It’s interesting to note that we can still get Aumann’s Agreement Theorem while abandoning the partition assumption (see Ignoring ignorance and agreeing to disagree, by Dov Samet). However, we still need Reflexivity and Transitivity for that result. Still, this gives some hope that we can do without the partition assumption without things getting too crazy.
I don’t quite get this paragraph. Do you suggest that the failure of Aumanns disagreement theorem would be “crazy”? I know his result has become widely accepted in some circles (including, I think, LessWrong) but
a) the conclusion of the theorem is highly counterintuitive, which should make us suspicious, and
b) it relies on Aumann’s own specific formalization of “common knowledge” (mentioned under “alternative accounts” in SEP) which may very well be fatally flawed and not be instantiated in rational agents, let alone in actual ones.
It has always baffled me that some people (including economists and LW style rationalists) celebrate a result which relies on the, as you argued, highly questionable, concept of common knowledge, or at least one specific formalization of it.
To be clear, rejecting Aumann’s account of common knowledge would make his proof unsound (albeit still valid), but it would not solve the general “disagreement paradox”, the counterintuitive conclusion that rational disagreements seem to be impossible: There are several other arguments which lead this conclusion, and which do not rely on any notion of common knowledge. (Such as this essay by Richard Feldman, which is quite well-known in philosophy and which makes only very weak assumptions.)
I was using “crazy” to mean something like “too different from what we are familiar with”, but I take your point. It’s not clear we should want to preserve Aumann.
To be clear, rejecting Aumann’s account of common knowledge would make his proof unsound (albeit still valid), but it would not solve the general “disagreement paradox”, the counterintuitive conclusion that rational disagreements seem to be impossible: There are several other arguments which lead this conclusion, and which do not rely on any notion of common knowledge.
I don’t quite get this paragraph. Do you suggest that the failure of Aumanns disagreement theorem would be “crazy”? I know his result has become widely accepted in some circles (including, I think, LessWrong) but
a) the conclusion of the theorem is highly counterintuitive, which should make us suspicious, and
b) it relies on Aumann’s own specific formalization of “common knowledge” (mentioned under “alternative accounts” in SEP) which may very well be fatally flawed and not be instantiated in rational agents, let alone in actual ones.
It has always baffled me that some people (including economists and LW style rationalists) celebrate a result which relies on the, as you argued, highly questionable, concept of common knowledge, or at least one specific formalization of it.
To be clear, rejecting Aumann’s account of common knowledge would make his proof unsound (albeit still valid), but it would not solve the general “disagreement paradox”, the counterintuitive conclusion that rational disagreements seem to be impossible: There are several other arguments which lead this conclusion, and which do not rely on any notion of common knowledge. (Such as this essay by Richard Feldman, which is quite well-known in philosophy and which makes only very weak assumptions.)
I was using “crazy” to mean something like “too different from what we are familiar with”, but I take your point. It’s not clear we should want to preserve Aumann.
Interesting, thanks for pointing this out!
Okay, this makes sense. (I also added a link to a paper at the end, which may have overlapped with your reply.)