The best insight into an interesting philosophy of persuasion comes from Imre Lakatos' philosophy of science.
In Science and Psuedoscience:
First, I claim that the typical descriptive unit of great scientific achievements is not an isolated hypothesis but rather a research programme. [Science is not simply trial and error, a series of conjectures and refutations.] 'All swans are white' may be falsified by the discovery of one black swan. But such trivial trial and error does not rank as science. Newtonian science, for instance, is not simply a set of four conjectures - the three laws of mechanics and the law of gravitation. These four laws constitute only the 'hard core' of the Newtonian programme. But this hard core is tenaciously protected from refutation by a vast 'protective belt' of auxiliary hypotheses. And, even more importantly, the research programme also has a 'heuristic', that is, a powerful problem-solving machinery, which, with the help of sophisticated mathematical techniques, digests anomalies and even turns them into positive evidence. For instance, if a planet does not move exactly as it should, the Newtonian scientist checks his conjectures concerning atmospheric refraction, concerning propagation of light in magnetic storms, and hundreds of other conjectures which are all part of the programme. He may even invent a hitherto unknown planet and calculate its position, mass and velocity in order to explain the anomaly.
It is the idea of the hard core, protective belt, and heuristic that we can adopt for interesting combat with ideas. Any given persuasive act will try to manipulate the opponent's protective belt or hard core. (More meta-persuasive acts will try to manipulate the heurstic) By modeling the opponent's arguments as combat, the player will have to strategically defeat elements of the protective belt before attacking the hard core. Successes and failures in any given persuasive test determines how well the opponent's protective belt holds up to attack.
In terms of mechanics, you have two options. The first option is to treat the belt like HP. Each player has HP, successes on persuasion reduce the opponent's HP. First one to 0 loses the argument.
The second option is to model the debate as combat (and this is why Lakatos is so useful here.) Have the players place representations of their arguments (power determined by the persuasion test) on whatever you use to perform combat. If you can decorate it, abstract/surreal art makes a perfect landscape. Have "minions" be the protective belt, heroically sacrificing themselves to protect the hard core. Over this combat, have players describe their verbal attacks and represent the efficacy of the attacks through movement and the combat mechanics of ubiquity.
Mechanically, for important persuasion attempts, I would use the second option. Simulated combat could be quite effective, especially set in a useful metaphorical arena. Defense would be (Willpower + Intelligence + Reasonability), Stun would be Willpower, Move (Charisma+Intelligence), and Health would be (Charisma + Willpower - Reasonability). Reasonability is an attribute that is treated similar to size, but is based from the distance from the center of the debate's Overton window.
Once all arguments are statted up and described within the metaphorical arena, grant them weapon equivalents based on the knowledge the person arguing them has about the topic. (Someone with high skill but little knowledge can still win, but facts make a remarkably simple and useful hammer.)
Apply combat as normal, but take note of the final Health score of all the arguments. For the audience's reaction, the higher the final health of an argument, the more the audience considers that argument to be "valid" as filtered by their own worldviews. The lower the health, the more compromises the specific argument must make with the opposing sides. If necessary, model this as the opposing sides being able to say "Yes, but..." for every lethal point of damage the argument has taken, restricting and confining the topic area where the argument applies.
Obviously, this should only apply to important persuasive attempts, but this could be easily modelled as an attempt to persuade a guard to open an important door (Different skills would necessarily be used) or a debate in front of a court.