Universal quantification

Universal quantification

Type	Quantifier
Field	Mathematical logic
Statement	$\forall xP(x)$is true when$P(x)$is true for all values of$x$.
Symbolic statement	$\forall xP(x)$

Inmathematical logic,auniversal quantificationis a type ofquantifier,alogical constantwhich isinterpretedas "given any","for all",or"for any".It expresses that apredicatecan besatisfiedby everymemberof adomain of discourse.In other words, it is thepredicationof apropertyorrelationto every member of the domain. Itassertsthat a predicate within thescopeof a universal quantifier is true of everyvalueof apredicate variable.

It is usually denoted by theturned A(∀)logical operatorsymbol,which, when used together with a predicate variable, is called auniversal quantifier( "∀x","∀(x)",or sometimes by"(x)"alone). Universal quantification is distinct fromexistentialquantification( "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 onquantification (logic).The universal quantifier is encoded asU+2200∀FOR ALLinUnicode,and as\forallinLaTeXand related formula editors.

Basics

Suppose it is given that

2·0 = 0 + 0, and 2·1 = 1 + 1, and2·2 = 2 + 2,etc.

This would seem to be alogical conjunctionbecause of the repeated use of "and". However, the "etc." cannot be interpreted as a conjunction informal logic.Instead, the statement must be rephrased:

For all natural numbersn,one has 2·n=n+n.

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 includesnatural numbers,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 istrue,because any natural number could be substituted fornand the statement "2·n=n+n"would be true. In contrast,

For all natural numbersn,one has 2·n> 2 +n

isfalse,because ifnis substituted with, for instance, 1, the statement "2·1 > 2 + 1" is false. It is immaterial that "2·n> 2 +n"is true formostnatural numbersn:even the existence of a singlecounterexampleis enough to prove the universal quantification false.

On the other hand, for allcomposite numbersn,one has 2·n> 2 +n is true, because none of the counterexamples are composite numbers. This indicates the importance of thedomain of discourse,which specifies which valuesncan take.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 alogical conditional.For example,

For all composite numbersn,one has 2·n> 2 +n

islogically equivalentto

For all natural numbersn,ifnis composite, then 2·n> 2 +n.

Here the "if... then" construction indicates the logical conditional.

Notation

Insymbolic logic,the universal quantifier symbol$\forall$(a turned "A"in asans-seriffont, Unicode U+2200) is used to indicate universal quantification. It was first used in this way byGerhard Gentzenin 1935, by analogy withGiuseppe Peano's$\exists$(turned E) notation forexistential quantificationand the later use of Peano's notation byBertrand Russell.

For example, ifP(n) is the predicate "2·n> 2 +n"andNis thesetof natural numbers, then

$\forall n\!\in \!\mathbb {N} \;P(n)$

is the (false) statement

"for all natural numbersn,one has 2·n> 2 +n".

Similarly, ifQ(n) is the predicate "nis composite ", then

$\forall n\!\in \!\mathbb {N} \;{\bigl (}Q(n)\rightarrow P(n){\bigr )}$

is the (true) statement

"for all natural numbersn,ifnis composite, then2·n> 2 + n".

Several variations in the notation for quantification (which apply to all forms) can be found in theQuantifierarticle.

Properties

Negation

The negation of a universally quantified function is obtained by changing the universal quantifier into anexistential quantifierand negating the quantified formula. That is,

$\lnot \forall x\;P(x)\quad {\text{is equivalent to}}\quad \exists x\;\lnot P(x)$

where$\lnot$denotesnegation.

For example, ifP(x)is thepropositional function"xis married ", then, for thesetXof all living human beings, the universal quantification

Given any living personx,that person is married

is written

$\forall x\in X\,P(x)$

This statement is false. Truthfully, it is stated that

It is not the case that, given any living personx,that person is married

or, symbolically:

$\lnot \ \forall x\in X\,P(x)$.

If the functionP(x)is not true foreveryelement ofX,then there must be at least one element for which the statement is false. That is, the negation of$\forall x\in X\,P(x)$is logically equivalent to "There exists a living personxwho is not married ", or:

$\exists x\in X\,\lnot P(x)$

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" ):

$\lnot \ \exists x\in X\,P(x)\equiv \ \forall x\in X\,\lnot P(x)\not \equiv \ \lnot \ \forall x\in X\,P(x)\equiv \ \exists x\in X\,\lnot P(x)$

Other connectives

The universal (and existential) quantifier moves unchanged across thelogical connectives∧,∨,→,and↚,as long as the other operand is not affected; that is:

${\begin{aligned}P(x)\land (\exists {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \exists {y}{\in }\mathbf {Y} \,(P(x)\land Q(y))\\P(x)\lor (\exists {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \exists {y}{\in }\mathbf {Y} \,(P(x)\lor Q(y)),&{\text{provided that }}\mathbf {Y} \neq \emptyset \\P(x)\to (\exists {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \exists {y}{\in }\mathbf {Y} \,(P(x)\to Q(y)),&{\text{provided that }}\mathbf {Y} \neq \emptyset \\P(x)\nleftarrow (\exists {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \exists {y}{\in }\mathbf {Y} \,(P(x)\nleftarrow Q(y))\\P(x)\land (\forall {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \forall {y}{\in }\mathbf {Y} \,(P(x)\land Q(y)),&{\text{provided that }}\mathbf {Y} \neq \emptyset \\P(x)\lor (\forall {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \forall {y}{\in }\mathbf {Y} \,(P(x)\lor Q(y))\\P(x)\to (\forall {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \forall {y}{\in }\mathbf {Y} \,(P(x)\to Q(y))\\P(x)\nleftarrow (\forall {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \forall {y}{\in }\mathbf {Y} \,(P(x)\nleftarrow Q(y)),&{\text{provided that }}\mathbf {Y} \neq \emptyset \end{aligned}}$

Conversely, for the logical connectives↑,↓,↛,and←,the quantifiers flip:

${\begin{aligned}P(x)\uparrow (\exists {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \forall {y}{\in }\mathbf {Y} \,(P(x)\uparrow Q(y))\\P(x)\downarrow (\exists {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \forall {y}{\in }\mathbf {Y} \,(P(x)\downarrow Q(y)),&{\text{provided that }}\mathbf {Y} \neq \emptyset \\P(x)\nrightarrow (\exists {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \forall {y}{\in }\mathbf {Y} \,(P(x)\nrightarrow Q(y)),&{\text{provided that }}\mathbf {Y} \neq \emptyset \\P(x)\gets (\exists {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \forall {y}{\in }\mathbf {Y} \,(P(x)\gets Q(y))\\P(x)\uparrow (\forall {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \exists {y}{\in }\mathbf {Y} \,(P(x)\uparrow Q(y)),&{\text{provided that }}\mathbf {Y} \neq \emptyset \\P(x)\downarrow (\forall {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \exists {y}{\in }\mathbf {Y} \,(P(x)\downarrow Q(y))\\P(x)\nrightarrow (\forall {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \exists {y}{\in }\mathbf {Y} \,(P(x)\nrightarrow Q(y))\\P(x)\gets (\forall {y}{\in }\mathbf {Y} \,Q(y))&\equiv \ \exists {y}{\in }\mathbf {Y} \,(P(x)\gets Q(y)),&{\text{provided that }}\mathbf {Y} \neq \emptyset \\\end{aligned}}$

Rules of inference

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

Universal instantiationconcludes 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

$\forall {x}{\in }\mathbf {X} \,P(x)\to P(c)$

wherecis a completely arbitrary element of the universe of discourse.

Universal generalizationconcludes the propositional function must be universally true if it is true for any arbitrary element of the universe of discourse. Symbolically, for an arbitraryc,

$P(c)\to \ \forall {x}{\in }\mathbf {X} \,P(x).$

The elementcmust be completely arbitrary; else, the logic does not follow: ifcis 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.

The empty set

By convention, the formula$\forall {x}{\in }\emptyset \,P(x)$is always true, regardless of the formulaP(x); seevacuous truth.

Universal closure

Theuniversal closureof a formula φ is the formula with nofree variablesobtained by adding a universal quantifier for every free variable in φ. For example, the universal closure of

$P(y)\land \exists xQ(x,z)$

is

$\forall y\forall z(P(y)\land \exists xQ(x,z))$.

As adjoint

Incategory theoryand the theory ofelementary topoi,the universal quantifier can be understood as theright adjointof afunctorbetweenpower sets,theinverse imagefunctor of a function between sets; likewise, theexistential quantifieris theleft adjoint.

For a set$X$,let${\mathcal {P}}X$denote itspowerset.For any function$f:X\to Y$between sets$X$and$Y$,there is aninverse imagefunctor$f^{*}:{\mathcal {P}}Y\to {\mathcal {P}}X$between powersets, that takes subsets of the codomain offback to subsets of its domain. The left adjoint of this functor is the existential quantifier$\exists _{f}$and the right adjoint is the universal quantifier$\forall _{f}$.

That is,$\exists _{f}\colon {\mathcal {P}}X\to {\mathcal {P}}Y$is a functor that, for each subset$S\subset X$,gives the subset$\exists _{f}S\subset Y$given by

$\exists _{f}S=\{y\in Y\;|\;\exists x\in X.\ f(x)=y\quad \land \quad x\in S\},$

those$y$in the image of$S$under$f$.Similarly, the universal quantifier$\forall _{f}\colon {\mathcal {P}}X\to {\mathcal {P}}Y$is a functor that, for each subset$S\subset X$,gives the subset$\forall _{f}S\subset Y$given by

$\forall _{f}S=\{y\in Y\;|\;\forall x\in X.\ f(x)=y\quad \implies \quad x\in S\},$

those$y$whose preimage under$f$is contained in$S$.

The more familiar form of the quantifiers as used infirst-order logicis obtained by taking the functionfto be the unique function${\displaystyle!:X\to 1}$so that${\mathcal {P}}(1)=\{T,F\}$is the two-element set holding the values true and false, a subsetSis that subset for which thepredicate$S(x)$holds, and

${\begin{array}{rl}{\mathcal {P}}(!)\colon {\mathcal {P}}(1)&\to {\mathcal {P}}(X)\\T&\mapsto X\\F&\mapsto \{\}\end{array}}$

$\exists _{!}S=\exists x.S(x),$

which is true if$S$is not empty, and

$\forall _{!}S=\forall x.S(x),$

which is false if S is not X.

The universal and existential quantifiers given above generalize to thepresheaf category.

See also

• Existential quantification
• First-order logic
• List of logic symbols—for the Unicode symbol ∀

References

• Hinman, P. (2005).Fundamentals of Mathematical Logic.A K Peters.ISBN1-56881-262-0.
• Franklin, J.and Daoud, A. (2011).Proof in Mathematics: An Introduction.Kew Books.ISBN978-0-646-54509-7.{{cite book}}:CS1 maint: multiple names: authors list (link)(ch. 2)

External links

• The dictionary definition ofeveryat Wiktionary