# 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?

• In ordinary English a NOT All statement expressed Some s is NOT P. There are no false instances of this. NOT ALL can express a possibility of two propositions: No s is p OR some s is not p. Not all men are married is equal to saying some men are not married. Not all birds are reptiles expresses the concept No birds are reptiles eventhough using some are not would also satisfy the truth value. The predicate quantifier you use can yield equivalent truth values. That is a not all would yield the same truth table as just using a Some quantifier with a negation in the correct position. Commented May 1, 2018 at 22:59

"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.

• There's also 'no' `~(∃x)` which also can be 0. How would it be different from "Not all"? Commented Nov 1, 2012 at 9:03
• No only allows one value - 0. Not all allows any value from 0 (inclusive) to the total number (exclusive).
– Ryno
Commented Nov 1, 2012 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. Commented Dec 1, 2013 at 14:54
• @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. Commented Apr 1, 2014 at 1:51
• @sindikat I was commenting on the 11 (now 10) points your totally incorrect response got. I found it striking. Surely you don't believe that the 'x' in ∃x refers to the number of instances of some quantity. But your error is easily corrected with education. I was truly struck by 11 viewers giving plus-points to your incorrect response. What do these 11 people have to say for themselves? Commented Apr 1, 2014 at 19:26

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.

Not all natural numbers equal -1 : TRUE.

Some natural number equals -1 : FALSE.

• I noticed a "low quality answer" flag. From my side I cleared it {I guess philosophers prefer more loquacity than mathematicians] But you could add the (pseudo) equation as clarification ∃ = ¬∀¬ Or more symmetrically ¬∃ = ∀¬ Commented Jan 21 at 7:40
• @Rushi, I am not sure why this is needed. The OP asked for an example when "not all" is different from "some". I provided such an example. Commented Jan 21 at 9:40

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.

• I can say not all birds are reptiles and this is equivalent to expressing NO birds are reptiles. I agree that not all is vague language but not all CAN express an E proposition or an O proposition. That is no s are p OR some s are not p. The phrase must be negative due to the HUGE NOT word. Commented May 1, 2018 at 22:43
• @Logikal: You can 'say' that as much as you like but that still won't make it true. Commented May 1, 2018 at 23:30
• @Logikal: Or make any sense. Commented May 1, 2018 at 23:37
• Well can you give me cases where my answer does not hold? All it takes is one exception to prove a proposition false. Commented May 1, 2018 at 23:51
• @logikal: your first sentence makes no sense. I'm not here to teach you logic. Commented May 1, 2018 at 23:53

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!

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.

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.

• 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. Commented Jan 18, 2016 at 17:48
• I would say NON-x is not equivalent to NOT x. What makes you think there is no distinction between a NON & NOT? If an employee is non-vested in the pension plan is that equal to someone NOT vested? Does the equation give identical answers in BOTH directions? I would say one direction give a different answer than if I reverse the order. Commented May 1, 2018 at 22:38
• @Logical what makes you think that what you say or dont say, change how quantifiers are used in the predicate calculus? Why do you assume that I claim a no distinction between non and not in generel? Please provide a proof of this. What equation are you referring to and what do you mean by a direction giving an answer? Commented May 2, 2018 at 3:08
• Your context in your answer males NO distinction between terms NOT & NON. The equation I refer to is any equation that has two sides such as 2x+1=8+1. Either way you calculate you get the same answer. Starting from the right side is actually faster in the example. Same answer no matter what direction. Your context indicates you just substitute the terms keep going. I said what I said because you don't cover every possible conclusion with your example. Hence the reasoning fails. Commented May 2, 2018 at 11:37
• If that is why you said it why dont you just contribute constructively by providing either a complete example on your own or sticking to the used example and simply state what possibilities are exactly are not covered? Commented May 2, 2018 at 12:42