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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 (without function symbols) is always of finite size, i.e. contains a finite number of formulas.

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