Consider the following premises. For all x_3, if x_3 sleeps, then x_3 is playing a game. For all x_3, x_3 will sleep. For all x_3, x_3 will not play a game. Can we infer the following from them? If we can, name the inference rule being used: For all x_3, x_3 is a politician.
 From the fact that for all x_3, if x_3 sleeps, then x_3 is playing a game, and that for all x_3, x_3 will not play a game we can infer that for all x_3, x_3 will not sleep via universal modus tollens. From the fact that for all x_3, x_3 will sleep we can infer that for all x_3, x_3 will sleep or x_3 is a politician via universal addition. Finally, from the fact that for all x_3, x_3 will sleep or x_3 is a politician, and that for all x_3, x_3 will not sleep we can infer that for all x_3, x_3 is a politician via universal disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.