Why "non-marriage"? Why not marriage? I ain't a shrink, but to me, it's common sense: If people who claim they are dedicated to each other refuse to make it official, they are afraid of how the official designation will change either their bedmate or themselves.
If marriage is "only a piece of paper", what is the deadly fear of it? That's actually a reason to GET married, not to refuse to marry.
I'm talking mainly about people who have been in previous marriages, especially those of us who still have children at home. Why can't you understand that some folks want to enjoy the company of another person without the tax and legal problems? Well those are just a start of the potential problems. If a mature couple wants to have a relationship, then what kind of problem could you have with that?