Branching quantifier

In logic a branching quantifier, also called a Henkin quantifier, finite partially ordered quantifier or even nonlinear quantifier, is a partial ordering

${\displaystyle \langle Qx_{1}\dots Qx_{n}\rangle }$

of quantifiers for Q ∈ {∀,∃}. It is a special case of generalized quantifier. In classical logic, quantifier prefixes are linearly ordered such that the value of a variable y_m bound by a quantifier Q_m depends on the value of the variables

y₁, ..., y_m−1

bound by quantifiers

Qy₁, ..., Qy_m−1

preceding Q_m. In a logic with (finite) partially ordered quantification this is not in general the case.

Branching quantification first appeared in a 1959 conference paper of Leon Henkin. Systems of partially ordered quantification are intermediate in strength between first-order logic and second-order logic. They are being used as a basis for Hintikka's and Gabriel Sandu's independence-friendly logic.