 A ranking function represents a belief state. More specifically, a ranking function \(\kappa\) assigns each possible world a non-negative natural number such that \(\kappa(w)=0\) for at least one possible world \(w\). Intuitively, the rank \(\kappa(w)\) of a possible world \(w\) expresses a degree of disbelief. \(\kappa(w)=0\) means that the degree of disbelief is zero. \(w\) is not disbelieved in this case. Otherwise, it is disbelieved with rank \(\kappa(w)=n &gt; 0\). A ranking function thus defines a Grovian system of spheres, used to represent AGM belief revision operators (Alchourrón, Gärdenfors, &amp; Makinson 1985; Grove 1988). A Grovian system of spheres is basically a set of nested sets. At the center of the system is its smallest member, a non-empty set of worlds. The smallest set is surrounded by its supersets, as shown in  Figure 3.