Consider the following premises. For all z, z will not listen to a song. For all z, z will play tennis. There is at least one z for which z will listen to a song or z is a janitor. For all z, z is a politician. Can we infer the following from them? If we can, name the inference rule being used: For all z, either z is a janitor or z will play tennis are not true.
 From the fact that there is at least one z for which z will listen to a song or z is a janitor, and that for all z, z will not listen to a song we can infer that there is at least one z for which z is a janitor via existential disjunctive syllogism. Finally, from the fact that for all z, z will play tennis, and that for all z, z is a politician we can infer that for all z, z will play tennis and z is a politician via universal conjunction, which contradicts that For all z, either z is a janitor or z will play tennis are not true. Therefore, the answer is no.