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comment Is a counterexample considered a rigorous proof that a property is not true?
When I read this I thought Qiaochu was making a mistake, because he says that the "negation of a universal statement is an existential statement". Better: the negation of a universal proposition can be expressed as an existential proposition. So, the negation of "All sheep are white" is not "Some sheep are white", it's "not all sheep are white", which can be rewritten as "some sheep are not white".