sorites paradox (from Greek soros ‘heap’), any of a number of paradoxes about heaps and their elements, and more broadly about gradations. A single grain of sand cannot be arranged so as to form a heap. Moreover, it seems that given a number of grains insufficient to form a heap, adding just one more grain still does not make a heap. (If a heap cannot be formed with one grain, it cannot be formed with two; if a heap cannot be formed with two, it cannot be formed with three; and so on.) But this seems to lead to the absurdity that however large the number of grains, it is not large enough to form a heap.
A similar paradox can be developed in the opposite direction. A million grains of sand can certainly be arranged so as to form a heap, and it is always possible to remove a grain from a heap in such a way that what is left is also a heap. This seems to lead to the absurdity that a heap can be formed even from just a single grain.
These paradoxes about heaps were known in antiquity (they are associated with Eubulides of Miletus, fourth century B.C.), and have since given their name to a number of similar paradoxes. The loss of a single hair does not make a man bald, and a man with a million hairs is certainly not bald. This seems to lead to the absurd conclusion that even a man with no hairs at all is not bald. Or consider a long painted wall (hundreds of yards or hundreds of miles long). The left-hand region is clearly painted red, but there is a subtle gradation of shades and the right-hand region is clearly yellow. A small double window exposes a small section of the wall at any one time. It is moved progressively rightward, in such a way that at each move after the initial position the left-hand segment of the window exposes just the area that was in the previous position exposed by the right-hand segment. The window is so small relative to the wall that in no position can you tell any difference in color between the exposed areas. When the window is at the extreme left, both exposed areas are certainly red. But as the window moves to the right, the area in the right segment looks just the same color as the area in the left, which you have already pronounced to be red. So it seems that one must call it red too. But then one is led to the absurdity of calling a clearly yellow area red.
As some of these cases suggest, there is a connection with dynamic processes. A tadpole turns gradually into a frog. Yet if you analyze a motion picture of the process, it seems that there are no two adjacent frames of which you can say the earlier shows a tadpole, the later a frog. So it seems that you could argue: if something is a tadpole at a given moment, it must also be a tadpole (and not a frog) a millionth of a second later, and this seems to lead to the absurd conclusion that a tadpole can never turn into a frog. Most responses to this paradox attempt to deny the ‘major premise,’ the one corresponding to the claim that if you cannot make a heap with n grains of sand then you cannot make a heap with n ! 1. The difficulty is that the negation of this premise is equivalent, in classical logic, to the proposition that there is a sharp cutoff: that, e.g., there is some number n of grains that are not enough to make a heap, where n ! 1 are enough to make a heap. The claim of a sharp cutoff may not be so very implausible for heaps (perhaps for things like grains of sand, four is the smallest number which can be formed into a heap) but is very implausible for colors and tadpoles. There are two main kinds of response to sorites paradoxes. One is to accept that there is in every such case a sharp cutoff, though typically we do not, and perhaps cannot, know where it is. Another kind of response is to evolve a non-classical logic within which one can refuse to accept the major premise without being committed to a sharp cutoff. At present, no such non-classical logic is entirely free of difficulties. So sorites paradoxes are still taken very seriously by contemporary philosophers. See also MANY-VALUED LOGIC, VAGUENES. R.M.S.