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.

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.

$\blacksquare$

Notes

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