Q&A: Refutation: Counterexample vs. Assuming the Claim and Deriving a Contradiction
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Refutation: Counterexample vs. Assuming the Claim and Deriving a Contradiction
Question
Hello!
Refuting a claim or theorem is possible in two ways: (a) bringing a counterexample, and (b) assuming the claim and arriving at a contradiction.
Is there any difference between them, or any advantage of the second over the first? It feels like the second one “explains more” about the flaw in the claim, but is there really a logical definition of an “explained proof” that something is false, as opposed to a “counterexample”? On the face of it, logically both are valid, so is one “more valid”?
I hope I made myself clear.
Answer
I don’t know how to answer that, and I don’t think there is one general, sweeping answer.