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