Existential Generalisation

Theorem
This is an extension of the Rule of Addition as follows:


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

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.

Proof
The propositional expansion of $\exists x: P \left({x}\right)$ is:


 * $P \left({\mathbf X_1}\right) \lor P \left({\mathbf X_2}\right) \lor P \left({\mathbf X_3}\right) \lor \ldots$

We have the fact that $P \left({\mathbf a}\right)$ where $\mathbf a$ is one of the above $\mathbf X_1, \mathbf X_2, \mathbf X_3, \ldots$, as it is by definition.

So the above statement $P \left({\mathbf X_1}\right) \lor P \left({\mathbf X_2}\right) \lor P \left({\mathbf X_3}\right) \lor \ldots$ follows by extension of the Rule of Addition.