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\section*{Introduction}
This group assignment wraps up the \textbf{Deduction Through the Ages}
project.  You will work, in your group, on some of the questions in
the readings and look at using modern notation with \emph{truth
  tables} to \textbf{prove} some logical equivalences.

\section*{Note}
\begin{itemize}
\item By \textbf{design}, this assignment includes questions about
  topics that you have not yet seen in lecture. This encourages
  \emph{creative} exploration and \emph{constructing} your own mental
  models.
\item Your group is expected to work together on \emph{each}
  question. I \textbf{must} hear the group discussing the questions
  and the answers. Remaining silent, even if you are following along,
  is choosing \emph{not} to participate.

  The goal is \textbf{not} to fill in the ``right'' answers as quickly
  as possible but rather to take the time to read, think, and discuss
  the material \textbf{together}. That requires communication and I
  must \emph{witness} the communication.
\item Constructing your own model is \textbf{short-circuited} by
  reading or parroting another thinker's answer. This means you
  should, as much as possible, use \emph{recall} of class material,
  looking up definitions in \emph{notes} or the \emph{textbook} when
  necessary, and \emph{avoid} just typing the question into a search
  engine on the Web. If you want more information, feel free to search
  \textbf{after} the group work is finished.
\item Each student in the group has more or less experience with the
  topics in the assignment. If you have greater familiarity
  with the topics, \emph{please} hold back a little bit so others can
  engage with building their own knowledge. You, too, must engage
  because there is always more to learn.
\end{itemize}

\section*{Assignment Goals}
\paragraph{Learning Outcomes} After completing this group assignment,
each student is expected to be able to
\begin{itemize}
  \item Be able to identify Wittgetstein's \emph{truth grounds} for
    logical expressions.
  \item Prove \textbf{deMorgan's Law} with a truth table.
  \item Rewrite an arbitrary \emph{implication} as an equivalent
    \emph{disjunction} and an equivalent \emph{conjunction}.
\end{itemize}

\section*{Procedure}

\textbf{Get out paper.} The group will turn in \emph{one}
document. Make sure all participating members' names are on the
page.

\noindent
\textbf{Copy each question before the answer.} This documents the
answer and makes the page a stand-alone study guide.

\noindent
\textbf{Assign Roles.} Students should take roles they
have not held recently (or, perhaps, ever):

\begin{quote}
\begin{description}
\item[Manager] Move discussion forward.
\item[Recorder] Writes the report that will be turned in.
\item[Reflector] Monitor that everyone gets heard and is caught
  up. (This is a \textbf{group} obligation, really.)
\item[Speaker] (Combine w/ \textbf{Reflector} if there are not four
  group members.) Asks the facilitator questions and communicates what
  the team has done.
\end{description}
\end{quote}
\noindent
\textbf{Answer these questions.} The numbers are from the
\textbf{Deduction Through the Ages} project.


\begin{questions}
\question{} Dr Ladd is a stickler about the formatting of a
  \emph{proof}. If you were asked to prove that $(a \land b) \lor (c
  \land b)$ is logically equivalent to $(a \lor c) \land b$, what
  would the \textbf{first} and the \textbf{last} lines of the proof
  be?
\question{} Comparing the equivalence in the previous question to
  rules for rewriting regular \emph{algebraic} expressions and
  assuming the equivalence is true, what name might you give to the
  rule?
\question{} Wittgenstein introduced the term \emph{truth-grounds} for
  a proposition to indicate the ``truth-possibilities of its
  truth-arguments that make it true''. (Wittgenstein's writing is nothing if not
  difficult to decipher.) What does \emph{truth-grounds} mean?

\question{} \label{question:truth-grounds} Find the truth-grounds for each of the following logic
  expressions, written in \emph{modern} notation.
  \begin{parts}
  \part{} $p \lor q$
  \part{} $p \land q$
  \part{} $\neg r \lor \neg s$
  \part{} $t \limplies u$
  \part{} $\neg t \lor u$
  \part{} $v \lor \neg w$
  \part{} $\neg (t \land \neg u)$
  \part{} $\neg (r \land s)$
  \end{parts}

\question{} Make note of which of the expressions in the above have
  the same truth-grounds.

\question{} Write a \emph{truth table} for each of the following
  expressions. Which compound propositions are \emph{logically equivalent}
  ($\equiv$)? Compare to problem~\ref{question:truth-grounds} above.
  \begin{parts}
  \part{} $r \land s$
  \part{} $\neg (r \land s)$
  \part{} $\neg r \lor \neg s$
  \end{parts}

\question{} Can an \emph{inclusive} \textbf{or} statement always be
  rewritten as a logically equivalent \textbf{if-then} statement?
  Justify your answer with a reference to appropriate truth tables.

\question{} \textbf{Prove} one of \emph{deMorgan's Laws}: the negation
  of a \emph{disjunction} is equivalent to the \emph{conjunction} of
  the negations of the terms. Write a full proof and use a truth table.

\end{questions}

\end{document}

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