Logic Puzzle (Valid and Invalid Arguments)

Question

I have been given a logic puzzle and I am having a tough time figuring out how to set it up and solve. Here is the puzzle:
a) The Statement "If Dr. Jones did not commit the murder then neither Ms. Scarlet nor Mr. Green committed the murder" is false. I interpret that as: $$ \lnot D \to (\lnot S \lor \lnot G)$$ Now isn't the ONLY way for conditional statements to be false is if the hypothesis is true and the condition is false? Thus leaving $ \lnot D $
b) Either Mr.Green did not commit the murder or the weapon was a candlestick. I read that as: $$ \lnot G\lor C $$
c) If the weapon was a candlestick then Dr. Jones commited the murder. That reads $$ C \to D $$ Question is who committed the murder? The thing that I am having most trouble with is figuring out what laws to use in order to solve this. A point in the right direction would be greatly appreciated.

Answer 1

Your deductions so far are correct, and you already know Dr. Jones did not commit the murder. Condition (c) then tells you the weapon was not a candlestick, and finally condition (b) tells you who committed the murder (since the other alternative, "the murder weapon was a candlestick", has already been shown to be false).

Answer 2

The correct interpretation of statement a is

$$\lnot (\lnot D \to (\lnot S \land \lnot G))$$ which is equivalent to $$\lnot D \land (S \lor G)$$. This means $S$ or $G$ committed the murder, not the Dr.; and by the contrapositive of c, that the candlestick was not used in the murder. Then by b, $S$ is responsible.