Universal Generalisation

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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