# Existential Generalisation

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## Theorem

### Informal Statement

- $P \left({\mathbf a}\right) \vdash \exists x: P \left({x}\right)$

Suppose we have the following:

- We can find an arbitrary object $\mathbf a$ in our universe of discourse which has the property $P$.

Then we may infer that:

- there exists in that universe
*at least one*object $x$ which has that property $P$.

This is called the **Rule of Existential Generalisation** and often appears in a proof with its abbreviation **EG**.

### Existential Generalisation 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 $\mathbf A \left({\tau}\right) \implies \exists x: \mathbf A \left({x}\right)$ is a tautology.

## Also known as

Some authors call this the **Rule of Existential Introduction** and it is then abbreviated **EI**.

However, beware the fact that other authors use **EI** to abbreviate the Rule of Existential Instantiation which is the antithesis of this one.

So make sure you know exactly what terminology is specified.