Sorry to necro this here, but I find this topic extremely interesting and I keep coming back to this page to stare at it and tie my brain in knots. Thanks for your notes on how it works in the logically uncertain case. I found a different objection based on the assumption of logical omniscience:
Regarding this you say:
Perhaps you think that the problem with the above version is that I assumed logical omniscience. It is unrealistic to suppose that agents have beliefs which perfectly respect logic. (Un)Fortunately, the argument doesn't really depend
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The proof doesn't work on a logically uncertain agent. The logic fails here:
Examining the source code of the agent, because we're assuming the agent crosses, either PA proved that crossing implies U=+10, or it proved that crossing implies U=0.
A logically uncertain agent does not need a proof of either of those things in order to cross, it simply needs a positive expectation of utility, for example a heuristic which says that there's a 99% chance crossing implies U=+10.
Though you did say there's a version which still works for logical ... (read more)
I've now edited the post to give the version which I claim works in the empirically uncertain case, and give more hints for how it still goes through in the fully logically uncertain case.
Seems to me that if an agent with a reasonable heuristic for logical uncertainty came upon this problem, and was confident but not certain of its consistency, it would simply cross because expected utility would be above zero, which is a reason that doesn't betray an inconsistency. (Besides, if it survived it would have good 3rd party validation of its own consistency, which would probably be pretty useful.)
I agree that "it seems that it should". I'll try and eventually edit the post to show why this is (at least) more difficult to achieve than it appears. The short version is that a proof is still a proof for a logically uncertain agent; so, if the Löbian proof did still work, then the agent would update to 100% believing it, eliminating its uncertainty; therefore, the proof still works (via its Löbian nature).