Seonglae ChoAll higher-order logics, which refer to a series of logical systems where universal and existential quantifiers can be applied to variables referring to relations or relations of relations, can be considered as subsystems of ω-order logic. For example, in second-order logic, variables can refer to individuals and predicates, and predicates can denote properties of sets of individuals or relations. Third-order logic includes second-order predicates as variables, and predicates can denote properties of second-order predicates.
Higher-order logics
Higher-order logic
In mathematics and logic, a higher-order logic (abbreviated HOL) is a form of logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more expressive, but their model-theoretic properties are less well-behaved than those of first-order logic.
https://en.wikipedia.org/wiki/Higher-order_logic