Consider the following premises. forall z_1: ~W_2(z_1). forall z_1: R_4(z_1) or P_4(z_1). forall z_1: W_2(z_1) <-> R_4(z_1). Can we infer exists z_1: ~P_4(z_1) from them?
 From forall z_1: W_2(z_1) <-> R_4(z_1), forall z_1: ~W_2(z_1) we can infer forall z_1: ~R_4(z_1). Finally, from forall z_1: R_4(z_1) or P_4(z_1), forall z_1: ~R_4(z_1) we can infer forall z_1: P_4(z_1), which contradicts exists z_1: ~P_4(z_1). Therefore, the answer is no.