\begin{frame}{Free Variables and Properties}
  \begin{goal}{}
  Formulas with free variables express properties and relations.
  \end{goal}
  \pause\medskip
  
  \begin{block}{}
    \begin{malign}
      \formula{\unpred{L}{x}} & \funin \sentence{\black{$x$} is a logician}
      & 
      \formula{\const{r}} & \funin \sentence{Rosalie}
      \\ 
      \formula{\binpred{K}{x}{y}} & \funin \sentence{\black{$x$} knows \black{$y$}}
      & 
      \formula{\const{j}} & \funin \sentence{Jan}
    \end{malign}
  \end{block}\vspace{-2ex}
  %
  \begin{talign}
    \uncover<2->{\formula{\unpred{L}{\freevar{x}}}}  
      & \hspace*{3ex} \uncover<3->{\sentence{$\black{x}$ is a logician}}
    \\[0.75ex]  
    \uncover<4->{\formula{\logand{\unpred{L}{\freevar{x}}}{\unpred{C}{\freevar{x}}}}}
      & \hspace*{3ex} \uncover<5->{\sentence{$\black{x}$ is a clever logician}}
    \\[0.75ex]  
    \uncover<6->{%
    \formula{\logand{\binpred{K}{\const{j}}{\freevar{y}}}
                    {\unpred{L}{\freevar{y}}}}}
      & \hspace*{3ex} \uncover<7->{\sentence{Jan knows $\black{y}$, and $\black{y}$ is a logician}} 
    \\[0.75ex] 
    \uncover<8->{%  
    \formula{\logand{\binpred{K}{\freevar{x}}{\freevar{y}}}
                    {\unpred{L}{\freevar{y}}}}}
      & \hspace*{3ex} \uncover<9->{\sentence{$\black{x}$ knows $\black{y}$, and $\black{y}$ is a logician}}  
    \\[0.75ex] 
    \uncover<10->{%  
    \formula{\existsst{x}{(\logand{\binpred{K}{\const{j}}{x}}
                                  {\unpred{L}{x}})}}}
      & \hspace*{3ex} \uncover<11->{\sentence{Jan knows a logician}}
    \\[0.75ex]  
    \uncover<12->{
    \formula{\existsst{x}{(\logand{\binpred{K}{\freevar{y}}{x}}
                                  {\unpred{L}{x}})}}}
      & \hspace*{3ex} \uncover<13->{\sentence{\black{$y$} knows a logician}}
    \\[0.75ex]  
    \uncover<14->{
    \formula{\forallst{y}{\existsst{x}{(\logand{\binpred{K}{y}{x}}
                                               {\unpred{L}{x}})}}}}
       & \hspace*{3ex} \uncover<15->{\sentence{Everybody knows a logician}}
  \end{talign}
\end{frame}