The Hierarchy of Disagreement
I found this illustration of how to order your arguments at the Starts With A Bang blog. This blog usually leads one to a first source. However, I had to do some digging to find the original creator of this image. This image is all over the web, but I think the first source is here, from the Create Debate blog in April 2008. This image (click on it to enlarge) was created to illustrate an earlier point by Paul Graham, whose text-only posts I've been (occasionally) reading for years. The premise is to always lead with your top level reasonable arguments, and never resort to the bottom layers. As Ethan Siegel (SWAB) put it,
It's sometimes tough to decipher what the central point of someone else's argument is, because most people don't argue clearly and logically. But if you can identify it, that's when you win. When someone else mucks around at the bottom of the pyramid, don't sink to their level; stay up high. Those top two levels are really the only way to ever change someone's mind, or to sway other intelligent, thinking people to your side.
This is an attitude that would serve us well on this site.