Generalized Predicate Logic and de Morgan's Law

Overview

• Predicate Logic with Universal and Existential quantifiers
• Determining the truth values of statements
• Generalized Predicate Logic
• De Morgan's Law application in Predicate Logic

Representation of Statements

Example

• Given statement: ∃X (X/(X²+1) > 1)
  • X takes values from the domain D (positive integers)
  • The statement is existentially quantified and seeks X such that X/(X²+1) > 1
• For all positive integers X, the statement X/(X²+1) > 1 is false because X² + 1 > X for all positive integers X
• Therefore, ∃X (X/(X²+1) > 1) is false
• Negation: ¬(X/(X²+1) > 1) becomes X/(X²+1) ≤ 1

Using Universal Quantifier

• The negation X/(X²+1) ≤ 1 is true for all values of X in D
• Relation: ∃X P(X) ↔ ∀X ¬P(X) and vice-versa
• Finding negation involves applying De Morgan's Law

De Morgan's Law in Predicate Logic

• Statements:
  1. ¬(∀X P(X)) ↔ ∃X ¬P(X)
  2. ¬(∃X P(X)) ↔ ∀X ¬P(X)

Proof of ¬(∀X P(X)) ↔ ∃X ¬P(X)

• Case 1: Assume ¬(∀X P(X)) is true
  • ∀X P(X) is false ↔ There is at least one X for which P(X) is false
  • Written as: ∃X (¬P(X)) holds true
• Case 2: Assume ¬(∀X P(X)) is false
  • ∀X P(X) is true ↔ For all values in D, P(X) is true
  • Negation `∃X (¬P(X)) is false \

Proof of ¬(∃X P(X)) ↔ ∀X ¬P(X)

• Similar proof by considering negation and the definitions of existential and universal quantifiers

Generalized Predicate Logic

Definition

• Using conjunctions and disjunctions:
  1. Universal Quantifier: ∀X P(X) ↔ P(D₁) ∧ P(D₂) ∧ ... ∧ P(Dn)
  2. Existential Quantifier: ∃X P(X) ↔ P(D₁) ∨ P(D₂) ∨ ... ∨ P(Dn)
• True Evaluation:
  • For ∀X P(X) Conjunction: True if all individual P(X) are true
  • For ∃X P(X) Disjunction: True if at least one P(X) is true
• False Evaluation:
  • For ∀X P(X) Conjunction: False if at least one P(X) is false
  • For ∃X P(X) Disjunction: False if all individual P(X) are false

Conclusion

• Generalizing the logic allows handling any number of predicates or quantifiers
• Applying De Morgan's Law effectively helps in finding negations of statements