A Positive Argument is an argument for your particular position. A Negative Argument is an argument against an opposing position.
It seems like everywhere I go, people want to point out what's wrong with the opposition's arguments. It does not matter if we are talking about political views, religious views, or any other view that is tied to a deep conviction. So many people focus so tightly on the opposition that they forget about their own point of view. This is not a very good strategy. The reason I say this is for one simple reason. Let's say you have a plan to accomplish something, and one of your teammates expresses great dissatisfaction with your plan and even provides every reason in the world not to use it. It would come natural to you to ask if your teammate had a better plan. If no other plan was proposed, then the team would have to stick with the original, no matter how many flaws it had.
I must point out that I am not about to defend a reason for only using positive arguments. The fact is that the negative arguments have their important place. They are used to convince your opposition that their idea is not as solid as they might believe. Depending on the person, and depending on how many holes are poked in the opposing idea, the person may be open to an alternative idea- your's, but only if you can show why it is superior to the original and does not suffer from the same problems (or create new ones) that the opposing idea had.
Of course, making negative arguments should never be abused. A person can only take so much negative information about their point of view before they start to believe you are not just attacking their point of view, but attacking them. Even though you may not intentionally make a personal attack, it may be perceived as one. You can recognize when they are starting to think this by their body language, before they say anything that would indicate it. If you don't notice this and continue with your negative argument, the person may "tune" you out and not "hear" anything else you have to say (this includes your positive argument). If you do notice the "offended" body language, ask them to provide a positive argument for their view. By doing that, you reinforce that you respect their view, and open doors to provide a positive argument for your point of view later in the conversation.
Providing a positive argument is what can ultimately convince someone to your point of view. Before you can do this, though, you need to investigate your point of view as deeply as you can. I like to start my positive argument by establishing some common ground. If there is no common ground then there will be nothing to build upon. If you decide to appeal to someone or something that you believe is an authority, make sure that the opposition recognizes your authority as you do. It makes no sense to appeal to something or someone they don't believe to be an authority. It also helps to use sources that may not be totally on your side, but do allow for your point of view as a possibility- this will let your opposition know that others with opposing points of view, at least, recognize the possibility that your point of view may be true. Defining sources is a great way to establish common ground.
Once common ground has been established, you can start making claims and backing them up with evidence. I would make sure to appeal specifically to things that the person would understand. If you are not sure what all they may understand, ask them about their interests, especially when they were in college. It does you no good to appeal to astronomy if the person is a paleontologist. It also does little good to appeal to the Bible if the person is not a Christian. Evidence based on unfamiliar disciplines can come in later, but should not be introduced immediately.
The danger in introducing evidence of a foreign nature is that the person will not understand why an argument is powerful or not. They will not know what questions to ask in order to understand the argument better. This may also cause a person to think that you are trying to talk down to them (unfamiliar technical terms), or elevate yourself over them. If you are unsure how a person will react to evidence from a certain discipline, ask them if they've considered the evidence from that discipline. If not, and they don't show interest in it, don't introduce it.
Of course, part of giving a positive argument is listening to and answering questions. You can expect many questions for two reasons: first, the person is skeptical of your argument and second, the person is unfamiliar with the evidence you have provided. If the person bombards you with many questions at one time, ask them to select one to focus on for the moment. Assure them that when you have answered their question to their satisfaction or when you determine that you need to do more research later, you will tackle the next. Don't try to answer too many questions at the same time. If you do, you and the other person may lose focus of a specific issue. This is extremely frustrating to the person if they are unfamiliar with the evidence you are providing.
The same Q&A technique goes the same for both sides. When you are providing a negative argument, don't bombard them with too many questions. Allow them to answer one question, then ask the next.
Something very important to remember is that which ever strategy you are using will be the opposite of what the other person is using (if you provide a positive argument for your view, they will present a negative argument against your view). A conversation will typically flip back and forth many times. The key to keep the conversation constructive is to treat the other view exactly how you expect your view to be treated.
If you haven't already, you should check out my posts "This Argument Is Full of Crap!" to discover some of the common reasoning fallacies that you should avoid as you are presenting both negative and positive arguments; and "The Importance of Learning to Communicate" to help guide you through providing a negative argument in a way that will be constructive.
Later on I will discuss an action that many people are uncomfortable with (it can be scary), but will ultimately help you in understanding your point of view and being able to defend it. That is: being able to question and challenge your own point of view with the strongest arguments against it (Why Should I Challenge My Own Views).
Tom Gilson at Thinking Christian posted a great series that demonstrates the interaction and relationship between positive and negative arguments: The Evolutionists Complaint: Its Wrong to Argue for ID by Arguing Against Evolution
Dr. Hugh Ross talks a bit about this whole issue on his podcast "More Than A Theory". He gives a scientist's perspective on positive and negative arguments, and the importance of each.
"What Are The Different Strategies?"
Download the MP3 here.