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by Alex Appel 992 days ago | Patrick LaVictoire likes this | link | parent

Interestingly enough, the approximate coherence condition doesn’t hold when there is a short proof of φ from ψ, but it does hold when there is a sufficiently short proof of ψ from φ. (Very roughly, the valuation of (φ∧¬ψ) is negligible, while the valuation of (φ∧ψ) is approximately equal to the valuation of φ.) So coherence only works one way.

On a mostly unrelated note, this sort of reasoning doesn’t seem to link well with the strategy of “add important mathematical facts that you discover into your pool of starting axioms to speed up the process of deriving things.” While a set of axioms, and a set of axioms plus a bunch of utility tool theorems (Godel’s 2nd incompleteness, Lob’s theorem, the fundamental theorem of calculus, etc..) may “span” the same space of theorems, the second one is much easier to quickly derive new interesting statements from, and seems to be how humans think about math. The inconsistency involved in getting 10 utility on the 5-and-10 problem is much easier to spot if the second incompleteness theorem is already in your pool of sentences to apply to a new problem.

As with the Russel’s Paradox example, in practice, counterfactuals seem to be mind-dependent, and vary depending on which of the many different lines of reasoning a mind heads down. If you define a subjective distance relative to a particular search algorithm, this objective valuation would just use the shortest possible subjective distance to the statement and a contradiction. The valuation using the human distance function of a statement in naive set theory would be high, because we were slow to notice the contradiction. So that aspect of counterfactual reasoning seems easy to capture.

Has anyone checked what this does on ASP problems?



by Patrick LaVictoire 981 days ago | link

ASP has only been formalized in ways that don’t translate to modal universes, so it’s only the analogous extensions that could apply there.

In the formulation here, it’s not enough that the agent one-box; it must do so provably in less than \(N\) steps. I think that both the valuations \(\nu_{\textsf{ZFC+Sound(ZFC)}+A()=1}(U()=1M)\) and \(\nu_{\textsf{ZFC+Sound(ZFC)}+A()=1}(U()=0)\) will probably be about \(1/2\), but I’m not sure.

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