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. ... In the jargon of the new mathematics of the 1960s, an open sentence is a sentence in which there are specific numbers which, when used to replace the variables, will allow the resulting expression to evaluate to true. ... Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ...

Quantification in general is covered in the article Quantification, while this article discusses existential quantification specifically. In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ...

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: Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

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 logic. Instead, we rephrase the statement as OR Logic Gate Logical disjunction (usual symbol or) is a logical operator that results in true if either of the operands is true. ... Logic (from ancient Greek λόγος (logos), meaning reason) is the study of arguments. ...

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 "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. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

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. Dissolving table salt in water In chemistry, a solution is one or more substance (the solute) dissolved in another substance (the solvent) forming a homogenous mixture. ... In mathematics, any integer (whole number) is either even or odd. ...

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 mathematics, any integer (whole number) is either even or odd. ... Logic (from ancient Greek λόγος (logos), meaning reason) is the study of arguments. ... In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ... AND Logic Gate Logical conjunction (usual symbol and) is a logical operator that results in true if both of the operands are true. ... In logic, statements p and q are logically equivalent if they have the same logical content. ...

In symbolic logic, we use the existential quantifier "∃" (a backwards letter "E" in a sans-serif font) to indicate existential quantification. Thus if P(a, b, c) is the predicate "a·b = c" and N is the set of natural numbers, then Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... The letter E is the fifth letter in the Latin alphabet. ... In typography, serifs are the small features at the end of strokes within letters. ... The notion of a set is one of the most important and fundamental concepts in modern mathematics. ...

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. In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ...

Properties

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