Principle of explosion

The principle of explosion is a logical principle which states that if we accept that a certain statement is true at the same time it is not true, then any other statement, no matter the truth value, can be proven. In other words, from a contradiction, any imaginable conclusion can be reached.

The Principle

The principle comes from an argument containing a proposition and its negation. By way of simplification and addition, anything can be proved deductively.

P1: A · ~A

P2: A (1, simplification)

P3: ~A (1, simplification)

P4: A v B (2, addition)

C: B, (3, 4 disjunctive syllogism)

In plain English, this means that we have proposition one (P1) as "A and not A". We can split these two variables apart because they're conjoined with the conjunctive "and", resulting in two new propositions that have the same strength and truth value as the old proposition "A and not A". These new propositions are (P2) "A" and (P3) "not A". We can add any proposition we like to the proposition "A", because by adding a proposition in a disjunctive, we're making our new proposition weaker than before. So P2 becomes P4 with addition, which now reads "Either A or B". Now that we have a disjunctive statement and a proposition (P3) that negates one of the alternates, we can prove that B.