# Definition:Quantifier

## Contents

## Definition

The universal quantifier $\forall$ and the existential quantifier $\exists$ are referred to collectively as **quantifiers**:

### The Universal Quantifier

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 $S$.

### The Existential Quantifier

The symbol $\exists$ is called the **existential quantifier**.

It expresses the fact that, in a particular universe of discourse, there exists (at least one) object having a particular property.

That is:

- $\exists x:$

means:

**There exists at least one object $x$ such that ...**

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

- $\exists x \in S: P \left({x}\right) := \left\{{x \in S: P \left({x}\right)}\right\} \ne \varnothing$

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

## Also known as

Some sources categorize the **quantifiers** as operators.

## Sources

- 1946: Alfred Tarski:
*Introduction to Logic and to the Methodology of Deductive Sciences*(2nd ed.) ... (previous) ... (next): $\S 1.4$: Universal and Existential Quantifiers - 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{III}$: The Logic of Predicates $(1): \ 3$: Quantifiers - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts