 Countermodel

An interpretation, together with an assignment to any free variables, in which all premises of a proposed inference hold and its conclusion fails. By soundness of the inference rules, a countermodel proves that the inference cannot be derived. For example, on $\{0,1\}$ with $P=Q=\{0\}$ and $x=0$, the formula $\forall y(P(y)\to Q(y))$ is true but $P(x)\to \forall yQ(y)$ is false. A pointwise assumption about $x$ cannot justify unrestricted universal quantification.