Is Batman a symbol of Vengeance or Justice??
Ra's Al Ghul: Your compassion ("justice") is a weakness your enemies will not share.
Bruce Wayne: That's why it's so important. It separates us from them.
So there have been debates of whether Batman is vengeance or justice as two of them are different yet the first one has been mentioned several times in his undying quote, "I am Vengeance! I am the Night! I am Batman!." But in the same time, Batman is a symbol of justice and he fights for it.
We noticed that his first years as the Batman, he is reckless in fighting, often going too far as to break the bones and possibly skulls of anyone who stand in his way. In Arkham Origins, he would even go as far as to hurt policemen (although they are corrupt), and would deny any help from other people because this was his quest for vengeance.
But I believe that turning point of his ideals was when he confronted the killer of his parents seen in an episode of Batman the Brave and the Bold, and we can see that he is conflicted with two choices: either to kill or let him live and let the law determine his fate.
However, in the cinematic universe (the Trilogy), his turning point was when he was about to kill Joe Chill, his parents' killer. He could not bring himself to it realizing that he will not kill because he is not like his enemies.
But in the end he chose justice symbolizing the start of Batman fighting for something greater than his personal quest for vengeance and to start being a guardian and the protector of his people by making sure what happened him in his childhood which led him to pursue this kind of life will never happen to anyone else ever again. By this, he starts to work with the police and even recruits potential youngsters to fight beside him such as Oracle and Robin.
True, he may still be encouraged to fight crime because of his parents' death but also to save Gotham from itself. He learns to control his vengeance in time. He not only becomes a symbol of justice, but an inspiration to others to fight for what is right no matter what.
So dear people, is Batman vengeance or justice?