- All horses are the same color
The horse paradox is a
falsidical paradoxthat arises from flawed demonstrations, which purport to use mathematical induction, of the statement "All horsesare the same color". The paradox does not truly exist, as these arguments have a crucial flaw that makes them incorrect. This example was used by George Pólyaas an example of the subtle errors that can occur in attempts to prove statements by induction.
The flawed argument claims to be based on
mathematical induction, and proceeds as follows.
Suppose that we have a set of five horses. We wish to prove that they are all the same colour. Suppose that we had a proof that all sets of four horses were the same colour. If that were true, we could prove that all five horses are the same colour by removing a horse to leave a group of four horses. Do this in two ways, and we have two different groups of four horses. By our supposed existing proof, since these are groups of four, all horses in them must be the same color. For example, the first, second, third and fourth horses constitute a group of four, and thus must all be the same colour; and the second, third, fourth and fifth horses also constitute a group of four and thus must also all be the same colour. For this to occur, all five horses in the group of five must be the same colour.
But how are we to get a proof that all sets of four horses are the same colour? We apply the same logic again. By the same process, a group of four horses could be broken down into groups of three, and then a group of three horses could be broken down into groups of two, and so on. Eventually we will reach a group size of one, and it is obvious that all horses in a group of one horse must be the same colour.
By the same logic we can also increase the group size. A group of five horses can be increased to a group of six, and so on upwards, so that all finite sized groups of horses must be the same colour.
The argument above makes the implicit assumption that the two
subsets of horses to which the induction assumption is applied have a common element. This is not true when "n" = 1, that is, when the original set only contains 2 horses.
Indeed, let the two horses be horse A and horse B. When horse A is removed, it is true that the remaining horses in the set are the same colour (only horse B remains). If horse B is removed instead, this leaves a different set containing only horse A, which may or may not be the same colour as horse B.
The problem in the argument is the assumption that because each of these two sets contains only one colour of horses, the original set also contained only one colour of horses. Because there are no common elements (horses) in the two sets, it is unknown whether the two horses share the same colour. The proof forms a
falsidical paradox; it seems to show something manifestly false by valid reasoning, but in fact the reasoning is flawed. The horse paradox exposes the pitfalls arising from failure to consider special cases for which a general statement may be false.
*"Enumerative Combinatorics" by George E. Martin, ISBN 0-387-95225-X
Wikimedia Foundation. 2010.
Look at other dictionaries:
The Country Mouse and the City Mouse Adventures — Also known as The Mouse Adventures (UK) Genre Animation Written by Patrick Granleese Caroline R. Maria Bruce Robb Voices of Julie Burroughs Terrence Scammell … Wikipedia
The Sims 3 — Developer(s) The Sims Studio Publisher(s) Electronic Arts … Wikipedia
The Chronicles of Thomas Covenant, the Unbeliever — is a trilogy of fantasy novels by Stephen R. Donaldson. It was followed by The Second Chronicles of Thomas Covenant , also a trilogy, and The Last Chronicles of Thomas Covenant , a planned tetralogy.The main character is Thomas Covenant, a… … Wikipedia
The Little Mermaid (franchise) — The Little Mermaid, The Little Mermaid II: Return to the Sea, and The Little Mermaid: Ariel s Beginning Little Mermaid Trilogy Gift Set Starring Jodi Benson Sam … Wikipedia
The Tale of the Heike — ( Heike monogatari , 平家物語) is an epic account of the struggle between the Taira and Minamoto clans for control of Japan at the end of the 12th century in the Genpei War (1180 1185). Heike (平家) refers to the Taira (平) clan; hei being an alternate… … Wikipedia
Color breed — A color breed is a term that refers to horses that are registered based primarily on their coat color, regardless of the horse s actual breed or breed type. Some color breeds only register horses with color that also meet specific pedigree… … Wikipedia
The Color Purple — For other uses, see The Color Purple (disambiguation). The Color Purple … Wikipedia
The Holocaust — Holocaust and Shoah redirect here. For other uses, see Holocaust (disambiguation) and Shoah (disambiguation). Selection on … Wikipedia
The Lone Ranger — Infobox actor imagesize = 175px caption = Clayton Moore as The Lone Ranger The Lone Ranger is an American, long running, old time radio and early television show created by George W. Trendle (with considerable input from station staff members),… … Wikipedia
The Wild Wild West — Infobox Television show name = The Wild Wild West caption = The Wild Wild West 1990s VHS release. Pictured: Robert Conrad (top) and Ross Martin genre = Western creator = Michael Garrison starring = Robert Conrad Ross Martin country = USA language … Wikipedia