# generalized quantifier

*Generalized quantifiers* are an abstract way of defining quantifiers^{}.

The underlying principle is that formulas^{} quantified by a generalized quantifier are true if the set of elements satisfying those formulas belong in some relation^{} associated with the quantifier.

Every generalized quantifier has an arity, which is the number of formulas it takes as arguments, and a type, which for an $n$-ary quantifier is a tuple of length $n$. The tuple represents the number of quantified variables for each argument.

The most common quantifiers are those of type $\u27e81\u27e9$, including $\forall $ and $\exists $. If $Q$ is a quantifier of type $\u27e81\u27e9$, $M$ is the universe^{} of a model, and ${Q}_{M}$ is the relation associated with $Q$ in that model, then $Qx\varphi (x)\leftrightarrow \{x\in M\mid \varphi (x)\}\in {Q}_{M}$.

So ${\forall}_{M}=\{M\}$, since the quantified formula is only true when all elements satisfy it. On the other hand ${\exists}_{M}=P(M)-\{\mathrm{\varnothing}\}$.

In general, the *monadic* quantifiers are those of type $\u27e81,\mathrm{\dots},1\u27e9$ and if $Q$ is an $n$-ary monadic quantifier then ${Q}_{M}\subseteq P{(M)}^{n}$. Härtig’s quantifier, for instance, is $\u27e81,1\u27e9$, and ${I}_{M}=\{\u27e8X,Y\u27e9\mid X,Y\subseteq M\wedge |X|=|Y|\}$.

A quantifier $Q$ is *polyadic* if it is of type $\u27e8{n}_{1},\mathrm{\dots},{n}_{n}\u27e9$ where each ${n}_{i}\in \mathbb{N}$. Then:

$${Q}_{M}\subseteq \prod _{i}P({M}^{{n}_{i}})$$ |

These can get quite elaborate; $Wxy\varphi (x,y)$ is a $\u27e82\u27e9$ quantifier where $X\in {W}_{M}\leftrightarrow X$ is a well-ordering. That is, it is true if the set of pairs making $\varphi $ true is a well-ordering.

Title | generalized quantifier |

Canonical name | GeneralizedQuantifier |

Date of creation | 2013-03-22 12:59:57 |

Last modified on | 2013-03-22 12:59:57 |

Owner | Henry (455) |

Last modified by | Henry (455) |

Numerical id | 5 |

Author | Henry (455) |

Entry type | Definition |

Classification | msc 03C80 |

Classification | msc 03B15 |

Classification | msc 03B10 |

Related topic | quantifier |

Related topic | Quantifier |

Defines | monadic |

Defines | polyadic |