Sorites paradox 231687 225153442 2008-07-12T04:28:55Z FrankTobia 267669 /* Variations */ copyedit {{expert-subject|Philosophy of language|article}} The '''Sorites paradox''' (''σωρός'' (''sōros'') being [[Greek language|Greek]] for "heap" and ''σωρίτης'' (''sōritēs'') the adjective) is a [[paradox]] that arises from [[vagueness|vague]] [[Predicate (grammar)|predicate]]s. 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 [[grain|grains]] are individually removed. One might construct the argument, using [[premise]]s, 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. This is Unger's proposal. 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. ===Variations=== This paradox can be reconstructed for a variety of predicates, for example, with "tall", "rich", "old", "blue" and so on. 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 [[Epistemology|Epistemicism]]) === [[Timothy Williamson|Williamson]] and Sorensen hold an approach that there are fixed boundaries but that they are necessarily unknowable. === Supervaluationism === {{incomplete|section}} 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. See Raffman, forthcoming. === 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. == See also == * [[Imprecise language]] * [[Continuum fallacy]] * [[Multi-valued logic]] * [[Ship of Theseus]] * [[Coastline paradox]] * [[Vagueness]] * [[Fuzzy logic]] * [[Philosophical Investigations]] * [[Loki's wager]] == 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. * [[Timothy Williamson|Williamson, T]]. 1994. ''Vagueness'' London: Routledge. ==External links== *{{sep entry|sorites-paradox}} by Dominic Hyde. [[Category:Paradoxes]] [[de:Paradoxie des Haufens]] [[el:Σόφισμα του μη σωρού]] [[es:Paradoja sorites]] [[fr:Paradoxe sorite]] [[he:פרדוקס הערימה]] [[ja:砂山のパラドックス]] [[no:Soritesparadokset]] [[fi:Sorites-paradoksi]] [[sv:Soritesparadoxen]]