Motexempel

Motexempel, commonly translated as counterexample, is a specific instance that disproves a general claim. In logic and mathematics, a motexempel is a concrete example that shows a universal proposition to be false or an argument to fail. If a theorem asserts that every X has property P, finding one X that does not have P serves as a motexempel and refutes the claim. Likewise, to refute an implication, one looks for a case where the antecedent is true but the consequent is false.

In practice, a motexempel must satisfy all stated conditions of the general claim. A single valid counterexample suffices to disprove a universal statement, whereas existential claims require different reasoning. Counterexamples are also used in philosophy and rhetoric to test arguments and generalizations, and in teaching to illustrate the limits of a rule.

Examples: The statement "All primes are odd" is false; the number 2 is a counterexample. The statement "The sum of two even numbers is odd" is false; for example, 2 + 4 = 6 is even, not odd. Another classic example is the statement "Every natural number is divisible by 2" which is false for numbers like 1 or 3.

Etymology and usage: The term is common in Swedish and Norwegian contexts and is closely related to the English term counterexample. In mathematical writing, clearly identifying a motexempel is essential to demonstrating that a proposed generalization does not hold.