 In Ripley 2009 it is suggested to use the neighbourhood semantics as a general framework for semantically capturing properties characteristic of negation connectives interpreted as a necessity operator \( \Box \), see also Yu 2010. Ripley points out, for example, that the contraposition rule  \[ A \vdash B \slashrel \Box B \vdash \Box A\]  is valid in a neighbourhood model \((W, N, v)\) iff for every \(w \in W\), \(N(w)\) is closed under subsets, i.e., if \(X \in N(w)\) and \(Y \subseteq X\), then \(Y \in N(w)\). It would be nice to have a convincing intuitive understanding of the neighbourhood function \(N\) in terms of a concept that explains some core aspects of negation. If \(\llbracket A\rrbracket \in N(w)\) is understood as saying that the proposition expressed by \(A\) is incompatible with world \(w\), then the above constraint emerges as reasonable because it says that if the set of worlds (the proposition) \(X\) is incompatible with \(w\) and proposition \(Y\) implies \(X\), then \(Y\) is incompatible with \(w\) as well. Whereas Ripley starts with a positive notion (\({\cal M}, w \models \Box A\) iff \(\llbracket A\rrbracket \in N(w)\)), in order to introduce a negation \(\osim \), one may also stipulate that \({\cal M}, w \models \osim A\) iff \(\llbracket A\rrbracket \not \in N(w)\), so as to obtain a connective that is more overtly a negative impossibility operator (although it is the clause for \(\Box A\) that is classically negated). The idea is that \(N(w)\) contains the sets of worlds compatible with \(w\), so that \(\llbracket A\rrbracket \not \in N(w)\) indicates that the proposition expressed by \(A\) is incompatible with \(w\). Negation as an “unnecessity” operator \(\neg\) in the sense of (“possibly not”) is then defined by \({\cal M}, w \models \neg A\) iff \(\overline{\llbracket A\rrbracket} \in N(w)\), where \(\overline{\llbracket A\rrbracket}\) is the complement of \(\llbracket A\rrbracket\) with respect to \(W\). As a result, \(\neg A\) is true at a state \(w\) iff the complement of the proposition expressed by \(A\) is compatible with \(w\).