Consider the following premises. forall x: W(x) -> S(x). forall x: W(x) or ~S(x). forall x: S(x) -> S_4(x). forall x: ~S_4(x). Can we infer exists x: W(x) from them? If possible, name the inference rules being used at each step.
 From forall x: W(x) or ~S(x) we can infer forall x: S(x) -> W(x) via universal rewriting. From forall x: S(x) -> S_4(x), forall x: ~S_4(x) we can infer forall x: ~S(x) via universal modus tollens. From forall x: W(x) -> S(x), forall x: S(x) -> W(x) we can infer forall x: W(x) <-> S(x) via universal biconditional introduction. Finally, from forall x: W(x) <-> S(x), forall x: ~S(x) we can infer forall x: ~W(x) via universal biconditional elimination, which contradicts exists x: W(x). Therefore, the answer is no.