Consider the following premises. forall x_5: P(x_5) -> S(x_5). exists x_5: P(x_5). forall x_5: S(x_5) -> P(x_5). Can we infer forall x_5: ~S(x_5) from them?
 From forall x_5: S(x_5) -> P(x_5), forall x_5: P(x_5) -> S(x_5) we can infer forall x_5: S(x_5) <-> P(x_5). From forall x_5: S(x_5) <-> P(x_5) we can infer forall x_5: P(x_5) -> S(x_5). Finally, from forall x_5: P(x_5) -> S(x_5), exists x_5: P(x_5) we can infer exists x_5: S(x_5), which contradicts forall x_5: ~S(x_5). Therefore, the answer is no.