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 predicate. In 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 say

0·0 = 25, or 1·1 = 25, or 2·2 = 25, etc.

This would seem to be a logical disjunction because of the repeated use of "or". But the "etc" can't be interpreted as a conjunction in formal logic. Instead, 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 one. It may seem obvious that the phrase "etc" 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 true. It 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 odd. This demonstrates the importance of the domain of discourse, which specifies which values the variable n is allowed to take. Further information on using domains of discourse with quantified statements can be found in the Quantification article. But 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 conjunction. For 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 quantification. Thus if P(n) is the predicate "n·n = 25" and N is the set of natural numbers, then

is the (true) statement

For some natural number n, n·n = 25.

Similarly, if Q(n) is the predicate "n is even", then

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.