Consider the following premises. forall x: P(x) -> S(x). forall x: Q(x) <-> W_4(x). exists x: ~Q(x). forall x: S(x) -> W_4(x). Can we infer exists x: ~P(x) from them?
 From forall x: Q(x) <-> W_4(x), exists x: ~Q(x) we can infer exists x: ~W_4(x). From forall x: P(x) -> S(x), forall x: S(x) -> W_4(x) we can infer forall x: P(x) -> W_4(x). Finally, from forall x: P(x) -> W_4(x), exists x: ~W_4(x) we can infer exists x: ~P(x). Therefore, the answer is yes.