\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}