 This semantics validates respective versions of congruence (\(\mathord{\vdash A \leftrightarrow B} \slashrel \mathord{\vdash \osim A \leftrightarrow \osim B}\) and \(\mathord{\vdash A \leftrightarrow B} \slashrel \mathord{\vdash \neg A \leftrightarrow \neg B}\)), but it does not yet impose any interesting constraints on negation. In order to exclude that for some world \(w\) and formula \(A\), both \(w \in \llbracket A\rrbracket\) and \(w \in \llbracket \osim A\rrbracket\), one has to stipulate that for every set of worlds \(X\), if \(w \in X\) then \(X \in N(w)\), which makes sense under the compatibility reading of the neighbourhood function \(N\) because it says that if \(X\) is true at \(w\), then \(X\) is compatible with \(w\). In order to validate contraposition, it has to be required that if \(X \subseteq Y\), then \(\{w\mid Y \not \in N(w)\}\) \(\subseteq\) \(\{w\mid X \not \in N(w)\}\). Under the compatibility reading of \(N\) this condition says that every world \(Y\) is incompatible with is also a world \(X\) is incompatible with, if proposition \(X\) implies proposition \(Y\). Similar conditions can be imposed for the dual negation that requires complementation.