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Unknowns comments on Self-modification is the correct justification for updateless decision theory - Less Wrong
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I've given such a logical fact before.
"After thinking about it for a sufficiently long time, the AI at some time or other will judge this statement to be false."
This might very well be a logical fact because it's truth or falsehood can be determined from the AI's programming, something quite logically determinate. But it is quite difficult for the AI to discover the truth of the matter.