These rules are conditionally true - i.e. if an entire clause matches EACH
premise, only then does the conclusion hold. They cannot be applied to
phrases inside a clause.

1. Modus Ponens (M.P.) | p ⇒q ∴ q |

2. Modus Tollens (M.T.) | p⇒q ~q ∴ ~p |

3. Hypothetical Syllogism (H.S.) | p⇒q q⇒r ∴ p⇒r |

4. Disjunctive Syllogism (D.S.) | p ∨ q ~ p ∴ q |

5. Constructive Dilemma (C.D. | (p⇒q) . (r⇒s) p ∨ r ∴ q v s |

6. Absorption (Abs.) | p⇒q ∴ p ⇒ (p∧q) |

7. Simplification (Simp.) | p∧q ∴ p |

8. Conjunction (Conj.) | p q ∴ p∧q |

9. Addition (Add.) | p ∴ p ∨ q |

Also called **Rules of Equivalence**, these are biconditional rules -
i.e. they apply from L to R and also from R to L. They are tautologies.
Hence, they can be applied to even parts of clauses (can be substituted).

10. De Morgan’s Theorem (De M.) | ~(p∧q) ≡ (~p∨~q) ~(p∨q) ≡ (~p∧~q) |

11. Commutation (Com.) | (p∨q) ≡ (q∨p) (p∧q) ≡ (q∧p) |

12. Association (Assoc.) | [p∨(q∨r)] ≡ [(p∨q)∨r] [p∧ (q∧r)] ≡ [(p∧q) ∧r] |

13. Distribution (Dist) | [p∧(q∨r)] ≡ [(p∧q)∨(p∧r)] [p∨(q∧r)] ≡ [(p∨q) ∧ (p∨r)] |

14. Double Negation (D.N.) | p ≡ ~ ~p |

15. Transposition (Trans.) | (p⇒q) ≡ (~q ⇒~p) |

16. Material Implication (M. Imp.) | (p⇒q) ≡ (~p∨q) |

17. Material Equivalence (M. Equiv.) | (p≡q) ≡ [(p⇒q) ∧ (q⇒p)] (p≡q) [(p∧q)∨(~p ∧ ~q)] |

18. Exportation (Exp.) | [(p∧q) r] ≡ [p ⇒ (q ⇒ r)] |

19. Tautology (Taut.) | p ≡ (p∨p) p ≡ (p∧p) |

One can clearly see that many of these rules can be proved based on other rules. For example, D.S. can be proved from M.T. and M.I. Similarly many operators are also excess - e.g. one need not have the implication operator at all. In fact, the system can be reduced to a single operator, the Peirce dagger or NOR, say). Thus, there is considerably redundancy in the structure. There are many equivalent rule sets that are exist for these connectives.

From: Introduction to Logic, Irving M. Copi. 3d ed. 1970.