\begin{frame}
  \frametitle{Predicate Logic}

  \begin{goal}{}
    In propositional logic there are:
    \begin{itemize}
      \item propositional variables $p,q,r,\ldots$ that can be $\T$ or $\F$
    \end{itemize}
    \medskip
    \pause

    In predicate logic there are:
    \begin{itemize}
    \pause
      \item variables $a,b,\ldots$ that represent \aemph{objects} (or \aemph{individuals}),
    \pause
      \item \aemph{predicates} $P(x)$ or \aemph{relations} $R(x,y)$ on the objects \\
    \end{itemize}
  \end{goal}
  \pause
  \smallskip
  
  \begin{exampleblock}{}
  \pause
    $1$-ary predicates like $P(\_)$ can express properties of objects:
    \begin{itemize}
      \item e.g. $P(x)$ can express that `$x$ is green'
    \end{itemize} 
  \pause
    $2$-ary predicates like $R(\_,\_)$ can express relations:
    \begin{itemize}
      \item e.g. $R(x,y)$ can express `$x$ knows $y$'.
    \end{itemize}
  \end{exampleblock}
  \pause
  \smallskip
  
  \begin{goal}{}
    Name natural examples of $3$-ary predicates?
    \pause
    \begin{itemize}
      \item $L(x,y,z)$ = $x,y,z$ are points on same line in the plane
    \end{itemize}
  \end{goal}
\end{frame}