that our ways of talking about pain, for instance. can evolve, despite the experiences being broadly constant through human history.
  • History of a Lie: The Stanford Prison Experiment
    ↪SophistiCat

    Thanks for posting this. I read all the links, and their links, and I'm still sorting through it. Something happened here, something we should probably be interested in, just not what Zimbardo says happened.
  • Mathematical Conundrum or Not? Number Five
    ↪Andrew M

    Just to clarify my position a little:

    According to the design of the experiment, there are two protocols:
    • a single interview on heads;
    • multiple interviews on tails.
    Beauty is asked for her degree of belief that the coin landed heads, that is, that the experiment is following the heads protocol. Chances of that are 1/2. As it happens, being told it's Monday provides no additional information because both protocols include a Monday interview. (If she were told it's Tuesday, she'd know it was the tails protocol.)

    Back when @Michael said it's just the difference between being asked once and being asked repeatedly, and that this makes no difference, he was absolutely right.
  • Mathematical Conundrum or Not? Number Five
    You also accept that conditionalization changes Beauty's probability of heads when she is told it is Monday — Andrew M

    No, I'm in the double halfer camp now. The post right above explains my current thinking.

    ((This is, I don't know, maybe the third time I've argued with @Michael about something and then concluded he was right all along.))
  • Donald Trump (All General Trump Conversations Here)
    In not reporting the crimes, the criminals are left to abuse without consequence, to walk and drive the same streets that I live on, that I am raising my kids on — ArguingWAristotleTiff

    Then your issue is not with the immigration status of the perpetrator, but the victim. Illegal immigrants, in this scenario, would also fail to report crimes committed against them by native-born citizens and legal immigrants. Is that what you intended to argue?

    I don't give a flying fig if someone is here legally or not, UNTIL they break the law. — ArguingWAristotleTiff

    See above.

    I believe there are a number of studies that claim to show immigrants, including illegal immigrants, commit crimes at a lower rate than the native-born. You can Google as well as I can. I agree that if there is an underreporting issue, which is plausible, it might be difficult, but not impossible, to correct for that.

    Also, I believe the Attorney General would take exception to your suggestion that entering the United States illegally is not in itself a crime.

    the immigrant presents a Social Security card. — ArguingWAristotleTiff

    Another social program illegal immigrants would pay into and receive no benefit from, as I understand it.
  • Donald Trump (All General Trump Conversations Here)
    For instance, if you are here illegally and your husband beats you and your children, who are you going to call? To call the authorities would be damning themselves in the process. What happens if you as an illegal immigrant are mugged and raped? Who do you turn to? — ArguingWAristotleTiff

    I'm confused. Are you describing here a taxpayer-funded service that illegal immigrants choose not to avail themselves of?
  • Donald Trump (All General Trump Conversations Here)
    So the taxes we pay are going not to improve our schools, to help our homeless or those who are hungry. No, they are being absorbed by the 'tax' that non legal citizens are putting on our social structure. — ArguingWAristotleTiff

    The economics is a little more complicated than that.
  • Mathematical Conundrum or Not? Number Five
    I think I've finally got the right model now, and it's nuts.

    Two urns, one with a one red marble, and one with one blue marble. One blue marble.

    You toss a fair coin to determine which urn you'll be taking a marble from. One coin toss with one outcome. The unchosen urn is removed.

    If we stop here, it's obvious that the chance of getting a red marble is 1/2, and the chance of getting a blue marble is the same. (In each case, there's a single marble with a 100% chance of getting chosen from the urn.) This is Monday.

    Now here's the wonky part: selection from the red urn is without replacement; selection from the blue urn is with replacement. You can go just twice, as in standard SB, or you can go a thousand times. Whenever you want, there's a single blue marble available.

    Looked at this way, you can see why there doesn't ever seem to be any discounting or conditioning -- not in the chances of a tails interview on Monday, not in the payoffs from wagering, not in the chances of a second interview, nowhere. Each marble inherits the full 1/2 probability of its urn because it is the only marble there. Tails doesn't show up as an event chopped up into parts, but as an event that repeats as often as you like.

    The coin toss determines whether you draw the one red marble or some number of blue marbles. The number of marbles drawn has nothing to do with the chances that you're drawing red or blue though.

    I'm not sure how to finish formalizing this, but I think it represents SB pretty well.
  • Mathematical Conundrum or Not? Number Five
    ↪Michael

    So Monday is independent of the coin toss but ¬Monday isn't? Aren't we just using ambiguous vocabulary here?

    Speaking of ambiguity, does when the coin toss happens affect Beauty's credences? Don't we want a solution that applies to a toss before the Monday interview as well as after?

    A fair coin will be tossed to determine which experimental procedure to undertake: if the coin comes up heads, Beauty will be awakened and interviewed on Monday only. If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday.

    The OP uses the toss-before model.
  • Mathematical Conundrum or Not? Number Five
    You are both ignoring part of the problem.

    I think the halfer reasoning should just be that it’s a 50:50 chance that it’s heads, whether unconditioned or conditioned to Monday. — Michael

    That's saying P(A | B) = P(A), and therefore A and B are independent events. Of course in one sense the coin toss and the day of the week are independent, but whether Beauty is interviewed on that day is clearly not independent of the coin toss: you yourself noted that if she's told it's Tuesday then her credence should be 0. Is that right? You have to decide whether you're using this table
       Mon  Tue
H  1/4  1/4
T  1/4  1/4
    or this table
       Mon  Tue
H  1/2  0
T  1/4  1/4
    Switching between them as needed is just equivocation.

    I think probability is a measure of self-locating uncertainty in a state space. It's not directly about coin outcomes, days, or even interviews at all, except in so far as they contribute to the construction of the state space. — Andrew M

    But Beauty is asked about the coin toss itself: it has one outcome which gives rise on average to 1.5 states for Beauty. Not discriminating between Beauty's states and the contribution of the coin toss to those states is another sort of equivocation. Beauty can make this distinction, just as you could in counting 1 red marble on heads and 2 blue marbles on tails, expecting a 2:1 ratio if the coin is fair.

    The thirder model relies explicitly on there being a single toss of a coin with heads and tails distributed 50:50. (And we've agreed you cannot construct an alternate model with a weighted coin.) How can Beauty take that as a premise and then be unable to reach the conclusion that the chances of heads were 1/2?
  • Mathematical Conundrum or Not? Number Five
    Yes, this is an important point. I think the intuitive comparison with a weighted coin is misleading since SB is just structured differently. Adding more interviews (and thus bets) on tails is not like increasing coin bias. — Andrew M

    Indeed, I think it means that the odds here are not truly 2:1 at all.

    I can't figure out how to make this into a normal wager of any kind. The closest I can come looks very similar to the problem itself: the bet is at even money, but on tails the wager is doubled. Now this is very strange. Conditional wagers are fine, but the one thing they would generally not condition on is the event whose outcome determines the outcome of the wager! Even your stake is unknown until the bet itself is resolved?! I don't know why people who find the wagering argument so attractive, as I once did, aren't more troubled by this.

    If you’re told it’s Monday then the probability is 1/2 that it’s heads and if you’re told it’s Tuesday then the probability is 0. — Michael

    This position is attractive, but I just don't understand how it works.

    Your expectation of heads is 1/2 before you know what day it is, right? So that includes the possibility that it's Tuesday. If you go with the way Lewis splits up the space, that's fine:

    .

    Learning that it's not Tuesday should increase your expectation of heads if you were taking the possibility of it being Tuesday into account, and that's how we end up with the 2/3.

    Is there a principled way to just ignore Tuesday, and any subsequent Tuesdays?

    I'd like to analyze the halfer's P(Heads|Monday) = 2/3 consequence further — Andrew M

    Michael is certainly right that this is the mirror image of the thirder problem -- we each seem committed to unreasonable confidence about heads or about tails as you increase the number of Tuesdays.

    I'm not convinced this is what the halfer position really implies, but the two are clearly related.

    That extra blue marble left in the hopper is not another outcome; it's just the rest of the outcome you already know about from the first blue marble. — Srap Tasmaner

    This is what I keep thinking about. You can add as many Tuesdays as you like to represent the tails outcome, but Beauty would only need to know about one of them to know the toss landed tails. All of the Tuesdays are clones of each other -- if you know the fact of the matter about one, you know them all.

    Still working on it. Maybe in another day or so I'll have something new to say.
  • Mathematical Conundrum or Not? Number Five
    Still another point:

    I have argued, as thirder, that there are three outcomes each of which has a 50% chance of occurring (the one heads interview and the two for tails). Taken conditional to the whole set {H1, T1, T2}, each would shift to a 33% chance.

    The thing is, this 2:1 proportion of interviews is right, but remember that SB does not payout like a wager on a 2:1 biased coin. It pays out like a 3:1 coin.

    It's not just the ratio of wins to losses, but the actual payouts that bothers me. If it were a genuine 2:1 deal, we'd expect a $1 wager on tails to pay out 2/3(1) - 1/3(1) = 1/3. It doesn't. It pays out 2*1/2(1) - 1/2(1) = 1/2.

    That means that scaling 50:50:50 to 33:33:33 never happens. The actual payouts represent a coin that has a 50% chance of heads and a 100% chance of tails, not a coin that is 33:67. That's pretty weird.
  • Mathematical Conundrum or Not? Number Five
    ↪Andrew M

    Here's another way to look at it.

    You could say it depends on what we take to be the event that must be predicted -- I've made that argument myself, and not long ago. Should Beauty predict the outcome of the coin toss or being asked about the coin toss?

    I used to think that being asked was itself evidence, but that's only true on two conditions: (a) Beauty doesn't know the rules and is unable to figure out the true chances of heads and tails; (b) she is willing to accept the absurdity that a single coin toss has, on average, 1.5 outcomes.

    It is certainly true that there are on average 1.5 interviews, but some of those interviews (the extra .5 on average) are about the same outcome. That extra blue marble left in the hopper is not another outcome; it's just the rest of the outcome you already know about from the first blue marble.
  • Mathematical Conundrum or Not? Number Five
    I keep saying Beauty can make a Dutch book, which is wrong of course and I'll quit it: the odds are coherent but they don't match the odds of the event being given odds on, so Beauty can expect a profit just by betting tails.
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