Definition:Propositional Expansion

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