Consider the following premises. forall x_12: P_12(x_12) -> P(x_12). forall x_12: P_12(x_12). exists x_12: ~P(x_12). Can we infer exists x_12: T_3(x_12) or S_2(x_12) from them? If possible, name the inference rules being used at each step.
 From forall x_12: P_12(x_12) -> P(x_12), forall x_12: P_12(x_12) we can infer forall x_12: P(x_12) via universal modus ponens. From forall x_12: P(x_12) we can infer forall x_12: P(x_12) or T_3(x_12) via universal addition. From forall x_12: P(x_12) or T_3(x_12), exists x_12: ~P(x_12) we can infer exists x_12: T_3(x_12) via existential disjunctive syllogism. Finally, from exists x_12: T_3(x_12) we can infer exists x_12: T_3(x_12) or S_2(x_12) via existential addition. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.