Confidence, entanglement, objectivity, social signaling, personal impact, and altruistic impact. Those are six properties of a belief where deep points of disagreement lie which can easily cause a discussion to degrade into a fruitless argument if you're not aware of them. And that's key - be aware of them - you don't necessarily want to explicitly mention these points in a discussion - in fact doing so can easily cause an argument!
For each of you, how confident are you that you're right? And really, what is that confidence based on? What is the thought process by which each of you acquire confidence in your belief? When the thought process itself diverges too much, any disagreement you have about the issue you're arguing about is only a surface level disagreement. The real disagreement lies with the thought process itself. If someone believes trusting their gut is a legitimate way to acquire a belief, and you're a rationalist, it's very unlikely you'll get anywhere in an argument. Abortion can be an impossible topic to argue because the confidence underlying the belief is often based on gut feelings.
For each of you, how entangled are your beliefs in your other beliefs? For example, a belief in God tends to be highly entangled with other beliefs. In comparison, the belief that Atkin's diet works, unless you've used it and now swear by it and run seminars on it, is probably something that you could acquire or drop, without really affecting any of your other beliefs. Entangled beliefs are far harder to discuss than unentangled beliefs, and are probably pointless to discuss if you differ in the thought processes by which you and the other person are confident in your beliefs.
For each of you, do you consider your belief to be universally true or just true for you? Broadly speaking, universal truths are generally scientific in nature, while subjective truths tend to be about ethics and aesthetics. Some beliefs entail a complex mixture of both. Now of course some people claim their ethical and aesthetic judgements are objective truths, and others claim scientific truths are subjective. Even then, the most extreme advocates of these positions will reluctantly acknowledge a "superficial" difference between such beliefs. That itself - a person's philosophical stance regarding objectivity - is key information to be aware of in an argument. If I'm arguing with someone about astrology, and they claim truth is relative - I can stop right there - we're never going to agree if I think the truth is objective and they think it's subjective.
For each of you, what does your belief signal socially? Does it portray you in a certain light, or align you with a particular group? The belief that Global Warming is real aligns you with environmentalism, while the belief that Global Warming is bunk aligns you with conservatism and skepticism. "How outrageous!" you might be thinking. Whether or not you or the other person likes or agrees with being aligned with a particular group based on a belief is itself something you have to be aware of in an argument. Politicians are masters of being aware of the signals a belief sends to others.
For each of you, what does your belief personally impact? Any entangled belief has a high personal impact because it permeates your world view. Beliefs that send strong signals define your identity, and also have a high personal impact. However, other beliefs might have a high personal impact, in that they can still strongly affect your decisions. For example, the belief that house prices are going to crash, could have a huge impact on the biggest purchase decision most of us ever make. Thinking about the personal impact a belief has can be a handy reality-check on a belief you might be taking too seriously. You might say you believe in life after death, but if on reflection you’re going to not act any differently regardless of the belief, then the belief is in nearly every sense immaterial.
For each of you, what positive impact does your belief have on the world? In real world arguments, some people tend to focus on idealistic impacts, while others tend to focus on realistic impacts. The idealistic impacts come from considering the best case scenario for your belief. Your vote changed the election, or your Hybrid tipped Earth's environmental balance. The martyr is the hero of the idealistic impact. In contrast, realistic impacts come from considering the most likely scenario for your belief. Perhaps your vote made no difference, and perhaps your Hybrid, even if it did have a small positive effect on the environment, was a foolish use of your money if you really wanted to help the environment. The economist is the hero of the realistic impact. Idealists focus on stories; realists on spreadsheets, idealists want to revolutionize politics; realists accept realpolitick as the name of the game.