How does Kant link the three subordinate formulations of the categorical imperative to the universal law?

Question

How does Kant derive The Formula of an End in Itself, The Formula of Autonomy, and The Formula of the Kingdom of Ends from the The Formula of the Universal Law?

So far, this is my understanding of it.

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The first formulation: Don't do anything that logic would not allow to be universalized.

For example, if everyone stole what they wanted to steal, then no one would recognize property rights so there would be no property, and if there is no property there can be no stealing. So, it is not logically possible to make stealing a universal imperative.

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The second formulation: Always treat everyone as an end in themselves, never just as a means to an end. 

• P1. Suppose an actor acts as though he were an end in himself, and as though others were not ends in themselves.

• P2. All acts must accord with universilizable principle.

• P3. An imperative is what one must or must not do.

• C1. Universalizing P1 produces a logically impossible principle.

For example, that impossible principle might read, 'All persons' actions ought to reflect the fact that they are ends in themselves, and that no other persons are ends in themselves'. 

• C2. Accordingly one must not act as though others are not ends in themselves, so it is an imperative that people act as though others are ends in themselves.

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The third formulation: Act that your will can regard itself at the same time as making universal law through its maxim.

• P1. An agent must act so that her will is free to express (regard) itself through action.
• P2. Suppose an agent acts in a way that constrains the expression of another agent's will.
• C1. Any universalization of P2 is inconsistent with P1.

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The fourth formulation: I don't see how the fourth formulation differs from the third formulation.

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Are any of these formulations mistaken? How does the third formulation differ from the fourth?

Answer 1

You're missing a formulation or two based on the way it is often parsed now. There are three groupings of claims that we often call the Categorical Imperative

Your sequencing is a little weird to me, so I will just reiterate the order most commonly used:

Universalization formulations: Formula of Universal Law = don't do what couldn't be universalized (G 4:421) (the question of whether he means "logically" is debated). The second half is as if it a Law of Nature --> do nothing that could not be made a universal law of nature. Its relatively undisputed that Kant sees these as two variants on the same theme. There is argument about what the "law of nature" part adds. A big piece of this confusion is if Kant in the Groundwork means by nature what we often mean by nature or what he uses it to mean in CPR.

Rational being formulas Formula of Ends in Themselves* = treat every rational being as an end in itself and not merely as a means (G 4:428-429). **Formula of Humanity = act such that humanity whether in your own person or any other is an end and not merely a means. There's a lot of disagreement about what humanity means in this case. The main source is that when Kant does define the term humanity elsewhere it ain't what we normally call humanity but rather a synonym for what he calls being a rational being.

The third formulation group is the Kingdom of Ends formulation = act such that you are a legislator in the kingdom of ends.

Sometimes considered part of the same group and sometimes not considered a formulationand sometimes considered its own group are the comments that follow about autonomy.

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You then ask: How does Kant derive the other formulations from the universalization formula? The answer is that this is a highly-debated question. It's not hard to get from the formulation of universalization to universal law of nature. But after that, there are several different theories.

I will share an abbreviated version of my own (you can probably buy my dissertation from ProQuest). For Kant, rationality is uniform such that a rational being given the same circumstances would always will what reason dictates -- except insofar as we are semi-rational beings capable of acting against reason (we have freedom and rationality -- we are not rationally necessitated). And reason is the universal. Thus, if we have a set of rational beings, each of them will pursue the universal correctly, and we should not impede their actions, because they are rational/universal. In my view, the word humanity merely repeats the same idea. The kingdom of ends then becomes a realm where we see all other humans as part of the same rational project.

There are some rather significantly different interpretations.