Consider the following premises. forall z_2: P(z_2) or P_3(z_2). forall z_2: P(z_2) -> T_2(z_2). forall z_2: P_3(z_2) -> T_2(z_2). forall z_2: T_2(z_2) -> P(z_2). Can we infer exists z_2: ~T_2(z_2) from them?
 From forall z_2: P(z_2) -> T_2(z_2), forall z_2: T_2(z_2) -> P(z_2) we can infer forall z_2: P(z_2) <-> T_2(z_2). From forall z_2: P(z_2) <-> T_2(z_2) we can infer forall z_2: P(z_2) -> T_2(z_2). Finally, from forall z_2: P(z_2) -> T_2(z_2), forall z_2: P_3(z_2) -> T_2(z_2), forall z_2: P(z_2) or P_3(z_2) we can infer forall z_2: T_2(z_2), which contradicts exists z_2: ~T_2(z_2). Therefore, the answer is no.