Some say the world is full of contradictions, like “parting is such sweet sorrow.” Parting is sweet, but parting is also sad; and sweetness and sadness are opposites. But logicians would say that this not a true contradiction; it’s just mixed emotions. A true contradiction would be like if Romeo was sad and not sad. Something would have to be both true and false at the same time.

Take the famous Liar Paradox. If I say, "I’m lying right now," I'm telling you I'm lying, so if I am actually lying the sentence is true—in which case I'm not lying. But if I am telling the truth, that means I'm lying. So either way, I'm both lying and telling the truth—and that's true contradiction.

Or is it? Could it be that sentence wasn’t both true and false, but that it was actually neither? That seems like a tempting solution, but it's easy to circumvent. What about this new sentence: "I’m not telling the truth right now"? Well, if that one is also neither true nor false, then I wasn’t telling the truth. But that’s what I said—which means I was telling the truth!

That’s a lot of logic to get yor head around. And some people maintain that however we handle liar paradoxes, true contradictions simply cannot exist. Nothing can be both true and false at once, because it would enable you to prove anything: that Santa Claus exists, that birds aren’t real, that the moon landing was faked, that ineapple belongs on pizza,—you name it.

So how did we get from the liar paradox to the moon landing? Think about proving things by a process of elimination, like Sherlock Holmes. He says, “When you have eliminated the impossible, whatever remains, however improbable, must be the truth.” Could true contradictions turn Holmes, the arch rationalist, into a supporter of conspiracy theories?

According to classical logic, they might. Here’s how it works. I want to use the process of elimination to prove that my enemy committed a murder. So I make a list of hypotheses. One of them is that my enemy committed a murder, and the other is the Liar Paradox. That may seem like a weird list, but we know that one of the hypotheses on the list is true—assuming we maintain that the Liar Paradox is both true and false.

You can see where I’m going: by process of elimination, I can get rid of all the false hypotheses on my list. And since the Liar Paradox is false, the only remaining possibility is that my enemy is the murderer! Or it's your enemy—it can be whatever you want!

Can we embrace the idea of true contradictions without giving up on reason and common sense? Our guest may convince you: it’s Graham Priest from the City University of New York and author Doubt Truth to be a Liar.