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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?

J. W. Tanner
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AGarza
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    Yes, they are the same, just as "for all" and "for every" and "for each" mean $\forall$. – Joe Jul 25 '21 at 20:05
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    They are logically equivalent, but the English language creates a difference, namely that "there exists" is followed by "such that", while "for some" isn't. Thus "there exists $x$ such that $P(x)$" but "for some $x$, $P(x)$." In this respect, the mathematical symbol $\exists$ behaves like "for some". So one should write "$\exists x,P(x)$" without any extra stuff like "s.t." or $\ni$ after $\exists x$. – Andreas Blass Jul 26 '21 at 01:26
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    "There exists an object where some property is true" .... "The property is true for some object" ..... "The property is true for some existing object".... Yeah, they all sound the same to me. – fleablood Jul 26 '21 at 07:42
  • @AndreasBlass, is the "such that" semantically important? Wouldn't it be just part of the syntax, without any effect on the meaning? In english, we say "$x$ is equal to $y$", but in mathematical notation, this is "$x = y$", where "$=$" is the equal sign, not the "is equal to" sign. Where did the "is ... to" go? "is ... to" is just part of the english syntax. From the semantic point of view, we don't need worry about it. – João Alves Jr. Jul 29 '21 at 01:54
  • @JoãoJúnior I agree that "such that", like "is ... to", is just part of English syntax. And I add that one should not import them into mathematical symbolism. That is, one should not write "$\exists x\text{ such that }P(x)$, just as one should not write "$x\text{ is }=\text{ to }y$." – Andreas Blass Jul 29 '21 at 03:18
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2 Answers2

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

João Alves Jr.
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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
ryang
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