Sorry for the confusion, but I couldn't recast your argument in any formal language whatsoever.

Sorry too. That was my risk of inventing a notion on the spot. The original comment started using excessive "believes X" which I basically replaced. I'm not aware of/trained in a notation to compactly write nested probability assertions.

I'm still trying to resolve the expansion and nesting issues I totally glossed over.

What is B[a][b]? b believes that a believes B or a believes that b believes that B?

[] is like a parameterized suffix. It could be bracketed as X[a][b] = (X[a])[b] if that is more clear. I just lent this from programming languages.

Note: There seems to be a theory of beliefs that might applicable but which uses a different notion (looks like X[a] == K_aX): http://en.wikipedia.org/wiki/Epistemic_modal_logic

So what does B[a] mean? B[a] means are are reasoning about the probability assignment Pa(B) of the actor a and we ask for variants of P(Pa(B)=p).

First: I glossed about a lot of required P(...) assuming (in my eagerness to address the issue) that that'd be clear from context. In general instead of writing e.g.

P((A & B)[p]) ~= P(A[p] & B[p])

I just wrote

(A & B)[P] ~= A[P] & B[P]

What is B0(a)? It is the same as B[a]?

No. the 0 was meant to indicate an apriori (which was hidden in the fragment "a) an aprioi B0 of a person"). Instead of writing the needed probability that Bobs prior probability of B is b (needed in the orig post) as

P_{Bob}(B) = b

I just wrote

B0(Bob)

That is informally I represented my belief in the prior p of another actor in some fact F as a fact in itself (calling it F0) instead of representing all beliefs of the represented actor as relative to that (P(F)=p).

This allowed me to simplify the never written out long form of P(B|X(Bob)). On this I'm still working.

What is X0(a)? It is the same as X[a], so that X is a relational variable?

Yes. For all prior belief expressions X0 it is plausible to approximate other persons prior probability to be less strict than your own.

Is X(a) different from X[a]?

Yes. X(a) is the X of person a. This is mostly releant for the priors.

What I now see after trying to clean up all the issues glossed over is that this possibly doesn't make sense. At least not in this incomplete form. Please stay tuned.