hominem
Explanation
This comic plays on the widespread misuse of the term "ad hominem" in everyday arguments, particularly in online discourse.
An ad hominem fallacy is a specific logical error in which someone attacks their opponent's character as a means of discrediting their argument -- for example, "You're a liar, so your data must be wrong." However, people frequently confuse this with simply insulting someone during an argument. A plain insult (e.g., calling someone "shit-for-brains") is rude, but it is not an ad hominem fallacy unless the speaker is using the insult as the basis for rejecting the other person's position.
In the comic, the bearded professor-like character (a recurring SMBC character) explains this distinction with perfect academic precision -- and then immediately calls the other person "shit-for-brains." He is demonstrating his point in real time: calling someone a name is not the same as committing the ad hominem fallacy, because he is not using the insult to discredit an argument. He is simply being insulting while also being logically correct.
The caption at the bottom -- "Learning the difference between an ad hominem and an insult has been life-changing" -- serves as the secondary punchline, implying that once someone understands this distinction, they feel liberated to insult people freely during intellectual debates, secure in the knowledge that they are not committing a logical fallacy. The humor comes from the gap between intellectual rigor and social decorum: being technically correct about logic while being absolutely terrible at human interaction.