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I'm not saying that non-arbitrariness is the whole of justice. But I am saying that it is an essential part of justice. No system that allocated benefits and burdens in an arbitrary manner would be considered just by anyone.

Think about it. Consider the passionate controversies we have had about justice. Look at what we are arguing about even today. Some gays would like to have the right to marry other gay people. They aren't just saying that they WANT this. They are saying that it is UNJUST for them to be denied it. They are claiming that there is no relevant difference between gays and straights on this point. Any law that restricts marriage to heterosexuals is therefore arbitrary in that it treats people differently on the basis of an irrelevant characteristic.

I have real doubts about the wisdom of legalizing gay marriage. But if I want to justify keeping it illegal, I have to claim that there is some relevant difference between gay marriage and heterosexual marriage. Otherwise, I would be asking the government to arbitrarily treat people differently.