Definition:Propositional Expansion

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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.