The **universal quantifier** is a symbol of symbolic logic which expresses that the statements within its scope are true for everything, or every instance of a specific thing.

The symbol ∀, which appears as a vertically inverted “A”, is used as the universal quantifier.

Universal quantifiers are normally used in logic in conjunction with predicate symbols, which say something about a variable or constant, in this case the variable being quantified.

The universal quantifier ∀ (which means “for all”), differs from the existential quantifier ∃ (which means “there exists”, or in contrast to ∀, “for at least one”).

*Note*: Some logicians and logic texts do not make use of the ∀ symbol, and simply use the notation (*x*) to indicate universal quantification. This site, however, will always use ∀.

For example, if the predicate symbol *Mx* is taken to mean “*x* is matter”, then we may formalize an expression using a universal quantifier:

∀*xMx*

Translated back into English, this reads as “for every *x*, *x* is matter”, or more simply, “everything is matter”.

If we wanted to say that everything is matter or energy, we could add the predicate symbol *Ex*, to refer to “*x* is energy”, and formalize:

∀*x*(*Mx* ∨ *Ex*)

The common logical phrase “All P are Q” may be similarly formulated using a universal quantifier and a conditional statement:

∀*x*(*Px* → *Qx*)

This reads as “For every *x*, if *x* is a P, then *x* is a Q”, which may be expressed simply as, “all P are Q”.