Universal Instantiation

Theorem
Suppose we have a universal statement:
 * $\forall x: P \left({x}\right)$

where $\forall$ is the universal quantifier and $P \left({x}\right)$ is a propositional function.

Then we can deduce:
 * $P \left({\mathbf a}\right)$

where $\mathbf a$ is any arbitrary object we care to choose in the universe of discourse.

In the language of symbolic logic:

In natural language:


 * Suppose $P$ is true of everything in the universe of discourse.


 * ''Let $\mathbf a$ be an element of the universe of discourse."


 * Then $P$ is true of $\mathbf a$.

Proof
We can express $\forall x$ using its propositional expansion:

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

where $\mathbf X_1, \mathbf X_2, \mathbf X_3 \ldots$ is the complete set of the objects in the universe of discourse.

We can now apply the Rule of Simplification and the result follows.

Also known as
Some authors call this the Rule of Universal Elimination and it is then abbreviated UE.