48/270
\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}
`