Consider the following premises. forall z: P(z) -> Q(z). exists z: Q(z) <-> S_4(z). forall z: P(z). Can we infer forall z: ~Q(z) from them? If possible, name the inference rules being used at each step.
 From exists z: Q(z) <-> S_4(z) we can infer exists z: S_4(z) -> Q(z) via existential biconditional elimination. From forall z: P(z) we can infer forall z: P(z) or S_4(z) via universal addition. Finally, from forall z: P(z) -> Q(z), exists z: S_4(z) -> Q(z), forall z: P(z) or S_4(z) we can infer exists z: Q(z) via existential disjunction elimination, which contradicts forall z: ~Q(z). Therefore, the answer is no.