Sorites paradox

The sorites paradox: If a heap is reduced by a single grain at a time, at what exact point does it cease to be considered a heap?

The sorites paradox (/soʊˈraɪtiːz/;^[1] sometimes known as the paradox of the heap) is a paradox that arises from vague predicates.^[2] A typical formulation involves a heap of sand, from which grains are individually removed. Under the assumption that removing a single grain does not turn a heap into a non-heap, the paradox is to consider what happens when the process is repeated enough times: is a single remaining grain still a heap? If not, when did it change from a heap to a non-heap?^[3]

The original formulation and variations

Paradox of the heap

The word "sorites" derives from the Greek word for heap.^[4] The paradox is so named because of its original characterization, attributed to Eubulides of Miletus.^[5] 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:^[3]

1000000 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 fewer grain) eventually forces one to accept the conclusion that a heap may be composed of just one grain of sand.^[6]). Read (1995) observes that "the argument is itself a heap, or sorites, of steps of modus ponens":^[7]

1000000 grains is a heap.

If 1000000 grains is a heap then 999999 grains is a heap.

So 999999 grains is a heap.

If 999999 grains is a heap then 999998 grains is a heap.

So 999998 grains is a heap.

If ...

... So 1 grain is a heap.

Variations

Color gradient illustrating a sorites paradox, any adjacent colors being indistinguishable by the human eye

Another formulation is to start with a grain of sand, which is clearly not a heap, and then assume that adding a single grain of sand to something that is not a heap does not turn it into a heap. Inductively, this process can be repeated as much as one wants without ever constructing a heap.^[2][3] A more natural formulation of this variant is to assume a set of colored chips exists such that two adjacent chips vary in color too little for human eyesight to be able to distinguish between them. Then by induction on this premise, humans would not be able to distinguish between any colors.^[2]

The removal of one drop from the ocean, will not make it 'not an ocean' (it is still an ocean), but since the volume of water in the ocean is finite, eventually, after enough removals, even a litre of water left is still an ocean.

This paradox can be reconstructed for a variety of predicates, for example, with "tall", "rich", "old", "blue", "bald", and so on. Bertrand Russell argued that all of natural language, even logical connectives, is vague; moreover, representations of propositions are vague.^[8]

Other similar paradoxes are:

• Argument of the beard^[9]
• The bald man paradox

Proposed resolutions

On the face of it, there are some ways to avoid this conclusion. One may object to the first premise by denying 1000000 grains of sand makes a heap. But 1000000 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.^[10] Alternatively, one may object to the second premise by stating that it is not true for all heaps of sand that removing one grain from it still makes a heap.

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, 10000 grains then one would claim that for fewer than 10000, it is not a heap; for 10000 or more, then it is a heap.

However, such solutions are unsatisfactory as there seems little significance to the difference between 9999 grains and 10000 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)

Timothy Williamson^[11][12][13] and Roy Sorensen^[14] hold an approach that there are fixed boundaries but that they are necessarily unknowable.

Supervaluationism

Supervaluationism is a semantics for dealing with irreferential singular terms and vagueness. It allows one to retain the usual tautological laws even when dealing with undefined truth values.^{[15][16][17][18]} As an example for a proposition about an irreferential singular term, consider the sentence "Pegasus likes licorice". Since the name "Pegasus" fails to refer, no truth value can be assigned to the sentence; there is nothing in the myth that would justify any such assignment. However, there are some statements about "Pegasus" which have definite truth values nevertheless, such as "Pegasus likes licorice or Pegasus doesn't like licorice". This sentence is an instance of the tautology "$p\vee \neg p$", i.e. the valid schema "$p$ or not-$p$". According to supervaluationism, it should be true regardless of whether or not its components have a truth value.

Similarly, "1000 grains of sand is a heap of sand" may be considered a border case having no truth value, but "1000 grains of sand is a heap of sand, or 1000 grains of sand is not a heap of sand" should be true.

Precisely, let $v$ be a classical valuation defined on every atomic sentence of the language $L$, and let $At(x)$ be the number of distinct atomic sentences in $x$. Then for every sentence $x$, at most $2^{At(x)}$ distinct classical valuations can exist. A supervaluation $V$ is a function from sentences to truth values such that, a sentence $x$ is super-true (i.e. $V(x) = \text{True}$) if and only if $v(x) = \text{True}$ for every classical valuation $v$; likewise for super-false. Otherwise, $V(x)$ is undefined—i.e. exactly when there are two classical valuations $v$ and $v'$ such that $v(x)=\text{True}$ and $v'(x) = \text{False}$.

For example, let $L \; p$ be the formal translation of "Pegasus likes licorice". Then there are exactly two classical valuations $v$ and $v'$ on $L \; p$, viz. $v(L \; p) = \text{True}$ and $v'(L \; p) = \text{False}$. So $L \; p$ is neither super-true nor super-false. However, the tautology $L \; p \lor \lnot L \; p$ is evaluated to $\text{True}$ by every classical valuation; it is hence super-true. Similarly, the formalization of the above heap proposition $H \; 1000$ is neither super-true nor super-false, but $H \; 1000 \lor \lnot H \; 1000$ is super-true.

Truth gaps, gluts, and many-valued logics

Another approach is to use a multi-valued logic. From this point of view, the problem is with the principle of bivalence: the sand is either a heap or is not a heap, without any shades of gray. Instead of two logical states, heap and not-heap, a three value system can be used, for example heap, indeterminate and not-heap. However, three valued systems do not truly resolve the paradox as there is still a dividing line between heap and indeterminate and also between indeterminate and not-heap. The third truth-value can be understood either as a truth-value gap or as a truth-value glut.^[19]

Alternatively, fuzzy logic offers a continuous spectrum of logical states represented in the unit interval of real numbers [0,1]—it is a many-valued logic with infinitely-many truth-values, and thus the sand moves smoothly from "definitely heap" to "definitely not heap", with shades in the intermediate region. 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. ^[20][21]

Hysteresis

Another approach, introduced by Raffman,^[22] 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" to a point, even as the actual amount of sand is reduced to a smaller number of grains. For example, suppose 500 grains is a pile and 1000 grains is a heap. There will be an overlap for these states. So if one is reducing it from a heap to a pile, it is a heap going down until, say, 750. At that point one would stop calling it a heap and start calling it a pile. But if one replaces one grain, it would not instantly turn back into a heap. When going up it would remain a pile until, say, 900 grains. The numbers picked are arbitrary; the point is, that the same amount can be either a heap or a pile depending on what it was before the change. A common use of hysteresis would be the thermostat for air conditioning: the AC is set at 77 °F and it then cools down to just below 77 °F, but does not turn on again instantly at 77.001 °F—it waits until almost 78 °F, to prevent instant change of state over and over again.^[23]

Group consensus

One can establish the meaning of the word "heap" by appealing to 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 considered 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". This can be considered an appeal to descriptive linguistics rather than prescriptive linguistics, as it resolves the issue of definition based on how the population uses natural language. Indeed, if a precise prescriptive definition of "heap" is available then the group consensus will always be unanimous and the paradox does not arise.

Modelling "X more or equally red than Y" as quasitransitive (Q) and as transitive (T) relation
X∖^Y	f01000	e02000	d03000	c04000	b05000	a06000
f01000	QT	QT	QTP	QTP	QTP	QTP
e02000	Q	QT	QT	QTP	QTP	QTP
d03000		Q	QT	QT	QTP	QTP
c04000			Q	QT	QT	QTP
b05000				Q	QT	QT
a06000					Q	QT

Dropping transitivity of the relations involved

In the above color example, the argument is tacitly based on considering the relation "for the human eye, color X is indistinguishable from Y" as an equivalence relation, in particular as transitive. To drop the transitivity assumption is another possibility to resolve the paradox.

Similarly, the paradox is based on considering the relation "for the human eye, color X looks more or equally red than Y" as a reflexive total ordering; dropping its transitivity again resolves the paradox. The latter relation can be described instead as a quasitransitive relation.

The table shows a simple example, with color differences overdone for readability. A "Q" and a "T" indicates that the row's color looks more or equally red than column's color in the quasitransitive and the transitive version of the relation, respectively. In the quasitransitive version, e.g. the colors f01000 and e02000 are modelled as indistinguishable, since a "Q" appears in both their intersection cells. A "P" indicates the asymmetric part of the quasitransitive version.

To resolve the original heap variation of the paradox with this approach, the relation "X grains are more a heap than Y grains" should be considered quasitransitive rather than transitive.

See also

• Ambiguity
• Boiling frog
• Beginning of human personhood
• Coastline paradox
• Continuum fallacy
• False dilemma
• I know it when I see it
• Loki's Wager
• Milo of Croton (on how he was able to lift a bull)
• Nirvana fallacy
• Philosophical Investigations
• Ring species
• Ship of Theseus
• Slippery slope
• Straw that broke the camel's back
• The Englishman Who Went Up a Hill But Came Down a Mountain

References

1. http://www.omnilexica.com/pronunciation/?q=Sorites#.UyL2hPldWSo
2. Barker, C. (2009). "Vagueness". In Allan, Keith. Concise Encyclopedia of Semantics. Elsevier. p. 1037. ISBN 978-0-08-095968-9.
3. Sorensen, Roy A. (2009). "sorites arguments". In Jaegwon Kim; Sosa, Ernest; Rosenkrantz, Gary S. A Companion to Metaphysics. John Wiley & Sons. p. 565. ISBN 978-1-4051-5298-3.
4. Bergmann, Merrie (2008). An Introduction to Many-Valued and Fuzzy Logic: Semantics, Algebras, and Derivation Systems. New York, NY: Cambridge University Press. p. 3. ISBN 978-0-521-88128-9.
5. (Barnes 1982), (Burnyeat 1982), (Williamson 1994)
6. Dolev, Y. (2004). "Why Induction Is No Cure For Baldness". Philosophical Investigations. 27 (4): 328–344. doi:10.1111/j.1467-9205.2004.t01-1-00230.x.
7. Read, Stephen (1995). Thinking About Logic, p.174. Oxford. ISBN 019289238X.
8. Russell, Bertrand (June 1923). "Vagueness". The Australasian Journal of Psychology and Philosophy. 1 (2): 84–92. doi:10.1080/00048402308540623. ISSN 1832-8660. Retrieved November 18, 2009. Shalizi's 1995 etext is archived at archive.org and at WebCite.
9. Thouless, Robert H. (1953), Straight and Crooked Thinking (PDF) (Revised ed.), London: Pan Books, p. 61
10. Unger, Peter (1979). "There Are No Ordinary Things". Synthese. 41: 117–154. doi:10.1007/bf00869568. Retrieved 19 July 2013. (Alternative: jstor.org)
11. Williamson, Timothy (1992). "Inexact Knowledge". Mind. 101: 218–242. doi:10.1093/mind/101.402.217. JSTOR 2254332.
12. Williamson, Timothy (1992). "Vagueness and Ignorance". Supplementary Proceedings of the Aristotelian Society. 66: 145–162. JSTOR 4106976.
13. Williamson, Timothy (1994). Vagueness. London: Routledge.
14. Sorensen, Roy (1988). Blindspots. Clarendeon Press.
15. Fine, Kit (Apr–May 1975). "Vagueness, Truth and Logic" (PDF). Synthese. 30 (3/4): 265–300. doi:10.1007/BF00485047.
16. van Fraassen, Bas C. (Sep 1966). "Singular Terms, Truth-Value Gaps, and Free Logic" (PDF). Journal of Philosophy. 63 (17): 481—495. JSTOR 2024549.
17. Kamp, Hans (1975). Keenan, E., ed. Two Theories about Adjectives. Cambridge University Press. pp. 123–155.
18. Dummett, Michael (1975). "Wang's Paradox" (PDF). Synthese. 30: 301–324. doi:10.1007/BF00485048.
19. http://plato.stanford.edu/entries/truth-values/
20. Zadeh, L. A. (1965). "Fuzzy Sets". Information and Control. 8: 338–353. doi:10.1016/s0019-9958(65)90241-x.
21. Goguen, J. A. (1969). "The Logic of Inexact Concepts". Synthese. 19 (3–4): 325–378. doi:10.1007/BF00485654.
22. Unruly Words: A Study of Vague Language (OUP, 2014)
23. Raffman, D. (2005). "How to understand contextualism about vagueness: reply to Stanley". Analysis. 65 (287): 244–248. doi:10.1111/j.1467-8284.2005.00558.x.

Bibliography

• Black, Max (1970). Margins of Precision. Ithaca, NY: Cornell University Press. ISBN 0-8014-0602-1.
• Barnes, J. (1982). "Medicine, Experience and Logic". In Barnes, J.; Brunschwig, J.; Burnyeat, M. F.; Schofield, M. Science and Speculation. Cambridge: Cambridge University Press.
• Burns (1991). Vagueness: An Investigation into Natural Languages and the Sorites Paradox. Dordrecht: Kluwer Academic Publishers. ISBN 0-7923-1489-1.
• Burnyeat, Myles (1982). "15. Gods and heaps". In Schofield, M.; Nussbaum, M. C. Language and Logos. Cambridge: Cambridge University Press. pp. 315–.
• Damir D. Dzhafarov, The sorites paradox: a behavioral approach (with E. N. Dzhafarov), in J. Valsiner and L. Dudolph (eds.).
• Gerla (2001). Fuzzy logic: Mathematical Tools for Approximate Reasoning. Dordrecht, Netherlands: Kluwer Academic Publishers. ISBN 0-7923-6941-6.
• Kirk Ludwig & Greg Ray, "Vagueness and the Sorites Paradox", Philosophical Perspectives 16, 2002.
• Nouwen, Rick; Rooij, Robert van; Sauerland, Uli; Schmitz, Hans-Christian (2009). International Workshop on Vagueness in Communication (ViC; held as part of ESSLLI). LNAI. 6517. Springer. ISBN 978-3-642-18445-1.
• Sainsbury, R. M. (2009). Paradoxes (3rd ed.). Cambridge University Press.; Sect.3

External links

• "Sorites Paradox". Stanford Encyclopedia of Philosophy. by Dominic Hyde.