In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any", "for all", "for every", or "given an arbitrary element".

It expresses that a predicate can be satisfied by every member of a domain of discourse.

In other words, it is the predication of a property or relation to every member of the domain.

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 only asserts that the property or relation holds for at least one member of the domain.

The universal quantifier is encoded as U+2200 ∀ FOR ALL in Unicode, and as \forall in LaTeX and related formula editors.

Suppose it is given that 2·0 = 0 + 0, and 2·1 = 1 + 1, and 2·2 = 2 + 2, ..., and 2 · 100 = 100 + 100, and ..., etc.This would seem to be an infinite logical conjunction because of the repeated use of "and".

cannot be interpreted as a conjunction in formal logic, Instead, the statement must be rephrased: For all natural numbers n, one has 2·n = n + n.This is a single statement using universal quantification.

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.

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.

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 numbers n, one has 2·n > 2 + nis logically equivalent to For all natural numbers n, if n is composite, then 2·n > 2 + n.Here the "if ... then" construction indicates the logical conditional.

(a turned "A" in a sans-serif font, Unicode 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

[1] For example, if P(n) is the predicate "2·n > 2 + n" and N is the set of natural numbers, then is the (false) statement Similarly, if Q(n) is the predicate "n is composite", then is the (true) statement Several variations in the notation for quantification (which apply to all forms) can be found in the Quantifier article.

For example, if P(x) is the propositional function "x is married", then, for the set X of all living human beings, the universal quantification Given any living person x, that person is marriedis written This statement is false.

Truthfully, it is stated that It is not the case that, given any living person x, that person is marriedor, symbolically: If the function P(x) is not true for every element of X, then there must be at least one element for which the statement is false.

is logically equivalent to "There exists a living person x who is not married", or: 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"): The universal (and existential) quantifier moves unchanged across the logical connectives ∧, ∨, →, and ↚, as long as the other operand is not affected;[2] that is: Conversely, for the logical connectives ↑, ↓, ↛, and ←, the quantifiers flip: 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 where c is a completely arbitrary element of the universe of discourse.

Symbolically, for an arbitrary c, 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.

is always true, regardless of the formula P(x); see vacuous truth.

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 is 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.

The left adjoint of this functor is the existential quantifier

The more familiar form of the quantifiers as used in first-order logic is obtained by taking the function f to be the unique function

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