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generalized quantifier

(Definition)

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 -ary quantifier is a tuple of length . 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 is a quantifier of type $\langle 1\rangle$ , is the universe of a model, and is the relation associated with 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 is an -ary monadic quantifier then $Q_M\subseteq P(M)^n$ . Härtig's quantifier, for instance, is $\langle 1,1\rangle$ , and $I_M=\{\langle X,Y\rangle\mid X,Y\subseteq M\wedge \vert X\vert=\vert Y\vert\}$ .

A quantifier is polyadic if it is of type $\langle n_1,\ldots, n_n\rangle$ where each $n_i\in\mathbb{N}$ . Then:

$\displaystyle Q_M\subseteq \prod_{i} P(M^{n_i})$

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.

"generalized quantifier" is owned by Henry. [ full author list (2) ]

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See Also: quantifier

Also defines:

monadic, polyadic

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Cross-references: well-ordering, Härtig's quantifier, satisfy, universe, variables, represents, length, tuple, type, arguments, number, arity, relation, formulas, quantifiers
There are 9 references to this entry.

This is version 2 of generalized quantifier, born on 2002-08-28, modified 2008-07-09.
Object id is 3377, canonical name is GeneralizedQuantifier.
Accessed 9483 times total.

Classification:

AMS MSC:	03B10 (Mathematical logic and foundations :: General logic :: Classical first-order logic)
	03B15 (Mathematical logic and foundations :: General logic :: Higher-order logic and type theory)
	03C80 (Mathematical logic and foundations :: Model theory :: Logic with extra quantifiers and operators)

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