 In weak second-order logic we have no function variables and the relation variables range over finite relations only. The resulting logic is in many ways similar to the extension of first order logic by the generalized quantifier \(Q_0\). Another way to limit the power of second-order logic is the following: Suppose \(\psi\) is a first order formula with an n-ary relation variable X and no non-logical symbols. For infinite A, we define \(\ma\models_s Q_{\psi}X\phi\) if and only if there is a relation \(P\subseteq M^n\) such that \(\ma\models_{s(P/X)}\phi\land\psi\). The quantifier \(Q_{\psi}\) allows second-order quantification over relations that satisfies \(\psi\). If \(\psi\) is \(\forall x(x=x)\) we obtain the usual second-order quantifier which we denote \(Q_{II}\). If moreover \( n=1\) we obtain the monadic second-order quantifier which we denote \(Q_{mon}\). If \(n=1\) and \(\psi\) says that X is a singleton, we obtain the usual first order quantifier which we now denote \(Q_I\). Finally, if \(\psi\) says X is the graph of a permutation of the model, we denote this quantifier by \(Q_{1-1}\). Rather surprisingly, it turns out that with the right concept of interpretability the quantifiers \(Q_I,Q_{mon},Q_{1-1}\) and \(Q_{II}\) are, up to biinterpretability, the only second-order quantifiers of the form \(Q_{\psi}\) (Shelah 1973).