“generalized quantifier”的概念、定义、翻译、参考文献-科学参考

generalized quantifier

Logic

An operator—first introduced by Per Lindström (1936–2009)—that generalizes the truth conditions associated with the universal and existential quantifiers in classical logic. Generalized quantifiers are useful because they may provide formal accounts of many quantifier expressions that are too fine-grained to be captured by combinations of such quantifiers, e.g.,
- • There are uncountably many points in the real line.
- • The number of spoons in the cabinet is odd.
- • Alice has $n$ more apples than Bob has oranges.
In the first two cases, the appropriate kind of quantifier should have truth conditions such as:
- • $Q_{0} x φ (x)$ is true in $M$ if there are uncountably many individuals of which $φ (x)$ is true
- • $Q_{1} y φ (y)$ is true in $M$ if the number of individuals of which $φ (y)$ is true is odd
where $Q_{0}$ and $Q_{1}$ bind instances of free variables $x$ and $y$ in the first and second subformulae, respectively. In the third case, however, an appropriate truth condition must involve more than one formula, as the number of elements satisfying ‘ $x$ is Alice’s apple’ and ‘ $y$ is Bob’s orange’ are being compared. A notion of type serves to classify generalized quantifiers, which is a tuple $〈 i_{0}, . . ., i_{n - 1} 〉$ where the quantifier accepts arguments $φ_{0}, . . ., φ_{n - 1}$ and binds $i_{j}$ many variables in formula $φ_{j}$ . The quantifier in the third case binds one variable in each of two formulae, and is thus type $〈 1, 1 〉$ . A well-known quantifier of type $〈 1, 1 〉$ is the Härtig quantifier $I$ named for Klaus Härtig (1936–2013),
- • $I [x] [y] (φ (x), ψ (y))$ is true in $M$ if and only if the number of elements satisfying $φ (x)$ is equal to the number of elements satisfying $ψ (y)$ .
$I$ is type $〈 1, 1 〉$ because it takes as arguments two formulae and binds a single variable in each.

单词	generalized quantifier
释义	generalized quantifier Logic An operator—first introduced by Per Lindström (1936–2009)—that generalizes the truth conditions associated with the universal and existential quantifiers in classical logic. Generalized quantifiers are useful because they may provide formal accounts of many quantifier expressions that are too fine-grained to be captured by combinations of such quantifiers, e.g., • There are uncountably many points in the real line. • The number of spoons in the cabinet is odd. • Alice has $n$ more apples than Bob has oranges. In the first two cases, the appropriate kind of quantifier should have truth conditions such as: • $Q_{0} x φ (x)$ is true in $M$ if there are uncountably many individuals of which $φ (x)$ is true • $Q_{1} y φ (y)$ is true in $M$ if the number of individuals of which $φ (y)$ is true is odd where $Q_{0}$ and $Q_{1}$ bind instances of free variables $x$ and $y$ in the first and second subformulae, respectively. In the third case, however, an appropriate truth condition must involve more than one formula, as the number of elements satisfying ‘ $x$ is Alice’s apple’ and ‘ $y$ is Bob’s orange’ are being compared. A notion of type serves to classify generalized quantifiers, which is a tuple $〈 i_{0}, . . ., i_{n - 1} 〉$ where the quantifier accepts arguments $φ_{0}, . . ., φ_{n - 1}$ and binds $i_{j}$ many variables in formula $φ_{j}$ . The quantifier in the third case binds one variable in each of two formulae, and is thus type $〈 1, 1 〉$ . A well-known quantifier of type $〈 1, 1 〉$ is the Härtig quantifier $I$ named for Klaus Härtig (1936–2013), • $I [x] [y] (φ (x), ψ (y))$ is true in $M$ if and only if the number of elements satisfying $φ (x)$ is equal to the number of elements satisfying $ψ (y)$ . $I$ is type $〈 1, 1 〉$ because it takes as arguments two formulae and binds a single variable in each.
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