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In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. [1] For example, the fact that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universal quantification "all students are ...
The Gettier problem, in the field of epistemology, is a landmark philosophical problem concerning the understanding of descriptive knowledge.Attributed to American philosopher Edmund Gettier, Gettier-type counterexamples (called "Gettier-cases") challenge the long-held justified true belief (JTB) account of knowledge.
The assumption that if there is a counterexample, there is a minimal counterexample, is based on a well-ordering of some kind. The usual ordering on the natural numbers is clearly possible, by the most usual formulation of mathematical induction; but the scope of the method can include well-ordered induction of any kind.
However, the failure to find a counterexample after extensive search does not constitute a proof that the conjecture is true—because the conjecture might be false but with a very large minimal counterexample. Nevertheless, mathematicians often regard a conjecture as strongly supported by evidence even though not yet proved.
For instance, an example of a first-countable space which is not second-countable is counterexample #3, the discrete topology on an uncountable set. This particular counterexample shows that second-countability does not follow from first-countability. Several other "Counterexamples in ..." books and papers have followed, with similar motivations.
Frankfurt's examples are significant because they suggest an alternative way to defend the compatibility of moral responsibility and determinism, in particular by rejecting the first premise of the argument. According to this view, responsibility is compatible with determinism because responsibility does not require the freedom to do otherwise.
The opening statement “In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule.” is inaccurate. An exception implies that the the rule under question is valid in some or most cases. The existence of a counterexample has no such implications.
If-by-whiskey: An example; Map-territory relation: Concept that words used to describe an underlying reality are arbitrary abstractions not to be confused with the reality itself; Mental reservation: A doctrine in moral theology; No true Scotsman: Changing a definition to exclude a counter-example; Persuasive definition: Skewed definition of term