# Universal Instantiation

## Theorem

### Informal Statement

Suppose we have a universal statement:

- $\forall x: \map P x$

where $\forall$ is the universal quantifier and $\map P x$ is a propositional function.

Then we can deduce:

- $\map P {\mathbf a}$

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

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 $\mathbf a$.*

### Universal Instantiation in Models

Let $\map {\mathbf A} x$ be a WFF of predicate logic.

Let $\tau$ be a term which is freely substitutable for $x$ in $\mathbf A$.

Then $\forall x: \map {\mathbf A} x \implies \map {\mathbf A} \tau$ is a tautology.

### Universal Instantiation in Proof Systems

Let $\map {\mathbf A} x$ be a WFF of predicate logic.

Let $\tau$ be a term which is freely substitutable for $x$ in $\mathbf A$.

Let $\mathscr H$ be Hilbert proof system instance 1 for predicate logic.

Then:

- $\forall x: \map {\mathbf A} x \vdash_{\mathscr H} \map {\mathbf A} \tau$

is a provable consequence in $\mathscr H$.

## Also known as

The **Rule of Universal Instantiation** can be abbreviated **UI**.

The **Rule of Universal Instantiation** can also be referred to as the **Rule of Universal Elimination** and it is then abbreviated **UE**.

The reasoning behind the name **Universal Elimination** is that it is used to "eliminate" the universal quantifier from a logical argument.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**logic** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**logic** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**universal instantiation**