Universal Generalisation

Theorem

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

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

This means that: if it can be shown that any arbitrarily selected object $\mathbf a$ in the universe has property $P$, we may infer that every $x$ in the universe has $P$.

This is called the Rule of Universal Generalisation and often appears in a proof with its abbreviation UG.

When using this rule of universal generalisation:

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

the instance of $P \left({\mathbf a}\right)$ is referred to as the typical conjunct.

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.

$\blacksquare$

Notes

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.