May I know which Greek or Roman philosopher is the first to define justice as "giving each his due"? And what arguments have he put forward to justify this definition?
This is an English translation of the Latin motto suum cuique, alternatively translated as "to each their own" or "may all get their due". The phrase was popularized by Cicero in De Natura Deorum ("iustitia suum cuique distribuit", justice renders to everyone his due) and later codified in the Justinian Corpus of Civil Law:"Justice is a habit whereby a man renders to each one his due with constant and perpetual will".
However, the idea goes back to Plato's Republic 4.433, the argument being appeal to tradition and authority that men must serve the state according to their nature, for that makes the state "virtuous". Socrates argues:
"For what we laid down in the beginning as a universal requirement when we were founding our city, this I think, or some form of this, is justice. And what we did lay down, and often said, you recall, was that each one man must perform one social service in the state for which his nature is best adapted.” “Yes, we said that.” “And again that to do one's own business and not to be a busybody is justice, is a saying that we have heard from many and have often repeated ourselves.” “We have.” “This, then,” I said, “my friend, if taken in a certain sense appears to be justice, this principle of doing one's own business.
“A thing, then, that in its contribution to the excellence of a state vies with and rivals its wisdom, its soberness, its bravery, is this principle of everyone in it doing his own task.” “It is indeed,” he said. “And is not justice the name you would have to give to the principle that rivals these as conducing to the virtue of state?” “By all means.” “Consider it in this wise too if so you will be convinced. Will you not assign the conduct of lawsuits in your state to the rulers?” “Of course.” “Will not this be the chief aim of their decisions, that no one shall have what belongs to others or be deprived of his own? Nothing else but this.” “On the assumption that this is just?” “Yes.”"
The latter idea of distributive justice is discussed by Aristotle in Nicomachean Ethics, who rephrases it in terms of distributive "equality by merit" (and is contrasted with the "rectificatory" justice). His argument is a bit more pragmatic, advancing social harmony and citing the existing consensus:
"If, then, the unjust is unequal, just is equal, as all men suppose it to be, even apart from argument... if they are not equal, they will not have what is equal, but this is the origin of quarrels and complaints-when either equals have and are awarded unequal shares, or unequals equal shares. Further, this is plain from the fact that awards should be 'according to merit'; for all men agree that what is just in distribution must be according to merit in some sense, though they do not all specify the same sort of merit, but democrats identify it with the status of freeman, supporters of oligarchy with wealth (or with noble birth), and supporters of aristocracy with excellence."
See also Kirk's Meaning of Justice.
I believe Cicero has said/written "justice renders to everyone his due" but Justinian comes closer having said/written "to give to each his own".
In short, both argued that this leads to a "just" society. Whether that conclusion is true or not is, of course, a matter of debate.
Assuming one takes these two as saying the same thing, Cicero (Roman politician) said it prior to Justinian (Byzantine Emperor).
I will say Aristotle. Plato views justice as harmony in the soul (or in the nation). To produce harmony, giving each his due might be needed, but harmony can be achieved in several other ways (e.g., noble lies) as Plato suggests in The Republic.
The Politics of Aristotle is to study the most beautiful form of government. According to him, such a gov must realize two kinds of justice: proportional justice and arithmetic justice.
Arithmetic justice, also called absolute justice, means that everyone deserves an equal share of some good in question if he/she satisfies certain criteria. For instance, all Athenian freemen (as opposed to slaves) have the right to a political say.
To explain proportional justice, Aristotle famously asks who should have the best flute. Neither the most handsome man, nor the most eloquent speaker, according to him. The best flute must be given to the best flute player, according to him. Giving each his due then is the essence of the proportional justice. Applying this view of justice to the case of political power sharing, Aristotle argues that those with political knowledge should have more power than those without.
Plato wrote this proposition in the Republic before both the Roman writers as well as before Aristotle.
Now the question of how opposing or antithetical claims for what is just are to be resolved. First, supposing both are rationally grounded with logical arguments, could there be any standard by which to decide in favor of one set of arguments over another?
What about the question of non-rational decision on such matters. Can one conception of justice be better than another not because it is more rational but rather because it favors or expresses values which are cherished above reason. Might we have the Nietzschean answer that one conception may be held to be more conducive to life, it's rational veracity not withstanding?