Consider the following premises. forall z: ~S_4(z). q <-> S_4(a_3). q. Can we infer w from them? If possible, name the inference rules being used at each step.
 From q we can infer q or w via addition. From forall z: ~S_4(z) we can infer ~S_4(a_3) via universal instantiation. From q <-> S_4(a_3), ~S_4(a_3) we can infer ~q via biconditional elimination. Finally, from q or w, ~q we can infer w via disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.