\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}