I wasn’t familiar with this description of “world states”, but it sounds interesting, yes. I take it that positing “I am a think that things” is the same as asserting K(E). In asserting K(K(E)), I assert that I know that I know that I am a thing that thinks. If this understanding is incorrect, my following logic doesn’t apply.
I would argue that K(K(E)) is actually a necessary condition for K(E). Because if I don’t know that I know proposition A, then I don’t know proposition A.
Edit/Revised: I think all you have to do is realize that “K(K(A)) false” permits “K(A) false”. At first I had a little proof but now it seems just redundant so I deleted it.
So I guess I disagree, I think the iterations K(K...) are actually weaker statements, which are necessary for K(A) to be achieved. Consequentially I don’t see how you can learn anything beyond K(A).
K(A) is always a stronger statement than A because if you know K(A) you necessarily know A. (To get the terms clear: a “strong” statement corresponds to a smaller set of world states than a “weak” one.) It is debatable whether K(K(A)) is always equivalent to K(A) for human beings. I need to think about it more.
Format definition of K(E)={s \in S | P(s) \subset E }, where P is partition of S, ensures that K(K(E))=K(E). It’s easy to see: if s \in K(E) then P(s) \subset e, thus s \in K(K(E)), and similarly for s \notin K(E).
As for informal sence, I don’t see much use of K(K(E)) where E is a plain fact, if I aware that I know E, introspecting on that awareness will provide as much K-s as I like and little more. If I don’t aware that I know E (deep buried memory?), I will be aware of it when I remember it. But If I know that I know some class of facts or rules, that is useful for planning. However I can’t come up with useful example for K(K(K())) and higher.
Addition: Aumann’s formalization have limitations: it can’t represent false knowledge, memory glitches (when I know that I know something, but I can’t remember it), meta-knowledge, knowledge of rules of any kind (I’m not completely sure about rules).
I wasn’t familiar with this description of “world states”, but it sounds interesting, yes. I take it that positing “I am a think that things” is the same as asserting K(E). In asserting K(K(E)), I assert that I know that I know that I am a thing that thinks. If this understanding is incorrect, my following logic doesn’t apply.
I would argue that K(K(E)) is actually a necessary condition for K(E). Because if I don’t know that I know proposition A, then I don’t know proposition A.
Edit/Revised: I think all you have to do is realize that “K(K(A)) false” permits “K(A) false”. At first I had a little proof but now it seems just redundant so I deleted it.
So I guess I disagree, I think the iterations K(K...) are actually weaker statements, which are necessary for K(A) to be achieved. Consequentially I don’t see how you can learn anything beyond K(A).
K(A) is always a stronger statement than A because if you know K(A) you necessarily know A. (To get the terms clear: a “strong” statement corresponds to a smaller set of world states than a “weak” one.) It is debatable whether K(K(A)) is always equivalent to K(A) for human beings. I need to think about it more.
Format definition of K(E)={s \in S | P(s) \subset E }, where P is partition of S, ensures that K(K(E))=K(E). It’s easy to see: if s \in K(E) then P(s) \subset e, thus s \in K(K(E)), and similarly for s \notin K(E).
As for informal sence, I don’t see much use of K(K(E)) where E is a plain fact, if I aware that I know E, introspecting on that awareness will provide as much K-s as I like and little more. If I don’t aware that I know E (deep buried memory?), I will be aware of it when I remember it. But If I know that I know some class of facts or rules, that is useful for planning. However I can’t come up with useful example for K(K(K())) and higher.
Addition: Aumann’s formalization have limitations: it can’t represent false knowledge, memory glitches (when I know that I know something, but I can’t remember it), meta-knowledge, knowledge of rules of any kind (I’m not completely sure about rules).