\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}