Sorites paradox

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The Sorites paradox (σωρός (sōros) being Greek for "heap" and σωρίτης (sōritēs) the adjective) is a paradox that arises from vague predicates. The paradox of the heap is an example of this paradox which arises when one considers a heap of sand, from which grains are individually removed. Is it still a "heap" when only one grain remains?

The problem is essentially one of philosophy of language, wherein terms may be relative and indefined, as opposed to problems in mathematics - wherein all terms by nature have some definition - even if it is only as a variable. The paradox is a normal aspect of any attempt to insert imprecise terms into mathematical-like logical formula, or likewise to apply logic to concepts which by definition are imprecise as to be undefinable.

Variations of the paradox

Paradox of the heap

Consider a heap of sand from which grains are individually removed. One might construct the argument, using premises, as follows:

A large collection of grains of sand makes a heap. (Premise 1)

A large collection of grains of sand minus one grain makes a heap. (Premise 2)

Repeated applications of Premise 2 (each time starting with one less number of grains), eventually forces one to accept the conclusion that a heap may be composed by just one grain of sand.

On the face of it, there are three ways to avoid this conclusion. One may object to the first premise by denying that a large collection of grains makes a heap (or more generally, by denying that there are heaps). One may object to the second premise by stating that it is not true for all collections of grains that removing one grain from it still makes a heap. Or one may reject the conclusion by insisting that a heap of sand can be composed of just one grain.

The paradox is tricky for philosophers because they must explain why one of the two premises, or the conclusion, is wrong even though they appear to be self-evident.

A man's height

An equivalent paradox can be constructed by considering a man who has a height of, say, seven feet. Although removing one inch from his height would still leave him tall, he would not be considered tall if this was repeated until he was four feet high. At what point is he no longer tall?

Proposed resolutions

Trivial solutions

A trivial solution is to deny that any number of grains will make a heap. In other words, one might say that the word "heap" is meaningless, since the precise conditions under which it can be verified cannot be produced. Taken to an extreme, this may naturally lead to mereological nihilism.

Other philosophers, like Bertrand Russell, simply deny that logic works with vague concepts.

Setting a fixed boundary

A common first response to the paradox is to call any set of grains, that has more than a certain number of grains in it, a heap. If one were to set the "fixed boundary" at, say, 10,000 grains then one would claim that for fewer than 10,000, it's not a heap; for 10,000 or more, then it is a heap.

However, such solutions are philosophically unsatisfactory, as there seems little significance to the difference between 9,999 grains and 10,001 grains. The boundary, wherever it may be set, remains as arbitrary and so its precision is misleading. Nevertheless, just such arbitrarily precise distinctions are often drawn in real world scenarios. For example, examiners will usually pick one score when setting the boundaries between exam grades. In other scenarios there seems to be a real threshold, such as the straw that broke the camel's back, but in reality an arbitrary choice was at some point made (such as the choice of that particular camel).

Multi-valued logic

Another approach is to use a multi-valued logic. Instead of two logical states, heap and not-heap, a three value system can be used, for example heap, unsure and not-heap.

However, three valued systems do not truly resolve the paradox as there is still a dividing line between heap and unsure and also between unsure and not-heap.

Visual Definition of Heap

Another approach is to define a "heap" visually instead of trying to define it by its number of grains. Someone looking at the collection of grains would probably find it difficult to tell if a grain has been removed (until the collection is very small).

But this approach has has a few problems. For one, from certain vantage points, the sand pile may be judged to look like a heap, from others, it may not. Even if one considers heap-ness to be a quality strictly dependent upon the area taken up by the pile of sand as it is projected on to the back of our eye (as in, how much of our field of view does the sand pile take) as seen from only one particular vantage point, one still has the same problem as before, since one needs to define a certain arbitrary threshold from which we separate the heaps from non heaps. This time, the quantity measured is area, not grains of sand.

Another argument against using human perception is that what a person sees as a heap depends on whether the mound is being reduced from a definite heap or increased from a non-heap. People, for example, have different responses to ambiguous images.

Group consensus

One can establish the meaning of the word "heap" by appealing to group consensus. This approach claims that a collection of grains is as much a "heap" as the proportion of people in a group who believe it to be so. In other words, the probability that any collection is a heap is the expected value of the distribution of the group's views.

A group may decide that:

One grain of sand on its own is not a heap
A large collection of grains of sand is a heap.

Between the two extremes, individual members of the group may disagree with each other over whether any particular collection can be labelled a "heap". The collection can then not be definitively claimed to be a "heap" or "not a heap", but rather it has a certain probability of being a heap, between zero and unity.

The usefulness of this approach is that it tightly defines the meaning of the terms.

Precise terms have a mechanism by which one can persuade others that a specific application of the term is valid. Vague terms have no such mechanism. If a person insists on calling a seven foot man short, one might suspect that their reference set includes professional basketball players. Vague terms are useful to the extent that we have consensus, but when used out of context, they risk generating confusion.

The Sorites paradox merely illustrates logical analysis of how one uses vague language. It indicates that it is a fallacy to assume that everybody agrees on the definition of a vague term. Some people may agree in its application to but not all members of the universe of discourse will as a matter of course. A consensus method essentially changes the definition of a heap from being a subjective definition to an objective one.

External links

References

"Margins of Precision" by Max Black
Boguslowski
Kit Fine
Peter Unger