Consider the following premises. Jessica is not driving a car. Jessica is driving a car if and only if Karen is a doctor. If Barbara writes a letter, then Karen is a doctor. If Karen were a doctor, then Barbara is writing a letter. Can we infer the following from them? Barbara is not writing a letter.
 From the fact that if Karen were a doctor, then Barbara is writing a letter, and that if Barbara writes a letter, then Karen is a doctor we can infer that Karen is a doctor if and only if Barbara is writing a letter. From the fact that Jessica is driving a car if and only if Karen is a doctor, and that Jessica is not driving a car we can infer that Karen is not a doctor. Finally, from the fact that Karen is a doctor if and only if Barbara is writing a letter, and that Karen is not a doctor we can infer that Barbara is not writing a letter. Therefore, the answer is yes.