Quantifier rank
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. Thus two logically equivalent formulae can have different quantifier ranks, when they express the same thing in different ways.Definition
Quantifier Rank of a Formula in First-order language
Let φ be a FO formula. The quantifier rank of φ, written qr, is defined as
Remarks
Quantifier Rank of a higher order Formula
...Examples
- A sentence of quantifier rank 2:
- A formula of quantifier rank 1:
- A formula of quantifier rank 0:
- A sentence in prenex normal form of quantifier rank 3:
- A sentence, equivalent to the previous, although of quantifier rank 2:
