When you want to colloquially express that you don’t care at all about something, you might say “I couldn’t care less.” This phrase first popped up in British English at the turn of the 20th century and is still popular today. In the 1960s, a controversial American variant of this phase entered popular usage: “I could care less.” Many native English speakers, both in the UK and US, find this expression to be logically flawed. If you couldn’t care less, then it’s impossible for you to care any less than you do. If you could care less, however (and you want to be taken literally), you’re saying that it is possible for you to care less than you care now. Those who take issue with this believe the later variant says very little about your level of caring, and so they fervidly avoid it.
Etymologists suggest that “I could care less” emerged as a sarcastic variant employing Yiddish humor. They point to the different intonations used in saying “I couldn’t care less” versus “I could care less.” The latter mirrors the intonation of the sarcastic Yiddish-English phrase “I should be so lucky!” where the verb is stressed.
The argument of logic falls apart when you consider the fact that both these phrases are idioms. In English, along with other languages, idioms aren’t required to follow logic, and to point out the lack of logic in one idiom and not all idioms is…illogical. Take the expression “head over heels,” which makes far less sense than “heels over head” when you think about the physics of a somersault. It turns out “heels over head” entered English around 1400, over 250 years before “head over heels,” however, the “logical” version of this idiom hasn’t been in popular usage since the late Victorian era.