Consider the following premises. forall x: U(x) -> S_4(x). forall x: S_4(x) -> U(x). forall x: U(x) or ~S_4(x). Can we infer exists x: ~(U(x) -> S_4(x)) from them?
 From forall x: U(x) -> S_4(x), forall x: S_4(x) -> U(x) we can infer forall x: U(x) <-> S_4(x). From forall x: U(x) <-> S_4(x) we can infer forall x: U(x) -> S_4(x). From forall x: U(x) or ~S_4(x) we can infer forall x: S_4(x) -> U(x). From forall x: U(x) -> S_4(x), forall x: S_4(x) -> U(x) we can infer forall x: U(x) <-> S_4(x). Finally, from forall x: U(x) <-> S_4(x) we can infer forall x: U(x) -> S_4(x), which contradicts exists x: ~(U(x) -> S_4(x)). Therefore, the answer is no.