Even if SEP was right about getting around the infinity problem and CK was easy to obtain before, it certainly isn’t now! (Because there is some chance that whoever you’re talking to has read this post, and whoever reads this post will have some doubt about whether the other believes that...)
Love this post overall! It’s hard to overstate the importance of (what is believed to be) common knowledge. Legitimacy is, as Vitalik notes[1], the most important scarce resource (not just in crypto) and is likely closer to whatever we usually intend to name when we say “common knowledge”, which this post argues (successfully IMO) is not actually common knowledge.
It does seem like legitimacy is possible to model with p-CK, but I’m not convinced.[2] Nor do I know how substitutable p-CK is with my old notion of CK, what it’s good for! Which theorems can be rescued with p-CK where they depended on CK? Does Aumann’s still hold with probability p upon p-CK of priors, or does the entire reasoning collapse? How careful do I have to be?
He seems to only talk about finitely many layers of higher order knowledge, as does Duncan (for explicitly pedagogical reasons) in his post on CK and Miasma. I think this can be right, but if so, for complicated reasons. And it still leaves some seemingly self-undermining “rationality” in our frameworks.
Mainly, I expect self-reference in legitimacy to be troublesome. A lot of things are legitimate because people think other people think they are legitimate, which seems enough like Lob’s theorem that I worry about the lobstacle.
Even if SEP was right about getting around the infinity problem and CK was easy to obtain before, it certainly isn’t now! (Because there is some chance that whoever you’re talking to has read this post, and whoever reads this post will have some doubt about whether the other believes that...)
Love this post overall! It’s hard to overstate the importance of (what is believed to be) common knowledge. Legitimacy is, as Vitalik notes[1], the most important scarce resource (not just in crypto) and is likely closer to whatever we usually intend to name when we say “common knowledge”, which this post argues (successfully IMO) is not actually common knowledge.
It does seem like legitimacy is possible to model with p-CK, but I’m not convinced.[2] Nor do I know how substitutable p-CK is with my old notion of CK, what it’s good for! Which theorems can be rescued with p-CK where they depended on CK? Does Aumann’s still hold with probability p upon p-CK of priors, or does the entire reasoning collapse? How careful do I have to be?
He seems to only talk about finitely many layers of higher order knowledge, as does Duncan (for explicitly pedagogical reasons) in his post on CK and Miasma. I think this can be right, but if so, for complicated reasons. And it still leaves some seemingly self-undermining “rationality” in our frameworks.
Mainly, I expect self-reference in legitimacy to be troublesome. A lot of things are legitimate because people think other people think they are legitimate, which seems enough like Lob’s theorem that I worry about the lobstacle.