We have, not all represented by ~(x) and some represented (∃x) For example if I say,
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?
"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.
~(∃x) which also can be 0. How would it be different from "Not all"?
– user963241
Nov 01 '12 at 09:03
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.
First, we should understand why these statements look the same.
They look the same because each one has a hidden clause that your brain fills in from informal english.
In the first case, if I say "Not all of these are animals", your brain adds a "Some, but" to the front. And in the second case, when you say "Some of these are animals" your brain adds a ", but not all" right after that some, so you end up with "Some, but not all of these are animals" in both statements.
We do this because in colloquial English, that's what those statements mean. So this is a problem of informal versus formal language. Informally speaking, if I say "not all" and "some", I'm saying the same thing. But in formal language, I'm saying something different. "not all" means "it isn't true that all" and "some" means "it is true that at least one".
So looking at the formal definitions, how are they different?
Say I give you two decks of cards. You have been told that each card has a picture on it, either an ear of corn or a mouse. For the first deck, I tell you "not all of these are mice" and for the other I say "some of these are mice". I want you to verify or refute those statements.
With the first deck, you get to stop at the very first ear of corn. One of these cards is an ear of corn, so not all of them are mice! With the second deck, you get to stop at the first mouse. One of these is a mouse, and one is enough for "some"!
Importantly, we don't have to look at any more cards in either case. The statements are true for each deck, even if there isn't a single mouse in deck 1 or a single ear of corn in deck 2, or if the cards are mixed, or if the first card is only example. We just don't care beyond that first ear of corn or mouse.
To refute the statements, I would have to look at every card. For the first deck I need to verify that there is no corn in the deck, for the second deck, I need to verify that there are no mice.
If I have a deck of all corn, it verifies the first statement and rejects the second. Different result, different statement.
There's a lesson here for all of us, though. This is an example of a really easy misunderstanding waiting to happen, because the language is ambiguous. If it's important that you mean "at least one", don't use "some". If it's important that you mean "but not all" add that as well!
Not all natural numbers equal -1 : TRUE.
Some natural number equals -1 : FALSE.
There is one king of the UK, and his name is Charles.
Not all kings of the UK are named Charles: False.
Some kings of the UK are named Charles: True.
For a set with one element, “not all” is the exact opposite of “some”.
For an empty set, “all elements of this set have property X” is always true, and “some elements have property X” is false. So for an empty set, “not all” and “some” are identical and false.
For larger sets, with two or more elements, “not all” and “some” are the same, unless either none or all elements of the set have a property X.
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.
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".
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.
