gaming halacha

I had a monetary dispute with my neighbor (not the one we like) yesterday. He's Russian and not Orthodox, though he believes that he is religious in his own way. About a year ago he built a wall in between our properties without discussing it with me and then asked me to help pay for it. According to Jewish law, I am obligated to pay for a minimal fence that is acceptable in our neighborhood, so I offered to pay for that and he said that was fine. I was going to pay him in installments, so when I gave him the first installment, I brought a receipt for him to sign. He insisted that there was no need for formalities like that between neighbors and since we are both honest there is no need for it. I figured that since there is no record of me owing him any money, if I paid him and he said I didn't he doesn't have a case anyways, so I agreed in the name of neighborly relations.
I paid him 3 installments and owed him a little bit more and he came to me yesterday to see if I had it, and when I mentioned how much I owed him he thought it was much more. He said that they kept track and they have written down that I paid him twice and they have no record of a third payment. I told my wife and she remembered me coming home and telling her that now I only owe them a little bit more.
So I went back and told them that I only owe them what I thought I did and not as much as they thought I did. I told him that I paid them and the fact that they don't have a record of it isn't my fault. I also reminded them that I asked for a receipt and they refused so they can't complain that I am not paying what I said I paid. I honestly don't believe they are trying to cheat me, I think I gave them the money at an inconvenient time and they forgot about it. In the end we decided to split the difference. They're very upset because they believe I owe them more moeny and I'm upset because I know I paid and was stupid for not demading the receipt.

I was just thinking about how Jewish law would play into this. If we had gone to a Jewish court and I said I owe him a little and he says I owe him a lot then I would have to swear and pay him what I said I owed him. Today they don't make you swear so they would work out a compromise similar to what we workied out. However, I know the law. If I went to court, I would say I don't owe him anything, and then get off scott free and then I would pay him what I thought I owed him. In other words, because I learned Jewish law, I can game the system and he can't because he never learned Jewish law. That being said, anyone who spent time learning gemara would know that if he goes to court with a claim of partial owing that he is going to lose. He even knows the reasons behind it - people are not so brazen to say they don't owe anything when they owe something. They instead claim they owe a little bit to get them off the hook and then if they ever get money they will pay. Understanding that logic will present a problem to the court because now there is no psychology involved. I weill now be brazen and say that I don't owe anything because I know that if I don't, the system will take advantage of my honesty.