Let ${\displaystyle \varphi }$  be a first-order formula. The quantifier rank of ${\displaystyle \varphi }$ , written ${\displaystyle \operatorname {qr} (\varphi )}$ , is defined as:

• ${\displaystyle \operatorname {qr} (\varphi )=0}$ , if ${\displaystyle \varphi }$  is atomic.
• ${\displaystyle \operatorname {qr} (\varphi _{1}\land \varphi _{2})=\operatorname {qr} (\varphi _{1}\lor \varphi _{2})=\max(\operatorname {qr} (\varphi _{1}),\operatorname {qr} (\varphi _{2}))}$ .
• ${\displaystyle \operatorname {qr} (\lnot \varphi )=\operatorname {qr} (\varphi )}$ .
• ${\displaystyle \operatorname {qr} (\exists _{x}\varphi )=\operatorname {qr} (\varphi )+1}$ .
• ${\displaystyle \operatorname {qr} (\forall _{x}\varphi )=\operatorname {qr} (\varphi )+1}$ .

Remarks

• We write ${\displaystyle \operatorname {FO} [n]}$  for the set of all first-order formulas ${\displaystyle \varphi }$  with ${\displaystyle \operatorname {qr} (\varphi )\leq n}$ .
• Relational ${\displaystyle \operatorname {FO} [n]}$  (without function symbols) is always of finite size, i.e. contains a finite number of formulas.
• In prenex normal form, the quantifier rank of ${\displaystyle \varphi }$  is exactly the number of quantifiers appearing in ${\displaystyle \varphi }$ .

For fixed-point logic, with a least fixed-point operator ${\displaystyle \operatorname {LFP} }$ : ${\displaystyle \operatorname {qr} ([\operatorname {LFP} _{\phi }]y)=1+\operatorname {qr} (\phi )}$ .

• A sentence of quantifier rank 2:

${\displaystyle \forall x\exists yR(x,y)}$ 

• A formula of quantifier rank 1:

${\displaystyle \forall xR(y,x)\wedge \exists xR(x,y)}$ 

• A formula of quantifier rank 0:

${\displaystyle R(x,y)\wedge x\neq y}$ 

• A sentence in prenex normal form of quantifier rank 3:

${\displaystyle \forall x\exists y\exists z((x\neq y\wedge xRy)\wedge (x\neq z\wedge zRx))}$ 

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

${\displaystyle \forall x(\exists y(x\neq y\wedge xRy))\wedge \exists z(x\neq z\wedge zRx))}$ 

• Ehrenfeucht–Fraïssé game

• 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.

• Quantifier Rank Spectrum of L-infinity-omega BA Thesis, 2000