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| The '''sorites paradox''' (sometimes translated as the '''paradox of the heap''' because in {{lang-grc|[[wikt:en:σωρίτης#Ancient Greek|σωρίτης]]}} ''sōritēs'' means "heaped up") is a [[paradox]] that arises from [[vagueness|vague]] [[Predicate (grammar)|predicate]]s.<ref name=Allan2009>{{cite book |last=Barker |first=C. |editor-last=Allan |editor-first=Keith |title=Concise Encyclopedia of Semantics |url=http://books.google.com/books?id=3_1snsgmqU8C&pg=PA1037 |year=2009 |publisher=Elsevier |isbn=978-0-08-095968-9 |chapter=Vagueness |page=1037}}</ref> 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? (Or are even no grains at all a heap?) If not, when did it change from a heap to a non-heap?<ref name=Bergmann2008>{{cite book |last=Bergmann |first=Merrie |title=An Introduction to Many-Valued and Fuzzy Logic: Semantics, Algebras, and Derivation Systems |url=http://books.google.com/books?id=zEwNfoAZEGoC&pg=PA3 |year=2008 |publisher=Cambridge University Press |isbn=978-0-521-88128-9 |page=3}}</ref><ref name=Sorensen2009>{{cite book |last=Sorensen |first=Roy A. |editor1=Jaegwon Kim |editor2-last=Sosa |editor2-first=Ernest |editor3-last=Rosenkrantz |editor3-first=Gary S. |title=A Companion to Metaphysics |chapter=sorites arguments |year=2009 |publisher=John Wiley & Sons |isbn=978-1-4051-5298-3 |url=http://books.google.com/books?id=i7PG-Vk824UC&pg=PA565 |page=565}}</ref>
| | If you have the desire to procedure settings immediately, loading files swiftly, however, the body is logy plus torpid, what would we do? If you are a giant "switchboard" that is deficiency of efficient management system and powerful housekeeper, what would we do? If you have send your exact commands to your mind, but the body cannot perform properly, what would we do? Yes! We want a full-featured repair registry!<br><br>Another solution would be to supply the computer program with a hot msvcr71 file. Frequently, once the file has been corrupted or damaged, it might no longer be able to function like it did before thus it's just all-natural to substitute the file. Simply download another msvcr71.dll file within the web. Frequently, the file comes in a zip formatting. Extract the files within the zip folder plus destination them accordingly in this location: C:\Windows\System32. Afterward, re-register the file. Click Start plus then choose Run. When the Run window appears, type "cmd". Press Enter and then type "regsvr32 -u msvcr71.dll" followed by "regsvr32 msvcr71.dll". Press Enter again plus the file ought to be registered accordingly.<br><br>The PC might also have a fragmented hard drive or the windows registry could have been corrupted. It could also be because of the dust plus dirt which should be cleaned. Whatever the issue, you are able to usually find a solution. Here are some strategies on how to make a PC run faster.<br><br>Handling intermittent mistakes - whenever there is a content to the impact which "memory or difficult disk is malfunctioning", you might place inside new hardware to replace the defective part till the actual issue is discovered. There are h/w diagnostic programs to identify the faulty portions.<br><br>Use a [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities]. This might look the Windows registry for 3 kinds of keys which really can hurt PC performance. These are: duplicate, missing, plus corrupted.<br><br>Files with the DOC extension are furthermore susceptible to viruses, yet this is solved by wise antivirus programs. Another problem is that .doc files can be corrupted, unreadable or damaged due to spyware, adware, and malware. These instances will prevent consumers from correctly opening DOC files. This is whenever powerful registry products become practical.<br><br>By restoring the state of the system to an earlier date, error 1721 could not appear in Windows 7, Vista and XP. There is a tool called System Restore which you have to use inside this process.<br><br>What I would suggest is to search on your for registry cleaners. You can do this with a Google search. If you find goods, look for ratings and reviews about the product. Then you are able to see how others like the product, and how effectively it works. |
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| ==The original formulation and variations==
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| ===Paradox of the heap===
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| The word "sorites" derives from the Greek word for heap. The paradox is so named because of its original characterization, attributed to [[Eubulides|Eubulides of Miletus]].<ref>{{Harv|Barnes|1982}}, {{Harv|Burnyeat|1982}}, {{Harv|Williamson|1994}}</ref> The paradox goes as follows: consider a [[Wiktionary:heap|heap]] of sand from which [[grain]]s are individually removed. One might construct the argument, using [[premise]]s, as follows:<ref name=Sorensen2009/>
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| :''1,000,000 grains of sand is a heap of sand'' (Premise 1)
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| :''A heap of sand minus one grain is still a heap.'' (Premise 2)
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| [[Mathematical induction|Repeated applications]] of Premise 2 (each time starting with one fewer grain) eventually forces one to accept the [[logical consequence|conclusion]] that a heap may be composed of just one grain of sand (and consequently, if one grain of sand is still a heap, then removing that one grain of sand to leave no grains at all still leaves a heap of sand; indeed a ''negative'' number of grains must also form a heap<ref>{{cite doi |10.1111/j.1467-9205.2004.t01-1-00230.x}}</ref>).
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| ===Variations===
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| [[File:Color gradient illustrating a sorites paradox.png|thumb|Color gradient illustrating a sorites paradox, any adjacent colors being indistinguishable by the human eye]]
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| 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.<ref name=Allan2009/><ref name=Sorensen2009/> 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.<ref name=Allan2009/> | |
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| This paradox can be reconstructed for a variety of predicates, for example, with "tall", "rich", "old", "blue", "bald", and so on. [[Bertrand Russell]] argued<ref>{{cite journal |last=Russell |first=Bertrand |authorlink=Bertrand Russell
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| |date=June 1923 |title=Vagueness |journal=The Australasian Journal of Psychology and Philosophy |volume=1 |issue=2 |pages=84–92 |issn=1832-8660 |doi=10.1080/00048402308540623 |url=http://cscs.umich.edu/~crshalizi/Russell/vagueness/ |accessdate=November 18, 2009 |ref=Russell1922}} [[Cosma Shalizi|Shalizi]]'s [http://cscs.umich.edu/~crshalizi/Russell/vagueness/ 1995 etext] is archived [http://web.archive.org/web/20080515181625/cscs.umich.edu/~crshalizi/Russell/vagueness/ at archive.org] and [http://www.webcitation.org/5lNl9fNiP at WebCite].</ref> that all of natural language, even logical connectives, is vague; moreover, representations of propositions are vague. However, most views do not go that far, but it is an open question.
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| ==Proposed resolutions==
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| {{Refimprove section|date=June 2009}}
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| {{Original research|section|date=August 2009}}
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| On the face of it, there are some ways to avoid this conclusion. One [[Denying the antecedent|may object to the first premise]] by denying 1,000,000 grains of sand makes a [[dict:heap|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 [[dict:deny|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 |pages=117–154 |url=http://link.springer.com/article/10.1007%2FBF00869568#page-1 |accessdate=19 July 2013}} (Alternative: [http://www.jstor.org/discover/10.2307/20115446?uid=3739672&uid=2&uid=4&uid=3739256&sid=21102521732917 jstor.org])</ref> 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. Or one may accept the conclusion by insisting that a heap of sand can be composed of just one grain, and solely deny the further conclusions regarding zero-grain or negative-grain-number heaps.
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| ===Setting a fixed boundary===
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| 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.
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| However, such solutions are unsatisfactory as there seems little significance to the difference between 9,999 grains and 10,000 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. A more acceptable solution is to call any collection of multiple grains (two or more) a heap, or to call a collection a heap if some grains of sand are supported solely by other grains of sand.
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| ===Unknowable boundaries (or Epistemicism)===
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| [[Timothy Williamson]]<ref>{{cite journal |last=Williamson |first=Timothy |title=Inexact Knowledge |journal=Mind |year=1992 |volume=101 |pages=218–242 |jstor=2254332}}</ref><ref>{{cite journal |last=Williamson |first=Timothy |title=Vagueness and Ignorance |journal=Supplementary Proceedings of the Aristotelian Society |year=1992 |volume=66 |pages=145–162 |jstor=4106976}}</ref><ref>{{cite book |last=Williamson |first=Timothy |authorlink=Timothy Williamson |title=Vagueness |year=1994 |publisher=Routledge |location=London |ref=harv}}</ref> and Roy Sorensen<ref>{{cite book |last=Sorensen |first=Roy |title=Blindspots |year=1988 |publisher=Clarendeon Press}}</ref> hold an approach that there are fixed boundaries but that they are necessarily unknowable.
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| ===Supervaluationism===
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| {{Main|Supervaluationism}}
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| Supervaluationism is a semantics 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 |year=1975 |volume=30 |pages=265–300 |doi=10.1007/BF00485047}}</ref><ref>{{cite journal |last=van Fraasen |first=B. C. |title=Singular Terms, Truth-Value Gaps, and Free Logic |journal=Journal of Philosophy |year=1966 |volume=53 |pages=481–485 |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 |pages=301–324 |doi=10.1007/BF00485048}}</ref>
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| As an example for a proposition about an irreferential singular term, consider the sentence "''[[Pegasus]] likes [[licorice]]''".
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| 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.
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| Similarly, "''1,000 grains of sand is a heap of sand''" may be considered a border case having no truth value, but "''1,000 grains of sand is a heap of sand, or 1,000 grains of sand is not a heap of sand''" should be true.
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| Precisely, 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>.
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| 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.
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| ===Truth gaps, gluts, and many-valued logics===
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| 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 gap]] or as a [[truth glut]].{{Clarify|date=March 2010}} | |
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| 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''.
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| <ref>{{cite journal |last=Zadeh |first=L. A. |title=Fuzzy Sets |journal=Information and Control |year=1965 |volume=8 |pages=338–353}}</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 |ref=harv}}</ref>
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| ===Hysteresis===
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| 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" to a point, even as the actual amount of sand is reduced to a smaller number of grains. For example, let's say 500 grains is a pile and 1,000 grains is a heap. There will be an overlap for these states. So if you are reducing it from a heap to a pile, it is a heap going down until, say, 750. At that point you would stop calling it a heap and start calling it a pile. But if you replace 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 degrees, to prevent instant change of state over and over again.<ref>{{Cite doi|10.1111/j.1467-8284.2005.00558.x}}</ref>
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| ===Group consensus===
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| 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 [[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 views.
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| A group may decide that:
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| *One grain of sand on its own is not a heap.
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| *A large collection of grains of sand is a heap.
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| 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.
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| ==See also==
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| {{colbegin}}
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| * [[Ambiguity]]
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| * [[Camel's nose]]
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| * [[Coastline paradox]]
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| * [[Continuum fallacy]]
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| * [[Eubulides]]
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| * [[False dilemma]]
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| * [[Fuzzy logic]]
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| * [[I know it when I see it]]
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| * [[Milo of Croton#Feats of strength|Milo of Croton]] (on how he was able to lift a bull)
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| * [[Multi-valued logic]]
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| * [[Nirvana fallacy]]
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| * [[Philosophical Investigations]]
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| * [[Principle of bivalence]]
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| * [[Ship of Theseus]]
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| * [[Slippery slope]]
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| * [[Straw that broke the camel's back]]
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| * [[The Englishman Who Went Up a Hill But Came Down a Mountain]]
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| * [[Three men make a tiger]]
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| * [[Vagueness]]
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| * [[Beginning of human personhood]]
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| {{colend}}
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| ==References==
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| {{reflist}}
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| {{refbegin}}
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| *{{cite book |last=Black |first=Max |title=Margins of Precision |isbn=0-8014-0602-1 |year=1970 |publisher=Cornell University Press |location=Ithaca, NY}}
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| *{{Cite book |last=Barnes |first=J. |year=1982 |chapter=Medicine, Experience and Logic |editor1-last=Barnes |editor1-first=J. |editor2-last=Brunschwig |editor2-first=J. |editor3-last=Burnyeat |editor3-first=M. F. |editor4-last=Schofield |editor4-first=M. |displayeditors=4 |title=Science and Speculation |location=Cambridge |publisher=[[Cambridge University Press]] |ref=harv |postscript=<!--None-->}}
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| *{{cite book |last=Burns |title=Vagueness: An Investigation into Natural Languages and the Sorites Paradox |isbn=0-7923-1489-1 |year=1991 |publisher=Kluwer Academic Publishers |location=Dordrecht}}
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| *{{Cite book | first=Myles |last=Burnyeat |authorlink=Myles Burnyeat |editor1-last=Schofield |editor1-first=M. |editor2-last=Nussbaum |editor2-first=M. C. |title=Language and Logos |year=1982 |chapter=15. Gods and heaps |location=Cambridge |publisher=[[Cambridge University Press]] |pages=[http://books.google.com/books?id=M8FjmaH–0V4C&pg=PA315 315–] |ref=harv |postscript=<!--None-->}}
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| *{{cite book |last=Gerla |title=Fuzzy logic: Mathematical Tools for Approximate Reasoning |year=2001 |isbn=0-7923-6941-6 |publisher=Kluwer Academic Publishers |location=Dordrecht, Netherlands}}
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| *{{cite conference |first1=Rick |last1=Nouwen |first2=Robert van |last2=Rooij |first3=Uli |last3=Sauerland |first4=Hans-Christian |last4=Schmitz |title=International Workshop on Vagueness in Communication (ViC; held as part of [[European Summer School in Logic, Language and Information|ESSLLI]]) |publisher=Springer |series=LNAI |volume=6517 |isbn=978-3-642-18445-1 |year=2009}}
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| *{{cite book|last=Sainsbury |first=R. M. |title=Paradoxes |year=2009 |publisher=Cambridge University Press |edition=3rd}}; Sect.3
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| {{refend}}
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| ==External links==
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| *{{sep entry|sorites-paradox}} by Dominic Hyde.
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| *{{cite web |last=North |first=Ryan |title=August 28th, 2009 - awesome fun times! |date=August 28, 2009 |work=[[Dinosaur Comics]] |url=http://www.qwantz.com/index.php?comic=1541 |accessdate=9 June 2010}}
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| {{DEFAULTSORT:Sorites Paradox}}
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| [[Category:Paradoxes]]
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| [[Category:Semantics]]
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If you have the desire to procedure settings immediately, loading files swiftly, however, the body is logy plus torpid, what would we do? If you are a giant "switchboard" that is deficiency of efficient management system and powerful housekeeper, what would we do? If you have send your exact commands to your mind, but the body cannot perform properly, what would we do? Yes! We want a full-featured repair registry!
Another solution would be to supply the computer program with a hot msvcr71 file. Frequently, once the file has been corrupted or damaged, it might no longer be able to function like it did before thus it's just all-natural to substitute the file. Simply download another msvcr71.dll file within the web. Frequently, the file comes in a zip formatting. Extract the files within the zip folder plus destination them accordingly in this location: C:\Windows\System32. Afterward, re-register the file. Click Start plus then choose Run. When the Run window appears, type "cmd". Press Enter and then type "regsvr32 -u msvcr71.dll" followed by "regsvr32 msvcr71.dll". Press Enter again plus the file ought to be registered accordingly.
The PC might also have a fragmented hard drive or the windows registry could have been corrupted. It could also be because of the dust plus dirt which should be cleaned. Whatever the issue, you are able to usually find a solution. Here are some strategies on how to make a PC run faster.
Handling intermittent mistakes - whenever there is a content to the impact which "memory or difficult disk is malfunctioning", you might place inside new hardware to replace the defective part till the actual issue is discovered. There are h/w diagnostic programs to identify the faulty portions.
Use a tuneup utilities. This might look the Windows registry for 3 kinds of keys which really can hurt PC performance. These are: duplicate, missing, plus corrupted.
Files with the DOC extension are furthermore susceptible to viruses, yet this is solved by wise antivirus programs. Another problem is that .doc files can be corrupted, unreadable or damaged due to spyware, adware, and malware. These instances will prevent consumers from correctly opening DOC files. This is whenever powerful registry products become practical.
By restoring the state of the system to an earlier date, error 1721 could not appear in Windows 7, Vista and XP. There is a tool called System Restore which you have to use inside this process.
What I would suggest is to search on your for registry cleaners. You can do this with a Google search. If you find goods, look for ratings and reviews about the product. Then you are able to see how others like the product, and how effectively it works.