Consider the following premises. forall x_8: P_8(x_8) or Q_8(x_8). forall x_8: ~Q_8(x_8). forall x_8: ~P_4(x_8). forall x_8: ~P_8(x_8) or P_4(x_8). Can we infer forall x_8: S(x_8) from them? If possible, name the inference rules being used at each step.
 From forall x_8: P_8(x_8) or Q_8(x_8), forall x_8: ~P_8(x_8) or P_4(x_8) we can infer forall x_8: Q_8(x_8) or P_4(x_8) via universal resolution. From forall x_8: Q_8(x_8) or P_4(x_8), forall x_8: ~Q_8(x_8) we can infer forall x_8: P_4(x_8) via universal disjunctive syllogism. From forall x_8: P_4(x_8) we can infer forall x_8: P_4(x_8) or S(x_8) via universal addition. Finally, from forall x_8: P_4(x_8) or S(x_8), forall x_8: ~P_4(x_8) we can infer forall x_8: S(x_8) via universal disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.