What's the difference between “not all” and “some” in logic?

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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"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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Given a number of things x we can sort all of them into two classes: Animals and Non-Animals.

(1) 'Not all x are animals' says that the class of non-animals are non-empty. There exists at least one x not being an animal and hence a non-animal.

(2) 'there exists an x that are animal' says that the class of animals are non-empty which is the same as not all x are non-animals.

The point of the above was to make the difference between the two statements clear: For sentence (1) the implied existence concerns non-animals as illustrated in figure 1 where the x's are meant as non-animals perhaps stones:

For sentence (2) the implied existence concerns animals as illustrated in figure 2 where the x's now represent the animals:

If we put one drawing on top of the other we can see that the two sentences are non-contradictory, they can both be true at the same same time, this merely requires a world where some x's are animals and some x's are non-animals as illustrated in figure 3:

And we also see that what the sentences have in common is that they imply existence hence both would be rendered false in case nothing exists, as in figure 4:

Here there are no animals hence all are non-animals but trivially so because there is not anything at all. Likewise there are no non-animals in which case all x's are animals but again this is trivially true because nothing is.

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I have made som edits hopefully sharing 'little more'. I do not pretend to give an argument justifying the standard use of logical quantifiers as much as merely providing an illustration of the difference between sentence (1) and (2) which I understood the as the main part of the question. – Jesper Hybel Jan 18 at 17:48