The styles of reason we use in political and philosophical discourse, these fields being divergent, are varied. If we wish to do structured thought, particularly with people with whom we know we share some basic intuitions, we can work from shared principles in a relatively structured way; legal analysis (and analogues like Fiqh or Halachkic reasoning) rely on common grounds, styles of discourse and analysis. When people already agree on these foundations (or agree to a frame that includes them), this is appropriate. When that frame is absent, one cannot do that.
When one is either arguing directly about the frames one might accept, or wants to argue specifics with people where one doesn't know of any relevant shared foundations, one usually relies on aesthetic arguments; arguments that are chosen because they presumably provide conclusions that appeal to unknown value-conclusions (perhaps common to most humanity, like empathy) or that have elegance in their tightness. These arguments are different in style from formal reason; people are swayed to them because of attraction rather than necessarily how well they fit with their existing frameworks. They will naturally be at their strongest for people who have not already had exposure to foundational ideas that might conflict with them, but even for people who do have such ideas, if the new ideas are compelling/attractive enough, they might stick around as alternative frames in someone else's head, perhaps as part of an alternative Weltanschauung.
This relates to the process towards reflective equilibrium like jumping someone else's car; the tension between specific intuitions and broad theory is met by suggesting new broad theories for others that help them make sense of their specific intuitions (or under some circumstances one can push on the other side; one rarely can push on both at the same time).
Aesthetic arguments generally are alien to the kinds of logic that are easily given rules. They easily resemble proof by assertion, although they don't aim to be proofs at all. In a less-than-friendly form they can be assertions of value-judgements, which can also be acceptable when recognised as such.
As a mechanism for consensus, aesthetic arguments are very useful despite their informal nature; without them it is impossible to sway people on foundations, and with them it is weakly possible. They may be dangerous in their lack of structure, but as instruments of consideration they are the most intellectual way to sway on foundations, other means (like access to friends based on accepting similar political views) are more disrespectful to philosophical process and development. The roots of any philosophy may be on unstable ground, but being able to inspect and work with those grounds requires these kinds of arguments if done honestly.