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
Remarks
Quantifier Rank of a higher order Formula