Counterexample
Encyclopedia
In logic
, and especially in its applications to mathematics
and philosophy
, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy".
Because this statement makes the claim that a certain property (laziness) holds for all students, even a single example of a diligent student will prove it false.
Thus, any hard-working student is a counterexample to "all students are lazy".
More precisely, a counterexample is a specific instance of the falsity of a universal quantification
(a "for all" statement).
In mathematics, this term is (by a slight abuse) also sometimes used for examples illustrating the necessity of the full hypothesis of a theorem, by considering a case where a part of the hypothesis is not verified, and where one can show that the conclusion does not hold. A counterexample may be local or global in an argument.
and shape
s, and she wishes to prove certain theorems about them. She conjecture
s that "All rectangles are squares
". She can either attempt to prove
the truth of this statement using deductive reasoning
, or if she suspects that her conjecture is false, she might attempt to find a counterexample. In this case, a counterexample would be a rectangle that is not a square, like 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 weaker than her original conjecture, since every square has four sides, even though not every four-sided shape is a square.
The previous paragraph explained how a mathematician might weaken her conjecture in the face of counterexamples, but counterexamples can also be used to show that the assumptions and hypothesis
are needed. Suppose that after a while the mathematician in question 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 'have four sides of equal length' and the mathematician would like to know if she can remove either assumption and still maintain the truth of her conjecture. So she needs to check the truth of the statements: (1) "All shapes that are rectangles are squares" and (2) "All shapes that have four sides of equal length are squares". A counterexample to (1) was already given, and a counterexample to (2) is a parallelogram
or a diamond. Thus the mathematician sees that both assumptions were necessary.
s are odd numbers" is the number 2, as it is a prime number but is not an odd number. Neither of the numbers 7 or 10 is a counterexample, as neither contradicts the statement. In this example, 2 is the only possible counterexample to the statement, but only a single example is needed to contradict "All prime numbers are odd numbers". Similarly the statement "All natural number
s are either prime
or composite
" has the number 1 as a counterexample as 1 is neither prime nor composite.
, counterexamples are usually used to argue that a certain philosophical position is wrong by showing that it does not apply in certain cases. Unlike mathematician
s, philosophers cannot prove their claims beyond any doubt, so other philosophers are free to disagree and try to find counterexamples in response. Of course, now the first philosopher can argue that the alleged counterexample does not really apply.
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, in Plato
's Gorgias
, Callicles
, trying to define what it means to say that some people are "better" than others, claims that those who are stronger are better.
But 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 are prima 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.
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
, and especially in its applications to mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy".
Because this statement makes the claim that a certain property (laziness) holds for all students, even a single example of a diligent student will prove it false.
Thus, any hard-working student is a counterexample to "all students are lazy".
More precisely, a counterexample is a specific instance of the falsity of a universal quantification
Universal quantification
In predicate logic, universal quantification formalizes the notion that something is true for everything, or every relevant thing....
(a "for all" statement).
In mathematics, this term is (by a slight abuse) also sometimes used for examples illustrating the necessity of the full hypothesis of a theorem, by considering a case where a part of the hypothesis is not verified, and where one can show that the conclusion does not hold. A counterexample may be local or global in an argument.
In mathematics
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 avoid going down blind alleys and learn how to modify conjectures to produce provable theorems.Rectangle example
Suppose that a mathematician is studying geometryGeometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
and shape
Shape
The shape of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as colour, content, and material...
s, and she wishes to prove certain theorems about them. She conjecture
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
s that "All rectangles are squares
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
". She can either attempt to prove
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
the truth of this statement using deductive reasoning
Deductive reasoning
Deductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis...
, or if she suspects that her conjecture is false, she might attempt to find a counterexample. In this case, a counterexample would be a rectangle that is not a square, like 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 weaker than her original conjecture, since every square has four sides, even though not every four-sided shape is a square.
The previous paragraph explained how a mathematician might weaken her conjecture in the face of counterexamples, but counterexamples can also be used to show that the assumptions and hypothesis
Hypothesis
A hypothesis is a proposed explanation for a phenomenon. The term derives from the Greek, ὑποτιθέναι – hypotithenai meaning "to put under" or "to suppose". For a hypothesis to be put forward as a scientific hypothesis, the scientific method requires that one can test it...
are needed. Suppose that after a while the mathematician in question 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 'have four sides of equal length' and the mathematician would like to know if she can remove either assumption and still maintain the truth of her conjecture. So she needs to check the truth of the statements: (1) "All shapes that are rectangles are squares" and (2) "All shapes that have four sides of equal length are squares". A counterexample to (1) was already given, and a counterexample to (2) is a parallelogram
Parallelogram
In Euclidean geometry, a parallelogram is a convex quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure...
or a diamond. Thus the mathematician sees that both assumptions were necessary.
Other mathematical examples
A counterexample to the statement "all prime numberPrime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s are odd numbers" is the number 2, as it is a prime number but is not an odd number. Neither of the numbers 7 or 10 is a counterexample, as neither contradicts the statement. In this example, 2 is the only possible counterexample to the statement, but only a single example is needed to contradict "All prime numbers are odd numbers". Similarly the statement "All natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s are either prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
or composite
Composite number
A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
" has the number 1 as a counterexample as 1 is neither prime nor composite.
In philosophy
In philosophyPhilosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
, counterexamples are usually used to argue that a certain philosophical position is wrong by showing that it does not apply in certain cases. Unlike mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
s, philosophers cannot prove their claims beyond any doubt, so other philosophers are free to disagree and try to find counterexamples in response. Of course, now the first philosopher can argue that the alleged counterexample does not really apply.
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, in Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
's Gorgias
Gorgias (dialogue)
Gorgias is a Socratic dialogue written by Plato around 380 BC. In this dialogue, Socrates seeks the true definition of rhetoric, attempting to pinpoint the essence of rhetoric and unveil the flaws of the sophistic oratory popular in Athens at this time...
, Callicles
Callicles
Callicles is a character in Plato’s dialogue Gorgias. He is an Athenian citizen, who is a student of the sophist Gorgias. In the dialogue, he argues the position of an oligarchic, proto-"Nietzschean" amoralism: it is natural and just for the strong to dominate the weak and that it is unfair for...
, trying to define what it means to say that some people are "better" than others, claims that those who are stronger are better.
But Socrates
Socrates
Socrates was a classical Greek Athenian philosopher. Credited as one of the founders of Western philosophy, he is an enigmatic figure known chiefly through the accounts of later classical writers, especially the writings of his students Plato and Xenophon, and the plays of his contemporary ...
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 are prima facie
Prima facie
Prima facie is a Latin expression meaning on its first encounter, first blush, or at first sight. The literal translation would be "at first face", from the feminine form of primus and facies , both in the ablative case. It is used in modern legal English to signify that on first examination, a...
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.
Further reading
Using counterexamples in this way proved to be so useful that there are several books collecting them:- Lynn Arthur Steen and J. Arthur Seebach, Jr.J. Arthur Seebach, Jr.J. Arthur Seebach, Jr was an American mathematician.He received his Ph.D. in 1968 from Northwestern University. He joined the faculty of Mathematics at St. Olaf College in 1965. He was Associate Editor of the American Mathematical Monthly from 1971 to 1986 and Editor of Mathematics Magazine from...
: Counterexamples in TopologyCounterexamples in TopologyCounterexamples in Topology is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr.In the process of working on problems like the metrization problem, topologists have defined a wide variety of topological properties...
, Springer, New York 1978, ISBN 0-486-68735-X. - Joseph P. Romano and Andrew F. Siegel: Counterexamples in Probability and Statistics, Chapman & Hall, New York, London 1986, ISBN 0-412-98901-8.
- Gary L. Wise and Eric B. Hall: Counterexamples in Probability and Real Analysis. Oxford University Press, New York 1993. ISBN 0-19-507068-2.
- Bernard R. Gelbaum, John M. H. Olmsted: Counterexamples in Analysis. Corrected reprint of the second (1965) edition, Dover Publications, Mineola, NY 2003, ISBN 0-486-42875-3.
- Jordan M. Stoyanov: Counterexamples in Probability. Second edition, Wiley, Chichester 1997, ISBN 0-471-96538-3.