##### 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