Higher-order logic
form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics
In mathematics and logic, a higher-order logic (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.
The term "higher-order logic" is commonly used to mean higher-order simple predicate logic. Here, "simple" indicates that the underlying type theory is the theory of simple types, also called the simple theory of types. Leon Chwistek and Frank P. Ramsey proposed this as a simplification of ramified theory of types specified in the Principia Mathematica by Alfred North Whitehead and Bertrand Russell. Simple types is sometimes also meant to exclude polymorphic and dependent types.
Quantification scope
First-order logic quantifies only variables that range over individuals; second-order logic, also quantifies over sets; third-order logic also quantifies over sets of sets, and so on.
Higher-order logic is the union of first-, second-, third-, ..., nth-order logic; i.e., higher-order logic admits quantification over sets that are nested arbitrarily deeply.
Semantics
There are two possible semantics for higher-order logic.
In the standard or full semantics, quantifiers over higher-type objects range over all possible objects of that type.
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