Peter Suber, "Glossary of First-Order Logic"

Predicate logic in which predicates take only individuals as arguments and quantifiers only bind individual variables. Predicate logic in which predicates take other predicates as arguments and quantifiers bind predicate variables. For example, second-order predicates take first-order predicates as arguments. Order n predicates take order n-1 predicates as arguments (n 1). Predicate logic that does not exclude interpretations with empty domains.