\problemdata                                                             
        {propositions} % latex-friendly label for the prob.
        {Translating english to symbols; making deductions } % The topic(s) of the problem content.  
        {Spring 97} % Source (if known)                              
        {S01, Tutorial 1, 1} % Usage (list ones you are aware of). 
        {Tina Wang, S02} % Last revision info (author, date).   
                                                                         
                                                                         
\begin{problem}           
\noindent Cast the following into propositional symbols, and verify or        
refute:\\                                                                     
                                                                              
{\em \noindent If Superman were able and willing to prevent evil, he would do 
  so. If Superman were unable to prevent evil, he would be impotent.          
  If he were unwilling to prevent evil, he would be malevolent.               
  Superman does not prevent evil. If Superman exists, he is neither           
  impotent nor malevolent. Therefore, Superman does not exist.}\\             
                                                                              
A: Superman were able to prevent evil. \\                                     
W: Superman were willing to prevent evil. \\                                  
I: Superman is impotent. \\                                                   
M: Superman is malevolent. \\                                                  
P: Superman prevents evil. \\                                                  
E: Superman exists. \\                                                         
                                                                               
\solution{                                                                     
\medskip                                                                       
\textbf{Solution:}\\                                                           
We have the following assumptions:\\                                           
\begin{tabular}{lll}                                                           
1 & $A \Land W \implies P$ &                                                
    If Superman were able and willing to prevent evil, he would do so.\\       
2 & $\bar A \implies I$ &                                                   
    If Superman were unable to prevent evil, he would be impotent. \\          
3 & $\bar W \implies M$ &                                                   
    If he were unwilling to prevent evil, he would be malevolent. \\           
4 & $\bar P$ & Superman does not prevent evil. \\                              
5 & $E \implies \bar (M \Lor I)$ &                                          
    If Superman exists, he is neither impotent nor malevolent.                 
\end{tabular}\\                                              
We now use these assumptions to prove $\bar E$, that Superman does not exist.\\
\begin{tabular}{lll}                                           
  (i)   & $\bar(A \Land W)$ & Assumptions 1 and 4; modus tollens. \\      
  (ii)  & $\bar A \Lor \bar W$ & DeMorgan. \\                        
  (iii) & $(\bar A \Lor \bar W) \implies (I \Lor M)$ & Assumptions 2 and 3. \\
  (iv) & $I \Lor M$ & (ii) and (iii); modus ponens. \\       
  (v)  & $\bar E$ & Assumption 5 together with (iv); modus tollens.  
\end{tabular}\\
                                         
Step (iii) above uses the fact that                                         
\begin{displaymath}                                                          
  \bigl((a \rightarrow b) \land (c \rightarrow d)\bigr)                      
  \rightarrow                                                                
  \bigl((a \lor c) \rightarrow (b \lor d)\bigr),     
\end{displaymath}                                                             
which should be proven as a separate lemma.}

\end{problem}     
                                                                           
                                 