Sorites paradox
[2] A typical formulation involves a heap of sand, from which grains are removed individually.[4] The paradox is so named because of its original characterization, attributed to Eubulides of Miletus.[5] The paradox is as follows: consider a heap of sand from which grains are removed individually.One might construct the argument from the following premises:[3] 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.[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.This paradox can be reconstructed for a variety of predicates, for example, with "tall", "rich", "old", "blue", "bald", and so on.for the material implication connective, so his argument originally ended with)[16] Jonathan Barnes has discovered the conditions for an argument of this general form to be soritical."[17] One may object to the first premise by denying that 1,000,000 grains of sand make a heap.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.[21] Timothy Williamson[22][23][24] and Roy Sorensen[25] claim that there are fixed boundaries but that they are necessarily unknowable.Supervaluationism is a method for dealing with irreferential singular terms and vagueness.It allows one to retain the usual tautological laws even when dealing with undefined truth values.[26][27][28][29] An example of a proposition about an irreferential singular term is the sentence "Pegasus likes licorice".By admitting sentences without defined truth values, supervaluationism avoids adjacent cases such that n grains of sand is a heap of sand, but n − 1 grains is not; for example, "1,000 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 "1,000 grains of sand is a heap or 1,000 grains of sand is not a heap" as a tautology, i.e. to assign it the value true.[30] 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.[31][32] Though the problem remains of where these borders occur; e.g. at what number of grains sand starts being definitely a heap.Another method, introduced by Raffman,[33] is to use hysteresis, that is, knowledge of what the collection of sand started as.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.[34] One can establish the meaning of the word heap by appealing to consensus.Williamson, in his epistemic solution to the paradox, assumes that the meaning of vague terms must be determined by group usage.[35] The consensus method typically 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 opinion.Indeed, if a precise prescriptive definition of "heap" is available then the group consensus will always be unanimous and the paradox does not occur.In the economics field of utility theory, the sorites paradox arises when a person's preferences patterns are investigated.Several kinds of relations were introduced to describe preference and indifference without running into the sorites paradox.Luce defined semi-orders and investigated their mathematical properties;[36] Amartya Sen performed a similar task for quasitransitive relations.Both fallacies cause one to erroneously reject a vague claim simply because it is not as precise as one would like it to be.For example, perhaps the addition of a grain of rice causes the total group of rice to be "slightly more" of a heap, and enough slight changes will certify the group's heap status – see fuzzy logic.
paradoxpredicatesAncient GreekEubulides of MiletusgrainspremisesRepeated applicationsconclusionmodus ponensBertrand Russellfirst-order logicJonathan Barnesmaterial implicationconnectiveCrispin Wrightobject to the first premisePeter UngerA. J. AyerDavid WigginsTimothy WilliamsonSupervaluationismsingular termsvaguenesstautological lawsPegasusfails to refertruth valuevaluationatomic sentencemulti-valued logicprinciple of bivalencefuzzy logicunit intervalhysteresisconsensusprobabilityexpected valuedescriptive linguisticsprescriptive linguisticsutility theoryRobert Duncan LuceComparativepositivetransitivesemi-ordersAmartya Senquasitransitive relationsinformal fallacydistinctcontinuumcounterexampleAmbiguityBoiling frogClosed conceptFuzzy conceptI know it when I see itImprecise languageList of fallaciesLoki's wagerRing speciesShip of TheseusSlippery slopeStraw that broke the camel's backJohn Wiley & SonsCambridge University PressPhilosophical InvestigationsRussell, BertrandThe Australasian Journal of Psychology and PhilosophyShaliziSyntheseAristotelian SocietyWilliamson, TimothyRoutledge Clarendon Pressvan Fraassen, Bas C.The Stanford Encyclopedia of PhilosophyStanford UniversityZadeh, L.A.Information and ControlWikidataAnalysisPeter C. FishburnUniversity of Haifapreference theorySen, AmartyaWayback MachineThouless, Robert H.Pan BooksBlack, MaxBurnyeat, MylesInternational Workshop on Vagueness in Communication (ViC; held as part of ESSLLI)Zalta, Edward N.Stanford Encyclopedia of PhilosophyparadoxesBuridan's bridgeDream argumentEpicureanFictionFitch's knowabilityFree willGoodman'sHedonismLiberalMeno'sMere additionMoore'sNewcomb'sNihilismOmnipotencePrefaceRule-followingTheseus' shipWhite horseZeno'sBarberBhartrhari'sBurali-FortiCrocodileCurry'sEpimenidesFree choice paradoxGrelling–NelsonKleene–RosserPinocchioQuine'sYablo's