\begin{frame} \frametitle{Usefulness of Semantics} Can we prove the following by natural deduction? \begin{center} $ \formula{\logimp{\logor{\prop{p}}{\prop{q}}}{\prop{r}}}, \;\; \formula{\logimp{\logand{\prop{q}}{\prop{r}}}{\lognot{\prop{p}}}} \;\;\derives\;\; \formula{\lognot{(\logand{\prop{r}}{\lognot{\prop{q}})}}} $ \end{center}\pause{} What if it is not derivable? \pause \alert{Might be difficult to show.} \pause\smallskip \begin{goal}{} Using the \emph{soundness and completeness theorem} we could concentrate on the \emph{equivalent semantic entailment}: \begin{center} $ \formula{\logimp{\logor{\prop{p}}{\prop{q}}}{\prop{r}}}, \;\; \formula{\logimp{\logand{\prop{q}}{\prop{r}}}{\lognot{\prop{p}}}} \;\;\satisfies\;\; \formula{\lognot{(\logand{\prop{r}}{\lognot{\prop{q}})}}} $ \end{center}\pause{} and actually demonstrate: \begin{center} $ \logimp{\logor{\prop{p}}{\prop{q}}}{\prop{r}}, \;\; \logimp{\logand{\prop{q}}{\prop{r}}}{\lognot{\prop{p}}} \;\;\satisfiesnot\;\; \lognot{(\logand{\prop{r}}{\lognot{\prop{q}})}} $ \end{center} \end{goal} \pause\smallskip That is, find a valuation $\saval$ such that \begin{align*} \aval{\formula{\logimp{\logor{\prop{p}}{\prop{q}}}{\prop{r}}}} & = \True \\ \aval{\formula{\logimp{\logand{\prop{q}}{\prop{r}}}{\lognot{\prop{p}}}}} & = \True \\ \aval{\formula{\lognot{(\logand{\prop{r}}{\lognot{\prop{q}})}}}} & = \False \end{align*} \end{frame}