 The idea behind the ternary relation semantics is rather simple. Consider C.I. Lewis’ attempt to avoid the paradoxes of material implication. He added a new connective to classical logic, that of strict implication. In post-Kripkean semantic terms, \(A \prurel B\) is true at a world \(w\) if and only if for all \(w'\) such that \(w'\) is accessible to \(w\), either \(A\) fails in \(w'\) or \(B\) obtains there. Now, in Kripke’s semantics for modal logic, the accessibility relation is a binary relation. It holds between pairs of worlds. Unfortunately, from a relevant point of view, the theory of strict implication is still irrelevant. That is, we still make valid formulae like \(p \prurel (q \prurel q)\). We can see quite easily that the Kripke truth condition forces this formula on us.