The Paradox of the Heap, from John L. Bell’s Oppositions and Paradoxes

In Oppositions and Paradoxes John L. Bell explores a variety of mathematical and scientific paradoxes with philosophical precision, while retaining a great sense of humour in his investigations. In this excerpt, Bell formulates and works through “The Problem of the Heap,” asking: how many grains of sand does one need to make a heap, exactly?

The paradox of the heap or sorites paradox (from the Greek sōritēs “heap”)—attributed to the ancient Greek philosopher Eubulides of Miletus—arises from the vagueness of certain predicates in ordinary language. In a typical formulation, we consider a heap of sand, from which grains are removed one by one. The paradox arises when one considers what happens when the process is repeated sufficiently many times. For suppose we make the natural assumption that, if we remove a single grain from a heap, we are still left with a heap. Then eventually just a single grain remains: is it still a heap? Or are even no grains at all a heap? If not, when did the heap change into to a non-heap?

We can turn the paradox on its head by starting with a totally bald man, and, noting that any man with just one more hair than a bald man is still bald, conclude that every man must be bald. For a man with no hair is bald, so a man with just one hair is bald, and thus a man with two hairs is bald, … whence a man with any number of hairs is bald.

A related formulation of the paradox is to suppose given a set of coloured chips such that the variation in colour of two adjacent chips is too small—a difference in wavelength of 1 nanometre say—for the human eye to be able to distinguish between them. Suppose that the first chip is coloured violet, which has a wavelength of about 400 nanometres, and the last chip is coloured red, with a wavelength of 650 nanometres. If we assume, as in the case of the bald man, that a chip whose colour differs in wavelength by one nanometre from a violet coloured chip would still be seen as violet, then the ‘bald man’ argument leads to the conclusion that the red chip would also have to be seen as violet.

The paradox can be reconstructed for a variety of predicates—all of which can be seen to be vague—for example, with “short,” “poor,” “young,” “red,” and so on.

A natural response to the paradox is to introduce a “fixed boundary” to the concept of heap by defining a “heap” to be a set of grains containing at least a certain fixed number—10000, say—of grains. In that case, a set of 9999 grains is not a heap but one of 10000 is. This seems unnatural since there would appear to be little significance to the difference between 9999 grains and 10000 grains. Wherever the boundary is set, it remains arbitrary. A more acceptable, if radical solution would be to call any collection of two or more a heap!

The paradox can be given a striking formulation by generalizing the colour example above. Suppose that we have a set S of things—colours, collections of grains of sand, people—on which is defined arelation I which we shall call indistinguishability. So for two elements a and b of S, a and b will be in the relation I—which we write as aIb—if and only if a is indistinguishable from b. We shall suppose that I is reflexive—for any a, aIa—and symmetric—for any a, b, aIb if and only if bIa. Let us call a property (or predicate) P defined on S vague if it is preserved under indistinguishability, that is, if aIb and P(a) then P(b) (in words: anything indistinguishable from something with the property P also has the property P). Let us say that two elements a, b of S are connected if there is a sequence a0, …, an of elements of S such that a0 = a, an = b and, for each i, ai,Iaii+1. Call S connected if each pair of elements of S are connected.

Suppose now that S is connected. Then, for any vague property P on S, if some element of S has P, then every element of S has P. To see this, suppose that a is an element of S such that a has the property P—we write this as P(a)—and let b be an arbitrary element of S. Then since S is connected, there is a sequence a0, …, an of elements of S such that a0 = a, an = b and, for each i, ai,Iaii+1. Now since P(a), i.e., P(a0) and a0Iai1, it follows from the vagueness of P that P(a1). From this it follows similarly that P(a2), whence P(a3) and so on. Finally we obtain P(an), i.e., P(b). Since b was arbitrary, we conclude that every element of S has P.

From this we infer that a vague property either applies to everything, or it applies to nothing. For example, consider the case of the vague predicate “bald” or better, baldish. Here S is the set of (heads of) men and I is the relation of differing by at most one hair. Then, if there is at least one baldish man, all men are baldish—including Brad Pitt. If, on the other hand, there is at least one nonbaldish man, then all men are nonbaldish—including Bruce Willis.