Consider the following premises. exists z: ~P_1(z). forall z: S(z) -> P_1(z). forall z: S(z) or ~P_1(z). Can we infer exists z: ~S(z) from them?
 From forall z: S(z) or ~P_1(z) we can infer forall z: P_1(z) -> S(z). From forall z: P_1(z) -> S(z), forall z: S(z) -> P_1(z) we can infer forall z: P_1(z) <-> S(z). Finally, from forall z: P_1(z) <-> S(z), exists z: ~P_1(z) we can infer exists z: ~S(z). Therefore, the answer is yes.