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