\begin{frame}\frametitle{Model Informally (2)} \begin{exampleblock}{} \begin{talign} \formula{\existsst{y}{\forallst{x}{\,\binpred{R}{x}{y}}}} \end{talign} \pause% If we interpret $\binpred{R}{x}{y}$ as `\sentence{\black{$x$} knows \black{$y$}}', this expresses: \begin{scenter} There is somebody who is known by everybody. \end{scenter} \pause\medskip This formula has a model: \begin{scenter} \begin{tikzpicture}[node distance=15mm, dot/.style={minimum size=4mm, circle, draw=none, fill=black, inner sep=0, outer sep=1mm, text=white}] \node [dot] (1) {1}; \node [dot,below right of=1] (2) {2}; \node [dot,above right of=2] (3) {3}; \begin{scope}[->,thick] \draw (1) -- (2); \draw (3) -- (2); \draw (2) to[out=-45,in=-45-90,looseness=5] (2); \end{scope} \draw [rounded corners=5mm,thick,dashed] ($(1) + (-6mm,5mm)$) rectangle ($(3) + (6mm,-20mm)$); \end{tikzpicture} \end{scenter} \pause Here the formula is true since for $y = 2$, we have: $\formula{\forallst{x}{\,\sunpred{R}(x,y)}}$ \begin{talign} \sbinpred{R}(1,2) && \sbinpred{R}(2,2) && \sbinpred{R}(3,2) \end{talign} \end{exampleblock} \end{frame}