 We will use \(\Pr(\cdot)\) to denote the probability function representing the agent’s partial beliefs. Of course, \(\Pr(\cdot)\) is defined on a \(\sigma\)-algebra of subsets of \(W\). In the usual case, when \(W\) is finite, we can take the powerset \(\mathcal{P}(W)\) to be the relevant \(\sigma\)-algebra. To update partial belief, we adopt the standard probabilistic modeling. For \(E\subseteq W\) such that \(\Pr(E)&gt;0,\) \(\Pr(\cdot|E)\) is the partial belief function resulting from learning \(E.\) We will sometimes use \(\Pr_E\) as a shorthand for \(\Pr(\cdot|E)\). Almost always, partial belief is updated via standard conditioning. As before, \({\bf A}^{\Pr}\) is the set of propositions with positive probability according to \(\Pr\).