What Is a Counterexample? One Case That Ends an Argument

Confirming a universal claim requires checking every case, which is usually impossible, while refuting one requires a single instance. That is why no number of white swans establishes that all swans are white and one black swan settles it, an example used so often that it has become the standard illustration and which happens to be historically accurate, since Europeans held the belief until encountering Australian swans. The asymmetry underpins falsificationism in the philosophy of science, on which a theory earns its status by making claims that could be refuted and surviving attempts to refute them, and it explains the emphasis on prediction rather than accumulation of confirming instances. It also explains a common frustration in argument, since someone defending a general claim can always retreat by narrowing it, and a claim narrowed enough to be immune to counterexample has usually given up most of what made it interesting. The productive response to a counterexample is generally to revise the claim precisely rather than to abandon or defend it.

The device operates in several fields with recognisably the same logic:

• •Mathematics, where a single counterexample disproves a conjecture outright, and where some conjectures have survived computer checks of billions of cases while remaining unproven because checking is not proof
• •Philosophy of knowledge, where cases constructed by Edmund Gettier in 1963 showed that justified true belief can fail to be knowledge, which redirected an entire field
• •Ethics, where constructed cases test whether a proposed principle gives acceptable verdicts, which is most of what trolley problems are for
• •Linguistics, where a single sentence a native speaker accepts can refute a proposed grammatical rule
• •Law, where a hypothetical applying a proposed rule to an unintended situation is the standard way of testing legislation and is exactly what appellate judges do in oral argument
• •Everyday argument, where a counterexample is the quickest way to show that a generalisation is doing less work than its user believes

Finding one is a skill with recognisable methods. Searching the extremes is the commonest, since a claim that works for typical cases frequently fails at the boundaries, so the smallest, largest, empty, zero and infinite cases are checked first. Varying one feature at a time isolates which part of the claim is doing the work. Constructing an artificial case rather than searching for a natural one is legitimate and is standard in philosophy, though it invites the objection that the case is too contrived to matter. Looking in adjacent domains helps, since a claim about one kind of thing frequently fails when applied to a neighbouring kind. In mathematics, computer search has become a substantial method, and several long-standing conjectures have been refuted by counterexamples with enormous numbers of digits that no person would have found, which raises its own questions about what a proof or refutation requires when no human can survey it.

Responses to a counterexample follow a limited set of patterns and only some of them are legitimate. Accepting it and narrowing the claim is the honest move, and the resulting narrower claim may still be valuable. Denying that the case falls under the claim is legitimate if the claim was genuinely ambiguous and is an evasion otherwise. Adding a qualification specifically to exclude the offending case, with no independent motivation, is the manoeuvre known as saving the hypothesis, and it is the move that falsificationists identify as the mark of an unfalsifiable theory, since a claim that can absorb any counterexample by amendment no longer forbids anything. Redefining a term so the case no longer counts is the no true Scotsman move, named for exactly that pattern. Dismissing the case as unrealistic is sometimes fair, since a claim about how the world works need not survive a case the world cannot produce, and it is frequently used to avoid an argument that the claim cannot answer.

A classic lesson in why patterns are not proof comes from the expression n squared plus n plus 41. Try n equals 0, 1, 2 and so on, and it produces 41, 43, 47, 53 and 61, all prime, and it keeps producing primes for every whole number up to 39. Anyone checking the first dozen cases would feel sure of the rule. At n equals 40, however, the result is 1,681, which is 41 multiplied by 41, and the claim that the formula always gives primes collapses. Trying n equals 41 would have exposed it at once, since every term then has 41 as a factor.

A more dramatic case is a conjecture made by Leonhard Euler in the eighteenth century, extending Fermat's famous claim: roughly, that you need at least n powers of degree n to add up to another power of degree n. It stood for about two hundred years until 1966, when two mathematicians running a computer search found that the fifth powers of 27, 84, 110 and 133 add up exactly to the fifth power of 144, so four fifth powers were enough where Euler's rule said five would be needed. Their refutation was one of the shortest papers ever published in a mathematics journal.

Euler noticed in the eighteenth century that for a solid with flat faces, the number of corners minus the number of edges plus the number of faces always comes to 2. A cube has 8 corners, 12 edges and 6 faces, and 8 minus 12 plus 6 is 2; a triangular pyramid gives 4 minus 6 plus 4, also 2. The rule held for every solid people tried, and several proofs were offered. Then counterexamples began to appear. A picture frame, a block with a square hole through the middle built from flat four-sided faces, has 16 corners, 32 edges and 16 faces, and the sum comes to 0.

The philosopher Imre Lakatos turned this history into a book, Proofs and Refutations, published in 1976 after his death and written as a classroom dialogue. Some of its characters respond to the picture frame by declaring that it is not a real polyhedron at all, a move Lakatos called monster-barring. Others study where the proof broke, find a hidden assumption, that the solid has no holes, and make it an explicit condition. The second response proved far more fruitful. It revealed that the sum depends on how many holes a solid has, 2 for none and 0 for one, an idea at the root of topology.

Some claims hold for every case anyone checks and still turn out to be false. In 1919 the mathematician George Polya made a conjecture about how numbers break down into prime factors: that among the numbers up to any point, those with an odd number of prime factors are never outnumbered by those with an even number. Checked by hand and early computer for a long way, it held up.

In 1958 the British mathematician C. Brian Haselgrove proved that a counterexample must exist, without finding it, and in 1980 the smallest one was found: 906,150,257. Examples like this explain why mathematicians insist on proof rather than evidence from many cases. A pattern that holds for the first hundred million numbers can still fail at the next.

A universal claim needs every case and a refutation needs one, which is why a single black swan settles what any number of white ones cannot. Gettier's constructed cases redirected the theory of knowledge with two pages. Searching extremes, varying one feature and building artificial cases are the standard methods. Adding a qualification purely to exclude the case that refuted you is the move that makes a theory unfalsifiable.