A Contradiction Entails All Things: Formal Proof

Contradictory assumptions permit an arbitrary conclusion.

        Given:  A

               ¬A

        Prove:  B

Step	Justification
A ∧ ¬A	Assumed
(A ∧ ¬A) ∨ B	The first clause is true, therefore the first clause or the second clause is true.
¬¬((A ∧ ¬A) ∨ B)	A double negative is an affirmation
¬(¬(A ∧ ¬A) ∧ ¬B)	DeMorgan's Laws
¬((¬A ∨ A) ∧ ¬B)	DeMorgan's Laws
¬((TRUE) ∧ ¬B)	Law of the Excluded Middle
¬(¬B)	The truth of the conjunction is equivalent to the second clause.
B	A double negative is an affirmation

Any line of argument is susceptible to error. More confidence may be placed in a conclusion supported by multiple lines of argument.