generalized quantifier

Generalized quantifiers are an abstract way of defining quantifiersMathworldPlanetmath.

The underlying principle is that formulasMathworldPlanetmathPlanetmath quantified by a generalized quantifier are true if the set of elements satisfying those formulas belong in some relationMathworldPlanetmathPlanetmath 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 1, including and . If Q is a quantifier of type 1, M is the universePlanetmathPlanetmath of a model, and QM is the relation associated with Q in that model, then Qxϕ(x){xMϕ(x)}QM.

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

In general, the monadic quantifiers are those of type 1,,1 and if Q is an n-ary monadic quantifier then QMP(M)n. Härtig’s quantifier, for instance, is 1,1, and IM={X,YX,YM|X|=|Y|}.

A quantifier Q is polyadic if it is of type n1,,nn where each ni. Then:


These can get quite elaborate; Wxyϕ(x,y) is a 2 quantifier where XWMX is a well-ordering. That is, it is true if the set of pairs making ϕ 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