Quantifiers and quantified expressions are a useful part of formal languages. They are useful because they let rigorous statements claim how widespread a criteria is. Two basic kinds of quantifiers used in predicate logic are universal and existential quantifiers. A universal quantifier states that all the elements considered fulfill the criteria. The universal quantifier is symbolized with "∀", an upside down "A", to stand for "all". An existence quantifier (symbolized with "∃") states that at least one element considered fits the criteria. The existential quantifier is symbolized with "∃", a backwards "E", to stand for "exists".
Quantifiers are also used in natural languages. Examples of quantifiers in English include "for all", "for some", "many", "few", "a lot", and "no".
This statement is infinitely long:
This is a problem for formal languages, since a formal statement needs to be finite in length. These problems can be avoided by using "universal quantification". This results in the following compact statement:
In the same way, we can shorten an infinite sequence of statements joined by "or" such as:
by rewriting it using "existential quantification":
The two quantifiers most widely used are the universal quantifier and the existence quantifier.
The universal quantifier is used to claim that for elements in a set, the elements all match some criteria. Usually, this statement "for all elements" is shortened to an "A" flipped upside down, which is "∀".
The existential quantifier is used to claim that for elements in a set, there exists at least one element that matches some criteria. Usually, this statement "there exists an element" is shortened to an "E" flipped upside down, which is "∃".
English statements can be often rewritten using symbols, predicates representing criteria, and quantifiers. One example is "Each of Peter's friends either likes to dance or likes to go to the beach". Let "X" be the set of all Peter's friends. Let "P"("x") be the predicate ""x" likes to dance". Let "Q"("x") be the predicate ""x" likes to go to the beach". We can rewrite the example using formal notation as formula_1. The statement can be read as "For every "x" that is a member of "X", "P" applies to "x" or "Q" applies to "x"."
There are other ways to use quantifiers in formal language. Each of the following statements below says the same thing as formula_2:
There are a few more ways to represent the universal quantifier:
Several statements above explicitly include "X", the set of elements that the quantifier applies to. This set of elements is also known as the range of quantification, or the universe of discourse. Some of the statements above do not include such a set. In this case, the set will have to be specified before the statement. For example, ""x" is an apple" must be stated before formula_12. In this case, we are making a statement that at least one apple fits the predicate "P".
Using quantifiers formally does not require using the symbol "x". In fact, other symbols, such as "y" and "z", can also be used. However, one must be careful not to use the same symbol to refer to two different things.
It is important to put quantifiers in the right order. Below is an example of an English sentence showing how meaning changes with order:
This statement is true. It states that every natural number has a square. However, if we turn the order of the quantifiers around:
then the statement is false. It claims that there is one natural number "s" that is the square of "every" natural number.
In certain circumstances, changing the order of quantifiers does not change the meaning of the statement. For instance:
There are also less common quantifiers used by mathematicians.
An example is the solution quantifier. It is used to state which elements solve a particular equation. The solution quantifier is represented by a § (section sign). For example, the following statement claims the squares of 0, 1 and 2 are smaller than 4. :formula_13
Other quantifiers include:
Term logic was developed by Aristotle. It was an early form of logic, and included quantification. The use of quantification was closer to that of natural language. This meant that statements in term logic with quantifiers were less suited for formal analysis. Term logic included quantifiers for "All", "Some" and "No" (none) in 4th century BC.
In 1879, Gottlob Frege created a notation for universal quantification. Unlike the modern notation, he would represent a universal quantification by writing a variable over a dimple in an otherwise straight line. Frege did not create a notation for existential quantification. Instead, he combined universal quantification and a number of negations to make an equivalent statement. Frege's use of quantification was not widely known until Bertrand Russell's 1903 "Principles of Mathematics".
In 1885, Charles Sanders Peirce and his student Oscar Howard Mitchell also created a notation for universal and existential quantifiers. They wrote Π and Σ where we now write ∀"x" and ∃"x". Pierce's notation was used by many mathematicians into the 1950s.
In 1897, William Ernest Johnson and Giuseppe Peano created another notation for universal and existential quantification. They were influenced by Pierce's previous quantification notation. Johnson and Peano used the simple ("x") for universal quantification, and ∃"x" for existential quantification. Peano's influence on math spread this notation across Europe.
In 1935, Gerhard Gentzen created the ∀ symbol for universal quantification. It was not widely used until the 1960s.
Universal quantifier
In mathematics and logic, the universal quantifier is a quantifier used to state that a proposition applies to all elements in the universe of discourse. An example that uses this quantifier would be the proposition "All men are mortal". Usually, a turned A (∀) is used to denote the universal quantifier, "for all x" is written as either "∀x", "∀(x)".
Predicate logic and syllogisms look at the properties of universal quantification.
Propositions can be falsified. To falsify a proposition which contains a universal quantifier, it is sufficient to find one element of the universe of discourse where the proposition is false. This element is known as a counterexample.
Existence quantifier
In mathematics and logic, the existence quantifier is a quantifier used to state that a proposition is true for at least one element in the universe of discourse. The existence quantifier is commonly written as formula_1 (a mirrored E), and is read as "there exists". An example involving an existence quantifier is the statement "some natural number is equal to 3+5", which can be written as formula_2.
In general, a statement of the form formula_3 is true if there is an "x" in the universe of discourse satisfying the predicate formula_4, and is false otherwise. An existence quantifier is different from a universal quantifier, which is used to state that a proposition is true for "all" elements in the universe of discourse.
Universe of discourse
In mathematics, the universe of discourse or domain of discourse is a set of all elements to which a function applies. A variable can take any of the values in its universe of discourse.
The term is also used informally.
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