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Härtig's quantifier (Definition)

Härtig's quantifier is a quantifier which takes two variables and two formulas, written $ Ixy\phi(x)\psi(y)$. It asserts that $ \vert\{x\mid \phi(x)\}\vert=\vert\{y\mid\psi(y)\}\vert$. That is, the cardinality of the values of $ x$ which make $ \phi$ is the same as the cardinality of the values which make $ \psi(x)$ true. Viewed as a generalized quantifier, $ I$ is a $ \langle 2\rangle$ quantifier.

Closely related is the Rescher quantifier, which also takes two variables and two formulas, is written $ Jxy\phi(x)\psi(y)$, and asserts that $ \vert\{x\mid \phi(x)\}\vert\leq\vert\{y\mid\psi(y)\vert$. The Rescher quantifier is sometimes defined instead to be a similar but different quantifier, $ Jx\phi(x)\leftrightarrow \vert\{x\mid\phi(x)\}\vert>\vert\{x\mid\neg\phi(x)\}\vert$. The first definition is a $ \langle 2\rangle$ quantifier while the second is a $ \langle 1\rangle$ quantifier.

Another similar quantifier is Chang's quantifier $ Q^C$, a $ \langle 1\rangle$ quantifier defined by $ Q^C_M=\{X\subseteq M\mid \vert X\vert=\vert M\vert\}$. That is, $ Q^Cx\phi(x)$ is true if the number of $ x$ satisfying $ \phi$ has the same cardinality as the universe; for finite models this is the same as $ \forall$, but for infinite ones it is not.



"Härtig's quantifier" is owned by Henry.
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See Also: quantifier

Also defines:  Rescher quantifier
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Cross-references: infinite, Models, finite, universe, number, similar, generalized quantifier, cardinality, formulas, variables, quantifier
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This is version 4 of Härtig's quantifier, born on 2002-08-25, modified 2005-03-07.
Object id is 3362, canonical name is HartigsQuantifier.
Accessed 3208 times total.

Classification:
AMS MSC03B15 (Mathematical logic and foundations :: General logic :: Higher-order logic and type theory)

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