Our aim has been to study the modal operators as functions on implication structures. There is nothing mysterious or arcane about the kind of functions that they are, and there is nothing recherché about the particular examples that qualify as modal. It should come as no surprise, then, that some of the operators commonly studied in elementary logic and mathematics should turn out to have modal character. This should help reinforce the idea that some modals are quite familiar, ordinary, and uncontroversial. With this in mind, we turn to some elementary examples.

Power sets

The power set of a set A is the set of all the subsets of A. Suppose S is a set of sets (1) that is closed under finite intersections and unions of its members, and (2) for any set A in S, P(A) is in S, and (3) there are at least two members, A* and B*, of S such that neither is a subset of the other.

Let I = 〈S, ⇒〉 be an implication structure, where S is as described above, and for any sets A1, …, An and B in S, A1, …, An ⇒ B if and only if A1 ∩ … ∩ An ⊆ B.