I know this seems like an obvious question, but I haven't been able to find any examples of sentences in logic higher than second order, so my intuition on how it's supposed to behave is failing me. There are descriptions describing third order logic as 'properties of properties' but without example syntax, I'm not sure if I'm on the wrong track or not.
Propositional logic sentences are simple propositions connected by logical connectives:
$\phi$ $\land$ $\psi$
$\phi \lor \psi$
$\lnot \phi \rightarrow \psi$
First order logic sentences are quantified objects with free predicates and functions:
$\exists$x $\forall$y P(f(x)) $\rightarrow$ Q(y)
Second order logic sentences don't just quantify the objects, but the functions and predicates:
$\forall$Q $\exists$P $\exists$f $\exists$x $\forall$y P(f(x)) $\rightarrow$ Q(y)
How do we go higher than second order? What are some examples of third, fourth, or fifth order logic sentences?