Universal Quantifier

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 ∀.

Examples

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(MxEx)

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

x(PxQx)

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”.