Anything follows from a contradiction. A moon made of cheese. Five-sided triangles. Decapod felines. Time flowing backwards. A non-existent universe.

Ex contradictione quodlibet.

No matter how absurd, anything at all is made true by a logical contradiction. This is the Principle of Explosion.

Okay, so the moon won’t really turn into cheese if you contradict yourself. But if you actually managed to make the contradiction true ,  it really would!

This is a post about the inner workings of the Laws of Logic, and how they should inform the way that we argue and debate about things. I wanted to write this for a couple of reasons. First, I’m really passionate about formal logic (yes, I know — don’t worry — I have some other marginally less dry interests too..!) and Ex Contradictione Quodlibet is my favourite proof. Second, it explains why proving that some kind of contradiction follows from somebody’s argument proves their argument is invalid.

And having a clear idea of how to structure a refutation of somebody’s argument is conducive to a fruitful variety of debate — something I’ve talked about in a previous blog post in relation to how to have useful political debates.

It seems like a pretty crazy claim, doesn’t it? If some — any — statement is both true and not true, then anything we can think of also becomes true. Let’s look under the hood of the proof and walk through its 4 steps.

1) Assume there’s a contradiction

Let’s start with a contradiction. In order to prove that anything follows from one, we need to assume one in the first place and see what happens!

PREMISE 1: A is true, and A is not true.

“Grass is green, and grass is not green.” From Premise 1, given that we assumed the contradiction is true, we must also accept that:

PREMISE 2: A is true (from Premise 1)

PREMISE 3: A is not true (from Premise 1)

“Grass is green” “Grass is not green”

2) “A true statement OR anything else” — Always true!

If it’s true that “Giraffes have necks”, then it is also true that “Giraffes have necks, or apples are orange”, and it’s also true that “Giraffes have necks, or all humans are made of chocolate”. Because only part of the sentence needs to be true in order for the disjunction (‘or’) to be true.

By extension, if ‘A’ is true, it’s also true that ‘A or B’ is true — or ‘A or Z’. So given that ‘A’ is true:

PREMISE 4: A is true, or B is true.

Remember… B could be ANYTHING.

“Grass is green, or the moon is made of cheese.” “Grass is green, or all triangles have five sides.” “Grass is green, or ANYTHING AT ALL!”

3) But we already said A was false..!

That was Premise 3. We already derived Premise 3 from our core assumption. Yet we just said that either A or B was true. Didn’t we?

So… If A is not true, B MUST be true. And since we said A was not true (Premise 3), that makes B true. Undeniably so, under the assumptions we’ve made.

CONCLUSION: B is true (from Premise 2 — A is not true — and 4 — A or B)

“The moon is made of cheese” “All triangles have five sides” “ANYTHING AT ALL!” So, how do we get out of this one? The notion of everything one could ever imagine being true all at once is unpalatable. So what we do is simply ban contradictions.

This little logical curiosity is enough, for the vast majority of philosophers, to justify the Aristotelian claim that:

There cannot be contradictions. This is the Principle of Non-Contradiction.

And this principle is why, in debate, proving that your opponents’ claims lead to a contradiction suffices to show their argument doesn’t work. In Philosophy, we call this Reductio Ad Absurdum, or ‘reduction to absurdity’.

One more complication. This proof works in most logics, including classical logic which is widely held. But there are other systems of logic, paraconsistent logics, which are more tolerant of contradiction. They reject the Principle of Non-Contradiction and maintain dialetheism: the idea that some statements can be both true and false. Now, that’s not necessarily a road down which I would suggest anyone go; but it’s interesting nevertheless that notable thinkers reject the paradigm of classical logic that any statement is either true or false.

I always enjoyed understanding how we can use logic to sift through arguments and draw interesting, rational conclusions. I hope looking under the hood of Ex Contradictione Quodlibet has made learning about the role of logic in debate that much more compelling!

Finally, if you are interested in logic, and are considering new careers — you could really thrive by learning to code..! I studied Philosophy, specialised in Formal Logic, and changed my life by learning to code. If you’re in the North of England, I studied at (and now work for) Northcoders, the Coding Bootcamp for the North. They’re fab.

Enjoy this? Follow me on Twitter @RuthYMNg!