\begin{frame} \frametitle{Exercises} \begin{alertblock}{} Assume you want to disprove \begin{talign} \phi \vdash \psi \end{talign} How can the soundness or completeness theorem help? \end{alertblock} \pause \begin{exampleblock}{} To show that there is no possible proof might be difficult. \pause\medskip It is easier to give a counter-model. \medskip\pause That is, a model $\mathcal{M}$ and environment $\ell$ such that \begin{talign} \mathcal{M} \models_\ell \phi &&\text{ and }&& \mathcal{M} \not\models_\ell \psi \end{talign} \pause Then we know that $\phi \not\models \psi$. \medskip\pause By the soundness we have \begin{talign} \phi \vdash \psi \;\implies\; \phi \models \psi \end{talign} \pause Hence we conclude $\phi \not\vdash \psi$. \end{exampleblock} \end{frame}