I agree that your theory could be understood as a less explicit version of p-common knowledge.
EG:
One reasonable steelman of the commonsense use of “knows” is to interpret “knowledge” as “true p-belief”, with “p” left unspecified, flexible to the situation. (Situations with higher risk naturally call for higher p.) We similarly interpret commonsense “certainty” as p-belief. “Very certain” is p-belief with even higher p, and so on.
We then naturally interpret the theory of “common knowledge” as code for p-common knowledge, by substituting iteration of “knows” with “correctly p-believes”.[1]
My problem with this approach is that it leads to a “missing stair” kind of problem, as I mentioned in my response to rpglover64. For example, the literature on common knowledge says that a public event (something everyone can see, simultaneously, and see that everyone else sees) is required. As I illustrated in the OP, p-common-knowledge doesn’t require anything like this; it’s much easier to establish.
So if everyone uses the term “common knowledge”, but in-the-know people privately mean “p-common-knowledge” and interpret others as meaning this, then not-in-the-know people run the risk of thinking this thing people call ‘common knowledge’ is really difficult and costly to establish. And this seems compatible with what people say, so they’re not so liable to notice the difference.
I would also point out that your and rpglover64′s interpretations were much less precise than p-common-knowledge; eg, you say things like “in some sense”, “kind of paradoxically”, “it seems like”. So I would say: why not use accurate language, rather than be confused? Why not correct inaccurate language, rather than leaving a missing stairstep?
As I mentioned in the post, I also have some doubts about whether p-common knowledge captures the real phenomenon people are actually getting at when they use “common knowledge” informally. So it also has the advantage of being falsifiable! So we might even learn something!! Leaving the theory at a vague “humans are only approximate instances of all these rationality concepts” is, of course, still true, but seems far less useful.
(Unless I’ve missed something, adding “correctly” doesn’t really change the definition of p-common knowledge, since it simply re-asserts the next-lower level.)
I agree that your theory could be understood as a less explicit version of p-common knowledge.
EG:
One reasonable steelman of the commonsense use of “knows” is to interpret “knowledge” as “true p-belief”, with “p” left unspecified, flexible to the situation. (Situations with higher risk naturally call for higher p.) We similarly interpret commonsense “certainty” as p-belief. “Very certain” is p-belief with even higher p, and so on.
We then naturally interpret the theory of “common knowledge” as code for p-common knowledge, by substituting iteration of “knows” with “correctly p-believes”.[1]
My problem with this approach is that it leads to a “missing stair” kind of problem, as I mentioned in my response to rpglover64. For example, the literature on common knowledge says that a public event (something everyone can see, simultaneously, and see that everyone else sees) is required. As I illustrated in the OP, p-common-knowledge doesn’t require anything like this; it’s much easier to establish.
So if everyone uses the term “common knowledge”, but in-the-know people privately mean “p-common-knowledge” and interpret others as meaning this, then not-in-the-know people run the risk of thinking this thing people call ‘common knowledge’ is really difficult and costly to establish. And this seems compatible with what people say, so they’re not so liable to notice the difference.
I would also point out that your and rpglover64′s interpretations were much less precise than p-common-knowledge; eg, you say things like “in some sense”, “kind of paradoxically”, “it seems like”. So I would say: why not use accurate language, rather than be confused? Why not correct inaccurate language, rather than leaving a missing stairstep?
As I mentioned in the post, I also have some doubts about whether p-common knowledge captures the real phenomenon people are actually getting at when they use “common knowledge” informally. So it also has the advantage of being falsifiable! So we might even learn something!! Leaving the theory at a vague “humans are only approximate instances of all these rationality concepts” is, of course, still true, but seems far less useful.
(Unless I’ve missed something, adding “correctly” doesn’t really change the definition of p-common knowledge, since it simply re-asserts the next-lower level.)