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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:

    • Q0xφ(x) is true in M if there are uncountably many individuals of which φ(x) is true

    • Q1yφ(y) is true in M if the number of individuals of which φ(y) is true is odd

    where Q0 and Q1 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 i0,...,in1 where the quantifier accepts arguments φ0,...,φn1 and binds ij many variables in formula φj. 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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