The term “deduction” has a special meaning for Kant. A deduction in his sense is a kind of proof, for a synthetic a priori proposition. The proposition to be proved by a deduction in Groundwork III is implied at the end of the quotation above (Kant 1997a, 58/4:454). Stated more completely, it is that The will of a sensible rational being is subject to the categorical imperative. This proposition is synthetic, because subjection to that imperative does not follow by analysis from the concept of the rational will of a sensible being. It is a priori because the necessity to act in a certain way, as the categorical imperative asserts, cannot be known by experience. Experience can indicate that someone acts a certain way, but it cannot show that he or she ought to act in any certain way.

Possibility of a categorical imperative. The deduction argument is found in the first paragraph of the fourth section of Groundwork III, entitled “How is a Categorical Imperative Possible?” The question of this title may seem a little “weak,” considering that earlier sections of the Groundwork had foreshadowed Kant’s intention to prove the categorical imperative’s validity, or its existence, not its possibility. But the title does match a procedure of Kant’s that can be recalled from his earlier writings. In the Prolegomena, for example, he had focused on questions like How is pure mathematics possible? and How is a pure science of nature possible? What Kant intends to show in the Groundwork deduction is the presupposition that makes a categorical imperative possible, and which then implies its validity, which means that it exists. Here is how he puts the point a few pages after the deduction:

Thus the question, how a categorical imperative is possible, can indeed be answered to the extent that one can furnish the sole presupposition on which alone it is possible, namely, the idea of freedom, and that one can also see the necessity of this presupposition, which is sufficient for the practical use of reason, that is, for the conviction of the validity of this imperative and so also of the moral law. (1997a, 64/4:461)

In other words, freedom is the presupposition that makes a categorical imperative possible; and assuming that one is free, one is subject to that imperative’s command.

Synthetic a priori statements. The necessity of the connection between the subject and predicate of a synthetic a priori statement is the reason these statements need a deduction. Kant’s usual procedure for a deduction is to point to some third thing that somehow encompasses both the subject and the predicate. For example: for the synthetic but necessary connection between any event and a cause, Kant points to nature as the third thing, and explains how the concept of nature can justify the assumption that every event (necessarily) has a cause. But nature will not do for the deduction of the categorical imperative, as Kant acknowledges (1997a, 53/4:447). Instead, it will be the idea of freedom, specifically, positive freedom—not just the will’s freedom from determination by natural causes, such as desires and inclinations. Positive freedom, which Kant connects with the concept of autonomy, is not freedom from something (negative), but freedom for something—freedom for acting always on maxims that conform to universal law.

The argument of the deduction. To “deduce” the synthetic a priori proposition that a sensible rational being is subject to the categorical imperative, Kant argues roughly as follows:

1. A rational being is a member of an intelligible world, or a world of understanding, where he is active through free will.

2. A rational being, as sensible, is a member of the natural world, and active there through desires and inclinations.

3. But “the world of understanding contains the ground of the world of sense and so too of its laws.”

4. Therefore, the rational being in the sensible world is subject to the categorical imperative; or as Kant puts it, “the laws of the world of understanding must be regarded as imperatives for me, and actions in conformity with these as duties.”

Comment on premise 1. The distinction between appearances and things in themselves, even as applied to our own self-awareness, is supposed to convince us that we belong to the intelligible world of things in themselves. On this basis we can infer that we are active there, through freedom. Why active? Because the connection of the members of a world is established through their activity—everything in the world makes a difference, affecting everything else. Why through freedom? Because the intelligible world is independent of the conditions of sensibility—principally time—the world-establishing activity of each member of the intelligible world is uncaused by anything prior.

Comment on premise 3. The “grounding” relation between the two worlds is supposed to give the laws of the intelligible world priority over the natural laws to which the rational being is subject as a member of the sensible world. If we regard these worlds as ontologically distinct, rather than merely as standpoints, then we can understand the nomic priority of the one world over the other—of the “real” world over its “appearance”: the sensible world. In that case, we are supposed to be able to understand how the sensible rational agent can be subject also to the law of the intelligible world.

Comment on the Conclusion. Kant does not explain why the content of the law for action in the intelligible world can be expressed by the categorical imperative. The assumption that it is a law for acting freely is supposed to convince us that it a law of autonomy, which as been presented as a categorical imperative. It is helpful to add, I think, that the world-establishing activities of the members of the intelligible world are reciprocal. That is, every rational member of that world considers the fact that his action there must affect everyone, at the same time that everyone’s action must affect him. Under that condition, it can easily be seen how the thinking behind the familiar “Golden Rule” applies: Do unto others as you would have them do unto you. In the timeless intelligible world you will not escape the effects of others’ actions; so as a rational agent this should influence your decision about how to act. The categorical imperative is of course not the “Golden Rule.” Its formula of universal law nevertheless supplies a rule for acting in the sensible world that is based on the law of reciprocity holding for active members of the intelligible world.

Knowledge of the intelligible world. When Kant presents the distinction between appearances and things in themselves (phenomena and noumena), he insists that human knowledge is limited to appearances, or to the phenomenal world (see Kant 1998, A235/B294). For this reason, some have charged that in the deduction Kant has violated his own restriction on knowledge of the intelligible, noumenal world. He would say in response, perhaps, that he has not done so, because nothing in his argument appeals to any material conditions of the intelligible world, or to any conditions that could be known only by experience of it. His argument appeals only to formal conditions of the idea of a world that serves as the “ground” of the sensible world (cf. Paton 1948, 272). For example, he has assumed that everything in the intelligible world stands in some form of causal relation with everything else. But this assumption follows from the very definition of a “world.” How intelligible things might be causally connected one with another (apart from time) is something that could be known only through experience of that world; and Kant insisted that human beings cannot have such experiences. But he did not deny that we can have any rationally justified beliefs about the noumenal realm. As he said near the end of the Groundwork: “the idea of a pure world of understanding as a whole of all intelligences, to which we ourselves belong as rational beings (though on the other side we are also members of the world of senses), remains always a useful and permitted idea for rational belief, even if all knowledge stops at its boundary” (1997, 66/4:462; see also 1997b, 43-44/5:47).