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Vaniver comments on Probability is in the Mind - Less Wrong
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Given the following propositions:
(P1) "My monitor is bamboozled."
(P2) "My monitor is not bamboozled."
(P3) "'My monitor is bamboozled' is not the sort of statement that has a binary truth value; monitors are neither bamboozled nor non-bamboozled."
...and knowing nothing at all about bamboozledness, never even having heard the word before, it seems I ought to assign high probability to P3 (since it's true of most statements that it's possible to construct) and consequently low probabilities to P1 and P2.
But when I read about bamboozledness on the Internet (or am asked whether my ball is blue), my confidence in P3 seems to go up [EDIT: I mean down] pretty quickly, based on my experience with people talking about stuff. (Which among other things suggests that my prior for P3 wasn't all that low [EDIT: I mean high].)
Having become convinced of NOT(P3) (despite still knowing nothing much about bamboozledness other than it's the sort of thing people talk about on the Internet), if I have very low confidence in P1, I have very high confidence in P2. If I have very low confidence in P2, I have very high confidence in P1. Very high confidence in either proposition seems unjustifiable... indeed, a lower probability for P1 than P2 or vice-versa seems unjustifiable... so I conclude 50%.
If I'm wrong to do so, it seems I'm wrong to reduce my confidence in P3 in the first place.
Which I guess is possible, though I do seem to do it quite naturally.
But given NOT(P3), I genuinely don't see why I should believe P(P2) > P(P1).
Just to be clear: you're offering me (300BCUs if P1, -100BCUs if P2)?
And you're suggesting I shouldn't take that bet, because P(P2) >> P(P1)?
It seems to follow from that reasoning that I ought to take (300BCUs if P2, -100BCUs if P1).
Would you suggest I take that bet?
Anyway, to answer your question: I wouldn't take either bet if offered, because of game-theoretical considerations... that is, the moment you offer me the bet, that's evidence that you expect to gain by the bet, which given my ignorance is enough to make me confident I'll lose by accepting it. But if I eliminate those concerns, and I am confident in P3, then I'll take either bet if offered. (Better yet, I'll take both bets, and walk away with 200 BCUs.)
I don't think the logic in this part follows. Some of it looks like precision: it's not clear to me that P1, P2, and P3 are mutually exclusive. What about cases where 'my monitor is bamboozled' and 'my monitor is not bamboozled' are both true, like sets that are both closed and open? Later, it looks like you want P3 to be the reverse of what you have it written as; there it looks like you want P3 to be the proposition that it is a well-formed statement with a binary truth value.
Blech; you're right, I incompletely transitioned from an earlier formulation and didn't shift signs all the way through. I think I fixed it now.
Your larger point about (p1 and p2) being just as plausible a priori is certainly true, and you're right that makes "and consequently low probabilities to P1 and P2" not follow from a properly constructed version of P3.
I'm not sure that makes a difference, though perhaps it does. It still seems that P(P1) > P(P2) is no more likely, given complete ignorance of the referent for "bamboozle", than P(P1) < P(P2)... and it still seems that knowing that otherwise sane people talk about whether monitors are bamboozled or not quickly makes me confident that P(P1 XOR P2) >> P((P1 AND P2) OR NOT(P1 OR P2))... though perhaps it ought not do so.