He thinks of nothing but himself.
Help! I’m confused by the combination of “しか” and “ない”, which seems to apply a double negation? To me the translation appears in this way: “He (彼は) doesn’t (ない) only (しか) think (考え) about his own matters. (自分の事)”
@Hoytesler
〜しかない exactly means “nothing but 〜” and the English translation is correct. It’s not a double negation. Don’t try to split a set phrase into each piece like “no + thing + but”. You just need to swallow the chunk as a set phrase.
I guess you mixed up だけ (only) with しか, but they are not interchangeable. しか is always followed by negation ない while だけ can be done by ある and ない.
For example,
私だけがこの部屋にいる (I am the only one who is in this room.)
—> 私しかこの部屋にいる is incorrect because だけ is not interchangeable with しか.
—> 私しかこの部屋にいない is correct, meaning “There is nobody but me in this room.”