Definition:Propositional Expansion

Context
Predicate Logic.

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
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 PropLog 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
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 PropLog 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 $$\forall x$$ as a disjunction is known as the propositional expansion of $$\exists x$$.