Testing a claim with a counterexample

Two ways to use a counterexample

A counterexample tests a claim by describing a case in which the claim fails. This study gives you practice with that idea from lesson 2. Before you look for one, decide what you are testing: a general statement, or the step from an argument’s premises to its conclusion.

To test the step, we use lesson 2’s definition of validity: can the premises be true while the conclusion is false? The cases below were invented for the course. [1]

First target: a general claim

Suppose a committee declares, “Every rule that applies equally to everyone is fair.” Now imagine that the garden offers every visitor the same stairway, including visitors who cannot climb stairs. Everyone is offered the same route, yet some people cannot get in.

The committee now has a specific problem to answer. It might reply that “applies equally” requires more than providing an identical stairway, or it might defend the rule despite the difference in access. The example does not settle the dispute on its own. What it does is show exactly where the committee must clarify its claim or defend it.

A good counterexample also keeps the meaning of its terms steady. We cannot use “equal” to mean an identical stairway at the start and then quietly switch to equal opportunity to enter, because the difference between those two meanings is the very thing we are examining.

Second target: an inference

Now take an argument: “All gardeners have a gate key. Ivo has a gate key. Therefore Ivo is a gardener.” To challenge the inference, we need a possible case in which both premises remain true and the conclusion is false.

Suppose all the gardeners have keys, and Ivo has one because he is a maintenance worker, not a gardener. Both premises remain true and the conclusion is false. The inference fails because the argument never said that only gardeners have keys.

Pointing out that “Some gardeners have lost their keys” would be a different move, because it challenges the first premise. That could be a useful criticism, but it is not the test we are running. We are asking whether the conclusion follows when both premises are true, so we have to leave them intact.

Repairing an argument fairly

One repair would add a premise: “Only gardeners have gate keys.” The argument then becomes valid, but it rests on a further claim that needs defending, and in our imagined arrangement that claim is false.

A more modest repair treats having a key as some evidence that a person is a gardener, evidence whose weight depends on who else receives keys. This version offers a tentative conclusion instead of a guarantee, and we should make that change explicit before attributing the revised argument to its speaker.

Understanding an argument fairly means making its reasons clearer, including reasons we may go on to reject. If we repair it so thoroughly that it becomes our own argument, we should say so.

Pause and try

Here is another argument: “Every visitor with a booking was admitted. Dalia was admitted. Therefore Dalia had a booking.” Give a counterexample to the inference. Then explain why your example does not prove Dalia actually lacked a booking.

Read an explanatory answer

Suppose Dalia entered without a booking, as a worker’s invited guest, while every visitor who had booked was also admitted. Both premises are then true and the conclusion is false, so the premises do not guarantee the conclusion. But the case is only a possibility: it shows that the inference fails, not that Dalia actually lacked a booking. To settle that, we would need evidence about her circumstances.

Return to the guided course

Keep track of what you have shown: a possible case can show that a conclusion need not follow, even if that case never happens. Return to lesson 3 to apply the same care to historical evidence: what does a source show, and what are we asking it to show?

Source and limit

[1] Timothy Shanahan, “Deductive and Inductive Arguments,” Internet Encyclopedia of Philosophy, §1 (section 1), for the definition of validity used in lesson 2. The garden, access and booking cases are original exercises. The ethical example invites you to examine a claim.