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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? If not, when did it change from a heap to a non-heap?

Variations of the paradox

Paradox of the heap

The name 'Sorites' derives from the Greek word for heap. The paradox is so-named because of its original characterization, attributed to Eubulides of Miletus. The paradox goes as follows: consider a heap of sand from which grains are individually removed. One might construct the argument, using premises, as follows:

1,000,000 grains of sand is a heap of sand. (Premise 1)

A heap of sand minus one grain is still 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 of just one grain of sand (and if you follow premise 2 again, composed of no grains at all!).

On the face of it, there are some ways to avoid this conclusion. One may object to the first premise by denying 1,000,000 grains of sand makes a heap. But 1,000,000 is just an arbitrarily large number, and the argument will go through with any such number. So the response must deny outright that there are such things as heaps. Peter Unger defends this solution. Alternatively, 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 accept the conclusion by insisting that a heap of sand can be composed of just one grain.

Alternatively, one may define a heap inductively instead of by reduction, and make the rules as follows:

100,000 grains of sand is a heap of sand.

A heap of sand plus one grain is still one heap of sand.

and adjust the number in the first premise to an arbitrary, but well-defined value for a heap.

Variations

This paradox can be reconstructed for a variety of predicates, for example, with "tall", "rich", "old", "blue" and so on. Bertrand Russell argues, in his paper titled 'Vagueness', that all of natural language, even logical connectives, are vague; most views do not go that far, but it is certainly an open question.

Proposed resolutions

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 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. It is objectionable on both philosophical and linguistic grounds: the former on account of its arbitrariness, and the latter on the ground that it is simply not how we use natural language.

Unknowable boundaries (or Epistemicism)

Williamson and Sorensen hold an approach that there are fixed boundaries but that they are necessarily unknowable.

Supervaluationism

See Fine (1975).

Truth gaps, gluts, and many-valued logics

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. The third truth-value can be understood either as a truth gap or as a truth glut. Fuzzy logic offers a continuous spectrum of logical states represented in the interval of real numbers [0,1] --- it is a many-valued logic with infinitely-many truth-values. Fuzzy hedges are used to divide the continuum into regions corresponding to classes like definitely heap, mostly heap, partly heap, slightly heap, and not heap.

Hysteresis

Another approach is to use hysteresis—that is, knowledge of what the collection of sand started as. Equivalent amounts of sand may be called heaps or not based on how they got there. If a large heap (indisputably described as a heap) is slowly diminished, it preserves its "heap status" even as the actual amount of sand is reduced to a small number of grains.^[1]

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.

References

Max Black (19nn) "Margins of Precision."
Boguslowski
Kit Fine
Peter Unger
Burns, L. (1991) Vagueness: An Investigation into Natural Languages and the Sorites Paradox. Springer. ISBN 0-792-314891 .
Gerla, G. (2001) Fuzzy logic: Mathematical Tools for Approximate Reasoning. Kluwer. ISBN 0-7923-6941-6.
Goguen, J. A. (1968/69) "The logic of inexact concepts," Synthese 19: 325-373.
Williamson, T. 1994. Vagueness London: Routledge.

^ Raffman, Diana (2005), "How to understand contextualism about vagueness: reply to Stanley", Analysis, 65 (3): 244–48, doi:10.1111/j.1467-8284.2005.00558.x {{citation}}: line feed character in |title= at position 49 (help)

External links

Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy https://plato.stanford.edu/entries/sorites-paradox/. {{cite encyclopedia}}: Missing or empty |title= (help) by Dominic Hyde.

[1] Raffman, Diana (2005), "How to understand contextualism about vagueness: reply to Stanley", Analysis, 65 (3): 244–48, doi:10.1111/j.1467-8284.2005.00558.x {{citation}}: line feed character in |title= at position 49 (help)

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@@ Line 8: / Line 8: @@
 :''A heap of sand minus one grain is still 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 of just one grain of sand (and if you follow premise 2 again, composed of no grains at all!).
+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 of just one grain of sand (and if you follow premise 2 again, composed of no grains at all!).<!-- And, if you follow premise 2 again, composed of a negative number of grains of sand, possibly antimatter. -->
 On the face of it, there are some ways to avoid this conclusion. One may object to the first premise by denying 1,000,000 grains of sand makes a heap. But 1,000,000 is just an arbitrarily large number, and the argument will go through with any such number. So the response must deny outright that there are such things as heaps. [[Peter Unger|Peter Unger]] defends this solution. Alternatively, 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 accept the conclusion by insisting that a heap of sand can be composed of just one grain.