Counterexamples

1. Counterexamples in the Real World

As we’ve already discussed, a counterexample is a hypothetical case used to test an argument. For example, many counterexamples are used to show that arguments are invalid by giving cases where the premise is true but the conclusion is false. We already learned that one way to show that an argument is invalid is to find at least one case where the premises are true and the conclusion is false. That one case is a counterexample. In the following argument, we showed how “the butler did it” presents a counterexample where the premises are still true but the conclusion is false.

1. If the driver did it, the maid didn’t do it.
2. The maid didn’t do it.
3. Therefore, the driver did it.

Counterexamples are really helpful beyond simply proving that an argument is invalid. We use counterexamples in mathematics, science, and philosophy more broadly. They are an integral part of critical thinking.

Think about a basic science experiment. If someone does an experiment and draws a general conclusion from their results, one way to question that conclusion is to come up with a possible counterexample. Is there a circumstance under which they would perform the same experiment and get a different result?

IN CONTEXT

Suppose a pair of forensic scientists run an experiment to detect whether or not blood is present in the house of a potential murder suspect. They use the standard Luminol procedure, a process in which a chemical is sprayed onto a surface where blood is suspected. The chemical in the spray reacts with blood traces, which produces a luminescence that can be seen using a black light. This procedure is done on the house and produces a positive result; the luminescence appears, indicating the presence of blood.

Initially, the scientists believed that this test provided a foolproof way to test for the trace presence of blood. They concluded “if luminescence is detected by the Luminol process, then there are trace amounts of blood at that location.”

However, this conclusion is too quick. There is a counterexample to this conclusion by way of scientific results. The counterexample is bleach. Bleach can also cause the luminescence result, in the same way trace amounts of blood do. Therefore, the conclusion is incorrect. A positive result in the Luminol process does not always imply the presence of blood.

Counterexamples also abound in mathematics and philosophy. Classically, in both disciplines (like in logic), a counterexample is a created circumstance in which the premises of the argument are true and the conclusion false. This circumstance can be true or as absurd (and even cartoonish) as it needs to be. Ultimately, if a set of premises or assumptions and a conclusion can be identified, then a counterexample can be searched for.

Let’s look at a brief example from ethical philosophy: the study of what is morally right and wrong. One theory of ethics is called consequentialism. This theory states (roughly) that an action is morally right if its positive consequences outweigh its negative consequences. If you’re trying to prove this theory wrong, you might first look for a counterexample. That is, come up with one or two cases where consequentialism deems an action morally right, but intuitively (according to our best judgment) it is morally wrong, or vice versa. For example, you can propose that to end human suffering and save the environment, you will release a supervirus that annihilates humankind. It’s hard to deny that this would end human suffering and curb pollution. But nobody would suggest it is a morally right thing to do, so as a counterexample, it would challenge the theory of consequentialism.

Constructing and deploying counterexamples is a really crucial skill to learn for critical thinking. Counterexamples arise in any case where an argument is deployed. Ensuring you have a strong argument is strengthening it against as many possible counterexamples as possible. Next, we will look a little further into the steps involved in constructing a counterexample.

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2. Constructing Counterexamples

Constructing a counterexample first involves identifying the premises and conclusion of the argument. Once these have been identified, the goal is to create a scenario in which the premises are true and the conclusion is false. Here are a few strategies for doing just that:

In ensuring that the premises are true in our counterexamples:

• Any simple premise (without terms like “and”, “or”, “if/then”, etc.) must be straightforwardly true in your counterexample.
• Any premise with a conditional statement (“if/then”) is true in your counterexample, if the antecedent (what comes after “if”) is false or both the antecedent and the consequent are true. (We will return to this in more depth in Unit 3).
• Any premise with an “or” is true in your counterexample if at least one statement connected by the “or” is true. Neither side has to be false.
• And-statements are true when the statements on both sides of the “and” are true. Thus, both parts of the and-statement must be true in your counterexample.

In ensuring that the conclusion is false in our counterexamples, keep in mind the following rules:

• If the conclusion is a simple declarative statement (without terms like “and”, “or”, “if/then”, etc.), it must be straightforwardly false in your counterexample.
• If the conclusion is a conditional (“if/then”), the antecedent (what comes after “if”) must be true and the consequent must be false. This is the only way to render a conditional false. (We will return to this in more depth in Unit 3).
• Or-statements are false if both of the two statements on either side of the “or” are false. Both sides must be false in your counterexample.
• And-statements are false when one of the two statements on either side of the “and” are false. Thus, only one part of the and-statement must be false in your counterexample.

In your counterexample…	Declarative statements	Or statements	And statements	If/then statements
For each premise in the argument	Must be true	At least one part must be true.	All parts must be true.	Either the “if” part must be false or both parts must be true.
For the conclusion of the argument	Must be false	All parts must be false.	At least one part must be false.	The “if” part must be true and the “then” part must be false.

Now let’s work through some concrete examples. Suppose we want to construct a counterexample to the following argument:

1. Either Alice is in Wonderland or Alice is in Great Britain.
2. Alice likes crumpets.
3. Therefore, Alice is in Great Britain.

Let’s walk through the steps of creating a counterexample for this argument to show that the argument is invalid.

First and foremost, we must make both the premises true. The second premise is the most straightforward. Our counterexample must stipulate that Alice likes crumpets.

Next (for just a minute), let’s jump to the conclusion. The conclusion must be false in our counterexample. So, our counterexample must stipulate that Alice is not in Great Britain.

Then, notice that if we stipulate Alice is not in Great Britain, the only way for the or-statement in premise 1 to be true is if Alice is in Wonderland.

Thus, our counterexample is as follows:

Counterexample: Alice likes crumpets and is currently in Wonderland.

Let’s look at one more.

1. If Alice eats the chocolate, then she will grow big.
2. Alice is big.
3. Therefore, Alice ate the chocolate.

Now, we have seen invalid arguments like this before. This argument commits the fallacy of affirming the consequent.

First, we ensure that the premises are true in our counterexample. We learned above that one way to make a conditional true is to make the antecedent false. Thus, we can specify that in our counterexample, Alice did not eat the chocolate. This actually serves two purposes at once. Why? Because we also need the conclusion to be false, and the conclusion is that Alice ate the chocolate.

Second, we affirm that the second premise is true—Alice is big. Thus, in our counterexample, Alice didn’t eat the chocolate and she is still big.

Counterexample: Alice didn’t eat the chocolate but something else she ate or drank made her grow big.

This is compatible with the premises of our argument and renders the conclusion false. Hence, the argument is invalid.

try it

Construct a counterexample for the following argument:

1. If an animal is a bird, then it can fly.
2. Flappy can fly.
3. Therefore, Flappy is a bird.

Check Your Answer

You may have suggested a case where Flappy is a bat or an insect, but you can even propose that Flappy is a flying elephant, and it would still show the argument is invalid.

big idea

The quality of an argument can be assessed in terms of whether the argument is valid and sound. Validity requires that the conclusion follows from the premises, with no other possible cases (or counterexample) where the premises are true and the conclusion is false. Soundness requires that the premises (and thus also conclusion) of an argument be actually true in the real world.

think about it

Is it possible to have a valid argument with a false conclusion?

Yes, the validity of an argument does not depend on the premises being true; they may be false, hypothetical, or unproven.

Is it possible for a valid argument to have a few rare exceptions where the conclusion is false even though the premises are true?

No, remember that a single exception (or counterexample) where the premises are true and the conclusion is false makes the argument invalid. In fact, the exception may be hypothetical, but if it is at least possible, then the argument is considered invalid.

Is it possible to have a sound argument with a false conclusion?

No, by definition of a sound argument, the conclusion must be true, because the premises are true and the conclusion follows from the premises. Put another way, if the conclusion is false, either the premises are false, or the argument is invalid.

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3. Limitations of Counterexamples

You may begin to see some problems with this method of testing validity. What if someone reads the sentence, “If the driver did it, the maid didn’t do it,” as meaning, “Either the driver or the maid did it”? They may feel that in the context of the argument and the speaker’s intent, the first sentence should be interpreted as stating either the driver or the maid did it. The problem here is that natural language permits for numerous readings of a single sentence. Take, for example, the sentence, “There is a bird in a cage that can talk.” This sentence again has two readings, the natural and the absurd. The absurd reading is that there is a bird in a cage and the cage can talk; the more natural reading is that there is a bird who can talk in a cage. This is just one example, but if you reflect on it, you’ll find numerous examples in natural language of sentences that are ambiguous in a variety of different ways, lending themselves to many different interpretations.

did you know

The entire field of Constitutional Law is about the different interpretations of statements in the Constitution and the intent of the authors.

Furthermore, there are no agreed-upon limits of what kind of hypothetical scenario is allowed. In the argument about the butler and the gardener, perhaps someone comes up with a scenario where the gardener did it, but only because the butler was using mind control, and the butler is actually responsible for the gardener’s crimes. We said that a single case where the premises are true and the conclusion is false proves the argument is invalid, but didn’t put any constraints on imagining far-fetched scenarios, loose interpretations of sentences, or going beyond the bounds of known science in those scenarios. Different people may therefore come to different determinations of whether the argument is valid. So, we are going to need to be precise about what the rules are for proposing counterexamples or cases in testing for validity. In the next tutorial, we will try to solve this problem, but for now we will have to settle for this imprecise test of validity.