When both the positive and negative volitional
auxiliary verbs are combined with a verb in Japanese, it has the nuance of 'whether or not (A)', the result will be (B)'. This is quite emphatic, and has a literal meaning similar to 'say it does (A) or say it doesn't (A), it will still be (B)'.
As a reminder, the positive
auxiliary verbs of volition are either う or よう, depending on what they are being connected to, while the negative
auxiliary verb of volition is まい. Either the
case marking particle が or と will appear after both the positive and the negative, with が being slightly more common.
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忘年会（ぼうねんかい）に出席（しゅっせき）しようが出席（しゅっせき）するまいが、会費（かいひ）はみんなからもらっています。
Whether you attend the year-end party or not, we will collect the admissions fee from everyone.
雨（あめ）が降（ふ）ろうが振（ふ）るまいが、明日（あした）は絶対（ぜったい）に釣（つ）りに行（い）くと決（き）めた。
Whether it rains or not, I have decided to go fishing tomorrow.
While this structure is primarily seen after verbs, it is important to remember that other
auxiliary verbs such as だ can be conjugated in the same way, as can い-Adjectives with the less common かろう structure. This means that both nouns, な-Adjectives and い-Adjectives can also appear using a similar grammar pattern. However, because the negative volitional まい cannot pair with these structures, an antonym will usually appear with them instead, with both words using the う or よう positive volitional.
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楽（たの）しかろうか楽（たの）しくなかろうか、学校（がっこう）は子供（こども）の仕事（しごと）だから学校（がっこう）には行（い）かなくてはだめだ。
Whether it is fun or not, school is a child's job, so you must go to school.
安全（あんぜん）だろうか安全（あんぜん）でなかろうか、２メートル以上（いじょう）の場所（ばしょ）に上（あ）がる場合（ばあい）は安全帯（あんぜんたい）を必（かなら）ず着用（あいちゃく）してください。
Whether it is safe or not, always wear a safety belt when you climb to a place higher than 2 meters.
犬（いぬ）であろうが猫（ねこ）であろうが、どちらにしても動物（どうぶつ）なので軽（かる）い気持（きも）ちで買（か）っては行（い）けない。
Whether it is a dog or a cat, they are both animals, so they must not be bought on a whim.
Based primarily on the use of volitionals, this grammar pattern will often indicate the speaker's strong will or opinion that will not change regardless of (A).