When you’re trying to convince someone of something the conversation will start in one of two places; either they know you don’t agree or they don’t know that yet. (Eg salesman know they don’t agree with your budget, but they want you to feel like they are on your side, so they don’t tell you they disagree).

But how do you go about convincing someone when there’s a known, articulated disagreement? 

You assume you don’t really understand what they’re saying. You ask them about their point of view, their idea, their world, and so on, until they stop you and start asking questions themselves. 

They might even start asking deeper questions about their own ideas.