Consider the following premises. ~p_2. p_4 -> p_2. p_4. Can we infer q_1 from them? If possible, name the inference rules being used at each step.
 From p_4 -> p_2, ~p_2 we can infer ~p_4 via modus tollens. From p_4 we can infer p_4 or q_1 via addition. Finally, from p_4 or q_1, ~p_4 we can infer q_1 via disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.