Higher-order logic
Properties

Higher-order logics include the offshoots of Church 's simple theory of types and the various forms of intuitionistic type theory. Gérard Huet has shown that unifiability is undecidable in a type-theoretic flavor of third-order logic, that is, there can be no algorithm to decide whether an arbitrary equation between second-order (let alone arbitrary higher-order) terms has a solution.

Up to a certain notion of isomorphism, the powerset operation is definable in second-order logic. Using this observation, Jaakko Hintikka established in 1955 that second-order logic can simulate higher-order logics in the sense that for every formula of a higher-order logic, one can find an equisatisfiable formula for it in second-order logic.

The term "higher-order logic" is assumed in some context to refer to classical higher-order logic. However, modal higher-order logic has been studied as well. According to several logicians, Gödel's ontological proof is best studied (from a technical perspective) in such a context.
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