Consider the following premises. If Joseph were a musician, then it is cloudy. If Karen writes a letter, then Joseph is a musician. It is not cloudy. If it is cloudy, then Joseph is a musician. Can we infer the following from them? Karen is not writing a letter.
 From the fact that if it is cloudy, then Joseph is a musician, and that if Joseph were a musician, then it is cloudy we can infer that it is cloudy if and only if Joseph is a musician. From the fact that it is cloudy if and only if Joseph is a musician, and that it is not cloudy we can infer that Joseph is not a musician. Finally, from the fact that if Karen writes a letter, then Joseph is a musician, and that Joseph is not a musician we can infer that Karen is not writing a letter. Therefore, the answer is yes.