Quick Take: People should not say the word “cruxy” when already there exists the word “crucial.” | Twitter
Crucial sometimes just means “important” but has a primary meaning of “decisive” or “pivotal” (it also derives from the word “crux”). This is what’s meant by a “crucial battle” or “crucial role” or “crucial game (in a tournament)” and so on.
So if Alice and Bob agree that Alice will work hard on her upcoming exam, but only Bob thinks that she will fail her exam—because he thinks that she will study the wrong topics (h/t @Saul Munn)—then they might have this conversation:
Bob: You’ll fail
Alice: I won’t, because I’ll study hard.
Bob: That’s not crucial to our disagreement.
If I understand the term “double crux” correctly, to say that something is a double crux is just to say that it is “crucial to our disagreement.”