# Principle of Explosion The **principle of explosion** is a concept in [Logic](Logic.md) stating that from a [Contradiction](Contradiction.md), you can derive *any statement*. In other words, it states: > "If $A$ and $\neg A$ are both true, then any statement $B$ can also be proven true." Here is an example: 1. Assume $A$: "It is raining 2. Assume $\neg A$: "It is not raining" 3. At this point we have a contradiction, because both $A$ and $\neg A$ cannot be true simultaneously. 4. Now, lets say we have $B$: "The moon is made of cheese". We can then use the [Rule of Disjunction](Rule%20of%20Disjunction.md) to introduce a disjunction (logical OR). Since we have assumed $A$ is true, we can write: $A \lor B$ 5. We can the apply a Disjunctive Syllogism. From (4) we have $A \lor B$, and from (2) we have $\neg A$. Therefore, $B$ must be true. 6. Thus we can conclude $B$ is true and that the moon must be made of cheese. And here is another example that I grabbed from wikipedia. As a demonstration of the principle, consider two contradictory statements—"All lemons are yellow" and "Not all lemons are yellow"—and suppose that both are true. If that is the case, anything can be proven, e.g., the assertion that "unicorns exist", by using the following argument: 1. We know that "Not all lemons are yellow", as it has been assumed to be true. 2. We know that "All lemons are yellow", as it has been assumed to be true. 3. Therefore, the two-part statement "All lemons are yellow _or_ unicorns exist" must also be true, since the first part of the statement ("All lemons are yellow") has already been assumed, and the use of "_or_" means that if even one part of the statement is true, the statement as a whole must be true as well. 4. However, since we also know that "Not all lemons are yellow" (as this has been assumed), the first part is false, and hence the second part must be true to ensure the two-part statement to be true, i.e., unicorns exist (this inference is known as the [Disjunctive syllogism](https://en.wikipedia.org/wiki/Disjunctive_syllogism "Disjunctive syllogism")). 5. The procedure may be repeated to prove that unicorns do _not_ exist (hence proving an additional contradiction where unicorns do and do not exist), as well as any other [well-formed formula](https://en.wikipedia.org/wiki/Well-formed_formula "Well-formed formula"). Thus, there is an _explosion_ of true statements. --- Date: 20241218 Links to: Tags: References: * [Principle of explosion - Wikipedia](https://en.wikipedia.org/wiki/Principle_of_explosion)