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Unicorns roam free in fantasy novels and children’s stories, not so much in the real world, much less the cold, analytical ones of math and philosophy. But it turns out that these logical disciplines are only one misstep away from proving the existence of the long-adored mythic creatures—or proving any absurdity.
To understand how unicorns could migrate into our most objective fields of study, we must first look to tenets laid down by Aristotle more than 2,300 years ago. Among his many impressive contributions, he is often credited with articulating the “three laws of thought”—self-evident statements that we must assume for any theory of logic to take flight. The one that matters for unicorn hunters is the law forbidding contradiction. That law says propositions cannot be both true and false. You can’t have A and not A. Square circles and married bachelors are simply unwelcome in a civilized logic.
Contradictions keep math and philosophy on course through negative feedback. Like dead ends in a maze, they signal “this is not the way forward” and demand that you retrace your steps and choose a different path. Contradictions also underpin all paradoxes. Consider the infamous liar paradox: “This sentence is false.” If it’s true, then we should take it at face value: the sentence is false. If it’s false then it is not the case that the sentence is false, i.e., it’s true. So if the statement is true, then we deduce that the statement is false and vice versa, a contradiction. Because of Aristotle’s law, the contradiction cannot stand, so the liar paradox and hundreds of other known paradoxes beg for resolutions. Reams of philosophical papers have been devoted to the impressively resilient liar paradox, all in an effort to purge the world of one contradiction.
But why are contradictions so unacceptable? Need we accept the law of noncontradiction? Maybe contradictions are akin to black holes. They’re weird, counterintuitive boundary objects that violate some accustomed rules, but we must make room for them in our description of reality. What would happen if we threw up our hands and accepted the liar paradox as a genuine contradiction? Aside from them being aesthetically unpalatable, inviting a contradiction into logic poses a major problem known as the principle of explosion. Once we admit even a single contradiction, we can prove anything, whether it’s true or not.
The argument that proves anything from a contradiction is remarkably straightforward. As a warm-up, suppose you know that the following statement is true.
True statement: Omar is married or Maria is five feet tall.
You know the above to be true. It doesn’t necessarily imply that Omar is married, nor does it imply that Maria is five feet tall. It only implies that at least one of those must be the case. Then you import an additional piece of knowledge.
True statement: Omar is not married.
What can you conclude from this pair of assertions? We conclude that Maria must be five feet tall. Because if she isn’t and Omar isn’t married either, then our original or-statement couldn’t have been true after all. With this example in mind, let’s assume a contradiction to be true and then derive something ridiculous from it. Philosophers love a married bachelor as a succinct example of a contradiction; so to honor that tradition, let’s assume the following:
True statement: Omar is married.
True statement: Omar is not married.
Using these as true statements, we’ll now prove that unicorns exist.
True statement: Omar is married or unicorns exist.
This is true because we know from our assumption that Omar is married and an or-statement as a whole is true whenever one of the claims on either side of the “or” is true.
True statement: Omar is not married.
Remember, we assumed this to be true.
Conclusion: Unicorns exist.
Just like we concluded that Maria must be five feet tall, once we accept that either Omar is married or unicorns exist and then add in that Omar is not married, we’re forced to admit the absurd. The simplicity of this argument can make it seem like sleight of hand, but the principle of explosion is fully sound and a key reason why contradictions cause intolerable destruction. If a single contradiction is true, then everything is true.
Some logicians find the principle of explosion so disturbing that they propose altering the rules of logic into a so-called paraconsistent logic, specifically designed to invalidate the arguments we’ve seen above. Proponents of this project argue that since unicorns have nothing to do with Omar’s marital status, we should not be able to learn anything about one from the other. Still, those in favor of paraconsistent logic have to bite some hearty bullets by rejecting seemingly obvious arguments as invalid, like the argument we used to conclude that Maria is five feet tall. Most philosophers decline to make that move.
Some advocates of paraconsistent logic take an even more radical stance called dialetheism, which asserts that some contradictions are actually true. Dialetheists reject the law of noncontradiction and claim that rather than expelling contradictions from every corner of rationality, we should embrace them as peculiar types of statements that are occasionally true and false simultaneously. Dialetheists boast that under their view, head-banging conundra like the liar paradox resolve themselves. They simply say that “this sentence is false” is both true and false, with no need for further debate. Although dialetheism has relatively few adherents, it has gained recognition as a respectable philosophical position, largely thanks to the extensive work of British philosopher Graham Priest.
Logic is also the foundation of mathematics, meaning that math is just as vulnerable to catastrophe if a contradiction arises. Spanning different eras and languages, mathematicians have erected a towering edifice of intricately tangled arguments that govern everything from the stuff you use to balance your checkbook to the calculations that make planes fly and nuclear reactors cook.
The principle of explosion ensures that unless we want to rewrite logic itself, a single contradiction would bring the whole field tumbling to the ground. It is remarkable to consider that among countless complicated arguments in logic and math, we’ve avoided collapse and not let one contradiction slip through the cracks—at least that we know of.
ABOUT THE AUTHOR(S)
Jack Murtagh writes about math and puzzles, including a series on mathematical curiosities at Scientific American and a weekly puzzle column at Gizmodo. His original puzzles have appeared in the New York Times, the Wall Street Journal and the Los Angeles Times, among other outlets. He holds a Ph.D. in theoretical computer science from Harvard University. Follow Murtagh on Twitter @JackPMurtagh
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