Consider the following premises. forall x_5: W(x_5) and Q_5(x_5). forall x_5: ~W(x_5). Can we infer forall x_5: R_4(x_5) from them? If possible, name the inference rules being used at each step.
 From forall x_5: W(x_5) and Q_5(x_5) we can infer forall x_5: W(x_5) via universal simplification. From forall x_5: W(x_5) we can infer forall x_5: W(x_5) or R_4(x_5) via universal addition. Finally, from forall x_5: W(x_5) or R_4(x_5), forall x_5: ~W(x_5) we can infer forall x_5: R_4(x_5) via universal disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.