A Modal Logic for Uncertainty: a Completeness Theorem

In the present paper, we axiomatize a logic that allows a general approach for reasoning about probability functions, belief functions, lower probabilities and their corresponding duals. The formal setting we consider arises from combining a modal S5 necessity operator square that applies to the formulas of the infinite-valued Lukasiewicz logic with the unary modality P that describes the behaviour of probability functions. The modality P together with an S5 modality square provides a language rich enough to characterise probability, belief and lower probability theories. For this logic, we provide an axiomatization and we prove that, once we restrict to suitable sublanguages, it turns out to be sound and complete with respect to belief functions and lower probability models.