Quiz. FOL - Closed, Open, and Ground Formulas [0010]

  • November 22, 2025

Consider the following formula

\[ \forall y.\ ((\forall x.\ p(x))) \to (\exists x.\ q(x, y)) \]

Is this formula closed, open, or ground?

Solution. 

  • November 22, 2025
 This formula is a closed formula because all variables within the formula are bound by quantifiers. The variable \(y\) is bound by the quantifier \(\forall y\), and both occurrences of \(x\) are bound by their respective quantifiers \(\forall x\) and \(\exists x\). There are no free variables in this formula. A nice example of a ground formula is something like this \[ p(a, f(b)) \to q(c) \] We can see here that there are no variables at all; \(a\), \(b\), and \(c\) are not bound by any quantifiers, hence the formula is ground since we aren't substituting/quantifying over any variables.