Consider the following premises. If Barbara climbs a mountain, then William is a janitor. Barbara is climbing a mountain. If Richard listens to a song, then William is a janitor. Can we infer the following from them? If we can, name the inference rule being used: William is not a janitor.
 From the fact that Barbara is climbing a mountain we can infer that Barbara is climbing a mountain or Richard is listening to a song via addition. Finally, from the fact that if Barbara climbs a mountain, then William is a janitor, that if Richard listens to a song, then William is a janitor, and that Barbara is climbing a mountain or Richard is listening to a song we can infer that William is a janitor via disjunction elimination, which contradicts that William is not a janitor. Therefore, the answer is no.