I guess subjective logic is also trying to handle this kind of thing. From Jøsang's book draft:

Subjective logic is a type of probabilistic logic that allows probability values to be expressed with degrees of uncertainty. The idea of probabilistic logic is to combine the strengths of logic and probability calculus, meaning that it has binary logic’s capacity to express structured argument models, and it has the power of probabilities to express degrees of truth of those arguments. The idea of subjective logic is to extend probabilistic logic by also expressing uncertainty about the probability values themselves, meaning that it is possible to reason with argument models in presence of uncertain or incomplete evidence.

Though maybe this particular formal system has really undesirable properties, I don't know.

Another attempt to do something like this thread: Viva la Books.