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Subsection 4.1.8 The Universe (Domain)

When we write quantified expressions, we write them with respect to some universe of discourse (sometimes called a domain).

Suppose I say:

Everyone must turn in the homework on Friday.

\(\forall \)x (MustTurnInHomework(x))

Surely I don’t mean these guys:

There are two standard ways to specify the universe:

  1. Define the universe at the beginning of a problem description.

  2. “Guard” each claim.

To make it clear that the cats Lucy and Agnes are not required to do homework, we could:

  1. Start by saying:

    Let the universe be the set of students in our class.

  2. Guard our claim by saying:

    Anyone who is in our class must turn in the homework on Friday.

    \(\forall \)x (InOurClass(x) \(\rightarrow \) MustTurnInHomework(x))

Exercises Exercises

1.

1. Suppose that all we cared about was representing the fact that Lucy and Agnes don’t sleep all the time. We could define two Boolean predicates:

L: Lucy (the calico) is awake sometimes.

A: Agnes (the tortoise shell) is awake sometimes.

Then we could write a simple Boolean expression:

[1] L ∧ A

But now suppose that we want to say something more general about cats. We’ll define two predicates:

C(x): True if x is a (living) cat.

W(x): True if x is awake sometimes.

Which of the following statements is/are true in the world in which we live:

  1. ∀x (W(x))

  2. ∀x (C(x) → W(x))

  3. ∃x (C(x) → W(x))

  1. Just I.

  2. Just II.

  3. Just III.

  4. Two of them.

  5. All of them.

Answer.
Correct answer is D.
Solution.
Explanation: I isn’t true because not everything in the world is sometimes awake. Chairs, for example, are not. But both II and III are. All cats (assuming that, by “cats” we mean living ones) are awake sometimes. And, since we know that some cats exist (for example, Lucy and Agnes), there must exist at least one object for which cat implies awake sometimes.