@sulfurx Nagel certainly wouldn’t say there is anything subjective about his approach. Indeed, he has spent most of his career attempting to defend the idea that there are objective moral values. Intentions come up because they are what determine what an action is aimed at. But Nagel would say what makes something good or bad/right or wrong is ultimately based on pure reason. Kant thinks this, too. The idea is that there are certain basic rules that we (as reasonable beings) cannot consistently deny, and that the rest of morality is built upon these rules.
Kant calls the rational basis of morality the “categorical imperative.” And while the categorical imperative cannot itself be put into words, there are various formulations of it that we can learn and follow. The first and most discussed formulation is the formulation of universal law: “act only according to that maxim whereby you can, at the same time, will that it should become a universal law.”
So what does that mean? It means that for any action you are thinking of taking, you have to figure out what the general principle behind your action is and then consider a world where everyone always acts on that principle. Is such a world even possible? If such a world is possible, is it logically consistent for someone who acts on that principle to want to live in a world where everyone acts on that principle? If the answer to either of these questions is “no,” then taking the action we are considering is contrary to reason and immoral.
One example that Kant gives is making false promises (that is, promising to do something even though you have no intention of keeping the promise). Now, consider a world where people always broke their promises. In such a world, there would be no such thing as promising because the whole practice of making and accepting promises is built on the edifice of people keeping their promises. Sure, promising can survive some people breaking their promises some of the time. But if nobody ever kept their promises, then the whole practice of promising would die out. So a world in which people always broke their promises is impossible because such a world would not contain any promises to break. Therefore, the action is contrary to reason and immoral.
But what about the second case, when the world is logically possible? One example that Kant uses here is selfishness. It is possible to have a world where everyone acted selfishly all the time. But we cannot consistently will that this be the case because our selfishness is best served by the altruism of others. That is to say, a selfish person ought to want a world in which others are not selfish. So it would be contrary to reason for someone who wanted to act selfishly all the time to wish for a world in which everyone else acted the same way since that would undermine his ability to act selfishly. The wish would be self-defeating. Thus it would be contrary to reason and immoral to act selfishly all he time.
Nagel agrees with this basic strategy of trying to find contradictions in people’s actions, but he also expands on it. One of his well known cases has to do with pain felt by people other than ourselves. We dislike pain and want to avoid it. Why? Because pain is bad. But is it just our pain, or everybody’s pain? Well, we don’t think our pain is bad just because it’s ours. We think it’s bad because it is pain, and pain is awful. It’s the badness of pain itself that makes us want to avoid it. But if we are therefore committed to saying that pain itself is bad, and if other people are similar to us such that they experience pain the same way that we do, then we cannot consistently think that the pain is not bad for them. And again, it’s bad for them because pain itself just is bad. So we should care about the pain of others because it would be contrary to reason to say “pain itself is just bad, but their pain isn’t bad.” Because if pain itself is bad, then it’s bad no matter who is experiencing it.
So in the end, the arguments of Kant and Nagel are supposed to be based on reason, and reason is supposed to be objective. I’m not sure I understand your last question, though. In the arguments I presented above, it’s not about A refuting B so much as Nagel’s argument not being enough to convince a consequentialist. The consequentialist should be unconvinced because in the case of the first premise, Nagel has a rather huge inconsistency to worry about (since his argument seems to prove too much—it works just as well against his own view as it does against consequentialism); and in the case of the second premise, Nagel has failed to engage the consequentialist on his own terms (and thus has made accusations regarding the view that a committed consequentialist has no reason to accept as valid).