I agree that there should not be a fundamental difference. Actually, I think that when an A.I. is reasoning about improving its reasoning ability some difficulties arise that are tricky to work out with probability theory, but similar to themes that have been explored in logic / recursion theory. But that only implies we haven’t worked out the versions of the logical results on reflectivity for uncertain reasoning, not that logical uncertainty is in general qualitatively different than probability. In the example you gave I think it is perfectly reasonable to use probabilities, because we have the tools to do this.
I agree that there should not be a fundamental difference. Actually, I think that when an A.I. is reasoning about improving its reasoning ability some difficulties arise that are tricky to work out with probability theory, but similar to themes that have been explored in logic / recursion theory. But that only implies we haven’t worked out the versions of the logical results on reflectivity for uncertain reasoning, not that logical uncertainty is in general qualitatively different than probability. In the example you gave I think it is perfectly reasonable to use probabilities, because we have the tools to do this.
See also my comment on a recent interesting post from Jessica Taylor: https://www.lesswrong.com/posts/Bi4yt7onyttmroRZb/executable-philosophy-as-a-failed-totalizing-meta-worldview?commentId=JYYqqpppE7sFfm9xs