Background

It is part of the lore of logical theory that sooner or later one comes to the study of implication. This study of the modal operators is a consequence of taking the old advice to heart by considering implication sooner rather than later. Modality, on our account, is a way of studying the question whether or not implication continues to be preserved when the elements related by implication are transformed by an operator. The basic idea is that a modal operator is any operator or function φ that transforms or maps the set S of an implication structure I = 〈S, ⇒〉 to itself in such a way that if A1, …, An ⇒ B, then φ(A1), φ(A2), …, φ(An) ⇒ φ(B). There is a second condition concerning the relation of φ to the dual implication relation “⇒̂,” which we shall introduce shortly. The two conditions will then specify the kind of functions that count as having modal character.

If we are correct about this, then the study of modal operators is a natural continuation of the study of implication itself. Whatever reservations a philosopher might have about the philosophical merit of such concepts as “necessity” and “possibility”, there is every reason for studying the modal operators, since they are among the operators that preserve implication. The key idea, then, is to think of a modal operator as modal relative to some implication relation.