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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 color (only horse B remains). The same is true when horse B is removed. However, the statement "the first horse that was excluded is of the same color as the non-excluded
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horses. By the same reasoning, these, too, must also be of the same color. Therefore, the first horse that was excluded is of the same color as the non-excluded horses, who in turn are of the same color as the other excluded horse. Hence, the first horse excluded, the non-excluded horses, and the
541:
horses, who in turn are of the same color as the other excluded horse" is meaningless, because there are no "non-excluded horses" (common elements (horses) in the two sets, since each horse is excluded once). Therefore, the above proof has a logical link broken. The proof forms a
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of horses to which the induction assumption is applied would necessarily share a common element. This is not true at the first step of induction, i.e., when
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The case with just one horse is trivial. If there is only one horse in the "group", then clearly all horses in that group have the same color.
45:. There is no actual contradiction, as these arguments have a crucial flaw that makes them incorrect. This example was originally raised by
57:
girls have eyes of the same color", as an exercise in mathematical induction. It has also been restated as "All cows have the same color".
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horses always are the same color. Likewise, exclude some other horse (not identical to the one first removed) and look only at the other
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545:; it seems to show by valid reasoning something that is manifestly false, but in fact the reasoning is flawed.
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We already saw in the base case that the rule ("all horses have the same color") was valid for
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The "horses" version of the paradox was presented in 1961 in a satirical article by
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Two differently colored horses, providing a counterexample to the general theorem.
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376:. The inductive step proved here implies that since the rule is valid for
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last horse excluded are all of the same color, and we have proven that:
19:"Horse paradox" redirects here. For a Chinese white horse paradox, see
630:, Franklin, Beedle and Associates, 2012, Section "Induction Gone Awry"
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The argument above makes the implicit assumption that the set of
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All horses are the same color paradox, induction step failing for
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Thus, in any group of horses, all horses must be the same color.
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horses always are the same color. Consider a group consisting of
675:. Harvey Mudd College Department of Mathematics. Archived from
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40:
68:, which in particular allowed the author to "prove" that
659:(R. L. Weber, ed.), Crane, Russak & Co., 1973, pp.
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horses has the size at least 3, so that the two proper
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did not exist, and he had an infinite number of limbs.
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First, exclude one horse and look only at the other
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643:(1961), "On the nature of mathematical proofs",
628:Discrete Mathematics and Functional Programming
255:horses; all these are the same color, since
49:in a 1954 book in different terms: "Are any
613:. Princeton University Press. p. 120.
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16:Paradox arising from an incorrect proof
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170:horses must also have the same color.
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346:horses will also have the same color.
611:Induction and Analogy in Mathematics
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565:When a white horse is not a horse
320:horses have the same color, then
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33:that arises from a flawed use of
21:When a white horse is not a horse
673:"All Horses are the Same Color"
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27:All horses are the same color
402:, it must also be valid for
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555:Unexpected hanging paradox
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715:Horses in popular culture
124:). We then prove that if
657:A Random Walk in Science
584:Łukowski, Piotr (2011).
53:numbers equal?" or "Any
37:to prove the statement
710:Mathematical paradoxes
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35:mathematical induction
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521:{\displaystyle n+1=2}
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174:Base case: One horse
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705:Inductive fallacies
484:{\displaystyle n+1}
447:{\displaystyle n=3}
421:{\displaystyle n=2}
395:{\displaystyle n=1}
369:{\displaystyle n=1}
339:{\displaystyle n+1}
225:{\displaystyle n+1}
163:{\displaystyle n+1}
117:{\displaystyle n=1}
70:Alexander the Great
725:Mathematical humor
626:Thomas VanDrunen,
543:falsidical paradox
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96:proof by induction
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43:are the same color
31:falsidical paradox
560:List of paradoxes
313:{\displaystyle n}
288:{\displaystyle n}
268:{\displaystyle n}
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679:on 12 April 2019
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655:. Reprinted in
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182:Inductive step
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681:. Retrieved
677:the original
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186:Assume that
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76:The argument
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47:George Pólya
38:
26:
25:
683:10 November
461:Explanation
454:and so on.
699:Categories
571:References
586:Paradoxes
609:(1954).
549:See also
232:horses.
493:subsets
41:horses
720:Color
661:34-36
66:lemma
29:is a
685:2023
39:All
653:(3)
651:III
300:If
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