How are morality and justice connected?
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It seems there’s a difference between justice and morality.  I’ve never quite come to a comfortable conclusion about the nature of the two concepts and their relationship, but it’s worth exploring.

Suppose you jump in someone else’s car parked in the valet entrance at a hotel and speed away to get your wife in for an emergency C-section.  You’ve saved the baby and possibly the mother.  It would be strange to call this immoral.  In fact, it might be very moral, even heroic.  But it also seems clear that the owner of the car has been wronged.  She was unable to make her meeting in time, some of her gas was used up, and maybe you even got a few dings in the door.  She has suffered an injustice.  So even though you acted morally, it’s possible you acted unjustly.

Let’s say you have a deep hatred for your neighbor.  One day an envious rage takes over so you pick up a rock and throw it at his new car, hoping to shatter the window.  You miss.  No one sees the action, and the rock rolls harmlessly into the weeds.  It seems likely you’ve acted immorally by trying to destroy his property.  But it would be odd to say any injustice was done.  Your neighbor hasn’t suffered a wit from your failed attempt at vandalism.

Justice is about living with other people, while morality is about living with yourself.  Justice is about right relation to others as measured against the mores of society, while morality is about right relation to right itself, as measured against your own beliefs.

Whether or not justice exists objectively or is entirely a social construct, it has an unmistakable universality.  The particulars, and the process of discovering and remedying injustice differ in each society, but the basic tenets are the same.  No society has ever praised or rewarded breaking a promise, stealing, or murder.  There are instances where such acts are called by other names or given a pass under special circumstances, but that’s just it; they always require justification.  The default human position is that coercion is bad, and social systems evolve to mitigate it.

What would justice demand from you in the car theft scenario?  The nice thing is, we don’t have to decide in the abstract.  Justice always takes place in a social context, and the process seems just as important as the outcome.  For productive cooperation, the systems that determine and deal with injustice are best when they are transparent, stable yet flexible, knowable in advance, and not applied preemptively.

Even though everyone may acknowledge that your theft of the car was unjust, if the process allows arbitrators to consider circumstances, they may let you off, or they may ask only that you pay the owner a small fee.  These contexts are rich, and the owner has a lot to consider as well.  Perhaps she hears your story and decides not to pursue any recompense.  Maybe she is really ticked and wants to, but realizes the social approbation she’ll get for doing so isn’t worth it, even though she would win her case.  Since justice exists only in a social context, and for the use and benefit of humans, even if it is violated, there needn’t be black and white, always-and-everywhere rules demanding uniform punishment.  Though a uniform and recognizable process is needed, uniform outcomes don’t seem to be.  This is why common law is so much more effective than legislation at maintaining peace.

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Hope this helps! :P