# 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

96 Related Articles

### 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**

### Semantics of logic

**formal semanticssemanticslogical 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.