 \(\bB\) is the set of all propositions the agent believes to some positive degree, or equivalently, whose complements she disbelieves to a positive degree. The belief set \(\bB\) induced by a (finitely/countable/completely minimitive) ranking function \(\varrho\) is consistent and deductively closed (in the finite/countably/complete sense). Since any proposition with \(\beta(A)&gt;0\) is believed, ranking theory can be seen as satisfying the Lockean thesis (see  Section 4.2.2)  without sacrificing deductive closure. Note, however, that we could have defined \(\bB\) to include all those propositions \(A\) with \(\beta(A) \gt t\) for some threshold \(t\) greater than zero. See Raidl (2019) for an exploration of these possibilities. 