\begin{frame}
  \frametitle{Formulas and Logic Connectives}

  In predicate logic, the role of propositional variables is taken over by
  the \aemph{atomic formulas} 
  with object/predicate-structure:
  \begin{goal}{}
  The \emph{atomic formulas} are predicates over objects:
  \begin{itemize}
    \item $P(x)$
    \item $R(x,y)$
  \end{itemize}
  \end{goal}
  \pause

  \begin{goal}{}
    Logic connectives $\;\neg,\; \vee,\; \wedge,\; \to\;$ keep their role.
  \end{goal}
  \pause
  
  \begin{block}{}
    We can build \emph{formulas} using propositional connectives,
    starting from the smallest building blocks of atomic formulas:
    \begin{itemize}
    \pause
      \item $P(x)$\quad\quad\pause\hint{($x$ is green)}
    \pause
      \item $P(x) \wedge R(x,y)$\quad\quad\pause\hint{($x$ is green and $x$ knows $y$)}
    \pause
      \item $R(x,y) \to \neg R(y,x)$\quad\quad\pause\hint{(if $x$ knows $y$ then $y$ does not know $x$)}
    \end{itemize}
  \end{block}
  \pause
  
  \begin{goal}{Predicate logic is more expressive}
    In propositional logic, we could only state $p$, $p \wedge q$, $p \to \neg q$.
  \end{goal}
\end{frame}