A crux for a belief B is another belief C such that if I change my mind about C, that will also change my mind a lot about B.
E.g., my cruxes for "it's raining" might include things like "I'm outdoors and can see and feel lots of water falling from the sky on me", "I'm not dreaming", "I don't think aliens are trying to trick me", and so on.
Related:...