# generalized 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 $\langle 1\rangle$, including $\forall$ and $\exists$. If $Q$ is a quantifier of type $\langle 1\rangle$, $M$ is the universe  of a model, and $Q_{M}$ is the relation associated with $Q$ in that model, then $Qx\phi(x)\leftrightarrow\{x\in M\mid\phi(x)\}\in Q_{M}$.

So $\forall_{M}=\{M\}$, since the quantified formula is only true when all elements satisfy it. On the other hand $\exists_{M}=P(M)-\{\emptyset\}$.

In general, the monadic quantifiers are those of type $\langle 1,\ldots,1\rangle$ and if $Q$ is an $n$-ary monadic quantifier then $Q_{M}\subseteq P(M)^{n}$. Härtig’s quantifier, for instance, is $\langle 1,1\rangle$, and $I_{M}=\{\langle X,Y\rangle\mid X,Y\subseteq M\wedge|X|=|Y|\}$.

A quantifier $Q$ is polyadic if it is of type $\langle n_{1},\ldots,n_{n}\rangle$ where each $n_{i}\in\mathbb{N}$. Then:

 $Q_{M}\subseteq\prod_{i}P(M^{n_{i}})$

These can get quite elaborate; $Wxy\phi(x,y)$ is a $\langle 2\rangle$ quantifier where $X\in W_{M}\leftrightarrow X$ is a well-ordering. That is, it is true if the set of pairs making $\phi$ 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