There is a question that naturally arises when modal operators are thought of as operators on implication structures. Suppose that there are several modals on a structure. How are they related, if at all? What are their comparative strengths?

Definition 34.1. Let I = 〈S, ⇒〉 be an implication structure, and let φ and φ* be two modal operators on I. Then φ is stronger than φ* if and only if φ(A) ⇒ φ*(A) for all A in S. We shall say that φ is definitely stronger than φ* if and only if φ is stronger than φ* and there is some A* in S such that φ*(A*) ⇏ φ(A*).

If φ and φ* are modal operators on some structure I = 〈S, ⇒〉, then φ and φ* are comparable (on I) (Definition 28.3) if and only if at least one of them is stronger than the other.

We have already observed (Section 27.3.1) that there are implication structures on which there are noncomparable modals. Thus, the comparability of modal operators is not to be expected in general, not even for very simple structures. Nevertheless, there are a few results available concerning the comparative strengths of modal operators, provided they are comparable.

We have already noted two results on the strength of modals that are comparable. The first concerned necessitation modals: If there are two comparable modals on a structure, one of which is a necessitation modal, and the other not, then the necessitation modal is the weaker of the two (Theorem 28.11).