 and it is shown that the logic obtained by adding the contraposition axiom \((A \rightarrow B) \rightarrow (\neg B \rightarrow \neg A)\) is characterized by the class of all frames that satisfy for all upwards closed sets \(X, X'\): if \(X \subseteq X'\) then \(N(X') \subseteq N(X)\). Moreover, it is shown that the negative ex contradictione axiom \((A \wedge \neg A) \rightarrow \neg B \) corresponds with the following frame property: \(\forall X \forall X' (X \cap N(X) \subseteq N(X'))\).