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What is a counterexample?

What is a counterexample? Here are some definitions.

Noun
  1. (logic) An exception to a proposed general rule; a specific instance of the falsity of a universally quantified statement.
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Examples
But it is far too early to declare China a counterexample to the Acemoglu-Robinson argument.
As is typical in analytic philosophy, he makes his point by means of a counterexample.
There are examples of much smaller armies beating larger ones, and one counterexample is enough to disconfirm a law.
To use myself as a counterexample, in psychological tests I always score off the charts in the rational measures.
This story serves as a counterexample, and demonstrates the correct way to design a security system.
And this latest work by the prolific and increasingly televisual Columbia University professor is no counterexample.

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