The famous political philosopher John Rawls describes a method known as "reflective equilibrium" for striving towards consistency in our moral beliefs.
We might depict the method with this diagram here :

Some terminology :
A moral intuition is some particular belief or feeling that we might have as to whether a certain action is right or wrong in some situation. For example, if we find out that a baby has been murdered, we might think that this is terrible and feel quite bad about it.
Given a moral intuition, the method of reflective equilibrium says that we should try to generalize and come up with a general moral principle that explains this intuition. This requires the formulation of a rule that explains why we think it is wrong for someone to kill the baby. In this particular case, perhaps we might propose this rule:
This rule explains our intuition because we also believe that babies are innocent, and together with the moral principle they explain our intuition.
According to the method of reflective equilibrium, we should try to ensure that our whole system of moral thinking is consistent, by coming up with many such moral principles, and checking that these principles are compatible with each other and consistent with the other intuitions we have. If they come into conflict then we should think carefully to see how they should be revised.
As an illustration, consider the moral principle above that we should never kill any innocent people. Is this principle always true? Suppose a group of terrorists has hijacked a plane and is about the crash the plane into a building where there are a few thousand people. These people in the building are most likely going to die because there is not enough time to evaculate them. The only way to avoid this consequence is to shoot down the plane. But the plane includes a few innocent people and we would kill them as well when we destroy the plane. So what is the right thing to do in this situation?
Mot people would probably agree that we should shoot down the plane if it is the only option available. It is of course terrible to have to kill the innocent victims on the plane, but this would be the right thing to do given the terrible situation. So we now have another moral intuition, but this time it is inconsistent with the principle that we should never kill the innocent. This tells us that the original principle would have to be modified. Perhaps a better formulation is as follows:
Note that in this particular example we have choosen to revise the principle in order to keep our moral intuitions, but sometimes we might opt the other way and accept a principle and revise our intuitions instead. Many of our intuitions are a product of culture and upbringing, and many of them are perhaps unjustified. This process of reflective equilibrium can continue on and on. The point is not just to maximize consistency in our moral beliefs. It also helps us gain a deeper insight into our own moral thinking and come up with better reasons for our actions.
U06.3 ExercisesConsider the principle just presented:? Question 2 - What if we reject the intuition?Can you think of a situation where it might conflict with our moral intuition? In other words, can you think of a case where it might be wrong to follow such a principle?
- We should never kill any innocent people, unless it is necessary to save a much larger number of innocent people, and there are no other alternatives available.
Consider the plane hijack example again. Suppose someone thinks it is indeed wrong to shoot down the plane, and that we should maintain the first principle that we should never kill any innocent people. Think about how you might try to convince this person to change his mind. If you do agree with this person, what other consequences should you be ready to accept?
Men are apt to mistake the strength of their feeling for the strength of their argument. The heated mind resents the chill touch and relentless scrutiny of logic.

William Gladstone