“quantifier”的概念、定义、翻译、参考文献-科学参考

quantifier

Mathematics

The two expressions ‘for all…’ and ‘there exists…’ are called quantifiers. A phrase such as ‘for all x’ or ‘there exists x’ may stand in front of a sentence involving a symbol x and thereby create a statement that makes sense and is either true or false. There are different ways in English of expressing the same sense as ‘for all x’, but it is sometimes useful to standardize the language to this particular form. This is known as a universal quantifier and is written in symbols as ‘∀x’. Similarly, ‘there exists x’ may be used as the standard form to replace any phrase with this meaning and is an existential quantifier, written in symbols as ‘∃x’.

For example, the statements ‘if x is any number greater than 3 then x is positive’ and ‘there is a real number satisfying x²=2’ can be written in more standard form: ‘for all x, if x is greater than 3 then x is positive’, and ‘there exists x such that x is real and x²=2’. These can be written, using the symbols of mathematical logic, as: (∀x)(x>3 ⇒ x>0), and (∃x)(x ϵ‎ ℝ ∧ x²=2).

Computer

One of the two symbols ∀ or ∃ used in predicate calculus. ∀ is the universal quantifier and is read ‘for all’. ∃ is the existential quantifier and is read ‘there exists’ or ‘for some’. In either case the reference is to possible values of the variable v that the quantifier introduces. ∀v. F means that the formula F is true for all values of v, while ∃v. F means that F is true for at least one value of v. As an example, suppose that P(x,y) is the predicate ‘x is less than or equal to y’. Then the following expression
$\exists x . \forall y . P (x, y)$
says that there exists an x that is less than or equal to all y. This statement is true if values range over the natural numbers, since x can be taken to be 0. It becomes false however if values are allowed to range over negative integers as well. Note also that it would be false even for natural numbers if the predicate were ‘x is less than y’. Other notations such as (∀v)F in place of ∀v . F are also found.

Logic

A sentential operator that takes a formula, binds some set of variables within its scope, and is evaluated in a model on the basis of the elements of its domain satisfying the formula that it modifies. The most frequently encountered quantifiers are the existential and universal quantifiers $\exists$ and $\forall$ , although more expressive quantifiers can be encountered if generalized quantifiers are considered.
Semantically, there are multiple ways to interpret a standard quantifier. On the account due to philosopher Gottlob Frege (1848–1925), quantifiers are second-order functions that take propositional functions as arguments and return truth values (or other propositional functions). For example, in the case of classical logic, a model $M$ can associate a propositional function $\hat{φ} (x)$ with any formula $φ (x)$ with free variable $x$ so that $\hat{φ} (x)$ maps every element $a$ of the domain $M$ to a truth value according to whether $a$ satisfies the formula $φ (x)$ or not. On the Fregean interpretation, a quantifier $\forall x$ is interpreted by a function that maps every propositional function $\hat{φ} (x)$ to a truth value, determined by the scheme:
- • $\forall x maps \hat{φ} (x) to {\begin{cases} truth if \hat{φ} (x) maps no element a to falsity \\ falsity if \hat{φ} (x) maps some element a to falsity \end{cases}$
In other words, $\forall x φ (x)$ is true in a model $M$ if the propositional function that $M$ associates with $φ (x)$ maps all elements of the domain to truth, i.e., $φ (x)$ is satisfied by all elements.
A second approach due to logician Walter Carnielli (1952– ) first defines a distribution of a formula $φ (x)$ in a model $M$ as the set of truth values to which the propositional function $\hat{φ} (x)$ maps elements of the domain. This approach interprets quantifiers as functions mapping such distributions to truth values, i.e., $f : ℘ (V) \\ \emptyset \to V$ when $V$ is a set of truth values. On the distribution account, the above conditions can be rephrased:
- • $\forall x maps D to {\begin{cases} truth if falsity is not a member of D \\ falsity if falsity is a member of D \end{cases}$
where $D$ is the distribution of a formula $φ (x)$ .
Many generalizations of quantification have been introduced, including logics that permit quantification over formulae (cf. propositional quantifiers), quantification over groups of individuals (cf. plural quantification), and accounts of quantification that eliminate dependence relations between variables (cf. branching quantifier). Even more variations are made possible by representing natural language quantifiers as generalized quantifiers.

Philosophy

Informally, a quantifier is an expression that reports a quantity of times that a predicate is satisfied in some class of things (i.e. in a ‘domain’). Thus, thinking about a class of children and their diets, one might report that some eat cake, or that all eat cake, or that not all eat cake, or that none eat cake. ‘Some’ and ‘all’ are represented in modern logic by the quantifiers. The important point is that the treatment fends off thinking of ‘something’, ‘nothing’, and their kin as kinds of names. In classical logic the two interdefinable quantifiers are the existential quantifier (∃x)…x, read as saying that something is…, and the universal quantifier (∀x)…x, read as saying that all things are.…Existential propositions, claiming that things of some kind exist, are represented by the existential quantifier. Less common quantifiers include the plurality quantifiers ‘many…’ and ‘few…’, and there are definable mathematical quantifiers such as ‘more than half…’, ‘exactly one…’.
More formally, a quantifier will bind a variable, turning an open sentence with n distinct free variables into one with n − 1 (an individual letter counts as one variable, although it may recur several times in a formula). When no variables remain free we have a closed sentence, i.e. one that can be evaluated as true or false within a domain. For example, from the open sentence Fx & Gx we can form (∃x)(Fx & Gx), meaning that something is both F and G. The one variable x is bound on each occurrence.

单词	quantifier
释义	quantifier Mathematics The two expressions ‘for all…’ and ‘there exists…’ are called quantifiers. A phrase such as ‘for all x’ or ‘there exists x’ may stand in front of a sentence involving a symbol x and thereby create a statement that makes sense and is either true or false. There are different ways in English of expressing the same sense as ‘for all x’, but it is sometimes useful to standardize the language to this particular form. This is known as a universal quantifier and is written in symbols as ‘∀x’. Similarly, ‘there exists x’ may be used as the standard form to replace any phrase with this meaning and is an existential quantifier, written in symbols as ‘∃x’. For example, the statements ‘if x is any number greater than 3 then x is positive’ and ‘there is a real number satisfying x²=2’ can be written in more standard form: ‘for all x, if x is greater than 3 then x is positive’, and ‘there exists x such that x is real and x²=2’. These can be written, using the symbols of mathematical logic, as: (∀x)(x>3 ⇒ x>0), and (∃x)(x ϵ‎ ℝ ∧ x²=2). Computer One of the two symbols ∀ or ∃ used in predicate calculus. ∀ is the universal quantifier and is read ‘for all’. ∃ is the existential quantifier and is read ‘there exists’ or ‘for some’. In either case the reference is to possible values of the variable v that the quantifier introduces. ∀v. F means that the formula F is true for all values of v, while ∃v. F means that F is true for at least one value of v. As an example, suppose that P(x,y) is the predicate ‘x is less than or equal to y’. Then the following expression $\exists x . \forall y . P (x, y)$ says that there exists an x that is less than or equal to all y. This statement is true if values range over the natural numbers, since x can be taken to be 0. It becomes false however if values are allowed to range over negative integers as well. Note also that it would be false even for natural numbers if the predicate were ‘x is less than y’. Other notations such as (∀v)F in place of ∀v . F are also found. Logic A sentential operator that takes a formula, binds some set of variables within its scope, and is evaluated in a model on the basis of the elements of its domain satisfying the formula that it modifies. The most frequently encountered quantifiers are the existential and universal quantifiers $\exists$ and $\forall$ , although more expressive quantifiers can be encountered if generalized quantifiers are considered. Semantically, there are multiple ways to interpret a standard quantifier. On the account due to philosopher Gottlob Frege (1848–1925), quantifiers are second-order functions that take propositional functions as arguments and return truth values (or other propositional functions). For example, in the case of classical logic, a model $M$ can associate a propositional function $\hat{φ} (x)$ with any formula $φ (x)$ with free variable $x$ so that $\hat{φ} (x)$ maps every element $a$ of the domain $M$ to a truth value according to whether $a$ satisfies the formula $φ (x)$ or not. On the Fregean interpretation, a quantifier $\forall x$ is interpreted by a function that maps every propositional function $\hat{φ} (x)$ to a truth value, determined by the scheme: • $\forall x maps \hat{φ} (x) to {\begin{cases} truth if \hat{φ} (x) maps no element a to falsity \\ falsity if \hat{φ} (x) maps some element a to falsity \end{cases}$ In other words, $\forall x φ (x)$ is true in a model $M$ if the propositional function that $M$ associates with $φ (x)$ maps all elements of the domain to truth, i.e., $φ (x)$ is satisfied by all elements. A second approach due to logician Walter Carnielli (1952– ) first defines a distribution of a formula $φ (x)$ in a model $M$ as the set of truth values to which the propositional function $\hat{φ} (x)$ maps elements of the domain. This approach interprets quantifiers as functions mapping such distributions to truth values, i.e., $f : ℘ (V) \\ \emptyset \to V$ when $V$ is a set of truth values. On the distribution account, the above conditions can be rephrased: • $\forall x maps D to {\begin{cases} truth if falsity is not a member of D \\ falsity if falsity is a member of D \end{cases}$ where $D$ is the distribution of a formula $φ (x)$ . Many generalizations of quantification have been introduced, including logics that permit quantification over formulae (cf. propositional quantifiers), quantification over groups of individuals (cf. plural quantification), and accounts of quantification that eliminate dependence relations between variables (cf. branching quantifier). Even more variations are made possible by representing natural language quantifiers as generalized quantifiers. Philosophy Informally, a quantifier is an expression that reports a quantity of times that a predicate is satisfied in some class of things (i.e. in a ‘domain’). Thus, thinking about a class of children and their diets, one might report that some eat cake, or that all eat cake, or that not all eat cake, or that none eat cake. ‘Some’ and ‘all’ are represented in modern logic by the quantifiers. The important point is that the treatment fends off thinking of ‘something’, ‘nothing’, and their kin as kinds of names. In classical logic the two interdefinable quantifiers are the existential quantifier (∃x)…x, read as saying that something is…, and the universal quantifier (∀x)…x, read as saying that all things are.…Existential propositions, claiming that things of some kind exist, are represented by the existential quantifier. Less common quantifiers include the plurality quantifiers ‘many…’ and ‘few…’, and there are definable mathematical quantifiers such as ‘more than half…’, ‘exactly one…’. More formally, a quantifier will bind a variable, turning an open sentence with n distinct free variables into one with n − 1 (an individual letter counts as one variable, although it may recur several times in a formula). When no variables remain free we have a closed sentence, i.e. one that can be evaluated as true or false within a domain. For example, from the open sentence Fx & Gx we can form (∃x)(Fx & Gx), meaning that something is both F and G. The one variable x is bound on each occurrence.
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