9/270
\begin{frame}
  \frametitle{Usefulness of Semantics}

  Find a valuation $\saval$ such that:
  \begin{align}
    \aval{\formula{\logimp{\logor{\prop{p}}{\prop{q}}}{\prop{r}}}} & = \True
    \label{v:1}\\
    \aval{\formula{\logimp{\logand{\prop{q}}{\prop{r}}}{\lognot{\prop{p}}}}} & = \True
    \label{v:2}\\
    \aval{\formula{\lognot{(\logand{\prop{r}}{\lognot{\prop{q}})}}}} & = \False
    \label{v:3}
  \end{align}
  \mpause[1]{For any such valuation $\saval$ we find by \emph{semantic reasoning}:}
  \begin{align}
    \mpause{\aval{\formula{\logand{\prop{r}}{\lognot{\prop{q}}}}} &= \True} \notag\\
    \mpause{\aval{\formula{\prop{r}}} & = \True \notag }\\
    \mpause{\aval{\formula{\lognot{\prop{q}}}} & = \True} \notag \\
    \mpause{\aval{\formula{\prop{q}}} & = \False \notag }\\[-2.75ex]\notag
  \end{align}%
  \pause[7]%
  Every valuation with $\aval{\prop{r}} = \True$ and $\aval{\prop{q}} = \False$
  fulfils \eqref{v:1}, \eqref{v:2}, \eqref{v:3}
\end{frame}