Accessibility relations and the diversity of modals

We have already observed (Section 28.3) how the nonempty theories of an implication structure can be used to provide a necessary and sufficient condition for an operator to be a necessitation modal. There we introduced the idea of an accessibility relation Rφ for each modal operator on the implication structure I (and, where σ is the dual of φ, Rσ is the appropriate accessibility relation). They are binary relations on the theories of the structure such that for all theories U, V, …,

1. (1) URφV if and only if φ−1U ⊆ V [where A is in φ−1U if and only if φ(A) is in U], and

2. (2) URσV if and only if V ⊆ σ−1U (where σ is the dual of φ).

Each accessibility relation is, according to these two conditions, tailored to the particular modal operator under study. The tightness of the relation between a modal and its accessibility relation was discussed, and need not be repeated here. The theories, as used in this chapter, are those subsets of the structure that are (weakly) closed under implication in this sense: If A is in U, and A ⇒ B, then B is in U as well.

Anyone who is even remotely aware of S. Kripke's ground-breaking work on the semantics of modal logic knows how the idea of accessibility relations and the possible worlds that they relate were used with startling effect to provide a series of completeness proofs of familiar modal systems and to distinguish the variety of modal laws from each other in a very simple and coherent manner.