Section 7.3: Quantifier Negation Rules

Aug 15, 2024

Section 7.3: Quantifier Negation Rules

Introduction

  • Quantifier Negation Rules - also referred to as QN rules or Change of Quantifier Rules
  • The term has changed from the older editions of the text to the current one.

Concept Overview

  • Example statement:
    • For all X, BX (e.g., Everything is B)
    • Question: What could make this a true statement?
    • Example: B = Banana is false since not everything is a banana.

Philosophical Context

  • Delving into philosophical ideas:
    • Some believe nothing can universally apply to all things (true for most).
    • Others, e.g., certain Hindus, believe in Brahman (everything is Brahman).
    • Modification: Belief that nothing exists which isn't Brahman.
      • This demonstrates logical equivalence.
      • Statements: ∀x Bx ↔ ¬∃x ¬Bx

Versions of QN Rules

  • Created by logically analyzing statements and their negations.
  • Example with Brahman:
    • It's not the case that everything is Brahman (¬∀x Bx)
    • Equivalent: There exists something that isn't Brahman (∃x ¬Bx)
  • Example with Martians:
    • Martians don't exist (¬∃x Mx)
    • Equivalent: Any chosen object isn't a Martian (∀x ¬Mx)

Breakdown of Four Versions

  1. Statement: ∀x Bx
    • Equivalent: ¬∃x ¬Bx
  2. Statement: ¬∀x Bx
    • Equivalent: ∃x ¬Bx
  3. Statement: ∃x Mx
    • Equivalent: ¬∀x ¬Mx
  4. Statement: ¬∃x Mx
    • Equivalent: ∀x ¬Mx

Application Rules

  • Logically equivalent statements can replace each other in proofs.
  • Important: Always change the quantifier when applying QN rules.
  • Example:
    • Single tilde can be moved to the other side of the quantifier, changing it.

Examples and Tips

  • If you remember this rule, it simplifies working with the QN rules.
  • Double negation: Two tildes can negate each other only if they are next to each other.
    • Not applicable if separated or on different parts of the statement.

Practical Use in Proofs

  • QN rules are replacement rules, can be used on parts of lines.
  • Example: Apply QN rules to one part of a statement while retaining the rest.

Conclusion

  • QN rules provide logical replacements ensuring valid transitions between equivalent statements.
  • Review these rules to ensure correct application during proofs and problem-solving.