Definition:Universal Quantifier

Definition

The symbol $\forall$ is called the universal quantifier.

It expresses the fact that, in a particular universe of discourse, all objects have a particular property.

That is:

$\forall x:$ means: For all objects $x$, it is true that ...

In the language of set theory, this can be formally defined:

$\forall x \in S: P \left({x}\right) := \left\{{x \in S: P \left({x}\right)}\right\} = S$

where $S$ is some set and $P \left({x}\right)$ is a propositional function on $S$.

Propositional Expansion

The universal quantifier can be considered as a repeated conjunction:

Let $\forall$ be the universal quantifier.

What $\forall x: P \left({x}\right)$ means is:

$\mathbf X_1$ has property $P$, and $\mathbf X_2$ has property $P$, and $\mathbf X_3$ has property $P$, and ...

This translates into propositional logic as:

$P \left({\mathbf X_1}\right) \land P \left({\mathbf X_2}\right) \land P \left({\mathbf X_3}\right) \land \ldots$

This expression of $\forall x$ as a conjunction is known as the propositional expansion of $\forall x$.

The propositional expansion for the universal quantifier can exist in actuality only when the number of objects in the universe is finite.

If the universe is infinite, then the propositional expansion can exist only conceptually, and the universal quantifier cannot be eliminated.

Variants

Some authors use $\left({x}\right)$ to mean $\forall x$, but the bespoke symbol is usually preferred as there is then no room for ambiguity.

Some authors use $\bigwedge$, which is appropriate when considering the propositional expansion.

Historical Note

The symbol $\forall$ was first used by Gerhard Gentzen in 1935: Untersuchungen über das logische Schließen. II (Math. Z. Vol. 39 (3): 405 – 431).

He invented it in analogy with the existential quantifier symbol $\exists$ which he borrowed from Bertrand Russell.

Russell himself used the notation $\left({x}\right)$ for for all $x$. See his 1908: Mathematical Logic as Based on the Theory of Types (Amer. J. Math. Vol. 30: 222 – 262).

Also see

• Universal statement

• Existential quantifier
• Existential statement

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 2$: The Axiom of Specification
• Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning (1964): $\text{III}: \S 1$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 4$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 3$
• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.1$
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): $\S 2.1$