We use the principle of mathematical induction. As the basis case, we note that in a set containing a single horse, all horses are clearly the same color. Now assume the truth of the statement for all sets of at most n horses. Let there be n+1 horses in a set. Remove the first horse to get a set of n horses. By the induction assumption, all horses in this set are the same color. It remains to show that this color is the same as that of the horse we removed. But this is easy: put back the first horse, take out a different horse and apply the induction principle to this set of n horses. Thus all horses in any set of n+1 horses are the same color. By the principle of induction, we have established that all horses are the same color.
The hole in the above "proof" is easy to spot with a little thought: it makes the implicit assumption that the two sets of horses to which we apply the induction assumption have a common element, but this fails when n=1.
Thus this "paradox" is merely the result of flawed reasoning; it exposes the pitfalls arising from failure to consider special cases for which a general statement may be false.
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2 A Related Paradox, Concerning Cats 3 Yet another horse paradox |
Somewhat related is a word-play by Raymond Smullyan, a "proof" that all horses have thirteen legs. First take all your horses and paint them red. Now look at the horses. If all the horses have thirteen legs, then we can stop. But what if one or more of the horses don't have thirteen legs? Well, that would be a horse of a different colour! However, we've painted all of them the same color, so there can't be any such horse: all horses have thirteen legs.
Here Smullyan is making a pun on the phrase "that would be a horse of a different colour", which means roughly "that would be a different situation".Another horse paradox