The Paradox of the heap reference article from the English Wikipedia on 24-Jul-2004
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Paradox of the heap

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The paradox of the heap (or the sorites paradox, sorites being Greek for "heap") is a paradox that arises when people apply "common sense" to certain vague concepts.

More specifically, the paradox is that, while common sense suggest that heaps of sand have the following properties, these properties are actually mutually inconsistent:

  1. Two or three grains of sand do not make a heap.
  2. A million grains do make a heap.
  3. If n grains of sand do not make a heap, neither do (n+1) grains.
  4. If n grains of sand make a heap, so do (n-1) grains.

Applying mathematical induction shows that the first property combined with the third imply that a million grains of sand do not make a heap, in contradiction with the second property. Similarly, a combination of the second and fourth properties shows that two or three grains do make a heap, in contradiction with the first property.

What gives rise to this contradiction? To find out, let's reexamine the above properties. The second two fairly clearly express the idea that there is no clear line between "is a heap" and "isn't a heap". Note, however, that the four taken together also imply that any pile of sand can non-problematically be classified as "heap" or "non-heap". (This again follows from mathematical induction.) What the paradox shows is that these two ideas are contradictory. That is, one cannot simultaneously claim, when classifying X's:

  1. that there's no clear line separating the X's that are Y from the X's that are not Y
  2. that every last X is either a Y or not a Y

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