Higher-order logic

In mathematics and logic, a 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 predicates of order n − 1 as arguments, where n > 1. A similar remark holds for higher-order functions.

Higher-order logic, abbreviated as HOL, is also commonly used to mean higher order simple predicate logic, that is the underlying type theory is simple, not polymorphic or dependent.^[1]

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 Church's Simple Theory of Types, Thierry Coquand's calculus of constructions, which allows for both dependent and polymorphic types, and of course HOL.

See also

• Higher-order grammar
• Typed lambda calculus
• Intuitionistic Type Theory
• Many-sorted logic

Notes

References

• Andrews, P.B., 2002. An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof, 2nd ed. Kluwer Academic Publishers, available from Springer.
• Stewart Shapiro, 1991, "Foundations Without Foundationalism: A Case for Second-Order Logic". Oxford University Press.
• Stewart Shapiro, 2001, "Classical Logic II: Higher Order Logic," in Lou Goble, ed., The Blackwell Guide to Philosophical Logic. Blackwell.
• Lambek, J. and Scott, P. J., 1986. Introduction to Higher Order Categorical Logic, Cambridge University Press.
• Jacobs, Bart (1999). Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics 141. North Holland, Elsevier. ISBN 0-444-50170-3. http://www.cs.ru.nl/B.Jacobs/CLT/bookinfo.html. 

External links

• Miller, Dale, 1991, "Logic: Higher-order," Encyclopedia of Artificial Intelligence, 2nd ed.
• Herbert B. Enderton, Second-order and Higher-order Logic in Stanford Encyclopedia of Philosophy, published Dec 20, 2007; substantive revision Mar 4, 2009.

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