Counterexamples (WritePhilosophy Guide)

Some of the most powerful philosophical criticisms provide counterexamples to a claim or an argument. In the 1930s, Sir Karl Popper, founder of the LSE Department of Philosophy, Logic & Scientific Method, build his whole philosophical system on the asymmetry between what it takes to prove vs. disprove a claim. For a universal generalisation like ‘All ravens are black’ or ‘Murder is morally impermissible’, it takes a substantial argument – or the supertask of checking every possible case – to prove the claim true. For the claims Popper cared about – scientific theories – it was not possible for mortal investigators to do. But disproving a generalisation takes just one example. It takes one non-black raven or one morally permissible murder to disprove those generalisations. That’s the power of the counterexample.


Gettier Cases

In 1963, Edmund Gettier published a short scrap of a paper entitled Is justified true belief knowledge? which, in a single blow, demolished over 2,000 years of epistemology. His paper provided counterexamples to Aristotle’s analysis of knowledge, which defined knowledge as a justified true belief. If you ever feel like you can’t accomplish anything of worth in a 1,500 word or 2,000 word philosophy paper, think of Gettier. His paper comes in at just 900 words – and most of those are superfluous!

Aristotle provided a conceptual analysis of knowledge, as follows:

A knows that X if and only if:
a) X is true.
b) A believes X.
b) A is justified in believing X.

To provide a counterexample and destroy this conceptual analysis, all we need to do is find either a case where A knows X but one of the three conditions is missing, or a case where the three conditions are satisfied but we would not be happy to say that A knows X. Gettier did the latter. He actually gave two counterexamples for good measure. Here’s his first:

Suppose that Smith and Jones have applied for a certain job. And suppose that Smith has strong evidence for the following conjunctive proposition:

(d) Jones is the man who will get the job, and Jones has ten coins in his pocket.

Smith’s evidence for (d) might be that the president of the company assured him that Jones would in the end be selected, and that he, Smith, had counted the coins in Jones’s pocket ten minutes ago. Proposition (d) entails:

(e) The man who will get the job has ten coins in his pocket.

Let us suppose that Smith sees the entailment from (d) to (e), and accepts (e) on the grounds of (d), for which he has strong evidence. In this case, Smith is clearly justified in believing that (e) is true.

But imagine, further, that unknown to Smith, he himself, not Jones, will get the job. And, also, unknown to Smith, he himself has ten coins in his pocket. Proposition (e) is then true, though proposition (d), from which Smith inferred (e), is false. In our example, then, all of the following are true: (i) (e) is true, (ii) Smith believes that (e) is true, and (iii) Smith is justified in believing that (e) is true. But it is equally clear that Smith does not KNOW that (e) is true; for (e) is true in virtue of the number of coins in Smith’s pocket, while Smith does not know how many coins are in Smith’s pocket, and bases his belief in (e) on a count of the coins in Jones’s pocket, whom he falsely believes to be the man who will get the job.

As long as we buy that Smith’s belief in (e) is a justified true belief, but that he does not know (e), in this case, Aristotle’s conceptual analysis must fall. The three criteria Aristotle presented might be necessary for knowledge, but are insufficient. At the very least, a fourth criterion will be needed.

How did you feel about Gettier’s counterexample? It’s natural to be a bit baffled by it. Philosophers who make breakthroughs like this often struggle to articulate themselves as clearly as possible the first time around. But once the format for Gettier’s counterexample was established, others could come along and think of far simpler ones. Consider, for example:

Aisha has a watch which has always been reliable. Last night at exactly 1:03am, her watch stopped. She didn’t notice. As it happens, at exactly 1:03pm this afternoon, Ben asked Aisha what time it was. Looking at her watch, she told him: “It’s 1:03.”

Aisha’s statement is true and she believes it. Her justification is inductive – every time she’s looked at the watch before, it has been right, and nothing had given her pause to suspect it was off now. She has a justified true belief that the time is 1:03. But we don’t want to say she knew it was 1:03. Intuitively, she got lucky in having a justified true belief. At any other moment in the day, she would have been wrong. Here’s another:

Boris looks into a field and sees what looks like a cow. He says: “There’s a cow in that field”. But what he saw wasn’t actually a cow – it was a very realistic cardboard cutout of a cow. But as it happened, there was a cow standing hidden behind the cardboard cutout. So Boris is correct.

Try to formulate your own Gettier counterexamples.

As a general rule, when formulating a counterexample try to keep it as simple as possible. Gettier muddies the waters when he talks about 10 coins in the pocket and so on. If at all possible, try to use something which could happen in everyday life, like a watch stopping, rather than something that’s well beyond the range of normal experience. Why? Because counterexamples often rely on our intuitions: we have to agree with Gettier that Smith did have a justified true belief and didn’t have knowledge. That agreement is harder to elicit if the case is unfamiliar. We don’t usually go about counting coins in pockets, and then we don’t make claims where we refer to people based on the coins in their pocket! But we do make claims about the time based on the hands of a watch.


Types of Counterexample

There are two broad types of counterexample: propositional and argumentative counterexamples. Gettier cases are an example of propositional counterexamples. Aristotle put forward a proposition, the claim that “Knowledge is justified true belief”. Gettier provided an exception to that generalisation, showing that the rule doesn’t hold.

Argumentative counterexamples follow a similar trajectory, but apply to argument schemata. An argument schema is the logical form which an argument takes. For example, suppose I make the argument:

1. If Plato is an idealist, then Berkeley is the inheritor of Plato.
2. Berkeley is the inheritor of Plato.

Therefore, Plato is an idealist.

The argument schema here is something like this:

1. If X, then Y.
2. Y.

Therefore, X.

This form of inference is logically invalid. In fact, it’s an example of the ‘affirming the consequent’ fallacy. How do we know this? Because we can provide an argumentative counterexample: a clear-cut case in which the premise is true but the conclusion is false that follows the exact same argument schema. Here’s one:

1. If unicorns exist, then the moon orbits the earth.
2. The moon orbits the earth.

Therefore, unicorns exist.

Here, it’s easy to see that the premises are true (the moon does orbit the Earth, and Premise 1 is trivially true), but the conclusion is false. So, we now know that all arguments of the same form are logically invalid. We have provided an argumentative counterexample. Try to provide a couple more examples of your own which show that this argument schema is invalid.


Responding to a counterexample

If someone formulates a counterexample to a view that you hold, or if you’ve formulated a counterexample to someone else’s position and want to anticipate their potential responses, there are a few things to consider. As a general rule, don’t quibble about the details of the case. It wouldn’t be helpful to respond to Gettier’s case of Smith, Jones and the 10 coins by objecting that, for instance, Smith wasn’t sure enough about how many coins Jones had in his pocket for us to consider this “justified”.

Because counterexamples are generated in the imagination and don’t need to be real-world cases, Gettier has an easy response to this: he can just make it more obvious. He could say that Jones had see-through pockets or that Smith had literally just placed the 10 coins into Jones’ empty pocket, and so on. It’s not a very effective tactic: almost always, the critic can formulate a slightly better counterexample which doesn’t suffer from the quibble you’ve identified, like our stopped-watch example.

Similarly don’t try to weasel out of it. It’s no good trying to say ‘but what if Jones had actually had a secret 11th coin in his pocket, so he didn’t actually have 10 coins’, or something like that. Accept the counterexample as given and if you’re not sure about it, improve it. This is partly a consequence of the Principle of Charity, which compels us to take the strongest possible version of an argument we’re criticising, but mainly because thinking of one case (or some cases) in which the counterexample wouldn’t work doesn’t show that the counterexample never works – and all the critic needs is for the counterexample to work once.

There are two viable routes to take in the face of a counterexample:

  1. Show that the apparent counterexample is not a true counterexample (and furthermore, that all similar counterexamples are not true counterexamples either!)
  2. Modify your claim/argument to avoid or accommodate the counterexample.

The first tactic involves showing that the counterexample involves some kind of mistake. This is often difficult to do, because it’s not enough to show that the specific counterexample the critic gave fails, we also need to show that it could not be adapted to work, and that no similar counterexample could be provided which would work. This sometimes happens when analogies are used as counterexamples. Analogies are often imperfect counterexamples. For instance, suppose I argue:

A woman has the right to do as she wishes with her reproductive system. Restrictions on abortion infringe on that right. So, abortion should not be restricted.

Now, consider the analogous argument:

A woman has the right to do as she wishes with her hand. Restrictions on punching people in the face infringe on that right. So, women should be allowed to punch people in the face.

Although the argument follows the same general schema, it fails as a counterexample. Committing to the claim that “A woman has the right to do as she wishes with her reproductive system” doesn’t commit me to the claim that “A woman has the right to do as she wishes with her hand”, which creates the absurdity in the counterexample. Arguably, the critic would need to provide a case in which it’s the right to do as she wishes with her reproductive system that creates an unacceptable consequence.

The more common tactic is to modify the thesis or argument in response. This may seem like a sign of weakness, but is usually actually a great way to make philosophical progress and refine your thinking.

For instance, suppose I claim “Killing is always immoral”. You provide the counterexample of killing hostile soldiers during war, so I weaken my scope to: “Killing civilians is always immoral”. This can often be an iterative process of refinement and new counterexamples. For example, you might now provide the counterexample of killing someone who is trying to kill you. Now I must modify my thesis again, to: “Killing civilians is always immoral except in defence of one’s own life.” Now you provide the counterexample, perhaps, of someone killing an intruder who is about to murder their family, but who doesn’t know that they are in the building. One’s own life is not threatened, but it may still be justified to kill to protect others (at least if we had no other option). We can see how this iterative process would lead to a much more refined thesis.

Another approach is to move from a universal generalisation to a weaker claim—for instance “Killing is usually immoral”. This can be a very problematic move, as claims about what often or usually happens are (usually!) defended only by empirical evidence, not by philosophical argument. These kinds of claims will (usually!) not do the job you need in a logical argument.

A third approach is to exclude the specific counterexample by regarding it as a special case in some way. For instance, I could change my thesis to: “Killing is always immoral, unless in the context of warfare”. It is important to provide clear reasons why the counterexample is exceptional. If you don’t provide good reasons, you might be accused of modifying the thesis purely to evade refutation. This is sometimes called ad hoc modification. While ad hoc modification is not necessarily wrong, it is considered poor philosophical practice. There should be some better rationale behind changing your thesis than merely evading the critcism. A particular form of fallacious thesis-modifying is called the no true Scotsman fallacy. This is when the thesis is subtly changed to deliberately exclude counterexamples, for instance:

“No Scotsman would ever live abroad.”
“But what about Sean Connery – he’s Scottish and he lived in the Bahamas.”
“Sean Connery wasn’t a true Scotsman. No true Scotsman would ever live abroad.”

Here, “true” is just a way to avoid counterexamples by defining true Scotsmen precisely as Scotsmen who don’t live abroad.


A return to Gettier

How would this play out in the case of Gettier? A philosopher who had previously accepted the justified-true-belief account of knowledge would need to provide some new criterion alongside the existing three to deal with Gettier-style cases – this is a way to modify the thesis to account for the new information. For example, it seems that in the Gettier cases we discussed above, the justification which Smith and Aisha had for their belief was based on false information. In Smith’s case, he had false information that Jones would get the job. In Aisha’s case, the false information was that her watch was reliable. Boris was misled by the cardboard cow cutout. So we might add a fourth clause to our thesis about knowledge:

4. The information which justifies A’s belief in X is all true.

This handles both of the Gettier cases, so we might be quite happy with ourselves. However, we should still consider whether we might be able to reformulate a new Gettier-style counterexample that doesn’t depend on any false information. See if you can think of one.

Here’s a possible counterexample that applies even despite (4):

Clara is driving through the countryside. As she drives, she passes what appear to be lots of cows in fields. But she doesn’t look out of the window. When she eventually looks out of the window, she sees a cow and says: “That’s a cow.” Unbeknownst to her, all of the other cows she had passed on her drive were cardboard cutouts left behind by a film crew. The cutouts were so realistic that had she looked at any of them, she would have believed it was a real cow. But the one cow she actually looked at happened to be the one real cow in the area.

Now, Clara saw a cow. That alone justified her belief that “That’s a cow” (justified, and based only on true information). It was indeed a cow (true), and she believes it (belief). But do we agree that Clara knows that’s a cow?

This is a more controversial counterexample than the Smith coins, Aisha’s broken watch or Boris’s cardboard cow. We might say that Clara did know that she’d seen a cow. But we’re aware that had she made that observation at any other point in her journey, she would have come to the same view, based on very similar observations, but would have been wrong. In that way, this seems to resemble Aisha’s broken watch, where she was only right because she happened to look at the watch at the exact right moment. We know that had the cow walked behind one of the cutouts, she would have made a mistaken claim just like Boris’s. So we might agree that Clara didn’t know she’d seen a cow, even though she never saw anything misleading.

To deal with the Clara case, we might try to add another criteria (or change our new criterion) to rule out any cases in which the person was lucky to be correct, but usually would have been wrong. That seems to cover at least the Clara and Aisha cases. Can you think of any Gettier cases where the protagonists aren’t lucky to be right? The progressive refinement of the criteria for knowledge goes on, and the quest for a “G criterion” to rule out Gettier-cases is an open project in philosophy.

Latest edit: 13/04/2021 by CJ Blunt