Principle of explosion

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The principle of explosion is a logical principle which states that if we accept that a proposition to be true at the same time it is not true, then any other proposition, 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. We'll then apply this to another example:

P1: Ice is cold and ice is not cold
P2: Ice is cold (1, simplification)
P3: Ice is not cold (1, simplification)
P4: Either ice is cold or my car is dry (2, addition)
C: My car is dry (3, 4 disjunctive syllogism)

Clearly, ice cannot be cold and not cold at the same time. But, if we're to accept the premises as true for the sake of the argument, it follows that my car must be dry, because we added the proposition "my car must be dry" to the proposition "ice is cold" to form a disjunctive statement. We then denied one of the alternates (we said that it wasn't the case that ice is cold), so my car must be dry.