The volitional form of the auxiliary verb was briefly introduced before when we looked at the volitional.
Form
Dictionary
Volitional
Polite
です
でしょう
Standard
だ
だろう
The volitional for verbs translates as “let’s” in English; however, for the auxiliary verb the nuance is actually one of supposition (“I bet”, “I think”, “It probably is”).
この問題を解決するのは無理だ。
Solving this problem is impossible.
この問題を解決するのは無理だろう。
This problem isn’t likely to be solved (I bet it’s impossible to solve).
Critical Connection Rules
Whether you are using the polite でしょう or the standard だろう, the word that comes immediately before it must be in the plain/casual form.
Verbs & i-Adjectives: Attach directly to the plain form (e.g., 行くでしょう, 難しいだろう). Never attach it to the polite masu/desu form (e.g., 行きますでしょう is wrong).
Nouns & na-Adjectives: Drop the だ and attach directly to the stem (e.g., 学生だろう, 無理でしょう). Never keep the だ (e.g., 学生だだろう is wrong).
彼女が頑張れば、できるだろう。
If she tried I reckon she could do it.
沙織が来ると言ったら彼も来るでしょう。
If Saori says she’ll come, I bet he will come, too.
それは難しいだろうな。
I bet that’s difficult.
The な particle here is a more masculine ending particle.
Asking Politely with でしょうか
Used as a question, でしょうか is much softer and more polite than ですか (again, it has the nuance of “wondering” if something might be okay, rather than directly demanding an answer).
個人的なことを聞いてもいいでしょうか？
I was wondering if I may ask something personal?
Seeking Agreement (でしょ / だろ)
In casual speech, we can seek agreement or confirmation from the listener by removing the う from both the standard and polite forms of the volitional copula. These are usually spoken with a rising intonation.
明日は学校に行くでしょ？
You’re going to school tomorrow, right?
やったのはお前だろ！
You’re the one who did it!
お前 is a very derogatory way of saying “you”. When this abbreviated form (だろ) is used with the standard form of the volitional it has almost an accusatory tone. In the last example, you know that he’s the one who did it, you just want him to admit it. It has quite a masculine nuance.