A Contradiction Entails All Things: Formal Proof

Contradictory assumptions permit an arbitrary conclusion.
        Given:  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.

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