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Subsection 4.2.4 Multiple Existential Quantifiers

There’s one other thing that we should say about using more than one existential quantifier in a single expression. Think of the objects that they each assert to exist as being chosen independently from each other. That means that it is possible (but not required) that they’ll be the same object. So, if we write:

∃x (∃y (P(x) ∧ P(y)))

we are not claiming that there are two different values for which P is true. We’ll see later how we can do that if we need to.

Suppose that I write:

\(\exists \) x (\(\exists \)y (IsStarving(x) \(\wedge \) HasLotsOfFood(y)))

While you might be tempted to conclude that there’s one hungry person and another mean hoarder, I didn’t say that. There could be just one person who’s got lots of food but is saving it or is on a diet or just doesn’t like the food he’s got.

Exercises Exercises

1.

1. Suppose that I write:

∃x (∃y (RedSoxFan(x) ∧ AstrosFan(y)))

What’s the smallest number of objects that must exist in my universe if this statement is true?

Answer.
Correct answer is 1.
Solution.
Explanation: We assert the existence of at least one object. Nothing prevents the same person from being a fan of two different teams.

2.

2. Suppose that we’d like to make the following claim about the social situation at the local school: There’s both a boy and a girl whom everyone likes. Assume the domain of people.

Define:

Boy(x): True if x is a boy.

Girl(x): True if x is a girl.

Likes(x, y): True if x likes y.

Consider the following statements:

I. ∀z (∃y (∃x (Boy(x) ∧ Girl(y) ∧ Likes(z, x) ∧ Likes(z, y)))

II. ∃x (∃y (∀z (Boy(x) ∧ Girl(y) ∧ Likes(z, x) ∧ Likes(z, y)))

III. ∃y (∃x (∀z (Boy(x) ∧ Girl(y) ∧ Likes(z, x) ∧ Likes(z, y)))

IV. ∃y (∀z (∃x (Boy(x) ∧ Girl(y) ∧ Likes(z, x) ∧ Likes(z, y)))

How many of these statements make our claim?

  1. 1

  2. 2

  3. 3

  4. 4

Answer.
Correct answer is B.
Solution.
Explanation: II and III are equivalent. It doesn’t matter what order the two existential quantifiers come in. I says that everyone likes at least one boy and one girl (but not necessarily the same ones for everyone). IV says that there exists a girl whom everyone likes, but each person can like a different boy.