Consider the following premises. forall z: S_1(z). exists z: P(z). Can we infer forall z: ~P(z) from them?
 From exists z: P(z), forall z: S_1(z) we can infer exists z: P(z) and S_1(z). Finally, from exists z: P(z) and S_1(z) we can infer exists z: P(z), which contradicts forall z: ~P(z). Therefore, the answer is no.