Logical Implication and Equivalence

Implication

Logical Equivalence

• Definition: Two statements are said to be logically equivalent if they always have the same truth value (i.e., both are true or both are false) in all possible scenarios.

Material Implication

• Definition: A material implication is a statement of the form "if p, then q" (p → q), which is false only when p is true and q is false.
• Truth Table:
  • p | q | p → q
  • --- | --- | -----
  • T | T | T
  • T | F | F
  • F | T | T
  • F | F | T

Logical Equivalence of Implications

• Rule of Detachment: If p → q and p are true, then q must be true.
• Rule of Syllogism: If p → q and q → r are true, then p → r is true.
• Transitive Property: If p → q and q → r are true, then p → r is true.

Logical Equivalence of Statements

• De Morgan's Laws:
  • ¬(p ∧ q) ≡ ¬p ∨ ¬q
  • ¬(p ∨ q) ≡ ¬p ∧ ¬q
• Distributive Property:
  • p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
  • p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)