This is exercise 22.6 from Boolos,Jeffrey,Burgess Computability and Logic 4th ed. You are asked to prove the validity of the following sentence (with second-order logic as the underlying logic and under standard semantics): $ \exists X \neg \exists x \forall y (Rxy \leftrightarrow Xx) $
For the life of me I cannot see how that can be true if I take a model $\mathcal{M}$ such that $R^{\mathcal{M}} = \vert \mathcal{M} \vert^2$, i.e. one in which the relation is true of any possible combination of elements.
Is there something I am missing about $R$ here? Surely I cannot then just take $X$ to be the empty set can I? I'm sure there's something obvious I am missing here, since this looks like a pretty straightforward exercise.