Consider the following premises. If Joseph were a doctor, then it is raining. Joseph is not a doctor. If it rains, then Joseph is a doctor. It is raining or it is late. Can we infer the following from them? It is late.
 From the fact that if it rains, then Joseph is a doctor, and that if Joseph were a doctor, then it is raining we can infer that it is raining if and only if Joseph is a doctor. From the fact that it is raining if and only if Joseph is a doctor, and that Joseph is not a doctor we can infer that it is not raining. Finally, from the fact that it is raining or it is late, and that it is not raining we can infer that it is late. Therefore, the answer is yes.