Concept

Higher-order logic

Overview

In mathematics and logic, a higher-order logic (HOL) is a form of 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" is commonly used to mean higher-order simple predicate logic. Here, "simple" indicates that the underlying type theory is the theory of simple types, also called the simple theory of types. Leon Chwistek and Frank P. Ramsey proposed this as a simplification of ramified theory of types specified in the Principia Mathematica by Alfred North Whitehead and Bertrand Russell. Simple types is sometimes also meant to exclude polymorphic and dependent types.

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

Possible passages

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1. Any such select set is called the set of postulates, or premises, from which the reasoning proceeds. The reasoning is nothing else than the exhibition of the whole pattern of general conditions involved in the pattern derived from the selected postulates.

   Alfred North Whitehead, Science and the modern world

   States what Higher-order logic is; the author is tied to Higher-order logic in the knowledge graph.

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2. All types of series are capable of formal definition, and their properties can be deduced from the principles of symbolic logic by means of the Algebra of Relatives. The notion of a limit, which is fundamental in the greater part of higher mathematics, used to be defined by means of quantity, as a term to which the terms of some series approximate as nearly as we please. But nowadays the limit is defined quite…

   Bertrand Russell, Mysticism and Logic and Other Essays

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3. Please remember that the names are technical names, meant, no doubt, to be suggestive, but used in strictly defined senses. We have suffered much from critics who consider it sufficient to criticise our procedure on the slender basis of a knowledge of the dictionary meanings of such terms. For example, a one-to-one correlation depends on the notion of a class with only one member, and this notion is defined without…

   Alfred North Whitehead, The organisation of thought, educational and scientific

   States what Higher-order logic is; the author is tied to Higher-order logic in the knowledge graph.

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Philosophers who discussed Higher-order logic

• Willard Van Orman Quine

Related ideas

• First-order logic
• Journal of Symbolic Logic
• Gödel's completeness theorem
• Kurt Gödel
• Jaakko Hintikka
• Real number
• Natural number
• Frank P. Ramsey
• Principia Mathematica
• Soundness
• Modal logic
• Second-order logic
• Intuitionistic type theory
• Type theory
• Bertrand Russell
• Alfred North Whitehead
• Alonzo Church
• Classical logic
• Leon Chwistek
• Gödel's ontological proof

Opposing perspectives

• First-order logic
• Mathematics
• Model theory
• Quantifier (logic)
• Logic

Place in the tradition

Part of Outline of philosophy.

Narrower topics:

• Intuitionistic type theory
• Leon Chwistek
• Gödel's ontological proof

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