Consider the following premises. u_2 -> S(e). forall x_4: S(x_4) -> ~P_1(x_4). forall x_4: P_1(x_4). Can we infer u_2 from them? If possible, name the inference rules being used at each step.
 From forall x_4: S(x_4) -> ~P_1(x_4), forall x_4: P_1(x_4) we can infer forall x_4: ~S(x_4) via universal modus tollens. From forall x_4: ~S(x_4) we can infer ~S(e) via universal instantiation. Finally, from u_2 -> S(e), ~S(e) we can infer ~u_2 via modus tollens, which contradicts u_2. Therefore, the answer is no.