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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.

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"My understanding is that there can only exist one antonym per word, but is it really binary?" This is incorrect. Language does not act like logic or mathematics and meanings are extremely fluid. –  MrHen Oct 6 '13 at 4:01
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There's a reason the objects you work with in your logic course are in an unfamiliar language with weird symbols; it's because natural language is ill-equipped for dealing with logic, so a formal language was invented. Your question highlights this ill-equippedness. As @MrHen points out, your understanding "that there can only exist one antonym per word" is overly simplistic. –  AakashM Oct 7 '13 at 15:54
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Antonyms (and synonyms, and hypernyms) are always, always defined by context. "The" antonym of man is woman, boy, god, robot, alien, animal — even though a man is an animal. Likewise, there is no such thing as the antonym of "all". There is no such thing as the antonym of anything. –  RegDwigнt Oct 7 '13 at 18:53

2 Answers 2

up vote 4 down vote accepted

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.

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This. What does that even mean: an antonym of a logical operator? –  iterums Oct 7 '13 at 15:50
    
Maybe my question was a little vague: this is not a mathematics question. I introduced mathematics are a guideline to paint the picture of my question. I'm asking what the antonym of "all" is; "not all" or "none"? I used math to illustrate that in mathematical logic, it's both. I wanted to know if it was also both in plain English. –  riista Oct 7 '13 at 17:03
    
An antonym is an "opposite word". –  Alkenrinnstet Oct 8 '13 at 0:48

"Everyone knows John" = "There is not a single person not knowing John"

The negation is then:

"No one knows John" = "There is not a single person knowing John"

and not : "Not everyone knows John", meaning that, possibly, some people however do.


"Someone knows John" = "Some people, or at least one person, know(s) John"

The negation is then :

"Not everyone knows John" = "Some people, or at least one person, do(es) not know John"


"¬Everyone knows John" means "Not everyone", and not "No one"


The negation of "all" is then "none", and not "not all"


Do not confound ¬(A is B) with ¬A is B

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Indeed. De Morgan's Laws should be posted somewhere visible. The thing is, "opposite" is not a logical word. Negation is logical (though negation in natural language is not truth-functional), but "opposition" harks back to Aristotle, and isn't really a well-defined concept. –  John Lawler Oct 6 '13 at 3:58
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I don't want to jump to conclusions since math logic is all too new to me, but I believe you're incorrect. ¬∀ does indeed mean "not all" or that some characteristic is making "all" false. This could be either "not everyone" or "everyone but one" or "only two" or etc. The negation of "Everyone knows John" can be both "Someone does not know John", and "No one knows John". Either would make it correct. As well as "Only one knows John" or "All but one know John", etc. –  riista Oct 6 '13 at 4:05
    
To better illustrate: ¬∀xV(x) is logically equivalent to ∃x¬V(x). The negation of All people know John is There exists a person who does not know John. –  riista Oct 6 '13 at 4:11
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'The negation of "all" is ... not "not all"' is obviously a contradiction, which implies that your initial assumptions were false. –  Alkenrinnstet Oct 6 '13 at 5:03
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@MarkThorin Mathematically this is wrong. See en.wikipedia.org/wiki/Universal_quantification#Negation –  iterums Oct 7 '13 at 15:56

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