EXISTENCE STATEMENTS

An existence statement is a claim that there is a value of a certain variable that makes a certain assertion true.

Examples

1. There is an integer $n$ that is bigger than $2$. The variable is $n$ and it varies over integers. The assertion is "$n$ is bigger than $2$." The statement is true.
2. There is a right triangle $T$ with sides $1$, $2$ and $3$. The variable is $T$ and it varies over triangles. The assertion is false by the Pythagorean Theorem.
3. There is a real number $x$ for which ${{x}^{2}}+x=1$. True. On the other hand, the statement "There is a real number $x$ for which ${{x}^{2}}+x=-1$" is false. (Proof: find the minimum of $x^2+x+1$.)

The existential quantifier symbol

For an assertion $P$, a statement of the form "$\exists xP(x)$" means that there is at least one mathematical object $c$ of the type of $x$ for which the assertion $P(c)$ is true. The symbol "$\exists$" is pronounced "there exist(s)" and is called the existential quantifier.  It is also discussed in the section on symbols.

Example

For the assertion (1) above, $P$ is the statement "$n\gt2$" and the statement "$P(42)$", in other words "$42\gt2$", is true, so "$\exists n(n\gt2)$" is true.

Witnesses

In the example just given, $42$ is said to be a witness that "$\exists n(n\gt2)$" is true. Note that $3$ is another witness but $1$ is not a witness.

The fact that "$1\gt2$" is false does not mean that "$\exists n(n\gt2)$" is false, it only means that $1$ is not a witness to its being true.

How existential statements are worded.

Let $n$ be of type integer and suppose $P(n)$ is the assertion "$n$ is divisible by $6$". Then the statement "$\exists nP(n)$" can be expressed in Mathematical English in these ways:

• There is (or "There exists") an integer $n$ that is divisible by $6$.
• For some integer $n$, $n$ is divisible by $6$.

You can also use statements that don't give the integer $n$ a name, such as:

• There is (or "There exists") an integer divisible by $6$.
• There are integers divisible by $6$.
• Some integers are divisible by $6$.

Warning: Cognitive Dissonance is immanent

If the assertion $\exists xP(x)$ is true, there may be only one object $c$ for which $P(c)$ is true, there may be many values of $x$ for which $P(x)$ is true, and in fact $P(x)$ may be true for every $x$ of the appropriate type.

These interpretations are different
from ordinary English usage
and can cause cognitive dissonance.

In particular, in mathematical discourse:

• The statement "Some primes are less than $3$" is true, even though there is exactly one prime less than $3$. The fact that "primes" is plural in the statement may lead you to say the statement is false, causing cognitive dissonance. In math, it is a true statement.
• The statement "Some primes are integers" is true, even though all primes are integers. In everyday English, you simply wouldn't say "some primes are integers" since all of them are.

Some phrases in Math English
don't have the same meaning as in ordinary English.

The Handbook (under existential quantifier) has more discussion about this and references.