Here we have a Strengthen with Necessary Premise question. Remember that a premise is necessary for a conclusion if the falsity of the premise guarantees or brings about the falsity of the conclusion. First, we check to see if the answer choice strengthens the passage, and then, if it does strengthen, we negate the answer choice to see if its negation makes the argument fall apart. If the answer choice does both those things, then it is our correct answer.
Our conclusion is that, "Hastings must have been disloyal." Why? Because we know that "if his dismissal was justified, then Hastings was either incompetent or else disloyal." We are also told that Hastings "had never been incompetent."
Let's diagram the principle rule in the premise.
PR1: J ==> I or D not I and not D ==> not J
We are then given the information: Hastings had never been incompetent.
P: not I
We then conclude, "Hastings must have been disloyal."
C: D
Let's look at answer choice (A): "Hasting's dismissal was justified."
Does this answer choice strengthen the passage? Yes! We know that if Hasting's dismissal was justified, then (according to the principle in the stimulus) he was either incompetent or else disloyal. We know that he was not incompetent; therefore, he must be disloyal. This supports the passage.
Does the negation of this answer choice make the argument fall apart? Well if Hasting's dismissal was not necessarily justified, then we do not have to have only two reasons for his dismissal, i.e. either being incompetent or disloyal. Perhaps someone threw Hasting's under the bus. If answer choice (A) is negated, then it is not necessarily true that Hastings must have been disloyal. Thus, answer choice (A) is our correct answer.
Now let's take a look at answer choice (C): "The dismissal of anyone who was disloyal would be justified."
Does this answer choice strengthen the passage? No. We do not know whether or not Hastings was dismissed because he was disloyal. Thus, knowing that the dismissal of one who is disloyal will be justified has no bearing on our argument.
Remember, we want to support the argument that because we know that a justified dismissal means one is either disloyal or incompetent and because Hastings is definitely not incompetent, he must, therefore, be disloyal. Answer choice (A) does this for us.
Hope that was helpful! Please let us know if you have any other questions.
