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Existential quantification

In predicate logic '''existential quantification''' is an attempt to formalize the notion that something (a logical predicate) is true for ''something'' or at least one relevant thing The resulting statement is an existentially quantified statement and we have existentially quantified over the predicateIn symbolic logic the existential quantifier (typically "∃") is the symbol used to denote existential quantification
Quantification in general is covered in the article Quantification while this article discusses existential quantification specifically

Basics

Suppose you wish to write a formula which is true if and only if some natural number multiplied by itself is 25. A naive approach you might try is the following:
0·0 = 25, or 1·1 = 25, or 2·2 = 25, or 3·3 = 25, and so on
This would seem to be a logical disjunction because of the repeated use of "or"However the "and so on" makes this impossible to interpret as a disjunction in formal logicInstead we rephrase the statement as
For some natural number n n·n = 25
This is a single statement using existential quantification
Notice that this statement is really more precise than the original oneIt may seem obvious that the phrase "and so on" is meant to include all natural numbers and nothing more but this wasn't explicitly stated which is essentially the reason that the phrase couldn't be interpreted formally In the quantified statement on the other hand the natural numbers are mentioned explicitly
This particular example is true because 5 is a natural number and when we substitute 5 for n we produce "5·5 = 25" which is trueIt does not matter that "n·n = 25" is false for most natural numbers n in fact false for all of them except 5; even the existence of a single solution is enough to prove the existential quantification true(Of course multiple solutions can only help!)In contrast "For some even number n n·n = 25" is false because there are no even solutions
On the other hand "For some odd number n n·n = 25" is true because the solution 5 is oddThis demonstrates the importance of the domain of discourse which specifies which values the variable n is allowed to takeFurther information on using domains of discourse with quantified statements can be found in the Quantification articleBut in particular note that if you wish to restrict the domain of discourse to consist only of those objects that satisfy a certain predicate then for existential quantification you do this with a logical conjunctionFor example "For some odd number n n·n = 25" is logically equivalent to "For some natural number n n is odd and n·n = 25"Here the "and" construction indicates the logical conjunction
In symbolic logic we use the existential quantifier "∃" (a backwards letter "E" in a sans-serif font) to indicate existential quantificationThus if P(a b c) is the predicate "a·b = c" and N is the set of natural numbers then
$exists\left\{n\right\}\left\{in\right\}mathbf\left\{N\right\} P\left(nn25\right)$
is the (true) statement
For some natural number n n·n = 25
Similarly if Q(n) is the predicate "n is even" then
$exists\left\{n\right\}\left\{in\right\}mathbf\left\{N\right\} big\left(Q\left(n\right);!;! \left\{wedge\right\};!;! P\left(nn25\right)big\right)$
is the (false) statement
For some even number n n·n = 25
Several variations in the notation for quantification (which apply to all forms) can be found in the Quantification article

Properties

We need a list of algebraic properties of existential quantification such as distributivity over disjunction and so on. Also rules of inference