Consider the following premises. forall y: P(y). forall y: P(y) -> S(y). forall y: S_4(y). Can we infer S_4(a_11) or w from them? If possible, name the inference rules being used at each step.
 From forall y: P(y) -> S(y), forall y: P(y) we can infer forall y: S(y) via universal modus ponens. From forall y: S_4(y), forall y: S(y) we can infer forall y: S_4(y) and S(y) via universal conjunction. From forall y: S_4(y) and S(y) we can infer S_4(a_11) and S(a_11) via universal instantiation. From S_4(a_11) and S(a_11) we can infer S_4(a_11) via simplification. Finally, from S_4(a_11) we can infer S_4(a_11) or w via addition. Therefore, the answer is yes.