Consider the following premises. If James were a carpenter, then it is overcast. If it is overcast, then James is a carpenter. If Thomas were a poet, then it is overcast. Thomas is a poet or James is a carpenter. Can we infer the following from them? If we can, name the inference rule being used: It is not overcast.
 From the fact that if James were a carpenter, then it is overcast, and that if it is overcast, then James is a carpenter we can infer that James is a carpenter if and only if it is overcast via biconditional introduction. From the fact that James is a carpenter if and only if it is overcast we can infer that if James were a carpenter, then it is overcast via biconditional elimination. Finally, from the fact that if Thomas were a poet, then it is overcast, that if James were a carpenter, then it is overcast, and that Thomas is a poet or James is a carpenter we can infer that it is overcast via disjunction elimination, which contradicts that It is not overcast. Therefore, the answer is no.