Definition. Closed, Open, and Ground Formulas (FOL) [000z]

  • November 22, 2025

We have 3 important classifications of formulas in First Order Logic (FOL) based on the nature of their variables:

  • Closed Formula: A formula with no free variables. All variables in the formula are bound by quantifiers. E.g., \(\forall x.\ \exists y.\ p(x, y)\) is a closed formula.
  • Open Formula: A formula with at least one free variable. E.g., \(p(x, y)\) is an open formula since both \(x\) and \(y\) are free.
  • Ground Formula: A formula with no variables at all. E.g., \(p(a, b)\) is a ground formula if \(a\) and \(b\) are constants.