 In the extended language, negative ex contradictione can be stated as \((A \wedge \osim A) \vdash \osim B\) and unrestricted ex contraditione as \((A \wedge \osim A) \vdash B\). It is also natural to assume a constantly true formula \(\top\) and a constantly untrue formula \(\bot\), so that the following sequents are valid: \(A\vdash \top\), \(\bot\vdash A\), and \(\top \vdash \osim \bot\). (With negation as unnecessity, \(\neg \top \vdash \bot\) is valid.) Whereas these sequents are indeed valid on any compatibility frame, the equally natural \(\osim \top \vdash \bot\) (as well as \(\top \vdash \neg \bot\)) corresponds with the seriality of \(C \mathrel{:} \forall x \exists y (xCy)\). In the extended vocabulary, unrestricted ex contradictione can be stated as \((A \wedge \osim A) \vdash \bot\), and in this form it is characterized by the reflexivity of the compatibility relation. The law of excluded middle \(\top \vdash (A \vee \osim A)\) corresponds to the mixed condition \(\forall x \forall y(xCy \Rightarrow y \leq x)\) but also to \(\forall x \forall y (xCy \Rightarrow (x =y))\).