The argument itself is formulated in relation to two different "cases", firstly "Case 2" and, secondly, "Case 3". Both, like the "agony" argument, concern mental states assessable in hedonic terms. "Case 2" formulates a conception of desiring a future period of agony although it provides no reason why the future period of agony is desired since it is explicitly set up in such a way as to rule out any specific telic desire (such as masochism) being instead chosen "for its own sake" alone. Parfit views this is as indicative of a kind of pure "subjective" model of desire though it is naturally seen by him as giving a picture of desire that is hardly credible.
Since the picture of desire in "Case 2" is so difficult to give credence to why is it that Parfit believes that, even on the "Humean" model, anyone would endorse it? The basis of his conception that it would be endorsed as a credible view of desire on this model appears simply to be that it includes no intentional element and, given that "subjective" theories of reasons can include no such element (on pain of admitting that "objective" factors are decisive in reasons for action) they can have no objection to such a purely subjective picture of their view. If the "subjective" view were correct, states Parfit, it wouldn't be difficult to imagine this case but we do find it difficult precisely because we intuitively include intentional factors in our conception of desire.
The basic rationale of the case turns on the view that "agony" is an objective factor for us and given that it has this status the purely subjective relation to agony in "Case 2" is one we do not find credible. However, there is another way of comprehending this reaction to "Case 2" which is that it does not represent a telic orientation in the simple sense of picturing a hypothetical imperative as the Humean model might be thought intended to do. To capture this it should rather include a reason why a desire has been incorporated into our reasons (to use the language of Henry Allison's "incorporation thesis").
Parfit incorporates a view like this (termed condition "F" by him) and this leads us from "Case 2" to "Case 3" in which, by contrast to "Case 2", I wish to avoid a future period of agony. Now, in this case, Parfit refers to a series of constraints that subjectivists might wish to place upon a desire prior to incorporating reference to it in their reasons for action. The first ("G") is that we have to have a present desire or aim that gives us a reason to have a particular desire or aim. (Note the way "desires" and "aims" are apparently run together here, another indication that it is the "Humean" model that is really pictured here.)
Now, one of the ways condition "G" can be met is if there are events that would produce its effects that we can use to achieve said aim and this reference to events is referred to further as condition "H". However these references are then related to "Case 3" in the sense that they are removed from it. In "Case 3" there is no instrumental reason to avoid future agony and so the reasons to avoid the agony in question are missing. Given this we replicate in "Case 3" the purity of conception of subjective "reasons" we had in "Case 2" with the same result of a presumed reductio of the "subjective" view.
However, just as the problem with "Case 2" could be said to have been precisely the absence of anything really telic in it, so the same objection can be made to "Case 3" as Parfit now presents it. Parfit considers other conditions that might present such telic components and again rules them out arriving at condition "J" in the process which would be were some present fully informed desire or aim is itself taken as a fact that itself justifies the presence of itself. This self-justifying move is taken by Parfit, correctly enough, as a desperate move since, on this conception, whatever was desired, however worthy or unworthy, would, ipso facto, be "justified".
This conception of the "subjectivist" view is arrived at simply by removing all instrumentality from the reason-desire model that is generally pictured as "Humean". It is, indeed, a reductio of such a view. Without necessarily wishing to defend the "Humean" view, however, I find this "victory" over it unconvincing as, in evacuating any reference to instrumentality it precisely prevents the model from giving any telic reasons and then faults it for failing to do what it has been prevented from doing. I fail to see how that provides an argument against it.
The reason why Parfit sees the case otherwise is that it is often true, on his view, that our relationship to hedonic states is regularly one in which there is no present desire or aim that would be fulfilled by the action that is open to us and this is because we want such things for their own sake, rather for some other end. So what Parfit wants to defend is the conception that some things are hedonically desired without reference to instrumental reasons and yet there are "objective" reasons present in such cases that the "subjectivist" cannot capture which is why they are unable to recognise any reasons at all in such cases.
Another way in which the argument to this result goes for Parfit is in terms of the regressive nature of reasons that appears to underly the "Humean" picture. If condition "F" is taken to be a key component of this picture then every desire refers to another apparently in infinitum and Parfit denies the coherence of this claiming, instead, that any such chain of reasons for action has to rest ultimately upon an end that cannot further be taken to be grounded instrumentally (and is therefore, for him, "objective"). This illuminates the difference between an instrumental theory like the "Humean" one and the conception Parfit is championing. It does not tell us, however, how it can simply appear to us when we incorporate a desire as worth adopting into our reasons, how that desire can be accepted when in other circumstances we would reject the same desire. Such a view requires the "objects" of Parfit's claim to be situationally evaluated suggesting that it is not correct, as his argument supposes, that it is purely their "objective" character that is valued.
This situational element of desire could, however, be appealed to as the basis of the chain of reasons by the Humean and, further, something like it simply has to be the basis of their argument. Rather than concede this Parfit instead attributes to them a need to give up condition "F" purely on the basis of the regress it apparently leads to (but which could be stopped by a situational constraint). This leads him directly to say that they have "no reason" to adopt the desires in question which is strictly a false view of the contention offered.
Similarly "Case 2" and "Case 3" are precisely faulty in not recognising the situational constraints under which the desire-reason model works. Parfit presents an easy case in this argument in which we either accept the difference between desires worthy of being incorporated into reasons and those not so worthy turn on "objective" features or in which there is no reason recognised at all. This is a false dichotomy and since the "all or none" argument turns on it, the argument is faulty and unconvincing. Parfit's ultimate conclusion that it is "telic value-based" reasons that give instrumental reasons their normative force can be accepted without having to agree that such reasons are "object-given".