D modals

Definition 28.1. Let I = 〈S, ⇒〉 be an implication structure. We shall say that a modal operator φ on I satisfies the D condition if and only if for all A in S, φ(A) ⇒ ¬φ(¬A).

One reason for studying modals that satisfy this special condition can be traced to the interest in what is obligatory and what is permissible. We shall not worry, for the moment, whether it is acts, or sentences, or statements upon which the corresponding operators O and P are defined. It is suggested that “P(A)” (“it is permissible that A”) conveys that it is not obligatory that “not A” (or it is not obligatory that you do “not A”), and conversely. Thus it looks as if “P” might be defined simply as “¬O¬,” and it is then argued that the condition O(A) ⇒ P(A) holds: It is obligatory that A implies that it is permissible that A. Nothing can be both obligatory and not permissible. Together with the definition of “P,” we have the particular condition O(A) ⇒ ¬O(¬A) for all A's in the relevant structure. This is, then, if one is persuaded, a particular case of a D modal.

There is another way of looking at the D condition that is more logical in flavor and in turn sheds some light on the plausibility of the D condition holding for obligation.