Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 6 - Set Theory - Exercise Set 6.4 - Page 382: 20

Answer

The paradox arises if Nixon says that everything Jones says about Watergate is true, and Jones says most of Nixon’s statements about Watergate are false. If both claims are taken at face value, they contradict each other, creating a circular paradox.

Work Step by Step

Let's examine the situation described by Saul Kripke: --- ### Given Sentences: 1. **(i)** Most of Nixon’s assertions about Watergate are false. 2. **(ii)** Everything Jones says about Watergate is true. --- ### Hint: Suppose **Nixon** says **(ii)**, and the **only** thing **Jones** says about Watergate is **(i)**. --- ### Step-by-Step Breakdown: - So Nixon says: **"Everything Jones says about Watergate is true."** — i.e., Nixon endorses **all** of Jones's Watergate claims. - But the **only** thing Jones says about Watergate is: **"Most of Nixon’s assertions about Watergate are false."** --- ### Now let’s analyze: Let’s assume **everything Jones says is true**, as Nixon claims. That means: - Jones’s claim — "Most of Nixon’s assertions about Watergate are false" — is **true**. - But Nixon’s statement **(ii)** is one of **his assertions about Watergate**. So if **most of Nixon’s assertions are false**, then his statement (ii) is **probably false**. But if (ii) is **false**, then **not everything Jones says is true**, which **contradicts** what Nixon asserted. --- ### 🔁 Circular Logic → Paradox: We end up in a loop: - Nixon says Jones is always right about Watergate. - Jones says Nixon is mostly wrong about Watergate. - But then Nixon’s statement about Jones must be **wrong**, which implies **Jones is not always right**, which contradicts Nixon. - But if Jones isn’t always right, then maybe Nixon’s statement is okay… and it goes in circles.
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