Consider the following premises. forall x_8: P_8(x_8) or Q_8(x_8). forall x_8: P_5(x_8) <-> W(x_8). forall x_8: ~Q_8(x_8) or S(x_8). forall x_8: ~P_5(x_8). forall x_8: ~P_8(x_8) or W(x_8). Can we infer exists 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 W(x_8) we can infer forall x_8: Q_8(x_8) or W(x_8) via universal resolution. From forall x_8: P_5(x_8) <-> W(x_8), forall x_8: ~P_5(x_8) we can infer forall x_8: ~W(x_8) via universal biconditional elimination. From forall x_8: Q_8(x_8) or W(x_8), forall x_8: ~Q_8(x_8) or S(x_8) we can infer forall x_8: W(x_8) or S(x_8) via universal resolution. Finally, from forall x_8: W(x_8) or S(x_8), forall x_8: ~W(x_8) we can infer forall x_8: S(x_8) via universal disjunctive syllogism, which contradicts exists x_8: ~S(x_8). Therefore, the answer is no.