Following are part of my notes of the famous Harvard open course Justice. Kant's theory is too hard to be understood in a shot time. I think the best way is to record his main idea and try to digest in the future.

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Part 1  Kant's Conception of Freedom

Kant thinks that the individual person, all human beings, have a certain dignity that commands our respect.

The reason the individual is sacred or the bearer of rights, doesn't stem from the idea that we own ourselves, but instead from the we are all rational beings, which means we are beings who are capable of reason.

We are also autonomous beings, which is to say that we are beings capable of acting and choosing freely.

The capacity of reason and freedom, isn't the only capacity we have. It is our rational capacity that makes our distinctive.

Freedom is the opposite of necessity.

Kant's Conception of Freedom
Autonomy:
To act freely (= to act autonomously =)
To act according to a law I give myself
Heteronomy:
To act according to desires I haven't chosen myself

To act freely is not to choose the best means to a given end; it's choose the end itself for its own sake. In so far we act on inclination or pursue pleasure, we act as means to the realization of ends given outside of us. We are instruments rather than  authors of the purposes we pursue, that's the heteronymous determination of the will.
On the other hand, in so fas as we act autonomously, according to a law we give ourselves, we do something for its own sake as an end in itself. When we act autonomously, wee seems to be instruments to purposes given outside us, we become as ends in ourselves. This capacity to act freely, is what gives human life its special dignity.
Respecting human dignity means regarding persons not just as means, but also as ends in themselves. This is why it is wrong to use people for the sake of other people's well-being or happiness. This is the real reason that utilitarianism goes wrong.

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Part 2. Kant's Conception of Morality

Kant's Conception of Morality
*moral worth of an action depends on motive (do the right thing for the right reason, or for the sake of duty)

What makes an action morally worthy, consists not in the consequences or in the results that flow from it, what makes an action morally worthy has to do with the motive, with the quality of the will, with the Intention for which the act is done. What matters is the motive, and the motive must be of a certain kind.

A good will isn't good because of what it effects or accomplishes, it's good in itself. Even if by utmost effort the good will accomplishes nothing it would still shine like a jewel for it's own sake as something which has it's full value in itself. -- Immanuel Kant
The idea is that the motive confers the moral worth on an action and the only kind of motive that can confer moral worth on an action is the motive of duty.
The opposite of acting out of duty, is all of the motives having to do with our inclinations. Inclinations refer to all of our desires, all of our contingently given wants, preferences, impulses, and the like.
Only actions done for the sake of the moral law, for the sake of duty, only these actions have moral duty.

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Part 3. Kant's Conception of Supreme Morality

What's the supreme principle of morality?
Only one kind of motive is consistent with morality, the motive of duty, doing the right thing for the right reason. Other motives are summed up by Kant in the category of inclination.
Every thine the motive for what we do is to satisfy a desire or a preference that we may have, to pursue some interest, we are acting of inclination.
It's fine to have sentiments and feelings that support the right thing provided they don't provide the reason for acting. One may have more than one motive, but it doesn't mean that the action is devoid of moral worth, just because he has one other motive -- the motive which involves duty is what gives the moral worth.

Kant's Three Contrasts:
(MORALITY) Motives:                         Duty vs. Inclination
(FREEDOM)  Determination of will :   autonomous  vs. Heteronomous
(REASON) Imperatives:                      Categorical vs. Hypothetical

If reason determines my will, then the will becomes a power to choose indecent of the dictates of nature, or inclination or circumstance. So, connected with Kant's demanding notions of morality and freedom, is a specially demanding notion of reason.

How can reason determine the will?
There are two ways. There are two different commands of reason. One is an imperative. An imperative is an *ought*.
One kind of imperative, perhaps the most familiar kind, is a *hypothetical* imperative. Hypothetical imperatives use instrumental reason. If want X then do y. It's means-ends reason.
If the action would be good solely as a means to something else, the imperative is hypothetical; if the action is represented as good in itself and therefore as necessary... for a will which of itself accords with reason, then the imperative is *categorical*.

What is the categorical imperative? What is the supreme principle of morality? What does it command of us?
THE CATEGORICAL IMPERATIVE
(1) The Formula of Universal Law
Act only on that maxim whereby you can at the same time will that is should become a universal law. "the maxim", he means a rule that explains the reason for what you are doing, a principle. For example, promise keeping.
And the test, the way we can determine that the false promise is at odds with the categorical imperative, is try to universalize it, universalize the maxim upon which you are about to act. If the maxim universalized would undermine itself, then it's not categorical.

(2) The Formula of Humanity as End
We can't base the categorical imperative on any particular interests, purposes, or ends, because then it would be only relative to the person whose ends they were. But suppose, however, there were something whose existence has in itself an absolute value.... an end in itself.... then in it, and in it alone, would there be the ground of a possible categorical imperative.
"I say that man, and in general every rational being, exists as an end in himself, not merely as a means for arbitrary used by this or that will." -- Kant

Act in such a way that you always treat humanity, whether in your own person or in the person of any other, never simply as a means, but always at the same time, as an end.

Respect is respect for humanity which is universal, for a rational capacity which is universal, and that's why violating it, in my own case, is as objectionable as violating it in the case of any other.

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Part 4. The Application of Kant' Categorical Imperative on Lying

How is a categorical imperative possible? How is morality possible?

"when we think of ourselves as free, we transfer ourselves into the intelligible world as members and recognize the autonomy of the will." --immanuel Kant
Only because the idea of freedom makes me a member of an intelligible world, the categorical imperative is possible.

Since we inhabits simultaneously the two standpoints, the two realms, the realm of freedom and the realm of necessity, there is always potentially a gap between what we do and what we ought to to, between is and ought.
Morality is not empirical. Whatever you see in the world, whatever you discover through science, can't decide moral questions. Morality stands at a certain distance from the world, from the empirical world. That's why no science could deliver moral truth.

From Kant's point of view, there actually is a world of difference between a lie and a misleading truth.

White lie 善意的谎言
Kant preferred leading truth(误导的真相) to a white lie.
Unlike a falsehood, unlike a lie, a misleading truth pays a certain homage to duty.

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Part 5  The Application of Kant's Categorical Imperative On Contract

Just laws arise from a certain kind of social contract, which is of an exceptional nature. What makes the contract exceptional is that it is not an actual contract, that happens when people come together and try to figure out what the constitution should be. That the contract generates justice is an idea of reason.

A contract that generates principles of right is merely an idea of reason, but it has undoubted practical reality, because it can oblige every legislator to frame his laws in such a way that they could have been produced by the united will of the whole nation.

"Each person possesses an inviolability founded on justice that even the welfare of society as a whole cannot override... The rights secured by justice are not subject to political bargaining or to the calculus of social interests." -- John Rawls

Moral Force of Actual Contracts
1. How do they bind or obligate?
(1) consent-based -> AUTONOMY(自律)
(2) benefit- based -> RECIPROCITY(互惠) : contracts sometimes bind us in so far as they are instruments of mutual benefit.

2. How do they justify the terms they produce?
Actual contracts are not self-sufficient moral instruments of any actual contract or agreement. The fact of the agreement never guarantees the fairness of the agreement.