Finding Counterexamples

When we think we have identified a set of individually necessary and jointly sufficient conditions to define a concept, it is important to test the definition by looking for counterexamples. A counterexample to a supposedly necessary condition is an example showing that the condition isn’t really necessary after all. Likewise, a counterexample to a putatively sufficient condition (or set of jointly sufficient conditions) is an example showing that the condition (or set) isn’t really sufficient after all. Thus, a counterexample to a definition shows either that the concept can still apply even when a supposedly necessary condition isn’t met, or that the concept doesn’t always apply even when the supposedly sufficient conditions are met.

For instance, suppose we are trying to come up with a set of individually necessary and jointly sufficient conditions to define the concept of a sport. After reflecting on various examples of sports and considering what they have in common, we might hypothesize that all sports involve some kind of competition. So, perhaps this is a necessary condition for something to be a sport: Is that really a necessary condition for sports, though? In order to test this hypothesis, we should try to think of a counterexample—i.e., an example of a sport that isn’t competitive. Skiing is considered a sport, yet it is often practiced non-competitively, purely for recreation. So, we have discovered a counterexample to the idea that being done competitively is a necessary condition for sports.

To avoid the counterexample, we may start over with a different hypothesis, or we may instead try to refine the condition in some way. Even skiing is done competitively sometimes. So, the necessary condition might be refined to avoid the counterexample, as follows:

Perhaps this still isn’t a genuinely necessary condition. There may be further counterexamples that we haven’t discovered, but let’s ignore that possibility for the moment and suppose we have identified a necessary condition for the concept of a sport.

Next, we can ask whether the condition is also sufficient. To test that idea, we need to consider whether it’s possible for something to be done competitively (sometimes) and yet not be a sport. Video games are often played competitively, yet the playing of video games doesn’t seem to count as a sport. Thus, we have a counterexample to the sufficiency of the above condition.

In order to succeed in defining sport, then, we need to add further necessary conditions, until we have a complete set of individually necessary and jointly sufficient conditions. For instance, perhaps the reason video games don’t count as sports is that they don’t involve physical exertion or exercise. So, we can add that requirement as an additional necessary condition. In other words, in order for something to be a sport, it must meet both of these conditions:

The addition of that second necessary condition eliminates the video game counterexample, but it might admit of new counterexamples. I’ll leave the reader to ponder whether those two conditions really are individually necessary and jointly sufficient for sports. The point of these examples is just to illustrate an important method of conceptual analysis.

It’s worth mentioning that some concepts may be too broad or too vague to define with necessary and sufficient conditions. For example, historical attempts to identify necessary and sufficient conditions for science have failed, and many philosophers of science have concluded that there simply is no single set of conditions satisfied by all and only the sciences. (Philosophers of science refer to this difficulty as the demarcation problem.) Other concepts in philosophy may be similarly intractable. Nevertheless, even when a proposed definition is refuted by counterexamples, we can gain a deeper understanding just by recognizing why the conditions we’ve identified are inadequate.

This last point about the value of counterexamples is nicely illustrated by a famous problem in epistemology, the philosophical study of knowledge. A traditional, formerly-popular theoretical definition of knowledge said that knowledge could be defined by three individually necessary and jointly sufficient conditions. You know some proposition P, according to this definition, if and only if:

  1. You believe P.
  2. P is true.
  3. You are justified in believing P. (That is, you have good evidence for P.)

The above three conditions are individually necessary because you can’t have knowledge if any one of them isn’t met: (1) you must believe P, and (2) your belief must be justified, and (3) P must be true. They are jointly sufficient, according to the traditional view, because meeting all three conditions is enough: if you believe P, and your belief is justified, and P is true, then you do have knowledge. Thus, propositional knowledge can be defined as justified, true belief. This traditional definition has been called the JTB theory of knowledge.

The “justified, true belief” (JTB) theory of knowledge was decisively refuted in a short paper by epistemologist Edmund Gettier in 1963. He offered counterexamples showing that the three conditions are not jointly sufficient: someone can meet all three conditions without really knowing P.Edmund L. Gettier, “Is Justified True Belief Knowledge?”, Analysis 23(6):121-123 (1963). Gettier’s paper deepened our understanding of knowledge not by proposing a new definition but by revealing a problem with the commonly-accepted definition.

To see what’s wrong with the JTB theory, consider the following counterexample, which Logician Bertrand Russell had raised fifteen years prior to Gettier’s influential paper. Russell’s counterexample is simpler than Gettier’s but suffices to illustrate the problem:

If you look at a clock which you believe to be going, but which in fact has stopped, and you happen to look at it at a moment when it is right, you will acquire a true belief as to the time of day, but you cannot be correctly said to have knowledge.Bertrand Russell, Human Knowledge: Its Scope and Limits (London: Routledge, 2009), 91. First published in 1948 by George Allen & Unwin Ltd., London.
Ordinarily, we take ourselves to know what time it is when we consult a clock, because clocks usually provide us with good evidence about the time. We are justified in believing what the clock says, so long as we have no reason to suspect it has malfunctioned. We don’t really know what time it is if the clock is wrong, obviously, since the belief is false in that instance. But what if a broken clock just happens to be right, showing the correct time at this moment by sheer coincidence? In that case, our belief about the time is true. Moreover, the belief is justified because it is based on good evidence (namely, looking at a clock which we have no reason to suspect is broken). So, in this unusual case, all three conditions of the JTB theory are met. Nevertheless, as Russell points out, it does not seem right to describe this as a genuine case of knowledge. Therefore, the JTB theory of knowledge is incorrect.

This counterexample reveals a shortcoming in the JTB definition: mere justified, true belief is not sufficient for knowledge. Something else is needed to ensure that the evidence is connected with the truth of the belief in an appropriate way, rather than by mere coincidence or luck. Perhaps we could correct the JTB definition by adding a fourth condition, as many epistemologists tried to do in subsequent years. Alternatively, we could start over with a different set of conditions. Epistemologists have tried that strategy as well. This is not the place to survey the history of such attempts to define knowledge. My point in recounting this important episode of epistemological thought is to illustrate how looking for counterexamples can lead to new insights, often by revealing limitations in our current understanding of a concept.