Modals as functions

In our characterization of modal operators we stressed the idea that they are certain kinds of functions or mappings on implication structures, I = 〈S, ⇒〉, mapping the set S of the structure to itself and preserving the implication relation “⇒,” but not its dual.

There are several types of questions that arise naturally if we focus on the modals as operators: (1) Given any implication structure I, are there any modals on it? That is, do any modals exist? We shall show that there is a simple necessary and sufficient condition for their existence. (2) If several modal operators exist on a structure, how are they related? For example, will any two, such as φ and φ*, have to be comparable in the sense that either φ(A) ⇒ φ*(A) for all A in S or else φ*(A) ⇒ φ(A) for all A in S. We shall see that the modals on some structures need not be comparable, but that there are interesting consequences when they are. (3) What about more refined descriptions of modals as special kinds of functions? That is, what can one say about those kinds of modals φ for which φ(A) ⇒ A for all A in S? Under what conditions do they exist? Furthermore, if φ is a modal operator, is its functional product with itself, φφ, always a modal operator as well? What can be said about those special modal operators that always map theses of the structure into theses?