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23.2 The Domain

Let’s say we’re still working at the animal shelter, and a new group of animals are dropped off that need homes.

They are all in a cardboard box. I peek in, and my eyes go wide.

You ask: “What’s in there?”

I reply: “They’re all snakes.”

AxSnake(x) is true just in case all objects are snakes.

Let’s say I’m right, the box is full of snakes and just snakes. The universal claim, AxSnake(x), says For all objects x, x is a snake.

That is true just in case there are no exceptions: no cats, no dogs, etc.

But what about the other animals in the shelter? We also have two cats and a dog.

Does AxSnake(x) claim that literally everything is a snake? Every object in the universe?

No. Otherwise FOL wouldn’t be very useful.

Domain: the set of objects a quantifier ranges over.

Here’s the solution. Whenever a quantifier is used, it has a set of objects it is talking about, called the Domain (or Domain of Quantification).

What AxSnake(x) claims is that every object in the domain is a snake.

Now see if you can figure this out.

The domain can be fixed in two ways. In English, we usually allow context and informal conventions to fix the domain.

When I look in the box and say they’re all snakes, normal informal rules of conversation dictate that I’m talking about the new animals in the box that just got dropped off.

But we can also fix the domain in another way: we can just specify what objects a quantifier ranges over.

It is common to use “D” for the domain. So I can say:

D = {animals in this box}.

Or in a different example, I might say:

D = {natural numbers}.

There is another key rule for domains that you must know. It’s not the case that every quantifier gets its own domain.

If we have multiple quantifiers in a sentence, or multiple sentences in an argument, etc., they all must have the same domain.

Otherwise all sorts of fallacious reasoning would be allowed.

But once we switch to a new context, or a new argument, then the domain can change too.

For example, let’s say that you reply: “Now we have cats, dogs, and snakes!”

My sentence and your sentence are both true.

But that means that your sentence has a different domain from mine.

Because if we assume that cats and dogs are not snakes (naturally), then your sentence would not be consistent with mine if they had the same domain.

In natural conversation, subtle conversational clues can shift the domain. In practice this is not a problem: as speakers of English, we are quite used to it and typically don’t have any problem understanding what is said.