How does the scope of not change a statement?

Negation denies a statement, but what it denies depends on its scope: the part of the statement to which not applies. Not all, none and not both do not express the same condition. Their differences become visible when the group and its records are fully specified.

The fictional record contains exactly Nia, Omar and Leah, with a known entry for whether each joined an event. In the initial version, Nia and Omar joined and Leah did not. All three joined is false, so not all joined is true. Not all means that at least one member did not join; it does not say that nobody joined.

None joined makes the stronger zero-participation claim. It is false in that version because two people joined. If no one joined, both none joined and not all joined would be true. These statements overlap in that situation without becoming equivalent across all possible records.

Not both Nia and Omar joined concerns the named pair, not all three people. It is false when both Nia and Omar joined, even if Leah did not. It becomes true when either or both of the pair did not join. It therefore does not require that neither joined, and it does not mean exactly one joined. The complete record fixes the facts so these are differences in statement meaning, not guesses about missing information.