Is there any difference in There exist and For some?

When the backwards E notation $\exists$ is shown I've been told that it can mean There exist but I've also been told that it could mean For some. Is there a difference between one or the other? Do I have to use one for a certain circumstance?

Solution 1:

Logically, they are the same. The sentences "There is $x$ such that $P(x)$" and "$P(x)$ for some $x$" are logically equivalent.

Solution 2:

Further to the existing comments and answer, do also be aware of the issue of hanging quantifiers: to prevent scope ambiguities, avoid writing "for some" behind the predicate when there are multiple quantifiers. (This issue doesn't exist when you use "there exists" instead.)

For example, do write

• for some $x,$ for all $y, P(x,y)$

instead of any of the following, each of which is ambiguous:

• for all $y, P(x,y)$ for some $x$
• $P(x,y)$ for all $y$ for some $x$
• $P(x,y)$ for some $x$ for all $y.$

Synonyms for "there exists… such that" include:

1. for some
2. there is some … such that
3. there is a … such that
4. there is at least one … such that