Produced by Charles Wells Revised 2015-09-03 Introduction to this website website TOC website index Head of Math Reasoning Chapter blog

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

**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.**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.**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 $x^2+x+1$.)

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.

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.

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.

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 $6$.

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.

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