Universal Generalisation

Context
PredLog.

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 $$P$$.

Some authors call this the Rule of Universal Introduction.

When using this rule of universal generalisation, 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.