\item \self Consider the propositional formula $\varphi = p \imp
    (q \imp r)$.

\begin{enumerate}
  \item Fill out the truth table for $\varphi$ and its
      subformulas.

\begin{tabular}{|c|c|c||c|c|c|}
\hline
$p$&$q$&$r$&$(q \imp r)$&$\varphi=p \imp (q \imp r)$\\
\hline
\hline
\textbf{F} &\textbf{F} &\textbf{F} & & \\
\hline
\textbf{F} &\textbf{F} &\textbf{T} & & \\
\hline
\textbf{F} &\textbf{T} &\textbf{F} & & \\
\hline
\textbf{F} &\textbf{T} &\textbf{T} & & \\
\hline
\textbf{T} &\textbf{F} &\textbf{F} & & \\
\hline
\textbf{T} &\textbf{F} &\textbf{T} & & \\
\hline
\textbf{T} &\textbf{T} &\textbf{F} & & \\
\hline
\textbf{T} &\textbf{T} &\textbf{T} & & \\
\hline
\end{tabular}

\item Is $\varphi$ satisfiable?
\item Give a formula $\psi$ that is semantically equivalent to
    $\varphi$, but does not use the ``$\imp$'' connective.
\item How can you check whether $\psi$ is semantically equivalent
    to $\varphi$?

\end{enumerate}
