I read the LtU thread as well, and began to wonder if Snively was stealing from me.

But no, you need a normalizing fragment to have a Prop sort. Or at least, you need more than this, particularly a solid notion of a probabilistic type theory (which linguists are at the beginning of forming) and Calude's work in Algorithmic Information Theory, to actually build self-verifying theorem-proving type-theories this way.