Consider the following premises. exists x_8: S_1(x_8) -> ~S(x_8). forall x_8: S(x_8) <-> Q_8(x_8). forall x_8: Q_8(x_8). Can we infer forall x_8: S_1(x_8) from them? If possible, name the inference rules being used at each step.
 From forall x_8: S(x_8) <-> Q_8(x_8) we can infer forall x_8: Q_8(x_8) -> S(x_8) via universal biconditional elimination. From forall x_8: Q_8(x_8) -> S(x_8), forall x_8: Q_8(x_8) we can infer forall x_8: S(x_8) via universal modus ponens. Finally, from exists x_8: S_1(x_8) -> ~S(x_8), forall x_8: S(x_8) we can infer exists x_8: ~S_1(x_8) via existential modus tollens, which contradicts forall x_8: S_1(x_8). Therefore, the answer is no.