View source for Universal quantification

{{Short description|Mathematical use of "for all"}}
{{for|"for every" in computing|Foreach loop}}
{{Infobox mathematical statement
| name = Universal quantification
| type = [[Quantification (logic)|Quantifier]]
| field = [[Mathematical logic]]
| statement = <math>\forall xP(x)</math> is true when <math>P(x)</math> is true for all values of <math>x</math>.
| symbolic statement = <math>\forall xP(x)</math>
}}

In [[mathematical logic]], a '''universal quantification''' is a type of [[Quantification (logic)|quantifier]], a [[logical constant]] which is [[interpretation (logic)|interpreted]] as "'''given any'''", "'''for all'''", "'''for every'''", or "'''given an [[Arbitrariness#Mathematics|arbitrary]] element'''". It expresses that a [[predicate (mathematical logic)|predicate]] can be [[satisfiability|satisfied]] by every [[element (mathematics)|member]] of a [[domain of discourse]]. In other words, it is the [[Predicate (mathematical logic)|predication]] of a [[property (philosophy)|property]] or [[binary relation|relation]] to every member of the domain. It [[logical assertion|asserts]] that a predicate within the [[scope (logic)|scope]] of a universal quantifier is true of every [[Valuation (logic)|value]] of a [[predicate variable]].

It is usually denoted by the [[turned A]] (∀) [[logical connective|logical operator]] [[Symbol (formal)|symbol]], which, when used together with a predicate variable, is called a '''universal quantifier''' ("{{math|∀''x''}}", "{{math|∀(''x'')}}", or sometimes by "{{math|(''x'')}}" alone). Universal quantification is distinct from [[existential quantification|''existential'' quantification]] ("there exists"), which only asserts that the property or relation holds for at least one member of the domain.

Quantification in general is covered in the article on [[quantification (logic)]]. The universal quantifier is encoded as {{unichar|2200|FOR ALL}} in [[Unicode]], and as <code>\forall</code> in [[LaTeX]] and related formula editors.

== Basics ==
Suppose it is given that
<blockquote>2·0 = 0 + 0, and 2·1 = 1 + 1, and {{nowrap|1=2·2 = 2 + 2}},  ..., and 2 · 100 = 100 + 100, and ..., etc.</blockquote>
This would seem to be an infinite [[logical conjunction]] because of the repeated use of "and".  However, the "etc." cannot be interpreted as a conjunction in [[formal logic]], Instead, the statement must be rephrased:
<blockquote>For all natural numbers ''n'', one has 2·''n'' = ''n'' + ''n''.</blockquote>
This is a single statement using universal quantification.

This statement can be said to be more precise than the original one. While the "etc." informally includes [[natural number]]s, and nothing more, this was not rigorously given. In the universal quantification, on the other hand, the natural numbers are mentioned explicitly.

This particular example is [[true (logic)|true]], because any natural number could be substituted for ''n'' and the statement "2·''n'' = ''n'' + ''n''" would be true. In contrast,
<blockquote>For all natural numbers ''n'', one has 2·''n'' > 2 + ''n''</blockquote>
is [[false (logic)|false]], because if ''n'' is substituted with, for instance, 1, the statement "2·1 > 2 + 1" is false. It is immaterial 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 number]]s ''n'', one has 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'' can take.<ref group="note">Further information on using domains of discourse with quantified statements can be found in the [[Quantification (logic)]] article.</ref> In particular, note that if the domain of discourse is restricted to consist only of those objects that satisfy a certain predicate, then for universal quantification this requires a [[logical conditional]]. For example,
<blockquote>For all composite numbers ''n'', one has 2·''n'' > 2 + ''n''</blockquote>
is [[logically equivalent]] to
<blockquote>For all natural numbers ''n'', if ''n'' is composite, then 2·''n'' > 2 + ''n''.</blockquote>
Here the "if ... then" construction indicates the logical conditional.

=== Notation ===
In [[First-order logic|symbolic logic]], the universal quantifier symbol <math> \forall </math> (a turned&nbsp;"[[A]]" in a [[sans-serif]] font, Unicode&nbsp;U+2200) is used to indicate universal quantification. It was first used in this way by [[Gerhard Gentzen]] in 1935, by analogy with [[Giuseppe Peano]]'s <math>\exists</math> (turned E) notation for [[existential quantification]] and the later use of Peano's notation by [[Bertrand Russell]].<ref>{{cite web|title=Earliest Uses of Symbols of Set Theory and Logic|url=http://jeff560.tripod.com/set.html|work=Earliest Uses of Various Mathematical Symbols|first=Jeff|last=Miller}}</ref>

For example, if ''P''(''n'') is the predicate "2·''n'' > 2 + ''n''" and '''N''' is the [[Set (mathematics)|set]] of natural numbers, then
: <math> \forall n\!\in\!\mathbb{N}\; P(n) </math>
is the (false) statement
:"for all natural numbers ''n'', one has 2·''n'' > 2 + ''n''".

Similarly, if ''Q''(''n'') is the predicate "''n'' is composite", then
: <math> \forall n\!\in\!\mathbb{N}\; \bigl( Q(n) \rightarrow  P(n) \bigr) </math>
is the (true) statement
:"for all natural numbers ''n'', if ''n'' is composite, then {{nowrap|2·''n'' > 2 + n}}".

Several variations in the notation for quantification (which apply to all forms) can be found in the ''[[Quantifier (logic)#Notation|Quantifier]]'' article.

== Properties ==

<!-- ''We need a list of algebraic properties of universal quantification, such as distributivity over conjunction, and so on. Also rules of inference.'' -->

=== Negation ===
The negation of a universally quantified function is obtained by changing the universal quantifier into an [[existential quantifier]] and negating the quantified formula. That is, 
:<math>\lnot \forall x\; P(x)\quad\text {is equivalent to}\quad \exists x\;\lnot P(x) </math>
where <math>\lnot</math> denotes [[negation]].

For example, if {{math|''P''(''x'')}} is the [[propositional function]] "{{math|''x''}} is married", then, for the [[set (mathematics)|set]] {{mvar|X}} of all living human beings, the universal quantification
<blockquote>Given any living person {{math|''x''}}, that person is married</blockquote>
is written
:<math>\forall x \in X\, P(x)</math>

This statement is false. Truthfully, it is stated that
<blockquote>It is not the case that, given any living person {{mvar|''x''}}, that person is married</blockquote>
or, symbolically:
:<math>\lnot\ \forall x \in X\, P(x)</math>.

If the function {{math|''P''(''x'')}} is not true for ''every'' element of {{mvar|X}}, then there must be at least one element for which the statement is false. That is, the negation of <math>\forall x \in X\, P(x)</math> is logically equivalent to "There exists a living person {{math|''x''}} who is not married", or:
:<math>\exists x \in X\, \lnot P(x)</math>

It is erroneous to confuse "all persons are not married" (i.e. "there exists no person who is married") with "not all persons are married" (i.e. "there exists a person who is not married"):
:<math>\lnot\ \exists x \in X\, P(x) \equiv\ \forall x \in X\, \lnot P(x) \not\equiv\ \lnot\ \forall x\in X\, P(x) \equiv\ \exists x \in X\, \lnot P(x)</math>

=== Other connectives ===
The universal (and existential) quantifier moves unchanged across the [[logical connective]]s [[logical conjunction|∧]], [[logical disjunction|∨]], [[material conditional|→]], and [[converse nonimplication|↚]], as long as the other operand is not affected;<ref>that is, if the variable <math>y</math> does not occur free in the formula <math>P(x)</math> in the equivalences below</ref> that is:

:<math>\begin{align}
P(x) \land (\exists{y}{\in}\mathbf{Y}\, Q(y)) &\equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \land Q(y)) \\
P(x) \lor  (\exists{y}{\in}\mathbf{Y}\, Q(y)) &\equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \lor Q(y)),& \text{provided that } \mathbf{Y}\neq \emptyset \\
P(x) \to   (\exists{y}{\in}\mathbf{Y}\, Q(y)) &\equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \to Q(y)),& \text{provided that } \mathbf{Y}\neq \emptyset \\
P(x) \nleftarrow (\exists{y}{\in}\mathbf{Y}\, Q(y)) &\equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \nleftarrow Q(y)) \\
P(x) \land (\forall{y}{\in}\mathbf{Y}\, Q(y)) &\equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \land Q(y)),& \text{provided that } \mathbf{Y}\neq \emptyset \\
P(x) \lor  (\forall{y}{\in}\mathbf{Y}\, Q(y)) &\equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \lor Q(y)) \\
P(x) \to   (\forall{y}{\in}\mathbf{Y}\, Q(y)) &\equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \to Q(y)) \\
P(x) \nleftarrow (\forall{y}{\in}\mathbf{Y}\, Q(y)) &\equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \nleftarrow Q(y)),& \text{provided that } \mathbf{Y}\neq \emptyset
\end{align}</math>

Conversely, for the logical connectives [[Sheffer stroke|↑]], [[Logical NOR|↓]], [[Material nonimplication|↛]], and [[converse implication|←]], the quantifiers flip:

:<math>\begin{align}
P(x) \uparrow (\exists{y}{\in}\mathbf{Y}\, Q(y)) & \equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \uparrow Q(y)) \\
P(x) \downarrow  (\exists{y}{\in}\mathbf{Y}\, Q(y)) & \equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \downarrow Q(y)),& \text{provided that } \mathbf{Y}\neq \emptyset \\
P(x) \nrightarrow (\exists{y}{\in}\mathbf{Y}\, Q(y)) & \equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \nrightarrow Q(y)),& \text{provided that } \mathbf{Y}\neq \emptyset \\
P(x) \gets (\exists{y}{\in}\mathbf{Y}\, Q(y)) & \equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \gets Q(y)) \\
P(x) \uparrow (\forall{y}{\in}\mathbf{Y}\, Q(y)) & \equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \uparrow Q(y)),& \text{provided that } \mathbf{Y}\neq \emptyset \\
P(x) \downarrow  (\forall{y}{\in}\mathbf{Y}\, Q(y)) & \equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \downarrow Q(y)) \\
P(x) \nrightarrow (\forall{y}{\in}\mathbf{Y}\, Q(y)) & \equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \nrightarrow Q(y)) \\
P(x) \gets (\forall{y}{\in}\mathbf{Y}\, Q(y)) & \equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \gets Q(y)),& \text{provided that } \mathbf{Y}\neq \emptyset \\
\end{align}</math>
<!-- What about:
*[[logical biconditional|Biconditional (if and only if) (xnor)]] (<math>\leftrightarrow</math>, <math>\equiv</math>, or <math>=</math>)
*[[Exclusive or|Exclusive disjunction (xor)]] (<math>\not\leftrightarrow</math>)
-->

=== 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

:<math> \forall{x}{\in}\mathbf{X}\, P(x) \to P(c)</math>

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'',

:<math> P(c) \to\ \forall{x}{\in}\mathbf{X}\, P(x).</math>

The element&nbsp;''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.
<!-- ''Discuss universally quantified types in [[type theory]].'' -->

=== The empty set ===

By convention, the formula <math>\forall{x}{\in}\emptyset \, P(x)</math> is always true, regardless of the formula ''P''(''x''); see [[vacuous truth]].

== Universal closure ==

The '''universal closure''' of a formula φ is the formula with no [[free variable]]s obtained by adding a universal quantifier for every free variable in φ. For example, the universal closure of
:<math>P(y) \land \exists x Q(x,z)</math>
is
:<math>\forall y \forall z ( P(y) \land \exists x Q(x,z))</math>.

== As adjoint ==
In [[category theory]] and the theory of [[elementary topos|elementary topoi]], the universal quantifier can be understood as the [[right adjoint]] of a [[functor]] between [[power set]]s, the [[inverse image]] functor of a function between sets; likewise, the [[existential quantifier]] is the [[left adjoint]].<ref>[[Saunders Mac Lane]], Ieke Moerdijk, (1992) ''Sheaves in Geometry and Logic'' Springer-Verlag. {{isbn|0-387-97710-4}} ''See page 58''</ref>

For a set <math>X</math>, let <math>\mathcal{P}X</math> denote its [[powerset]].  For any function <math>f:X\to Y</math> between sets <math>X</math> and <math>Y</math>, there is an [[inverse image]] functor <math>f^*:\mathcal{P}Y\to \mathcal{P}X</math> between powersets, that takes subsets of the codomain of ''f'' back to subsets of its domain. The left adjoint of this functor is the existential quantifier <math>\exists_f</math> and the right adjoint is the universal quantifier <math>\forall_f</math>.

That is, <math>\exists_f\colon \mathcal{P}X\to \mathcal{P}Y</math> is a functor that, for each subset <math>S \subset X</math>, gives the subset <math>\exists_f S \subset Y</math> given by
:<math>\exists_f S =\{ y\in Y \;|\; \exists x\in X.\ f(x)=y \quad\land\quad x\in S \},</math>
those <math>y</math> in the image of <math>S</math> under <math>f</math>.  Similarly, the universal quantifier <math>\forall_f\colon \mathcal{P}X\to \mathcal{P}Y</math> is a functor that, for each subset <math>S \subset X</math>, gives the subset <math>\forall_f S \subset Y</math> given by
:<math>\forall_f S =\{ y\in Y \;|\; \forall x\in X.\ f(x)=y \quad\implies\quad x\in S \},</math>
those <math>y</math> whose preimage under <math>f</math> is contained in <math>S</math>.

The more familiar form of the quantifiers as used in [[first-order logic]] is obtained by taking the function ''f'' to be the unique function <math>!:X \to 1</math> so that <math>\mathcal{P}(1) = \{T,F\}</math> is the two-element set holding the values true and false, a subset ''S'' is that subset for which the [[predicate (mathematical logic)|predicate]] <math>S(x)</math> holds, and
:<math>\begin{array}{rl}\mathcal{P}(!)\colon \mathcal{P}(1) & \to \mathcal{P}(X)\\ T &\mapsto X \\ F &\mapsto \{\}\end{array}</math>
:<math>\exists_! S = \exists x. S(x),</math>
which is true if <math>S</math> is not empty, and
:<math>\forall_! S = \forall x. S(x),</math>
which is false if S is not X.

The universal and existential quantifiers given above generalize to the [[presheaf category]].

== See also ==
* [[Existential quantification]]
* [[First-order logic]]
* [[List of logic symbols]]—for the Unicode symbol ∀

== Notes ==
<references group="note" />

== References ==
{{Reflist}}

*{{cite book | author = Hinman, P. | title = Fundamentals of Mathematical Logic | publisher = [[A K Peters]] | year = 2005 | isbn = 1-56881-262-0}}
*{{cite book | author = [[James Franklin (philosopher)|Franklin, J.]] and Daoud, A. | title = Proof in Mathematics: An Introduction | url = http://www.maths.unsw.edu.au/~jim/proofs.html | publisher = Kew Books | year = 2011 | isbn = 978-0-646-54509-7}} (ch. 2)

== External links ==
* {{Wiktionary-inline|every}}

{{Common logical symbols}}
{{Mathematical logic}}

[[Category:Logic symbols]]
[[Category:Logical expressions]]
[[Category:Quantifier (logic)]]