30.4 Existential Instantiation
The last proof method for quantifiers that we need to learn is how to reason from an existential.
Say we know that (1) all the pets are mammals, and (2) there exists a pet. We want to be able to prove that it follows that there is a mammal.
How do we do that?
The key premise is the second one: there exists a pet. We need a way to talk about that thing, even though we don’t know what it is or what it’s name is.
Here’s how we proceed.
Proof. Assume “a” is an arbitrary pet (for prem 2). Then we know it is a mammal (prem 2). So there is some mammal. Done.
We call this method Existential Instantiation, since we use the arbitrary name “a” to talk about the object we know exists from an existential quantifier.