Carl works Monday through Friday in the European Rain Recording Society in Berlin. He records daily data from two field agents: Colin in London, and Carlos in Madrid.
On Sunday Night, the temporary janitor in his office is careless with his cigarette, and accidentally sets fire to some papers on Carl’s desk. Most are totally destroyed; just the bottom half of one piece of paper remains. It says “It rained here today.” Knowing nothing of the work that is performed in the office, he thinks this is a very odd message. “Today” and “here” are indexicals, and cannot have meaning at face value alone. So the note seems to convey no information. Still, he leaves it on Carl’s desk.
When Carl arrives at work on Monday, and the accident is explained to him, he knows some details that can provide context. Each report he receives from his agents is a single piece of paper with a header at the top that states the date and the name of the reporting agent.
Case A: His agents only report on days when it rained. So he knows that it rained at least once, in at least one of the two cities. This isn’t very useful to Carl, but it isn’t “nothing.”
Case B: His agents report every day, rain or shine. So he knows that four reports were one his desk when the fire started. The burned papers included three reports, and the half-burned one is the fourth. In addition to the information in case A, he can now place a 25% probability on each of the propositions that the report referred to Saturday in London, Saturday in Madrid, Sunday in London, and Sunday in Madrid, respectively.
In other words, when the words “today” and “here” can only be used in a context that resolves the issues linguistics has with indexicals, there is no such issue present. This is different from knowing that context, which is where probability is used.
+++++
“Indexical” is an adjective meaning “of or pertaining to an index.” As used in linguistics, and discussed in the article, it also means “without sufficient context to determine the value that is indexed.” Much like the pronouns “he, she, it, that, …” when no antecedent is evident. With the added meaning, it refers to the usage of the word, not to the word itself.
If an index’s context is not supplied, as in the janitor’s reading of the half-burned paper, it conveys no meaning. But if the context is explicit, as in “Colin Cumberbund; London, England; Saturday June 2, 2018: It rained here today,” then the meaning is explicit despite the fact that by itself the word “today” conveys no information.
So it is a non sequitur to say that such words cannot be used in a logic problem. They can, if context is supplied for the index. And in a probability problem, they can be used if the context narrows the range to a set of values—a sample space—to which you can attach a probability distribution.
Example: On Sunday, Beauty is put to sleep. A six-sided die is rolled. Beauty is wakened on whichever day, over the next six days, that is indexed by the die roll (1=Monday, 2=Tuesday, etc.). She is asked “What is the probability that today is Wednesday?”
The word “today” does not refer to the entire range of days Monday thru Saturday. It is used on only one day, and it has one unchanging value over the period when Beauty is awake. The fact that she does not know the value makes it a random variable, not an unfathomable reference. The answer to the question is 1⁄6.
The answer is the same if she is woken twice (with the amnesia drug), based on rolling two dice until the result is not doubles. Or if she is woken once or twice by accepting doubles. It can be used because, even if she is awake on another day, its usage refers to a fixed index into the range. The answer is 1⁄6 because no day is preferred over the others, even when the number of awakenings is uncertain.
+++++
I agree with the above analysis, about betting arguments. But not about the rest.
The error in the above argument, is that the details of the experiment do provide context for “today.” But as a random variable, not an explicit value. This is complicated by the coin toss, but using the non sequitur “‘today’ is an indexical so we can’t evaluate it” is a placebo used to avoid analyzing the context. “Raising technical challenges” does not mean “challenges that can’t be met,” it actually means they can.
Still, there are ways to avoid using an indexical in a solution. I suggested one in a comment to part 1: use four Beauties, where each is left asleep under a different combination of {Coin,Day}. Three are wakened each day of the experiment. One of those three will be awakened only once during the experiment. They are asked, essentially, “what is the probability that you will be awake only once?” It was agreed that this question is equivalent the original problem. Since there is no information that makes any of the three more, or less, likely to be the one, the answer is 1⁄3.
Another is to use identical twins Tim and Tom as interviewers. They play “Rock, Paper, Scissors” on Sunday night to see who will interview Beauty first. Beauty can describe the possible outcomes of the experiment based on the two propositions C=”Heads” and I=”Tim first.” Each outcome in the sample space {TT, TF, FT, FF} has a prior (that is, on Sunday) probability of 1⁄4.
Case A: They wear nametags to the interview. If Beauty sees that she is being interviewed by Tim, she knows that the outcome is not TF. That is, that it is not possible that the coin was/will be Heads and Tom interviews first. From this, she can update her probability for the proposition C=True from 1⁄2 to 1⁄3. She can do the same if she is interviewed by Tom.
Case B: They conceal their names. She can assign a probability Q to the proposition that her interviewer is TIm, and 1-Q to the proposition that it is Tom. Regardless of what value Q has, the Law of Total Probability says the answer is 1⁄3.
...the details of the experiment do provide context for “today.” But as a random variable, not an explicit value.
You seem to think that “random” variables are special in some way that avoids the problems of indexicals. They are not. When dealing with epistemic probabilities, a “random” variable is any variable whose precise value is not known with complete certainty.
Still, there are ways to avoid using an indexical in a solution. I suggested one in a comment to part 1: use four Beauties, where each is left asleep under a different combination of {Coin,Day}. Three are wakened each day of the experiment. One of those three will be awakened only once during the experiment. They are asked, essentially, “what is the probability that you will be awake only once?” It was agreed that this question is equivalent the original problem.
This is not what I understood you to be proposing. As described here, I would say that this is not the same as the original question, and does not avoid using an indexical. You have simply camouflaged the indexical by omission when you write that “One of those three [who are awake today] will be awakened only once during the experiment.”
The situation with indexicals is similar to the situation with “irrelevant” information. If there is any dispute over whether some information is irrelevant, you condition on it and see if it changes the answer. If it does, the judgment that the information was irrelevant was wrong.
Same thing with indexicals. You may claim that use of an indexical in a proposition is unambiguous. The only way to prove this is to actually remove the ambiguity—replace it with a more explicit statement that lacks indexicals—and see that this doesn’t change anything. So for your burned paper analogy, “today” and “here” are replaced by “the day on which Carl wrote this note” and “the city of origin for the call for which Carl took this note”. For the dice-throwing example, “What is the probability that today is Wednesday?” can be replaced by “What is the probability that the day on Beauty experiences y is Wednesday” because there can only be one such day, in which her last memories before her last sleep were from Sunday.
When we try this for the SB problem, however, a nonzero probability of ambiguity remains. Neal gives one way of removing the ambiguity in terms of information to which Beauty actually has access, that is, her memories and experiences. Doing that leads to an answer that is close to, but not quite the same as, treating “today” as unambiguous. If Beauty has exactly the same experiences y on both Monday and Tuesday, she cannot disambiguate “today”.
That leaves you with a choice: either you must agree that “today” is ambiguous in this problem, or you need to propose a different way of rephrasing the statement “today is Monday” into a form that removes the indexicals and then condition on information Beauty actually has.
Carl works Monday through Friday in the European Rain Recording Society in Berlin. He records daily data from two field agents: Colin in London, and Carlos in Madrid.
On Sunday Night, the temporary janitor in his office is careless with his cigarette, and accidentally sets fire to some papers on Carl’s desk. Most are totally destroyed; just the bottom half of one piece of paper remains. It says “It rained here today.” Knowing nothing of the work that is performed in the office, he thinks this is a very odd message. “Today” and “here” are indexicals, and cannot have meaning at face value alone. So the note seems to convey no information. Still, he leaves it on Carl’s desk.
When Carl arrives at work on Monday, and the accident is explained to him, he knows some details that can provide context. Each report he receives from his agents is a single piece of paper with a header at the top that states the date and the name of the reporting agent.
Case A: His agents only report on days when it rained. So he knows that it rained at least once, in at least one of the two cities. This isn’t very useful to Carl, but it isn’t “nothing.”
Case B: His agents report every day, rain or shine. So he knows that four reports were one his desk when the fire started. The burned papers included three reports, and the half-burned one is the fourth. In addition to the information in case A, he can now place a 25% probability on each of the propositions that the report referred to Saturday in London, Saturday in Madrid, Sunday in London, and Sunday in Madrid, respectively.
In other words, when the words “today” and “here” can only be used in a context that resolves the issues linguistics has with indexicals, there is no such issue present. This is different from knowing that context, which is where probability is used.
+++++
“Indexical” is an adjective meaning “of or pertaining to an index.” As used in linguistics, and discussed in the article, it also means “without sufficient context to determine the value that is indexed.” Much like the pronouns “he, she, it, that, …” when no antecedent is evident. With the added meaning, it refers to the usage of the word, not to the word itself.
If an index’s context is not supplied, as in the janitor’s reading of the half-burned paper, it conveys no meaning. But if the context is explicit, as in “Colin Cumberbund; London, England; Saturday June 2, 2018: It rained here today,” then the meaning is explicit despite the fact that by itself the word “today” conveys no information.
So it is a non sequitur to say that such words cannot be used in a logic problem. They can, if context is supplied for the index. And in a probability problem, they can be used if the context narrows the range to a set of values—a sample space—to which you can attach a probability distribution.
Example: On Sunday, Beauty is put to sleep. A six-sided die is rolled. Beauty is wakened on whichever day, over the next six days, that is indexed by the die roll (1=Monday, 2=Tuesday, etc.). She is asked “What is the probability that today is Wednesday?”
The word “today” does not refer to the entire range of days Monday thru Saturday. It is used on only one day, and it has one unchanging value over the period when Beauty is awake. The fact that she does not know the value makes it a random variable, not an unfathomable reference. The answer to the question is 1⁄6.
The answer is the same if she is woken twice (with the amnesia drug), based on rolling two dice until the result is not doubles. Or if she is woken once or twice by accepting doubles. It can be used because, even if she is awake on another day, its usage refers to a fixed index into the range. The answer is 1⁄6 because no day is preferred over the others, even when the number of awakenings is uncertain.
+++++
I agree with the above analysis, about betting arguments. But not about the rest.
The error in the above argument, is that the details of the experiment do provide context for “today.” But as a random variable, not an explicit value. This is complicated by the coin toss, but using the non sequitur “‘today’ is an indexical so we can’t evaluate it” is a placebo used to avoid analyzing the context. “Raising technical challenges” does not mean “challenges that can’t be met,” it actually means they can.
Still, there are ways to avoid using an indexical in a solution. I suggested one in a comment to part 1: use four Beauties, where each is left asleep under a different combination of {Coin,Day}. Three are wakened each day of the experiment. One of those three will be awakened only once during the experiment. They are asked, essentially, “what is the probability that you will be awake only once?” It was agreed that this question is equivalent the original problem. Since there is no information that makes any of the three more, or less, likely to be the one, the answer is 1⁄3.
Another is to use identical twins Tim and Tom as interviewers. They play “Rock, Paper, Scissors” on Sunday night to see who will interview Beauty first. Beauty can describe the possible outcomes of the experiment based on the two propositions C=”Heads” and I=”Tim first.” Each outcome in the sample space {TT, TF, FT, FF} has a prior (that is, on Sunday) probability of 1⁄4.
Case A: They wear nametags to the interview. If Beauty sees that she is being interviewed by Tim, she knows that the outcome is not TF. That is, that it is not possible that the coin was/will be Heads and Tom interviews first. From this, she can update her probability for the proposition C=True from 1⁄2 to 1⁄3. She can do the same if she is interviewed by Tom.
Case B: They conceal their names. She can assign a probability Q to the proposition that her interviewer is TIm, and 1-Q to the proposition that it is Tom. Regardless of what value Q has, the Law of Total Probability says the answer is 1⁄3.
You seem to think that “random” variables are special in some way that avoids the problems of indexicals. They are not. When dealing with epistemic probabilities, a “random” variable is any variable whose precise value is not known with complete certainty.
This is not what I understood you to be proposing. As described here, I would say that this is not the same as the original question, and does not avoid using an indexical. You have simply camouflaged the indexical by omission when you write that “One of those three [who are awake today] will be awakened only once during the experiment.”
The situation with indexicals is similar to the situation with “irrelevant” information. If there is any dispute over whether some information is irrelevant, you condition on it and see if it changes the answer. If it does, the judgment that the information was irrelevant was wrong.
Same thing with indexicals. You may claim that use of an indexical in a proposition is unambiguous. The only way to prove this is to actually remove the ambiguity—replace it with a more explicit statement that lacks indexicals—and see that this doesn’t change anything. So for your burned paper analogy, “today” and “here” are replaced by “the day on which Carl wrote this note” and “the city of origin for the call for which Carl took this note”. For the dice-throwing example, “What is the probability that today is Wednesday?” can be replaced by “What is the probability that the day on Beauty experiences y is Wednesday” because there can only be one such day, in which her last memories before her last sleep were from Sunday.
When we try this for the SB problem, however, a nonzero probability of ambiguity remains. Neal gives one way of removing the ambiguity in terms of information to which Beauty actually has access, that is, her memories and experiences. Doing that leads to an answer that is close to, but not quite the same as, treating “today” as unambiguous. If Beauty has exactly the same experiences y on both Monday and Tuesday, she cannot disambiguate “today”.
That leaves you with a choice: either you must agree that “today” is ambiguous in this problem, or you need to propose a different way of rephrasing the statement “today is Monday” into a form that removes the indexicals and then condition on information Beauty actually has.