Consider the following premises. exists x: Q(x) -> R(x). forall x: Q(x). forall x: P(x) -> R(x). forall x: S_2(x) -> P(x). Can we infer forall x: ~R(x) from them? If possible, name the inference rules being used at each step.
 From forall x: S_2(x) -> P(x), forall x: P(x) -> R(x) we can infer forall x: S_2(x) -> R(x) via universal transitivity. From forall x: Q(x) we can infer forall x: Q(x) or S_2(x) via universal addition. Finally, from exists x: Q(x) -> R(x), forall x: S_2(x) -> R(x), forall x: Q(x) or S_2(x) we can infer exists x: R(x) via existential disjunction elimination, which contradicts forall x: ~R(x). Therefore, the answer is no.