Suppose you have a heap of sand. You remove one grain. Is there still a heap? You remove another, until you get down to a heap with three grains, a heap with two grains, a heap with one grain, and finally a heap with no grains at all. But that’s ridiculous. There must be something wrong? Does removing one grain turn a heap into not-a-heap? But that's ridiculous too. How can one grain make so much difference?
Is there a solution to this problem?
The solution is to realise that the problem as posed is based on a false assumption that there is always a clear dividing line between two opposing classifications of degree. Take long and short, heavy and light, wide and narrow, too salty and not too salty, etc etc. It is impossible to define an unambiguous boundary between pairs of terms like that. The terms 'a long piece of string' and 'a short piece of string' are overlapping with blurred boundaries, so a piece of string can be both long and short depending on the context.
One solution would be to say that even 1 grain is a heap. That would be defining "heap" more precisely than its informal, intuitive meaning. What does "heap" precisely mean anyway? The whole "paradox" is built on the lack of precision in the informal understanding of "heap", so it doesn't seem to be such a formidable "paradox".
The answer is in the definition of a heap.
I offer the definition that a heap must have at least one layer stacked upon a base layer.
For grains of sand you need at least 3 grains in a base layer to support 1 grain of sand stacked upon it (thus forming a tetrahedral stack). Thus, properly arranged, a heap could be as little as 4 grains of sand; any fewer is no longer a heap.
So I would say that two possible answers to the paradox are
The existing answers focus on solution number 1, but I don't believe anybody has discussed option 2 which I find is a more intuitive resolution.
Consider the chance of a given person at a given point in time considering a given amount of grains of sand as a 'heap' as some probability function depending on the number of grains, starting at 0 (or very close to it) and growing to nearly 1. Removing grains one by one does affect the probability something will be classified as a heap, just not by much.
Of course, other factors will be at play - have they been primed, whether they are a construction worker or a child (as per Mark Andrew's answer), whether they are in a heap-finding mood, or any number of other things.
The solution, of course, is to learn to cook:
Therefore, how many grains must be removed to "turn a heap into not-a-heap" depends on the spoon.
There is no paradox - there is a fallacy
Let us compare with integers.
If I take an integer (Let's say 100) and subtract 1, I get 99, a positive integer; I subtract one again and get 98, another positive integer. From that Sorites would presumably argue that whenever I subtract 1 from any integer I still have a positive integer. However that is patently not true. When I reach 1, the subtraction will yield zero which does not qualify.
The above argument is no different in structure than that of the 'paradox'. All we need to do is define boundary conditions.
Example
Definition: A heap of sand is a quantity of sand such that a path can be traced from any single grain to all others without intervening gaps and that has at least one grain that is elevated from the underlying supporting surface by resting on one or more of the other grains.
Boundary condition: When I remove a grain of sand from a quantity of grains that constitute a heap, I will be left with another heap unless there are no elevated grains.
Note that the definitions are arbitrary and can be adjusted as required. All that is required in rigorous discourse is that terms should be agreed ahead of time. It is not sufficient to rely on folk terminology.
The solution is to acknowledge that the common notion of a "heap" does not have a rigorous definition. Like all terms in everyday language it is defined by context.
What’s the solution to the Sorites paradox?
Most responders have offered the same answer: the solution depends on the definition of "heap".
I agree, but I add that the definition itself depends on what you intend to do. To a road construction crew, a heap of sand might be several tons. To a child making a sand castle, a heap might be only three plastic buckets full. To a molecular scientist, three grains might be enough.
Edited by Ashlin Harris:
A certain number of grains is not necessarily a heap. For instance, they could be arranged in a row.
A heap it is something without any artful structure; when a heap gets any order it is already not a heap. An ordered structure can have elements removed one by one and still be the same(*) structure. If you take an element from a heap, it might become something different(**).
*but incomplete, at that time a heap is always complete to a heap.
**by the structure, or you can make a new heap.
Old poor version:
i named it "the method of the ordering elements"
all sets* contained more then 2 grains are the heaps. But possible to remove all grains from the heap and create the string on numerated grains that not a heap. 2 grains is the heap and is the string at the same time.
So, all that you need something that not a heap, but consists of same grains.
easy.
*non-ordered.
**one grain is not a heap and not a string, it is the single element of a set or string.
