 Now that the semantics of second-order logic is defined we can define what it means for a formula \(\phi\) to be valid and for two formulas \(\phi\) and \(\psi\) to be logically equivalent. As in logic in general, we say that \(\phi\) is (logically) valid if \(\mm\models_s\phi\) holds for all \(\mm\) and all s. Likewise, we define \(\phi\) and \(\psi\) to be logically equivalent, \(\phi\equiv\psi\), if \(\phi\leftrightarrow\psi\) is valid. Two models \(\mm\) and \(\mn\) are said to be second-order equivalent, in symbols \(\mm\equiv_{L^2}\mn\) if for all sentences \(\phi\) we have \(\mm\models\phi\iff\mn\models\psi\).