I love the first verse in this reading, it is one that I often use when I am evaluating the appropriateness of actions or the truthfulness of statements; but I have never really thought in depth about what this verse communicated. Paul identifies the need to love one another as a debt, it is this fact that I find so interesting.
Debt is, many times, a negative thing. We try to stay out of financial debt as much as possible, we do not like owing other people favors, and the idea that another person has the right to expect us to give them something is uncomfortable in most situations. So why would Paul equate loving others with a debt? I would suggest that he uses this word for two reasons. Most importantly is the fact that love is something that others can expect from us as Christians.
If we are Christians we believe that Jesus is the son of God and that He died for our sins, He saved us from an eternity apart from Him, and we can never repay Him, so if we accept His salvation we are forever indebted to him. Jesus told us that the greatest commandment is to love God and the second is to love others, and so, if we are his followers forever indebted to Him it is our duty to follow His command. We can never pay off the debt we owe, but we can live our lives trying to pay the debt by following His commands.
The second reason I think Paul uses the word debt is because he wants to challenge our concept of what a debt is, especially a debt within a relationship. When we enter into relationships we acknowledge that the other person, or persons, can expect something from us. If we take a class we acknowledge that we owe the teacher our time and efforts to learn what is being taught, and if we enter into a friendship we acknowledge that we owe the other a large emotional investment. By thinking of this we can realize that a debt can be a deeply rewarding experience, and when we work to acknowledge and pay our debt to God by loving others we can only further enrich all of our relationships.