You can informally disagree with someone by saying:
I disagree!
I don't think so.
You must be kidding/joking!
If you are in a business setting it is important to be polite and respectful.
You can formally (politely) disagree by saying:
I would have to disagree with you.
I'm sorry but I don't agree with you.
I respectfully disagree.
インフォーマルに反対の意見を言うなら：
I disagree!（そうは思わない）
I don't think so.（そうは思わない）
You must be kidding/joking!（冗談よしてよ）
ビジネスシーンでは、丁寧に、相手を尊重することが大切です。
フォーマル（丁寧）に反対するなら：
I would have to disagree with you.（私はそうは思いません）
I'm sorry but I don't agree with you.（申し訳ありませんが、そうは思いません）
I respectfully disagree.（私はそうは思いません）
The first two example sentences are polite and reserved.
The third example sentence is an exclamation and outright statement that you do
not believe either the content or the implications of a statement someone has
made.
Eg A: "I was a tortoise in my previous life, you know."
B: "That's pie in the sky!"
Pie in the sky is a false hope or an illusion.
はじめの2つの例文は、丁寧で控えめです。
3つ目は、誰かが作った内容や意味合いを信じていないことに驚き、率直な表現です。
例：
A: "I was a tortoise in my previous life, you know."
ねぇ、私前世はカメだったんだ。
B: "That's pie in the sky!"
そんなわけないよ！
Pie in the skyとは、嘘の思いや幻想のことです。
『私はそうは思わない』のみを英語で言おうとしたら
この様な言い方になるのですが、
この言い方のみですと少々つっけんどんな言い方になってしまいますので
Umm... I don't think so. But you may be right.
ええと……私はそうは思わないけれどなぁ。でも、あなたの言っている事が正しいのかもしれないし……。
と言う風に濁してみたり
Sorry, I can't agree with you on that.
ごめん、それについては賛成できないよ。
と言った風にクッション言葉を入れておいた方が無難かもしれません。
The first phrase states your disagreement loud and clear. With the second phrase it shows that you respect and understand the other person's opinion but you don't agree with them.
While there is nothing wrong with using the first phrase, I personally lean towards the second one. It shows maturity and respectfulness that is always needed in conversations and could perhaps make the other person see your point of view.
Hope this helps!
When you're trying to explain that you disagree with someone's opinion, you can say:
-I disagree with what you are saying.
-I don't agree with what you are saying.
-I am of a different view.
誰かの意見に同意しないことを説明したいとき、以下のように言えます：
-I disagree with what you are saying.
（私のあなたとは意見が違います。）
-I don't agree with what you are saying.
（私のあなたとは意見が違います。）
-I am of a different view.
（私の意見は違います。）
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