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, $Q^Cx\phi(x)$ 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.