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Produced by Charles Wells     Revised 2017-02-14
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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 prime $p$ for which $p+2$ is also a prime. This is true. One example that makes the assertion true is $p=11$.
  3. 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.
  4. There is a real number $x$ for which ${{x}^{2}}+x=1$. True (use the quadratic formula).
  5. There is a real number $x$ for which ${{x}^{2}}+x=-1$". This statement 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.

Example

For the assertion (1) above, $P$ is the statement "$n\gt2$" and the statement "$P(42)$" 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:

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

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:

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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