I might have missed your point here. In fact, SB does know the truth value of “C appears in the experiment”, because C is, by definition, the symbol that appears on the calendar on the day in which she is engaged in this reasoning. So, for that specific symbol C, she knows for a fact that it appears during the experiment.
I would have agreed with you that “that can’t be extrapolated from ‘I’m seeing C on the calendar’”, but only for the technical reason that the statement “I’m seeing C on the calendar” couldn’t even be given meaning, much less used as a premise for further deductions.
My point was that it’s not good enough to “know whether it appears during the experiment.” She has to know whether or not it appears during the experiment, to calculate other odds based on the truth or falsehood of that fact. Because if she can’t accurately detect all cases where the statement is true, she can’t calculate the odds using math that’s based on *all *the cases where it’s true.
Just to be clear, I’m not using C to represent a variable. It is the literal symbol that appears on the calendar on the day that she is doing the reasoning that we are discussing. That is, the display literally shows a semi-circular arc that is open on the right-hand side.
On another day of the experiment (if there is another day), she’ll reason similarly, but with a different symbol. Of course, the probabilities will work out the same.
Whereas the thirder argument you presented assumes that she will know it regardless of whether it appears in the experiment on the day in question or not, in order to arrive at a certainty level of 1/3. Because if you look at the actual cases, across the cases that any given symbol appears on the day in question, enabling her to know about that particular symbol, the odds of heads are 1/2. Only by explicitly including cases where she will not know that that specific symbol is in the experiment will you arrive at 1/3.
Claiming that she knows about it becuase she can see it, and then including cases where she won’t know about it because she’s seeing the other symbol and claiming they work too because the other symbol is now in play, is sort of the mathematical equivalent of moving the goalposts - except when you move them to cover symbol B, you uncover the cases covered by symbol C.
That’s just not true. Look again at the computation in post #140. Remember that C is literally the symbol that appears on the calendar on the day that she is carrying out that computation. She uses only the fact that C appears during the experiment. There is no assumption that any symbol appears during the experiment other than the one symbol about which she has direct knowledge.
Contrariwise, look at the list of actual possibilities and odds I gave in post 142 as a rebuttal to 140. It is quite clear that if you look at only the cases where she actually sees the literal symbol C, the odds of heads are 1/2 across them. Your post 140 explanation does explicitly map also to cases where C is in the experiment and SB does not know it, and requires their inclusion to reach the thirder conclusion.
Put another way - your math is equavalent to the math for “B is not in the experiment” - to the point that it could basically be assumed as being synonymous to “C is in the experiment” as far as the math is concerned. But SB can never know that B is not in the experiment. This should be a clue that your math assumes a more complete knowledge of the situation than is actually available.
We’re having a failure to communicate here, but I don’t know what the problem is. Could you point to the first sentence in my post 140 that you think doesn’t follow from what precedes it? I’ll proceed with a line-by-line response to your last post, in the hopes that it will make my position clear.
I’m afraid that it’s not “quite clear” to me. You appear to me to be begging the question.
No, it doesn’t, because when she doesn’t know that C is in the experiment, she won’t be computing
p(coin lands heads | C appears during the experiment)
to determine her credence that the coin landed heads. Instead, she’ll computing
p(coin lands heads | B appears during the experiment)
(supposing that B is the literal symbol that appears on that day). But the probability will again come out to be 1/3.
No, it’s not. If it were, you would be able to derive
p(B appears during the experiment | X) = 0
from my equations, where X is some knowledge that she has at the time that she is performing this computation. But you will find that you won’t be able to do this. If you believe that you can, please show the explicit manipulations of the equations that give you this result.
It would be a clue if my math made that assumption. But it doesn’t.
If anything, my math assumes less knowledge than it should, because I was disallowing the statement “C appears today”, which is intuitively known to SB and stronger than merely "C appears during the experiment. I was disallowing the stronger statement because I believed that it couldn’t be given meaning at all. I now consider that to be an unacceptable dodge, but I don’t yet know how to take the matter head on.
Agree that scenario 2 is a better analogy for the SB situation. Unfortunately, given that, it leads us right back to the same disagreement: the chance that he put in the low-scoring game is 50%, but each particular awakening is probably the result of the high-scoring game, which can be easily shown if he repeats the identical procedure on every Sunday of the season.
Ok, at least we’re making a *little *bit of progress, as I now understand how you get from 8/14 Mondays to 50% heads.
Maybe the problem is that we’re trying to answer different questions. It seems like you’re answering: “How likely is it that the coin landed heads?” Whereas, I’m answering: “How likely is it that this interview was preceded by heads?” The answer to your question must be 1/2, because it’s a fair coin and nothing done after the fact can change that. The answer to the second question, however, must be less than 1/2, because there are more tails-interviews and SB has no way to tell one interview from another.
Not in a given experiment it’s not. In a given experiment it’s certainly the case that if you’re in the case where there’s any possibility at all of seeing the low-scoring game’s touchdown, the odds of seeing a high-scoring game’s touchdown is 0. And vice versa.
Two points: the question I’m asking is the one dictated by the problem, and secondly, in a single given experiment, your question becomes my question (and assumes the probability results associated with my question).
All rights reserved; got it.
Okay, Tyrrell McAllister, I’ve thought of a whole new approach that I think will solve this on its own, preventing me from even having to explain the logic behind my prior post.
When doing probabilities, you have to take into account all the information you have available, right? For example if you did the Monty Hall problem while taking into account that you know Monty opened a door (with a goat), and fail to take into account that you also know that he isn’t picking randomly, you’d get probabilities for the doors at 1/2 and 1/2, not 1/3 and 2/3 with is the final correct answer.
SB knows that if the experiment landed on tails, that there is another day besides the one she is answering any given question on, correct? And she also knows that no symbol appears on both days.
So, when she looks at the calendar and sees C, she not only learns “C appears in the experiment”, she also learns that it is not the case that “C appears on another day in this experiment”.
You have to take that into account when calculating your probabilities to get the correct result. I’m too rusty on my probabalistic math to do this (subtracting probabilities conditioned on facts that have a nonzero correlation with one another? Umm…), but I bet if you do it, you get a halfer result.
If I understand you, you’re saying exactly what I said in the last paragraph of my last post, except that I hesitate to condition on “C appears today (rather than some other day of the experiment)” before I have a satisfactory way (to me) to understand the meaning of the word “today”.
At the time that I wrote the OP, I was denying that the word “today” had any meaning except for the meaning provided by the random calendar, according to which “today” means “the day that C appears on the calendar.” (This is precisely why I introduced the random calendar: to give the word “today” meaning.)
I know realize that “today” should be meaningful apart from the random calendar, but I don’t yet know how exactly to do that. And until I understand how to give “today” meaning (apart from the random calendar), I don’t know what happens when you condition on statements that contain that word.
The upshot is that I will probably settle on being either a thirder or a halfer (instead of the halfer-in-some-cases position that I had before). But I don’t yet know which.
“C appears on another day in this experiment” doesn’t include the word “today”.
However, I’ll try to spread my lack of confusion. Here’s how to define “another day” without referring to “today”.
SB knows that if tails is flipped, there will be two interviews. She can, reasonably, label them as Interview One and Interview Two in an arbitrary manner, so as to be able to mentally distinguish between them.
SB also knows that when she is interviewed, that interview will either be Interview One, or Interview Two (assuming the coin landed on tails).
She can, based only on the above information, define “another day” as “The set of days including the single day on which interview 2 occurs, if the current interview is interview 1 and the coin was tails. Or the set of days including the single day on which interview 1 occurs, if the current interview is interview 2 and the coin was tails. Or the empty set, if the coin was heads”.
That can be defined based only on information explicitly given in the experiment setup and the ability to recognize that the interview that is happening to her is one of the interviews in the set of experiment interviews. If she can’t figure this much out, then she certainly doesn’t have the ability to calculate any probabilities, in part because she’ll be unable to connect the symbol on the calendar with the interview she’s experiencing or either of those with the experiment itself.
(I think I’ll label the interview if the coin was heads as Interview Zero.)
She’d be accomplishing this by noticing that she is being interviewed, and applying the knowledge that the interview is one of the interviews of the experiment, and thus must be a single member of the set [Interview Zero, Interview One, Interview Two]. The ‘current’ label is defined as an arbitrary label on the interview she is experiencing (as opposed to the other interviews, which she is clearly not currently experiencing).
As noted, this association must be made to deduce that her current experience -which notably includes the calender with it’s C symbol- has anything at all to do with the experiment. Thus, in order to be capable of knowing that “C appears in the experiment” (which is necessary for the thirder conclusion), the only way to reach that knowledge is by first identifying the interview in question as a single specific (if unknown) member of the set of interviews in the experiment. And after that identification is made, “the current interview” is as valid an arbitrary label for it as any.
So if she has the ability to learn “C appears in the experiment”, then she does indeed also learn that it is not the case that “C appears on another day in this experiment”.
I don’t think that’s a given; the question can be parsed different ways. “What is your credence now for the proposition that our coin landed heads?” For one thing, the inclusion of the word “now” suggests the relevance of SB’s temporal location vis-à-vis the coin flip and the experiment. For another, the phrase “your credence” is somewhat ambiguous, suggesting, like the word “now,” that what is being asked about is the probability relative to SB’s current predicament, not the probability as calculated by an objective observer. Presumably, there’s a reason why the question in the paradox is never phrased as simply as: “What are the odds that the coin we flipped came up heads?”, which would be the the most obvious and least confusing way to phrase it if that was the piece of information being sought.
For an example of someone demonstrating that SB’s “credence” is 1/3, see Elga, here, arguing the thirder position. Now, nominally he’s answering the question: “When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads?”, but if you read the article it becomes pretty clear that he would consider that question synonymous with the one in the OP.
It’s worth reading the following section (it’s brief), but I won’t quote it here as doing so would mean reproducing the larger part of the article.
I’m not sure exactly what you mean here. Are you differentiating between an experiment and an interview? If so, why, since my question specifically addresses an interview?
I think that you’re just not getting why I’m unhappy with just considering the meaning of “today” to be already self-evident. It seems to be one of those things where one person’s self-evident fact is another person’s paradoxical swamp of confusion.
My problem is with your use of the word “current”. If I didn’t have that problem, she could have just defined “today” to mean “the current day”
Now, you get close to my thinking when you propose considering the word “current” to be an “arbitrary label”. That, indeed, is exactly what I was trying to make explicit with my random calendar. The display on the calendar provided the “arbitrary label” for “today” (or, equivalently, “the current day” or “the current interview”). But, as you saw, that approach did not lead to the halfer conclusion.
That would be one way to do it, except that, from my perspective, you’re deducing a non-problematic statement (“C appears during the experiment”) from a problematic one (“C appears during today/the current day/the current interview”)
If I restrict myself only to statements I’m comfortable with, I can only say this:
If she learns “C appears in the experiment”, then she does indeed also learn that it is not the case that “C appears on another day in this experiment than the day on which it appears”.
Let’s look at this. He points out, correctly, that if it’s tails that monday and tuesday are equally likely to occur. Then he points out, correctly, that if it’s monday heads and tails are equally likely to occur. And then from this, he deduces that, if it’s monday, heads and tails are notequally likely to occur. Hmm, deducing X = 1/3 from X = 1/2 doesn’t sound right. What could the problem be?
He’s taking the odds of it being tuesday and the odds of it being heads, and then equating apples to oranges.
I mean that, in a single experiment, my answer is true for each and every interview. The odds are based on the fact that the number of times you are asked the question doesn’t influence the answer.
I maintain that if she can’t correlate C with a specific day in the experiment, she can’t correlate it to the experiment at all. Because the experiment is composed of nothing but specific days - and your math specifically correlates C with specific days of the experiment.
As for your calendar’s C being an eplicit arbitrary label for “today”, that only works if you correctly account for the fact that your calendar also conveys the information “C doesn’t appear on another day in the experiment.”
To review:
Options Odds (* is the day of interrogation)
1 H B* .25
2 H C* .25
3 T B* C .125
4 T B C* .125
5 T C* B .125
6 T C B* .125
Cases where C is in the experiment: 2 3 4 5 6
Cases where C is “today”: 2 4 5
Cases where C is another day in the experiment: 3 6
Notice how nicely the cases work out when you include all the available information.
That’s the only way to get to the non-problematic experiement. Your non-problematic statement has a foundation of sand (according to you).
How does she recognize that C appears in the experiment if she can’t connect that she is experiencing one specific single interview in the experiment? If she can’t do that, she can’t connect the symbol C on the calender to the experiment at all, because she won’t be able to connect her current experience with any of the specific separate cases that she could be in if she was in the experiment.
