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31.3 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?

Existential Instantiation: how to reason from an existential claim.

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 its name is.

Here’s how we proceed.

Proof. Assume “n” is an arbitrary pet (for prem 2). Then we know it is a mammal (prem 1). So there is some mammal. Done.

We call this method Existential Instantiation (EI), since we use the arbitrary name “n” to talk about the object we know exists from an existential quantifier.

Notice that this method is in some ways similar and in some ways different from the universal generalization (UG) or AIntro idea we learned earlier in this chapter.

It is similar because we want to talk about some object or objects in the domain, without talking about any specific object. That is why it uses the same arbitrary name idea.

When we know an existential claim, we don’t want to talk about a completely arbitrary object in the domain.

But here is the difference. When we know an existential claim, we don’t want to talk about a completely arbitrary object in the domain.

For example, in the proof above we know that there is a pet. To complete the proof, we need some way of talking about it. (Or, since there could be many pets, we need some way of talking about an arbitrary one of them.)

But perhaps the domain is all the animals on earth, many of which are not pets. If we are using premise (2) to refer to a pet, we don’t want “n” to be one of the non-pets in the domain.

So when we do existential instantiation (EI), we want to talk about an arbitrary pet. Or, to put it another way, we want to talk about an arbitrary n that has the pet property.