  The operator \(\Diamond\) (for ‘possibly’) can be defined from \(\Box\) by letting \(\Diamond A = {\sim}\Box{\sim}A\). In \(\bK\), the operators \(\Box\) and \(\Diamond\) behave very much like the quantifiers \(\forall\) (all) and \(\exists\) (some). For example, the definition of \(\Diamond\) from \(\Box\) mirrors the equivalence of \(\forall xA\) with \({\sim}\exists x{\sim}A\) in predicate logic. Furthermore, \(\Box(A \amp B)\) entails \(\Box A \amp \Box B\) and vice versa; while \(\Box A\vee \Box B\) entails \(\Box (A\vee B)\), but not vice versa. This reflects the patterns exhibited by the universal quantifier: \(\forall x(A \amp B)\) entails \(\forall xA \amp \forall xB\) and vice versa, while \(\forall xA \vee \forall xB\) entails \(\forall x(A \vee B)\) but not vice versa. Similar parallels between \(\Diamond\) and \(\exists\) can be drawn. The basis for this correspondence between the modal operators and the quantifiers will emerge more clearly in the section on  Possible Worlds Semantics.