Mathematical Thinking

Logical Implication
  • Implication involves causality.
  • Implication has a truth part and a causality part.
  • Mathematics only deals with truth part or conditional.
  • φ => ψ
    • φ is antecedent.
    • ψ is consequence.
  • φ => ψ means
    • If φ, then ψ
    • φ is sufficient for ψ
    • φ only if ψ
    • ψ if φ
    • ψ whenever φ
    • ψ is necessary for φ
φ ψ φ => ψ
T T T
T F F
F T T
F F T
Logical equivalence.
  • Equivalence is similar to equations/equality.
  • Biconditional.
  • φ <=> ψ if (φ => ψ) ^ (ψ => φ)
  • φ <=> ψ means
    • φ is necessary and sufficient for ψ
    • φ if and only if ψ