Existential Generalisation
Theorem
Informal Statement
- $\map P {\mathbf a} \vdash \exists x: \map P x$
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 $\text {EG}$.
Existential Generalisation in Models
Let $\map {\mathbf A} x$ be a WFF of predicate logic.
Let $\tau$ be a term which is freely substitutable for $x$ in $\mathbf A$.
Then $\map {\mathbf A} \tau \implies \exists x: \map {\mathbf A} x$ is a tautology.
Existential Generalisation in Proof Systems
Let $\map {\mathbf A} x$ be a WFF of predicate Logic of $\LL$.
Let $\tau$ be a term which is freely substitutable for $x$ in $\mathbf A$.
Let $\mathscr H$ be Hilbert proof system instance 1 for predicate logic.
Then:
- $\map {\mathbf A} \tau \vdash_{\mathscr H} \exists x: \map {\mathbf A} x$
is a provable consequence in $\mathscr H$.
Also known as
Some authors call this the Rule of Existential Introduction and it is then abbreviated $\text {EI}$.
However, beware the fact that other authors use $\text {EI}$ to abbreviate the Rule of Existential Instantiation which is the antithesis of this one.
So make sure you know exactly what terminology is specified.