～ましょうか vs ～ませんか

Question

I have been working on these types of sentences, but I am having trouble figuring out the difference between:

In this context it is an invitation to go eat.

[食]{た}べましょうか。

and

[食]{た}べませんか。

What is the difference?

Answer 1

One key difference is the context in which the suggestion is made, and the assumption(s) the asker has about the listener's response. Take the following situation where a man is asking a woman on a date (note that I'll use the plain forms ～ない and ～よう for such a familiar situation):

お茶にでも[行かない]{LLLH}？　→　"Won't we go get tea or something?"　→　"Would you like to go have tea (or whatever) with me?"

From this, the most likely context is that the man isn't certain of the woman's response, and thus, he's probably asking her out for the first time. Now compare that to:

お茶にでも行こうか？　→　"Should/Shall we go get tea or something?"

In this case, we can probably infer either, 1) He has already asked her out at some time prior, she agreed, and they are just trying to solidify the plan, or 2) They're already a couple, and he is making a suggestion of something to do together.

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So I think whether the topic of the suggestion has been previously discussed or not makes a big difference of which one you should use. If it's a "brand new" topic, you'd use ～ませんか; if the topic has already been approached and you're suggesting one possible option, then you use ～ましょうか.

So in your case, 食べませんか tells me that the suggestion is "brand new", out-of-the-blue to the listener, while 食べましょうか tells me that they're already planning to do something, so "How about let's go eat?"

Answer 2

Well I'm kinda lacking context, but for

食べましょうか。

食べましょう is volitional form of 食べる expressing volition, Let's eat for example. With ka at the end I guess it becomes

shall (we?) eat?

As for

食べませんか。

It's polite negative form of the same verb: (You/I/He) are/am/is not eating or doesn't eat. With the question bit at the end:

You aren't eating?

Would be one possible interpretation I guess.

Again it all depends on context, but I would imagine the first one is someone asking shell they start eating dinner or something, while the later is someone politely asking if/why someone is not eating.

Edit:

Yes, this form can apparently be used for invitations.

Wouldn't you eat (lunch with me)?

The difference I would gather would lie in the fact that I guess ませんか is more about inviting someone, and asking if they want to eat, while しょうか is more of a suggestion, shall we eat.

But more definite guide on the nuance should probably be provided by someone more competent and fluent than me.

EDIT 2:

Ah, according to this,

When you use the volitional form with other people, your speech sounds like an invitation and can be translated as “let’s” in English. This is a strong invitation and is suitable for when you would like him or her to accept your invitation regardless of his or her will.

When you use negative questions with other people, your speech sounds like an invitation and can be translated as “won’t you” in English. This is less strong than the above (volitional) and is suitable for when you are willing to accept his or her will.