Quiz. Semantic Argument method (FOL) [0019]

  • November 22, 2025

Consider the following formula:

\[ F \triangleq (\forall x.\ p(x)) \to (\forall y.\ p(y)) \]

Using the semantic argument method, determine whether the formula \(F\) is valid.

Solution. 

  • November 22, 2025
 We begin by assuming that \(\exists S, \sigma \nvDash F\) \[ \begin {align*} & S, \sigma \nvDash (\forall x.\ p(x)) \to (\forall y.\ p(y)) \\ & \therefore S, \sigma \models \forall x.\ p(x) \land S, \sigma \nvDash \forall y.\ p(y) \\ & \therefore S, \sigma \models \forall x.\ p(x) \land S, \sigma \models \neg \forall y.\ p(y) \\ \end {align*} \] Since we have \(p\) and its negation both holding under the same interpretation and variable assignment, we can derive a contradiction using the Contradiction rule from the semantic argument method. Therefore, our initial assumption that \(F\) is not valid must be false. Hence, we conclude that the formula \(F\) is valid.