Consider the following premises. forall x_2: R(x_2) -> S_1(x_2). forall x_2: ~R(x_2). exists x_2: S_1(x_2) -> R(x_2). forall x_2: R(x_2) or ~S_1(x_2). Can we infer exists x_2: ~S_1(x_2) from them?
 From exists x_2: S_1(x_2) -> R(x_2), forall x_2: R(x_2) -> S_1(x_2) we can infer exists x_2: S_1(x_2) <-> R(x_2). From forall x_2: R(x_2) or ~S_1(x_2) we can infer forall x_2: S_1(x_2) -> R(x_2). From exists x_2: S_1(x_2) <-> R(x_2) we can infer exists x_2: R(x_2) -> S_1(x_2). From exists x_2: R(x_2) -> S_1(x_2), forall x_2: S_1(x_2) -> R(x_2) we can infer exists x_2: R(x_2) <-> S_1(x_2). Finally, from exists x_2: R(x_2) <-> S_1(x_2), forall x_2: ~R(x_2) we can infer exists x_2: ~S_1(x_2). Therefore, the answer is yes.