In mathematics, higher-order logic is distinguished from first-order logic in a number of ways.

One of these is the type of variables appearing in quantifications; in first-order logic, roughly speaking, it is forbidden to quantify over predicates. See second-order logic for systems in which this is permitted.

Another way in which higher-order logic differs from first-order logic is in the constructions allowed in the underlying type theory. A higher-order predicate is a predicate that takes one or more other predicates as arguments. In general, a higher-order predicate of order n takes one or more (n − 1)th-order predicates as arguments, where n > 1. A similar remark holds for higher-order functions.

Higher-order logics are more expressive, but their properties, in particular with respect to model theory, make them less well-behaved for many applications. By a result of Gödel, classical higher-order logic does not admit a (recursively axiomatized) sound and complete proof calculus; however, such a proof calculus does exist which is sound and complete with respect to Henkin models.

Examples of higher order logics include the Church's Simple Theory of Types and calculus of constructions.

Reference:
http://en.wikipedia.org/wiki/Higher_order_logic

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