Distinction between "not everyone" and "no-one"

I recently learnt that だれも means "everyone" with an affirmative verb and "no-one" with a negative verb. So, the literal translation of the Japanese "No-one is here" would be "Everyone is not here". Then how would one go about saying "Not everyone is here?"

Not everyone is here.' is translated into
すべての人がここにいるのではない。

Here 「すべて～ではない」is a partial negation.

Everyne is not here' is translated into
すべての人がここにいない。(i.e. 誰もいない)

Here「すべて～ない」is a total negation.

If you are familiar with formal language representations :-),
We can interpret the above situation as below:

When P(x)≡[x is here],
Not everyone is here.
⇔ ￢(∀xP(x))
⇔ すべての人がここにいるのではない。

⇔ No one is here.
⇔ ￢(∃xP(x))
⇔ ∀x(￢P(x))
⇔ Everyone is not here.
⇔ すべての人がここにいない。

In contract to the expresson 「すべての人がここにいない」(total negation),
the expression 「すべての人はここにいない」is somewhat ambiguous.
this can be interpreted as both partial negation and total negation.

• I actually think Japanese's も construction has a much cleaner translation into formal logic: whatever interrogative that has も after it is simply turned into a forall quantified variable with the type of the interrogative: 誰もいない = ∀x:人 ￢いる(x) Oct 18, 2013 at 0:11
• And of course 何もない = ∀x:物 ￢ある(x) Oct 18, 2013 at 0:12

"Not everything is X" is the same logically as "Some things (exist which) are not X", so in the general case you can do something like

there also exist things that are not blue
= some of them aren't blue
= not every one of them is blue

Unfortunately, for the "is here" case, where our verb is いる, that would give us something like いない人もいる, which is at best confusing, so we might want to look for a rephrase.

Depending on how you want the emphasis you can also negate the clause "everything is X" directly by inserting a noun like わけ:

it is not that everyone is here

• 誰もいる, with or without (という)わけではない, sounds archaic (if not incorrect) to me. More generally, 誰も without a case particle with an affirmative verb sounds archaic. 誰もがいる sounds much better to me (again, with or without (という)わけではない). Oct 17, 2013 at 23:46