# The Fallacy of the Heap

Alias:

• The Argument of the Beard1
• The Fallacy of the Beard2
• The Fallacy of the Continuum3
• Slippery Slope4

Etymology: This fallacy takes its name from a particular example called "the paradox of the heap5", or the "sorites"6 paradox. See the Example, below. The fallacy derives its name from the paradox because the paradox rests on a fallacious argument.

Taxonomy: Logical Fallacy > Informal Fallacy > Vagueness > The Fallacy of the Heap7

### Forms:

1. A differs from Z by a continuum of insignificant changes, and there is no non-arbitrary place at which a sharp line between the two can be drawn.
Therefore, there is really no difference between A and Z.
2. A differs from Z by a continuum of insignificant changes with no non-arbitrary line between the two.
Therefore, A doesn't exist.

### Example:

A single grain of sand does not make a heap of sand. Also, a single grain of sand won't turn a non-heap into a heap. Therefore, there are no heaps of sand8.

### Exposition9:

The Fallacy of the Heap plays upon the vagueness of the distinction between two terms that lie on a continuum. For instance, "bald" is a vague word, and a man who is a borderline case of baldness is a familiar sight: it isn't clear whether he is bald or not, so we say that he is "balding".

"Bald" and "non-bald" lie at opposite ends of a spectrum of hairiness, and there is no precise point where baldness turns into non-baldness. We could, of course, decide to count, say, 10,000 hairs or less as the definition of "bald", but this would be arbitrary. Why not 10,001 or 9,999? I know of no answer other than the fact that we prefer round numbers10.

So, there is no non-arbitrary line between baldness and hairiness, but it does not follow from this fact that there really is no difference between the two. A difference in degree is still a difference, and a big enough difference in degree can amount to a difference in kind. For instance, according to the theory of evolution, the difference between species is a difference in degree.

Similarly, the lack of a bright line between contrary concepts does not mean that one of the concepts is a myth―that is, there is nothing to which it refers. For example, some people have argued that there is no such thing as life, since the line between animate and inanimate thing is fuzzy. However, we can all easily identify many living things and nonliving things, and the fact that there are some things which fall into a gray area―viruses, for instance―does not mean that the concept of life is without reference.

### Exposure:

• The mathematically-inclined reader may notice that the heap and beard arguments are reminiscent of proofs by mathematical induction. As a matter of fact, a modern version of these arguments is known as "Wang's paradox"11:

One is a small number.
If you add one to a small number, the resulting number is small.
Therefore, all numbers are small.

The conclusion follows from the premisses by mathematical induction. However, you should realize by now that the problem is that "small" is vague, and the moral of the paradox is that we must be careful when reasoning mathematically with imprecise concepts.

• The Fallacy of the Heap is primarily a philosopher's fallacy, though it occasionally arises in legal or moral contexts. A great deal of ink has been spilled in fruitless philosophical debates over exactly where to draw the line between concepts that lie on continua. This might be called the "legalistic" side of philosophy, for it is primarily in the law that we are forced to decide hard cases that lie in gray areas. For instance, if the legislature were to decide that baldness is a disability deserving of certain benefits, then the courts might be forced to decide the issue of whether a certain person is bald. In everyday life, we are seldom faced with decisions of this kind, and we continue to use the concept of baldness without worrying about borderline cases. When someone falls into the fuzzy area between bald and hairy, we just say that he is "balding", thus avoiding the issue of whether he is now bald.

One reason that so many philosophical debates are seemingly endless and undecidable is because they involve a search for a mythical entity, namely, a non-arbitrary distinction between concepts that lie upon continua in conceptual space. The logical attitude towards such problems is to avoid them if at all possible; but if a decision cannot be avoided, then draw an arbitrary line in the gray zone and stick with it. Don't be drawn into defending the decision against the charge that it is arbitrary; of course it's arbitary, for any such decision will be arbitrary. For this reason, it is not a criticism of such decisions to point out their arbitrariness. Philosophers, naturally, are uneasy about arbitrariness, but when we are dealing with conceptual continua, it is an unavoidable fact of life. Where there is only gray, there are no black-and-white distinctions to be made.

Notes:

1. Robert H. Thouless, Straight and Crooked Thinking (1974), pp. 104-8. This name comes from an argument similar to that giving rise to the heap argument: One hair on your chin does not make a beard. Adding only one hair to an unbearded chin does not turn it into a bearded one. Therefore, there are no beards, or there is no difference between having a beard and not having one.
2. Robert J. Gula, Nonsense: A Handbook of Logical Fallacies (2002), pp. 100-1.
3. T. Edward Damer, Attacking Faulty Reasoning: A Practical Guide to Fallacy-Free Arguments (3rd edition, 1994), pp. 88-90.
4. The name "slippery slope" is most often used for a causal fallacy: see the entry under that name. However, the continuum between two concepts that gives rise to the Fallacy of the Heap is sometimes called a "slippery slope", especially by philosophers. Though these two fallacies are distinct, and the fact that they share a name is unfortunate, fallacies of the heap sometimes form a basis for slippery slopes in the causal sense. People may think that a causal slide from A to Z is unavoidable because there is no precise, non-arbitrary dividing line between the two concepts. For instance, opponents of abortion often believe that the legality of abortion will lead causally to the legality of infanticide, and one reason for this belief is that the only precise dividing line between an embryo and a newborn baby is the morally arbitrary one of birth. For this reason, slopes are sometimes slippery because there is no non-arbitrary point on the slope to stop sliding down it. For a well-known example of this usage, see: Judith Jarvis Thomson, "A Defense of Abortion", Philosophy & Public Affairs, Volume 1, Number 1 (Fall, 1971), p. 47.
5. Nicholas Falletta, The Paradoxicon: A Collection of Contradictory Challenges, Problematical Puzzles, and Impossible Illustrations (1990), pp. 11-5.
6. "Heaped", classical Greek. See: A. R. Lacey, A Dictionary of Philosophy (3rd Revised Edition, 1999), under: "Heap (paradox of)".
7. Eduard Zeller, Outlines of the History of Greek Philosophy (13th Edition, 1965), p. 124.
8. This is a version of the paradox of the heap. There are alternative versions of the paradox, some with different conclusions, such as: A million grains of sand together in one place would not make a heap. Another version runs the argument backwards: A million grains of sand is a heap of sand. Taking away a single grain of sand from a heap will leave a heap. Therefore, even a single grain of sand is a heap. Or, for an even more absurd conclusion: an empty patch of ground where a heap once stood is still a heap.
9. Thanks to Kevin Donaldson for a criticism which led to some revisions to this section.
10. A "round" number is one whose numeral ends in one or more zeroes, but this depends upon what base is used in the numbering system. For example, 32 is not round in the decimal (base 10) system, but is round in binary and hexadecimal (base 16). However, what base we use is arbitrary. Human beings like the decimal system because we have ten fingers, but what if we had only eight?
11. For a brief discussion, see: Wang's Paradox, 9/2/2006.