\begin{frame}{Disproving Semantic Entailment: Counter Models} \begin{exampleblock}{} \begin{malign} \formula{\forallst{x}{\existsst{y}{\,\binpred{R}{x}{y}}}} \satisfiesnot \formula{\existsst{y}{\forallst{x}{\,\binpred{R}{x}{y}}}} \end{malign} \pause We can give a model $\model{\amodel}$ such that: \begin{center} $\model{\amodel} \satisfies \formula{\forallst{x}{\existsst{y}{\,\binpred{R}{x}{y}}}}\,$, \hspace*{3ex} $\model{\amodel} \satisfiesnot \formula{\existsst{y}{\forallst{x}{\,\binpred{R}{x}{y}}}}\,$. \end{center} \pause We choose $\model{\amodel}$ as follows: \begin{center} \scalebox{0.8}{ \begin{tikzpicture}[point/.style={circle,fill=black!60!white,draw=none,inner sep=2pt}, node distance=26mm] \node (1) [point] {}; \node (2) [point, right of=1] {}; \node at (1) [yshift=4mm] {$a_1$}; \node at (2) [yshift=4mm] {$a_2$}; \begin{scope}[shorten <= 2mm, shorten >= 2mm, very thick, >=stealth] \draw [->] (1) to[out=-140,in=-40,looseness=30] (1); \draw [->] (2) to[out=-140,in=-40,looseness=30] (2); \end{scope} \node [right of=2,node distance=20mm,yshift=10mm] {\Large $\model{\model{\amodel}}$}; \draw [rounded corners=5mm,thick,dashed] ($(1) + (-14mm,12mm)$) rectangle ($(2) + (14mm,-15mm)$); \end{tikzpicture} } \end{center} \end{exampleblock} \pause\smallskip \begin{goal}{} This is a \emph{counter model}:\\ it satisfies the premise, but not the conclusion. \end{goal} \end{frame}