\documentclass[11pt]{article}

\usepackage{homework,amssymb}

\newcommand{\ruleone}[2]{}
\newcommand{\rulethree}[4]{}

\begin{document}

\dotitle{11}{Wednesday, November 14}

\begin{list1} 

\item Read section 3.1 in van Dalen. Start reading Section 3.2, using
  the class notes to determine which parts we will focus on.

% \item Do parts (i), (ii), and (vi) of problem 1 on page 96.

\item Do problem 1 on page 96.

\item \addstar Do the following.
\begin{list2}
\item Prove $\lnot (p \liff \lnot p)$ in propositional logic.
\item Use this to prove $\lnot \ex y \fa x (S(y,x) \liff \lnot
  S(x,x))$ in first-order logic.
\end{list2}
You can take $\liff$ to be a defined in terms of $\land$ and
$\limplies$, or to be a new symbol with the rules introduced in class,
as you prefer. Hint for part (b): if suffices to show that $\fa x
(S(y,x) \liff \lnot S(x,x))$ leads to a contradiction. This is a
formalization of the Barber paradox: in a given town there is a (male)
barber who shaves every man that does not shave himself. Who shaves
the barber?

\item \addstar Do problem 1 on page 99 of van Dalen.

\item Do problems 2 and 3 on page 99.

\item \addcirc Do problem 4 on page 99. The $\limplies$ direction is
difficult. (Because the implication is {\em not} intuitionistically
valid, you will need to use RAA.) 

% By the same token, proving $\lnot \varphi \lor \lnot \psi$ from $\lnot
% (\varphi \land \psi)$ is also difficult.

\item \addstar Do problem 5 on page 99.

% \item \addcirc Do problem 6 on page 99. Here the $\leftarrow$
% direction is difficult.

% ; turn in a solution for extra credit.
%
% Note that the $\limplies$ direction is needed to prove the
% completeness theorem.

\item Do other problems on page 99 of van Dalen, for practice.

\item \addstar An old song goes, ``Everybody loves my baby, but my
baby don't love nobody but me.'' Prove that if this is true, I am my baby.

More precisely: Let $L(x,y)$ stand for ``$x$ loves $y$,'' let $b$ be a
constant denoting ``my baby,'' and $m$ be a constant denoting me. From
assumptions $\fa x L(x,b)$ and $\fa x (L(b,x) \limplies x=m)$, prove
$b=m$.

\item \addstar Do problem 1 on page 103. Here, $I_2$ is the axiom 
\[
\fa x \fa y (x=y \limplies y = x)
\]
and $I_3$ is the axiom
\[
\fa x \fa y \fa z (x=y \land y = z \limplies x = z).
\]
Do not use the equality rules! The point of the exercise is to show
that you can replace the three basic axioms of equality (reflexivity,
symmetry, and transitivity) by two axioms.

\item Do problem 3 on page 103.

\item Do other problems on page 103 for practice.
  
\item \addstar Assuming $\ph$ is any formula, prove $\varphi \lor
  \lnot \varphi$, using the $\lor$-rules on page 50 in van Dalen.
  (Hint: use a proof by contradiction.)
 
\end{list1}

\end{document}