The first thing to remember: most often people ask questions and argue, when they are interested. That's why don't get offended by contradictions and doubts! Use them to reveal the theme deeper and include more listeners in the process.The key to the constructive dialogue is an understandable structure of speech. If every element of the logical chain is opened in detail, the arguments will be simpler.
For example, when there is an argument, and a person puts a question to the concrete block,
it is more comfortable to move the discussion to that small piece of logics (but not to discuss everything on the whole). In that way you will quicker come to the explanation of your rightness, or, on the contrary, you will see the contradiction and you'll be able to correct the situation.
If you want to create a really strong argumentation structure, here's a tip.