# Dictionary Definition

every adj

1 each and all of a series of entities or
intervals as specified; "every third seat"; "every two hours" [syn:
every(a)]

2 (used of count nouns) each and all of the
members of a group considered singly and without exception; "every
person is mortal"; "every party is welcome"; "had every hope of
success"; "every chance of winning" [syn: every(a)]

# User Contributed Dictionary

## English

### Determiner

- All of a countable group, without exception.
- Every person in the room stood and cheered.

- Used with ordinal
numbers to denote those items whose position is divisible by
the corresponding cardinal
number, or a portion of equal size to that set.
- Every third bead was red, and the rest were blue. The sequence was thus red, blue, blue, red, blue, blue etc.
- Decimation originally meant the execution of every tenth soldier in a unit.

#### Synonyms

#### Antonyms

#### Related terms

#### Translations

- Albanian: cilido
- Arabic: (kull)
- trreq Armenian
- Azerbaijani: hər
- Bulgarian: всеки (vseki)
- Catalan: cada
- Chinese: 每 (měi)
- trreq Croatian
- Czech: každý (1) (2)
- Danish: hver
- Dutch: elk, ieder
- Esperanto: ĉiu
- Estonian: kõik
- Finnish: jokainen (1.), joka (2.)
- French: chaque
- Georgian: ყველა (qvela)
- German: jeder, jede, jedes
- Greek: κάθε (káthe)
- trreq Hebrew
- trreq Hindi
- Hungarian: minden
- trreq Icelandic
- trreq Ido
- Indonesian: setiap
- Irish: gach
- Italian: ogni
- Japanese: あらゆる (arayuru), 全ての (すべての, subete no)
- Korean: 각 (gak)
- Kurdish:
- Latin: quisque, omnis
- trreq Latvian
- trreq Lithuanian
- Latvian: katrs
- Lithuanian: kiekvienas
- trreq Mongolian
- Norwegian: hver
- trreq Old English
- Persian: هر
- Polish: każdy, każda, każde
- Portuguese: cada
- Romanian: fiecare
- Russian: каждый (každyj)
- Scottish Gaelic: a h-uile
- Slovak: každý
- Slovene: vsak , vsaka , vsako
- Spanish: cada
- Swahili: kila
- Swedish: varje (1.), var (2.)
- Thai: (tóok)
- Turkish: her
- Vietnamese: mỗi
- Welsh: pob
- Yiddish: יעדער (yeder)

### See also

# Extensive Definition

In predicate
logic, universal quantification is an attempt to formalize the
notion that something (a logical
predicate) is true for everything, or every relevant thing. The
resulting statement is a universally quantified statement, and we
have universally quantified over the predicate. In symbolic
logic, the universal quantifier (typically ∀) is the symbol
used to denote universal quantification, and is often informally
read as "given any" or "for all".

Quantification in general is covered in the
article on quantification, while
this article discusses universal quantification specifically.

Compare this with existential
quantification, which says that something is true for at least
one thing.

## Basics

Suppose you wish to say- 2·0 = 0 + 0, and 2·1 = 1 + 1, and 2·2 = 2 + 2, etc.

- For all natural numbers n, 2·n = n + n.

Notice that this statement is really more precise
than the original one. It may seem obvious that the phrase "etc" 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 universal quantification, on the other
hand, the natural numbers are mentioned explicitly.

This particular example is true,
because you could put any natural number in for n and the statement
"2·n = n + n" would be true. In contrast, "For all natural numbers
n, 2·n > 2 + n" is false,
because if you replace n with, say, 1 you get the false statement
"2·1 > 2 + 1". It doesn't matter that "2·n > 2 + n" is true
for most natural numbers n: even the existence of a single counterexample is enough
to prove the universal quantification false.

On the other hand, "For all composite
numbers n, 2·n > 2 + n" is true, because none of the
counterexamples are composite numbers. This indicates the
importance of the domain
of discourse, which specifies which values 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 universal quantification, you do this
with a logical
conditional. For example, "For all composite numbers n, 2·n
> 2 + n" is logically
equivalent to "For all natural numbers n, if n is composite,
then 2·n > 2 + n". Here the "if ... then" construction indicates
the logical conditional.

In symbolic
logic, we use the universal quantifier symbol \forall (an
upside-down
letter "A" in
a sans-serif font,
Unicode 2200) to indicate universal quantification. Thus if P(n) is
the predicate "2·n > 2 + n" and N is the set of natural numbers, then

- \forall n\!\in\!\mathbf\; P(n)

is the (false) statement

- For all natural numbers n, 2·n > 2 + n.

Similarly, if Q(n) is the predicate "n is
composite", then

- \forall n\!\in\!\mathbf\; \bigl( Q(n) \rightarrow P(n) \bigr)

is the (true) statement

- For all natural numbers n, if n is composite, then 2·n > 2 + n,

- For all composite numbers n, 2·n > 2 + n.

Several variations in the notation for
quantification (which apply to all forms) can be found in the
quantification
article. But there is a special notation used only for universal
quantification, which we also give here:

- (n\mathbf)\, P(n)

The parentheses indicate universal quantification
by default.

## Properties

### Negation

Note that a quantified propositional function is
a statement; thus, like statements, quantified functions can be
negated. The notation mathematicians and logicians utilize to
denote negation is: \lnot\ .

For example, let P(x) be the propositional
function "x is married"; then, for a Universe of Discourse X of all
living human beings, consider the universal quantification "Given
any living person x, that person is married":

- \forall\mathbf\, P(x)

A few seconds' thought demonstrates this as
irrevocably false; then, truthfully, we may say, "It is not the
case that, given any living person x, that person is married", or,
symbolically:

- \lnot\ \forall\mathbf\, P(x).

Take a moment and consider what, exactly,
negating the universal quantifier means: if the statement is not
true for every element of the Universe of Discourse, then there
must be at least one element for which the statement is false. That
is, the negation of \forall\mathbf\, P(x) is logically equivalent
to "There exists a living person x such that he is not married",
or:

- \exists\mathbf\, \lnot P(x)

Generally, then, the negation of a propositional
function's universal quantification is an existential
quantification of that propositional function's negation;
symbolically,

- \lnot\ \forall\mathbf\, P(x) \equiv\ \exists\mathbf\, \lnot P(x)

A common error is writing "all persons are not
married" (i.e. "there exists no person who is married") when one
means "not all persons are married" (i.e. "there exists a person
who is not married"):

- \lnot\ \exists\mathbf\, P(x) \equiv\ \forall\mathbf\, \lnot P(x) \not\equiv\ \lnot\ \forall\mathbf\, P(x) \equiv\ \exists\mathbf\, \lnot P(x)

### Rules of inference

A rule of
inference is a rule justifying a logical step from hypothesis
to conclusion. There are several rules of inference which utilize
the universal quantifier.

Universal
instantiation concludes that, if the propositional function is
known to be universally true, then it must be true for any
arbitrary element of the Universe of Discourse. Symbolically, this
is represented as

- \forall\mathbf\, P(x) \to\ P(c)

where c is a completely arbitrary element of the
Universe of Discourse.

Universal
generalization concludes the propositional function must be
universally true if it is true for any arbitrary element of the
Universe of Discourse. Symbolically, for an arbitrary c,

- P(c) \to\ \forall\mathbf\, P(x).

It is especially important to note c must be
completely arbitrary; else, the logic does not follow: if c is not
arbitrary, and is instead a specific element of the Universe of
Discourse, then P(c) only implies an existential quantification of
the propositional function.

## References

- Fundamentals of Mathematical Logic
- Proof in Mathematics: An Introduction (ch. 2)

every in Czech: Univerzální kvantifikátor

every in Danish: Alkvantor

every in German: Quantor

every in Spanish: Cuantificador universal

every in Italian: Quantificatore universale
(simbolo)

every in Hungarian: Univerzális
kvantifikáció

every in Dutch: Universaliteit

every in Japanese: 全称記号

every in Polish: Kwantyfikator ogólny

every in Portuguese: Quantificação
universal

every in Russian: Квантор всеобщности

every in Finnish: Universaalikvanttori

every in Swedish: Allkvantifikator

every in Chinese: 全称量化