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Paul_Gowder 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: Paul_Gowder 10 January 2008 07:39:57AM 1 point [-]

hmm... I feel even more confident about the existence of probability-zero statements than I feel about the existence of probability-1 statements. Because not only do we have logical contradictions, but we also have incoherent statements (like Husserl's "the green is either").

Can one form subjective probabilities over the truth of "the green is either" at all? I don't think so, but I remember a some-months-ago suggestion of Robin's about "impossible possible worlds," which might also imply the ability to form probability estimates over incoherencies. (Why not incoherent worlds? One might ask.) So the idea is at least potentially on the table.

And then it seems obvious that we will forever, across all space and time, have no evidence to support an incoherent proposition. That's as good an approximation of infinite lack of evidence as I can come up with. P("the green is either")=0?

Comment author: RobbBB 21 November 2012 06:58:52PM *  13 points [-]

If you assign 0 to logical contradictions, you should assign 1 to the negations of logical contradictions. (Particularly since your confidence in bivalence and the power of negation is what allowed you to doubt the truth of the contradiction in the first place.) So it's strange to say that you feel safer appealing to 0s than to 1s.

For my part, I have a hard time convincing myself that there's simply no (epistemic) chance that Graham Priest is right. On the other hand, assigning any value but 1 to the sentence "All bachelors are bachelors" just seems perverse. It seems as though I could only get that sentence wrong if I misunderstand it. But what am I assigning a probability to, if not the truth of the sentence as I understand it?

Another way of saying this is that I feel queasy assigning a nonzero probability to "Not all bachelors are bachelors," (i.e., ¬(p → p)) even though I think it probably makes some sense to entertain as a vanishingly small possibility "All bachelors are non-bachelors" (i.e., p → ¬p, all bachelors are contradictory objects).