Laws of , , and  Excluded middle law Contradiction law P   P  T P   P  F NameLaw Identity laws P  F  P P  T  P Domination laws P  T  T P  F  F Idempotent laws P  P  P P  P  P Double-negation law  (  P)  P

Commutative laws P  Q  Q  P P  Q  Q  P NameLaw Associative laws (P  Q)  R  P  (Q  R) (P  Q)  R  P  (Q  R) Distributive laws (P  Q)  (P  R)  P  (Q  R) (P  Q)  (P  R)  P  (Q  R) De Morgan’s laws  (P  Q)   P   Q  (P  Q)   P   Q Absorption laws P  (P  Q)  P P  (P  Q)  P

Laws of  and  Implication Law P  Q   P  Q NameLaw Equivalence Law P  Q  P  Q  Q  P Contrapositive Law P  Q   Q   P P  Q  P   Q  F Contradiction Law

Main Rules of Inference A, B |= A  B Law of combination A  B |= B Law of simplification A  B |= A Variant of law of simplification A |= A  B Law of addition B |= A  B Variant of law of addition A, A  B |= B Modus ponens  B, A  B |=  A Modus tollens A  B, B  C |= A  C Hypothetical syllogism A  B,  A |= B Disjunctive syllogism A  B,  B |= A Variant of disjunctive syllogism A  B,  A  B |= B Law of cases A  B |= A  B Equivalence elimination A  B |= B  A Variant of equivalence elimination A  B, B  A |= A  B Equivalence introduction A,  A |= B Inconsistency law