====== Universal quantification ======

A [[glossary:proposition|proposition]] that is valid for all indicated objects.

For example:

> //All humans are mortal.//

Universal quantifications can be positive (as in the previous example) or negative, as the following:

> //No human lives forever.//


===== Description =====

A //positive// universal quantification is a proposition that is //true// for //all// elements that it refers to:

> //All// X //are// Y.

Or as logical formula:

> ''∀ 𝑥 ∈ 𝕏 : Y''
> (For all 𝑥 in the set 𝕏, Y is true)

==== Negation ====

A //negative// universal quantification, is a (true) statement describing a property that is //not true// for any of the elements of the set:

> //No// X //is// Y.

In a formula, this is expressed as follows:

> ''∄ 𝑥 ∈ 𝕏 : Y''
> (There exists //no// 𝑥 in the set 𝕏, which is Y)

<aside box>
**Note:** There are other forms of //negating// a universal quantification. These are however not //negative universal quantifications// in the sense that they can be used as such, e.g. in [[glossary:syllogism|syllogisms]].

For example, a positive universal quantification can be negated to signify that it is not true. This can look as follows (the brackets are optional and were only added for clarification):

> <u questionable "negated universal quantification">''¬(∀ 𝑥 ∈ 𝕏 : Y)''</u>
> (it is not [true], that for all 𝑥 in the set 𝕏, Y is true)

There is also a negative [[glossary:existential_quantification|existential quantification]], which looks like this:

> <u questionable "negative existential quantification">''∃ 𝑛 ∈ 𝕄 : ¬X''</u>
> (there exists [at least] one 𝑛 in the set 𝕄 which is //not// X)

</aside>

==== Vacuous truths ====

In principle, any //universal quantification// statement which has an [[glossary:antecedent|antecedent]] that refers to an [[wp>Empty set|empty set]] always holds (//vacuously//) //true//. For example:

> All //unicorns// (mythical creatures) are //immortal//.

Assuming that none of the mythical creatures that we refer to as "unicorns" exist, the statement is //true// because there are no unicorns that could actually die. As such, this statement is a <span maniculus "go to:">[[glossary:vacuous_truth|vacuous truth]]</span>.

==== Relation to existential quantifications ====

At first glance, a general statement of the form "all A are B" also seems to imply an existence statement of the form "some A are B". However, the latter also imply actual existence of any item to refers to. In order to be able to transform a universal quantification into an existential one, it must first be proven that it statement does not refer to an empty set.

> All A are B.
> [and at least one A exists].
> Then the following holds: Some A are B.

Such constraints can be found, for example, in the forms [[logic:inferences:modus_barbara:modus_barbari|Modus Barbari]] or [[logic:inferences:modus_celarent:modus_calemos|Modus Calemos]].

==== Falsification ====

To disprove a //universal quantification// statement, it is sufficient to find //a single// counterexample. This applies to both positive and negative propositions.

Because they are in principle easier to disprove, universal quantifications are more meaningful than existential ones - provided, of course, that it has not yet been disproved.

==== Verification ====

To verify //universal quantifications//, every single elements of the extension set must be examined. This is of course only possible with limited and relatively small sets or within formal systems like mathematics (e.g. by [[wp>Mathematical induction]]).

==== Distributivity ====

The //subject// of a //positive// universal quantification is [[glossary:distributivity|distributed]], i.e. it can be replaced by a subset of any set it describes. For example:

> All //cats// are //mammals//.

Also implies:

> All //long-hair cats// are mammals.
> All //cats in the neighbourhood// are mammals.
> //Findus// is a mammal.

In contrast, the //predicate// (here: "mammals") is //not// distributed: a statement like "<s invalid>all cats are dolphins</s>" (with //dolphins// being a subset of //mammals//) can obviously //not// be inferred.

In //negative// universal quantification statements, both //subject and predicate// are distributed:

> No //cat// is a //dog//.

Also implies:

> No //long-hair cat// is a //dachshund//.
> None of //the cats in my neighbourhood// is a //greyhound//.
> //Findus// is not //a guide dog//.

==== Relation of the whole to its parts ====

The principle "<i :la>[[logic:glossary:dictum_de_omni_et_nullo|dictum de omni et nullo]]</i>" (Lat.: "the maxim of all and none") states that what holds true in a (positive or negative) universal quantification is also true for every single element that the statement [[glossary:extension|extends]] to.

Obviously, a statement like "all horses are mammals" also implies that every single horse is a mammal, as likewise "no pig can fly" extends to every single pig (<span maniculus "see:"></span>[[glossary:distributivity|distributivity]]). However, there are specific situations where this transfer does not apply.

  * There are certain properties of groups that only [[glossary:emergence|emerge]] through the composition of the group or the interaction of the group members themselves. For example, a statement like "all the horses in this herd are differently coloured" only makes sense if the herd comprises more than one horse (<span maniculus "see:">[[logic:emergence:index|fallacies of emergence]]</span>).

  * Properties of a complex system are not necessarily (recognizably) inherent in any of its components (<span maniculus "see:">[[generalization:mereological_fallacy|mereological fallacy]]</span>).

  * If a population has a certain //statistical property//, it can not be inferred that every member of a population has this particular property. For example, even if a statement like "Americans like hamburgers" is true, it does not follow that every specific American will like hamburgers (<span maniculus "see:">[[mathematics:statistics:interpretation:ecological_fallacy|ecological fallacy]]</span>).

==== Symbols ====

Both in logic and mathematics, the symbol <span "for all …">''∀''</span> is normally used to denote the //universal quantifier//. This is pronounced as "for all … holds true". For example:

> ''∀ 𝑛 ∈ ℕ: 2·𝑛 = 𝑛 + 𝑛''
> (for all 𝑛, which are elements of the set of [[wp>Natural number|natural numbers]], it is true that 2·𝑛 is equal to 𝑛 + 𝑛).

For negative universal quantifiers, the exists symbol (''∃'') is used instead: either in the crossed-out variant (''∄'') or in combination with a "not" symbol (''¬∃''). In both cases, it is pronounced as: "there exists no...".

> ''∄ 𝑛 ∈ ℕ: 2·𝑛 < 𝑛''
> (there exists no element 𝑛 in the set of natural numbers for which 2·𝑛 is less than 𝑛).

FIXME **This article is still under construction.**

==== See also ====

  * <i :la>[[logic:glossary:dictum_de_omni_et_nullo|Dictum de omni et nullo]]</i>
  * [[glossary:distributivity|Distributivity]]
  * [[glossary:existential_quantification|Existential quantification]]
  * [[glossary:syllogism|Syllogism]]

==== More information ====

  * [[wp> Universal quantification]] on //Wikipedia//