# Intermediate Level COUNTER EXAMPLES Counter examples A counter

- Slides: 15

Intermediate Level COUNTER EXAMPLES

Counter examples A counter example is an example which demonstrates that a general claim is false.

Counter examples In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule or law.

Counter examples In philosophy, counterexamples are usually used to argue that a certain philosophical position is wrong by showing that it does not apply in certain cases.

Counter examples Of course, now the first philosopher can argue that the alleged counterexample does not really apply.

Counter examples A counterexample is reasoning of the same form as the original, showing that form of reasoning to be invalid because in the parallel example the premises are known to be true and the conclusion to be false.

Counter examples 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.

Counter examples 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.

Counter examples 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.

Counter examples Young children can learn to supply counterexamples

Counter examples For example, take the following two general claims to which young children should be able to offer counter examples. • A bird is a creature that flies • Mammals have either two legs or four legs

Counter examples All students need to do is to find an example of the thing is questions that does not have the attributed feature. A counterexample to the first would be any type of flightless bird, such as a penguin or an emu. ; or for the second we might cite mammals such as a whale or a dolphin.

Counter examples Note that for a counterexample we want something that is an example of the kind of thing mentioned in the generalisation that does not have the feature in question.

Counter examples Here a couple of examples of the kind of reasoning that could be used to help students learn to construct counterexamples. All houses have walls and a roof. Grass is green.

Counter examples Bibliogrphy 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. Imre Lakatos, Proofs and Refutations Cambridge University Press, 1976, ISBN 0521290384 Gary L. Wise and Eric B. Hall: Counterexamples in Probability and Real Analysis. Oxford University Press, New York 1993. ISBN 0 -19 -507068 -2. Lynn Arthur Steen and J. Arthur Seebach, Jr. : Counterexamples in Topology, 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. Lander, Parkin (1966). "Counterexample to Euler's conjecture on sums of like powers" (PDF). Bulletin of the American Mathematical Society. Americal Mathematical Society. 72: 1079. ISSN 0273 -0979. Retrieved 2 August 2018. Jordan M. Stoyanov: Counterexamples in Probability. Second edition, Wiley, Chichester 1997, ISBN 0 -471 -96538 -3. Michael Copobianco & John Mulluzzo (1978) Examples and Counterexamples in Graph Theory, Elsevier North-Holland ISBN 0 -444 -00255 -3.

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