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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 (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

  • , if φ is atomic.
  • .
  • .
  • .
  • .


  • We write FO[n] for the set of all first-order formulas φ with .
  • Relational FO[n] (without function symbols) is always of finite size, i.e. contains a finite number of formulas
  • Notice that in Prenex normal form the Quantifier Rank of φ is exactly the number of quantifiers appearing in φ.

Quantifier Rank of a higher order Formula

  • For Fixpoint logic, with a least fix point operator LFP:


  • A sentence of quantifier rank 2:
  • A formula of quantifier rank 1:
  • A formula of quantifier rank 0:
  • A sentence, equivalent to the previous, although of quantifier rank 2:

See also


  • Ebbinghaus, Heinz-Dieter; Flum, Jörg (1995), Finite Model Theory, Springer, ISBN 978-3-540-60149-4.
  • Grädel, Erich; Kolaitis, Phokion G.; Libkin, Leonid; Maarten, Marx; Spencer, Joel; Vardi, Moshe Y.; Venema, Yde; Weinstein, Scott (2007), Finite model theory and its applications, Texts in Theoretical Computer Science. An EATCS Series, Berlin: Springer-Verlag, p. 133, ISBN 978-3-540-00428-8, Zbl 1133.03001.
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Quantifier rank
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