Editing Universal quantification (section)

== 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.