Can we achieve a just society solely through the exercise of our reason? Since justice, being a matter of how we treat people, is a moral issue, the question is really whether reason alone can provide a basis for morality.
The famous 18th century Scottish philosopher David Hume didn't think it was possible since he believed that morality is simply a matter of good feeling others have about what you do.
In his An Enquiry Concerning the Principles of Morals, he writes, "Morality is determined by sentiment. It defines virtue to be whatever mental action or quality gives to a spectator the pleasing sentiment of approbation, and vice the contrary." (italics his)
In other words, if society approves of what you do then it's moral. If they disapprove, it's vice. Or, as we might say today, morality is culturally constructed. If that's so - if Hume is correct - then justice is whatever treatment we dispense to others that society approves of, and injustice is whatever it disapproves of.
A little thought will show that this view of justice is completely inadequate. After all, societies have approved of all sorts of things we consider to be unjust - chattel slavery, the oppression of women, infant sacrifice, the Hindu caste system, torturing enemies, to name a few.
If we insist with Hume that these things are just or acceptable because society approved of them then there'd never be any motive or reason to change them. They are morally right and abolishing them would be morally wrong.
We might go one step further and point out that if justice (morality) is a matter of the subjective whims of society, then there can be no objective human rights since such rights are predicated upon a standard of justice that transcends societal norms.
The 19th century atheist philosopher Friedrich Nietzsche argued that, because the Christian God doesn't exist there's no basis for belief in human equality, rights nor dignity, and that all of his contemporary atheists and liberals who prize such values are really piggybacking on Christianity without acknowledging it and probably without realizing it.
In order for there to be human rights there has to be justice, and in order for there to be justice there has to be an objective obligation to do it. The problem for modern man, however, is that having abandoned God he has left himself no objective obligation to do justice and no objective standard for defining it.
He has cast himself adrift at sea in a raft with no anchor nor compass, and oddly enough, he seems scarcely aware of his predicament.