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nshepperd comments on Walkthrough of "Definability of Truth in Probabilistic Logic" - Less Wrong
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Yes, the axioms seem incomplete, or perhaps it was simply meant to be implied that "P(p) = P(q & p) + P(¬q & p)" also. Otherwise as far as I can tell there's no axiom that lets you relate any expression containing "P(p" to an expression containing "P(q", unless the arguments of P(·) are each a tautology or contradiction (which is unhelpful).
Well, a tautology can be made up of non-tautological things; we could conceivably have some sentence phi(p, q) that's a tautology if p <=> q, such that P(p) = f(P(phi(p,q))) = f(P(phi(q,p))) = P(p). I think this is what ygert is trying to do. I don't have much hope for this approach, though.