quantifier

quan·ti·fi·er

Q0014450 (kwŏn′tə-fī′ər)n.1. Logic An operator that limits the variables of a proposition, as some or all.2. Linguistics A word or other constituent that expresses a quantity or contrast in quantity, as some, all, or many.

quantifier

(ˈkwɒntɪˌfaɪə) n1. (Logic) logic a. a symbol including a variable that indicates the degree of generality of the expression in which that variable occurs, as (∃x) in (∃x)Fx, rendered "something is an F", (x) in (x)(Fx→Gx), rendered "all Fs are Gs"b. any other symbol with an analogous interpretation: the existential quantifier, (∃x), corresponds to the words "there is something, x, such that …". 2. (Grammar) grammar a word or phrase in a natural language having this role, such as some, all, or many in English

quan•ti•fi•er

(ˈkwɒn təˌfaɪ ər)

n. 1. Logic. an expression, as “all” or “some,” that indicates the quantity of a proposition. Compare existential quantifier, universal quantifier. 2. a word or phrase, usu. modifying a noun, that indicates quantity, as much or few. [1875–80]

Thesaurus

Noun	1.	quantifier - (logic) a word (such as `some' or `all' or `no') that binds the variables in a logical propositionlogical quantifierlogic - the branch of philosophy that analyzes inferenceword - a unit of language that native speakers can identify; "words are the blocks from which sentences are made"; "he hardly said ten words all morning"existential operator, existential quantifier - a logical quantifier of a proposition that asserts the existence of at least one thing for which the proposition is true
	2.	quantifier - (grammar) a word that expresses a quantity (as `fifteen' or `many')grammar - the branch of linguistics that deals with syntax and morphology (and sometimes also deals with semantics)word - a unit of language that native speakers can identify; "words are the blocks from which sentences are made"; "he hardly said ten words all morning"universal quantifier - a logical quantifier of a proposition that asserts that the proposition is true for all members of a class of things

Translations

quantificateur

quantifier

[′kwän·tə‚fī·ər] (mathematics) Either of the phrases “for all” and “there exists”; these are symbolized respectively by an inverted A and a backward E.

Quantifier

(from the Latin quantum, “how much”), a logical operation that gives the quantitative character of the range of objects with which the expression obtained as a result of its application is concerned. In ordinary language, words of the type “all,” “each,” “some,” “there exists,” “there is,” “any,” “every,” “unique,” “several,” “infinitely many,” “a finite number,” as well as all cardinal numbers, serve as conveyors of these characteristics. In formal languages in which predicate calculus is a constituent part, two kinds of quantifier turn out to be sufficient for the expression of all such characteristics: the universal quantifier (“for all x,” denoted by ∀x, (∀x), (x), (Ax), ) and the existential quantifier (“for some x,” denoted by, ). With the aid of quantifiers it is possible to write down the four fundamental forms of judgment of traditional logic: “all A are B” is written as ∀x [A(x) ⊃ B(x)], “no A is B” as ∀x[A(x)⊃ ∼ B(x)], “some A are B” as ∃ x [A (x) & B(x)], and “some A are not B” as ∃x[A(x)]; here A(x) denotes that x possesses the property A, ⊃ is the implication sign, ∼ is the negation sign, and & is the conjunction sign.

The part of a formula over which the operation of any quantifier is distributed is called the scope of operation of that quantifier (it may be indicated by parentheses). The entry of any variable into the formula directly after the quantifier or within the scope of the quantifier after which the variable stands is called its bound entry. All remaining variable entries are called free. A formula containing free variable entries is dependent on them (is a function of them), but the bound entries may be “renamed”; for example, the expressions ∃x (x = 2y) and ∃z(z = 2y) denote one and the same thing, but the same cannot be said of ∃x(x =2y) and ∃x(x = 2t). The use of quantifiers reduces the number of free variables in logical expressions and, if the quantifier is not a “dummy” (that is, if it is related to a variable actually entering into the formula), it transforms a three-place predicate into a two-place one, a two-place one into a single-place one, and single-place one into a proposition. The use of quantifiers is codified by special “quantification postulates” (whose addition to propositional calculus essentially implies an expansion of the latter into predicate calculus)—for example, the “Bernays’ postulates”: the axioms A(t)⊃ ∃xA(x) and ∀xA(x) ∃ A(t) and the rules of deduction “if it has been proved that C⊃ A(x), then it may be considered also proved that C∀ xA(x)” and “if it has been proved that A(x)⊃ C” then it may be considered also proved that ∃xA(x)⊃ C” (here x does not enter freely into C).

Other kinds of quantifiers may be reduced to the universal and existential quantifiers. For example, in place of the uniqueness quantifier ∃!x (“there exists a unique x such that”) it is possible to write “ordinary” quantifiers, replacing ∃!xA(x) with

∃xA(x) & ∀y∀z [A(y) &A(z) ⊃ y = z]

Analogously, quantifiers “bound” to a single-place predicate P(x) —∃x_px) (“there exists an x satisfying the property P and such that”) and ∀x_(p)x (“for all x satisfying the property P it is true that”)—can easily be expressed by means of the universal and existential quantifiers and the implication and conjunction operators:

∃x_P(x)A(x) ≡ ∃x[P(x) & A(x)]]

and

∀x_P(x)A(x) ≡ ∀x[P(x) ⊃ A(x)]

REFERENCES

Kleene, S. C. Vvedenie v metamatematiku. Moscow, 1957. Pages 72–80, 130–38. (Translated from English.)
Church, A. Vvedenie v matematicheskuiu logiku, vol. 1. Moscow, 1960. Pages 42–48. (Translated from English.)

IU. A. GASTEV

quantifier

(logic)An operator in predicate logic specifying for whichvalues of a variable a formula is true. Universallyquantified means "for all values" (written with an inverted A,LaTeX \\forall) and existentially quantified means "thereexists some value" (written with a reversed E, LaTeX\\exists). To be unambiguous, the set to which the values ofthe variable belong should be specified, though this is oftenomitted when it is clear from the context (the "universe ofdiscourse"). E.g.

Forall x . P(x) <=> not (Exists x . not P(x))

meaning that any x (in some unspecified set) has property Pwhich is equivalent to saying that there does not exist any xwhich does not have the property.

If a variable is not quantified then it is a free variable.In logic programming this usually means that it is actuallyuniversally quantified.

See also first order logic.

quantifier

quan·ti·fi·er

quantifier

quan•ti•fi•er

quantifier

quantifier

Quantifier

REFERENCES

quantifier

quantifier

Synonyms for quantifier

noun (logic) a word (such as 'some' or 'all' or 'no') that binds the variables in a logical proposition

Synonyms

Related Words

noun (grammar) a word that expresses a quantity (as 'fifteen' or 'many')

Related Words