Logic of predicates over predicates?

Question

Is there logic, that allow to form predicates over predicates (or even formulas), I.e. derive is-interesting-statment(is(I, liar)). Of course, the model operators in the modal logic can take predicates and formulas as "arguments", but what about generalization of modal operators to the full expressiveness of arbitrary predicates? As I understand, then higher order logic is generalization about quantifiers and not the generalization that I am seeking. Generally, there is trend to express modalities using predicates, so, such generalization that I am seeking, should be possible, at least in the domain "modalities as predicates".