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We have, not all represented by ~(x) and some represented (∃x) For example if I say,

  1. Not all are animals.
  2. Some are animals.

Because we aren't considering all the animal nor we are disregarding all the animal. What would be difference between the two statements and how do we use them?

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4 Answers 4

up vote 10 down vote accepted

"Some", (∃x), is left-open, right-closed interval - the number of animals is in (0, x] or 0 < n ≤ x

"Not all", ~(x), is right-open, left-closed interval - the number of animals is in [0, x) or 0 ≤ n < x.

"Some" means at least one (can't be 0), "not all" can be 0.

"No", ~(∃x), allows only number 0.

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There's also 'no' ~(∃x) which also can be 0. How would it be different from "Not all"? – user963241 Nov 1 '12 at 9:03
No only allows one value - 0. Not all allows any value from 0 (inclusive) to the total number (exclusive). – Ryno Nov 1 '12 at 9:11
Just saying, this is a pretty confusing answer, and cryptic to anyone not familiar with your interval notation. It seems to me that someone who isn't familiar with the basics of logic (either term logic of predicate logic) will have an equally hard time with your answer. – Kevin Holmes Dec 1 '13 at 14:54
A totally incorrect answer with 11 points. Nice work folks. Do people think that ~(x) has something to do with an interval with x as an endpoint? What on earth are people voting for here? – user4894 Mar 30 '14 at 19:41
@user4894, can you suggest improvements or write your answer? I'm not a mathematician, so i thought using metaphor of intervals is appropriate as illustration. – Mirzhan Irkegulov Apr 1 '14 at 1:51

This may be clearer in first order logic. Let P be the relevant property:

"Some x are P" is ∃x(P(x))

"Not all x are P" is ∃x(~P(x)), or equivalently, ~(∀x P(x))

The practical difference between some and not all is in contradictions. If P(x) is never true, ∃x(P(x)) is false but ∃x(~P(x)) is true.

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In mathematics it is usual to say not all as it is a combination of two mathematical logic operators: not and all. One could introduce a new operator called some and define it as this.

But what does this operator allow? It certainly doesn't allow everything, as one specifically says not all. So some is always a part. Can it allow nothing at all? Yes, because nothing is definitely not all.

Now in ordinary language usage it is much more usual to say some rather than say not all. Is there any differences here from the above? Yes, if someone offered you some potatoes in a bag and when you looked in the bag you discovered that there were no potatoes in the bag, you would be right to feel cheated. Here some definitely means not nothing; now if a friend offered you some cake and gave you the whole cake you would rightly feel surprised, so it means not all; but you will also probably feel surprised if you were offered three-quarters or even half the cake, so it also means a few or not much.

Going back to mathematics it is actually usual to say there exists some - which means that there is at least one, it may be a few or even all but it cannot be nothing.

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The first statement is equivalent to "some are not animals". The second statement explicitly says "some are animals".

That should make the difference clear.

Or did you mean to ask about the difference between "not all or animals" and "some are not animals"?

In that case, the answer to your second question would be "carefully to avoid statements that mean something quite different from what we intended".

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