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Kawoomba comments on Public Service Announcement Collection - Less Wrong
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Does each god know which god is which? And can I ask the same question twice to the same god?
Yes, True and False have to be omniscient to be able to answer consistently correctly or incorrectly, for any arbitrary binary question. There's a version of the answer which (spoiler) relies on asking unanswerable questions, which only Random would answer. There's also solution that doesn't rely on such gimmicks, however.
Do True and False know what answer Random would give, or are they required to say "I don't know?"
I interpreted it to mean that the question must be answerable with yes or no.
There are questions for which you don't know the answerability, so either the rules must be that questions asked are provably answerable, or else you are allowed to glean information from whether the god answers it or not.
Assuming that True and False do not know the future results of questions to Random, an example is a question to A (True) of "Would B say 1 + 1 = 2?" If B is False, it is answerable (with a 'no'). If B is Random, it is unanswerable.
Provably answerable from your own knowledge.
There's nothing in your wording that suggests random is not able to refuse an unanswerable question as one of it's potential random responses.