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.
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.