Consider the following premises. forall x_4: P_4(x_4) -> ~Q_4(x_4). exists x_4: Q_4(x_4). forall x_4: S(x_4) -> P_4(x_4). Can we infer exists x_4: ~S(x_4) from them?
 From forall x_4: P_4(x_4) -> ~Q_4(x_4), exists x_4: Q_4(x_4) we can infer exists x_4: ~P_4(x_4). Finally, from forall x_4: S(x_4) -> P_4(x_4), exists x_4: ~P_4(x_4) we can infer exists x_4: ~S(x_4). Therefore, the answer is yes.