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First, I know that "not all" is partial negation. For example, "not all the numbers are 0" means there may be a number that is nonzero.

And to me, a non-native speaker of English, it seems that "all not" should be used for complete negation. However, I remember distinctly from high-school English lessons that "all not" is also partial negation.

Exactly which is it?

More specifically, suppose I say "all the numbers are not zero", does it mean every single number cannot be zero, or that there may be one or more numbers that are zero?

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The construction you are asking about, "all-not" can be a partial negation or a complete negation depending on the context.

That is:

"All X are not Y"

could mean either,

"Some X are Y, and some aren't",

or

"No X are Y".

Pathetic, I know, but it all depends on the situation.

Regular people, who may be native speakers, sometimes (but not always!) use all-not as a partial negation in everyday speech, even though that goes counter to what you would figure from a logic point of view.

The common folk-expression "all that glitters is not gold" is an example of that. It just means "some [but not all] apparently valuable things are worthless".

People who are writing or talking about zero and non-zero numbers, as in your example, or Xs and Ys, in mine, would probably be more careful to word their statements clearly, unambiguously, and/or in accordance with logic. In that sort of context, statements like, "... but all members of the group aren't like that" should be avoided in favor of something like, "...but not all members of the group are like that".

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    No, this is wrong. If you say all X are not Y, it's not possible for even one to be Y. You might be confusing this with not all X are Y, in which case at least one is not Y, but the actual makeup, aside from that, is ambiguous. – Jason Bassford Aug 7 '19 at 17:11
  • Not confusing them. It is the way some folks phrase things, sometimes, when they are speaking this language. I am not talking about logicians, computer programmers, "coders", or mathematicians, but regular people. How do you interpret, "All that glitters is not gold"? --- That gold doesn't glitter? – Lorel C. Aug 7 '19 at 17:22
  • The sentence all that glitters is not gold explicitly means that gold cannot glitter. (Because if it did glitter, it couldn't be gold.) Just as the sentence all that glitters is gold explicitly means that everything that glitters is gold. (However, the sentence all that glitters is gold does not exclude things that don't glitter as also being gold.) But this is an idiom and it's not meant to be taken in a literal sense in the first place—regardless of the specific syntax. – Jason Bassford Aug 7 '19 at 17:29
  • Where-as Strider was described as "all that is gold does not glitter." But it's a poem. By a wizard. In a hurry. So you can forgive that "not" being moved over to give the poem a pleasing form. – puppetsock Jan 14 at 15:04
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not all the numbers are 0

means that there is at least one non-zero number. It suggests, but does not say, that most of the numbers are zero, otherwise a different form would have been used, such as "only a few of the numbers are zeros".

All the numbers are not zero.

means that there are no zero values in the group of numbers being discussed. This would be more likely to be written as:

None of the numbers are zero.

It could also be written as:

All the numbers are non-zero.

However

All the numbers are greater than zero.

is a much more likely phrase. I suspect that people prefer to avoid "not zero". For a person with experience in math, science, or computers, "non-zero" seems very natural, but it may be less likely outside people with technical experience.

There can be other contexts in which an "all not" construction means "some" rather than "none" -- it will depend on more specific context.

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  • So in general, "not all" indicates complete negation rather than partial negation (not sure whether these are standard terms, but I am sure you know what I mean)? – trisct Aug 7 '19 at 15:41
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    @trisct I carefully did not say that -- that is too general a rule to pronounce. It will depend on the construction, and perhaps on idioms or common usage in a particular field. I'm sorry, but English has fewer clear and definite rules than one might wish. – David Siegel Aug 7 '19 at 15:43
  • Thanks for the clarification. I think I am gonna wait a little longer in case someone else comes with different examples. – trisct Aug 7 '19 at 15:47
  • @trisct In general "Not all X are Y" means that at least one X is non-Y. "All X are not Y" most often means that no X are Y, that is all X are non-Y. But it could be used in a particular context to mean "some X are non-Y". – David Siegel Aug 7 '19 at 15:47
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    Here's a typical example: all migrants are not refugees or asylum seekers. Where obviously the writer means that although some migrants are refugees or asylum seekers, not all of them are. – FumbleFingers Reinstate Monica Aug 7 '19 at 15:59

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