Consider the following premises. forall z: S_3(z) <-> W(z). forall z: W(z). exists z: ~S_3(z). Can we infer exists z: Q_4(z) from them? If possible, name the inference rules being used at each step.
 From forall z: S_3(z) <-> W(z), exists z: ~S_3(z) we can infer exists z: ~W(z) via existential biconditional elimination. From forall z: W(z) we can infer forall z: W(z) or Q_4(z) via universal addition. Finally, from forall z: W(z) or Q_4(z), exists z: ~W(z) we can infer exists z: Q_4(z) via existential disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.