 The truth conditions of (\(\ref{boygirl2}\)), (\(\ref{boygirl3}\)), (\(\ref{boygirl4}\)) and (\(\ref{boygirl5}\)) are precisely the same: any model that satisfies one of these expressions (in the respective languages) satisfies all four. More in general, as we will see, the expressive powers of existential second-order logic, independence-friendly logic and dependence logic with respect to definability of classes of models are precisely the same. This is, however, not the case for formulas with free variables; and furthermore, these logics can be extended and modified along markedly different lines.