I assume you are asking about the Objectivist definition of Justice and since all knowledge, concepts and values are objective in Objectivism, we reject the subjective in those areas. So:
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Justice is an objective evaluation (judgment) based on objective values that the judger objectively holds.
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Values are objectively based on reality.
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Judgment is an objective value that promotes your life and Justice is the virtue by which it is attained.
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Justice is connected to Rights in that they are both objective values that promote your life.
I know values are chosen by the judge, and he determines how rational they are.
Values are chosen and it is how they are chosen that determines whether they are rational or not.
I also know rights are connected with reality, so if these values are connected to rights then justice is connected to reality given this chain of relationships.
As I said, all objective values and concepts are connected to reality.
That got me thinking what is the objective base for justice?
The objective base of justice is that: we must judge in order to live. We must judge whether a berry is edible or not, we must judge where to build our house, we must judge when to cross the street, we must judge what is good and bad, we must judge what is true, we must judge whether someone is telling the truth. We must “judge and be prepared to be judged.” – AR (paraphrase)
The virtue of justice is “that one must never seek or grant the unearned and undeserved, neither in matter nor in spirit” – Ayn Rand, VOS pg. 28
Justice is getting what you deserve.
In general, whenever you seek a definition of an Objectivist concept the Lexicon is a great place to start. Here is a great entry on Justice from same:
What fact of reality gave rise to the concept “justice”? The fact that man must draw conclusions about the things, people and events around him, i.e., must judge and evaluate them. Is his judgment automatically right? No. What causes his judgment to be wrong? The lack of sufficient evidence, or his evasion of the evidence, or his inclusion of considerations other than the facts of the case. How, then, is he to arrive at the right judgment? By basing it exclusively on the factual evidence and by considering all the relevant evidence available. But isn’t this a description of “objectivity”? Yes, “objective judgment” is one of the wider categories to which the concept “justice” belongs. What distinguishes “justice” from other instances of objective judgment? When one evaluates the nature or actions of inanimate objects, the criterion of judgment is determined by the particular purpose for which one evaluates them. But how does one determine a criterion for evaluating the character and actions of men, in view of the fact that men possess the faculty of volition? What science can provide an objective criterion of evaluation in regard to volitional matters? Ethics. Now, do I need a concept to designate the act of judging a man’s character and/or actions exclusively on the basis of all the factual evidence available, and of evaluating it by means of an objective moral criterion? Yes. That concept is “justice.” – Originally from ITOE pg. 51
Here is an excellent lecture on Justice given by Tara Smith.
EDIT: Oops, me and Zoid and khaight make three, good job, nice quote.