Consider the following premises. If it is cloudy, then John is a poet. Karen is playing tennis if and only if Patricia is writing a letter. John is a poet if and only if Karen is playing tennis. Patricia is not writing a letter. Can we infer the following from them? It is not cloudy.
 From the fact that Karen is playing tennis if and only if Patricia is writing a letter, and that Patricia is not writing a letter we can infer that Karen is not playing tennis. From the fact that John is a poet if and only if Karen is playing tennis, and that Karen is not playing tennis we can infer that John is not a poet. Finally, from the fact that if it is cloudy, then John is a poet, and that John is not a poet we can infer that it is not cloudy. Therefore, the answer is yes.