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