In mathematical logic, the quantifier rank of a formula is the depth of nesting of its quantifiers. It plays an essential role in model theory.

Notice that the quantifier rank is a property of the formula itself (i.e. the expression in a language). Thus two logically equivalent formulae can have different quantifier ranks, when they express the same thing in different ways.


Quantifier Rank of a Formula in First-order language (FO)

Let φ be a FO formula. The quantifier rank of φ, written qr(φ), is defined as


Quantifier Rank of a higher order Formula

qr([LFPφ]y) = 1 + qr( φ)



See also