Counterexample

A counterexample in mathematics or in philosophy , especially in logic, is an empirical or constructed state of affairs that refutes a certain hypothesis . Since Karl Popper's demand for falsifiability , only those statements are considered scientific for which counterexamples are in principle possible.

In mathematics one proves propositions of the form "If A, then B". The proof excludes the existence of counterexamples in principle, so that the concept of falsifiability does not make sense here. For the non-existence of such an implication "If A, then B" it is sufficient to give an example that A satisfies but not B. Such an example is called a counterexample. Furthermore, when introducing mathematical properties, one speaks of counterexamples if one gives an example of something that does not have this property. Typical applications of the term counterexample are therefore:

• It applies "If A, then B". The converse "If B, then A" does not hold, as the counterexample x shows.
• Introduction of a property E. Examples of E are x and y. z is a counterexample (i.e., an example that does not have E).

Examples

• Gettier's counterexamples to the assertion that knowledge is "justified true belief ".
• Statement: All prime numbers are odd . This statement is wrong, as counterexample 2 shows. (The existence of a single counterexample is sufficient for falsehood.)
• A group is called Abelian if the connection is commutative. Examples are or . The counterexample S 3 shows that not all groups are Abelian.${\ displaystyle (\ mathbb {Z}, +)}$ ${\ displaystyle \ mathbb {Z} / n \ mathbb {Z}}$ 