can 食べよう mean
- lets eat / shall (we) eat (suggesting)
- should eat (expressing an opinion)
?
eg;
- 昼食を食べよう lets have lunch
- お昼は、どこで食べようか? where should we eat for lunch?
Which is correct meaning for ~よう/こう/おう/ろう....(verb volitional form)
If you are a beginner, you should learn the following three usages.
しよう (with a flat or falling intonation)
This pattern is used when you suggest that you and the listener do something together. It is often translated as "Let's ..."
昼食を食べよう。
Let’s have lunch.
しようか with a falling intonation
This pattern is used when you invite the listener to decide something with you about something you are going to do together. It is often used with a question word (interrogative) such as なに, どこ, and いつ.
お昼は、どこで食べようか。(↓)
Where are we going to eat for lunch?
予約をしようか。(↓)
Why don’t we make a reservation?
しようか with a rising intonation
This pattern is used when you offer to do something for the listener.
予約をしようか。(↑)
Shall I make a reservation for you?
You cannot expect a one-to-one correspondence for such basic function words. The volitional form can be translated into English as will, let's, shall, should and so on depending on the context. And should can be translated into Japanese as (よ)う, べき, はず, (れ)ばいい and so on depending on the intended meaning.
In English, should in a question can express someone's volition or casual suggestion, so these translations are both fine. But when you say something like "You should stop smoking", this should is stronger, and you usually need something stronger than the (よ)う-form.