Consider the following premises. s -> P_3(j_3). forall x: P_3(x) -> P(x). p_1 or s. p_1 -> P_3(j_3). Can we infer ~P(j_3) from them? If possible, name the inference rules being used at each step.
 From forall x: P_3(x) -> P(x) we can infer P_3(j_3) -> P(j_3) via universal instantiation. From p_1 -> P_3(j_3), s -> P_3(j_3), p_1 or s we can infer P_3(j_3) via disjunction elimination. Finally, from P_3(j_3) -> P(j_3), P_3(j_3) we can infer P(j_3) via modus ponens, which contradicts ~P(j_3). Therefore, the answer is no.