\begin{frame} \frametitle{Logical Reasoning} \begin{goal}{} \emph{From} $\alpha_1,\ldots,\alpha_n$ \emph{follows} $\beta$. \medskip This is denoted as: \begin{talign} \alpha_1,\ldots,\alpha_n \;\therefore\; \beta \end{talign} \end{goal} \pause \begin{exampleblock}{} \vspace{-1ex} \begin{tgather} p \wedge \neg q \;\therefore\; \neg q\\[.5ex] \mpause[1] { p \to (q \to r), \; p, \; \neg r \;\therefore\;} \mpause{ \neg q } \end{tgather} \end{exampleblock} \pause\pause\pause \bigskip \begin{goal}{Two Important Variants} \vspace{-1ex} \begin{talign} \text{semantic} & \quad\quad \alpha_1,\ldots,\alpha_n \;\models \; \beta \\ \text{syntactic} & \quad\quad \alpha_1,\ldots,\alpha_n \;\vdash \; \beta \end{talign} \end{goal} \end{frame}