Question

A detective has interviewed four witnesses to a crime. From the stories of the witnesses the detective has conclued that if the butler is telling the truth then so is the cook; the cook and the gardener cannot both be telling the truth; the gardener and the handyman are not both lying; and if the handyman is telling the truth then the cook is lying. For each of the four witnesses, can the detective deermine whether that person is telling the truth or lying? Explain your reasoning.

Short answer

The butler and the cook are lying, and it is unknown if gardener and handyman tell truth.

Step-by-step solution

Tips

Detective a person, especially a police officer, whose occupation is to investigate and solve crimes. Crimean illegal act for which someone can be punished by the government especially: a gross violation of law.

Gardener a person who tends and cultivates a garden as a pastime or for a living.

Find the person is lying or truth

Find the person is lying or truth:

According to the provided conditions, we can build truth table with only 3 possible rows in it; any other combination will contradict conditions of the task.

$\mathrm{B}$	$\mathrm{C}$	$\mathrm{G}$	$\mathrm{H}$
$\mathrm{F}$	$\mathrm{F}$	$\mathrm{F}$	$\mathrm{T}$
$\mathrm{F}$	$\mathrm{F}$	$\mathrm{F}$	$\mathrm{T}$
$\mathrm{F}$	$\mathrm{F}$	$\mathrm{T}$	$\mathrm{F}$

From the information provided in the task and respectively from the truth table we cannot determine if person telling the truth or lying for all 4 witnesses, because 3 different solutions are possible. However, we can definitely say that the butler and cock are lying.