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Universal quantification

A 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)

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 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):

¬(∀ 𝑥 ∈ 𝕏 : Y)
(it is not [true], that for all 𝑥 in the set 𝕏, Y is true)

There is also a negative existential quantification, which looks like this:

∃ 𝑛 ∈ 𝕄 : ¬X
(there exists [at least] one 𝑛 in the set 𝕄 which is not X)

Vacuous truths

In principle, any universal quantification statement which has an antecedent that refers to an 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  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 are B.

Such constraints can be found, for example, in the forms Modus Barbari or 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 Mathematical induction).

Distributivity

The subject of a positive universal quantification is 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 “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 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 ( distributivity). However, there are specific situations where this transfer does not apply.

  • There are certain properties of groups that only 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 ( fallacies of emergence).
  • Properties of a complex system are not necessarily (recognizably) inherent in any of its components ( 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 ( ecological fallacy).

Symbols

Both in logic and mathematics, the symbol 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 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

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