# Universal Instantiation

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## Contents

## Theorem

### Informal Statement

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 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 $\mathbf A \left({x}\right)$ be a WFF of predicate logic.

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

Then $\forall x: \mathbf A \left({x}\right) \implies \mathbf A \left({\tau}\right)$ is a tautology.

## Also known as

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

## Sources

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