An Apparent Paradox In Probability

I believe it was Gödel who made the statement, “S = S is false”, famous, through his work on the foundations of mathematics, but I’m not completely certain of its history. In any case, it is an interesting statement to consider, because it plainly demonstrates the limitations of logic, which are known elsewhere, e.g., in the case of Turing’s Halting Problem, which similarly demonstrates the limits of computation. You can resolve S by simply barring self-reference, and requiring ultimate reference to some physical system. This would connect logic and reality, and since reality cannot realistically be inconsistent, you shouldn’t have any paradoxical statements in such a system. The Halting Problem however does not have a resolution, it is instead a fundamental limit on computing using UTMs. I discovered something similar that relates to the Uniform Distribution in work my on uncertainty, specifically, this is in Footnote 4 of my paper, Information, Knowledge, and Uncertainty:

“Imagine … you’re told there is no distribution [for some source], in that the distribution is unstable, and changes over time. Even in this case, you have no reason to assume that one [outcome is more] likely than any other. You simply have additional information, that over time, recording the frequency with which each [outcome occurs] will not produce any stable distribution. As a result, ex ante, your expectation is that each [outcome] is equally likely, [since you have no basis to differentiate between outcomes], producing a uniform distribution, despite knowing that the actual observed distribution cannot be a uniform distribution, since you are told beforehand that there is no stable distribution at all.”

From the perspective of pure logic, which is not addressed in the paper, this is an example where the only answer to a problem, is known to be wrong. We can resolve this apparent paradox by saying that there is no answer to the problem, since the only possible answer is also known to be wrong. Nonetheless, it is a bit unnerving, because logic leaves you with exactly one possibility, that is wrong, suggesting the possibility of other problems that have superficially correct answers, that are nonetheless wrong, because of non-obvious, and possibly logically independent considerations. This is not such a problem, because it is actually resolved by simply saying there is no distribution, which is perfectly consistent with the description of the problem, that assumes the distribution is not stable. The point being, that you could have superficially correct answers to some other problem, that are ultimately wrong due to information that you simply don’t have access to.


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