# 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 $\u27e82\u27e9$ 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 $\u27e82\u27e9$ quantifier while the second is a $\u27e81\u27e9$ quantifier.

Another similar quantifier is Chang’s quantifier ${Q}^{C}$, a $\u27e81\u27e9$ 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.

Title | Härtig’s quantifier |
---|---|

Canonical name | HartigsQuantifier |

Date of creation | 2013-03-22 12:59:16 |

Last modified on | 2013-03-22 12:59:16 |

Owner | Henry (455) |

Last modified by | Henry (455) |

Numerical id | 7 |

Author | Henry (455) |

Entry type | Definition |

Classification | msc 03B15 |

Related topic | Quantifier |

Defines | Rescher quantifier |