Comment on Irrationality Game II

Wed, 04 Jul 2012 21:59:59 GMT

even non-mathematical propositions do end up having probabilities of 0 or 1 ... Tautologies

Tautologies are true for mathematical reasons and there is little difference - as far as probability assessment goes - between "P ∨ ∽P" and "Yding Skovhøj is the highest peak of Egypt or Yding Skovhøj is not the highest peak of Egypt". Thus, tautologies (and pseudologies, or how do we call their false counterparts) don't really make a category distinct from mathematical statements.

Conditional probabilities

I am not sure what you mean here. Of course there are conditional probabilities of form "if X, then X", but they already belong to the tautology group.

Regarding mathematical statements it's nevertheless important to notice that there are two meanings of "probability". First, there is what I would call "idealised" or "mathematical probability", formally defined inside a mathematical theory. One typically defines probability as a measure over some abstract space and usually is able to prove that there exist sets of probability 1 or 0. This is, more or less, the sort of probability relevant to the probabilistic method you have linked to. Second, there is the "psychological" probability which has the intuitive meaning of "degree of belief", where 1 and 0 refer to absolute certainty. This is, more or less, the sort of probability spoken about in "1 and 0 aren't probabilities".

These two kinds of probabilities may correspond to each other more or less closely, but aren't the same: having a formal proof of a proposition isn't the same as being absolutely certain about it; people make mistakes when checking proofs.

Plus in practice accepting "0 and 1 are not probabilities" rhetorically or otherwise just means that you stop writing 1 and start writing 1-epsilon. Whose belief is it really that doesn't affect anything?

If believing P doesn't affect anything, then naturally believing non-P doesn't affect anything either. So, if you agree that "1 and 0 aren't probabilities" is an inconsequential belief, does it mean that your answer to my original question is "yes"?

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