I agree with you entirely that justice and law go hand in hand. The law does not always give a just result. In a lot of situations, people do the wrong thing but are not punished in any way unless the said individual is to be punished by a consideration of law, opinion and or one’s conscience. A moral obligation is considered to be someone’s duty, a conscience that one should do the correct thing since they are compelled to do so. There the thought that one should be punished or not be punished is the only the essence of moral thinking.
Mills argument on how justice can be distinguished from other forms of moralities is very valid by looking at the perfect and imperfect obligations. Looking at the situation of an imperfect obligation is that which cannot be expected from another person. Perfect ones are the ones that a person can honestly expect from the other person. The notion of an ideal right: When it comes to justice, for instance, a given individual who has been wronged will feel good when the other person who hurt them has been punished; if they have been mistreated upon badly it would be a good idea to make sure that they seek justice since their moral right has been impinged upon; therefore the person feels that the obligation of seeking restitution. I have learned a lot from your post especially this portion where you regard that people should seek justice since it brings a sense of security. It is a significant aspect as even Mill holds it to be so.
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