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 $\forall x: P \left({x}\right)$ 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 $\forall x$.

Existential Quantifier
Let $\exists$ be the existential quantifier.

What $\exists x: P \left({x}\right)$ 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:


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

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