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One of these is the type of Variables appearing in Quantification s; in first-order logic, roughly speaking, it is forbidden to quantify over Predicate s. 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. SEE ALSO EXTERNAL LINKS LITERATURE
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