negation of "must"

Question

You mustn’t eat it all. [i]
You needn’t eat it all. [ii]

CGEL says “In [i] the negative applies to the eating, not the modal obligation: “It is necessary that you not eat it all”; it is thus internal, within the scope of the modal. In [ii] the negative applies to the obligation: “It isn’t necessary for you to eat it all; here then it is external, outside the scope of the modal.” (The Cambridge Grammar of the English Language, p.176)

For [ii]’s explanation, I’m not confused. But for [i] it’s not easy to understand the explanation, for my native tongue has three ways of negation for the obligation –– applying to eating, or must, or it’s sort of ambiguous. So what I want to know is whether there isn't no arguments against the explanation of [i].

Answer 1

It might be easier to understand if we use a syntactically simpler action, initially without negation...

1: You must eat (it is necessary that you eat)
2: You need to eat (note that to is grammatically required here)

For most purposes, #1 and #2 are semantically equivalent (it is necessary that you eat). But things change if we introduce negation. There's only one valid way to negate #1...

3: You must not eat (it is necessary that you do not eat)

But there are two ways to negate #2, and they have completely different meanings...

4: You need to not eat
5: You need not eat

In #4 the negation attaches to eat (i.e. - to the activity of eating), so it just means the same as #3 above.
But in #5, the negation attaches to need (i.e. - it's not actually necessary that you do not eat). You can either eat or not eat - it's a matter of personal choice, not constrained by any obligatory need.

Note that although it's perfectly grammatical, and not exactly an "unusual" form, #4 is a far less likely construction than #5. Probably in order to avoid any possible confusion, native speakers tend to use must rather than need in such constructions (or introduce "Do-support" per @hellion's comment below).

Answer 2

You have two quite elaborate answers. To contrast, I'll try saying it as briefly as I can. The affix -n't has different semantics when it's attached to different modals. If P stands for a logical proposition, then:

   mustn't(P) = MUST(NOT(P))
   needn't(P) = NOT(NEED(P))

Answer 3

"Must" alone doesn't have a negation, but there are two possible ways of negating "must [verb]". The logical negation of "must [verb]" is "not (to) have to [verb]". Whatever you are not required to do is optional. The other negation is "must not [verb]". Note that there is no "not must [verb]"; it is not grammatical. We must replace the word "must" with "have to" in this construction.

The auxiliary word "do" is required with "not have to [verb]": it often appears as "{do|does|did} not have to".

You {do not|don't} have to go to work today; it's a holiday.

But not always:

I did my homework while still at school, so as not to have to do it later at home.

"Need not [verb]" is essentially equivalent to "not have to [verb]". I suspect that this usage is more popular in British English than in North American. The "needn't" contraction is seldom heard in North America.

Also note that "need" has a possible meaning of "must":

You need to learn to stay out of my way!

What the CEGL is saying in the negative applies to the eating, not the modal obligation: "It is necessary that you not eat it all" is that "must not eat" means "must (not eat) and not "(must not) eat". I.e. you are obliged to refrain from eating, rather than not obliged to eat. The negative really is on the verb "eat", and not on the modal obligation "must". In fact there is no "must not" or "not must" which negates the must. Even if "must not" is used without a verb, it is understood that there is a verb, and it is that verb which is negated.

A: Can I connect these wires backwards?

B: No, you must not!

Here, "No, you must not!" is understood from context as "No, you must not connect these wires backwards!", and the negation actually applies to the verb "connect", not to "must". Not to connect the wires backwards is what you must.

In short, no combination of the words not and must means "freedom from obligation". Freedom from obligation is only expressed using "not have to", "not need to" or "need not".