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Janos2 comments on 0 And 1 Are Not Probabilities - Less Wrong

34 Post author: Eliezer_Yudkowsky 10 January 2008 06:58AM

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Comment author: Janos2 10 January 2008 04:58:57PM 5 points [-]

I agree with cumulant. The mathematical subject of probability is based on measure theory, which loses a ton of convergence theorems if we exclude 0 and 1. We can agree that things that are not known a priori can't have probability 0 or 1, but I think we must also agree that "an impossible thing will happen soon" has probability 0, because it's a contradiction. An alternate universe in which the number 7 (in the same kind of number system as ours, etc.) is prime is damn-near inconceivable, but an alternate universe in which impossible things are possible is purely absurd.

If our mathematical reasoning is coherent enough for it to be meaningful to make probability assignments then certainly we are not so fundamentally flawed that what we consider tautologies could be false. If you are willing to accept that maybe 0 is 1, then you can't do any of your probability adjustments, or use Bayes' Theorem, or anything of the sort without having a (possibly unstated) caveat that probability theory might be complete nonsense. But what's the probability that probability theory is nonsense (i.e. false or inconsistent)? What does that even mean? We can only assign a probability if that makes sense, so conditioned on the sentence making sense, probability theory must be nonsense with probability 0, no? So averaged over all possible universes (those where probability theory makes sense, and those where it doesn't) the sentence "probability makes sense with probability 1" better approximates the truth value of probability making sense than "probability makes sense with probability p" for p<1, assuming the probability of probability making sense is >0. If it's not, it's still not worse, but what the hell are we even saying?