In FOL, can we define equality for two predicate symbols?

Question

In FOL, I think equality is always used for two variable or constant symbols. Can we define equality for two predicate symbols? If not, why? (Do we need higher order logic to use such a concept?)

Answer 1

In FOL, can we define equality for two predicate symbols?

NO. In first-order logic, equality is defined between "objects" (referred by terms, i.e. individual variables and constants) and not between "properties" (referred by predicate symbols).

In Second-order logic, instead, we can define equality between predicates :

$P = Q \leftrightarrow \forall x (P(x) \leftrightarrow Q(x))$.