# Why the principle of explosion works? [duplicate]

I was reading about the principle of explosion and there is one thing I don't get.

Based on my understanding, the principle of explosion goes as follows: Given two statements $$P$$ and $$Q$$, and suppose $$P$$ and $$\neg P$$ are both true. Then we assert that $$P\lor Q$$ is true since $$P$$ is true. Finally, since $$\neg P$$ is true, it follows that $$Q$$ must be true for $$P\lor Q$$ to be true.

How does this form of deduction make sense? What I don't get is that if both $$P$$ and $$\neg P$$ are true, then it is a contradiction. At this point, we cannot conclude whether $$P$$ is true or false. So, it would be meaningless to determine whether $$P\lor Q$$ is true.

• The point is from a contradiction you can deduce any $Q$ in your formal proof system. Oct 8 at 2:42
• If $P$ and $\lnot P$ have both been derived at a point, then you can conclude that $P$ is true - because, you have done so. Also that it is false; simultaneously. Oct 8 at 2:48

There are many ways to see why the principle of explosion "should" be true. Here's one which I think should be particularly approachable (even if it's ultimately not "the right way" to see why explosion should be true):

Surely you agree that from $$P$$ and $$\lnot P$$ we can derive false, $$\bot$$. But now for any statement $$Q$$ at all, $$\bot \to Q$$ is a tautology. If you like, this is because

$$\bot \to Q \equiv \lnot \bot \lor q \equiv \top \lor q \equiv \top$$

So if we have $$P$$ and $$\lnot P$$ we can get $$\bot$$, and from $$\bot$$ we can get any $$Q$$ we like.

I hope this helps ^_^

• Thanks, what do you mean by getting $\bot$ from $P$ and $\neg P$? Contradiction implies $P$ is false? Oct 8 at 3:02
• You can think of $\lnot P$ as an abbreviation for $P \to \bot$. So from $P$ and $P \to \bot$ we can derive $\bot$. Oct 8 at 3:47
• I don't understand the point of deriving $\bot$... Oct 8 at 5:39
• The point is that, for any $Q$, $\bot \to Q \equiv \top$ (by the proof I gave in the body of the answer). So as soon as we've derived $\bot$, we know that we can derive any $Q$ we want! Oct 8 at 6:17