\begin{frame} \frametitle{Modalities} \begin{goal}{} Modal logic introduces \emph{modalities} \begin{itemize} \item box $\all$ \item diamond $\some$ \smallskip \end{itemize} \end{goal} \pause \vspace{-2ex} \begin{eqnarray*} \DARKRED{\all} & &\DARKGREEN{\some}\\ \DARKRED{\mbox{\em{``Box''}}} && \DARKGREEN{\mbox{\em``Diamond''}}\\ \hline \\[-.5ex] \DARKRED{sure} && \DARKGREEN{possibly} \\ \DARKRED{always} && \DARKGREEN{sometimes} \\ \DARKRED{has\;to\;be} && \DARKGREEN{maybe} \\ \DARKRED{knows} && \DARKGREEN{believes\;is\;possible} \\ \DARKRED{guaranteed\;result} && \DARKGREEN{possible\;result} \end{eqnarray*} \end{frame}