Härtig’s quantifier

Härtig’s quantifierMathworldPlanetmath is a quantifier which takes two variables and two formulasMathworldPlanetmathPlanetmath, written Ixyϕ(x)ψ(y). It asserts that |{xϕ(x)}|=|{yψ(y)}|. That is, the cardinality of the values of x which make ϕ is the same as the cardinality of the values which make ψ(x) true. Viewed as a generalized quantifier, I is a 2 quantifier.

Closely related is the Rescher quantifier, which also takes two variables and two formulas, is written Jxyϕ(x)ψ(y), and asserts that |{xϕ(x)}||{yψ(y)|. The Rescher quantifier is sometimes defined instead to be a similar but different quantifier, Jxϕ(x)|{xϕ(x)}|>|{x¬ϕ(x)}|. The first definition is a 2 quantifier while the second is a 1 quantifier.

Another similar quantifier is Chang’s quantifier QC, a 1 quantifier defined by QMC={XM|X|=|M|}. That is, QCxϕ(x) is true if the number of x satisfying ϕ has the same cardinality as the universePlanetmathPlanetmath; for finite models this is the same as , but for infiniteMathworldPlanetmath ones it is not.

Title Härtig’s quantifier
Canonical name HartigsQuantifier
Date of creation 2013-03-22 12:59:16
Last modified on 2013-03-22 12:59:16
Owner Henry (455)
Last modified by Henry (455)
Numerical id 7
Author Henry (455)
Entry type Definition
Classification msc 03B15
Related topic Quantifier
Defines Rescher quantifier