Don’t Forget about Black Swans
On the Importance of Counterexamples in Your Intellectual Life
If you have ever watched a courtroom drama, you know that in each episode the case often hinges on a single piece of evidence. This vital piece of evidence is usually not revealed until the last possible minute. When all seems lost for the heroes, the evidence arrives and undermines the entirety of the opposition’s argument, allowing justice to be served.
While in TV shows this is done largely for dramatic effect, it reveals something about the nature of arguments — a single counterexample can undermine any argument no matter how strong. Counterexamples have the power to overturn mountains of positive evidence and humble even the most rigorous theories.
Despite this destructive power, counterexamples are a central part of rigorous thinking. They are something to be sought after and not avoided. For those who know how to use them, counterexamples are not just annoyances or exceptions, they are often the engines of intellectual progress.
In this article, we will look at the two primary ways that counterexamples help our intellectual lives. First, we will look at how counterexamples serve as a kind of check against truth. Then, we will look at how counterexamples can help us clarify our ideas. Finally, we will conclude by looking at some ways that we can find and include counterexamples in our arguments.
Trying to Prove Yourself Wrong is Rational
Often when we construct arguments we are focused on trying to be right. We want the truth and to convince others that what we say is true. To this end, we stack our arguments with examples, evidence, and reasons to back up our claims. However, we usually don’t check to see if the argument might be false.
Perhaps this is because of our human tendency to seek proof of our views rather than contradictions. Yet spending time to make sure we are not wrong is as important as making sure that we are telling the truth. To leave out the pursuit of possible contradictions is to make our position weaker rather than stronger.
This is because, just like legal TV dramas, a single piece of contradictory evidence is more powerful than an infinite amount of positive evidence. This ‘rule’ is best explained by Karl Popper and his work in the philosophy of science. In his book The Logic of Scientific Discovery (1934), Popper established the concept of ‘falsifiability’ — the idea that in order for a scientific theory to be accepted it must be possible for it to be proven wrong. In science this means that a valid scientific theory must be able to be proven wrong by empirical evidence. The example he used was that of black swans.
For centuries, as far as western science was concerned, all swans were white. Scientists have tens of thousands of examples of swans being white and no other color. They claimed that being white was a defining part of what it was to be a swan. There is even an old Roman saying — “a rare bird upon earth, and exceedingly like a black swan” (rara avis in terris nigroque simillima cygno) — which was used in the same way we say “when pigs fly” today. However, as Popper points out:
“No matter how many instances of white swans we may have observed, this does not justify the conclusion that all swans are white” (Karl Popper, The Logic of Scientific Discovery, 4)
In fact, everything changed in 1697 when the Dutch explorer Willem Vlamingh discovered black swans in Australia and caused science to rethink everything it knew about swans.
At the heart of the power of counterexamples is the nature of universal claims. When we use words like all, none, or every, we are making the strongest claims possible. We are claiming that 100 percent of something is or is not a certain way. Thus, even a single instance to the contrary destroys the pure 100 percent nature of the claim making it 99.999 repeating and thus false.
Take for example the world being flat. Conspiracy theorists have a long list of reasons to believe that the world is flat. Yet one simple counterexample, a picture from space, disproves all other evidence and reasoning.
This is perhaps the hardest thing to do. It can take years or even centuries before someone can think of a way to show that something is false.
Clarifying Concepts
The other major contribution that counterexamples make is in helping us clarify what exactly it is we are talking about. If you have ever had a friend nag you about every possible strange thing they could get you to eat for a million dollars, you know what I am talking about. With each increasingly weird item they ask you to eat, the concept of what you will do for a million dollars becomes clearer.
The same is true in philosophical arguments. Counterexamples test the boundaries of concepts and help make those boundaries visible. For example, does a dirty urinal in a gallery still count as art? How about a person just sitting on a chair? With each case we consider if it is or is not art and get one step closer to knowing what art is.
Philosophy is full of examples where counterexamples forced us to reexamine concepts we thought we understood. Perhaps the most famous example is the Gettier problem which exposed the traditional account of knowledge as justified true belief. Proposed by Edmund Gettier, after whom it is named, offers the following scenario:
Suppose that Smith and Jones have applied for a certain job. And suppose that Smith has strong evidence for the following conjunctive proposition: Jones is the man who will get the job, and Jones has ten coins in his pocket. (Gettier, “Is Justified True Belief Knowledge?,” 122)
Gettier argues that, according to the traditional account of knowledge, Smith can be said to know the following: “The man who will get the job has ten coins in his pocket” (Gettier, “Is Justified True Belief Knowledge?,” 122).
Smith is justified because the boss told him that Jones will get the job, and he saw Jones count ten coins and put them in his pocket. He believes it because it was his boss (not a random employee) who told him about the job and Smith himself saw Jones put the coins in his pocket.
The examples hinge then on the truth of the statement. Gettier says suppose that the boss was mistaken that Jones would get the job and that, just like Jones, Smith also has ten coins in his pocket but forgot about them. Thus, the statement above is true and should then count as knowledge.
But our instincts tell us that Smith does not know that the man who will get the job (i.e. him) has ten coins in his pocket. Gettier says that Smith doesn’t know it; he is just lucky, and that we need something else besides justified true belief to have knowledge (e.g. we need a bit of luck).
Finding Counterexamples
So how do we come up with good counterexamples? There are many ways to do this, but three of the easiest are to come up with powerful counterexamples to test your arguments and those of others.
First, to create a counterexample all you need to do is to negate the claim — just say the opposite. The opposite of the claim ‘Democracy is the best form of government’ is ‘Democracy is not the best form of government’. If the opposite of the claim is true, then it disproves the original claim.
Second, an easy way to find counterexamples is to look at extreme cases. We have all heard that you should get 10,000 steps a day to stay healthy. To find a counterexample, do not look for cases of 9,000 or 11,000 steps. Instead, what about cases of ten steps a day? If someone can be healthy and not take more than ten steps a day, then it would disprove the claim that we need to take around 10,000 steps a day to stay healthy.
Finally, we can simply move around or swap out key terms in the claim to construct a counterexample. This helps us look at the inferences and logical structure of the claim from a different point of view and thus better understand and critique it.
For example, the claim “Successful people are rich” says that there is a connection between success and being rich. To find a counterexample, we can replace the key terms (successful and rich) with other words to see if the connection still holds. We could replace “successful” with someone or a group that we think is successful but is not rich like a monk who finds inner peace by giving up all his worldly belongings. We can even swap the key terms around and ask if there are any rich people who are not successful. If one of the altered claims is true, then it is a counterexample to the original claim.
Conclusion
In conclusion, counterexamples are not obstacles to be overcome but essential tools for intellectual progress. They expose vague or hidden assumptions in our concepts and arguments making it possible for us to improve them. More importantly, counterexamples also help ensure that a position or argument is not just true but also that it is not false. Thus, the pursuit of counterexamples helps us develop both our arguments and intellectual lives.
The next time you present an argument or critique someone else’s, be sure to discuss possible counterexamples. Ask yourself, “What could prove me wrong?” and put as much effort into answering that question as you do into answering “What could prove me right?”
Remember, an exception doesn’t just prove the rule — it often rewrites it.