The argument itself is stated in a brief and simple form by Parfit. It is a discussion of the reference to reflective endorsement of desires and aims and argues that when such endorsement is properly understood it cannot be coupled with the "subjective" conception of reasons that theorists have tried to link it to. The first step is to understand the process of such reflective endorsement as requiring that "desires and aims" are related to "relevant facts" in the process of reflection. This reference to "relevant facts" is then related to the claim "N" that we try to learn more about "the different possible outcomes of our acts". The combination of concern with "relevant facts" with claim "N" is, states Parfit, to implicitly commit us to viewing possible outcomes as including "intrinsic features that would give us object-given reasons to want either to produce or to prevent these outcomes, if we can". But this commitment is a commitment to an objectivist view of reasons so the practice of reflection itself undercuts the subjective view of reasons on which it is alleged to rely.
The basis of the argument is concealed in the view that commitment to a "subjectivist" view of reasons means that we have to deny that "facts give us reasons", something that Parfit later associates with a remark cited from Korsgaard that it is not necessary on her view to identify "especially rational ends". Whilst Parfit does not give the source of Korsgaard's comment it is fairly clear that what she is here rejecting is the kind of view that identifies the "value" of something with an independent feature of the thing in question (independent, that is, of the process of endorsing its value). And Parfit implies that such a view is effectively equivalent to the "Humean" denial that "facts give us reasons" to value something.
In replying to Parfit a number of points can be made. Firstly, in partial defence of the "Humean", it can be pointed out that, for them, "facts do give us reasons" in all kinds of ways. It is evident to a "Humean" that the fact that I perceive someone running towards me carrying a deadly weapon provides me with a reason to run away. In this basic sense it is evident that "facts give us reasons". But what is involved in the appreciation that this "fact" gives us a reason to act in the way we do? Parfit claims it is the "intrinsic" fact of the nature of the threat in question but Hume would deny this and on this point I am sure Kant would agree. Hume's denial would consist in the claim that it is not the simple "fact" itself that leads to provision of the "reason" as it is rather the perception of this fact conjoined with the "natural inclination" that preserves life that gives me the basis of running away. Such an "inclination" is not a "reason" in one sense, which is that it may not fit Hume's theoretical comprehension of "reasons" but it is a reason in another which is that it fits his naturalistic understanding of us.
For Kant the case would be different given that he does not suppose the naturalist view of reasons Hume possesses and Korsgaard points, in a different passage to the one Parfit cites, to what is at work in Kant's view when she claims that "salience" is determined by adoption of an end which is to "come to perceive the world in the way that having the end requires". On Korsgaard's rendition, the "incoherence argument" gets things the wrong way around. Let's see how that can be put.
First of all, Parfit assumes that the process of reflective endorsement requires fitting "desires and aims" to the "relevant facts". Now, before turning to assumption "N", it is first necessary to understand what the status of these "relevant facts" is. On Korsgaard's construal these facts have the status that they do in relation to the perception we have of the world by means of adoption of an end. It is through the adoption of this end that we can see that the "facts" are, indeed, "relevant". They are not "relevant" in some general sense but relevant instead to the aim chosen. So it is the aim that gives them their salience.
At this point we can see that Parfit's introduction of "N" performs a change of status on the "relevant facts" as these are now viewed in relation to the "different possible outcomes" of our acts, something that assumes consequentialism. But the general consequentialist assumption appears to give outcomes a value independently of the nature of our choice so it requires the adherent of reflective endorsement to automatically view the kind of choice they have undertaken in endorsing an end to have the objectivist structure Parfit then produces to convict the "subjective" theorist of incoherence.
A different type of response would fail to concede the "objectivist" construal of consequentialism and instead see the reference to "different possible outcomes" as part of the "salience" involved in selecting the end. So it is by means of the end selected that the outcomes are viewed rather than the outcomes having an independent status. After all, if outcomes were merely such as affect the end adopted without being also something that was judged by reference to the end then why would we include them in our view of the end? If we view outcomes like this, however, then they are no more "intrinsic" in their features than are ends themselves. So this construal of the consequentialist view would not require us to reject it necessarily but would instead interpret reference to "outcomes" in ways that don't require commit to the objectivist construal of ends.
At this point I think we can also see what is generally wrong with Parfit's comprehension of "values" as intentional objects. If "value" has such an intentional objectivity it is in relation to the coherence of things valued in the willing thereof. It is not that "value" has some status completely independent of willing but rather that its status is internally related to willing. The means by which this internality is to be understood requires the Kantian process of reasoning to be undergone, reasoning that refers, for example, to the notions of universality and the sense of humanity as the one form of end-in-itself. That would be quite a different story to Parfit's but his "Incoherence Argument" does nothing to undermine it.