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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
There are examples of much smaller armies beating larger ones, and one counterexample is enough to disconfirm a law.
The addition of a new rule for a counterexample can jeopardize the whole system.
To use myself as a counterexample, in psychological tests I always score off the charts in the rational measures.
In all the articles maligning students of the past two decades for apathy, the media rarely deign to mention this counterexample.
He did however hold that theories could be falsified, and that falsifiability, or the liability of a theory to counterexample, was a virtue.
This story serves as a counterexample, and demonstrates the correct way to design a security system.

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