Answer
The detective cannot determine whether $G$ and $H$ are telling the truth or lying
Work Step by Step
$B$= The butler is telling the truth
$C$ = The cook is telling the truth
$G$ = The gardener is telling the truth
$H$ - The handyman is telling the truth
Lets list the statements
1) If the butler is telling the truth, so is the cook.
2) The cook and the gardener cannot both be telling the truth.
3) The gardener and the handyman are not both lying.
4)If the handyman is telling the truth, the cook is lying.
Now, rewrite the statements.
1) $ B \to C$
2) $\neg C \lor \neg G$
3) $G \lor H$
4) $H \to \neg C$
All these four statements have to be true. Lets try to see if we can conclude whether we can conclude who is telling the truth and who is lying.
Lets start from $C$. We will first assume $C$ is true, and then that $C$ is false and check what this tells us about other variables.
I) Assume $C$ is true
If $C$ is true, H has to be false (in order for statement 4 to be true)
If $H$ is false, $G$ has to be true (in order for statement 3 to be true)
If $G$ is true, $C$ cannot be true as well, because then statement 2 will be false. Therefore $C$ would have to be false. However, since the assumption was $C$ is true, this cannot be the case.
We conclude $C$ cannot be true and all the listed statements valid.
II) Assume $C$ is false
If $C$ is false, $B$ has to be false in order for statement 1 to be true.
However, since $C$ is false, statements 2 and 4 will be true whether $H$ and $G$ are true or false.
Therefore the only thing we can conclude is $G$ and $H$ cannot both be false (in order for statement 3 to be valid).
To sum up, the possible solutions are
1) $C$ false, $B$ false, $G$ true, $H$ true.
2) $C$ false, $B$ false, $G$ true, $H$ false.
3) $C$ false, $B$ false, $G$ false, $H$ true.
Therefore, the detective cannot determine whether $G$ and $H$ are telling the truth or lying.
