むしろ
rather, if anything
(A より) むしろ B
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むしろ corrects an expectation by saying the opposite, or something else, is more accurate: 今日は寒くない。むしろ暑いくらいだ, "it's not cold today. If anything, it's hot".
It often follows より: 都会よりむしろ田舎に住みたい, "I'd rather live in the countryside than the city". It says B is preferable, or truer, than the obvious choice A.
It's common after a negative statement, to flip it: 失敗は悪いことではない。むしろいい経験だ, "failure isn't bad. If anything, it's good experience".
Compare もっと ("more"), which only adds degree. むしろ switches to a different answer.
Examples
今日は寒くない。むしろ暑いくらいだ。
It's not cold today. If anything, it's hot.
失敗は悪いことではない。むしろいい経験だ。
Failure isn't a bad thing. If anything, it's good experience.
謝るより、むしろお礼を言いたい。
Rather than apologise, I'd like to say thank you.
都会よりむしろ田舎に住みたい。
I'd rather live in the countryside than the city.
彼は怒るどころか、むしろ喜んでいた。
Far from being angry, he actually seemed pleased.