 Let us start by noting that there are some trivial consequences of second-order logic being able to quantify over relations. For example, the Craig Interpolation Theorem holds for second-order logic for trivial reasons: If the finite relational vocabulary of \(\phi\) is L, the finite relational vocabulary of \(\phi'\) is \(L'\) and \(\models\phi\to\phi',\) then there is \(\theta\) such that \(\models\phi\to\theta\text{ and } \models\theta\to\phi'\) and the vocabulary of \(\theta\) is \(L\cap L'\), because we can let \(\theta\) be the sentence \(\exists X_1\ldots\exists X_n\,\phi\), where \(L\setminus L'=\{X_1,\ldots, X_n\}\). We can additionally demand that every predicate symbol occurring positively in \(\theta\) occurs positively in \(\phi\) and \(\psi\), and every predicate symbol occurring negatively in \(\theta\) occurs negatively in \(\phi\) and \(\psi\) (the Lyndon Interpolation Theorem [Lyndon 1959]). In the same way the Beth Definability Theorem holds for trivial reasons, also some preservation  results[7]  have a trivial proof. For a negative result, see Craig 1965.