====== 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//Β Y.
Or as logical formula:
> ''β π₯ β πβ―: Y''
> (For all π₯ in the set π, 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//Β Y.
In a formula, this is expressed as follows:
> ''β π₯ β πβ―: Y''
> (There exists //no// π₯ in the set π, which is Y)
==== 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 [[glossary:vacuous_truth|vacuous truth]].
==== 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 areΒ 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 "all cats are dolphins" (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 "[[logic:glossary:dictum_de_omni_et_nullo|dictum de omni et nullo]]" (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]]Β 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 ([[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 ([[logic:emergence:index|fallacies of emergence]]).
* Properties of a complex system are not necessarily (recognizably) inherent in any of its components ([[generalization:mereological_fallacy|mereological fallacy]]).
* 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 ([[mathematics:statistics:interpretation:ecological_fallacy|ecological fallacy]]).
==== Symbols ====
Both in logic and mathematics, the symbol ''β'' is normally used to denote the //universal quantifier//. This is pronounced as "for all β¦ holds true". For example:
> ''β π β β: 2Β·π = π + π''
> (for all π, which are elements of the set of [[wp>Natural number|natural numbers]], it is true that 2Β·π is equal to π + π).
For negative universal quantifiers, the exists symbol (''β'') is used instead: either in the crossed-out variant (''β'') or in combination with a "not" symbol (''Β¬β''). In both cases, it is pronounced as: "there exists no...".
> ''β π β β: 2Β·π < π''
> (there exists no element π in the set of natural numbers for which 2Β·π is less than π).
FIXME **This article is still under construction.**
==== See also ====
* [[logic:glossary:dictum_de_omni_et_nullo|Dictum de omni et nullo]]
* [[glossary:distributivity|Distributivity]]
* [[glossary:existential_quantification|Existential quantification]]
* [[glossary:syllogism|Syllogism]]
==== More information ====
* [[wp> Universal quantification]] on //Wikipedia//