Consider the following premises. exists x: S(x) -> S_4(x). forall x: S(x) <-> P_3(x). forall x: ~S_4(x). forall x: P_3(x) <-> U(x). Can we infer exists x: ~U(x) from them?
 From exists x: S(x) -> S_4(x), forall x: ~S_4(x) we can infer exists x: ~S(x). From forall x: S(x) <-> P_3(x), exists x: ~S(x) we can infer exists x: ~P_3(x). Finally, from forall x: P_3(x) <-> U(x), exists x: ~P_3(x) we can infer exists x: ~U(x). Therefore, the answer is yes.