Consider the following premises. forall x_14: T_1(x_14) -> ~Q_14(x_14). forall x_14: Q_12(x_14) -> T_1(x_14). forall x_14: P_12(x_14) -> T_1(x_14). exists x_14: P_12(x_14) or Q_12(x_14). forall x_14: Q_14(x_14). Can we infer exists x_14: Q_1(x_14) from them? If possible, name the inference rules being used at each step.
 From forall x_14: T_1(x_14) -> ~Q_14(x_14), forall x_14: Q_14(x_14) we can infer forall x_14: ~T_1(x_14) via universal modus tollens. From forall x_14: P_12(x_14) -> T_1(x_14), forall x_14: Q_12(x_14) -> T_1(x_14), exists x_14: P_12(x_14) or Q_12(x_14) we can infer exists x_14: T_1(x_14) via existential disjunction elimination. From exists x_14: T_1(x_14) we can infer exists x_14: T_1(x_14) or Q_1(x_14) via existential addition. Finally, from exists x_14: T_1(x_14) or Q_1(x_14), forall x_14: ~T_1(x_14) we can infer exists x_14: Q_1(x_14) via existential disjunctive syllogism. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.