\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}