What is the sorites paradox of the heap?

The sorites paradox, also called the paradox of the heap, arises from vague words, meaning words with unclear boundaries. If you remove one grain at a time from a heap of sand, it seems that no single removal can turn a heap into a non-heap. Yet repeating the step enough times leaves one grain, or none, which is plainly not a heap. Apparently sound reasoning from apparently sound premises ends in a falsehood. [1] [2]

The word sorites comes from the Greek word for heap, and the puzzle is attributed to Eubulides of Miletus. One version starts with the premise that 1,000,000 grains make a heap, and adds the premise that a heap minus one grain is still a heap. Applying the second premise again and again leads to the conclusion that one grain is a heap. As one source notes, the argument is itself a heap of steps of modus ponens, a basic valid form of inference. [1]

The argument also runs in reverse. Starting from the premise that one grain does not make a heap, and that adding one grain never creates a heap, you reach the conclusion that no number of grains makes a heap. So the reasoning appears to show both that nothing is a heap and that everything is. [2]

The Stanford Encyclopedia of Philosophy lists three conditions for a sorites argument. There must be an ordered series along a relevant dimension, such as number of grains for 'heap' or height for 'tall'. Neighboring members must differ only slightly. The predicate must be true of the first member and false of the last. The idea that a predicate applies equally across tiny differences is often called tolerance. Other vague terms such as 'bald', 'old' and 'blue' generate the same puzzle. [2]

A precise definition would remove the problem, since it would say exactly how many grains are needed. No such definition exists for 'heap', and ordinary use of the word does not require one. [3]

Most theorists of vagueness assume the paradox can be solved, because we use such words successfully every day and the argument must therefore be defective somewhere. They disagree about where the defect lies. [2]

Timothy Williamson holds that standard logic is not broken and that vagueness is a problem about knowledge, not logic. On his view there really is a point at which removing one grain stops a collection from being a heap, but we cannot know where it falls. He describes proposals to revise logic to accommodate vagueness as attempts to fix something that is not broken. [3]

Another approach, in a computer science text by Elaine Rich and Alan Cline, says there is no logical problem at all. They argue the trouble comes from choosing premises, such as that one grain never makes a difference, that do not accurately describe what heapness is. The Stanford entry also surveys other proposed solutions, which the search results do not describe in detail, so this article does not say which view specialists most widely hold. [4] [2]