Existential Generalisation

Context
Predicate Logic.

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.