wouldn’t it be

“I have seen the sentences in books where wouldn’t seems to have been used in the meaning didn’t want, and I wonder if such a rule exists. For instance, I wanted to participate, but he wouldn’t take me on. What kind of rule is applied in the sentence above?”

という文章をこちらで見かけた。つまりなんでhe DIDN’T take me onではなくhe WOULDN’T take me onなのかってこと。答えは

Would is a modal verb with several uses, of which one is to express past intention or willingness. As you suggested, in your example, it expresses willingness, or rather, because of the negative n’t, unwillingness. You could say but he didn’t want to take me on, but the use of wouldn’t makes the statement a little less explicit.

modal verbはググると「法動詞」って意味らしいんだけど俺の解釈でいうと助動詞のもっと(というかむしろ)話し手の意志に関わる使われ方を専門とする分野。例えばwouldってwillの過去形以外に使われ方があってそれら全てを法動詞としてタグ付けできる。

もう一つの回答で一つ目の回答とは全く違う視点から意見を言っていて面白かった。反対意見とまではいかないが的を得ていると思う。

It doesn’t state that “he” didn’t want to “take me on”, it says that this is a thing that he was not going to do. It could be that “he” didn’t want to, it could be that he didn’t have time, there was a full roster of people participating and no more could be added, that as much as he really did want to include the speaker they lacked the prerequisite skill at the task in question, or anything else. We’re informed of the lack of action, not of the motivation. We can sometimes infer that the reason someone wouldn’t do something is that they don’t want to, but not always, and not here (in the bare sentence at least, perhaps in the full passage we can).

まとめると「wouldは確かにwillingnessとintentionを意味するしこの文脈ではそうなるかもしれないけど、質問者の少ない情報だけでそれだと判断できるの?もしかしたら本当に時間がないとか(その他どうしても無理な理由)で入れなかったのかもしれないやん」と言ってるわけです。そして

We’re informed of the lack of action, not of the motivation.

といいます。つまりactionというのは事実。その事実というのは事柄が何であれ彼が質問者を入れなかった理由。ただここで回答者1の言うようにdidn’t want toならwouldn’tでもいいかもしれない(=これが回答者2の言うmotivation)ただそれは実際のところは誰にも分からない。(心の中:僕はここに英語、(特に母国語として)英語を使う人たちの英語に関する議論を読むのが好きな理由がここにある!!!!そしてそれを本気になって議論しあう人たち!!!!)

We can sometimes infer that the reason someone wouldn’t do something is that they don’t want to, but not always, and not here.

うん。まあ早い話実際に話してたら分かるんだけどね。。ということでもう一つ見つけた記事を紹介:

What does “Wouldn’t it be nice if you could…?” mean? Does “Wouldn’t it be nice if you could …?” imply “If you could …, it would be nice.”?

まあ質問はwouldn’t it be niceとit would be niceの違いを知りたがっている。

To answer your second question first: not quite. When you say “Wouldn’t it be nice if you could ride a giraffe?” what you are saying is this: “I think it would be nice if you could ride a giraffe. Don’t you?” When you say “Don’t you?” instead of “Do you?” you are implying that you expect the person you are asking to agree.This addition of the negative in a question has more general application, and it usually means you are encouraging agreement with your question. For example:
Do you think Joe is acting strangely?
Don’t you think Joe is acting strangely?
In the first case, you are simply asking whether someone thinks Joe is acting strangely. In the second case, you are saying that you think Joe is acting strangely, and you are expressing the expectation that the person you are asking will agree with you.
A little different example:
You: Joe is acting strangely.
Me: Isn’t he?
Here, I am saying that I agree with you. And:
You: It would be nice to be able to ride a giraffe.
Me: Wouldn’t it?
Again, I’m agreeing with you.
It’s a little complicated, isn’t it? :)

まあ一言でまとめれば you are implying that you expect the person you are asking to agreeってことかな。逆に相手の意見をそのままbe not it?で返せば賛成になるという。ちょっと大変でしょ?ってきいてる。

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