# Definition:Propositional Expansion

## Definition

Suppose our universe of discourse consists of the objects $\mathbf X_1, \mathbf X_2, \mathbf X_3, \ldots$ and so on.

(There may be an infinite number of objects in this universe.)

### Universal Quantifier

Let $\forall$ be the universal quantifier.

What $\forall x: \map P x$ 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:

- $\map P {\mathbf X_1} \land \map P {\mathbf X_2} \land \map P {\mathbf X_3} \land \ldots$

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.

### Existential Quantifier

Let $\exists$ be the existential quantifier.

What $\exists x: \map P x$ means is:

- At least one of $\mathbf X_1, \mathbf X_2, \mathbf X_3, \ldots$ has property $P$.

This means:

- Either $\mathbf X_1$ has property $P$, or $\mathbf X_2$ has property $P$, or $\mathbf X_3$ has property $P$, or ...

This translates into propositional logic as:

- $\map P {\mathbf X_1} \lor \map P {\mathbf X_2} \lor \map P {\mathbf X_3} \lor \ldots$

This expression of $\exists x$ as a disjunction is known as the **propositional expansion** of $\exists x$.

The **propositional expansion** for the existential 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 existential quantifier cannot be eliminated.

## Also see

- Results about
**propositional expansions**can be found**here**.