Härtig’s quantifier

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

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

Another similar quantifier is Chang’s quantifier $Q^C$, a $⟨1⟩$ quantifier defined by $Q_M^C=\{X\subseteq M\mid |X|=|M|\}$. That is, $Q^{C}x\varphi (x)$ is true if the number of $x$ satisfying $\varphi$ has the same cardinality as the universe; for finite models this is the same as $\forall$, but for infinite ones it is not.