\begin{frame} \frametitle{Semantic Entailment} \begin{goal}{Semantic Entailment / Consequence} \vspace{-1ex} \begin{talign} \quad\quad \alpha_1,\ldots,\alpha_n \;\models \; \beta \end{talign} means \begin{center} Whenever $\alpha_1,\ldots,\alpha_n$ are all true, $\beta$ is also true. \end{center} \end{goal} \bigskip \pause \begin{exampleblock}{} Do we have \quad $q \;\models\; p \to q$ \quad ? \pause \begin{center} \begin{tabular}{|c|c|c|} \hline \thd $p$ & \thd $q$ & \thd $p \to q$ \\ \hline $\F$ & $\F$ & $\T$\\ \hline $\F$ & \malert{1}{4}{$\T$} & \malert{2}{3}{$\T$}\\ \hline $\T$ & $\F$ & $\F$\\ \hline $\T$ & \malert{1}{4}{$\T$} & \malert{2}{3}{$\T$}\\ \hline \end{tabular} \end{center} \pause\pause\pause Whenever $q$ is $\T$ also $p \to q$ is $\T$. \pause Hence: $q \;\models\; p \to q$\;. \end{exampleblock} \end{frame}