\begin{frame}
  \frametitle{Tautologies and Semantic Equivalence}
  
  \begin{block}{Tautology}
    A formula $\phi$ is a tautology if it holds without premises:
    \begin{talign}
      \models \phi \quad \iff\quad \text{$\phi$ is a tautology}
    \end{talign}
  \end{block}
  \pause
  \begin{exampleblock}{}
    \begin{malign}
      \models p \vee \neg p
    \end{malign}
  \end{exampleblock}
  \pause\medskip
  
  \begin{block}{Semantic Equivalence}
    \vspace{-1.5ex}
    \begin{talign}
      \alpha \equiv \beta \quad \iff\quad \alpha \models \beta \;\text{ and }\; \beta \models \alpha
    \end{talign}
    (In other words: $\alpha$ and $\beta$ have the same truth table.)
  \end{block}
  \pause
  Note that $\equiv$ is an equivalence relation:
  \begin{itemize}
  \pause
    \item reflexive,
  \pause
    \item symmetric,
  \pause
    \item transitive.
  \end{itemize}
\end{frame}