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Existential quantificationIn 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.
Suppose you wish to say
Suppose you wish to say
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 put 5 in for n, we get "5·5 = 25", which is true. It doesn't 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 universal 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 then n·n = 25". Here the "and" construction indicates the logical conjunction.
In symbolic logic, we use the existential quantifier "∃" (an upside-down 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
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