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Subsection 7.3.4 Interpreting Sentences that Carry Presuppositions

What truth value shall we assign to sentences like “The largest prime number ends in 1.” This sentence carries the false presupposition that there is a largest prime number. Here are two possible approaches:

  1. If a sentence S carries a false presupposition, assign the truth value False to both S and ¬S.

  2. If a sentence S carries a false presupposition, then assign no truth value to either S or ¬S.

But have we now identified a challenge to the Law of the Excluded Middle (if we take approach 2) or the Principle of Noncontradiction (if we take approach 1)?

No.

Both the Law of the Excluded Middle and the Principle of Noncontradiction are claims about logical statements. They are not claims about English sentences.

There is, however, one thing that we can say about English sentences: The relationship between them and useful logical statements can be complex. The problem of assigning meaning to English sentences is hard. Centuries of philosophers and linguists have worked on it. We can only scratch the surface of that problem here.

Recall that presuppositions are unstated assumptions. So a good first step, in attempting to map English sentences into logical expressions is to make those assumptions explicit.

Consider again: “The last decimal digit of the largest prime number is 1.”

We can rewrite it as: “There exists a largest prime number and the last decimal digit of that number is 1.”

Or, in our logical language: x (Prime(x)  (y (Prime(y)  x > y))  LastDigitOf(x, 1))

Now there is no confusion. Since there exists no x such that (y (Prime(y)  x > y)), this entire statement is simply false.

Big Idea

It’s critical to keep in mind the difference between English sentences and logical ones. The mapping from the former to the latter is not always straightforward.

Exercises Exercises

1.

Consider: “Taylor’s brother’s son likes ice cream.”

Assume that the referent of Taylor is clear, so there is no issue there.

We want to encode this English sentence as a logical claim that will be either true or false. So we need to make its presuppositions explicit. Which one or more of the following logical expressions correctly does that? (Read BrotherOf(x, y) as x is the brother of y, SonOf(x, y) as x is the son of y, and Likes(x, y) as x likes y.)

  1. ∃x (∃y (BrotherOf(x, Taylor) ∧ SonOf(y, x) ∧ Likes(y, ice cream)))

  2. ∃y (Likes(y, ice cream) ∧ ∃x (BrotherOf(x, Taylor) ∧ SonOf(y, x)))

  3. ∃y (Likes(y, ice cream) ∧ SonOf(y, x) ∧ BrotherOf(x, Taylor))

  1. Just I.

  2. Just II.

  3. Just III.

  4. Just I and II.

  5. Just I and III.

  6. Just IIand III.

  7. All three.

Answer.

Correct answer is D

Solution.

Explantion: III is wrong because x isn’t bound by any quantifier.

2.

Let the domain be the reals. Consider:

[1] \(\sqrt{- 2}\) > 0.

Which of the following is true:

  1. [1] carries no presuppositions.

  2. [1] carries one or more presuppositions but they are all true.

  3. [1] carries one or more false presuppositions.

Answer.

Solution.

3.

Let the domain be the reals. Consider:

[1] \(\sqrt{- 2}\) > 0.

We want to encode this claim as a logical claim that will be either true or false. So we need to make any presuppositions explicit. Which one or more of the following is a well-formed logical expressions (assuming the domain is the reals) that correctly does that?

I. \(\sqrt{- 2}\) > 0

II. ∃x (x = \(\sqrt{2}\) ∧ x > 0)

III. ∃x (\(\sqrt{- 2}\) > 0)

Answer.

Solution.