In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy.
It is sometimes said that mathematical development consists primarily in finding (and proving) theorems and counterexamples.
[3] Suppose that a mathematician is studying geometry and shapes, and she wishes to prove certain theorems about them.
For example, suppose that after a while, the mathematician above settled on the new conjecture "All shapes that are rectangles and have four sides of equal length are squares".
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.
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.
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.