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\begin{frame}
\frametitle{Examples: Semantics Intuitive}

\begin{goal}{Semantics Intuitive}
The semantics of a predicate logic formula are \emph{models}.
\medskip\pause

A \emph{model} for a formula consists of
\begin{itemize}
\item a \emph{universe} of objects/individuals
\item \emph{predicates} / \emph{relations} over the universe
\end{itemize}
such that the formula holds in this model.
\end{goal}
\pause

\begin{exampleblock}{}
\vspace{-1ex}
\begin{talign}
\myex{x}{P(x)}
\end{talign}
\pause
This formula has a \emph{model}; for example
\begin{itemize}
\item universe $\{1,2,3\}$
\item $P(1)$,\; $\neg P(2)$,\; $\neg P(3)$
\end{itemize}
\medskip
\pause

\begin{minipage}{.6\textwidth}
In a picture, this looks as follows:
\begin{itemize}
\item a \aemph{green dot} indicates that $P$ is $\T$
\end{itemize}
\bigskip
\end{minipage}
\begin{minipage}{.39\textwidth}
\centerline{
\begin{tikzpicture}[dot/.style={minimum size=4mm, circle, draw=none, fill=black, inner sep=0, outer sep=1mm, text=white}]
\node [dot,fill=mgreen] (1) {1};
\node [dot,below right of=1] (2) {2};
\node [dot,above right of=2] (3) {3};
\end{tikzpicture}
}
\end{minipage}
\end{exampleblock}
\end{frame}