Consider the following premises. forall x: ~S_1(x). forall x: W(x) -> ~U(x). forall x: S_1(x) or U(x). Can we infer exists x: W(x) from them? If possible, name the inference rules being used at each step.
 From forall x: S_1(x) or U(x), forall x: ~S_1(x) we can infer forall x: U(x) via universal disjunctive syllogism. Finally, from forall x: W(x) -> ~U(x), forall x: U(x) we can infer forall x: ~W(x) via universal modus tollens, which contradicts exists x: W(x). Therefore, the answer is no.