Acounterexample is any exception to ageneralization. Inlogic a counterexample disproves the generalization, and does sorigorously in the fields ofmathematics andphilosophy.[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, theuniversal quantification "all students are lazy."[2]
In mathematics, counterexamples are often used to prove the boundaries of possible theorems. By using counterexamples to show that certain conjectures are false, mathematical researchers can then avoid going down blind alleys and learn to modify conjectures to produce provable theorems. It is sometimes said that mathematical development consists primarily in finding (and proving) theorems and counterexamples.[3]
Suppose that a mathematician is studyinggeometry andshapes, and she wishes to prove certain theorems about them. Sheconjectures that "Allrectangles aresquares", and she is interested in knowing whether this statement is true or false.
In this case, she can either attempt toprove the truth of the statement usingdeductive reasoning, or she can attempt to find a counterexample of the statement if she suspects it to be false. In the latter case, a counterexample would be a rectangle that is not a square, such as a rectangle with two sides of length 5 and two sides of length 7. However, despite having found rectangles that were not squares, all the rectangles she did find had four sides. She then makes the new conjecture "All rectangles have four sides". This is logically weaker than her original conjecture, since every square has four sides, but not every four-sided shape is a square.
The above example explained — in a simplified way — how a mathematician might weaken her conjecture in the face of counterexamples, but counterexamples can also be used to demonstrate the necessity of certain assumptions andhypothesis. For example, suppose that after a while, the mathematician above settled on the new conjecture "All shapes that are rectangles and have four sides of equal length are squares". This conjecture has two parts to the hypothesis: the shape must be 'a rectangle' and must have 'four sides of equal length'. The mathematician then would like to know if she can remove either assumption, and still maintain the truth of her conjecture. This means that she needs to check the truth of the following two statements:
A counterexample to (1) was already given above, and a counterexample to (2) is a non-squarerhombus. Thus, the mathematician now knows that each assumption by itself is insufficient.
A counterexample to the statement "allprime numbers areodd numbers" is the number 2, as it is a prime number but is not an odd number.[1] Neither of the numbers 7 or 10 is a counterexample, as neither of them are enough to contradict the statement. In this example, 2 is in fact the only possible counterexample to the statement, even though that alone is enough to contradict the statement. In a similar manner, the statement "Allnatural numbers are eitherprime orcomposite" has the number 1 as a counterexample, as 1 is neither prime nor composite.
Euler's sum of powers conjecture was disproved by counterexample. It asserted that at leastnnth powers were necessary to sum to anothernth power. This conjecture was disproved in 1966,[4] with a counterexample involvingn = 5; othern = 5 counterexamples are now known, as well as somen = 4 counterexamples.[5]
Witsenhausen's counterexample shows that it is not always true (forcontrol problems) that a quadraticloss function and a linear equation of evolution of thestate variable imply optimal control laws that are linear.
AllEuclidean plane isometries are mappings that preservearea, but theconverse is false as shown by counterexamplesshear mapping andsqueeze mapping.
Other examples include the disproofs of theSeifert conjecture, thePólya conjecture, the conjecture ofHilbert's fourteenth problem,Tait's conjecture, and theGanea conjecture.
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Inphilosophy, counterexamples are usually used to argue that a certain philosophical position is wrong by showing that it does not apply in certain cases. Alternatively, the first philosopher can modify their claim so that the counterexample no longer applies; this is analogous to when a mathematician modifies a conjecture because of a counterexample.
For example, inPlato'sGorgias,Callicles, trying to define what it means to say that some people are "better" than others, claims that those who are stronger are better.Socrates replies that, because of their strength of numbers, the class of common rabble is stronger than the propertied class of nobles, even though the masses areprima facie of worse character. Thus Socrates has proposed a counterexample to Callicles' claim, by looking in an area that Callicles perhaps did not expect — groups of people rather than individual persons.
Callicles might challenge Socrates' counterexample, arguing perhaps that the common rabble really are better than the nobles, or that even in their large numbers, they still are not stronger. But if Callicles accepts the counterexample, then he must either withdraw his claim, or modify it so that the counterexample no longer applies. For example, he might modify his claim to refer only to individual persons, requiring him to think of the common people as a collection of individuals rather than as a mob. As it happens, he modifies his claim to say "wiser" instead of "stronger", arguing that no amount of numerical superiority can make people wiser.
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