Editing Universal quantification (section)

{{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.