Existential Instantiation
Theorem
Informal Statement
- $\exists x: \map P x, \map P {\mathbf a} \implies y \vdash y$
Suppose we have the following:
- From our universe of discourse, any arbitrarily selected object $\mathbf a$ which has the property $P$ implies a conclusion $y$
- $\mathbf a$ is not free in $y$
- It is known that there does actually exists an object that has $P$.
Then we may infer $y$.
This is called the Rule of Existential Instantiation and often appears in a proof with its abbreviation $\text {EI}$.
When using this rule of existential instantiation:
- $\exists x: \map P x, \map P {\mathbf a} \implies y \vdash y$
the instance of $\map P {\mathbf a}$ is referred to as the typical disjunct.
Existential Instantiation in Proof Systems
Let $\LL$ be a specific signature for the language of predicate logic.
Let $\mathscr H$ be Hilbert proof system instance 1 for predicate logic.
Let $\map {\mathbf A} x, \mathbf B$ be WFFs of $\LL$.
Let $\FF$ be a collection of WFFs of $\LL$.
Let $c$ be an arbitrary constant symbol which is not in $\LL$.
Let $\LL'$ be the signature $\LL$ extended with the constant symbol $c$.
Suppose that we have the provable consequence (in $\LL'$):
- $\FF, \map {\mathbf A} c \vdash_{\mathscr H} \mathbf B$
Then we may infer (in $\LL$):
- $\FF, \exists x: \map {\mathbf A} x \vdash_{\mathscr H} \mathbf B$
Also known as
Some authors call this the Rule of Existential Elimination and it is then abbreviated $\text {EE}$.