21.1 All (A) and Exists (E)
We will learn how to refer to quantities of objects in FOL.
For example, to say that every object is a dog, we write: AxDog(x)
In English, “every” or “all” refer in a way to a quantity of things; not a specific number, but the idea that a property is universally shared among everything.
For example, if we’re taking inventory at an animal shelter, and I say that all the animals are dogs, I’m making a claim about the totality of the animals there: each and every one is a dog.
We call “all” or “every” a quantifier, a part of language for making quantity claims.
There is a standard logical symbol “all”, which is an upside down letter “A”: ∀.
So we would express the idea that all the objects are dogs, we’d write: ∀xDog(x).
There’s no easy way to write ∀ with a standard keyboard, though, so we’ll also use A for the same thing.
Now, if I claim every animal is a dog, you might disagree and say: No, there is a cat. In other words: there exists a cat, which we symbolize: ExCat(x).
Saying that a cat or a dog exists is also a kind of quantity claim: there is at least one of them.
We’ll call “E” or the Existential Quantifier, since it makes a claim of existence.
The standard symbol for “all” was an upside down A, ∀. An upside down E though, would still be an E. So the standard symbol for “exists” is a backwards E: ∃.
But since like ∀ that is difficult to write with a standard keyboard, we will use “E” in addition to ∃. We use both because it is essential to understand the ∃ notation, but on practice and homework problems just use E.
You might have noticed that whenever we write a quantifier in a sentence, it has a variable with it, like ExCat(x). So next we need to learn about variables.