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The veil of ignorance is a concept introduced by John Rawls in A Theory of Justice. It is a method of determining the morality of a certain issue (e.g. slavery) based upon the following principle: imagine that societal roles were completely re-fashioned and redistributed, and that from behind your veil of ignorance you do not know what role you will be reassigned. Only then can you truly consider the morality of an issue. For example, whites in the pre-Civil War south did indeed condone slavery, but they most likely would not have done so had there been a re-fashioning of society because of which they would not know if they would be the ones enslaved. It is a philosophical idea related to the method of two people dividing a cake: one cuts, the other chooses first (see pie method).
An important feature of this thought-experiment is that you don't get to keep any aspects of your current role, even aspects that are an integral part of your self. For example, in the imaginary society, you might not be intelligent and have a good sense of humor. That you currently possess these gifts is no more than good luck, which you must be willing to relinquish when building the society from behind the veil.
One could argue that society needs to promote some desirable qualities, even if that increases the disparity between roles (for example, hard work). On the other hand, it's the natural inclination of the world (and of markets in particular) to reward these qualities, and all reasonable efforts to help the disadvantaged won't raise their roles higher than the roles with natural advantage.
The veil of ignorance is part of the long tradition of thinking in terms of a social contract. See John Locke, Jean Jacques Rousseau, Thomas Jefferson and The Federalist Papers for examples of this tradition.
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