$[10$ Pts$]$ Use logical reasoning to solve the following puzzle:

A detective has interviewed four witnesses to a crime. From the stories of the witnesses

the detective has concluded that if the butler is telling the truth then so is the cook. The

cook and the gardener are either both telling the truth or they are both lying. Either

the gardener or the handyman is telling the truth, but not both. If the handyman is

lying then so is the cook. Determine which witnesses are lying and which are telling

the truth. Define four basic propositions and use these to describe the four statements

above. Then use deductive reasoning, not truth tables, to derive the truth of each of

those basic propositions.