The third of Parfit's 2002 lectures concerns contractualism and involves an extended discussion of what he calls Kant's "contractualist formula". Since the formula in question is not one that Kant himself specifically states and since Parfit's derivation of this formula occurs in the concluding part of the second of the 2002 lectures it is necessary to return to this derivation before the argument of the third lecture can be considered. This is what I will do in this posting, which is devoted only to uncovering the argument by which Parfit arrives at and justifies this formula.
In the final section of the second 2002 lecture Parfit turns to the question of whether, in the course of this lecture, he can be said to have misinterpreted Kant's understanding of universal law. Whilst considering this possible objection Parfit points to the way in which Kant discusses, in the case of beneficence, the application of the law of nature formula by means of appeal not to what "everyone" could will but rather to the nature of the specific person's willing. Parfit also rejects the view that Kant's universal law test should be seen in terms of Rawlsian notions of the veil of ignorance. Parfit presents both Scanlon and Rawls as offering not interpretations of Kantian notions of universal law but as instead providing ways of revising it. This leads to Parfit presenting a complicated diagram of possible ways the universal law formula could be understood. Prominent amongst the possible understandings for Parfit is the treatment given by Scanlon which leads Scanlon to state that an act is wrong "unless everyone could rationally will that everyone acts in this way".
However whilst Parfit is favourably impressed by Scanlon's formula he does not leave the matter there. Scanlon's formula is one that Parfit thinks is impartial but that it goes further than we have to since some acts are right without being ones that everyone could perform. This leads Parfit to revising Scanlon's formula so that it becomes, as Parfit terms it, the "formula of universally willed acts" which states: "an act is wrong unless it could be rationally willed by everyone". This formula is now understood to be equivalent to the first half of the Formula of Humanity on Parfit's construal of the latter, which he has earlier termed "the Consent Principle". On the basis of this we treat people only in ways that they could rationally consent to. The wider formula of this that has become the formula of universally willed acts now adds that an act is wrong unless everyone, if they had the choice, could rationally choose the act in question as one that would be done.
However despite the argument having thus given us a formula that is at least loosely related to something Kant said Parfit is not finished yet. Parfit had earlier invoked the quite different standard that he termed Kant's "moral belief" formula which involved stating that an act is wrong unless we could rationally will it to be true that everyone believed it was permissible. This formula introduces a lot of complicated epistemic issues that Parfit does not really discuss but at this point all he does is argue that the so-called "moral belief" formula is one that can be converted into Scanlonian terms so that it becomes a claim to the effect that an act is wrong unless everyone could rationally will it to be true that everyone believed such acts permissible. This version of the "moral belief" formula interprets the claim about belief in a better way in a sense since it is no longer merely a claim about a given person's belief states and this is surely preferable as otherwise we seem mired in subjectivism. This version of the moral belief claim is meant to be understood as a way of addressing the problem of what general principles of action we could all will.
Not only is the Scanlonian notion of moral belief a reply to a genuine Kantian question it also has the advantage, according to Parfit, of being suggested by Kant's formulas of autonomy and the realm of ends. The point about the formula of autonomy is that it involves a claim about rational beings giving themselves universal law through the maxims of their will. The understanding of this that Parfit wishes to champion is in terms of the Scanlonian version of "moral belief". The Scanlonian version is also termed by Parfit a "formula of universally willed moral beliefs" but it is shortened and given the name "Kant's Contractualist Formula" so that it becomes: "we ought to act on the principles whose universal acceptance everyone could rationally will". Parfit concludes the second lecture by contrasting this formula with a Rawlsian formula that involves, instead, reference to choice under conditions where we know nothing about the circumstances of others. The contrast between these formulas will motivate the structure of the third 2002 lecture. I am making no comment here on the argument Parfit has given as I wanted merely to draw it out as the basis on which the formula that Parfit consistently in the third lecture refers to as "Kant's" formula, despite not being stated anywhere by Kant, is arrived at.