Grading, Sorting, and the Sorites

Grading, Sorting, and the SoritesTIM MAUDLIN

Vague predicates admit of borderline cases. One is commonly inclined toregard certain claims about the borderline cases as violating bivalence: If John

has a borderline case of baldness, then it is neither correct nor incorrect to call himbald, and so the claim that he is bald is neither true nor false. Vague predicates arealso commonly supposed to lack sharp boundaries.There is no precise point in timewhen a balding man becomes bald, or when a tall person becomes tall. There wasno exact nanosecond when Muhammad Ali became famous, even though at sometime he was not famous and at some later time he was. These notions are evidentlylinked: In the process which took Ali from not being famous to being famous, hespent some (vaguely defined) time as a borderline case of being famous.

The usual approach to vagueness treats these features of vague predicates assemantic, rather than epistemic, matters. In a borderline case of baldness, there maynothing relevant that is unknown, either about the state of the head in question orabout the meaning of the term “bald.” Rather, that meaning simply fails to deter-mine a classical truth value in the case at hand. The epistemic view of vagueness, incontrast, maintains that even in borderline cases, a claim like “John is bald” is eithertrue or false, and that if John becomes bald, there is a perfectly exact moment in thecourse of his hair loss at which he became bald. According to the epistemic view,John is a borderline case of baldness only because it is in principle unknowablewhether or not he is bald, but every borderline bald person is either a person whois, in fact, bald or a person who is not.

There are three main lines of defense of the epistemic view. One is a purelylogical argument contending that the denial of bivalence entails a contradiction,

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© 2008 Copyright the Authors. Journal compilation © 2008 Wiley Periodicals, Inc.

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and so cannot be consistently maintained. The second adverts to the Sorites argu-ment, which is taken to show that, contrary to the naive view, there must be anabsolutely exact boundary that separates the bald from the not-bald, the famousfrom the not-famous. The third argument is by elimination: Various alternatives toepistemicism are examined and found to be untenable. At the end of the investi-gation, only epistemicism is remains as a viable alternative.

The logical argument attempts to show that the denial of bivalence leads tocontradiction:The claim that some meaningful utterance or sentence is neither truenor false is supposed to imply a straightforward contradiction, in the form of theconjunction of a sentence with its negation. I have dealt with this argument else-where (Maudlin 2004, 196–199), and will not repeat the details here. Suffice it to saythat the argument begs the question at hand. Anyone who denies bivalence willautomatically be committed to the existence of several distinct extensions of clas-sical negation, which differ with respect to the truth value of the negation ofsentences that are neither true nor false. The more distinct semantic values onerecognizes (e.g., if one recognizes not only borderline cases, but borderline border-line cases, and so on), the more forms of truth-functional negation one will becommitted to. The “logical” argument, as constructed by, for example, TimothyWilliamson (1992, 145–147; 1994, 187–89) and Paul Horwich (1990, 90), presumesthat there is only one form of negation.1 Once the natural multiple forms areallowed, the denial of bivalence can be framed so that no contradiction validlyfollows from it.

But this observation does nothing to defuse the Sorites, nor does it provideany positive account of semantics from which the failure of bivalence follows.Whatis wanted is a theory of the meaning of vague terms from which the possibility ofborderline cases can be derived. Such a theory ought also to explain both themanifest appeal and the ultimate resolution of the Sorites argument.

“Meaning” is a notoriously obscure notion, so asking after the meaning of aterm like “heap” or “bald” is not yet to set an entirely clear problem. Slogans suchas “meaning is use” or “meanings determine truth conditions” are of little help inthis regard. So what I propose to do is simply ignore the question of meaning at theoutset of this paper. Instead, I would like to examine in some detail a particularpractice, whose workings are sufficiently well-known and transparent to be uncon-troversial. Then I will suggest that the use of terms like “bald” and “heap” aresufficiently similar to this practice to be illuminated by it. I will not be arguing thatthis is a useful way to regard meaning in general, but that the structure of thispractice does help us understand “heap” and “bald” and kindred terms. The prac-tice I have in mind is the practice of grading.

At the end of a semester, the instructor of a class is obliged to assign thestudents in the class grades. There is typically a relatively small set of gradesavailable:A, B, C, D, and F, supplemented, perhaps, by pluses and minuses. It is also

1. While some approaches to vagueness, such as that of Williamson, simply overlook thepossibility of distinct extensions of classical negation to cover a multivalent semantics, others, suchas that of Kit Fine, rule it out by means of a formal principle. In Fine’s (1975), the condition calledStability rules out the form of negation which maps borderline sentences to true sentences. Fine’sdefense of the Stability principle (p. 275), however, is rather poorly motivated.

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not unusual to have, as the basis of assigning the grade, an aggregate numericalscore ranging from 0 to 100, a weighted sum of numerical scores achieved onvarious exams and quizzes.The task of the instructor is then to assign grades on thebasis of that aggregate numerical score.

For the purposes of this paper, I will idealize somewhat and assume that theaggregate score is the only piece of information used to assign the grade. Inpractice, of course, one often takes into account other factors: class participation,attendance, and so on, but that would needlessly complicate the account, so I willomit them. We can therefore call the aggregate numerical score the objective basisfor the assignment of the final grade.That is not to say that the numerical score wasitself arrived at objectively (in some sense) but that the score is the only input fromthe side of the student into the final decision process.

Assignment of final grades given the objective basis is a procedure that hasnormative rules associated with it. Our first task is to try to be very precise aboutwhat those rules are. I will be describing the rules I myself use in this sort ofsituation.

To begin with, the rules have some Absolute Paradigms. To take an obviouscase, anyone whose aggregate score is 95 or above must get an A (we do not havethe grade A+ available) and anyone whose aggregate score is 50 or below must getan F. Anyone who matches an Absolute Paradigm can be assigned a grade imme-diately, without regard to the rest of the class. We could obviously set the firstparadigm higher and the second lower, but they are more informative this way.

There is another absolute rule: any student who gets a better grade thananother must have a higher aggregate score than the other. There follows a sort of“supervenience” principle for grades: students who get the same aggregate scoremust get the same grade. The grades “supervene” on the objective basis. I have putscare quotes around “supervenience” as an alert that the exact content of thisprinciple will be further articulated. Let us call this principle Dominance, since itsays that if one student dominates another in final grade, she must dominate theother in the objective basis.

Dominance and Absolute Paradigms are the only unbendable rules govern-ing the assignment of final grades. Were either of these to be violated, a studentwould have, ipso facto, legitimate grounds to appeal the final grade. What followthem are softer rules that can be violated, and whose violation can be defended inparticular instances, but which one would prefer to respect.

With the highest and lowest grades, one can have absolute conditions fortheir application, but with the intermediate grades one has instead a Soft Target.For example, I think of an 85 as a sort of paradigm B:Absent some other mitigatingfactor, a student who gets an aggregate of 85 will get a B. An example of a possiblemitigating factor is when 85 is the highest aggregate score in the class, so one“grades on a curve.” One can deviate from a soft target, but only when there issome particular circumstance that can be used to justify the deviation.

Another soft constraint, one which will much concern us, we may call theEpsilon Principle. The Epsilon Principle states that students whose aggregatescores are very close to each other (within some vaguely indicated small amountepsilon) ought to get the same final grade. In a typical course, epsilon is often

Grading, Sorting, and the Sorites 143

around a point in the aggregate score. One dislikes, and tries to avoid, giving acertain grade to one student and a worse grade to a student whose aggregate iswithin a point of the first.

The justification of the Epsilon Principle is straightforward. The purpose ofgiving final grades is to convey information about the students’ performance in theclass. That performance, we may suppose, is quantified by the value each studenthas in the objective basis.The final grades constitute a sort of coarse-graining of theobjective basis, and in the process of coarse-graining, information is lost: Studentswho performed differently in the class are nonetheless assigned the same finalgrade. The loss of information seems particularly unjust when the performance oftwo students is nearly the same, but they nonetheless they are given different finalgrades. The small difference is magnified by the coarse-graining. There is theappearance of a kind of injustice here, although not the form that would automati-cally invalidate the grades, as a violation of the absolute principles would. So onetries, insofar as is possible and consistent with the other soft and absolute con-straints, so satisfy the Epsilon Principle in assigning final grades.

Let us call the four principles listed above, Absolute Paradigms, Dominance,Soft Targets and the Epsilon Principle, the Ideal for Grading. The most satisfactoryoutcome, when assigning grades, is when all of the principles in the Ideal can besatisfied. There is no logical guarantee that the Ideal can be satisfied in any givenclass: That depends on the actual distribution of aggregate scores. It may happenthat no way of assigning grades will meet the Ideal. It may also happen that severalways of assigning grades can meet all of the principles in the Ideal. That too issomewhat problematic, since the Ideal gives no further guidance about which ofthose assignations to make.We could, of course, expand the Ideal to include furtherprinciples, in hopes of reducing these sorts of ties, but we won’t. We therefore havea ranking of possible situations, depending on the distribution of aggregate scores.In the best of all possible worlds, the Ideal can be satisfied, and only in one way. Inthe next best case, the Ideal can be satisfied in multiple ways. One will have tochoose among these different solutions, and the choice will be somewhat arbitrary,leading to a certain sense of injustice with respect to students whose grades dependon that arbitrary choice. Below these are the situations in which the Ideal cannotbe satisfied:At least one principle must be violated. In my own case, there is a clearorder ranking here: I will more easily give up the Soft Targets than the EpsilonPrinciple. Of course, the choice between giving up Soft Targets or the EpsilonPrinciple is itself a matter of degree: I will allow the grades to shift from the idealSoft Targets to some degree in order to satisfy the Epsilon Principle, but there is a(vague) limit to how far I would allow them to shift just to save that principle.

In practice, then, here is how I assign grades at the end of term. I calculate anaggregate score for each student. I then write on a piece of paper, in descendingorder, the numbers from 100 to 50 (less than 50 is always F), and make a tally markfor each student next to their aggregate score. This gives me a distribution of tallymarks ranged down the paper. I then look for clumps of marks with gaps betweenthe clumps. If, for example, there is a clump of marks centered near 85, I look fora gap above it so that I can have the Bs go up to the gap, with the B+s starting aboveit, and a gap below so the Bs go down to the gap. with the C+s starting below it

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(we don’t have the minus grades). In a class of 40 or so students, there are usuallysuch gaps in the distribution of aggregate scores.There is evidently no guarantee ofthis—each score from 60 to 100 could be represented by exactly one student—butas a matter of fact the scores tend to cluster and the clusters leave gaps. Byseparating the grades by the gaps, I can satisfy the Epsilon Principle (for the valueepsilon equals one point), and I let the placement of the separations be influencedby the score distribution in this way. I imagine that the way I assign grades is fairlytypical.

If the distribution of aggregate scores is full, with every possible score occu-pied, then there is no way to satisfy the Epsilon principle, and students with almostidentical scores will end up with different grades. One might have Very Soft TargetBoundaries for this case: precisely defined division lines to use for separating thegrades. And if there are hundreds of students, rather than just 40, the chances ofgaps is almost nil. But even when there are hundreds of students, appeal to targetboundaries is a last resort. If the distribution of grades is not flat, there will be hillsand valleys in the distribution of aggregates scores, and running the grade bound-aries through the valleys will minimize the violation of the Epsilon Principle, andmaximize the amount of information about the objective basis conveyed by thegrades. So the only case where target boundaries would need to be invoked is whenthe distribution is both full and flat—no gaps, no hills, no valleys- and that practi-cally never occurs.

The procedure I have just described is pedestrian. But there are aspects of itwhich on reflection are rather surprising.

The first point has to do with the scope of situations to which the principlesapply.The Ideal for Grading is used in a particular grading situation—when assign-ing grades at the end of one semester to the students in that class. The principlesapply to that class only. So, for example, giving one grade to a score of 90 andanother to a score of 89 in the same class violates the Epsilon Principle, but givingone grade for 90 in one semester and another for 89 for the same course taught inanother semester violates no principle at all. Even more striking, Dominance onlyapplies to students in the same class. A student who gets an 89 in a course onesemester can get an A, while a student who took the same course a year earlier andgot 90 could get a B+. There is no doubt a certain uneasiness about this, but inpractice the second student would have no grounds for complaint. If the studentshad been in the same class, the grading policy could not be defended.

Doubtless, one reason for this is that there is no guarantee that the classeswere taught exactly the same way, or the exams equally difficult. But on the otherhand, the courses might have been identical, and the first student could benefitsimply from having duller classmates. In any case, the principles apply only intra-class, not interclass.That is why we had to put scare quotes around “supervenience”above: Dominance guarantees only that students in the same class with the sameobjective basis get the same grades, not students in different classes.

It is essential that the scope of the principles be confined to a single class: Ifone had to aggregate all the students one ever taught (even in the same course), thechance of gaps in the objective basis would disappear, and the distribution wouldbe more likely to lack the sorts of hills and valleys the Epsilon Principle needs.And


if one were to consider merely possible students, then the practice would collapseentirely. In the first place, there would be no distribution in the objective basis atall, and in the second, pesky students would constantly try to slip their way up theslope to a better grade by filling in gaps between their score and the lowest scorethat got a higher grade with merely possible students. The gaps and distributionsthat make the Epsilon Principle useful are contingent matters, and contemplationof counterfactual grades given to non-actual students will tend to obscure thepractical utility of the Principle. In the worst case, contemplation of merely possiblestudents will deflect one’s attention from the actual task at hand to the worst casescenario: flat distributions with no gaps on which one needs to draw boundaries.

At this point, connections to the Sorites arguments are manifest, so it isworthwhile to get a bit more exact.What is the logical form of the Epsilon Principleas an ideal that we would like to come out true? The Principle states that any pairof students whose aggregate scores are within epsilon of each other should get thesame grade. If we represent the relation of having scores within epsilon of eachother as E(x,y), and the relation of having the same grade as S(x,y), the Principlebecomes

∀ ∀ ( ) ⊃ ( )( )x y E x y S x y, , .

In cases where the Ideal can be satisfied, the sentence above will be true. Thequantifier in the sentence ranges only over members of the class at issue.

If we introduce predicates that correspond to the possible scores in theobjective basis, for example, 100(x), 99(x), 98(x), etc., then the Epsilon Principlecan be written in a logically equivalent way as the conjunction of a series ofconditionals:

∃ ( ) ⊃ ∀ ( ) ⊃ ( )( )( ) ∃ ( ) ⊃ ∀ ( ) ⊃ ( )( )( )x x y y S x y x x y y S x y100 99 99 98, & , & . . .

The conditionals appear similar to those commonly used in constructing a Sorites,but the quantificational structure renders them harmless if there happen to be theright sorts of gaps in the distribution.A student with a 95 can get an A and a studentwith a 50 an F, as the Absolute Paradigms require, without rendering any of theconditionals false. If there happens to be at least one student with each possibleaggregate score, then at least one conditional must be false, but that is a contingentmatter.

Let’s now consider some examples from the taxonomy of cases given above.The best case is when there is a unique assignment of grades that satisfies theIdeal. If, for example, the grade distribution has clusters of aggregate scores thatare confined to intervals 94–96, 84–86, 74–76 and 35–50, then the only way tosatisfy the Ideal is to assign all of the first group As, the second Bs, the third Cs andthe last Fs. Anyone trying to satisfy the Ideal would come up with the same finalassignment.

The next best case is when the Ideal can be completely satisfied, but inseveral different ways. If one only has the straight letter grades to assign (withoutpluses or minuses), then adding a single student with an aggregate score of 90 to the

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distribution above produces such a case. The student could be given either an A ora B without violating the Ideal, so something beyond the Ideal must be employedto decide the case. Such a student is a benign borderline case: borderline becauseequally acceptable grade assignments assign him different grades, and benignbecause no deviation from the Ideal is required.

It is important to note that the task at hand is simply to assign grades to all thestudents. The Ideal concerns evaluation of such assignments. The task is not to“draw boundaries” for each grade. In the first example discussed above, one simplyhas to decide what grade each actual student gets, not which aggregate score willseparate the As from the Bs. If one had to draw boundaries, then even in the firstcase many distinct ways of drawing the boundaries would be equally acceptable.But since we are engaged simply in sorting the actual cases, not in drawing exactboundaries, the solution that satisfies the Ideal is unique.

In the first case described above, there is a clear sense in which the gradesassigned to the students are objectively determined. The objective basis togetherwith the Ideal determines a unique grade for each student, so the role of theteacher evaporates into an entirely mechanical task. Similarly, in the second case,the grades of the non-borderline students are objectively determined, since theyare required to get the grades they do if the Ideal is to be satisfied.We can even saythat in the second case the student with 90 is objectively borderline, since there arealternative ways to satisfy the Ideal that assign him different grades. The instructorwill play an ineliminable role in assigning the borderline case a particular grade,but plays no role in making the borderline case a borderline case.

So if the Ideal can be met at all, every student in the class will either have anobjectively determined grade or will be objectively borderline. And anyone whoengages in the practice or grading in accordance with such an Ideal can employ thenotion of a “borderline case” unproblematically in these circumstances. Clearly,being an objective borderline case in the sense defined above has nothing to dowith ignorance of anything. One may be ignorant of something, for example,ignorant of which grade will ultimately be assigned to a borderline case, but thedefinition of a borderline case does not advert to any ignorance.

In real life, the existence of objective borderline cases presents a naturaltemptation to expand the relevant taxonomy. Since an objective borderline case isseparated by at least epsilon from the clear cases both above it and below it, if oneinvents a new taxonomic category, one can then satisfy the Ideal uniquely. If onewere grading papers, rather than assigning from a fixed set of final grades, theborderline case above would get an A/B or an A- or a B+. Taxonomic innovationof this sort often wears its origin on its sleeve.There is a utensil that has both a bowland several prongs called a spork; my daughter sometime wears apparel with legslike shorts but a front like a skirt called a skort, and everyone is familiar with theoddly place meal called brunch. Of course, any new taxonomic structure will itselfbe liable to objective borderline cases: one of Homer Simpson’s claims to fame isthat he “discovered a meal between breakfast and brunch.”

Expansion of a taxonomy would likely be accompanied by a reduction in thesize of epsilon: the more pigeonholes there are to sort things into, the less of a gapone needs between the objects sorted. Epsilon as a precise magnitude is in any case


an evidently false idealization: rather than have a fixed value for epsilon, one looksfor both a taxonomy and a value of epsilon that renders the sorting objectivelydetermined. This puts pressures on epsilon in both directions: set epsilon too largeand no sorting will satisfy the principle, set epsilon too small and many will.

So far, we have normative standards than can govern a method of sortingindividuals into a taxonomy. We have seen that in a particular judgment situation,if the distribution of the individuals in the objective basis has the best form, thesorting can be objectively determined. In the second best case, the sorting for someindividuals will be objectively determined and the rest will be objectively border-line.There is no guarantee that either the best or second best case will obtain: thereare distributions such that the epsilon principle will be violated for any reasonablesetting of epsilon. But let’s leave these worst-case scenarios aside for a moment andmake contact with semantics.

Suppose a student is objectively determined to get an A in a class. Then itis tempting to say, even before the final grades have been officially assigned, thatthe student was an A student: the instructor, upon examining the distribution ofaggregate scores discovers rather than decides what the grade is. Since any otherassignment of a grade would violate the Ideal, the instructor has no choice in thematter.

Of course, all of this talk about the discovery of a “pre-existent fact” carriesno ontological weight. The only objective contribution on the side of the student isthe aggregate score: there is no property that any student has that corresponds to“being an A student.” There is a sense in which the grades given a class reflectnothing over and above their aggregate scores, since the scores are the only con-tribution from the side of the students into the process and since (in the favorablecase) those scores, together with the Ideal, determine a unique sorting. But onemust be careful about exactly how this ontological point is made.

When a student’s grade is objectively determined it is tempting to say that astudent’s grade supervenes on her aggregate score, but for most of the technicalsenses of supervenience this is false. For students in the same class, there is asupervenience thesis: difference in grade implies difference in aggregate score. Ifthis were to fail, then there would be a violation of Dominance. But for students indifferent classes, no such guarantee holds. An 89 in one class could be a B, and inanother an A. In this obvious sense, the assignment of grades can be contextual, butmay still be perfectly objective in a given context.

We can make the relevant metaphysical point without recourse to the notionof supervenience: it is not that the grade supervenes on the aggregate score, butthat from the point of view of ontology all that exists is a distribution of aggregatescores and a sorting procedure. The input to the sorting procedure is a judgmentsituation: a specification of a collection of students and a value for each student inthe objective basis. In the most favorable cases, there is only one sorting of thestudents that completely satisfies the Ideal, so every student gets a grade. In thenext most favorable situation, there are several sortings that satisfy the Ideal, sosome get grades and some are objectively borderline. But the result of this sortingprocedure, even in these favorable cases, does not imply the existence of anythingat all in the world beside the students and their aggregate scores.

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In favorable cases, not only does the distribution of aggregate grades and the“meaning” of the grading system determine which students get particular gradesand which are borderline, it also determines truth values for some counterfactuals.Suppose, in the favorable case described above, Sue’s grade falls in the clusteraround 85, and she gets a B. Had she gotten ten points more, she would have gottenan A. If Sue lost 10 points on one question, she can rightly say: those 10 points madethe difference between an A and a B. In such a case, for the counterfactual to beclearly true, the point difference being considered must be greater than epsilon;otherwise the counterfactual distribution of aggregate scores will not objectivelydetermine that Sue would have gotten a different grade. So a sufficiently large jumpin a score can objectively determine a change in grade, while a sufficiently smallone cannot.

If one happens to be thinking of changes in grades that are objectivelydetermined, then it is correct, given the Ideal, that a difference in aggregate scoreless than epsilon cannot “make the difference” between grades. This is surely thesort of thing one has in mind in typical Sorites cases when one affirms that loss ofone hair cannot make one bald, or gain of a penny make one rich.The locution “theloss of one hair makes the difference” directs one’s attention to cases where theobjective basis (in this case, hair distribution) objectively determines the sorting.Indeed, one can even say that it is analytic, that is, follows from the meaning, thatis, a consequence of the Ideal, that a difference in grade less than epsilon cannotmake an objectively determined difference in a grade, while a difference greaterthan epsilon can.2

So far, the only place we have mentioned truth values is with regard to certaincounterfactuals about grades, but we are evidently very close to making directcontact with semantics. Before we take that last step, though, let’s quickly reviewwhat has and what has not been done.

We have seen so far that, given a certain grading procedure, in some favor-able judgment situations it is objectively determined what grades some studentsget or whether they are objectively borderline. We have not, as yet, had anydiscussion at all of the unfavorable situations: situations where, for example, theEpsilon Principle cannot be satisfied. The usual Sorites problem presents us withexactly such a situation, so we have not yet begun to address those problematiccases. But the strategy at this point is just to convince ourselves that there areunproblematic cases for sorting in accord with the Ideal: this explains why the Idealis useful, and why the procedure need not be abandoned even if there are someproblematic cases for it.

We should also note that although some account has been given of sorting infavorable situations, we have not delineated the favorable situations in completelyprecise language. If one were given the task of sorting judgment situations into the

2. In very special circumstances, when Sue is the outlier in a distribution that employs gapsjust above epsilon, one can construct a case where changing her grade by less than epsilon wouldshift the sorting and therefore change her grade. But such cases rely on the fiction that epsilon isa precisely defined magnitude and the rules are applied in some ironclad fashion. In practice, all ofthese things are themselves vague.


categories favorable and unfavorable, no doubt one would confront borderlinecases or other problematic cases. So if we follow usual usage, and call any taxonomycum sorting procedure vague if the sorting procedure sometimes recognizes bor-derline cases, then not only is the grading system vague, but the distinction ofjudgment situations into favorable and unfavorable and the distinction betweenthe most favorable and next most favorable cases are also vague. There is, in thissense, second-order (and higher order) vagueness.

It is almost an autonomic reaction among philosophers to point out this sortof second-order vagueness. At the moment, we merely need to note it, and also tonote that there is nothing particularly threatening or worrying about it. Since ouroverall strategy is to emphasize the utility, coherence, and objectivity of the pro-cedure in the favorable cases, it does not matter that the favorable cases beprecisely defined, just that there be favorable cases in which it is unproblematicthat we have a favorable case. Anyone who thinks that we are likely to come togrips with vagueness by analyzing vague language in completely precise terms isadvised to consult the founding documents of the Neurath Ship RefurbishingCorporation.

Bearing all this in mind, let’s make the connection to semantics proper. Thesuggestion is obvious. Suppose one is given a particular judgment situation, includ-ing the student Sue. If it is objectively determined that Sue should get a B in thesituation, let us say that “Sue is a B student” is true. If it is objectively determinedthat Sue should get a grade other than a B, or is objectively a borderline casebetween two grades neither of which is a B, then let us say that “Sue is a B student”is false. And if it objectively determined that Sue is a borderline case between Aand B or B and C, let us say that “Sue is a B student” is borderline. So in either ofthe favorable judgment situations, “Sue is a B student” is assigned one of threepossible semantic values.

If we accept this suggestion, then we have already established that it issometimes true, sometimes false, and sometimes neither true nor false to assert thatSue is a B student, so bivalence fails. We have also established that “is a B student”is a vague predicate since it admits of borderline cases. We have also establishedthat the vagueness is not epistemic. The Ideal for Grading has been made explicit,and even if there are vague elements in it, the vagueness of those elements plays norole in establishing the existence of benign borderline cases. We get the benigncases not because we are unsure of how to understand the Ideal, but because wesee that, in some circumstances, the Ideal can be equally satisfied in more than oneway.The existence of such cases may prompt the recommendation that the Ideal beamended by adding procedures for breaking ties in this sort of case, but we arenot concerned here about criticizing the Ideal. The Ideal as it stands does a job:It guides the assignment of grades in a judgment situation. As a practical matter,it may do its job well enough for our purposes in most cases.

An epistemicist about vagueness, or any defender of bivalence, must rejectthe suggestion made above for assigning semantic values in the favorable cases. Ido not see how it can be disputed that, in the favorable cases discussed above, theassignment of a grade or the recognition of borderline status is determined by theIdeal. So any objection must be made at the point where these facts are used as a

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basis for assigning semantic values to sentences such as “Sue is a B student.” Whatsorts of objections are likely to arise there?

Here is one possible objection. In order to have a semantic value at all, asentence must express a proposition. But a proposition is a set of possible worlds,or a total function from possible worlds to semantic values, or a total set of truthconditions, framed to cover every possible situation. The account of the “semanticvalue” of “Sue is a B student,” though, is not complete in the way it would have tobe to specify which proposition that sentence expresses. For although someaccount is given of what semantic value a sentence gets in a favorable judgmentsituation, not every possible judgment situation is favorable. In particular, there areSorites sorts of situations, such as a completely flat and gapless distribution ofaggregate grades. No assignment of grades in such a situation can satisfy the Ideal.Since no account has yet been given of how to deal with those cases, no propositionhas been associated with the sentence, so it can’t be true. (This sort of argument hasbeen defended by Ted Sider and David Braun [2007].)

In its own terms, this objection is perfectly clear.The only question is why oneshould accept the animating proposal that a sentence must express a proposition(in the sense explicated) in order to be true in any circumstances. Indeed, theproposal is in serious tension with the idea that vague language is vague exactlybecause the applicability of terms has not been determined for all circumstances,but that nonetheless vague language can be used to express truths. That is, theproposal seems to rather seriously beg all of the foundational questions aboutvagueness.

Why should the truth conditions of a sentence be determined in all possiblecircumstances in order for it to be determined in some possible circumstances? Weadmit that the Ideal for Grading does not display perfect universal decisiveness: insome cases it underdetermines the sorting into grades because its conditions can bemet in several different ways, while in other circumstances (which we have yet todiscuss) its conditions cannot be met at all. But just because a normative standardhas some problematic cases, it does not follow that every case is problematic. Soagain, what can be said against the proposal that the sentence “Sue is a B student”is true when Sue is objectively determined to get a B?

A rather convoluted sort of objection takes a long detour through metaphys-ics. It contends that “Sue is a B student” predicates the property of being a Bstudent of Sue. And the truth conditions of such a sentence are just that Sue havethe property so predicated. But as a matter of metaphysics, any object either has orfails to have a given property. So it is incoherent to suppose that a sentence like“Sue is a B student” could be other than true or false: it is true if she has theproperty and false if she does not.

The answer to this metaphysical gambit has already been given.We explicitlyreject the idea that as a matter of ontology there is any property of being a Bstudent that Sue could even possibly have. On Sue’s side of the metaphysicalequation, all she has is an aggregate score. Her score together with the scores of theother students in the judgment situation form the complete ontological input intothe grading process. Whether that process manages to determine a specific gradefor her or not makes no metaphysical difference at all to her or her classmates.


Lying behind this sort of an objection is a rather seductive picture of therelation between semantics and metaphysics that deserves some attention. Whenwe do semantics for a formal language, as in first-order logic, we explicate how anatomic sentence in subject/predicate form gets a truth value as follows. First, onehas to supply the language with an interpretation. This means specifying a domain,which is just a set of elements. One then associates with each individual constant anelement in the domain, and with each predicate a subset of the domain. An atomicsentence of subject/predicate form is true if the object associated with the indi-vidual constant is an element of the set associated with the predicate, and falseotherwise.

This picture of semantics encourages a method by which we try to discoverthe correct ontological account of the world by first collecting together sentenceswe take (for one reason or another) to be true and then considering what objectsmust be in the world and what subsets must be associated with predicates in orderfor the sentences to come out true. And if one adds that the most obvious way toassociate a subset with a predicate is to associate a property to it, then we are wellon our way to trying to discern both what objects and what properties the worldcontains by reflection on sentences we accept as true together with the semanticsof the predicate calculus.

No doubt, if an individual term somehow denotes an object in the world, andif a predicate somehow denotes a property, then the truth conditions for a sentencewhich predicates the predicate of the term should be that the corresponding objecthave the corresponding property. But there is no reason to suppose that every truesentence of subject/predicate form has truth conditions of this kind. What is on thetable now is a different proposal for understanding how sentences like “Sue is a Bstudent” can be assigned a truth value in a particular judgment situation, eventhough the predicate does not denote any property. Some such proposal must be onoffer if, on the one hand, we want to maintain that sentences involving vaguepredicates can sometimes be true, and on the other, we don’t want to cram ourontology with properties that correspond to each vague predicate.

The method for assigning truth values offered above secures a connectionbetween the truth value assigned to “Sue is a B student” and the truth value of thesubjunctive conditional “If someone following the Ideal were to assign grades tothis class, he would assign Sue a B.” If Sue is objectively determined to get a B, thenthe subjunctive conditional is true; if she is objectively determined not to get a B,then it is false. If Sue is objectively borderline, then the truth value of the subjunc-tive conditional depends on how the semantics for modal discourse is constructed.If we treat all assignments of grades that satisfy the Ideal as possible worlds, allequally accessible and equally distant from the actual world, then there are somesuch worlds in which Sue is assigned a B and some in which she isn’t. We mightargue that in such a situation the subjunctive conditional is neither true nor false,in which case the conditional fails to have a classical truth value whenever Sue isobjectively borderline. But it is exactly because the semantics of the subjunctiveconditional is controversial that I do not want to analyze the truth value of “Sue isa B student” in terms of the associated subjunctive conditional. We have alreadygiven semantic conditions for “Sue is a B student” that do not advert to what

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anyone would do. In the favorable cases, the role of the person assigning the gradesdisappears from view, and there is no reason to gratuitously let it back in via thesubjunctive conditional.

If the world were guaranteed only to contain favorable judgment situations,our work would be done. The Sorites argument, however, is the starkest possiblereminder that the world need not be so accommodating. So it is time to turn to theunfavorable situations.

SORTING WITHOUT GAPS

Unfortunately, we sometimes have to assign grades even when the distribution inthe objective basis presents no epsilon-sized gaps. If the distribution is both gaplessand flat, the objective side of the sorting process provides no useful input abouthow to sort, so the decision about where to stop giving one grade and start givinganother must be motivated by something other than the Ideal, just as the finaldecision in the case of objective borderline cases is not dictated by the Ideal. Whatsort of semantics is appropriate to this situation?

As a warm-up to this question, let’s first engage in a bit of fiction. We havesaid that the four conditions listed in the Ideal are ranked: Most important tosatisfy are Absolute Paradigms and Dominance, less important are the Soft Targetsand the Epsilon Principle. Suppose the ranking were otherwise, and the EpsilonPrinciple were more important to maintain than Absolute Paradigms. Then whenfaced with a flat, gapless aggregate score distribution, one would be required togive all the students in the class the same grade. Since the Ideal does not putrequirements on what that grade should be, every member of the class would beobjectively borderline. Knowledge of the Ideal and the distribution in the objectivebasis would not allow one to predict what any grade given to any student will be.

If the Ideal were arranged in this way, then the Epsilon Principle wouldalways be satisfied in any judgment situation, but the “Absolute” Paradigms wouldno longer be Absolute: They would be defeasible. In some classes, either studentswith an aggregate score of 95 would not be assigned an A, or students with a 50would not be assigned an F, or both. And the answer to the Sorites paradox wouldbe straightforward.

The paradox arises from the desire to satisfy the Absolute Paradigms andalso to satisfy the Epsilon Principle in a flat gapless judgment situation. That, ofcourse, is impossible: One or the other must be violated. If the Absolute Paradigmswere violated, then there would be no problem about “hidden lines” or “unknow-able boundaries,” since no such boundary (between different assigned grades)would exist in the problematic situation. One would have to concede that in somesituations, Sue’s getting an aggregate score of 95 does not make “Sue is an Astudent” true, and does not constitute a guarantee that she will be assigned an A.But that does not mean that “Sue is an A student” is never true in any judgmentsituation. Nor would it render the whole practice of assigning grades in accordancewith the Ideal empty or pointless, since many judgment situations are not prob-lematic. Where the Ideal can be fully satisfied, it does not matter how the differentprinciples mentioned in the Ideal are ranked, so having an Absolute Epsilon


Principle and Defeasible Paradigms would make no material difference in favor-able situations. The Sorites situation, no matter how it is dealt with, cannot beparlayed into a threat against the use of the Ideal in all situations.

The reason that the Sorites situation might appear to be a general threat tothe practice of assigning grades is because one can easily lose sight of the importantrole that the judgment situation plays. Presented with Sue, who has an aggregategrade of 95, one is inclined to say: if this isn’t an A student then no one is. Moreformally, one might mistakenly believe that the truth value of “Sue is an A student”must strongly supervene on her aggregate score, so whatever goes for her goes forall students with that score. If so, then the problems that arise in the Soritessituation will export to all situations: If all the students in that situation areborderline, then every student in every situation is borderline.

But as we have seen, the truth value of “Sue is an A student” need notsupervene on Sue’s aggregate score. The only relevant supervenience thesisgoverns complete judgment situations, not individuals: If two judgment situationsare identical with respect to the objective basis, then a student in one class will beobjectively determined to have a grade, or to be borderline, if and only if thecorresponding student in the other class is. Two students with identical aggregategrades can be objectively determined to have different grades if they happen toinhabit different classes.

So the Sorites would not present much of an analytical problem under therevised Ideal: one would simply point out that in certain situations, the DefeasibleParadigms are, in fact, defeated. How exactly they are defeated will depend on thearbitrary decision of the grader. No more can be said.

But although the revised Ideal would make life easy for philosophical analy-sis, actual grading adheres to the original Ideal, and we must deal with it. Theproblem with the revised Ideal is that it defeats the very purpose of grading in theproblematic situation. Grades are supposed to provide coarse-grained informationabout the distribution of aggregate scores in the judgment situation. If everystudent in a class with a flat, gapless distribution gets the same grade, then thegrades convey no information at all about the objective basis.The Epsilon Principleis defended, but only by undermining the whole point of the exercise.

So in practice, it is the Epsilon Principle that is defeasible and the Paradigmsthat are absolute. Even in a flat, gapless distribution, anyone who has a 95 aggre-gate score will get an A and anyone with a 50 will get an F. And therefore,somewhere or other, students with scores that differ by less than Epsilon willnonetheless get different grades.And the Ideal does not specify where those breakswill come: they are not determined by the “meaning.”

What about semantics in unfavorable situations? Suppose we have 201 stu-dents in the class, each conveniently with a different integer or half-integer aggre-gate score. As usual, we use the score as a name of the student. What then shouldbe the semantic value of “100 is an A student” or “92 is an A student” or “3 is anA student”?

So far we have only provided semantic conditions for sentences of this formfor favorable judgment situations: “92 is an A student” is to be true if 92 isobjectively determined to get an A, that is, if the Ideal can be satisfied and if every

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way it can be satisfied assigns 92 an A. In a Sorites situation, though, no one isobjectively determined to be anything, since the Ideal cannot be satisfied. Still, wewould like it to come out, even in this situation, that “100 is an A student” is trueand “3 is an A student” is false. This can be done in several ways.

The simplest way is this. Even where the Ideal cannot be fully satisfied, itdoes not become irrelevant to the assignment of grades. We have specified thatwhen not all of the conditions in the Ideal can be met, Dominance and AbsoluteParadigms take precedence over the Epsilon Principle. So even in the Soritessituation, everyone at 95 and above must get and A, and everyone at 50 and belowmust get an F.The first suggestion, then is this: let “n is an A student” be true in thissituation iff n’s aggregate score is 95 or above. And let “n is an A student” be falsein this situation if it is true that n is a student with some other grade, for example,if “n is an F student” is true. And similarly, let “n is an F student” be true if n’saggregate score is 50 or lower.Then it will be true that 100 is an A student and falsethat 3 is, as we wished.

Here’s a second suggestion. Since 95 is a paradigm A, and since the EpsilonPrinciple reflects our desire to treat differences less than epsilon as insignificant forgrading purposes, let’s let “n is an A student” be true if n’s score is within epsilonof a paradigm A score, and similarly for F. Then “n is an A student” will be true forall students with scores 94 and above, and “n is an F student” true for all scores 51and below.This “stretching” of the truth range must not, of course, be iterated.Thatis, one must not argue that since it is true that 94 is an A student, it follows that 94is a paradigm A student, so it must be true that 93 is an A student.The method doesnot condone such iteration, which only results from confusing the truth value of“n is an A student” with the status of n as a paradigm.

It would be natural, in a Sorites situation, to apply the same technique to theintermediate grades, using the Soft Targets rather than the Absolute Paradigms.Since the distribution is both gapless and flat, the Epsilon Principle will be equallyviolated no matter where the boundary between different grades is put, so there isno pressure to shift the Soft Targets. (If the distribution were gapless but not flat,with scores piled up more in some regions than others, then there would bepressure to run the boundaries through the valleys in the distribution, as we haveseen.) So presumably any student who exactly hits the Soft Target for a B will beassigned a B, and anyone who is within epsilon of that soft target will also get a B.With epsilon set at 1 and the soft target for a B at 85, then, “n is a B student” willbe true for any student with a score between 84 and 86. “n is an A student” willtherefore be false for students in the region, and also for any students with lowerscores, since by Dominance they can’t get A’s if the students above them get B’s.

If we only have the grades A, B, C, D, and F available, and we set the AbsoluteParadigms for A and F at 95 and 50 respectively, and the Soft Targets for B, C, andD at 85, 75 and 65 respectively, it is easy to calculate the results. “n is an A student”is true for student 94 but neither true nor false for student 93.5.The borderline A/Bcases run from 93.5 to 86.5, while it is true to say that 86 is a B student and false tosay that he is an A student. And so on.

Once again, a sentence of the form “n is an A student” can have either ofthree truth values: true, false, and borderline. And once again, the truth values of


the sentence will presumably have predictive power when it comes to guessing howthe final distribution of grades will go. When the instructor finally settles on grade,all of the students for which “n is an A student” is true will get A’s, and theborderline between the A’s and the B’s will run somewhere in the region whereboth “n is an A student” and “n is an B student” are borderline.

There is an obvious sense in which this semantics has sharp cutoffs: thesemantic value of “94 is an A student” differs from that of “93.5 is an A student,” andif there had been even more students, whose scores were separated by even smallergaps, there would be students whose aggregate scores are as close together as onelikes, but still for whom the corresponding sentences get different truth values. Suchsharp cutoffs in the semantics are generally thought to be objectionable in anaccount of vagueness, and we will turn to those objections presently. But for themoment, let’s just leave the semantics as specified and analyze the Sorites Paradox.

We can think of the Sorites Paradox as having a certain canonical form.Thereis some predicate P, there is some conditional connective fi, and there is some(usually finite) series of individuals 1, 2, . . . N who differ from each other in onlyvery small amounts in the objective basis. For heaps, the objective basis is usuallynumbers of grains, for baldness, number of hairs, for richness net worth. (These areobvious simplifications: It is both number and distribution and length of hair thatdetermines baldness, both number and distribution of grains heapishness, andso on.) One then considers the set of sentence: {P(1), P(1) fi P(2), P(2) fiP(3), . . . P(N - 1) fi P(N), P(N)}. It is impossible for these four conditions tojointly hold: P(1) is true, P(N) is not true, all the conditionals are true, and theconditional supports modus ponens as a valid (i.e. truth-preserving) rule of infer-ence. For any such Sorites series, then, at least one of the conditions must berejected.

We get a particular Sorites argument by specifying both the relevant predi-cate P and by specifying the relevant conditional fi. In our case, there is a sub-stantial risk of confusion since there are various predicates and variousconditionals from which a Sorites argument can be constructed. In particular, thereare three different sorts predicate that we have made use of:“x is an A student,”“xis objectively determined to get an A,” and “x is assigned an A.” The truth condi-tions for these predicates are obviously different: a student can be assigned an A ina class even though she was not objectively determined to get an A (maybe she wasobjectively borderline), and it can be true that a student is an A student eventhough she is not objectively determined to get an A (e.g., in a Sorites situation).And as a matter of logic, a student can be assigned any grade no matter what gradeshe is objectively determined to get, since the instructor can refuse to abide by theIdeal. If we restrict our attention to instructors who do not needlessly violatethe Ideal, then the truth of “n is objectively determined to get an A” guaranteesthe truth of “n is assigned an A” as well as “n is an A student.”

On the side of the connective, there are two obvious candidates. We can usea material conditional of some sort, or a subjunctive conditional. So all told, thereare six different possible Sorites arguments we can construct, depending on choiceof predicate and conditional. We need to consider all six, because the resolution ofdifferent arguments is different.

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Let’s begin with the arguments using the material conditional. The firstpredicate to consider is “x is an A student.” This gives us a Sorites series that startswith “100 is an A student,” end with “0 is an A student,” and has conditionals of theform “If 95 is an A student, then 94.5 is an A student.”The first sentence is true, thelast sentence is false, and modus ponens for the material conditional is valid. So atleast one of the conditionals must not be true. And, indeed, several are not true. Inparticular, “If 94 is an A student, then 93.5 is an A student” is not true, so theargument becomes unsound when it employs that conditional. Of course, “If 94 isan A student, then 93.5 is an A student” is not false either: its semantic value isborderline. So there is no problem resolving this Sorites argument. One might, ofcourse, have been misled if one thought that every untrue sentence is false, for thenthere would have to be a sharp cutoff between students of whom it is true to saythey are A students and students of whom it is false to say that. There is no suchsharp boundary, exactly because the true sentences are separated from the falseones by a buffer of borderline sentences.

We were only able to identify the first premise that fails to be true becausethe semantics provides a sharp cutoff.And, as noted above, this sharp cutoff may beobjectionable. But we are holding off that objection for the moment, so let’s go on.

The second Sorites argument uses the predicate “x is objectively determinedto get an A.” In a Sorites situation, “x is objectively determined to get an A” isalways false: since no sorting can fully satisfy the Ideal, no student is objectivelydetermined to get any grade. So the resolution of the Sorites is easy: All theconditionals are true, and two unconditional sentences are false. The argument isagain valid but unsound.

The third argument uses the predicate “x is assigned an A.” Now we haveto assume that the unlucky instructor, faced with this difficult situation, has infact given out the grades. The grades will presumably satisfy Absolute Paradigmsand Dominance, but violate the Epsilon Principle: Some students will getdifferent grades even though their aggregate scores differ by less than a singlepoint.

The resolution of this Sorites is also trivial: “100 is assigned an A” is true, “0is assigned an A” is false, and exactly one of the conditional premises is false. Thefalse premise will fall wherever the instructor decided to put the boundary betweengrades, and we are not in a position to identify that location. Different instructors,faced with the same grading situation, will put the boundaries in different locations,and all equally legitimately.The instructor could even flip a coin to decide where inthe borderline region the boundary should run, and no one would have a complaintagainst the method.

Unlike the first case, there is not merely an untrue conditional sentence, buta false one. But also unlike the first case, the location of the false premise isdetermined by extra-semantic facts. Nothing in the meaning (i.e. in the rules gov-erning the grading procedure) determines where the boundary runs. So if oneunderstands the claim that the grading system is vague as the claim that the rulesof the system do not determine, in every instance, exactly who gets what grade, thenthe system is still vague. If the system directed the instructor in a case like thisexactly where to draw the line, the system would not be vague.


The resolutions to these paradoxes are perhaps a bit too straightforward tobe entirely satisfying. After all, a paradox is only a paradox if one feels a stronginclination to accept all the premises, but an equally strong inclination to reject theconclusion. In the first and last cases, the resolution comes by rejecting one of theconditional premises. In the second case, all of the conditional premises are true,but only trivially so: they are true just because their antecedents are uniformlyfalse. So the question is: Why would anyone have been strongly attracted to thecollection of conditional premises in the first place? Surely not because, on onereading of the predicate, they are all trivially true.

There is already some explanation for our attraction to the conditionals, evenin the first and third forms of the paradox.As we have noted, in the first form, noneof the conditionals is false, so we might feel inclined to regard them all as true.Andin the third, exactly one conditional will be false, but we have no means to tell apriori which it is. So in either case, rejecting a particular conditional might beuneasy.

But I don’t think these explanations go to the heart of the matter. Ourattachment to the conditionals is more robust, and has a deeper source, than theseexplanations indicate. In order to understand that source, we need to look at theforms of the arguments that employ the subjunctive conditional rather than thematerial conditional.

In particular, we need to consider the form that uses the predicate “x isobjectively determined to get an A” together with the subjunctive conditional. Theunconditional premises remain the same, but the conditionals have forms like: “If94 were objectively determined to get an A, then 93.5 would be objectively deter-mined to get an A,” and so on.

Once again, in the Sorites situation the unconditional premises are false,since no one in this situation is objectively determined to get any grade. But thesubjunctive conditionals in this case are all true, and not for trivial reasons. Thesubjunctive conditionals are all true exactly because the Ideal contains the EpsilonPrinciple.

If 94 had been objectively determined to get an A, then there would havebeen some way to completely satisfy the Ideal, and every such way would assign 94an A. And since 93.5 is within epsilon of 94, every such way of satisfying the Idealwould also have assigned 93.5 an A. So 93.5 would also have been objectivelydetermined to get an A.

In this sense, the truth of these subjunctive conditionals is “analytic”: itfollows from the very content of the Ideal. Our insistence on (ambiguous) condi-tionals like “If 94 is an A then surely 93.5 is,” or “if this guy is bald, then surely thenext guy, with only one more hair is,” arises from the role that the Epsilon Principleplays in the Ideal.And properly understood, they are all true.As we survey the longSorites series, we recognize not only that we do not know where to put a linebetween, for example, those who are bald and those who are not, but also that thevery practice that governs the use of “bald” (defeasibly) enjoins us not to make adistinction between men who differ in only one hair.

Furthermore, if most judgments situations are favorable, we are typicallyable to obey the injunction. In most cases, we do not make a distinction in final

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grade between two students whose aggregate grades differ only by half a point.When we can, we take positive measures to avoid such a result, by letting theboundaries between grades vary from class to class.We also become acutely aware,when such a distinction must be made, that it is we ourselves, without any directionfrom the Ideal, who must decide where the ax is to fall. We give the grades with anuneasy conscience, since students who performed so similarly are being treated sodifferently. The arbitrariness of the distinction is manifest.

On the other hand, as we have seen, in favorable judgment situations, theIdeal does all the work: The students “sort themselves” into their grades. Andbecause of the Epsilon Principle, in these situations, students with nearly identicalaggregate scores never sort themselves into different grades. So as we survey thelong Sorites series, with no noticeable gaps, we feel at each location that we wouldnot make a distinction here if we could avoid it. Of course, if forced to sort into thebald and the non-bald, or the heaps and the non-heaps, then we cannot avoidmaking a cut somewhere, and we do.

The other two Sorites arguments, using the subjunctive conditional and theother predicates, do not yield much new. For in those cases, the first unconditionalpremise, “100 is an A student” or “100 is assigned an A,” is true, and a subjunctiveconditional with a true antecedent is usually considered to have the same truthconditions as the material conditional. So the series of subjective conditionals willdepart from being true at exactly the same point that the series of material con-ditionals did.

Even so, the subjunctive conditional allows us to use conditionals to get atelements in the Ideal. Consider, for example, the subjunctive conditionals “If 34were an A student, then 35 would be an A student,”“If 34 were assigned an A, then35 would be assigned an A” and “If 34 were objectively determined to be an A, then35 would be objectively determined to be an A.” All of these subjunctive condi-tionals have impossible antecedents: in no circumstance could the antecedent betrue. Nonetheless, all three subjunctive conditionals are unproblematically true,and not because the antecedents are impossible. They are all unproblematicallytrue because the Ideal contains Dominance as an absolute principle. Similarly fora case of two borderline A/B students with scores of 89 and 90: if the 89 were an A(in any sense), so too would the 90 be.

These sorts of subjunctive conditionals are what Kit Fine calls “penumbraltruths” (Fine 1975, 270). Fine has noted that there are “penumbral connections”among vague terms, and has tried to express the content of those connections insome penumbral truths. He then complains that approaches to vagueness that usemultiple truth values and truth-functional connectives cannot get the penumbraltruths right. And that is right: the material conditional “If 90 is an A student then89 is” is borderline, since the antecedent and consequent each are. The rightconclusion to draw is that penumbral connections are reflected in (non-truth-functional) subjunctive conditionals, not material conditionals.

The deeper conclusion is that the right way to get at penumbral connectionsis not through penumbral truths in the first place.What backs the penumbral truthsis the structure of the Ideal. It is the Ideal that specifies how different vague terms,and judgments employing them, are to be related to one another. The truths are


signs of the structure of the Ideal, but the direct route to the connections is throughthe Ideal itself.

The advantage to this way of understanding things is that there can bepenumbral connections, elements of the Ideal, that do not always give rise to truths.Such is the Epsilon Principle: It does constitute a penumbral connection, since itspecifies a constraint on how nearby borderline cases are to be resolved. But it isa defeasible condition, and so it does not always guarantee the truth of the claimthat nearby borderline cases are resolved the same way. By focusing on the Idealrather than the semantic status of sentences governed by the Ideal (such as thepenumbral truths), finer distinctions can be drawn.

We might finally note that the existence of the various predicates comportswell with some terminology that almost spontaneously appears when discussingvagueness. Attempts to construct a formal language suitable for vague predicatesoften introduce an operator read “determinately,” or make a distinction betweensentences that are true and those that are determinately true. The function of theoperator, and the distinction between the two kinds of truth, however, can beobscure.

A different understanding of the intuitive meaning of “determinately,”though, is available to us. The adverb is used not as a sentence operator, but as anindication that the predicate under consideration concerns what is objectivelydetermined. Thus “Determinately, Sue is either an A or a B, but she is neitherdeterminately A nor determinately B” can be reexpressed as “Sue is objectivelydetermined to get either an A or a B, but not objectively determined to get an Anor objectively determined to get a B.” This sentence can be true if Sue is objec-tively borderline between A and B. Similarly, “John is bald but not determinatelybald” could be rendered “John is bald, but not objectively determined to be bald.”If John is, for example, only slightly more hairy than a paradigm bald man in aSorites series, this could be true. The supposed distinction between truth anddeterminate truth would be an illusion: to say that a sentence is determinately trueis just to say that the sentence prefixed by “Determinately” is true, which is to saythat the sentence is true when the predicates are understood to refer to what isobjectively determined.

SHARP SEMANTIC CUTOFFS AND HIGHER-ORDER VAGUENESS

Our account so far has provided conditions under which a sentence can fail to havea classical truth value. In the Sorites situation, the account has even specified aplace in the Sorites sequence where the classical truth values stop and the truthvalue borderline begins. This is almost universally regarded as an objectionablefeature of any account of vagueness. Let’s begin by clearly separating three sorts ofobjections.

The first objection is the most radical. It contends that the existence ofsharp cutoffs indicates that what is been proposed is not an account of vaguenessat all. The language described has no vague terms: It has perfectly precise terms,albeit with nonclassical truth values. A vague language simply contains no sharp

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boundaries anywhere, so the very existence of such sharp semantic boundariesshows the language isn’t vague.

This objection is just a non sequitur. The account above describes a languagewith something vague in it: The method for assigning final grades on the basis ofaggregate scores is vague. That method does not determine, in all cases, exactlywhat grade a student with a perfectly determinate aggregate score, in a class ofstudents with perfectly determinate aggregate scores, should get. That methodadmits of borderline cases, in which different final grades can be assigned withequal (and maximal) legitimacy. It is the existence of such borderline cases thatsignals vagueness:Whether there is a sharp boundary between the borderline casesand non-borderline cases is neither here nor there. That rather concerns the ques-tion of whether the predicate “borderline” is itself vague.

Furthermore, the semantics does not require that there by any “sharp bound-aries” between the borderline and non-borderline cases. In a favorable situation ofthe second sort, there are objectively borderline cases, but no such exact boundary.The sorting of students into the categories “borderline” and “non-borderline” isobjective, but that sorting nowhere draws a line in the continuum of possibleaggregate scores. It is only when dealing with non-favorable cases, like the Soritessituation, that we have introduced machinery that allows such a line to be drawn.

The second sort of objection to sharp cutoffs is not really an argumentagainst them: It is a tu quoque argument meant to support the epistemicist. Theepistemicist seizes upon the existence of sharp semantic cutoffs between, say, truesentences and borderline sentences to argue by analogy for the acceptability ofa sharp cutoff between true sentences and false sentences. After all, the epistemi-cist complains, if you are allowed your sharp semantic divisions why aren’t Iallowed such sharp cutoffs? The main objection to epistemicism is the postula-tion of a line that separates the bald from the non-bald: What advantage is therein rejecting this in favor of a sharp line that separates the bald from the border-line bald?

Timothy Williamson uses this sort of tu quoque repeatedly. He even imaginesa very fanciful scenario, with several omniscient agents who are instructed toresolve all possible vagueness in a certain way, simply to argue that any account ofvagueness must be committed to some sort of sharp cutoff between something andsomething else. As Williamson states it: “Thus, if all are instructed to be conserva-tive, all will stop at the same point [i.e. all will stop answering ‘yes’ to ‘Is this a heap’when asked sequentially about a Sorites series].You do not know in advance whereit will come. It marks some previously hidden boundary, although it may be adelicate matter to say just what it is a boundary between” (Williamson 1994, 200).And then, in conclusion, “Once hidden lines are admitted, why should a linebetween truth and falsity not be one of them?” (Williamson 1994, 201).

Well, the account above does postulate a line between true sentences andborderline sentences, but the reason that gives us no reason to accept a linebetween true sentences and false sentences is that, in that account, it is not a linebetween true sentences and false sentences. And in the account above, there is nosuch line between true sentences and false sentences, between the students ofwhich it is true to say they are A students and the students of which it is false to say


they are A students. And furthermore, the account given above, which rejectsbivalence and epistemicism, is actually an account of how the language works.Williamson offers no such account, but only the bare assertions that maybe,somehow or other, our usage determines a meaning that draws a sharp linebetween true and false. And he certainly can’t help himself to our account indefense of his supposed sharp boundaries.

The third objection against sharp cutoffs is of an entirely different tenor. Itobjects not that the sharp cutoffs render the language precise, rather than vague, orthat they render the account no more plausible than epistemicism, but that thesharp cutoffs are simply implausible. In the case, we have discussed, the objectionis perfectly valid. It turns out, however, not to be particularly significant.

Why did we get, in the Sorites series, a perfectly sharp cutoff between stu-dents of whom “is an A student” is true and those of whom it is borderline?Because the terms in which the relevant part of Ideal was formulated were supposedto be perfectly precise. We have been imagining that the Ideal supplies an exactnumber, in the objective basis, for the Absolute Paradigms, and also that the valueof epsilon is perfectly precise. Since we used these values to determine where thetrue sentences end and the borderline sentences begin, we could draw a preciseboundary. We got precision out in the semantics because we put precision in in theIdeal. And this is clearly inaccurate, even in the very artificial case of assigninggrades.

When I assign grades, I use the system outlined here, but I do not have anyprecise values in the objective basis for the Absolute Paradigms, nor any precisevalue for epsilon. Both of these are themselves vague. And since these are vague,to try to draw a boundary between the true sentences and the borderline sentencesby subtracting the value of epsilon from the value assigned to the paradigm is notto succeed in drawing a sharp boundary. The vagueness in the language used toframe the Ideal leads to second order vagueness in the system governed by thatIdeal. Not only may there be (first-order) borderline cases, where the Ideal doesnot determine a grade, there can be (second-order) borderline cases where thesystem does determine whether or not a case is a (first-order) borderline case.

All of this is perfectly correct. But the overriding question that arises is: Sowhat? What difference does it make if there is second-order, or third-order, orhigher-order vagueness?

The existence of higher order vagueness certainly does not undercut theaccount we have given of vagueness. There is vagueness because there are border-line cases. We have seen how borderline cases can arise even if the language usedto “give the meaning” (state the Ideal) is perfectly precise. Making that languageitself vague rather than precise will not make the first-order vagueness go away.

There are cases where is it objectively determined that a student gets an A.The actuality, or mere possibility, of borderline A’s does not threaten this. Similarly,there are cases where a value for epsilon is appropriate. In the case of grading, if wecan satisfy the Epsilon Principle with epsilon equal to two, that epsilon is clearlyappropriate. No one could complain that epsilon was set too low. And similarly,there are values of epsilon that are clearly inappropriate. If epsilon is set at 0.01,and the grading boundaries are shifted to run them through gaps of only just that

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size, that is clearly inappropriate. But in some situations, there will be a wide rangeof settings for epsilon all of which are appropriate, and all of which would give thesame objective sorting. In those cases, the vagueness in setting epsilon makes nodifference. The machinery works perfectly well even though no exact value forepsilon has been set.

The point of the account so far is to show how there can be unproblematicsituations where grades are assigned even though the grading system is vague, andcan run into cases where arbitrary decisions must be made. If that account iscorrect, then there can be situations where the application of the Ideal is unprob-lematic even though, being vague, the Ideal may have its own problematic situa-tions. As a matter of empirical fact, my own grading often confronts first-ordervagueness and almost never second-order. I often have to decide borderlinegrades, but almost never find myself wondering whether a gap in the gradingdistribution is large enough to be used as a break between grades.

How many orders of vagueness are there? We have seen that a perfectlyprecise Ideal would give rise to first order but not second order vagueness. In thegrading situation, there is certainly second-order vagueness because the terms ofIdeal are themselves vague.What if we try to formulate an Ideal for, say, setting thevalue of Epsilon? Will it also be vague, leading to third-order vagueness, or not?

It all depends on the language used to specify that Ideal. If it is perfectlyprecise, then there will be second-order vagueness in the grading system but notthird-order. If it is vague, there will be more orders.

At this point, we have two choices. Either we think that some perfectlyprecise fragment of language is possible, or we do not. If we do think it possible,then how many orders of vagueness there are in any given case is contingent.We start explicating the vague terms by means of Ideals. If those have vague terms,we do the same. If we eventually manage to reach a point where an Ideal can bespecified in the precise fragment of the language, then the iteration stops. It stopswhere it does, at the fourth or eighteenth level, because that’s where the precisefragment suffices.

Alternatively, the iterations may never end, either because there is no precisefragment of the language at all, or because it never provides the resources tospecify the relevant Ideal. If one is inclined to think that a precise language is inprinciple impossible, that all language is vague, then one will obviously concludethat there are unending orders of vagueness. This would be an interesting result,but not an especially important one. Or at least, the account of vagueness we haveoffered is not hostage to this question. However, it turns out, the basic account ofvagueness on offer will be unchanged.

JUDGMENT SITUATIONS

We have used the practical problem of assigning grades on the basis of aggregatescores as an instructive example of how vagueness may arise in a system. Inoutlining the normative rules governing the assignment of grades, we initially alsomade an unrealistic idealization: We imagined that various elements of theIdeal were themselves precisely defined when in reality they are vague. But the


institution of grading involves yet another peculiarity that we have made use of,and which has no exact analog when it comes to sorting collections of grains intoheaps and non-heaps, or sorting men into the bald and the non-bald. For in grading,there is always a precisely defined judgment situation, a specific set of students withspecific aggregate scores who must be sorted into the different final grades. Thejudgment situation has played an important role in our account: The very samestudent, with the very same aggregate score, can be objectively determined to getan A in one class and objectively determined to get a B in another. By analogy, wewould have that a particular man could be objectively determined to be bald in onejudgment situation (where the only salient break between the bald and the non-bald occurs above him in hair-distribution space) and objectively determined to benot bald in another (where the only salient break is below him).3 So whether it istrue or false to say he is bald would depend on the judgment situation in which heis considered.

There is a bad reason to be worried about this that we have already noted. Itis a consequence of this account that “Sam is bald” can change from true to false(or borderline to true or to false) simply by relocating Sam from one judgmentsituation to another. But, the worry goes, how can “Sam is bald” change its truthvalue without the hair on Sam’s head changing? The truth value of “Sam is bald”must supervene on the state of his head, and so cannot change just because he isbeing considered in different contexts.

If baldness were any sort of property at all, this would be a real problem. Forif baldness was any sort of property at all, then it would presumably be an intrinsicproperty of Sam’s head, and hence Sam could not lose the property without hishead changing its physical state. And if baldness were any sort of property at allthen “Sam is bald” would be true just in case Sam had the property and false justin case he didn’t. So the sentence could not change truth value without an intrinsicchange to Sam.

We have already rejected the claim that baldness is a property of any sort,and a fortiori that it is an intrinsic property. We therefore have no commitment tothe supervenience of the truth value of “Sam is bald” on the state of Sam’s head.But one still may worry that in real life (unlike in grading) we are not confrontedwith clear-cut judgment situations, so the whole machinery we have advocated canget no purchase. The only completely objective judgment situation relevant to thebaldness of men, for example, is the situation that includes all actual men from alltimes. But that distribution is clearly a Sorites distribution: It will contain no salientgaps that could be used to separate the bald from the non-bald.

The problem of the judgment situation is a problem, but it ought not to beoverstated. In many real-life situations, the relevant class to be sorted is uncontro-versial. If I have to sort my silverware into the little bins in the drawer, theexistence of sporks somewhere in the universe need not detain me. If the shoe storemanager tells an employee to put the shoes on one side and the boots on the other,

3. The man in question would have to be hairier than an Absolute Paradigm bald man, andbalder than an Absolute Paradigm non-bald man.

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the existence elsewhere of borderline cases between the two, or even worse, themerely possible existence of borderline cases, will not keep “This is a shoe” frombeing objectively determined to be true.

Sorites situations occur when the judgment situation contains too manyitems that are too closely spaced in the objective basis. One might worry about theopposite problem: Isn’t a shoe sitting alone in the desert, with nothing else tocompare it with, still truly a shoe? This is less of a problem: Absolute Paradigmshoes, we think, mark out a region in “shoe space” which, by Dominance, guaran-tees that many things are truly shoes. It is important to note in this regard that thegreatest utility comes from Absolute Paradigms that are not at extremes in theobjective basis. The student who gets a 100 is not a useful Absolute Paradigm A,since few grades will be settled by comparison to her. What one wants is the lowestgrade that can count as a paradigm A, and the highest that can count as an F.These,plus Dominance, will settle the most cases and leave the smallest range of border-line cases. (There is, however, some counter-pressure here: the smaller the gapbetween a Paradigm bald and Paradigm non-bald person, the less likely it is that anepsilon-sized gap in the distribution will run through the region between them thatcan serve to objectively determine the sorting.)

So it is not clear after all that vagueness about the judgment situation will beof much material consequence in usual situations. There is, however, one circum-stance where the option of changing the judgment situation seems quite important.Suppose one is confronted with a Sorites series. Suppose, to vary the example, oneis confronted with a very long line of color tiles which change, by insensiblegradations, from fire-engine red to a pastel pink. Starting at the red end of theseries, one is to walk along and call out the color of the tiles.The only colors one cancall are “red” and “pink.”

The situation has been described so that the judgment situation includes allof the color tiles. But still, as one stands at one end of the series there is a stronginclination to include only the first part of the series, the part in view, in thejudgment situation. Relative to that judgment situation, all of the tiles are objec-tively determined to be red: the Epsilon Principle can be satisfied by counting allthe tiles as the same color, and the only color consistent with the Absolute Para-digms in that situation is red. Once the judgment situation expands to includeparadigm pink tiles, there will be no way to satisfy the Ideal, but at one end of theseries there is little pressure to include the distant pink tiles in the consideration.

As we walk down the series of tiles, there will be a strong inclination to treatthe series as a sequence of small judgment situations rather than as a single largejudgment situation. Suppose we include in the situation only the next ten or twentytiles, along with the last twenty that we have already passed. At the beginning,all the tiles are objectively red. Once the lightest paradigm red tile is far enoughbehind us not to be included in the situation, all the tiles will be objectivelyborderline, so we could properly either judge them all to be red or all pink. Thereis probably some mental inertia that will induce us to continue calling the tiles red,but that is a matter of contingent psychology. As we pass through the borderregion, well beyond the last paradigm red and well before the first paradigm pink,there will be increasing pressure to call all of the tiles in the present judgment


situation pink, and when the first paradigm pink tile comes into consideration, allthe tiles in the restricted judgment situation will be objectively determined to bepink. So at some time we will stop calling the tiles red and start classing them pink.But if this is how we treat the situation, the switch from “red” to “pink” will not beoccasioned by ever making the judgment of two adjacent tiles that one is red andthe next is pink. Rather, the shift will be associated with a change in psychological“set”: the tile before us will appear to be pink, but so will many of the tiles behindus, which we had already denominated red. If we were allowed to, we would changeour earlier denominations given our present judgment situation.

So given the constraints of the task, there will be a particular point at whichwe switch from saying “red” to “pink,” but that change will not come because weever have the impression that the Epsilon Principle is being violated. It neverseems to us that a tile is a different color from an adjacent tile, it rather seems, allof a sudden, that a whole set of tiles (the present small judgment situation) are adifferent shade from an immediately previous set even though the two sets havemost of their tiles in common. Indeed, it is perfectly possible that we will suddenlyfeel as though the color of a set of tiles has changed even though it is the very sameset:The change in appearance is a subjective change of the set as a whole, much likethe subjective change of appearance in a Necker cube. And just as we know, whenwe study the Necker cube, that the change in its appearance to us does not reflectany change in the drawing, so too we know that a change in psychological set is achange in us, not a change in the objects being observed.

This is another reason why we are so loath to reject any of the conditionalsused in the Sorites argument. For not only is it never the case that one tile isobjectively determined to be red and the adjacent tile objectively determined to bepink, and never the case that it is true that one tile is red and true that the adjacenttile is pink, it is also never the case that one tile will appear to us to be red andsimultaneously the adjacent tile appear to us to be pink. It seems to follow that ifthe first tile appears to us to be red, then they will all appear to be red. Like the frogin the slowly heated pot who boils to death rather than jumping out because thetemperature rises so slowly, it seems as though we will be induced to call even thepale pink tile “red” since we could never distinguish between two adjacent tiles.But the conclusion does not follow: It only follows that when we finally change ourmind about the color, we will at the same time change our mind about some tileswe had already called “red.”

So yet another diagnoses of the appeal of the conditionals used in the Soritesis this: The conditionals are each individually appealing because for each condi-tional of the form “If n is red, then n + 1 is red” there is a judgment situation inwhich it is, unproblematically, true. Furthermore, these judgments situations are allproper subsets of the complete Sorites situation:They can be constructed by simplydeleting some of the cases to be judged. But although there is, for each conditional,a situation in which it is true, there is no situation in which they are all simulta-neously true. We balk at rejecting any particular conditional because we can easilyfind a judgment situation in which that conditional ought not to be rejected.

According to this analysis, what fails in the Sorites argument is its validity.Since the truth value of some claims may depend on the particular judgment

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situation, the claims themselves have to be indexed to the situation. The validity ofmodus ponens should be expressed: If in some judgment situation A is true, and inthe same judgment situation A ⊃ B is true, then in that judgment situation B is true.To argue that the truth of A in one judgment situation and the truth of A ⊃ B inanother judgment situation entails the truth of B in any judgment situation is tocommit something like a fallacy of ambiguity.

In the Sorites argument, the first unconditional premise, “Tile 1 is red,” maybe objectively determined to be true (in some judgment situation), and every oneof the conditionals “If tile n is red then tile n + 1 is red” objectively determined tobe true (in some other judgment situations)4, and the final unconditional premise“Tile N is red” objectively determined to be false (in yet another judgment situa-tion). It is not quite accurate to say that this is because “red” is ambiguous: “red”may be governed by a single Ideal, just as the grading system is. Nor is it right to saythat “red” is indexical:The truth value is not determined by the particular tokening.But the dependence of the semantic value on a judgment situation can give rise toinvalid arguments that look, on the surface, to be valid.

The usual Sorites paradox, then, arises because we are asked to confront asort of judgment situation that we are commonly able to avoid. Since we are usuallysomewhat free to delimit a judgment situation, we can restrict our attention to asubset of cases that admit of a sorting that satisfies the Ideal. Although the Idealcannot be satisfied in every judgment situation, latitude in determining the relevantjudgment situation at a given time can often allow us to avoid violating the Ideal.And since we are not often required to make judgments that violate the Ideal, wecan easily come to feel that all of the principles mentioned in the Ideal are equallyessential to it: We can easily overlook the fact that the Epsilon Principle is adefeasible constraint on sorting while Dominance and Absolute Paradigms are not.

In being asked to sort all the members in a Sorites series at once, we are beingasked to apply the norms governing the sorting system to the least favorablejudgment situation.This is an obvious source of distress, and we are likely to remainsomewhat dissatisfied no matter how we sort the case, since the arbitrary nature ofthe boundaries used in the sorting will be manifest. But it is not warranted toconclude from this circumstance that the sorting system, or the Ideal governing it,

4. There is a small technical detail about how to understand the semantic value of, say,“If tileM is red then tile M + 1 is red” in a judgment situation where both tile M and tile M + 1 areobjectively borderline. Suppose, for example, the judgment situation contains only those two tiles,and they fall between the lightest paradigm red and the darkest paradigm pink (not within epsilonof either). And suppose the shades of the two tiles are within epsilon of each other. Then they areboth objectively borderline: denominating them both “red” or both “pink” would satisfy the Ideal.If the connective in “If tile M is red then tile M + 1 is red” is treated truth-functionally, then theconditional may be borderline: at least some truth-functional extensions of the classical connectivewill give this result. But on the other hand, one could treat the molecular sentence supervaluation-ally. Since in every sorting into red and pink that satisfies the Ideal, “If tile M is red then tile M + 1is red” comes out true, we can say that the conditional is objectively true, and similarly for “Tile Mis red if and only if tile M + 1 is red”. I see no reason to choose between these ways of ascribing asemantic value to the conditional.As an abstract matter, both the truth-functional connectives andthe method of supervaluation exist. In a particular instance, it may be clear that one or the otheris meant.


is somehow incoherent or fatally flawed. In most judgment situations, the sortingprocedure works fine, and renders objectively determined results. To concludefrom a Sorites situation that there is something fatally defective in the predicate“heap” or “bald” would be as much of an overreaction as concluding that there isnever any coherent way to assign grades just because sometimes arbitrary decisionsmust be made in assigning them.

In favorable judgment situations our methods for sorting students into finalgrades, or collections of grain into heaps and non-heaps, or silverware into forksand spoons, works flawlessly.The sorting, given the Ideal, is objectively determinedby the distribution in the objective basis. And even in unfavorable situations, somecollections of grain are unproblematically heaps and some men are unproblemati-cally bald. I see no reason, in such favorable circumstances, to withhold the seman-tic status of truth from the corresponding sentences: If someone is objectivelydetermined to be bald, then it is true to say he is bald.And if someone is objectivelydetermined not to be bald, then it is false to say he is bald. The price of assigningclassical truth values in these favorable cases, though, is that it leaves us with theunfavorable cases: The objectively borderline cases and the situations where theIdeal cannot be met. I see no reason not to introduce a third semantic value forthese cases, and even more semantic values if there is higher-order vagueness. Thisreflects our preanalytic intuition that in some cases it is neither true nor false to sayof someone that he is bald. Nothing in the Sorites situations forces us to abandonthis commonsense view.

REFERENCES

Braun, David, and Theodore Sider. 2007. “Vague, so Untrue.” Nous 41: 133–56.Fine, Kit. 1975. “ Vagueness, Truth and Logic.” Synthese 30: 265–300.Horwich, Paul. 1990. Truth. Oxford: Basil Blackwell.Maudlin, Tim. 2004. Truth and Paradox. Oxford: Oxford University Press.Williamson, Timothy. 1992. “Vagueness and Ignorance.” Proceedings of the Aristotelian Society

Supp. 66: 145–62.———. 1994. Vagueness. London: Routledge.

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Grading, Sorting, and the Sorites

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