 If \(\phi\) is a sentence of \(\mathbf{FO}\), \(\phi\) is false in a model \(M\) iff \({\sim}\phi\) is true in \(M\) iff \(\phi\) is not true in \(M\). By contrast, in \(\mathbf{IFL}\) falsity and non-truth do not coincide. An extension of \(\mathbf{IFL}\) can be introduced, where it is possible to speak of the non-truth of sentences. To this end, let us introduce a new negation symbol, \(\neg\), referred to as weak negation, contradictory negation or classical  negation.[29]  The set of formulas of extended IF first-order logic (to be denoted \(\mathbf{EIFL})\) is obtained from the set of formulas of \(\mathbf{IFL}\) by closing it under the operations \(\neg\), \(\wedge\), and  \(\vee\):[30]