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A probabilistic version of the liar paradox.

mind = blown. (from @richardwiseman's twitter)

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- E) All of the above.Oct 30, 2011
- The question is not a mathematically valid question (first order logic and set theory does not allow self-referring). Still, it's fun to play around with. :)Nov 1, 2011
- What is the probability of answering THIS question correctly? ..... I always get this right (and wrong)Nov 2, 2011
- Oh, and if you find a correct answer, then maybe you can also answer THIS?Nov 2, 2011
- 0% seems like the correct answer! BUT as Tao mentioned this is really a probabilistic version of liar paradox. The problem is not really a well-posed probability problem. Its self referencing sentence is not a LOGICAL statement. This will be more clear if we replace 60% with 0%. The contradiction would be more clear in this case but even having 60%, the problem IS still ill-posed although less apparent and it "looks logical" in this case.Nov 5, 2011
- The self-referential character of the question does not preclude its being logically well-formed. There are all sorts of logical frameworks that permit this sort of self-reference. Notably, any first-order logical system containing a bit of arithmetic allows for the "arithmetization of syntax" that Gödel invented to prove his famous incompleteness theorems, wherein the formulas of the language of the system are coded as natural numbers, and this permits a rather robust sort of self-reference. There are other perfectly sound logical systems that use less oblique devices such as the down arrow of Jon Barwise and +John Etchemendy's book The Liar that basically formalizes the "this sentence" operator of English.

My knowledge of probability theory is terrible, so with some trepidation I'll put forward my understanding of the issue. Suppose first we reframe the question (equivalently, I think) thus: If you have a 4-sided die whose sides are marked "A", "B", "C", and "D", what is the probability that, when you throw it, the letter that comes up labels the correct answer to this question? It seems to me that we get a paradox only on the assumption that

(*) The correct answer to this question is among the options listed.

On that assumption, exactly one of the listed answers, 25%, 50%, and 60%, is correct. (There are obviously so-called "grounding" issues here, but never mind.) Suppose then the answer is 25%. But, as the probability you will roll an A or a D is 50%, the probability you will get the right answer is 50%. So the answer can't be 25%. So suppose the correct answer is 50%. But the probability you will roll a B is 25%. So 50% can't be the correct answer either. By the same reasoning the correct answer can't be 60%. So, given (*), we have a paradox.

But this "paradox" is easily avoided simply by rejecting the dubious assumption (*) — obviously, there are multiple choice questions that fail to list a correct answer. So it appears to me that all we've got in the problem as stated is such a multiple choice question. But as +Terence Tao notes, a genuine paradox does seem to emerge if we replace answer C with 0%. For, by the reasoning above, the problem as stated DOES have a correct answer, namely, 0% — there is no positive probability that you can roll a correct answer. So, with the change to answer C noted, our possible correct answers are now 0%, 25%, and 50%. By the reasoning above, 25% and 50% cannot be correct. But suppose the correct answer is 0%. As before, we have a 25% chance of rolling a C and, hence, a 25% chance of getting the correct answer. But then 0% ISN'T the correct answer. But this means, once again, that there was no correct answer, i.e., that the probability of rolling a correct answer was 0%. So that IS the correct answer after all.

This paradox (if I've got it right) doesn't rely on the dubious assumption (*) above — we can of course assume that there are multiple choice questions that don't list a correct answer. The paradox here is that, if you assume there is a correct answer, there isn't one; and if you assume there isn't one, there is.Mar 14, 2014