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Kunstformen der Natur (1904) by Ernst Haeckel
Kunstformen der Natur (1904) by Ernst Haeckel

In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy". Because this statement makes the claim that a certain property (laziness) holds for all students, even a single example of a diligent student will prove it false. Thus, any hard-working student is a counterexample to "all students are lazy". More precisely, a counterexample is a specific instance of the falsity of a universal quantification (a "for all" statement).

In philosophy

In philosophy, counterexamples are usually used to argue that a certain philosophical position is wrong by showing that it does not apply in certain cases. Unlike mathematicians, philosophers cannot prove their claims beyond any doubt, so other philosophers are free to disagree and try to find counterexamples in response. Of course, now the first philosopher can argue that the alleged counterexample does not really apply.

Alternatively, the first philosopher can modify their claim so that the counterexample no longer applies; this is analogous to when a mathematician modifies a conjecture because of a counterexample.

For example, in Plato's Gorgias, Callicles, trying to define what it means to say that some people are "better" than others, claims that those who are stronger are better.

But Socrates replies that, because of their strength of numbers, the class of common rabble is stronger than the propertied class of nobles, even though the masses are prima facie of worse character. Thus Socrates has proposed a counterexample to Callicles' claim, by looking in an area that Callicles perhaps did not expect — groups of people rather than individual persons.

Callicles might challenge Socrates' counterexample, arguing perhaps that the common rabble really are better than the nobles, or that even in their large numbers, they still are not stronger. But if Callicles accepts the counterexample, then he must either withdraw his claim or modify it so that the counterexample no longer applies. For example, he might modify his claim to refer only to individual persons, requiring him to think of the common people as a collection of individuals rather than as a mob.

As it happens, he modifies his claim to say "wiser" instead of "stronger", arguing that no amount of numerical superiority can make people wiser.

See also

Unless indicated otherwise, the text in this article is either based on Wikipedia article "Counterexample" or another language Wikipedia page thereof used under the terms of the GNU Free Documentation License; or on original research by Jahsonic and friends. See Art and Popular Culture's copyright notice.

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