### For all

In predicate logic, a **universal quantification** is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a propositional function can be satisfied by every member of a domain of discourse. In other terms, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable.

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

Quantification in general is covered in the article on quantification. Symbols are encoded U+2200 ∀ for all (HTML: `∀`

`∀`

as a mathematical symbol).

## Contents

## Basics

Suppose it is given that

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

This would seem to be a 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:

For all natural numbersn, 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 includes natural 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 is true, because any natural number could be substituted for *n* and the statement "2·*n* = *n* + *n*" would be true. In contrast,

For all natural numbersn, 2·n> 2 +n

is 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 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* can take.^{[note 1]} 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,

For all composite numbersn, 2·n> 2 +n

is logically equivalent to

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

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

### Notation

In symbolic logic, the universal quantifier symbol $\backslash forall$ (an inverted "A" in a sans-serif font, Unicode 0x2200) is used to indicate universal quantification.^{[1]}

For example, if *P*(*n*) is the predicate "2·*n* > 2 + *n*" and **N** is the set of natural numbers, then:

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

is the (false) statement:

For all natural numbersn, 2·n> 2 +n.

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

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

is the (true) statement:

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

and since "*n* is composite" implies that *n* must already be a natural number, we can shorten this statement to the equivalent:

- $\backslash forall\; n\backslash ;\; \backslash bigl(\; Q(n)\; \backslash rightarrow\; P(n)\; \backslash bigr)$

For all composite numbersn, 2·n> 2 +n.

Several variations in the notation for quantification (which apply to all forms) can be found in the quantification article. There is a special notation used only for universal quantification, which is given:

- $(n\{\backslash in\}\backslash mathbb\{N\})\backslash ,\; P(n)$

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 most mathematicians and logicians utilize to denote negation is: $\backslash lnot\backslash $. However, some (such as Douglas Hofstadter) use the tilde (~).

For example, if P(*x*) is the propositional function "x is married", then, for a Universe of Discourse X of all living human beings, the universal quantification

Given any living personx, that person is married

is given:

- $\backslash forall\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)$

It can be seen that this is irrevocably false. Truthfully, it is stated that

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

or, symbolically:

- $\backslash lnot\backslash \; \backslash forall\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)$.

If the statement is not true for *every* element of the Universe of Discourse, then, presuming the universe of discourse is non-empty, there must be at least one element for which the statement is false. That is, the negation of $\backslash forall\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)$ is logically equivalent to "There exists a living person *x* such that he is not married", or:

- $\backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; \backslash lnot\; P(x)$

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

- $\backslash lnot\backslash \; \backslash forall\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)\; \backslash equiv\backslash \; \backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; \backslash lnot\; P(x)$

It is erroneous to state "all persons are not married" (i.e. "there exists no person who is married") when it is meant that "not all persons are married" (i.e. "there exists a person who is not married"):

- $\backslash lnot\backslash \; \backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)\; \backslash equiv\backslash \; \backslash forall\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; \backslash lnot\; P(x)\; \backslash not\backslash equiv\backslash \; \backslash lnot\backslash \; \backslash forall\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)\; \backslash equiv\backslash \; \backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; \backslash lnot\; P(x)$

### Other connectives

The universal (and existential) quantifier moves unchanged across the logical connectives ∧, ∨, →, and $\backslash nleftarrow$, as long as the other operand is not affected; that is:

- $P(x)\; \backslash land\; (\backslash exists\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; Q(y))\; \backslash equiv\backslash \; \backslash exists\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; (P(x)\; \backslash land\; Q(y))$
- $P(x)\; \backslash lor\; (\backslash exists\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; Q(y))\; \backslash equiv\backslash \; \backslash exists\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; (P(x)\; \backslash lor\; Q(y)),~\backslash mathrm\{provided~that\}~\backslash mathbf\{Y\}\backslash neq\; \backslash emptyset$
- $P(x)\; \backslash to\; (\backslash exists\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; Q(y))\; \backslash equiv\backslash \; \backslash exists\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; (P(x)\; \backslash to\; Q(y)),~\backslash mathrm\{provided~that\}~\backslash mathbf\{Y\}\backslash neq\; \backslash emptyset$
- $P(x)\; \backslash nleftarrow\; (\backslash exists\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; Q(y))\; \backslash equiv\backslash \; \backslash exists\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; (P(x)\; \backslash nleftarrow\; Q(y))$
- $P(x)\; \backslash land\; (\backslash forall\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; Q(y))\; \backslash equiv\backslash \; \backslash forall\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; (P(x)\; \backslash land\; Q(y)),~\backslash mathrm\{provided~that\}~\backslash mathbf\{Y\}\backslash neq\; \backslash emptyset$
- $P(x)\; \backslash lor\; (\backslash forall\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; Q(y))\; \backslash equiv\backslash \; \backslash forall\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; (P(x)\; \backslash lor\; Q(y))$
- $P(x)\; \backslash to\; (\backslash forall\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; Q(y))\; \backslash equiv\backslash \; \backslash forall\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; (P(x)\; \backslash to\; Q(y))$
- $P(x)\; \backslash nleftarrow\; (\backslash forall\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; Q(y))\; \backslash equiv\backslash \; \backslash forall\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; (P(x)\; \backslash nleftarrow\; Q(y)),~\backslash mathrm\{provided~that\}~\backslash mathbf\{Y\}\backslash neq\; \backslash emptyset$

Conversely, for the logical connectives ↑, ↓, $\backslash nrightarrow$, and ←, the quantifiers flip:

- $P(x)\; \backslash uparrow\; (\backslash exists\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; Q(y))\; \backslash equiv\backslash \; \backslash forall\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; (P(x)\; \backslash uparrow\; Q(y))$
- $P(x)\; \backslash downarrow\; (\backslash exists\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; Q(y))\; \backslash equiv\backslash \; \backslash forall\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; (P(x)\; \backslash downarrow\; Q(y)),~\backslash mathrm\{provided~that\}~\backslash mathbf\{Y\}\backslash neq\; \backslash emptyset$
- $P(x)\; \backslash nrightarrow\; (\backslash exists\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; Q(y))\; \backslash equiv\backslash \; \backslash forall\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; (P(x)\; \backslash nrightarrow\; Q(y)),~\backslash mathrm\{provided~that\}~\backslash mathbf\{Y\}\backslash neq\; \backslash emptyset$
- $P(x)\; \backslash gets\; (\backslash exists\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; Q(y))\; \backslash equiv\backslash \; \backslash forall\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; (P(x)\; \backslash gets\; Q(y))$
- $P(x)\; \backslash uparrow\; (\backslash forall\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; Q(y))\; \backslash equiv\backslash \; \backslash exists\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; (P(x)\; \backslash uparrow\; Q(y)),~\backslash mathrm\{provided~that\}~\backslash mathbf\{Y\}\backslash neq\; \backslash emptyset$
- $P(x)\; \backslash downarrow\; (\backslash forall\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; Q(y))\; \backslash equiv\backslash \; \backslash exists\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; (P(x)\; \backslash downarrow\; Q(y))$
- $P(x)\; \backslash nrightarrow\; (\backslash forall\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; Q(y))\; \backslash equiv\backslash \; \backslash exists\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; (P(x)\; \backslash nrightarrow\; Q(y))$
- $P(x)\; \backslash gets\; (\backslash forall\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; Q(y))\; \backslash equiv\backslash \; \backslash exists\{y\}\{\backslash in\}\backslash mathbf\{Y\}\backslash ,\; (P(x)\; \backslash gets\; Q(y)),~\backslash mathrm\{provided~that\}~\backslash mathbf\{Y\}\backslash neq\; \backslash emptyset$

### 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

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

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*,

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

The element *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.

### The empty set

By convention, the formula $\backslash forall\{x\}\{\backslash in\}\backslash emptyset\; \backslash ,\; P(x)$ 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 variables obtained by adding a universal quantifier for every free variable in φ. For example, the universal closure of

- $P(y)\; \backslash land\; \backslash exists\; x\; Q(x,z)$

is

- $\backslash forall\; y\; \backslash forall\; z\; (\; P(y)\; \backslash land\; \backslash exists\; x\; Q(x,z))$.

## As adjoint

In category theory and the theory of elementary topoi, the universal quantifier can be understood as the right adjoint of a functor between power sets, the inverse image functor of a function between sets; likewise, the existential quantifier is the left adjoint.^{[2]}

For a set $X$, let $\backslash mathcal\{P\}X$ denote its powerset. For any function $f:X\backslash to\; Y$ between sets $X$ and $Y$, there is an inverse image functor $f^*:\backslash mathcal\{P\}Y\backslash to\; \backslash mathcal\{P\}X$ 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 $\backslash exists\_f$ and the right adjoint is the universal quantifier $\backslash forall\_f$.

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

- $\backslash exists\_f\; S\; =\backslash \{\; y\backslash in\; Y\; |\; \backslash mbox\{\; there\; exists\; \}\; x\backslash in\; X\; \backslash mbox\{\; s.t.\; \}\; f(x)=y\; \backslash \}$.

Likewise, the universal quantifier $\backslash forall\_f\backslash colon\; \backslash mathcal\{P\}X\backslash to\; \backslash mathcal\{P\}Y$ is given by

- $\backslash forall\_f\; S\; =\backslash \{\; y\backslash in\; Y\; |\; f(x)=y\; \backslash mbox\{\; for\; all\; \}\; x\backslash in\; X\; \backslash \}$.

The more familiar form of the quantifiers as used in first-order logic is obtained by taking the function *f* to be the projection operator $\backslash pi:X\; \backslash times\; \backslash \{T,F\backslash \}\backslash to\; \backslash \{T,F\backslash \}$ where $\backslash \{T,F\backslash \}$ is the two-element set holding the values true, false, and subsets *S* to be predicates $S\backslash subset\; X\backslash times\; \backslash \{T,F\backslash \}$, so that

- $\backslash exists\_\backslash pi\; S\; =\; \backslash \{y\backslash ,|\backslash ,\backslash exists\; x\backslash ,\; S(x,y)\backslash \}$

which is either a one-element set (false) or a two-element set (true).

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

## See also

Look up in , the free dictionary.every |

- Existential quantification
- Quantifiers
- First-order logic
- List of logic symbols - for the unicode symbol ∀

## Notes

## References

**(ch. 2)**