A simple "debate" or "argument" can be set up where as the number of points "won" is equal to the complexity of the debate. Consider the points, the number of bases or connected ideas between to opposing ideas.
Its a simple game of contents for each point; ideally all points have to be won. If not, then it is an ambiguous win. Every argument can be role-played, -ideally the GM is familiar with logical fallacies/cognitive bias and objections, can adjudicate. Since this is not a legal debate, where the other player can interrupt with an objection it is a contest roll for each side.
in the end, like my previous discussion, no one is really convinced. what ever decision is just for Face-Value, when an arbitration is needed. an Appeal is an opportunity to retry, and follows the rules on retrying- that it should be a new argument with a different set of circumstances.
Since its only for Face Value and no one is really convinced, in the game world this decision affects the opinion and the knowledge of the denizens. More role-playing gold can be found when the players RP how this affects them.