Editing Sorites paradox (section)

==Proposed resolutions==
===Denying the existence of heaps===
One may [[Denying the antecedent|object to the first premise]] by denying that {{val|1000000|fmt=commas}} grains of sand make a heap. But {{val|1000000|fmt=commas}} is just an arbitrary large number, and the argument will apply with any such number. So the response must deny outright that there are such things as heaps. [[Peter Unger]] defends this solution.<ref>{{cite journal |last=Unger |first=Peter |title=There Are No Ordinary Things |journal=[[Synthese]] |year=1979 |volume=41 |issue=2 |pages=117–154 |doi=10.1007/bf00869568 |jstor=20115446|s2cid=46956605 }}</ref> However, [[A. J. Ayer]] repudiated it when presented with it by Unger: "If we regard everything as being composed of atoms, and think of Unger as consisting not of cells but of the atoms which compose the cells, then, as [[David Wiggins]] has pointed out to me, a similar argument could be used to prove that Unger, so far from being non-existent, is identical with everything that there is. We have only to substitute for the premise that the subtraction of one atom from Unger's body never makes any difference to his existence the premise that the addition of one atom to it never makes any difference either."<ref>{{cite book |last1=Ayer |first1=A.J. |title=Perception and identity: Essays presented to A. J. Ayer, With His Replies. |date=1979 |publisher=Cornell University Press. |location=Ithica, NY |page=324}}</ref>

===Setting a fixed boundary===
A common first response to the paradox is to term any set of grains that has more than a certain number of grains in it a heap. If one were to define the "fixed boundary" at {{val|10000|fmt=commas}} grains then one would claim that for fewer than {{val|10000|fmt=commas}}, it is not a heap; for {{val|10000|fmt=commas}} or more, then it is a heap.{{sfn|Collins|2018|p=32}}

Collins argues that such solutions are unsatisfactory as there seems little significance to the difference between {{val|9999|fmt=commas}} grains and {{val|10000|fmt=commas}} grains. The boundary, wherever it may be set, remains 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 natural language is used.{{sfn|Collins|2018|p=35}}

===Unknowable boundaries (or epistemicism)===
[[Timothy Williamson]]<ref>{{cite journal |last=Williamson |first=Timothy |title=Inexact Knowledge |journal=Mind |year=1992 |volume=101 |issue=402 |pages=218–242 |jstor=2254332 |doi=10.1093/mind/101.402.217}}</ref><ref>{{cite journal |last=Williamson |first=Timothy |title=Vagueness and Ignorance |journal=[[Supplementary Proceedings of the Aristotelian Society]] |publisher=[[Aristotelian Society]] |year=1992 |volume=66 |pages=145–162 |jstor=4106976|doi=10.1093/aristoteliansupp/66.1.145 }}</ref><ref>{{cite book |last=Williamson |first=Timothy |author-link=Timothy Williamson |title=Vagueness |year=1994 |publisher=[[Routledge ]]|location=London }}</ref> and Roy Sorensen<ref>{{cite book |last=Sorensen |first=Roy |title=Blindspots |year=1988 |publisher=[[Clarendon Press]] |isbn=9780198249818 }}</ref> claim that there are fixed boundaries but that they are necessarily unknowable.

===Supervaluationism===
[[Supervaluationism]] is a method for dealing with irreferential [[singular term]]s and [[vagueness]]. It allows one to retain the usual [[tautology (logic)|tautological laws]] even when dealing with undefined truth values.<ref>{{cite journal |last=Fine |first=Kit |title=Vagueness, Truth and Logic |journal=Synthese |date=Apr–May 1975 |volume=30 | number= 3/4|pages=265–300 |doi=10.1007/BF00485047 |jstor=20115033 |s2cid=17544558 |url=http://www.niu.edu/~gpynn/Fine_Vagueness_Truth%26Logic.pdf |archive-url=https://web.archive.org/web/20150608192009/http://www.niu.edu/~gpynn/Fine_Vagueness_Truth%26Logic.pdf |url-status=dead |archive-date=2015-06-08 }}</ref><ref>{{cite journal | url=https://ontologievorlesung.files.wordpress.com/2011/11/van-fraassen-gaps-and-singular-terms.pdf | first=Bas C. |last=van Fraassen |author-link=Bas van Fraassen| title=Singular Terms, Truth-Value Gaps, and Free Logic | journal=Journal of Philosophy | volume=63 | number=17 | pages=481–495 | year=1966 | doi=10.2307/2024549 | jstor=2024549}}</ref><ref>{{cite book |last=Kamp |first=Hans |editor-last=Keenan |editor-first=E. |title=Two Theories about Adjectives |year=1975 |pages=123–155 |publisher=Cambridge University Press}}</ref><ref>{{cite journal |last=Dummett |first=Michael |title=Wang's Paradox |journal=[[Synthese]] |year=1975 |volume=30 |issue=3/4 |pages=301–324 |doi=10.1007/BF00485048 |jstor=20115034 |s2cid=46956702 |url=http://wylieb.com/Philosophy/ElectronicTexts/Transitivity/Dummett1975.pdf |archive-url=https://web.archive.org/web/20160422061349/http://wylieb.com/Philosophy/ElectronicTexts/Transitivity/Dummett1975.pdf |url-status=dead |archive-date=2016-04-22 }}</ref>
An example of a proposition about an irreferential singular term is the sentence "''[[Pegasus]] likes licorice''".
Since the name "''Pegasus''" [[failure of reference|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 "<math>p \vee \neg p</math>", i.e. the valid schema "''<math>p</math> or not-<math>p</math>''". According to supervaluationism, it should be true regardless of whether or not its components have a truth value.

By admitting sentences without defined truth values, supervaluationism avoids adjacent cases such that {{mvar|n}} grains of sand is a heap of sand, but {{math|''n'' − 1}} grains is not;
for example, "''{{val|1000|fmt=commas}} grains of sand is a heap''" may be considered a border case having no defined truth value. Nevertheless, supervaluationism is able to handle a sentence like "''{{val|1000|fmt=commas}} grains of sand is a heap or {{val|1000|fmt=commas}} grains of sand is not a heap''" as a tautology, i.e. to assign it the value ''true''.{{citation needed|date=January 2020}}

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

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

===Truth gaps, gluts, and multi-valued logics===
Another method is to use a [[multi-valued logic]]. In this context, 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''. A response to this proposed solution is that 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''.<ref>{{Cite book | chapter-url=http://plato.stanford.edu/entries/truth-values/ | title=[[The Stanford Encyclopedia of Philosophy]]| chapter=Truth Values| publisher=Metaphysics Research Lab, [[Stanford University]]| year=2018}}</ref>

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 transitions gradually 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''.<ref>{{cite q | Q25938993 |last1=Zadeh |first1=L.A. | author-link1 = Lotfi A. Zadeh | | journal = [[Information and Computation|Information and Control]] | doi-access = free }}</ref><ref>{{cite journal |last=Goguen |first=J. A. |title=The Logic of Inexact Concepts |journal=Synthese |year=1969 |volume=19 |issue=3–4 |pages=325–378 |doi=10.1007/BF00485654 |jstor=20114646 |s2cid=46965639 }}</ref> Though the problem remains of where these borders occur; e.g. at what number of grains sand starts being ''definitely'' a heap.

===Hysteresis===
Another method, introduced by Raffman,<ref>{{Cite book | doi=10.1093/acprof:oso/9780199915101.001.0001 |title = Unruly Words: A Study of Vague Language|year = 2014|last1 = Raffman|first1 = Diana|isbn = 9780199915101|publisher=OUP|url=https://books.google.com/books?id=NVfSAQAAQBAJ&pg=PA136|pages=136ff}}</ref> is to use [[hysteresis]], that is, knowledge of what the collection of sand started as. Equivalent amounts of sand may be termed heaps or not based on how they got there. If a large heap (indisputably described as a heap) is  diminished slowly, 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, {{val|500}} grains is a pile and {{val|1000|fmt=commas}} 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 {{val|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 {{val|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&nbsp;°F and it then cools the air to just below 77&nbsp;°F, but does not activate again instantly when the air warms to 77.001&nbsp;°F—it waits until almost 78&nbsp;°F, to prevent instant change of state over and over again.<ref>{{Cite journal| doi = 10.1111/j.1467-8284.2005.00558.x |jstor=3329033 | title = How to understand contextualism about vagueness: reply to Stanley| year = 2005| last1 = Raffman | first1 = D.| journal =[[Analysis (journal)|Analysis]]| volume = 65| issue = 287| pages = 244–248 }}</ref>

===Group consensus===
One can establish the meaning of the word ''heap'' by appealing to [[Consensus decision-making|consensus]]. Williamson, in his epistemic solution to the paradox, assumes that the meaning of vague terms must be determined by group usage.{{sfn|Collins|2018|p=33}} The consensus method typically claims that a collection of grains is as much a "heap" as the proportion of people in a [[social group|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 opinion.

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 occur.

===Resolutions in utility theory===
{|class="wikitable collapsible" align="right"
|- 
! colspan=7 | "''X'' more or equally red than ''Y''"<br>modelled as quasitransitive relation<br>≈ : indistinguishable, > : clearly more red
|-
! {{diagonal split header|<sub>''X''</sub>|<sup>''Y''</sup>}}
! {{color|#f10|f10}}
! {{color|#e20|e20}}
! {{color|#d30|d30}}
! {{color|#c40|c40}}
! {{color|#b50|b50}}
! {{color|#a60|a60}}
|-
! {{color|#f10|f10}}
! ≈
! ≈
! >
! >
! >
! >
|-
! {{color|#e20|e20}}
! ≈
! ≈
! ≈
! >
! >
! >
|-
! {{color|#d30|d30}}
!
! ≈
! ≈
! ≈
! >
! >
|-
! {{color|#c40|c40}}
!
!
! ≈
! ≈
! ≈
! >
|-
! {{color|#b50|b50}}
!
!
!
! ≈
! ≈
! ≈
|-
! {{color|#a60|a60}}
!
!
!
!
! ≈
! ≈
|}

In the economics field of [[utility theory]], the sorites paradox arises when a person's preferences patterns are investigated.
As an example by [[Robert Duncan Luce]], it is easy to find a person, say, Peggy, who prefers in her coffee 3 grams (that is, 1 cube) of sugar to 15 grams (5 cubes), however, she will usually be indifferent between 3.00 and 3.03 grams, as well as between 3.03 and 3.06 grams, and so on, as well as finally between 14.97 and 15.00 grams.<ref name="Luce.1956">{{cite journal | url=https://www.imbs.uci.edu/files/personnel/luce/pre1990/1956/Luce_Econometrica_1956.pdf | author=Robert Duncan Luce | author-link=Robert Duncan Luce | title=Semiorders and a Theory of Utility Discrimination | journal=Econometrica | volume=24 | number=2 | pages=178–191 | date=Apr 1956 | doi=10.2307/1905751 | jstor=1905751 }} Here: p.179</ref>

Two measures were taken by economists to avoid the sorites paradox in such a setting.

* [[Comparative (linguistics)|Comparative]], rather than [[positive (linguistics)|positive]], forms of properties are used. The above example deliberately does not make a statement like "Peggy likes a cup of coffee with 3 grams of sugar", or "Peggy does not like a cup of coffee with 15 grams of sugar". Instead, it states "Peggy likes a cup of coffee with 3 grams of sugar more than one with 15 grams of sugar".{{#tag:ref|The comparative form was found in all economics publications investigated so far.<ref name="Armstrong.1948">{{cite journal | url=https://www.jstor.org/stable/2226342 | author=Wallace E. Armstrong | title=Uncertainty and the Utility Function | journal=Economic Journal | volume=58 | number=229 | pages=1&ndash;10 | date=Mar 1948 | doi=10.2307/2226342 | jstor=2226342 | url-access=subscription }}</ref><ref>{{cite journal | url=https://www.jstor.org/stable/1909554 | author=Peter C. Fishburn | author-link=Peter C. Fishburn |title=Intransitive Individual Indifference and Transitive Majorities | journal=Econometrica | volume=38 | number=3 | pages=482&ndash;489 | date=May 1970 | doi=10.2307/1909554 | jstor=1909554 | url-access=subscription }}</ref><ref>{{cite report | url=http://econ.haifa.ac.il/~admiller/ArrowWithoutTransitivity.pdf | author1=Alan D. Miller | author2=  Shiran Rachmilevitch | title=Arrow's Theorem Without Transitivity | institution=[[University of Haifa]] | type=Working paper | pages=11 | date=Feb 2014 }}</ref> Apparently it is entailed by the object of investigations in utility theory.}}
* Economists distinguish preference ("Peggy likes ... more than ...") from indifference ("Peggy likes ... as much as ... "), and do not consider the latter relation to be [[transitive relation|transitive]].{{#tag:ref|According to Armstrong (1948), indifference ''was'' considered transitive in [[preference theory]],<ref name="Armstrong.1948"/>{{rp|2}} the latter was challenged in 1939 for this very reason,<ref>{{cite journal | url=https://www.jstor.org/stable/2224802 | author=Wallace E. Armstrong | title=The Determinateness of the Utility Function | journal=Economic Journal | volume=49 | number=195 | pages=453&ndash;467 | date=Sep 1939 | doi=10.2307/2224802 | jstor=2224802 | url-access=subscription }}</ref>{{rp|463}} and succeeded by utility theory.}} In the above example, abbreviating "a cup of coffee with x grams of sugar" by "''c''<sub>''x''</sub>", and "Peggy is indifferent between ''c''<sub>''x''</sub> and ''c''<sub>''y''</sub>" as {{nowrap|"''c''<sub>''x''</sub> ≈ ''c''<sub>''y''</sub>",}} the facts {{nowrap|''c''<sub>3.00</sub> ≈ ''c''<sub>3.03</sub>}} and {{nowrap|''c''<sub>3.03</sub> ≈ ''c''<sub>3.06</sub>}} and ... and {{nowrap|''c''<sub>14.97</sub> ≈ ''c''<sub>15.00</sub>}} do not imply {{nowrap|''c''<sub>3.00</sub> ≈ ''c''<sub>15.00</sub>.}}

Several kinds of relations were introduced to describe preference and indifference without running into the sorites paradox.
Luce defined [[semi-order]]s and investigated their mathematical properties;<ref name="Luce.1956"/> 
[[Amartya Sen]] performed a similar task for [[quasitransitive relation]]s.<ref>{{cite journal | last=Sen | first=Amartya | author-link=Amartya Sen | title=Quasi-transitivity, rational choice and collective decisions | zbl=0181.47302 | journal=The Review of Economic Studies | volume=36 | issue=3 | pages=381–393 | year=1969 | doi=10.2307/2296434 | jstor=2296434 }}</ref>
Abbreviating "Peggy likes ''c''<sub>''x''</sub> more than ''c''<sub>''y''</sub>" as {{nowrap|"''c''<sub>''x''</sub> > ''c''<sub>''y''</sub>",}} and abbreviating {{nowrap|"''c''<sub>''x''</sub> > ''c''<sub>''y''</sub>}} or {{nowrap|''c''<sub>''x''</sub> ≈ ''c''<sub>''y''</sub>"}} by {{nowrap|"''c''<sub>''x''</sub> ≥ ''c''<sub>''y''</sub>",}} it is reasonable that the relation ">" is a semi-order while ≥ is quasitransitive.
Conversely, from a given semi-order > the indifference relation ≈ can be reconstructed by defining {{nowrap|''c''<sub>''x''</sub> ≈ ''c''<sub>''y''</sub>}} if neither {{nowrap|''c''<sub>''x''</sub> > ''c''<sub>''y''</sub>}} nor {{nowrap|''c''<sub>''y''</sub> > ''c''<sub>''x''</sub>.}}
Similarly, from a given quasitransitive relation ≥ the indifference relation ≈ can be reconstructed by defining {{nowrap|''c''<sub>''x''</sub> ≈ ''c''<sub>''y''</sub>}} if both {{nowrap|''c''<sub>''x''</sub> ≥ ''c''<sub>''y''</sub>}} and {{nowrap|''c''<sub>''y''</sub> ≥ ''c''<sub>''x''</sub>.}}
These reconstructed ≈ relations are usually not transitive.

The table to the right shows how the above color example can be modelled as a quasi-transitive relation ≥. Color differences overdone for readability. A color ''X'' is said to be more or equally red than a color ''Y'' if the table cell in row ''X'' and column ''Y'' is not empty. In that case, if it holds a "≈", then ''X'' and ''Y'' look indistinguishably equal, and if it holds a ">", then ''X'' looks clearly more red than ''Y''. The relation ≥ is the disjoint union of the symmetric relation ≈ and the transitive relation >. Using the transitivity of >, the knowledge of both {{color|#f10|f10}} > {{color|#d30|d30}} and {{color|#d30|d30}} > {{color|#b50|b50}} allows one to infer that {{color|#f10|f10}} > {{color|#b50|b50}}. However, since ≥ is not transitive, a "paradoxical" inference like "{{color|#d30|d30}} ≥ {{color|#e20|e20}} and {{color|#e20|e20}} ≥ {{color|#f10|f10}}, hence  {{color|#d30|d30}} ≥ {{color|#f10|f10}}" is no longer possible. For the same reason, e.g. "{{color|#d30|d30}} ≈ {{color|#e20|e20}} and {{color|#e20|e20}} ≈ {{color|#f10|f10}}, hence  {{color|#d30|d30}} ≈ {{color|#f10|f10}}" is no longer a valid inference. Similarly, to resolve the original heap variation of the paradox with this approach, the relation "''X'' grains are more a heap than ''Y'' grains" could be considered quasitransitive rather than transitive.