Imagine you’re in the lunch line with your friend Omar. The man serving your food is especially grumpy and scolds Omar for taking too many french fries. “Ugh,” Omar tells you, “All the cafeteria workers are so mean.” Your mind immediately jumps to Maxine, the sweet lady that works at the cash register. She wears hand-knitted sweaters and always asks how your day is going. “Maxine isn’t mean,” you remind Omar. “Yeah, I guess you’re right,” Omar reconsiders. “I guess not all the cafeteria workers are mean.”
Omar’s statement about cafeteria workers was a universal statement because it used the word “all.” Therefore, by citing a particular cafeteria worker who did not fit with Omar’s general statement, you disproved its universality. Not all cafeteria workers are mean. Maxine isn’t.
Definition of Counterexample
Your reference to Maxine as a response to Omar is a counterexample. A counterexample is a specific example that proves a general statement or argument is incorrect. Counterexamples are used as a logical device in a wide range of disciplines, including mathematics, philosophy, and public policy.
How It Works
A counterexample usually arises when one party in a debate or field poses a universal statement, such as “All squirrels are gray,” “Any even number is divisible by 2,” or “No horses can talk.” All of these statements are universal because they take a group of things and attempt to apply a particular property to every single thing in that group. Universal statements can be identified by the words “all,” “any,” “always,” “every,” “no,” “none,” and “never.” Counterexamples are a way to disprove a universal statement. As long as the opposition can identify a single thing in that group that does not possess the property in the universalization, then it is an exception to the rule. For example, not all squirrels are gray because there are albino squirrels!
Some universal statements, especially ones about abstract concepts like math, do not have counterexamples. All even numbers are divisible by 2, and there is no even number that one can appeal to as an exception to this rule. Other universals are trickier. No horses in real life can talk, but there are fictional talking horses in television shows and books. With these kinds of universals, it is helpful to identify the scope of the statement to see if a counterexample is available.
Counterexamples are also useful in disproving the validity of an argument. An argument is valid when the conclusion cannot be false if all the premises are true. Consider the validity of this argument: Some birds swim. All ducks are birds. Therefore, all ducks swim. Both the premises and the conclusion of this argument are true, but it is not valid. To prove this, we can construct a counterexample with the same argument form that has true premises and a false conclusion. For example: Some people have black hair. All women are people. Therefore, all women have black hair. Although both premises are true, the conclusion is false, meaning this argument form is invalid. This argument operates as a counterexample to disprove the original argument.
Be on the lookout for universal statements in debates and persuasive arguments. Even if most things in a group have that particular property, there is often at least one exception. Keep in mind the essential qualities of the thing you are discussing. Would it still be that thing if it didn’t have the property in contention? If yes, you can probably find a counterexample to disprove the universal. After all, the ability to talk would not make a horse less of a horse, but divisibility by 2 is a defining feature of even numbers.
At the same time, exceptions and counterexamples shouldn’t be used to downgrade the importance of general statements. For example, Black exceptionalism is a common response to discussions about systemic racism. When one party points out that Black people make less money on average and are disproportionately less likely to hold positions of power because of institutionalized racial oppression, another may respond with a counterexample of Oprah or Obama. “These Black people are successful,” they may say, “Therefore, racism isn’t a problem in our country.” Yet, these successful exceptions are no indication of a post-racial society. Notice that the original claims were not universal but rather described statistical evidence.
Lastly, even if there is a valid counterexample to a certain claim, the claim is still worth consideration. Consider the warning, “Everyone who smokes cigarettes develops lung cancer.” This is a hyperbole, and some people who smoke do not develop lung cancer. However, this counterexample does not invalidate the message of the claim. Smoking greatly increases one’s chances of developing lung cancer. This fact has no counterexample!