Counterexample

In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i.e., a specific instance of the falsity of a universal quantification (a "for all" statement).

Related Topics:
Logic - Mathematics - Philosophy - Universal quantification

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For example, consider the proposition "all students are lazy".

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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.

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Thus, any hard-working student is a counterexample to "all students are lazy".

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In mathematics, this term is (by a slight abuse) also frequently used for examples illustrating the necessity of the full hypothesis of a theorem, by considering a case where a part of the hypothesis is not verified, and where one can show that the conclusion does not hold.

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