# Härtig’s quantifier

Härtig’s quantifier is a quantifier which takes two variables and two formulas, written $Ixy\phi(x)\psi(y)$. It asserts that $|\{x\mid\phi(x)\}|=|\{y\mid\psi(y)\}|$. 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 $|\{x\mid\phi(x)\}|\leq|\{y\mid\psi(y)|$. The Rescher quantifier is sometimes defined instead to be a similar but different quantifier, $Jx\phi(x)\leftrightarrow|\{x\mid\phi(x)\}|>|\{x\mid\neg\phi(x)\}|$. 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 $Q^{C}_{M}=\{X\subseteq M\mid|X|=|M|\}$. That is, $Q^{C}x\phi(x)$ 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.

Title Härtig’s quantifier HartigsQuantifier 2013-03-22 12:59:16 2013-03-22 12:59:16 Henry (455) Henry (455) 7 Henry (455) Definition msc 03B15 Quantifier Rescher quantifier