# How to use “some” and “not all” in logic?

As asked here about the difference between "some" and "not all". I'm looking for a practical example in real world where these two can be applied. Do we have a situation where we can use either of them? Are these two merely based on how we infer things? I came up with this example:

Suppose, we enter a hall where we find that there are five individuals. We suppose that one of them is American and four of them are Italians. So, Should we conclude that "Some are American" or "Not all are American" since both would be true, right? Now, if later on we find that the one person we met wasn't an American then would our conclusion "Not all are American" still be true?

-
Related post: How do quantifiers work in predicate logic? – Niel de Beaudrap Mar 15 '13 at 14:02

Yes, "not all are American" would still be true if we found out the one we assumed was actually wasn't. As Sindikat already pointed out:

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

Which means that n, the number of individuals that have the attribute x, can be 0 or positive.

But in this case (we find out he actually wasn't American) your second premise "stops" being true, because:

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

Which means, there must be one individual with that attribute, it can't be 0.

-
I hope it is alright I used Sindikat's excellent answer... – iphigenie Nov 8 '12 at 14:28
Can we change that to "no one is American" when we found out that the one we assumed wasn't American? So, Does it mean it would be equivalent to the statement "not all are American"? – cpx Nov 8 '12 at 19:58
Yes, we can change it, but no, it doesn't mean it's equivalent. What we know of the facts changed, right? "Not all are American" can mean there are Americans or not, "No one is American" clearly means that - well, there is exactly, not +/- 0 Americans! – iphigenie Nov 8 '12 at 20:20
Now if we have "All are Italians" can we conclude "Some are Italians" from that as I did here? Since "some" as you said above means that "there must be one individual with that attribute", right? – cpx Nov 11 '12 at 7:34
Yes, if all are "some are" is necessarily right. – iphigenie Nov 11 '12 at 14:58

Allow me to play around a little with the situation you have proposed.

We walk into a room with 5 individuals, and ask them one at a time: "Are you American?"

We may assume (of the people in the room):
1) "Not all are American" as soon as we get a "No" Answer.
2) "Some are American" as soon as we get a "Yes" Answer.

The real difference comes in the "Some are Not American" situation. We can only assume this when we get a "No" answer, just like "Not all are American". However, if a random number of people are now removed from the room, Americans first, we can assume "Not all are American" as long as we had a "No" answer originally. We can NOT assume "Some are not American" unless we know that someone is left in the room.

A more silly, but more straightforward set of hypotheses:
1) Not all Americans are Martians.
2) Some Americans are not Martians.
3) Not all Martians are Americans.
4) Some Martians are not Americans.

1, 2, 3 are all true, but 4 is false, as we are proposing an existing member of an empty set.

Hope that helps!

-
Arg, can anyone tell me how to get a line break that isn't a paragraph separator in answers and comments? – Ryno Nov 8 '12 at 15:34
<br> will do it in an answer or question (but not a comment--comments can't have any sort of line break). – Rex Kerr Nov 9 '12 at 17:03
thanks, that looks much more readable now @RexKerr – Ryno Nov 12 '12 at 12:42