Quantifier (logic)

quantifierquantifiersquantificationquantifiedlogical quantifierquantified variableslogical quantifiersquantifiesquantifiablequantification of the predicate
In natural languages, a quantifier turns a sentence about something having some property into a sentence about the number (quantity) of things having the property.wikipedia
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Mathematical logic

formal logicsymbolic logiclogic
In mathematical logic, in particular in first-order logic, a quantifier achieves a similar task, operating on a mathematical formula rather than an English sentence.
Its syntax involves only finite expressions as well-formed formulas, while its semantics are characterized by the limitation of all quantifiers to a fixed domain of discourse.

Domain of discourse

universe of discoursedomainarea of interest
More precisely, a quantifier specifies the quantity of specimens in the domain of discourse that satisfy an open formula. Gottlob Frege, in his 1879 Begriffsschrift, was the first to employ a quantifier to bind a variable ranging over a domain of discourse and appearing in predicates.
For example, in an interpretation of first-order logic, the domain of discourse is the set of individuals over which the quantifiers range.

Well-formed formula

formulamathematical formulaformulas
In mathematical logic, in particular in first-order logic, a quantifier achieves a similar task, operating on a mathematical formula rather than an English sentence.
An atomic formula is a formula that contains no logical connectives nor quantifiers, or equivalently a formula that has no strict subformulas.

Universal quantification

universal quantifieruniversally quantifieduniversal
The two most common formal quantifiers are "for each" (traditionally symbolized by "∀"), and "there exists some" .
In predicate logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all".

Existential quantification

existential quantifierthere exists
The two most common formal quantifiers are "for each" (traditionally symbolized by "∀"), and "there exists some" .
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some".

Cylindric algebra

cylindric logic
Indeed, cylindric algebras are Boolean algebras equipped with additional cylindrification operations that model quantification and equality.

Ernst Schröder

SchröderErnst SchroderErnst Schroeder
Peirce's notation can be found in the writings of Ernst Schröder, Leopold Loewenheim, Thoralf Skolem, and Polish logicians into the 1950s.
To their work he subsequently added several important concepts due to Charles Sanders Peirce, including subsumption and quantification.

Uniqueness quantification

uniqueuniquenessone and only one
The semantics for uniqueness quantification requires first-order predicate calculus with equality.
In mathematical logic, this sort of quantification is known as uniqueness quantification or unique existential quantification.

First-order logic

predicate logicfirst-orderpredicate calculus
In mathematical logic, in particular in first-order logic, a quantifier achieves a similar task, operating on a mathematical formula rather than an English sentence. Most notably, it is the notation of Kurt Gödel's landmark 1930 paper on the completeness of first-order logic, and 1931 paper on the incompleteness of Peano arithmetic.
First-order logic uses quantified variables over non-logical objects and allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form "there exists x such that x is Socrates and x is a man" and there exists is a quantifier while x is a variable.

Peano axioms

Peano arithmeticfirst-order arithmeticPeano's axioms
Most notably, it is the notation of Kurt Gödel's landmark 1930 paper on the completeness of first-order logic, and 1931 paper on the incompleteness of Peano arithmetic.
The axiom of induction is in second-order, since it quantifies over predicates (equivalently, sets of natural numbers rather than natural numbers), but it can be transformed into a first-order axiom schema of induction.

Open formula

Open sentencelogical satisfactionSentential function
More precisely, a quantifier specifies the quantity of specimens in the domain of discourse that satisfy an open formula.

Relation algebra

relational
Hence RA is, in effect, a way of algebraizing nearly all mathematics, while dispensing with FOL and its connectives, quantifiers, turnstiles, and modus ponens.

Monadic predicate calculus

monadicmonadic predicate logicMonadic first-order logic
For a finite domain of discourse D = {a 1,...a n } the universal quantifier is equivalent to a logical conjunction of propositions with singular terms a i having the form Pa i for monadic predicates.
Each formula in the monadic predicate calculus is equivalent to a formula in which quantifiers appear only in closed subformulas of the form

Uniform continuity

uniformly continuousuniformly continuous functionuniform
Continuity of a function for every point x of an interval can thus be expressed by a formula starting with the quantification

Quantifier rank

The maximum depth of nesting of quantifiers in a formula is called its quantifier rank.
In mathematical logic, the quantifier rank of a formula is the depth of nesting of its quantifiers.

Scope (logic)

scope
A quantifier has a scope, and an occurrence of a variable x is free if it is not within the scope of a quantification for that variable.
In logic, the scope of a quantifier or a quantification is the range in the formula where the quantifier "engages in".

Gottlob Frege

FregeFregeanFrege, Gottlob
Gottlob Frege, in his 1879 Begriffsschrift, was the first to employ a quantifier to bind a variable ranging over a domain of discourse and appearing in predicates.
In effect, Frege invented axiomatic predicate logic, in large part thanks to his invention of quantified variables, which eventually became ubiquitous in mathematics and logic, and which solved the problem of multiple generality.

Free variables and bound variables

free variablesbound variablebound
A formal quantifier requires a variable, which is said to be bound by it, and a subformula specifying a property of that variable.
z is a free variable and x and y are bound variables, associated with logical quantifiers; consequently the logical value of this expression depends on the value of z, but there is nothing called x or y on which it could depend.

George Bentham

Benth.BenthamBenth
In 1827, George Bentham published his Outline of a new system of logic, with a critical examination of Dr Whately's Elements of Logic, describing the principle of the quantifier, but the book was not widely circulated.
In this the principle of the quantification of the predicate was first explicitly stated.

Quantifier elimination

elimination of quantifierseliminates quantifiers
One way of classifying formulas is by the amount of quantification.

Existential graph

alpha graph
Around 1895, Peirce began developing his existential graphs, whose variables can be seen as tacitly quantified.
There are no literal variables or quantifiers in the sense of first-order logic.

Willard Van Orman Quine

QuineW. V. O. QuineW. V. Quine
Peano's notation was adopted by the Principia Mathematica of Whitehead and Russell, Quine, and Alonzo Church.
Most of Quine's original work in formal logic from 1960 onwards was on variants of his predicate functor logic, one of several ways that have been proposed for doing logic without quantifiers.

Term logic

Aristotelian logicscholastic logictraditional logic
Term logic, also called Aristotelian logic, treats quantification in a manner that is closer to natural language, and also less suited to formal analysis.

Natural language

linguisticnaturalnatural languages
In natural languages, a quantifier turns a sentence about something having some property into a sentence about the number (quantity) of things having the property.