# Universal Generalisation

## Contents

## Theorem

Let $\mathbf a$ be any arbitrarily selected object in the universe of discourse.

Then:

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

In natural language:

*Suppose $P$ is true of any arbitrarily selected $\mathbf a$*in the universe of discourse.

*Then $P$ is true of everything 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.

The fact that any object we care to choose has the property in question means that they *all* must have this property.

The result then follows by generalising the Rule of Conjunction.

$\blacksquare$

## Also known as

Some authors call this the **Rule of Universal Introduction** and it is then abbreviated **UI**.

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

So make sure you know exactly what terminology is specified.

## Sources

- 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $4.2$: Preliminary Quantification Rules - 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{IV}$: The Logic of Predicates $(2): \ 2$: Universal Instantiation