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

Question

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?

Answer 1

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

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

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

"No", ~(∃x), allows only number 0.

Answer 2

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.