### Proof: Existential Instantiation

Existential instantiation (EI) is the reverse of existential generalization. It has the following form:

(x) (...x...)
_____________
...a...

This does not appear to be a valid inference. The premise tells us that something in the world fits a certain description, but if a is the name of a specific individual with which we are already familiar, we cannot infer that a fits the description.

Nevertheless, it is valid to reason as follows: At least one thing in the world is both an actor and a waiter. Let's call that thing a. We can now assert that Aa Wa.

Existential instantiation is valid if we replace the variable with a name we introduce solely for the purpose of standing for the particular thing, whatever (or whoever) it may be, that makes the premise true.

This means that when we use EI in a proof, the name we use must be one that has not been used previously in the proof and that does not occur in the conclusion.

Equivalence rule |
Universal instantiation | Existential generalization |
Existential instantiation | Universal generalization

Proof