2 + 2 = 5 implies Loch Ness

One of the things I like about studying logic and computation is using some of the rigorous mathematical ideas as a grounding for some everyday ‘intuitive’ ideas. One example of this is “ex falso quodlibet” – the principle that “from a contradiction, anything follows”.

Let’s say someone believed something that’s false, call it C (for ‘contradiction’), and they are using C in an argument. In other words, they claim that C is true, when it is actually false. Intuitively, this doesn’t seem ‘good’, but why? What’s wrong with a ‘true contradiction’?

Ex falso quodlibet helps answer this question.

Ex falso quodlibet is a principle of classical logic stating that anything can be proven by using a contradiction as a premise. In our example, ex falso quodlibet says that we can use the ‘true contradiction’ C to prove any statement to be true, regardless of whether the statement is actually true or not.

But wait, this means that we can prove that all “false” things are “true”, and then … everything is true! Ex falso quodlibet shows us that letting a contradiction slip into a logical framework eliminates any way of distinguishing between truth and falsehood. In a way, allowing that single contradiction in as a truth causes the entire framework to ‘explode’ (ex falso quodlibet is also called the principle of explosion).

But besides being useful in proofs and in bending minds, what real-world significance does this have? It may help defend the claims that one probably shouldn’t believe false things, and that it’s probably a good idea to correct inconsistencies in things that one believes. Because otherwise, those contradictions could be used as evidence to ‘prove’ other false ideas. Easier said than done, but ex falso quodlibet illustrates some downsides to keeping C around.

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xkcd 704

It also aids in understanding the structure of some snarky comments. Let’s say you’re arguing with a friend about whether Lance Armstrong used performance enhancing drugs. You’re arguing that he did, and you say something like

“Well, if Lance was clean then I’m the queen of England”

In other words, you’re saying,

“Well, if [falsehood is true], then [insert-statement-here is also true]”

A subtle ex falso quodlibet reference.

Or, let’s say your friend was making an argument that the Loch Ness monster exists, using “2+2=5” as a premise. His or her argument is structured as

“2+2=5 implies that the Loch Ness monster exists”

which, by ex falso quodlibet, translates into

“[A contradiction] implies [insert-statement-here]”

So you could respond by saying,

“Well yeah, but due to ex falso quodlibet, 2+2=5 also implies that the Loch Ness monster does not exist”

An extreme example, but as mathematician Terence Tao pointed out, it may show up from time to time.

As a takeaway one-liner for remembering the principle, “absurdity begets absurdity”.

Further reading