Higher-order logic

In mathematics and logic, a higher-order logic is a form of predicate 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", abbreviated as HOL, is commonly used to mean higher order simple predicate logic. Here "simple" indicates that the underlying type theory is simple, not polymorphic or dependent.

First-order logic quantifies only variables that range over individuals; second-order logic, in addition, also quantifies over sets; third-order logic also quantifies over sets of sets, and so on. For example, the second-order sentence

expresses the principle of mathematical induction. 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.

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. For example, a quantifier over sets of individuals ranges over the entire powerset of the set of individuals. Thus, in standard semantics, once the set of individuals is specified, this is enough to specify all the quantifiers. HOL with standard semantics is more expressive than first-order logic. For example, HOL admits categorical axiomatizations of the natural numbers, and of the real numbers, which are impossible with first-order logic. However, by a result of Gödel, HOL with standard semantics does not admit an effective, sound, and complete proof calculus.

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