Consider the following premises. It is not late. If Joseph were a bartender, then William is not a nurse. William is a nurse. Patricia is writing a letter if and only if it is late. Can we infer the following from them? If we can, name the inference rule being used: Joseph is a bartender.
 From the fact that Patricia is writing a letter if and only if it is late, and that it is not late we can infer that Patricia is not writing a letter via biconditional elimination. Finally, from the fact that if Joseph were a bartender, then William is not a nurse, and that William is a nurse we can infer that Joseph is not a bartender via modus tollens, which contradicts that Joseph is a bartender. Therefore, the answer is no.