Despite the growing interest in using a combination of qualitative and quantitative methods, probabilistic epistemic logics have received little attention. This paper presents a follow-up to the work by Fagin and Halpern [1] on probabilistic knowledge logic by developing a belief logic of additive probability.

The main contribution of the paper is a sound and complete axiomatization for a modal operator B_i(a,P), whose interpretation is “agent i believes that the probability of P is no less than a,” with respect to Aumann semantics. Most of the paper is dedicated to the proof of soundness (straightforward) and completeness. The completeness proof is based on canonical models; it requires some ingenuity to show that every consistent set can be extended to a maximally consistent one. It is not clear from the proof whether all the axioms are actually needed, thus one might wonder whether the proposed axiomatization is minimal.

The work is motivated by the relevance of reasoning about knowledge and belief in fields such as economics and artificial intelligence, but the paper does not discuss possible applications.

Examples of the main constructions are provided, and axioms and inference rules are presented, along with their intuitive readings. The explanation of Rule 7, the main rule of the system, is not very clear, and no examples are provided.

While the logic is appropriate for the intended reading of the modal operator, it fails to provide a satisfactory characterization of belief about additive probability. In fact, Axiom 7 reduces the probability of two possibly independent and nonexclusive events occurring together to the probability of incompatible events. This is technically correct, given the interpretation, but it seems counterintuitive.