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Subsection 4.2.2 More on Scope

The details really matter when we write logical expressions that use quantifiers. So let’s look at one more example of the kind of distinction that it’s important that we make.

Every time we write a predicate, we need to be careful: Should we introduce a new quantifier to bind one of its variables? Or should we write the predicate inside the scope of a quantifier that we’ve already written?

Assume a universe that corresponds to a group of people who are planning to go on a trip. Consider the following statements:

[1] \(\forall \) x (HasArrived(x) \(\rightarrow \) MayBoardBus(x))

[2] (\(\forall \) x (HasArrived(x))) \(\rightarrow \) \(\forall \) x (MayBoardBus(x))

Let’s read [1] carefully: It says that the rule is, for any person x, if x has arrived then x may board the bus. Good deal for the early birds.

Now let’s read [2]: It says that once everyone has arrived, everyone may board the bus. The early birds will now want to strangle any stragglers.

By the way, notice that [2], while correct (assuming that it says what we’re trying to say), violates our rule of thumb that tells us, that to avoid mistakes, we should use a different variable name each time we introduce a new quantifier. So, what we really should have written, just to be on the safe side, is:

[3] (\(\forall \) x (HasArrived(x))) \(\rightarrow \) \(\forall \) y (MayBoardBus(y))

Note, by the way, that in [2] and [3], above, we’ve actually used one more set of parentheses than is strictly necessary. We enclosed the complete quantified expression on the left of the implication in parentheses. We didn’t really need to do that. The quantifier and its scope (already enclosed in parentheses) form a complete expression. But we must be particularly careful when a fully quantified expression is the antecedent of an implication. So we’ll use the extra parentheses to help us make sure that we’ve written what we mean.

It may be tempting to get annoyed with our notation when it forces us to be so picky about every little parenthesis. But that’s actually why the notation is great: Yes, we must be picky. But someone reading what we write knows exactly what we mean.

Big Idea

Parentheses really matter.

Exercises Exercises

Exercise Group.

1. Consider the following logical expressions:

I. ∃x (HasLostWallet(x) → Frantic(x))

II. ∃x (HasLostWallet(x)) → ∃x (Frantic(x))

III. ∀x (HasLostWallet(x) → Frantic(x))

1.

(Part 1) Which (one or more of) these those expressions correspond(s) to the claim that if there’s anyone around who’s lost a wallet then there is definitely someone who’s frantic:

  1. Just I.

  2. Just II.

  3. Just III.

  4. Just I and II.

  5. Just II and III.

Answer.
Correct answer is B.
Solution.
Explanation: Notice that the English sentence that we’re trying to translate doesn’t claim that the wallet loser is the same person as the frantic one. So we don’t want to say that either. II is correct. It says exactly what the sentence says: If there exists a wallet loser then there exists a frantic person.
2.

(Part 2) Which (one or more of) of those expressions correspond(s) to the claim that anyone who has lost a wallet will be frantic?

  1. Just I.

  2. Just II.

  3. Just III.

  4. Just I and II.

  5. Just II and III.

Answer.
Correct answer is C.
Solution.
Explanation: Now there’s a connection between the wallet loser and the frantic person. III is correct. It asserts that every individual has the property that if she has lost her wallet then she’ll be frantic.
3.

(Part 3) Which (one or more of) those expressions correspond(s) to the claim that there exist panic-prone people who will become frantic if they lose their wallets?

  1. Just I.

  2. Just II.

  3. Just III.

  4. Just I and II.

  5. Just II and III.

Answer.
Correct answer is A.
Solution.
Explanation: Again there’s a connection between the wallet loser and the frantic person. But now we’re not claiming that everyone would panic; just that there exists someone who would. I is correct.