More on Lambda Logic
I've read more about Lambda Logic. The Lambda Logic that I found is actually a bit different than the one that I was considering on my own. First, it is not referentially strong-- the logic has a complete inference method, which means it can only talk about provable things. So, it can't really reference lambda-computations in the way I want, because there are unprovable facts about computations.
Another interesting point-- Lambda Logic is inconsistent with the Axiom of Choice!