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
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:


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

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:


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

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