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 $\langle 1\rangle$, 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