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Kaj_Sotala comments on Open Thread: March 2010 - Less Wrong
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I have two basic questions that I am confused about. This is probably a good place to ask them.
What probability should you assign as a Bayesian to the answer of a yes/no question being yes if you have absolutely no clue about what the answer should be? For example, let's say you are suddenly sent to the planet Progsta and a Sillpruk comes and asks you whether the game of Doldun will be won by the team Strigli.
Consider the following very interesting game. You have been given a person who will respond to all your yes/no questions by assigning a probability to 'yes' and a probability to 'no'. What's the smallest sequence of questions you can ask him to decide for sure that a) he is not a rationalist, b) he is not a Bayesian?
1: If you have no information to support either alternative more than the other, you should assign them both equal credence. So, fifty-fifty. Note that yes-no questions are the easiest possible case, as you have exactly two options. Things get much trickier once it's not obvious what things should be classified as the alternatives that should be considered equally plausible.
Though I would say that in this situation, the most rational approach would be to tell the Sillpruk, "I'm sorry, I'm not from around here. Before I answer, does this planet have a custom of killing people who give the wrong answer to this question, or is there anything else I should be aware of before replying?"
2: This depends a lot how we define a rationalist and a Bayesian. A question like "is the Bible literally true" could reveal a lot of irrational people, but I'm not certain of the amount of questions that'd need to be asked before we could know for sure that they were irrational. (Well, since 1 and 0 aren't probabilities, the strict answer to this question is "it can't be done", but I'm assuming you mean "before we know with such a certainty that in practice we can say it's for sure".)
Yes, I should be more specific about 2.
So let's say the following are the first three questions you ask and their answers -
Q1. Do you think A is true? A. Yes. Q2. Do you think A=>B is true? A. Yes. Q3. Do you think B is true? A. No.
At this point, will you conclude that the person you are talking to is not rational? Or will you first want to ask him the following question.
Q4. Do you believe in Modus Ponens?
or in other words,
Q4. Do you think that if A and A=>B are both true then B should also be true?
If you think you should ask this question before deciding whether the person is rational or not, then why stop here? You should continue and ask him the following question as well.
Q5. Do you think that if you believe in Modus Ponens and if you also think that A and A=>B are true, then you should also believe that B is true as well?
And I can go on and on...
So the point is, if you think asking all these questions is necessary to decide whether the person is rational or not, then in effect any given person can have any arbitrary set of beliefs and he can still claim to be rational by adding a few extra beliefs to his belief system that say the n^th level of "Modus Ponens is wrong" for some suitably chosen n.
I think that belief in modus ponens is a part of the definition of "rational", at least practically. So Q1 is enough. However, there are not much tortoises among the general public, so this type of question isn't probably much helpful.