Propositional logic problem (getting wrong answer)

Question

Q3. Sharky, a leader of the underworld, was killed by one of his own band of four henchmen. Detective Sharp interviewed the men and determined that all were lying except for one. He deduced who killed Sharky on the basis of the following statements:

a. Socko: Lefty killed Sharky.

b. Fats: Muscles didn't kill Sharky.

c. Lefty: Muscles were shooting craps with Socko when Sharky was knocked off.

d. Muscles: Lefty didn't kill Sharky.

Who did kill Sharky?

Solution:

P= Socko: Lefty killed Sharky

Q= Fats: Muscles kill Sharky.

R= Lefty: Muscles were shooting craps with Socko when Sharky was knocked off.

S= Muscles: Lefty killed Sharky.

p

~q

conclusion: p and ~q (conjunction rule )

r

~s

conclusion: r and ~s (conjunction rule )

What and where I am doing wrong?

Answer 1

If we can use the fact that all but one of the henchmen are lying, then it is possible to solve this problem in the following way. First we translate it to more mathematical terms. Let:

• $A$ be the proposition that Socko did it
• $B$ be the proposition that Fats did it
• $C$ be the proposition that Lefty did it
• $D$ be the proposition that Muscles did it

Then the four propositions of the four henchmen state:

• $C$
• $\neg D$
• $\neg A\wedge \neg D$
• $\neg C$

Now we do casework on which of the four propositions is true (which means we negate the other three):

1. $C,D,A\vee D,C$
2. $\neg C,\neg D,A\vee D,C$
3. $\neg C,D,\neg A\wedge\neg D,C$
4. $\neg C,D,A\vee D,\neg C$

The second and third possibilities have the contradictions $C$ and $\neg C$ so they are not possible. The first list is impossible because it means $C$ and $D$ both did it. So it must be the last possibility, where we find that $D$ (Muscles) did it.

This is a rather contrived problem though because in real life we would not know that all but one are lying without knowing which one is lying, and we would not know that several of the henchmen were not involved instead of just one.