 Inexpressibility of contradictory negation. In \(\mathbf{IFL}\) the strong negation \({\sim}\) and the contradictory negation \(\neg\) do not coincide: we may have \(M \vDash \neg \phi\) without having \(M \vDash{\sim}\phi\). This fact by itself still leaves open the possibility that the contradictory negation of every sentence \(\phi\) of \(\mathbf{IFL}\) could be defined in \(\mathbf{IFL}\), i.e., that there was a sentence \(neg(\phi)\) of \(\mathbf{IFL}\) such that \(M \vDash\) \(neg(\phi)\) iff \(M \not\vDash \phi\), for all models \(M\). All we know by the failure of the law of excluded middle is that not in all cases can \(neg(\phi)\) be chosen to be \({\sim}\phi\). However, as a matter of fact contradictory negation is inexpressible in \(\mathbf{IFL}\). There are sentences \(\phi\) of \(\mathbf{IFL}\) such that \(\neg \phi\) (which is a sentence of \(\mathbf{EIFL})\) is not truth equivalent to any sentence of \(\mathbf{IFL}\). This follows from the well-known fact that \(\mathbf{ESO}\) is not closed under negation and \(\mathbf{IFL}\) has the same expressive power as  \(\mathbf{ESO}\).[41]