How so? Can you point to the line where it does that? But perhaps this is another philosophical issue; see below.
The calendar conveys the information that C doesn’t appear on any day other than the day on which it appears. That fact is used in the math from my post 140. (The prior probability that C appears is computed using the assumption that each symbol has at most one opportunity to appear.)
I don’t mean to seem dense, but I don’t understand this argument. Can you formulate it in terms of the probability calculus as I did in post 140?
What can I say :)? On this point, I have no problem attributing meaning directly, but you insist that the statement is meaningless without further analysis. Oh how the tables have turned ;). (Okay, not really. I understand that you’re not calling the statement meaningless. You’re just arguing that it’s baseless unless she knows that the symbol appears today, where “today” is meaningful without appeal to the calendar.)
I just have no problem with saying that some element of a set has a particular property even if I am unable to indicate that element as anything other than “the element having that property”. Nonconstructive existence-proofs are pretty common in mathematics, especially when one can appeal to the Axiom of Choice. So I’m fine with saying that C appears on some day of the experiment, even if I have no way of indicating that day, except as “the day on which C appears”.
(Bolding mine; the remainder of the section included for context.)
You are presuming that she is able to infer that the current state of the calendar has any correlation with the experiment - this presupposes that she is able to determine that she is currently experiencing one of the specific days in the experiment. Which in turn presupposes that she can define an arbitrary term (we’ll pick “Yadot”, for no particular reason) to label the day in the set of the possible days that she is currently experiencing. This in turn means that she has all the context she needs to define and comprehend the term “another day of the experiment” (by defining it as the set of all days in the experiment period that she will experience as experment days and are not the the day labeled Yadot). And thus she has all the terms necessary to state and know the other fact the calendar symbol tells her: that C is not on another day of the experiment.
Given that she knows that, solving for “P(C appears during the experiment)” is only half the battle. She needs to then calculate p(C appears during the experiment AND C is not on another day of the experiment), and then calculate p(coin lands heads | C appears during the experiment AND C is not on another day of the experiment).
Assuming a tails flip, every symbol appears on the calendar. But the SB doesn’t see every symbol; and she doesn’t see every symbol appearing on the day it appears. She sees only the symbol that appears on the day that she is experiencing, which she necessarily has to have already singled out and uniquely label an arbitrary day “Yadot” as a precondition of being able to connect the symbol C with any day of the expermint.
I shall ramble further on that last point. It’s one thing to know based on the abstract problem definition that C will appear on the day it appears. She knows that before she looks at the calendar, in fact. What she learns upon looking at the calendar is that if she has recognized that she is experiencing some specific day of the experiment arbitrarily labeled “Yadot”, that the symbol C is then associated with Yadot, and based on that fact, she knows that C appears on one of the days in the set of days within the experiment. If she can’t associate C with Yadot, she can’t associate it with the experiment.
A failure to be able to identify the day being currently experienced as Yadot is equivalent to drawing a blue ball from a bag with a blue and maybe a red ball in it, and failing to comprehend that the ball you’re holding has any relation to the bag. It’s a critical failure of connection that precludes any data gained from your current ball-holding state or observations thereof from being correllated to the original problem.
The equivalent of deducing that C isn’t on another day in the experiment is noticing that since the ball in your hand is blue, the ball remaining in the bag must not be.
The equivalent of accepting that C is in the experiment, and not that C isn’t associated with non-Yadot days (if any) is managing to notice that the ball in your hand is blue, and failing to deduce that the ball in the bag (if any) must be red.
You need to calculate p(coin lands heads | C appears during the experiment AND C is not on another day of the experiment). Unforunately, I can’t do that in terms of symbolic probability calculus. I think of probabilities intuitively, not symolically - it’s been a long time since I actually was in a math class.
All that P(A) means is “the probability of A”.
All that P(A | B) means is “the probability of A limiting yourself to all the cases where B is true”.
Each of the six lines lists the probability of that given case occuring. To deduce the odds of any P(A | some set of conditions), you take all the lines that match the set of conditions, and take the ratio of the summed probabilities of the lines with A occuring, to the summed probabilities of the lines without A occuring.
Lines 2-5 satisfy the condition “C appears in the experiment”. Across them, it’s .25 to .5 that it’s heads.
Lines 2, 4, and 5 satisfy the condition “C is associated with Yadot”. Across them, it’s .25 to .25 that it’s heads.
Lines 3 and 6 satisfy the condition “C is assocated with a non-Yadot day”. Across them, it’s 0 to .25 that it’s heads.
For rather obvious reasons, the second is equal to the first minus the third - though I don’t know how to demonstrate that using probability calculus.
The problem is that to associate the visible calender symbol with some arbitrary one of the of the possible days in the abstractly understood experiment, she has to first define an Yadot within the abstractly understood experiment to assocate the calendar symbol with. SB doesn’t experience the whole experiment at once - she understands it abstractly, which is fine. Bringing knowledge from observed reality is also fine - though she obviously has to first mentally separate out an arbitrary day (Yadot) in the abstract experiment before she can think of a day in the abstract experiment as having that observed property.
You are, of course, doing this implicity without thinking about it. The error you’re making is not taking it to its conclusion and recognizing and including *all *the things she must be able to deduce in order to be able to deduce the one deducible thing you’re currently taking into consideration.
Basically I don’t care if you let her mentally separate out and label Yadot or not; however you can’t accept one statement that requires that separation and then fail to account for all the other facts that that separation implies - not if you want to get a correct answer.