Consider the following premises. forall z_2: T(z_2) -> P_4(z_2). exists z_2: W_2(z_2) -> P(z_2). forall z_2: P_4(z_2) -> W_2(z_2). forall z_2: P(z_2) -> T(z_2). forall z_2: ~P_4(z_2). Can we infer exists z_2: ~P(z_2) from them?
 From forall z_2: P(z_2) -> T(z_2), forall z_2: T(z_2) -> P_4(z_2) we can infer forall z_2: P(z_2) -> P_4(z_2). From forall z_2: P_4(z_2) -> W_2(z_2), exists z_2: W_2(z_2) -> P(z_2) we can infer exists z_2: P_4(z_2) -> P(z_2). From forall z_2: P(z_2) -> P_4(z_2), exists z_2: P_4(z_2) -> P(z_2) we can infer exists z_2: P(z_2) <-> P_4(z_2). From exists z_2: P(z_2) <-> P_4(z_2) we can infer exists z_2: P(z_2) -> P_4(z_2). Finally, from exists z_2: P(z_2) -> P_4(z_2), forall z_2: ~P_4(z_2) we can infer exists z_2: ~P(z_2). Therefore, the answer is yes.