I have an idea about the basics, but I'm fuzzy on the details, and it's been awhile since I've studied the subject in depth. Anyone have some answers?
恩 - favor or kindness from others that one should be feeling a sense of gratitude toward
義務 - duty rooted in social systems, usually associated with legal senses, such as liability or military service
義理 - duty rooted in personal relations usually friendship. family relations never or rarely come in this category.
人情 - pity or sympathy one feels when facing someone else's undeserved accident/accusation
Note
義理がある/ない - describes a person or people or their relation's being or not being observing such duties
かれは義理がある He is a faithful person.
かれは義理がない He is a unreliable person.
人情がある/ない - similar to 義理, describing subject's property.
かれは人情がある He is easily moved by compassion.
かれは人情がない He is cold-hearted.
義務がある/ない - describes such duties' being put on or waived from someone or a group
かれは兵役の義務がある He has to serve his country.
かれは兵役の義務がない He is exempted from military service.
恩がある/ない - not a valid expression see comments
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1Great answer but 義理 in Japanese examples under the 人情がある/ない section seems to be wrong copy-and-paste. Also, we do say AにB(の)恩がある in the way of "owe B to A" in English. – broken laptop May 1 '15 at 12:10
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@broccoliforest I was making examples based on かれは(X)がある form, and thought かれは恩がある is not valid. But you're right. It is perfectly legitimate to say かれはかのじょに恩がある when she did a favor of him and he owes to her. – krim May 1 '15 at 13:31