Learn English – Antonym of all: none, not all, both

antonymsetymologymathematics

If you ask someone what the opposite of "all" was, most times the answer will be "none", such as the example of "no one" is the opposite of "everyone". There are three antonyms for "all" on Thesaurus.com: none, zero, and incompletely. I'm most concerned about the last one, "incompletely".

As a math student, I am taking a maths logic course, and a couple of logical quantifiers are frequently brought up: ∃ (there exists), and ∀ (all/every). The negation of these is specifically defined as "there does not exist" and "not all/every" respectively.

Everyone knows John becomes Not everyone knows John

Someone knows John becomes No one knows John

Which would be more appropriate to define as the opposite of "all" in English? Is the antonym "not all" or "none"? My understanding is that there can only exist one antonym per word, but is it really binary? To refer back to mathematics:

¬Everyone knows John can mean both No one knows John and Not everyone knows John. Both of these fall outside the domain of which Everyone knows John would be true.

Maybe I'm reading too into it.

edit; this isn't a mathematics question. I only introduced mathematical logic to have some kind of reference to what the "antonym" (actually negation) of "all" is. This question is essentially asking whether the antonym of "all" is "not all", "none", or both.

Best Answer

The problem here is that the idea of "opposite" is not analogous to a logical negation.

Consider a scale from −10 to 10, and let us define the property good as anything greater than or equal to 9. The negation of goodnot good—would of course be anything less than 9. On the other hand, "opposite" suggests the inverse of the property; the same property reflected symmetrically about some pivot onto the opposite end of the spectrum. Hence the "opposite" of good in this case would be anything less than or equal to −9.