Universal quantification

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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 $\forall$ ) 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.

1 Basics
2 Properties
- 2.1 Negation
- 2.2 Rules of Inference
3 See also
4 Reference

[edit] Basics

Suppose you wish to say

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." But the "etc" can't be interpreted as a conjunction in formal logic. Instead, rephrase the statement as

For any natural number n, 2·n = n + n.

This is a single statement using universal quantification.

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 any natural number 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 any composite number 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 any composite number n, 2·n > 2 + n" is logically equivalent to "For any natural number 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 $\forall$ (an upside-down letter "A" in a sans-serif font) to indicate universal quantification. Thus if P(n) is the predicate "2·n > 2 + n" and N is the set of natural numbers, then

$\forall{n}{\in}\mathbf{N}\, P(n)$

is the (false) statement

For any natural number n, 2·n > 2 + n.

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

$\forall{n}{\in}\mathbf{N}\, Q(n)\;\!\;\! {\rightarrow}\;\!\;\! P(n)$

is the (true) statement

For any composite number 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:

$(n{\in}\mathbf{N})\, P(n)$

The parentheses indicate universal quantification by default.

[edit] Properties

[edit] 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":

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

A few second's 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:

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

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 $\forall{x}{\in}\mathbf{X}\, P(x)$ is logically equivalent to "There exists a living person x such that he is not married", or:

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

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

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

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"):

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

[edit] 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

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

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,

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

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.