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.
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

In first-order logic

Let be a first-order formula. The quantifier rank of, written, is defined as:

, if is atomic.
.
.
.
.

Remarks

We write for the set of all first-order formulas with.
Relational is always of finite size, i.e. contains a finite number of formulas.
In prenex normal form, the quantifier rank of is exactly the number of quantifiers appearing in.

In higher-order logic

For fixed-point logic, with a least fixed-point operator :.

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: