| CONCEPTE VAGI SI PARADOXURI SORITEVague Concepts and Sorites ParadoxesVlad VieruMachine Intelligence Research LaboratoryOttawaCANADAe-mail: wladw@hotmail.com "Whatever we may want to say, we won’t probably say exactlythat"Marvin Minski"It is…easy to be certain. One has only to be sufficientlyvague."Charles S. Peirce"Subjectivity is the truth. By virtue of the relationshipsubsisting betweenthe external truth and the existing individual, the paradoxcame into being."Soren Kierkegaard Abstract: The purpose of thisarticle is the study of certain aspects regarding the indetermination introducedby vague concepts and, in particular, the explanation of sorites paradoxes.In the first part of the article, I will show that any theory on vagueconcepts must consider not only the relations between concepts and objects,but also the attitude of the users regarding these relations. Consistentto this point of view, the containing vague concepts will not be regardedas abstract descriptions. We will suggest instead an approach focusingon the individual descriptions belonging to various users at differentmoments in time. Based on this formal model and the associated semantics,we will show that sorites paradoxes are only apparently logic paradoxes.In fact, any sorites judgement speculates on our psychological difficultiesin classifying objects versus one property or another.1. The Nature of Vague Concepts1.1 Terminology UsedDefinitionsThe term "user" shall refer to any human subject, as userof natural language and mental capabilities.We will refer to something as "(In)dependent ofusers" if its existence and/or characteristics are (in)dependentof the users’ perceptions/representations/attitudes of it. By (in)dependencewe will mean the necessary, not the contingent."Real" or "objective" shall refer to anyfact or thing completely independent of users. Exterior realityor world shall refer to all real things and facts."Subjective" shall refer to any fact or thingdependent on users. Accordingly, we shall refer to interiorreality or world."Objective concept" shall refer to any conceptwhose intension is completely independent of users, while exclusively determinedby the exterior reality. Objective property shall refer to a propertywhich a certain thing really possesses or not."Subjective concept" shall refer to any conceptwhose intension is (partially) dependent on users. Accordingly, we shallrefer to subjective properties or predicates.When referring to "users working (subjectively)with concepts", we mean the associations which the users make (subjectively)between concepts and objects."Positive extension" and "Negative extension"of a concept shall refer to the multitude of objects to which the conceptapplies, respectively the multitude of objects to which the concept doesnot apply. By "extension", we implicitly mean "positive extension".RemarksThroughout the article, any person is implicitly considereda user.Throughout the article the word "concept" will beused with preference. In some places, we will refer to concepts,although we would rather use conceptual terms. However, astried to be suggested in Section 1.7, the distinction between conceptsand terms designating concepts is not essential under the circumstances.Similar reasons justify the flexible use of "property" and "predicate".The objectivity or subjectivity of a concept does not dependon the nature of objects to which the concept applies.Although concepts have been classified with respect to theirintensions, the objective or subjective nature of a concept is reflectedin its extension as well. The extension of an objective concept is completelydetermined by the exterior reality, while the extension of a subjectiveconcept is determined, at least partially, by the users’ perceptions andattitudes.We prefer to discuss objective concepts, and not realones, in order not to leave the impression that we are approaching realism.From the very beginning, we consider that at least certain concepts(e.g. numbers, geometrical figures, physical measure units) are objective.For example, the mathematical concept of triangle is objective:something is or is not a triangle, regardless of anyone’s opinion. Thisdoes not mean, however, that triangularity is an universal, in thesense used by the realist theories.When we refer to certain properties or concepts as subjective,we almost always mean, even if not explicitly, that they are partiallysubjective. In other words, the intension of subjective concepts dependson the users’ attitudes only to a certain extent. For some concepts, (e.g.the concept of live being), the users’ subjectivity plays a smallrole. However, as long as a concept implies a certain dose of inherentsubjectivity, we shall refer to the respective concept as being subjective.Our intention is not to defend relativism in this article, just to showthat, to a certain degree, it is inevitable.We will frequently use the term "object". The meaningof the term varies according to the context in which it appears. Thus,"object" can be either an object stricto sensu, or a thingor fact making the object of a discourse or an intentional (i.e. faith,conviction, outlook, etc.). In its most general sense, object shallbe any concrete or abstract entity (case, instance, moment of time) towhich one concept or another can apply.1.2 The Issue ProperTogether with the liar’s paradox or Zenon’s famous paradoxes,the sorites paradoxes are among the most lasting logic "games" ever imagined.Originally from ancient Greece, this type of paradoxes is still an objectof joy and debate for many modern logicians. The name comes from the Greekword soros, meaning "heap". Actually, the best known sorites paradoxremains the paradox of the heap. Initially, it was meant as an exercise:"Can one grain of wheat be described as a heap? No. Can two grains of wheatbe described as forming a heap? No. …Still, we must admit that, at onepoint, several grains of wheat do make a heap. The question is: when doseveral grains of wheat become a heap?". The heap paradox, as we know ittoday, conceals the same problem in the form of a logic reasoning:"A grain of wheat does not make a heap.If a grain of wheat doesn’t make a heap,Then neither do two grains of wheat.If two grains of wheat don’t make a heap,Then neither do 3 grains of wheat.----------------------------------------------------If 9999 grains of wheat do not make a heap,Then neither do 10.000 grains of wheat.Therefore, 10000 grains of wheat do not make a heap."The argument can go on forever, so that no number of wheatis enough to make a heap. Although the logic argument seems flawless, itsconclusion violates the most basic intuition.Formally speaking, the heap paradox can be transcribedas a logic argument with an unconditional premise ("1 wheat grain doesnot make a heap"), a general conditional premise ("If i wheat grainsdo not make a heap, then neither do i+1 wheat grains") and a conclusionlike "10000 wheat grains do not make a heap" (10000 can be replaced withno matter how large a number). The only inference rules used are modusponens and cut.The formal aspects of the paradox will be discussed indetail later on. Right now, we shall only notice that the same type ofreasoning can be applied to a wide range of cases, and is not specificto the heap paradox only. For example, in the same manner it can be shownthat one person having 1 billion hairs is bald, that a person 2.20m inheight is a tall person, etc. Paradoxes of this type form the class ofsorites. The logical reasoning underlying them bears the generic name ofsorites argument. At the same time, the sentences lending themselves tosorites arguments - for example, "x is a heap", "x is tall", "x is bald",etc. - are sorites predicates or sentences.In the following, we shall be referring less to "soritessentences" and more to "vague concepts". Although the two syntagmsare quite close in meaning, the term "vague concepts" is more general.Discussing vague concepts does not imply an explicit reference to soritesparadoxes. However, the problem of vague concepts indetermination and thatof paradoxes overlap to a large extent. This results clearly from the followinginterpretation of vague concepts:"A term, or a concept, is vague if there are particularcases (instances, things, etc.) which do not fall clearly inside or outsidethe range of applicability of the term. At what point exactly does an objectcease being red? How much hair exactly must a man lose to be judged bald?In what manner and to what degree must a person hold theological beliefsand practice the dictates of some religion to be regarded as religious?Such questions admit of no precise answers and the concepts redness, baldness,and religious must be regarded as vague." (Norman Swartz - "BeyondExperience").Norman Swartz suggests as a (pseudo) definition of vagueconcepts the so-called notion of"borderline cases". Borderlinesare those cases in which the applicability of a concept cannot be clearlyaccepted or clearly rejected. For example, a person 1.75m tall representsa borderline case versus the concept of "tall person". A concept’s shadowyarea consists of all its borderline cases. According to several studies,a concept is vague to the extent it possesses borderline cases, namelyto the extent to which its shadowy area is full. Any vague concept’s mainfeature is the impossibility of drawing a firm line between its positiveand negative extension. Thus defined, vague concepts coincide with whatwe called sorites sentences. The essence of sorites paradoxes is preciselythe impossibility of determining the limits within which sorites sentencesapply. Therefore any investigation of the nature of vague concepts willhave as implicit result a certain interpretation of sorites paradoxes.1.3 Types of Possible SolutionsFaced with sorites paradoxes, we can take one of the followingfour positions:Deny the possibility of applying logic to sorites sentencesDeny one or more premisesDeny the logic reasoning validityAccept the paradoxOne answer to the sorites paradoxes evolves from the fundamentalidea that the indetermination is of a linguistic nature. More logicianshave suggested that, given the ambiguity and inaccuracy inherent to naturallanguage, it goes beyond the purpose of logic to deal with linguistic structuresas sorites paradoxes. According to this point of view, sorites paradoxeswould result from an illegitimate attempt to apply the reasoning of classicallogic to vague concepts. In order to eliminate the linguistic indetermination,a number of philosophers and logicians - among them Frege and Quine - proposedthe creation of a so-called ideal language which would have to eliminatefrom the natural language any form of ambiguity and vagueness, enablingany phrase to be subject of logic investigations. Currently, the ideallanguage doctrine was practically abandoned. If we want logic to be a powerfulenough instrument, then its application to the usual language must be possible.Therefore it is generally considered that logic can be applied to naturallanguage, but that the sorites sentence is wrong, for one reason or another.The type (2) answers accept classical logic and semanticsas instruments of study, but consider the mistake to result from one ormore incorrect premises. Since it seems impossible to deny the unconditionalpremise, the error consists in accepting a general unconditional premise.One of the most remarkable approaches of this type is known as epistemology.According to it, there would be a multitude of wheat grains not makinga heap, but which would become a heap once a single wheat grain is addedto it. Epistemologists state that there are precise dividing lines betweenheap and not heap, tall and not tall, red and not red, but that it is uswho do not have the knowledge of distinguishing them. Therefore indeterminationwould be a form of ignorance and, accordingly, sorites paradoxes wouldnot be logical, but epistemological. The main fault of the epistemologicalhypothesis resides in its counter-intuitive nature.Most attempts to explain sorites paradoxes probably sprangfrom a type (3) position, namely from the feeling that there is somethingwrong with the logic reasoning or with the associated semantics. Generally,it is considered that the indetermination of vague concepts representsa semantic phenomenon. That is the reason why logicians focused on theattempt to revise classical semantics and with it the classical bivalentlogic. The result was a relatively large number of non-classical logics(e.g. intuitionism, semantics supervaluationist semantics, paraconsistentlogics) - also called non-standard or deviate logics. They generally operatewith the idea that the simple notions of truth and false - as inheritedfrom Aristotle - are not enough to approach the vagueness aspect. The solutionsuggested is almost always a multivalued logic (usually with 3 values).During the ’60s and ’70s, Zadeh and Goguen developed fuzzy logic with aninfinity of truth values within the continuous interval of [0 1]. The mainprinciple underlying fuzzy logic is the following: there is no absolutetruth or falsehood, but degrees of truth. For example, just as a man canbe bald to a higher or lower degree, so can the sentences to which theword "bald" applies have a higher or lower degree of truth. The adeptsof fuzzy logic hold that the transition from a property to its oppositeis generally continuous and gradual, hence it makes no sense to speak ofan exact threshold between the extensions of the property (i.e. the positiveand negative extensions). In particular, they assert that the transitionfrom a non-heap to a heap takes place by a series of imperceptible qualitativechanges. On a formal level, the paradox of the heap is explained by thefact that the word "heap" cannot be applied to a collection of iwheat grains in the same degree it can be applied to one of i+1grains. Despite the claims to the solution to the problem of imprecisionand the sorites paradoxes, as well as the fantastic success it enjoys,fuzzy logic begs a series of questions. For instance, it is not obviousthat all statements that can be expressed in natural language may be comparedas to their truth. What is more, fuzzy logic operates with an uncountableinfinity of degrees of truth, while there is at most a countable infinityof possible sentences.A last and desperate option is the unconditioned acceptanceof the paradox. We can end up by accepting the idea that neither 10000nor any other number of grains of wheat make up a heap. Setting aside theextremely counter-intuitive character of such a conclusion, we are leftwith an additional problem. Any sorites paradox admits a positive and anegative version. For instance, the paradox of the heap can be reformulated,in its positive version, as follows: Ten thousand wheat grains forma heap. If 10000 grains form a heap, so do 9999. Finally, we reach theconclusion that a single grain of wheat forms a heap as well. Thereexits no special reason for accepting the positive version of a paradoxover its negative one, or the other way around. Therefore it must be concludedthat words such as "heap", "bald", "tall" apply either everywhere or nowhereat all. In any case, their use is either redundant or self contradictory.Such a vision of natural language would be, we must admit, at least bizzare.Taking exception from all these attempts to reformulatelanguage, a small group of philosophers consider that objects themselvesare vague. We want to exclude this metaphysical supposition from the scratch.Throughout the paper we shall use the fundamental premise that outsidereality is precise. We do not believe there is a simple proof, or, indeed,any proof of this. Moreover, we do not know if the meaning of the statementoutside reality is precise can be made perfectly clear. In any case,the premise of an outside reality does not play a crucial role in thispaper. This is not essential for supporting the theory proposed,but for refuting the hypothesis of ontological indetermination asan explanation for sorites paradoxes. Therefore we do not believe thatall readers believing in an imprecise reality should leave off at thispoint. I invite them to read further, at least those who do not believethat our problems in dealing with vague concepts are caused by a chaoticand vague reality.As for the other approaches: we do not believe that thesolution to the problem of indetermination or to the sorites paradox canbe found in any of the previously mentioned ones. In our view, the "key"can be found only through an investigation that covers linguistic, semantic,logical, psychological and, not least, pragmatic aspects of the phenomenon.1.4 Are Vague Concepts ReallyVague?In the previous Section we have explicitly underlinedthat objects are not intrinsically vague. What can be said, however, ofvague concepts? Are vague concepts really vague?Although rhetorically formulated, this question raisesa very important point: is imprecision an essential feature of certainobjects, or is it due to the way we use these concepts? In otherwords is a concept vague because it does not admit a precise applicationin certain cases, or because we do not know precisely how to apply it?Do borderline cases really exist, or are they just cases in which our imperfectperception of reality prevents us from discerning the applicability ofconcepts? Is classical logic really being tested, or is it just our abilityto apply it to borderline cases? Is the extension of vague concepts reallyimprecise, or is it that we just do not know how to delimit it?This series of questions could continue. But all thesequestions have, in the end, a common denominator. They question, on theone hand, the relation between concepts and objects, and on the other therole played by the user in this "equation". The following Sections willspecifically address these problems. The declared goal is that, leavingaside the discussion of the epistemological hypothesis, the side effectsof the investigation should reflect themselves in a plausible explanationof the phenomenon of imprecision.1.5 Objective Concepts VersusSubjective ConceptsThe borderline conception illustrated by the quotationfrom Norman Swartz does not point out the essence of imprecision, but onlyits symptoms. What does it really mean to say "particular cases whichdo not fall clearly inside or outside the range of applicability of theterm"? This apparently unproblematic phrasing allows two distinct (thoughnot disjoint) interpretations:I1: Users are unable to decide whether acertain vague concept can be applied in some special cases.I2: The possibility of applying a vagueconcept is inherently undecidable in some special cases.Interpretation I1 only expresses an unquestionable fact.Besides being intuitively obvious, it is empirically confirmed by the followingremarks referring to any vague concept:at any time, for any limit case, some users consider theconcept applicable, and others do not;users change their opinion in time when dealing with limitcases.These remarks prove, for now, only that we do notknow how to establish the applicability of a concept to borderline cases.We can say that the way in which we apply vague words to objectsis partially subjective.But, leaving aside our incapacity, are the intensionsand extensions of the concepts clearly delimited? Can we speak of a realapplicability of concepts to things? Does it make sense to say that, inreality, any thing is either tall or not tall, either red or not red, etc.irrespective of our attitudes? The answer depends on whether we acceptor reject the second interpretation, which states that imprecision residesin the intrinsic nature of the relation between concepts and objects, independentlyof the users.The two interpretations are not independent, but the differencebetween them is essential. I1 could be considered a weak version of theinterpretation of vague concepts, while I2 could be a strong one. The relationbetween them is I2 ® I1.The reciprocal implication is, however, invalid.Indeed, if the applicability of vague concepts is, atleast in some cases, meaningless, then users can deal with them only subjectively.But from our inability of finding an objective way of settling the applicabilityof a concept it does not follow that this applicability is not, in fact,uniquely determined by outside reality.There is an analogy with the terms decidable andundecidable. When we wish to prove or refute a statement insidea formal system, it sometimes happens that we do not know how to do it(e.g. trying to prove Goldbach’s conjecture). This does not mean that thestatement is inherently undecidable in the respective formal system. Onthe other hand, if the system is incomplete, there are inherently undecidablesystems. Any attempt to prove or refute such a statement inside the formalsystem is in principle doomed to failure.Despite the analogy, there are remarkable differences.Problematic formal statements generally refer to the universality of aproperty - they state that a certain property is applied to every elementof a certain infinite set. Our difficulty comes from the absence of a methodto show that the property holds for all elements. Still, we can,in general, check the statement for a practically unlimited number of elements.When dealing with vague concepts, things stand differently. Not only dowe not possess a recipe for their general use, but we do not even knowhow to test them objectively in borderline cases.Returning to our two interpretations. I1 is practicallya truism, I2 is debatable. We may choose to accept both, or just acceptI1 and reject I2. Depending on our choice, we shall subscribe to one oftwo mutually incompatible points of view:Io : Vague concepts are objectiveIs : Vague concepts are partially subjectiveThe choice between these two alternatives, which we callthe objectivist hypothesis and the subjectivist hypothesis respectively,holds, in our opinion, the key to understanding vague concepts and thesorites paradoxes.One major difficulty is presented by the very classificationof concepts in objective and subjective. The possibility of actually makingthis distinction and the criteria to use in making such a distinction arequestionable. This is, in fact, the source of many open philosophical problems.However, we should not draw the conclusion that any attemptto investigate the nature of vague concepts is doomed to failure. Quitethe opposite: anyone interested in this question will answer it. But theanswer, we believe, will not have universal validity. In last instance,any answer will represent the choice of one paradigm over another.In the next Sections we shall attempt a personal choicefor the subjectivist point of view.1.6 A Pragmatic Argumentin Favor of the Subjectivist HypothesisBefore going into deeper arguments, we would like to pointout the simplest argument in favor of the objectivist thesis, viz. "Occam’sRazor": the more simple hypothesis is the likelier . In this case, giventhat users apply vague concepts subjectively, it is likelier that theseare themselves subjective.But in our opinion there are a lot more arguments in favorof the subjective thesis. In the following we shall present an essentiallypragmatic argument, inspired by C. S. Peirce. Going against traditionalmetaphysics, Peirce says: "It must be shown that almost every sentenceof ontological metaphysics is either meaningless talk - in which a wordis defined through another, and this one by others, without ever reachinga real concept - or is simply absurd." Against such a sterile approach,Peirce suggested an "operational" theory of meaning, making practical verificationinto the criterion of truth. His point of view is that the entire meaningof an idea can be revealed by considering the practical implications ofthat idea."Our idea of anything is our idea of its sensibleeffects; and if we fancy that we have any other we deceive ourselves, andmistake a mere sensation accompanying the thought for a part of the thoughtitself. […] Consider what effects, which might conceivably havepractical bearing, we conceive the object of our conception to have. Thenour conception of those effects is the whole of our conception of the object."(C.S.Peirce - "How To Make Our Ideas Clear")In order to see how Peirce’s principle can be appliedto vague concepts, let us remember the weak version of their interpretation:users do not know how to apply the term in some special cases. Thismeans that users have no objective criteria to determine the extensionof vague concepts. To be more precise, this means that when consideringthe applicability of a vague concept to a borderline case, we are confrontedwith one of the following problems:we cannot determine it a prioriwe cannot determine it by empirical methodswe cannot establish it by an algorithmWe doubt anyone can seriously reject any of these problems.The first states, for instance, that we cannot speak ofa priori knowledge about the applicability of the concept "tall"to a man of height 1.75m. One should remark that we do not exclude thepossibility of an interesting theory about tallness that starts by takingany man of 1.75m is tall as an a priori truth; we just donot think that such a theory could be accepted as a satisfying conceptualanalysis of the word "tall".The second remark is a generalization of the fact thatno one has ever imagined - or will ever imagine - an empirical test toestablish whether a 1.75m tall man is "tall". By empirical test we referto observation and experiments performed on the object, not to the moreor less accidental way that users report how objects seem to them relativeto one concept or another. It is perfectly possible to conduct a surveyamong the users in order to determine their opinion on the applicabilityof a certain concept to a certain object at a particular time, but theresults of this survey would show us not the relation between the objectand the concept, but the attitudes of users to this matter. This paperattempts to show that investigating the attitude of the users is the onlypractical way of discussing the applicability of vague concepts, and thatit is, therefore, the only one that is relevant. But for now, we are contentwith underlining that the extension of a vague concept cannot be determinedsolely by studying facts of the outside world.Finally, in the last remark we take the algorithm to bedescribable using only precise concepts. It seems obvious that in thiscase there is no algorithm that can tell us whether, for instance, a 1.75mtall man is tall or not.We have explained in detail what was meant by lack ofobjective criteria in order to point out two things:the objectivist thesis stands no chance of a practical verification;adopting the objectivist position can have no practical effecton the way we operate with vague concepts.For these reasons, once we apply Peirce’s principles thesubjectivist hypothesis imposes itself. It is absurd to consider vagueconcepts objective as long as no user has objective criteria for theirapplicability.1.7 Is Red An Objectiveor A Subjective Concept?In the following we will try to anticipate and refutean objection that might be made against the previous argument.When talking about the lack of objective criteria we took"tall" as an example. But, one might reply, there are objectivecriteria for some vague concepts, and therefore our generalization wasfaulty. Consider, for instance, the concept of red.An objectivist could consider that applying a soritesparadox to red is wrong precisely because it presupposes redto be a partially subjective concept. The objectivist might support thisby quoting from optical physics, which states that behind the sensationof redness lies a precise physical property: a wavelength between620 and 760 micrometers. In other words, an object is either red or notred depending on the wavelength of the light it reflects. If this wavelengthis 759 m m it is red, while if it is 761 mm it is not, no matter what the user thinks of it.In our opinion, this line of argument is essentially flawed,because it supposes that the concept of red with which we operate in everydaylife is, or should be, identical to the one used in optical physics. Toshow why this idea is false, we shall invoke a completely reasonable pointof view:"It is not essential that we try to get very clearwhat a concept is. That exercise may be left for books on the philosophyof language and of mind. Let me say only this: persons have a concept of- let us take as an example - redness, if they are able, for most part,to use correctly the word "redness" or some other word, in some other language,which means pretty much what "redness" does in English." (Norman Swartz- "Beyond Experience").Apparently, discussing concepts from the point of viewof their use only troubles the waters. Objectivists will say that the useshould be judged exclusively in reference to the state of facts in theoutside world. On the other hand, subjectivists will hold that the standardof correctness in use is to be found not only in the outside world, butalso in our perception of it. The weak point in Swartz’s statement is thatit assumes we know what the correct use of a word is, which is unfortunatelynot the case. It would seem that the problem has only been rephrased interms of correct use of words.In our opinion, the only way out is, again, judging thecorrectness of the use in a pragmatic fashion. In other words, give upthe attempt to search, or impose, theoretical criteria independent of theusers for the correctness of the use of words. The approach I suggest issummed up in the following pragmatic principle: "Whatever works is correct."Natural language works fine just the way it is. It fulfils its essentialmission of facilitating the communication between its users. We can thereforeassume that we use natural language correctly, at least most of the time.Thus, the correctness of the use of words must be lookedfor in the capacity of the users to communicate something through them.This is independent of an agreement on what is being communicated. Forinstance, if one says a pie is tasty, everyone understands what is meant,irrespective of whether they agree that the pie is tasty or not. The conceptof "tasty" is clear, what is questionable is its applications to particularcases.Let us return to the color red. There are two differentconceptions:red designates a subjective visual experience;red designates an objective physical property.Let us take a user, say John, who says "I’m eating a redapple". His statement is perfectly clear. All users will understand thatJohn’s apple looks in a certain way. That is, the apple causes a specificvisual sensation. This is the only plausible meaning of John’s statement.It would be absurd to interpret John’s message as "the apple I’m just eatingreflects light with a wavelength between 620 and 760 mm". In fact, neither John nor anyone else need know anything about wavelengthsat all.We want to make ourselves perfectly clear. Even if everybodyknew optical physics, no one had ever applied, or would ever apply, physicalmeasurements in order to use "red" correctly. The property of wavelengthsis simply irrelevant in understanding and using the phrase "red apple".Red is essentially a subjective sensation, not an objective property.Correspondingly, the use of the word red in phrases suchas "Hubble’s red shift" supposes the understanding of it as an intervalof wavelengths. In such a context, the interpretation of red as a visualsensation completely loses its relevance. Once we have decided to representby "red" a set of wavelengths, it no longer matters whether we have ornot a visual representation of red. The fact that, as it happens, we dohave such a representation, affords a very intuitive interpretation. Butthe only meaning red has in such a specialized situation is that of a qualifierfor a set of wavelengths.It follows that in certain contexts we understand redto mean a visual sensation, and in others to be a set of wavelengths. Thereis, however, no context that requires us to understand red as both. Thekey to understanding red is to be found, always, in one interpretationor the other. We can safely say that, in this sense, the two meanings ofred are logically independent.As such, it is useless and counter-productive to referto red as a unique concept. We can talk about two distinct concepts: physicalred and sensorial red respectively. While physical red’s extensioncan be clearly delimited from that of physical non-red, the same cannotbe done for sensorial red. That is why using the wavelength definitionof red does not solve the sorites paradox applied to colors.1.8 The Failure of the ObjectivistThesis. Intermediary ConclusionsWe have seen that physical measurements, no matter howprecise, cannot quench our fascination with the impossibility of establishingan exact boundary for vague concepts. That is why the inherent indeterminationin dealing with vague concepts, besides being logical or linguistic, isabove all of psychological nature. Any attempt at solving the sorites paradoxesby examining only the outside world, or by reforming language, is doomedto failure. Objectivists can prove things about outside reality and aboutour theoretical possibility of representing it objectively, but they cannotprove that there is a unique way - lacking any imprecision - of thinking,that is, of representing reality in the mind. The simple fact thatwe are able to imagine vague concepts (such as tall, red, heap) which areinherently subjective and partially dependent on our mind is a proof ofthe objectivists’ failure.In conclusion, the objectivists have a flawed vision ofvague concepts and as a consequence offer a false solution to the soritesparadoxes. In our opinion, any semantics of vague concepts must take intoaccount not only outside reality, but also inner reality, i.e. the attitudesand conceptions of the users. In the second Part of this paper we attemptto show the implications of this fact at the level of logical formalism.2. The Model of SubjectiveDescriptions2.1 IntroductionBefore studying the purely formal aspects of sentencesabout vague concepts, it is necessary to answer the following question:what does it mean for a sentence to be true? The Tarskian theoryof correspondence-truth ("The snow is white" if and only if the snowis white) cannot be used here, since vague concepts are partially subjectiveand therefore it makes no sense to say, for instance, that "John is tall"is true if and only if John is in reality tall.In the case of vague concepts, the criterion of truthdefined by the relation between concepts and objects must be replaced byanother: the attitude of the users regarding the relation betweenconcepts and objects. To this end, we shall give a formal model built onthe notion of subjective description.2.2 Descriptions and Bets(Pseudo-)definitionsA subjective description is a description that reflectsthe attitude of the user, at a certain moment, towards the applicabilityof a predicate to a specific object. Let P be a predicate: we call P-descriptionany description referring to the relation between an object x and a predicateP.Say that user u gives a positive (negative) P-descriptionof object x at time t if u considers at time t that x has (has not) propertyP. Alternatively, say that x elicits a positive (negative) P-descriptionfrom u at time t.RemarksSuppose that users decide the applicability of concepts toobjects on the basis of bivalent logic, i.e. a description can be eitherpositive or negative. This is simply a working hypothesis adopted for thesake of clarity. The formal model and its associated semantics can be extendedwith little effort to any multivalued descriptions.Users give descriptions practically all the time. When saying"I am eating a red apple" one associates the object one is holding withthe word apple, then describes the object apple by using the word red andfinally associates the action on the thus described object with the verbalphrase "am eating".There is no one-to-one relation between descriptions andsentences. Not every sentence expresses a real description, and not everymental description is communicated.In order to gain an intuition of descriptions, we can imaginethat any P-description given by a user u of an object x is equivalent toa bet on whether concept P applies to object x. We denote the bet by ?[P(x)]and the positive P-description by True[P(x)] and the negative P-descriptionby False[P(x)]. We assume that u chooses the version that best reflectshis attitude; if x is a borderline case, u does not have a well definedattitude towards either description, but he has to make a choice.I take the moment u calls out his choice to be the moment the descriptionis made. For instance, let us take as P the predicate "tall". If u givesa positive P-description of John, then he is betting on the sentence "Johnis tall", denoted by True[P(John)].The betting interpretation of P-descriptions allows us toconsider simultaneous P-descriptions. For instance, u can take part simultaneouslyin two bets ?[P(x)]and ?[P(x’)]. The results of these bets will naturallysatisfy certain compatibility conditions (for instance if John is sensiblyshorter than Robin, it would be absurd for u to bet simultaneously on thesentences "John is tall" and "Robin is not tall").NotationsConsider an arbitrary tuple (P, u, x, t). I shall usethe following notations:P (u, x, t) if u gives at time t a positive P-descriptionof object x~P (u, x, t) if u gives at time t a negative P-descriptionof object x#P (u, x, t) if u gives at time t no P-description ofobject xRemarksUsing the betting interpretation:P (u, x, t) means that u bets on True[P(x)]~P (u, x, t) means that u bets on False[P(x)]#P (u, x, t) means that u takes no part in bet ?[P(x)]Notations P (u, x, t) and ~P ( u, x, t) refer to actual descriptions,while #P (u, x, t) shows that it is meaningless to ask ourselves what ufeels about applying P to x at time t. Actual descriptions are subjectto the laws of bivalent logic (as per our assumption), but in order tofaithfully describe the attitudes of users towards relations between conceptsand objects we need this kind of trivalent logic.For instance, the law of non-contradiction remains the same~ [ P (u, x, t) & ~P (u, x, t) ]But the law of the excluded middle is changed toP (u, x, t) v ~P (u, x, t) v #P (u, x, t)DefinitionTwo P-descriptions are identical P (u, x, t) = P (u’,x’, t’) iff:( P (u, x, t) & P (u’, x’, t’) ) v( ~P (u, x, t) & ~P (u’, x’, t’) )Otherwise the two P-descriptions are opposed:P (u, x, t) ¹ P (u’,x’, t’)RemarksIn particular, it is possible to have: u = u’ and/or x =x’ and/or t = t’.The identity of P-descriptions is not concerned with theidentity of users or moments of time. For instance the descriptions expressedby "Milk is white" and "Snow is white" are identical because milk and snoware described in the same way relative to the predicate "white".DefinitionObject x elicits a stable positive (negative) P-descriptionfrom user u if u never gives a negative (positive) P-description ofx.We use the notation:P (u, x) if x elicits a stable positive P-descriptionfrom u~P (u, x) if x elicits a stable negative P-descriptionfrom u#P (u, x) if x does not elicit a stable P-descriptionfrom u.RemarksUsing the betting interpretation:P (u, x) means that u always chooses True[P(x)]~P (u, x) means that u always chooses False[P(x)]#P (u, x) means that u does not always make the same choice.Formally, the conditions for positive stability, negativestability and instability can be written as follows:P (u, x) « [ ("t) ( #P (u, x, t) vP (u, x, t) ) ]~P (u, x) « [ ("t) ( #P (u, x, t) v~P (u, x, t) ) ]~P (u, x) « [ ($t) ($ t’) ( (t ¹t’) &P (u, x, t) & ~P (u, x, t’) ) ]In general, the attitude of a user towards the relation betweena predicate and an object changes in time. That is why P-descriptions arevalid only for moments in time.Instead, stable P-descriptions are invariantin time.An object may elicit a stable P-description from one user,but not from another. On the other hand, there are objects that elicitstable P-descriptions from all users. The next definition refers to theseobjects and their associated P-descriptions.DefinitionObject x elicits a universal positive (negative) P-descriptionif no user ever gives a negative (positive) P-description of x.We use the notation:P (x) if x elicits a universal positive P-description~P (x) if x elicits a universal negative P-description#P (x) if x does not elicit a universal P-description.RemarksIn the betting interpretation:P (x) means that no one ever bets on False[P(x)]~P (x) means that no one ever bets on True[P(x)]#P (x) means that there exists at least one user who doesnot always choose the same bet as the others.Formally, we can write the following relations:P (x) « [ ("u) P (u, x) ]~P (x) « [ ("u) ~P (u, x) ]#P (x) « [ ($u) ($ u’) ($ t) ($t’)( P (u, x, t) & ~P (u’, x, t’) ]Universal P-descriptions are specially important: they representthe formal instrument needed to avoid the trap of an absolute relativism.They are statements that transcend the quality of being simple unlastingopinions. Therefore, we could return to Tarski’s truth as a correspondencetheory, and say that truth is in the correspondence of the statement toreality. But we will also add an empirical criterion: a statement agreeswith reality when there is a consensus on this matter. Once this pragmaticinterpretation of truth is established, P-descriptions are equivalent totrue statements. The sentence "10000 grains of wheat make up a heap" isno longer a subjective description reflecting an individual opinion, buta universally valid statement. The philosophy is : since everyone considersthat 10000 grains of wheat make up a heap, that we can simply considerthat 10000 grains of wheat do, in fact, make up a heap.2.3 Dissonance and IndiscernabilityDefinitionWe call two objects x and x’ indiscernable relative topredicate P if whenever a user gives two simultaneous P-descriptions forx and x’ they are identical. Otherwise, we call x and x’ discernable relativeto P.RemarksFormally, the indiscernability condition is written as follows:Ind (P, x, x’) « { ("u) (" y, y ¹x, y ¹ x’) ("t) [ #P (u, y, t) ® ( P (u, x, t) = P (u,x’, t) ) ] }Two objects can be discernable relative to a property, butindiscernable relative to another. For instance, two twin brothers mightbe indiscernable relative to "tall" but discernable relative to "generous".This also serves to show that indiscernability is not always sensorialin nature.Indiscernability is a binary tolerance relation (i.e. reflexive,symmetrical and transitive).Indiscernability appears to be a vague notion. Supposingwe define a distance between objects, we cannot find the maximum distancethat satisfies:(" x)("y) [ ( dist (x, y) < d ) ®Ind (P, x, y) )Indetermination is not semantical, but epistemological.The limits of indiscernability cannot be set a priori because,although it is expressed as a logical relation, indiscernability is essentiallydefined on empirical grounds.The axiom of indiscernable objectsIf x and x’ are two indiscernable objects relative toP, they cannot elicit opposite stable P-descriptions from any one user. RemarksFormally, the axiom of indiscernable objects can be written:Ind (P, x, x’) ® { ("u) ~[ ( P (u, x) & ~P (u, x’) ) v ( ~P (u, x) & P (u, x’) ) ] }Although very natural, this axiom cannot be proved (thereis no axiom that says that two indiscernable objects have to be consideredsimultaneously at least once).DefinitionAn empirically continuous series relative to a predicateP is any ordered collection of objects X having the following property:any two consecutive objects of X are indiscernable relative to P.RemarksLet X = {x1, x2,…,xn} bean ordered collection. The fact that X is an empirically continuous seriesrelative to P may be written formally:ECS (X, P) « [ ("k, 0<k<n) Ind (P, xk, xk+1) ]DefinitionA slippery slope relative to P is any collectionX = {x1, x2, …, xn} satisfying the followingtwo conditions:i) it represents an empirically continuous series relativeto Pii) x1 elicits a universal negative (positive)P-description and xn elicits a universal positive (negative)P-description.RemarksLet X = {x1, x2,…,xn}. Thefact that X is a slippery slope can be formally written as:SS (X, P) « { ECS (X,P) & [ ( ~P (x1) & P (xn) ) v ( P (x1)& ~P (xn) ) ] }Sorites paradoxes are built on the grounds of a relationbetween objects forming a slippery slope (the sorites argument is, in fact,also called the slippery slope argument). For instance, the series of quantitiesmade up of one, two, … ten thousand grains of wheat is a slippery sloperelative to the property of being a heap.DefinitionsA P-classification of a set X at time t is the operationthereby a user classifies the objects of X at time t by predicate P.A P-classification results in two sets X1 andX2 such that:X1 U X2 = X(" xk ÎX1) P (u, xk, t)(" xl ÎX2) ~P (u, xl, t)Subsets X1 and X2 are called equivalenceclasses relative to P.RemarksIn the betting interpretation, the operation of P-classificationis equivalent to taking part simultaneously in all the bets ?P[x], withx Î X. The two equivalence classes aremade up of all the objects xk on which the user decides to bet True[P(xk)],respectively of all the objects xl on which the user decidesto bet False[P(xl)].Formally, we can describe a P-classification of X by u attime t as: (" x Î X) ~#P(u, x, t).DefinitionsConsider a user u and two objects x and x’. Supposingthat u gives simultaneous P-descriptions for the two objects, we call theP-descriptions dissonant if the two objects are indiscernable. Otherwise,we call them consonant.A P-classification of a set X is consonant if all P-descriptionsof objects in X are mutually consonant. Otherwise, the P-classificationis dissonant.We shall say that u’s attitudes at time t are consonant(or that u is consistent at time t) if all the P-descriptions given byu at time t are consonant. Otherwise, we shall say that u’s attitudes attime t are dissonant (or that u is inconsistent at time t).We use the notation:Cons (u, t) if u is consistent at time t~Cons (u, t) if u is inconsistent at time tRemarksIn the betting interpretation, dissonant descriptions correspondto cases when the user bets simultaneously on True[P(x)] and False[P(x’)],or False[P(x)] and True[P(x’)]. In this case, u’s bets are practicallymutually contradictory.The consonance condition can be formally written :Cons (u, t) « [ (Vx)(Vy) ( Ind (x, y) & ~# P (u, x, t) & ~#P (u, y, t) ) ®( P (u, x, t) = P (u, y, t) ) ]The consonance condition is practically the principle ofidentity of indiscernable objects applied to simultaneous descriptions.The request for simultaneity is important, for there is no contradictionin applying a vague concept in a different way to a borderline case ontwo different occasions (in fact, as already discussed, this is an essentialfeature of vague concepts). It is important to distinguish between consonantattitudes and stable attitudes.2.4 The Standard Version ofthe Paradox of the HeapWe now have all the formal instruments necessary to returnto the sorites paradoxes. We shall start by stating once more the paradoxof the heap:"10000 grains of wheat make up a heap.If 10000 grains make up a heap,then so do 9999.--------------------------------------------------If 2 grains make up a heap,then so does one.Therefore one grain of wheat makes up a heap."Let us denote by xk a collection of kgrains of wheat and by P the property of making up a heap. Formally, theparadox can be written:P (x10000)P (x10000) ®P (x9999)P (x9999) ®P (x9998)-------------------------P (x2) ® P (x1)===============P (x1)The sorites argument is extremely puzzling, because ofthe evidence of the premises and the simplicity of reasoning, on the onehand, and the enormity of the conclusion, on the other.Despite appearances, the sorites argument is not correct.Its error consists in implicitly adopting an objectivist interpretationof the concept of heap. When it says "k grains form a heap" it implicitlyassumes that this statement is true independently of individual opinions.Or, we have seen that the truth as correspondence theory is not valid forvague concepts such as "heap". Vague concepts are partially subjective.We cannot consider that "k grains make up a heap" is an absolutely validfact. "Heap" is, after all, only a label that someone attaches to a collectionof grains at some time.On the other hand, the formal transcription of the argumentallows its interpretation by way of the model of subjective descriptions.The ordered collection X = {x1, x2, …, x9999,x10000} forms a slippery slope. By P(xk), we mean that all theusers bet on True[P(xk)] ). Reading the sorites paradox in themodel of descriptions does not eliminate the paradox, it just restatesit. It continues to start from a true premise ("all users always describe10000 grains of wheat as a heap") and reaches a false conclusion ("allusers always describe one grain of wheat as a heap"). The paradox is builtexclusively through logical relations between universal P-descriptions(a fact not surprising if thinking that universal P-descriptions representthe semantic equivalent of the objective truth).In order to discover the logical flaw of the argumentit is sufficient to analyze its fundamental premise:If x and y are indiscernable relative to P and if xelicits a positive universal P-description, then y also elicits a positiveuniversal P-description.Using the betting interpretation this can be restatedas:If x and y are indiscernable and if all users alwaysbet on True[P(x)], then all users always bet on True[P(y)]No matter how counter-intuitive it may seem, this hypothesisis false. The statement will be rigorously accounted for, both empiricallyand logically, when giving the theorem of stability thresholds. For now,we will just remark that the classical version of the paradox is flawedby the assumption that any statement is or is not in agreement with reality,irrespective of what the users may think of. We can agree on this onlythere where precise conventions exist regarding the correspondence betweencertain concepts and the outside world. No such precise conventions existfor vague concepts. There are of course implicit rules governing the useof vague concepts which guarantee the viability of the communication betweenthe users, but they are ad hoc, and their interpretation, at leastfor borderline cases, is left to the subjective appreciation of the users.2.5 The Subjectivist Versionof the Paradox of the HeapThe previous explanation, as good as it might appear,lacks an altogether convincing substance. Our feeling is that the soritesargument keeps inside a more general reasoning paradigm than ever guessedbefore. This fact made us venture not too far and keep close to what wecalled the classical version of the sorites paradox, i.e. an argumentsolely accountable in terms of universal P-descriptions.We shall now try to study how the sorites reasoning maybe extrapolated from universal P-descriptions (i.e. objective descriptions)to arbitrary P-descriptions (i.e. subjective descriptions). Thisnew form of the sorites argument was called the subjectivist version.Consider, as above, predicate P and the slippery sloperelative to P {x1, x2, …,x10000}. Additionally,consider a user u and a moment of time t. Assume that at time t, u wishesto simultaneously describe a grain of wheat and a collection of 10000 grainsof wheat. If u describes the 10000 grains as making up a heap, then thesubjectivist version "proves" that u will also describe one grain of wheatas making up a heap! Here is how the argument goes:At time t, u considers that 10000 grains of wheat makeup a heapIf u considers at time t that 10000 grains make upa heap, and if u is consistent at time t, then u considers at time t that9999 grains make up a heapIf u considers at time t that 2 grains make up a heap,and if u is consistent at time t, then u considers that one grain of wheatmakes up a heap.Therefore either u considers at time t that one grainmakes up a heap, or u is inconsistent at time t.Although the subjective character of vague concepts istaken into account, the new version of the sorites argument reaches a conclusionno less paradoxical: in order to be consistent, a user cannot describesimultaneously two quantities of grains but identically, otherwise hisattitudes are dissonant. This is obviously absurd. Try as we may, we couldcannot find no dissonance in our decisions of describing a grain of wheatas not being a heap and 10000 grains as being one.Any sorites paradox in subjectivist version uses the followingpremise:"If x and y are two indiscernable objects, and u givesat time t a positive P-description of x, then u is consistent only if theP-description that u gives of y is also positive."The betting interpretation runs as follows:"If x and y are indiscernable objects, and u bets attime t on True[P(x)], then u is consistent only if he bets at time t onTrue[P(y)] "Although it may seem correct, the premise assumes impliciltythat if u gives at time t a P-description of x, then u must give simultaneouslya P-description of y.Nevertheless this hypothesis is wrong. Indeed, the factthat the user has at time t an attitude towards an object x is independentof whether the user has an attitude toward any other specific object. Inparticular, if u gives at time t a P-description of object xk,then u may or may not give, at the same time t, a P-description of objectxk-1 (i.e. the fact that u takes the bet ?[P(xk)]does not entail the fact that he also takes the bet ?[P(xk-1)]).All which consistency requires is that if u contingently has two simultaneousattitudes these must conform to certain compatibility conditions.Assume that we have a grain of wheat on our right and10000 grains on our left, when saying "there is a heap to my left and nota heap to my right" we only describe two entities. We have no attitudetowards the application of the concept "heap" to other quantities of grains.In particular, we have no attitude toward the quantities x2,x3, …, x9999. At time t, only the problem of describingquantities x1 and x10000 exists for us.These remarks find a perfect correspondence at a formallevel. The subjectivist version of the paradox of the heap is based onthe following reasoning:Cons (u, t)(" k, 1<k<10.000) Ind(xk, xk+1)P (u, x10000, t)P (u, x10000, t) ®P (u, x9999, t)--------------------------------------P (u, x2, t) ®P (u, x1, t)=======================P (u, x1, t)Formally, the consonance condition is written:Cons (u, t) ® [ ( Ind (x,x’) & ~#P (u, x, t) &~#P (u, x’, t) ) ®( P (u, x, t) = P (u, x’, t) )The problem is that in the given situation the conditions~#P (u, xk, t), 1<k<10000 are not satisfied, so that theconditionals P (u, xk, t) ® P(u, xk-1, t) are invalid. The P-description of object x10000at time t does not have to be identical to the P-description of objectx9999 at time t for the simple reason that, at time t, the usergives no P-description of object x9999. Therefore modus ponenscannot be applied and the entire formal edifice crashes.When u classifies a grain as not being a heap and 10000grains as being a heap, he gives a P-description of the collection thatholds only objects x1 and x10000. Since x1and x10000 are discernable relative to the concept "heap", theirclassification in opposite equivalence classes is quite consonant and natural.That is why the subjectivist version of the paradox of the heap is justas fallacious as the classical one.2.6 The Classification ofSlippery SlopesIn the previous Section we have shown that giving oppositedescriptions to objects x1 and x10000 implies nodissonance. But what happens if we want to simultaneously classify allobjects of the set {x1, x2, …, x10000}? The answer is found in the following results.LemmaAny P-classification of an empirically continuous seriesis either dissonant or results in the assignment of all the objects tothe same equivalence class.Dissonance TheoremAny P-classification of a slippery slope is inherentlydissonant.Formally, the theorem can be stated as follows:SS (X) ® { ("u) (" t) [ ( (" xe X) ~#P (u, x, t) ) ® ~Cons (u, t) ] }We shall prove the theorem by reductio ad absurdum.Let X = {x1, x2, …, xn}be a slippery slope such that ~P (x1) & P (xn).Let u be a user that at time t gives a P-classificationof X.Supposing Cons (u, t) we apply modus ponens forthe slippery slope condition and for the consistency condition.(" k, 1<k<n) ( Ind (xk,xk+1) & ~#P (u, xk, t) & ~#P (u, xk+1,t) )(" k, 1<k<n) ( Ind (xk,xk+1) & ~#P (u, xk, t) & ~#P (u, xk+1,t) ) ® ( P (u, xk, t) = P (u,xk+1, t) )==================================================================(" k, 1<k<n) ( P (u,xk, t) = P (u, xk+1, t) )We can now correctly apply the sorites argument.P (u, xn, t)(" k) ( P (u, xk,t) = P (u, xk+1, t) )=========================P (u, x1, t)The conclusion reached, P (u, x1, t), contradictsthe fact that x1 elicits a universal negative P-description.Therefore the hypothesis Cons (u, t) is false. q.e.d.An analogous proof can be given for the slippery slopeshaving P (x1) & ~P (xn).RemarkIf u is inconsistent at time t then u has to give at timet two dissonant P-descriptions. Therefore:($ p, 1<p<n) ( ~P (u,xp, t) & P (u, xp+1, t) )We call the corresponding pair of objects (xp,xp+1) the (subjective) separation threshold between theequivalence classes that result following the P-classification of seriesX.The dissonance theorem is in full agreement with the personalexperience of any one of us. Not only is there no objective, universallyvalid threshold between tall and not tall, heap and not heap, etc., buteven establishing a subjective separation threshold gives rise to an inevitableinternal inconsistency. Assume we wish to classify the various quantitiesof grains of wheat into heaps and non-heaps. Where can we draw the line?Between 599 and 600? Why not between 598 and 599 or between 600 and 601?The decision to be made will be eventually arbitrary, not only as pertainingto outside reality but also to our innermost convictions.We do not want that the psychological issues of the dissonancetheorem give the impression of justifying the adoption of an extremal skepticalor relativistic position towards vague concepts. Indeed, any classificationof a slippery slope is arbitrary, but none is completely arbitrary.Although subjective separation thresholds vary from one classificationto another – in relation to different users and different moments of time– they never exceed certain limits. These limits are the subject of thenext two theorems.The stability thresholds theoremLet X = {x1, x2, … xn}be a slippery slope such that ~P (x1) & P (xn).Let u be a user that gives, at several moments of time, P-classificationsof objects in X.Then there is a stable pair of indiscernable objects xp,xp+1 Î X such that xpdoes not elicit a stable P-description from u, but xp+1 elicitsa stable positive P-description from u. Say that (xp, xp+1)represents the individual positive stable threshold of u. A similardefinition holds for the individual negative stable threshold of u.RemarkUnder the betting interpretation the theorem reads:There is a pair of indiscernable objects xp,xp+1 Î X such that u does notalways bet True[P(xp)], but does always bet True[P(xp+1)](and similarly for the symmetrical case).Formally, the theorem can be written:SS (X) ® [ ($p, 1<p<n) ( Ind (xp, xp+1) & #P (u, xp)& P (u, xp+1) ) ]SS (X) ® [ ($q, 1<q<n) ( Ind (xq, xq+1) & ~P (u, xq)& #P (u, xq+1) ) ]Suppose, as a reductio ad absurdum, that(" k, 1<k<n) ~ [ #P (u,xk) & P (u, xk+1) ] (1)Relation (1) can be rewritten(" k, 1<k<n) [ P (u,xk+1) ® ( P (u, xk)v ~P (u, xk) ) ] (2)Now we can apply the sorites argument:P (u, xn) (slippery slope condition)(" k, 1<k<n) [ P (u,xk+1) ® P (u, xk)] ((2) + indiscernability axiom)=============================P (u, x1)This conclusion contradicts the assumption that x1elicits a universally negative P-description. Therefore relation (1) isfalse and the existence of the positive stability threshold is proven.A similar proof can be given for the symmetrical threshold.In order to make things clearer, we shall present thefollowing example. Consider once again the collection X = {x1,x2,…,x10000}, where xk represents kgrains of wheat, and the concept of "heap" denoted by P. Imagine an experimentin which u gives, at different moments of time, P-classifications of differentquantities of grains. Then, the theorem of stability thresholds assertsthat there exist two indiscernable quantities xp and xp+1such that xp+1 is always classified as a heap, while xpis at least once classified as not being a heap.Assume that there are 100 P-classifications throughoutthe experiment. It is obvious that there are quantities u will describeas being a heap exactly 100 times, and others that u will describeas being a heap for almost 100 times. No matter what the resultof the experiment is, there will always be two indiscernable quantitiessuch that one is described as being a heap for 100 times, and the otherfor less than 100 times (for instance, for 99 times).Let us now assume that the experiment consists of 1000P-classifications. The previous statements remain valid. The only thingthat changes is the effective position of the threshold. The number ofP-classifications is immaterial. The more classifications, the fartherto the right goes the positive stability threshold (i.e. the shadowy areaof the concept of heap tends to grow). But no matter how long the experimentthere is a limit that will never be exceeded. In other words, there existtwo quantities xp and xp+1 such that, even ifthe experiment were to consist of an infinite number of steps, xp+1would always be described as a heap, and xp would at least oncebe described as not being a heap. Since no user ever makes an infinitenumber of classifications, we can call (xp, xp+1)the absolute stability threshold of u.In spite of the formal proof and the empirical considerations,we doubt that there will be many readers who will accept the existenceof precise stable thresholds.The individual stability thresholds theoreminduces its own kind of cognitive dissonance. The first temptation of auser is to argue as follows:"Assume the theorem holds and my own stabilitythreshold is defined by the pair (xp, xp+1).But,since xp and xp+1 are indiscernable quantities, howcan I hold different attitudes toward them? If I have even once describedxp as not being a heap, why would I not ever make the same judgementfor xp+1?"As in so many other cases this line of reasoning is confusingbecause it is based on a false hypothesis. In this case, it is incorrectto believe that if the positive stability threshold exits, then it canalso be determined. In fact, any attempt u makes to identify through introspectionhis stability thresholds is doomed to failure. Whenever u tries to checkwhether (xp, xp+1) represent his positive stabilitythreshold, he gives an ad hoc P-classification of the two objects.But since xp and xp+1 are indiscernable objects,u holds at that moment identical attitudes towards them. Thus, u has norational method of deciding on whether (xp, xp+1)is his positive stability threshold. The fact that a specific pair (xp,xp+1) defines the positive stability threshold is, after all,a chance matter.This last remark might seem to open up the way for a probabilisticinterpretation of the stability of descriptions. For instance, it mightbe considered that an object x elicits a positive stable P-descriptionfrom user u of the probability that u gives a negative P-description ofx be zero. Absolute stability thresholds would represent precise linesof delimitation between events having zero probability and events havingnon-zero probabilities of occurring. But such a purely theoretical probabilisticinterpretation is inadequate. It would be impossible, for instance, toexplain what "the probability that u describes 1 billion grains of wheatas not being a heap is rigorously zero".If x is an object that elicits a positive stable P-description,then a negative P-description of x is not an event having a mathematicallyzero probability of occurring, but an event which only contingentlynever occurs. The existence of a negative P-description is not a possibilitythat is contradicted by theory, but one that is never confirmed by practice.As an analogy, let us imagine a potentially infinite number of throws ofa die and the event that the face marked 6 comes up for 1000 times in arow. The probability that this event occurs is 1/61000, notzero. This event may be a perfectly good example of an eventthat is theoretically possible but which never happens. We can wellimagine a world in which someone throws the die and the face makes 6 comesup for 1000 times in a row. But in our world, this will never happen. Similarly,it is theoretically possible for someone to describe a quantityof one billion grains of wheat as not being a heap, but this will neverhappen.Let us now suppose that the above experiment is extendedto an arbitrary number of users. The positive stability thresholds willobviously vary from one user to another (although we would expect the differencebetween them be small). However, it is clear that there are two indiscernablequantities xi and xi+1 such that xi+1is always considered as being a heap by all the users, while xiis for once, if not more, classified, at least by one user, as not beinga heap. Therefore, we can extend the individual stability thresholds theoremto a universal stability thresholds theorem.The consensus thresholds theoremConsider a slippery slope X = {x1, x2,… xn} such that ~P (x1) & P (xn).There exists a pair of indiscernable objects xp,xp+1 Î X such that xpdoes not elicit a universal P-description, but xp+1 does elicita universal positive P-description. We shall call (xp, xp+1)the threshold between the shadowy area and the positive extension of P(an analogous theorem and definition holds in the symmetrical case). RemarkUnder the betting interpretation the theorem reads:There exists a pair of indiscernable objects xp,xp+1 Î X such that not allusers bet True[P(xp)], but all users do bet on True[P(xp+1)](and similarly for the symmetrical case).Formally, the theorem can be written:SS (X) ® [ ($p, 1<p<n) ( Ind (xp, xp+1) & #P (xp)& P (xp+1) ) ]SS (X) ® [ ($q, 1<q<n) ( Ind (xq, xq+1) & ~P (xq)& #P (xq+1) ) ]The proof is similar to that given for the stability thresholdstheorem.2.7 Final ConsiderationsIn general, logicians consider that the indeterminationof vague concepts is recursive. Not only is the distinction between thepositive and the negative extensions of a vague predicate blurred, butit is also unclear where this blurring starts and where it ends. In thissense, one talks about higher order indeterminations. The majority of studiestreat exclusively the problem of order one indetermination. Instead , oneof the declared strong suits of fuzzy logic is its claim of being a oneshot solution to the types of indetermination.The author, for one, considers that it does not make anysense to speak of a primary and a secondary indetermination. First orderindetermination is of semantic character, and is manifested in the inherentsubjectivity of deciding the separation threshold. Second order indeterminationrefers to the boundaries of subjectivity in establishing the separationthresholds. In our opinion this indetermination is mainly epistemological.Although we do not know the delimiting lines between consensus and subjectivity,they surely exist. These lines change with every new subjective that shattersthe absolute consensus that exits at one time. Still, we can talk aboutabsolute limits in the sense of the above absolute thresholds.As a result, we can say that any vague predicate P determinesa threefold divison of the universe of objects: the positive extension(objects to which P applies by absolute consensus), the negative extension(objects to which P , by absolute consensus, does not apply), and the shadowyarea of P (objects for which there is no absolute consensus). The threeregions correspond to:{ x | P (x) } – positive extension{ x | #P (x) } – shadowy area{ x | ~P (x) } – negative extensionThe problem is that this threefold universal classificationrequires a higher point of vantage from which to analyze all the bivalentclassifications made by each individual, which is practically an impossibility.Fortunately, in practice we rarely need absolute consensus and thereforewe can use approximations.Since we have admitted the practical impossibility ofestablishing a precise universal classification of objects in relationto a vague property, it might seem that indetermination cannot be donewith by using the assumption of exclusively bivalent descriptions leadingto a threefold universal classification. But this needs not favor multi-valuedlogics. First, the logical and psychological requirements of identicallydescribing indiscernable objects do not depend on the number of availablealternatives. Second, it is much more difficult to estimate the degreeof applicability of a concept than to judge whether the concept isapplicable or not. It is our opinion that any attempt at classifying objectsin relation to a predicate in a higher (possibly infinite) number of classesof equivalence will not simplify the issue, but only make it more complex(i.e. it will not reduce the cognitive dissonance but rather amplify it).BIBLIOGRAPHY Ph. P. Wiener (Ed.) Values in A Universe of Chance. SelectedWritings of Charles S. Peirce, Doubleday Anchor Books, 1958.CLARK, A., A Physicalist Theory of Qualia, The Monist,Vol. 68, No. 4, October 1985, pp. 491-506.GOGUEN, J. A., The Logic of Inexact Concepts,SynthesE, 19, 1968, pp. 325-273.HYDE, D., Sorites Paradox , Stanford Encyclopediaof Philosophy, 1996, http://plato.stanford.edu/entries/sorites-paradox/KYBURG, H. E. and SMOKLER, H. E., Studies in SubjectiveProbability , John Wiley & Sons Inc., 1964.NADIN, M., The Logic of Vagueness and the Category ofSynechism, The Monist, Vol. 63, No. 3, July 1980, pp. 351-363.POUNDSTONE, W., Labyrinths of Reason: Paradox, Puzzlesand The Frailty of Knowledge , Penguin Books, 1968.RUSSELL, B., Mysticism and Logic, Doubleday AnchorBooks, 1957.RUSSELL, B., Our Knowledge of the External World,The New American LIBRARY,1956.RUSSELL, B., Vagueness, The Australasian Journal ofPsychology and PhilosoPhy 1, June 1923, pp. 84-92.SORENSEN, R., Metaphysics of Words, PhilosophicalStudies, 81/2-3 March 1996, pp. 193-214.SORENSEN, R., Vagueness, Stanford Encyclopedia ofPhilosophy,1997, http://plato.stanford.edu/entries/vagueness/SWARTZ, N., Beyond Experience: Metaphysical Theoriesand Philosophical Constraints, University of Toronto Press, 1991.WITTIG, A. F. and WILLIAMS, G., Psychology: An Introduction,McGraw-Hill, 1984. |
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