The axioms of calculational propositional logic C are listed in the order in which they are usually presented and taught. Note that equivalence comes first. Note also that, after the first axiom, we take advantage of associativity of equivalence and write sequences of equivalences without parentheses. We use == for equivalence, | for disjunction, & for conjunction, ~ for negation (not), => for implication, and <= for consequence.

• Associativity of ==: ((p == q) == r) == (p == (q == r))
• Symmetry of ==: p == q == q == p
• Identity of ==: true == q == q
  
  
• Definition of false: false == ~true
• Distributivity of not: ~(p == q) == ~p == q
• Definition of =/=: (p =/= q) == ~(p == q)
  
  
• Associativity of |: (p | q) & r == p | (q | r)
• Symmetry of |: p | q == q | p
• Idempotency of |: p | p == p
• Distributivity of |: p |(q == r) == p | q == p | r
• Excluded Middle: p | ~p
  
  
• Golden rule: p & q == p == q == p | q
  
  
• Implication: p => q == p | q == q
• Consequence: p <= q == q => p