95:, 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.
103:. 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:
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144:" 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 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 "All
261:, 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.
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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
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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
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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 and
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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.
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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.
43:. 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
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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
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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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110:"All shapes that have four sides of equal length are squares".
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was disproved by counterexample. It asserted that at least
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a counterexample disproves the generalization, and does so
571:Counterexamples in Probability and Real Analysis
616:Michael Copobianco & John Mulluzzo (1978)
584:Bernard R. Gelbaum, John M. H. Olmsted (2003)
555:Counterexamples in Probability and Statistics
552:Joseph P. Romano and Andrew F. Siegel (1986)
420:Bulletin of the American Mathematical Society
107:"All shapes that are rectangles are squares."
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618:Examples and Counterexamples in Graph Theory
229:Other examples include the disproofs of the
203:imply optimal control laws that are linear.
426:(6). Americal Mathematical Society: 1079.
199:and a linear equation of evolution of the
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64:Suppose that a mathematician is studying
169:powers were necessary to sum to another
87:In this case, she can either attempt to
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136:A counterexample to the statement "all
191:shows that it is not always true (for
569:Gary L. Wise and Eric B. Hall (1993)
558:Chapman & Hall, New York, London
515:Proof in Mathematics: An Introduction
218:is false as shown by counterexamples
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16:Exception to a proposed general rule
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500:(1976) Cambridge University Press
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160:Euler's sum of powers conjecture
433:10.1090/s0002-9904-1966-11654-3
602:Counterexamples in Probability
311:Exception that proves the rule
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462:Elkies, Noam (October 1988).
189:Witsenhausen's counterexample
239:Hilbert's fourteenth problem
671:Interpretation (philosophy)
586:Counterexamples in Analysis
538:Counterexamples in Topology
339:"Mathwords: Counterexample"
210:are mappings that preserve
128:Counterexamples in topology
122:Other mathematical examples
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599:Jordan M. Stoyanov (1997)
471:Mathematics of Computation
208:Euclidean plane isometries
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620:, Elsevier North-Holland
388:"What Is Counterexample?"
47:"all students are lazy."
661:Mathematical terminology
513:and Albert Daoud (2011)
45:universal quantification
411:Lander, Parkin (1966).
643:Quotations related to
533:J. Arthur Seebach, Jr.
497:Proofs and Refutations
464:"On A4 + B4 + C4 = D4"
316:Minimal counterexample
177: = 5; other
132:Minimal counterexample
23:is any exception to a
541:, Springer, New York
368:mathworld.wolfram.com
392:www.cut-the-knot.org
237:, the conjecture of
362:Weisstein, Eric W.
195:) that a quadratic
93:deductive reasoning
231:Seifert conjecture
529:Lynn Arthur Steen
523:978-0-646-54509-7
343:www.mathwords.com
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397:2019-11-28
373:2019-11-28
348:2019-11-28
322:References
259:philosophy
245:, and the
214:, but the
126:See also:
101:hypothesis
78:rectangles
76:that "All
41:philosophy
33:rigorously
442:0273-0979
276:Callicles
154:composite
525:, ch. 6.
447:2 August
300:See also
283:Socrates
216:converse
66:geometry
535:(1978)
271:Gorgias
116:rhombus
82:squares
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233:, the
70:shapes
27:. In
666:Logic
467:(PDF)
416:(PDF)
266:Plato
150:prime
89:prove
29:logic
622:ISBN
607:ISBN
590:ISBN
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531:and
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449:2018
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281:But
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