Higher-order logic

higher order logichigher-orderHigher Orderhigher-order predicateHigher-order quantification(higher-order)higher order logicshigher order quantificationhigher-order predicate logicHOL
In mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics.wikipedia
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First-order logic

predicate logicfirst-orderpredicate calculus
In mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics.
The adjective "first-order" distinguishes first-order logic from higher-order logic in which there are predicates having predicates or functions as arguments, or in which one or both of predicate quantifiers or function quantifiers are permitted.

Second-order logic

second-ordersecond order logicexistential second-order logic
First-order logic quantifies only variables that range over individuals; second-order logic, in addition, also quantifies over sets; third-order logic also quantifies over sets of sets, and so on. For example, the Löwenheim number of second-order logic is already larger than the first measurable cardinal, if such a cardinal exists.
Second-order logic is in turn extended by higher-order logic and type theory.

Model theory

modelmodelsmodel-theoretic
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.
Model theory in higher-order logics or infinitary logics is hampered by the fact that completeness and compactness do not in general hold for these logics.

Type theory

typestheory of typestype
The term "higher-order logic", abbreviated as HOL, is commonly used to mean higher-order simple predicate logic. Here "simple" indicates that the underlying type theory is simple, not polymorphic or dependent.
Church demonstrated that it could serve as a foundation of mathematics and it was referred to as a higher-order logic.

Categorical theory

categoricalMorley's theoremuncountably categorical
For example, HOL admits categorical axiomatizations of the natural numbers, and of the real numbers, which are impossible with first-order logic.
Higher-order logic contains categorical theories with an infinite model.

Löwenheim number

For example, the Löwenheim number of second-order logic is already larger than the first measurable cardinal, if such a cardinal exists.
It thus applies to a very broad collection of logics, including first-order logic, higher-order logics, and infinitary logics.

Higher order grammar

higher-order grammar
Higher-order grammar
Higher order grammar (HOG) is a grammar theory based on higher-order logic.

Alonzo Church

ChurchChurch, Alonzo Church, Alonzo
Higher order logics include the offshoots of Church's Simple theory of types and the various forms of Intuitionistic type theory.
Higher-order logic

Gödel's ontological proof

his ontological proof
According to several logicians, Gödel's ontological proof is best studied (from a technical perspective) in such a context.
Furthermore, the proof uses higher-order (modal) logic because the definition of God employs an explicit quantification over properties.

Classical logic

classicallaws of logicalternative logical systems
The term "higher-order logic" is assumed in some context to refer to classical higher-order logic.
Willard Van Orman Quine insisted on classical, first-order logic as the true logic, saying higher-order logic was "set theory in disguise".

Categorical logic

internal languagecategorical semanticsCategorical
Lambek, J. and Scott, P. J., 1986. Introduction to Higher Order Categorical Logic, Cambridge University Press, ISBN: 0-521-35653-9
Lambek, J. and Scott, P. J., 1986. Introduction to Higher Order Categorical Logic. Fairly accessible introduction, but somewhat dated. The categorical approach to higher-order logics over polymorphic and dependent types was developed largely after this book was published.

Mathematics

mathematicalmathmathematician
In mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics.

Logic

logicianlogicallogics
In mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics.

Quantifier (logic)

quantifierquantifiersquantification
In mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics.

Semantics of logic

formal semanticssemanticslogical semantics
In mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics.

Polymorphism (computer science)

polymorphismpolymorphictype polymorphism
The term "higher-order logic", abbreviated as HOL, is commonly used to mean higher-order simple predicate logic. Here "simple" indicates that the underlying type theory is simple, not polymorphic or dependent.

Dependent type

dependent typesdependently typeddependent
The term "higher-order logic", abbreviated as HOL, is commonly used to mean higher-order simple predicate logic. Here "simple" indicates that the underlying type theory is simple, not polymorphic or dependent.

Power set

powerset2all subsets
For example, a quantifier over sets of individuals ranges over the entire powerset of the set of individuals.

Kurt Gödel

GödelGödel, Kurt Gödel, Kurt
However, by a result of Gödel, HOL with standard semantics does not admit an effective, sound, and complete proof calculus.

Gödel's completeness theorem

completeness theoremcompletenesscomplete
However, by a result of Gödel, HOL with standard semantics does not admit an effective, sound, and complete proof calculus.

Proof calculus

proof calculiproof systemcalculi
However, by a result of Gödel, HOL with standard semantics does not admit an effective, sound, and complete proof calculus.

Measurable cardinal

measurablemeasurabilitymeasures
For example, the Löwenheim number of second-order logic is already larger than the first measurable cardinal, if such a cardinal exists.

Aleph number

aleph-null\aleph_0aleph
The Löwenheim number of first-order logic, in contrast, is ℵ 0, the smallest infinite cardinal.

Gérard Huet

Gérard Pierre Huet
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 third-order (let alone arbitrary higher-order) terms has a solution.

Unification (computer science)

unificationmost general unifierunification algorithm
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 third-order (let alone arbitrary higher-order) terms has a solution.