Consider the following premises. forall x_12: P_12(x_12). ~P_12(a_12). Can we infer w from them? If possible, name the inference rules being used at each step.
 From forall x_12: P_12(x_12) we can infer P_12(a_12) via universal instantiation. From P_12(a_12) we can infer P_12(a_12) or w via addition. Finally, from P_12(a_12) or w, ~P_12(a_12) 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.