Cook_Vagueness and Mathematical Precision

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Mind, Vol. 111 . 442 . April 2002 Oxford University Press 2002

Vagueness and Mathematical PrecisionRoy T. Cook

One of the main reasons for providing formal semantics for languages is that themathematical precision afforded by such semantics allows us to study and manipu-late the formalization much more easily than if we were to study the relevant naturallanguages directly. Michael Tye and R. M. Sainsbury have argued that traditional set-theoretic semantics for vague languages are all but useless, however, since this math-ematical precision eliminates the very phenomenon (vagueness) that we are tryingto capture. Here we meet this objection by viewing formalization as a process ofbuilding models, not providing descriptions. When we are constructing models, asopposed to accurate descriptions, we often include in the model extra machinery ofsome sort in order to facilitate our manipulation of the model. In other words, whilesome parts of a model accurately represent actual aspects of the phenomenon beingmodelled, other parts might be merely artefacts of the particular model. With thisdistinction in place, the criticisms of Sainsbury and Tye are easily dealt withtheprecision of the semantics is artefactual and does not represent any real precision invague discourse. Although this solution to this problem is independent of any par-

ticular semantics, a detailed account of how we would distinguish between represen-tor and artefact within Dorothy Edgingtons degree-theoretic semantics is presented.

1. Introduction

One of the greatest advantages of formal semantics is that it allows us tostudy sometimes messy and imprecise chunks of natural language indi-rectly by examining the clean, precise mathematics provided by thesemantics. Roughly, the logician first provides a chunk of mathematicsthat is supposed to represent the semantics of the language, and he thenstudies the mathematical construction instead of getting involved withthe possibly irrelevant complexities of the natural language itself. Were

it not for the precision of mathematics, it is unlikely that this projectwould have got offthe ground, as this precision is what allows us toprove general theorems and draw far-reaching conclusions from ourmetalogical work.

Recently, however, R. M. Sainsbury and Michael Tye have argued thatthe mathematical precision of formal semantics is actually a drawbackin certain cases. In particular, they argue that the precision afforded bythe best available semantics for vagueness eliminates the essential fea-


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226 Roy T. Cook

ture of vague discourse its imprecision. They believe that this preci-sion is not a result of idealizing away unnecessary messiness but comesinstead from idealizing away the vagueness we are trying to understandin the first place. If this is right, they argue, then we need to incorporatevagueness into the semantics itself, thereby greatly increasing the diffi-culty of studying vague languages.1

I argue that Sainsbury and Tye fail to distinguish between two dis-tinct theses. They do convincingly argue that vagueness must be incor-porated into our account somehow, otherwise we risk replacing thevagueness with unwanted and unwarranted precision. They are unsuc-

cessful, however, in arguing that the mathematicsof the account mustbe vague. I suggest instead that the idea that logic does not providedescriptions of linguistic phenomena, but instead provides us withmodelsof those phenomena, allows the relation between the languagebeing formalized and the formal semantics to involve vagueness in anessential way.2I use degree-theoretic semantics as an example of howsuch an account might proceed, showing how the logic as modellingapproach allows us to have a mathematically precise formal semanticswhile avoiding the charge that we have left vagueness out of the accountaltogether.

2. Sorites and the degree-theoretic approachThe difficulties with vagueness are strikingly illustrated by the Soritesparadox. Consider the predicate bald. Certainly someone with no hairon his head is bald. The predicate bald, however, is what CrispinWright () calls tolerant: A difference of one hair more or lessshould not transform a man who is bald into a man who is not bald orvice versa. If this is right, then we can construct the following classicallyvalid argument:3

1 A semantics that is mathematically difficult is not one that is useless. In fact, many semantics,including much of the work on counterfactuals, utilize vague notions without thereby rendering

the mathematics intractable. The point here is merely that a semantics that does not invoke vagueconcepts will, prima facie, be more perspicuous than one that does.

2 Of course the relation between the language being modelled and the formalization cannot bevague, since vagueness, at least as traditionally conceived, is a property of predicates, not relations.The vagueness will come into play when we carefully articulate the relation between model andmodelled. In a sense, the relation will be defined, or at least described, in terms of vague predi-cates, in a manner similar to the way we can arrive at the vaguerelation is cleverer thanby a con-struction building on the vague predicate is clever.

3 Even if we restrict ourselves to intuitionistic or relevance logics we still have a problem, sinceall that is needed for the Sorites argument is repeated applications of modus ponens.


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Vagueness and Mathematical Precision 227

(P) A man with hairs on his head is bald.

(P) If a man with hairs is bald, then a man with hair is bald.

(P) If a man with hair is bald, then a man with hairs is bald.

: : : :

: : : :

(P) If a man with hairs is bald then a man with hairs is

bald

(C) A man with

hairs on his head is bald.

hairs on a head are more than enough to prevent correct applicationof the predicate bald, however.

In Vagueness by Degrees() Dorothy Edgington proposes adegree-theoretic semantics that explains how the conclusion of theSorites argument can be false yet nevertheless vindicates classical logic.She begins with the idea that, instead of sentences being either true orfalse, there are different degrees of truth that sentences can have.4Edg-ington calls the degree of truth of a sentence P its verity, symbolized asv(P). v(P)is a real number in the interval [,].

Edgington examines the similarities between degrees of belief anddegrees of truth.5Since the two notions behave similarly, and degrees ofbelief can be fruitfully formalized as obeying rules identical in form tothe rules of the probability calculus, Edgington concludes that degreesof truth should be formalized in the same way. Thus, we obtain the fol-lowing rules for the propositional connectives:

v(A) = v(A)

v(A!B) = v(A) v(BgivenA)

v(A"B) = v(A) + v(B) v(A!B)

4 Actually, Edgington speaks in terms of degree of closeness to clear truth, not degrees of truth(see Edgington , pp. ), and she accepts that every sentence has one of two truth values.Nevertheless, talk of degrees of truth seems appropriate here, since it is within the realm of thesentences that are near to (but are not) clear truths that all of the relevant action takes place(thanks go to an anonymous referee who pointed this out to me).

5 It is not actual degrees of belief, but rational degrees of belief, that are usefully formalized us-ing the rules of the probability calculus. It is possible that someones actual degree of belief insome conjunction could greater than his degree of belief in either conjunct, but one rationallyought not to be in such a state. This observation only strengthens Edgingtons analogy between de-grees of belief and degrees of truth, however, since it seems more likely that degrees of truth willbehave similarly to rational degrees of belief than to the actual, often irrational, degree of certaintywe assign to our beliefs.


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228 Roy T. Cook

The use of the locution BgivenAprevents the semantics from beingdegree-functional,6since the verity of B givenAmay not be a functionsolely of the verities ofAand of B. Edgington argues that v(BgivenA)can be computed for most cases of the form Fxgiven Fyaccording tothe following rule:

v(Fxgiven Fy) = if v(Fx)v(Fy)

v(Fxgiven Fy) = v(Fx) v(Fy) otherwise.

Notice that for this particular form of BgivenAstatement, where the

same predicate occurs twice, we have regained degree-functionality.Interestingly, Sorites arguments, which motivated the formulation ofdegree-theoretic semantics in the first place, consist of sentences thatwill be handled degree-functionally in this way.

There are other specific cases where we regain degree functionalitythat are convenient to notice now, since they will be used below. 7First,ifA andB are independent, in the sense that v(A givenB)=v(A) andv(B givenA) =v(B), then the rule for conjunction simplifies to:

v(A!B) = v(A) v(B)

And as a result the rule for disjunction becomes:

v(A"B) = v(A) + v(B) [v(A) v(B)]Similarly, ifAand Bare mutually incompatible, that is, if v(Agiven B)= v(BgivenA) = ,8then the rule for disjunction is even simpler:

6 Most other degree-theoretic semantics for vagueness, such as Kenton Machinas in Truth, Be-lief, and Vagueness() are not only degree-theoretic (i.e. based on assigning degrees of truth tosentences) but are also degree-functional, since the verities of compound statements are functionsof the verities of the component sentences. For example, Machina gives the following rule for con-

junction (recast into the notation used here):

v(P !Q) = min(v(P), v(Q)), (Machina , p. )

For more on the degree-theoreticity versus degree functionality issue, see Edgington ().

7 One might be tempted to argue that we do not necessarily have true degree functionality here,as we need to know the relationships between the sentencesAand Bbefore we can determinewhether or not the simpler, degree-functional rules apply. Although the definition of incompati-bility and independence do contain instances of the problematic, non-degree-functional Bgiven

Aexpressions, it is important to notice that often we can determine whether two sentences are in-dependent or incompatible based solely on their meaning and logical form. Thus, we can often de-termine that these rules are the correct ones to apply without having to actually evaluate anyexpression of the form BgivenA.

8 Notice that the inclusion of both v(Agiven B) = and v(BgivenA) = is not redundant, aswe might have one without necessarily having the other. For example, given two sentences FaandFbsuch that v(Fa) = and v(Fb) = , the rule for the verities of givenstatements for this type ofcase tells us that v(Fagiven Fb) = , yet v(Fbgiven Fa) = .


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v(A"B) = v(A) + v(B)

On Edgingtons rules every classical tautology gets a verity of , andevery classical contradiction gets a verity of . We get even more, how-ever, as the following theorem, proved by Edgington and called theConstraining Property, demonstrates:

Let uv(A), the unverity ofA, be v(A).

If P, P, Pn#Q, then uv(Q)uv(Pj) (jn)

In other words, given a classical valid argument, the degree of falsity ofthe conclusion is no more than the sum of the degrees of falsity of thepremisses. The Constraining Propertyallows us to rely on classical rea-soning when the premisses do not have a verity of but neverthelesshave high verity. As long as there are not too many premisses, and theyare not too far from absolute truth, then the conclusion of the argu-ment will have relatively high verity.

Edgington uses the Constraining Propertyto solve the Sorites Para-dox. Even if each premiss of the form If a man with nhairs is bald thena man with n+hairs is baldhas a verity close to , this does not implythat the conclusion should have verity close to , as the degrees of fal-sity, especially in long arguments, add up. Consider the version of theSorites discussed earlier.

(P) A man with hairs on his head is bald.

(P) If a man with hairs is bald, then a man with hair is bald.

(P) If a man with hair is bald, then a man with hairs is bald.

: : : :

: : : :

(P) If a man with hairs is bald then a man with hairs is

bald

(C) A man with hairs on his head is bald.

If, for each value of n, we assign A man with nhairs is balda verity of ( ), and we define the conditional in terms of negation and con-junction, then it follows that the verity of each of the ,,,conditional premisses is ., yet the verity of the conclusion is.

n


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230 Roy T. Cook

3. Sainsburys criticism: against set theoryIn Concepts Without Boundaries() R. M. Sainsbury argues thatset theory is incapable of providing an adequate semantics for vague-ness on the grounds that sets, by their very nature, impose sharpboundaries, yet vagueness is essentially the lack of the same sort ofsharp boundary. Thus the nature of set theory is fundamentally at oddswith the nature of vagueness, and it is implausible that set theory canprovide an adequate account of vagueness.

Sainsbury begins by pointing out that classical semantics identifiesthe extension of a predicate with a set, but he observes that this

approach will not work for vague language:Suppose there were a set of things of which redis true: it would be the setof red things. However, redis vague: there are object of which it is neitherthe case that redis (definitely) true nor the case that redis (definitely) nottrue. Such an object would neither definitely belong to the set of red thingsnor definitely fail to belong to this set. But this is impossible, by the very na-ture of sets. Hence there is no set of red things. (Sainsbury , p. )

Of course almost no one claims that classical semantics adequatelyaccounts for the truth conditions of vague sentences. Sainsbury, how-ever, thinks the root of the problem is that the semantics draws a sharpline between truth and falsity, and any account that allows the collec-

tion of definite truths to be a set, with sharp boundaries, is just as inad-equate. He argues that this objection can be generalized to anysemantics based on set-theoretic constructions, since the addition ofmore finely grained divisions does nothing to alleviate the problem. Hewrites that:

you do not improve upon a bad idea by iterating it. In more detail, sup-pose we have a finished account of a predicate, associating it with some pos-sibly infinite number of boundaries, and some possibly infinite number ofsets. Given the aims of the description, we must be able to organize the setsin the following threefold way: one of them is the set supposedly correspond-ing to the things of which the predicate is absolutely definitely and unim-pugnably true, the things to which the predicates application is untainted by

the shadow of vagueness; one of them is the set supposedly corresponding tothe things of which the predicate is absolutely definitely and unimpugnablyfalse, the things to which the predicates non-application is untainted by theshadow of vagueness; the union of the remaining sets would correspond toone or another kind of borderline case. So the old problem re-emerges: nosharp cut-offto the shadow of vagueness is marked in our linguistic practice,so to attribute it to the predicate is to misdescribe it. (Sainsbury , p. )

Sainsburys overall strategy is clear. Any semantics that utilizes set the-


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ory will, because of the precise nature of sets, impose sharp boundariesof some sort. Since no sharp boundaries are present in our actual lin-guistic practice involving vague predicates, a semantics that imposessuch sharp boundaries is a misdescription of the phenomena. It is thisemphasis on description that I will capitalize on below.

Sainsburys objection, if correct, is devastating to the practice ofstudying languages by constructing formal semantics. If set theory isruled out, and any construction isomorphic to a set-theoretic construc-tion is also eliminated by similar arguments, then it is unclear that thereare any resources left. There are certainly no other mathematical

resources, since all of modern mathematics can be embedded into settheory using well known constructions. If mathematics is banned fromour account of vagueness, then Sainsbury owes us some explanation ofwhat resources can replace it. He addresses this problem when he asks:

If standard set-theoretic descriptions are incorrect for boundaryless con-cepts, what kind of semantics are appropriate? A generalization of the con-siderations so far suggests that there is no precise description of vagueness.So what kind of description should be offered? More pointedly, I hear a cer-tain kind of objector say: We cant even tell what boundarylessness is untilyou give us your semantics. (Sainsbury , p. )

Sainsbury goes on to sketch a semantics that he calls homophonic. Theidea is that the same vague expressions we are concerned with in theobject language can occur in the metalanguage, allowing the formula-tion of truth conditions such as:

redis true of something iffthat thing is red. (Sainsbury , p.)

Of course, once we allow vague predicates such as redto occur explic-itly in our semantics for vagueness we have abandoned the mathemati-cal precision that has been so useful elsewhere.

4. Tyes criticism: against precise degrees

In Vagueness: Welcome to the Quicksand ( a) Michael Tyepresents a criticism that is more pointed than Sainsburys general com-

plaints regarding set theory. He begins along the same lines, however,objecting to the precise borderlines that degree-theoretic semanticsprovides:

One serious objection to this view is that it really replaces vagueness with themost refined and incredible precision. Set membership, as viewed by the de-grees of truth theorist, comes in precise degrees, as does predicate applica-tion and truth. The result is a commitment to precise dividing lines that isnot only unbelievable but also thoroughly contrary to what I [call] ro-


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232 Roy T. Cook

bustor resilientvagueness. For it seems an essential part of the resilientvagueness of ordinary terms such as bald, tall, and overweightthat in So-rites sequences there is indeterminacy with respect to the division be-tween the conditionals that have the value , and those that have the nexthighest value, whatever it might be. It is this central feature of vaguenesswhich the degrees of truth approach, in its standard form, fails to accommo-date, regardless of how many truth-values it introduces. (Tye a, p. )

There is nothing here that was not also in Sainsburys objection: Thedegree-theoretic approach introduces sharp boundaries where there arenone in the actual linguistic phenomena. Tye goes on to consider an

objection more specific to the degree-theoretic approach, however.Tye first points out that, as a man grows an inch, the value assignedto him as an instance of tallshould increase. He argues that if this iscorrect, then the degree-theoretic approach:

does have a very counterintuitive aspect. On the above view, it is notwholly true that the worlds heaviest man is overweight, or that the worldstallest man is tall. This seems preposterous, however. The worlds heaviestman, now deceased was Robert Earl Hughes, who weighed no less thanpounds The worlds tallest man, again deceased, was Robert Ward-low, with a height of feet inches. There could have been someone heavieror taller, and likely one day there will be, but so what? Intuitively, it surelydoes not follow that it is not quite true that Mr. Hughes was overweight or

that Mr. Wardlow was tall. Moreover, at an intuitive level, both of theseclaims are surely just plain true. But what could this possibly mean, on thedegrees of truth approach, if they are not wholly true? (Tye a, p. )

In other words, it is possible that there could be a man taller than the foot, inch Wardlow, and it is likely that someday there will be some-one taller, say feet. If this is right, then the degree of truth assigned tothe claim that a foot person is tall should be higher than that given toWardlow as an instance of tall. is the highest value, so it follows thatWardlow must get a value less than , and so, contrary to our intuitions,it is not wholly true that he is tall. Since there is no precise upper limitto height,9no human will get a value of for the claim that he is tall.10

9 This statement needs clarification. Of course, feet is certainly an acceptable upper limiton the height of a human being, and a precise one. The point, however, is that the distinction be-tween heights that humans could possibly reach and heights that are not possible is vague. If someheight is within the range of possibility, then presumably a height only one billionth of an inchtaller would be possible as well.

10 It should be noted that this argument is not an objection to degree-theoretic semantics basedon its excessive precision, but is an objection to the fact that the degrees of truth are ordered insuch a way that there is a precise maximal value ( ) that predications of xis tallcan never reach.


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Like Sainsbury, Tye concludes that these problems stem from the factthat we have not incorporated vagueness into the semantics itself. InSorites Paradoxes and the Semantics of Vagueness(Tye b) he pro-vides a semantics that incorporated vagueness directly into the meta-mathematics, formulating an alternative set theory where the sets arewhat he calls genuinely vague items. The details of his solution to theparadox utilizing this vague set theory need not concern us here. Whatis important is that Tye, like Sainsbury, argues that there is a mismatchbetween the precision found in degree-theoretic semantics and the lackof precision in vague natural language. Tye, like Sainsbury, concludes

that this mismatch represents a fatal flaw in the degree-theoreticapproach, and he provides an alternative semantics that abandons pre-cision, opting instead for a purposely vague metalanguage. As a result,we get a semantics that, while doing justice to the intuition that vague-ness must come into the account somehow, saddles us with a semanticsthat seems impossible to study with any rigour. An alternative approachthat allows us to respect Sainsbury and Tyes insights, yet also allows usto formalize a semantics amenable to precise mathematical study,would be preferable.

5. The solution: logic as modelling

On the traditional view of the role of formalization, where the goal is togive a description of the workings of the language in question, the criti-cisms discussed in and are devastating.11Fortunately, the tradi-tional view of formalization is not the only available stance. The mainquestion is what exactly the connection is between the formal seman-tics and the actual everyday discourse that we are attempting to expli-cate and/or understand. A number of answers are possible:

11 Of course, semanticists have in the past had many different goals when presenting formal se-mantics for various languages, and they have not always had accurate descriptions of truth condi-tions, etc., in mind. In the case of vague language, however, presumably the goal of much of thework is to provide something like an accurate description of the truth conditions of sentences ofthe language, since we are formulating the semantics with the express purpose of gaining insightsinto, and learning how to eliminate or avoid, the Sorites paradox.

Since this paper is as much a study of the role of formalization in general as it is a defence of a par-

ticular formalism against charges of inappropriate mathematical precision, however, the inclusionof this problem (and its eventual solution) is apropos. At any rate, Tye himself thinks both of hicriticisms stem from the same problem, the failure to incorporate vagueness into the semantics it-self.


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234 Roy T. Cook

(1) The Traditional View: Logic as Description

Degree-theoretic semantics is an attempt to describe what is reallygoing on vis--visthe truth conditions, meaning, etc., of the variousassertions involved in Sorites-type arguments, or talk involving vaguepredicates in general. On this view, every aspect of the formalism corre-sponds (at least roughly) to something actually occurring in the phe-nomenon being formalized, and the semantics is really just an accountof what has been going on all along.

(2) The Logic As Modelling View:

Degree-theoretic semantics provides a good model of how vague lan-guage behaves. On this view, the formalism is not a description of whatis really occurring but is instead a fruitful way to represent the phenom-enon, that is, it is merely one tool among many that can further ourunderstanding of the discourse in question. In particular, not everyaspect of the model need correspond to actual aspects of the phenome-non being modelled.

(3) The Instrumentalist View of Logic:

Degree-theoretic semantics might mirror little or nothing going on inthe real world. Rather, then entire mathematical machinery behind theaccount is a fiction, representing nothing actually occurring and givingus no real explanation of the behaviour of the language being studied.Other than the language of the formalism roughly matching up withthe natural language being investigated, no aspect of the formalizationhas any connection to anything really involved in the truth conditionsof the discourse.12

It should be noted that this tripartite division is a caricature, simplify-ing the reality of the situation in two ways. First, the attitudes one couldtake towards the connection between formalization and the subjectmatter being formalized do not split cleanly into three categories, butinstead are located on a continuum with () and () as endpoints. Sec-

ond, these endpoints are themselves caricatures, representing extremeversions of either end of the spectrum. It is unlikely that anyone hasendorsed a view as extreme as either of these options, at least not as

12 Notice that the instrumentalist approach is consistent with the claim that the semantics,while actually matching up with nothing in the world, could still serve as a formalization of theheuristics we use when reasoning. For example, one could argue that possible world semantics formodal logic, while not actually corresponding to any possible worlds other than the actual one, isnevertheless a useful formulation because we think as if there were other possible worlds when weevaluate modal claims.


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they are expressed here. Nevertheless, most philosophers who haveendorsed an opinion on the connection between formalization andnatural language take something similar to () or () (usually ()) astheir view.13

Option () renders our formalizations philosophically unilluminat-ing. If this is all that we get, then we should pay less attention to thesemantics and devote more time to finding an explanation of what goeswrong in the Sorites paradox. In short, a semantics that tells us whatinferences to accept and reject but fails to provide any insight into whywe ought to reason in this fashion might be helpful as a practical tool

but is next to useless philosophically.Option () seems to be what Sainsbury and Tye have in mind. In fact,their objections stand or fall with the correctness of the traditionalview. They are in good company here, however, as the historical atti-tude towards formalization has been that it provides the real founda-tions of mathematical inference, truth, and knowledge.

Degree-theoretic semantics in immune to Sainsbury and Tyes criti-cisms if we view it not as a description but as a fruitful model of vague-ness. Before undertaking this defence of degree-theoretic semantics,however, a bit more discussion of the notion of modelling is in order.

The view that (many of) the various activities that fall under theterm logicare instances of model building can be compared to thehobby of building model ships. Although the hobbyist might not con-struct his models for this reason, there are facts about ships that are eas-ier to discover by looking at the model than by exploring the actualship. Examples include the number of lifeboats and the ratio betweenthe length of the ship and its width. Similarly, logicians construct mod-els of various discourses in order to investigate, explain, or understandthem more easily.

Unlike model ship building, however, building models of linguisticphenomena is primarily a mathematical exercise. One constructs amodel by identifying a mathematical structure that one hopes matchesupwith the system being modelled in an illuminating way. At the out-

set we require that a model, even a bad model, is a model at all only if atleast some of the objects and relations in the model correspond at leastroughly to some of the objects and relations in the phenomenon beingmodelled.

13 It is possible that one could take different stances (vis--visthe three options) towards differ-ent formalizations. Nothing here is meant to rule out that option (), or even the rather unappeal-ing option (), is the right way to view some particular case of formalization. The project here isinstead to argue that option (), the logic as modelling approach, is often, and in fact usually, themost fruitful way to go.


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236 Roy T. Cook

The idea that the correspondence between the model and the phe-nomenon can be somewhat rough leads us to the first differencebetween constructing a model of a particular phenomenon and giving adescription of the phenomenon. In building models it is often advanta-geous (and sometimes unavoidable) to introduce some simplification.The idea is that we can eliminate, or at least reduce in complexity,aspects of the phenomenon that we find less interesting in order toexamine more easily aspects we do wish to investigate. We can comparethis to the fact that model ships often simplify aspects of the originalship, such as detailed moulding around door frames or fancy calligra-

phy on signs labelling the doors. Thus, some aspects of the model thatare intended to represent real aspects of the phenomenon may do so ina less than perfect way, ignoring the finer details in favour of a moreeasily manipulated account of the whole.

There is a second, and more important, difference between providinga model and providing a description. At least some parts of any modelare intended to be, in some sense, important, representing (in a per-haps simplified way) real aspects of the phenomenon being modelled.Other parts of the model, however, might not be intended to match upwith anything real. In other words, although we require that some partof the model must match up with some aspects of the phenomenon,not all of them have to do so. Returning to the analogy with model shipbuilding, a model ship might have, deep in its interior, supports situ-ated where the engine room is located in the actual ship. Although thesupports do not represent anything real on the actual ship, they are notnecessarily useless or eliminable as a result, since they might be crucialto the structural integrity of the model. Along similar lines, parts of alogical model, including objects and relations intimately involved in thesemantics, might be there just to facilitate the mathematics or to sim-plify our manipulations of the model.

The point is further illustrated by considering one of the set-theo-retic reconstructions of the natural numbers, viewed as a model of thenaturals. In the construction that represents the naturals as von Neu-

mann ordinals, for example, the fact that there is no von Neumannordinal between {} and {, {}} accurately represents the fact thatthere is no natural number between and . On the other hand, the factthat {} has one less member than {,{}} does not reflect any signifi-cant relation between and . Call those aspects of the model that areintended to correspond to real aspects of the phenomenon being mod-


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elled representors, and those that are not intended to so correspondartefacts.14

This example highlights an important fact that was less clear in themodel ship analogy. It is not just the objects in the model that are rep-resentors and artefacts. In addition, properties of and relations betweenthe objects can be representors or artefacts. Thus, just as the three mastson a model ship are representors, indicative of the three masts on theactual ship, the fact that the relation no von Neumann ordinal comesbetweenholds of {} and {, {}} is a representor.

We can avoid Sainsbury and Tyes criticism by carefully distinguish-

ing between representors and artefacts. In essence, the idea is to treatthe problematic parts of the degree-theoretic picture, namely theassignment of particular real numbers to sentences, as mere artefacts.Since only the representors are intended to reflect anything occurringin the vague natural language, the account becomes immune fromSainsburys observation that:

No sharp cut-offto the shadow of vagueness is marked in our linguistic prac-tice, so to attribute it to the [vague] predicate is to misdescribeit. (Sainsbury, p. , emphasis added)

Since we are not attempting to describe fully and accurately everyaspect of vague language, but only to model it, we are safe from thiscriticism. We are misdescribing our linguistic practice only if we assertthat the sharp cut-offs provided by assignments of real numbers to sen-tences represent real qualities of the phenomenon, but this is exactlywhat is denied on the logic as model picture. If the problematic parts ofthe account are not intended actually to describe anything occurring inthe phenomenon in the first place, then they certainly cannot be misde-scribing.

Timothy Williamson seems to be hinting at this sort of approach tothe problem of how precise mathematics can account for an essentiallyimprecise phenomenon when he writes that:

The use of numerical degrees of truth may appear to be a denial of higher-order vagueness, for numbers are associated with precision. However, the

appearance is deceptive. A degree-theorist can and should regard the assign-ment of numerical degrees of truth to sentences of natural language as avague matter. (Williamson , p. )

14 The terminology here is borrowed from Shapiro (), although I have diverged a bit fromhis own usage of the terms. Shapiro treats those aspects of the model that actually correspond toaspects of the phenomenon as representors, regardless of whether we recognize this correspon-dence or not. The importance of the distinction between those aspects that do correspond but arenot recognized as doing so and those correspondences that we recognize seems to warrant themodification, however.


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238 Roy T. Cook

Since Williamsons purpose is to demonstrate the deficiencies of thedegree-theoretic approach, and not to defend it, he understandablydoes not follow up on this suggestion.

One of the first philosophers to pursue seriously the idea that someparts of the semantics might not match up to real aspects of the lan-guage being formalized (and the first to do so in the context of seman-tics for vagueness) is Dorothy Edgington (). She first characterizesthe degree-theoretic account of degrees of belief as:

A well known framework for theorizing about uncertainty yield[ing] a plau-sible account of deduction from uncertain premisses. (Edgington ,

p.)Notice that there are no claims that the degree-theoretic account ofbelief that inspires her own account of verity is a correct description.Rather, it is a useful framework for theorizing. After arguing for herown degree-theoretic semantics for vagueness, she characterizes thesemantics as a structurally similar framework for vagueness(Edging-ton , p. ). She writes that:

The numbers serve a useful purpose as a theoretical tool, even if there is noperfect mapping between them and the phenomena; they give us a way ofrepresenting significant and insignificant differences, and the logical struc-ture and combination of these The results may still be approximately cor-rect. (Edgington , p. )

In other words, the semantics allows for some inexactness as a result ofsimplification. Nevertheless, it is still a useful tool.

Edgington has not yet explicitly endorsed the logic as modellingviewpoint, however. Models, as we are understanding the term here, arecharacterized by two things, simplification and artefacts. It is possible,however, to give an account of some phenomenon in which aspects ofthe phenomenon have been simplified but all parts of the account aremeant to represent, at least roughly, real aspects of the phenomenon. Itis useful to distinguish this sort of approach from what we might calltrue modelling, which involves artefacts. So far Edgington hasendorsed the idea that her semantics involves some simplification, but

without the representor/artefact distinction it is tempting to concludethat what she has given is an approximation, not a model.

Edgington states without argument (in a footnote) that viewingdegree-theoretic semantics as an idealization allows here to avoidMichael Tyes claim that One serious objection is that it replaces vague-ness by the most refined and incredible precision(Tye a, p.). Ifher account is merely an approximation, and not a model, then thisclaim is unconvincing. Merely introducing the real numbers as a sim-


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plification that approximately mirrored what was really occurring invague discourse would leave her open to the objections canvassed ear-lier. Even if the use of the real numbers is only an approximation, theprecision of the real numbers would still correspond to somethingapproximating the precision of the reals in the discourse. As the criticsof degree theories have pointed out, however, vague discourse is notonly characterized by the lack of precise borderlines but is in essencethe lack of anything even remotely like precision. Surely borderlinesthat are almost precise or approximate the precision of the reals are justas objectionable.

Although Edgington does not draw the representor/artefact distinc-tion, her comments regarding how we should view various aspects ofthe semantics make it clear that she has something like this distinctionin mind. She claims that:

The relation between the representation and the reality is still vague: thereneed be no fact of the matter exactly what number to assign. Worthwhile re-sults generated by the idealisation must be robust enough to be independentof small numerical differences. (Edgington , p. )

We can interpret Edgington here as claiming that the account beingoffered is a model, and that the assignment of a particular number to asentence is an artefact, since any number sufficiently close to the origi-nal one would be equally legitimate. Although she does not follow upon the details, this attitude towards the use of real numbers in thesemantics allows us effectively to deflect the criticisms of Sainsbury andTye.

We can flesh out this idea as follows. Truth comes in degrees. Thus,the fact that degree-theoretic semantics represents truth as coming inmore varieties than the traditional two (absolute truth and absolute fal-sity) is representative; in other words, the assignment of verities is arepresentor, and there are real verities in the world. We use the realnumbers to model these verities, however, as a matter of convenience,and many (but not all) of the properties holding of them are artefac-tual. The real numbers are used merely because their ordering and/or

density and/or other properties make them a useful surrogate for theactual verities.

Before moving on, a terminological clarification is in order. We havebeen calling the real number assigned to a sentence its verity. Nowthat we have the representor/artefact distinction in place, and haveconcluded that although sentences do have real verities, these veritiesare not real numbers but are only modelledby real numbers, we needto be more careful. In what follows, the term veritywill be reserved


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240 Roy T. Cook

for the actual degree of truth of a sentence, which is not a real number,and the value assigned to a sentence to model its verity will be indi-cated by the voperator or phrases such as the real number assignedto the sentence.

There is already a vexing issue arising from the application of thelogic as model framework to degree-theoretic semantics. If sentencesdo have actual verities, but these are not real numbers, then what sortsof objects are they? I will not attempt to answer the question fully here,but a few things can be said. The situation can be usefully compared tothe classical logicians use of and as surrogates for the classical truth

values. We are positing a new sort of entity, verities, that stand to sen-tences in vague discourses as the classical truth values stand to sen-tences in non-vague discourses. Just as the classical logician uses and to serve as truth and falsity yet presumably does not think that true sen-tences really have some special relationship with the number and falsesentences with , the degree-theorist uses the real numbers to stand forverities. Actual verities are just a generalized notion of truth values andshould no more be identified with real numbers than the classical truthvalues should be identified with the first two natural numbers.15If thisis right, then the natural question to ask is which properties of the realscorrespond to actual properties of verities and which do not.

First, in order to deflect the criticisms of Sainsbury and Tye, the pre-cision of the real numbers and, in particular, the precise boundariesthey provide must be artefactual. We have already seen the basic strat-egy. The assignment of aparticularreal number is not representative, asthe idealization must be robust enough to be independent of smallnumerical differences. If there is no single correct real number to assignto a particular sentence, then the worries about precision evaporate. Wecan no longer draw the sharp boundaries that bothered the critics sincethere can be sentences for which it is indeterminate which side of theboundary they are on.

Of course, saying that the account is meant to be a model, and thusthat certain unattractive parts of the semantics are artefactual, is not

enough. We have yet to determine in general which aspects of themodel are artefacts and which are representors, other than pointing outthat those parts of the model that Sainsbury and Tye find problematic

15 There is, however, a difference between the two cases. The properties of and relations be-tween and seem to match up with the properties of and relations between actual truth and fal-sity better than the properties of the reals match up with those of the actual verities. Of course, thisis not a problem on the position being argued for here, since we can just accept that (at the presenttime) our mathematical model of the classical truth values is better than our model of degree-the-oretic verities.


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must be artefacts. Without knowing in more detail what is representorand what is artefact we cannot draw any useful insights from the model,since we do not know what parts of it are intended to provide suchinformation.

There are aspects of the real numbers that are representative of actualproperties of the verities, however. Trivially, the fact that there are morethan two reals is representative, since verities were introduced to give usintermediate truth values other than the traditional two. In addition,some part of the ordering must be representative, as the guiding insightbehind degree-theoretic semantics is not just that sentences can be

assigned verities other than the traditional true and false but also thatsome sentences are more or less true than others.As we have already seen, small changes in the real numbers assigned

to sentences are often artefactual, and will not affect the relations, logi-cal or otherwise, between the sentences. Clearly, however, large changesin these assignments will change these relations. Following mathemati-cal practice (loosely), we can represent one real number being strictlysmaller than another using the standard inequality symbol


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242 Roy T. Cook

We can utilize the fact that small changes in real number assignmentare not necessarily representative to answer this objection. If the dis-tinction between and assignment of and an assignment of for suf-ficiently small is not representative, then we can assign RobertWardlow is talla real number close enough to that the differencebetween this value and is an artefact. We obtain the desired result thatRobert Wardlow is wholly and utterly tall, yet there will still be room togive the predication of tallto a foot man a slightly higher value.

The second consequence of distinguishing between small, artefactualdifferences and large, representative differences in real number assign-

ment is that this allows us to bring vagueness into the account, albeit inan indirect manner. Intuitively, if we are given two values aand bsuchthat the difference between aand bis small, and if we add a suitablysmall amount to b, then the difference between aand b+should stillbe small. This is enough to run a Sorites on small, however, demon-stratingthat all differences in real number assignment are in fact small.16

In other words, the difference between large differences and small onesdepends in an essential way on the notion of small, which is vague ifanything is. Thus, we have brought vagueness into the picture in animportant way, since the distinction between representors and artefactsdepends crucially on notions that are vague.

One might think that we have reintroduced the same sort of metalin-guistic vagueness found in both Sainsbury and Tyes semantics. Notice,however, that we formulated the degree-theoretic semantics withoutmentioning artefacts, representors, or the much-less-than relation . Itis only when we described the connection betweenthe object languageand the metalanguage within which the model is formulated that themuch-less-than relation, and the vagueness that comes with it, occurs.In effect, we have pushed the vagueness up a level, from the formalsemantics itself to the description of its role. While this distinctionbetween the metatheory and its connection to the object language is inall likelihood somewhat artificial, it gives us a means to distinguishbetween the strategies of Tye and Sainsbury and the logic as modelling

approach taken here.At first glance, the vagueness that arises when explicating the repre-

sentor/artefact distinction might appear unwelcome. We should want,

16 Assuming that the suitably small sdo not get smaller as the number they are being addedto gets larger. If the suitably small sdo get smaller as the number they are being added to getslarger, we would need to add the additional assumption that the least upper bound of a set of smallreal numbers is itself a small real number. At any rate, the claim that the distinction between < andinvolves vagueness in an essential way is compelling on its own, independent of the argumentgiven above.


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however, and even expect vagueness to occur somewhere in our expla-nation of how vague language works. Although we might disagree withtheir conclusions regarding the need for imprecise, vague mathematics,it is hard to ignore the core of truth in Sainsbury and Tye s arguments.In adequately accounting for the semantics of a vague discourse, a dis-course characterized by its lack of both precision and sharp boundaries,we should expect imprecision to come into the account at some point.Sainsbury and Tye both made the mistake of thinking that the vague-ness had to appear in the mathematics, and thus replaced the precisionof the degree-theoretic account with vague sets or other mathematically

imprecise messiness. We do not need to bring this requisite vaguenessinto the formal semantics directly, however, and instead it arises in therelation between the mathematics and the natural language. As a result,we retain precise and easily manipulated mathematics while also doingjustice to the insight that vagueness must come into play somewhere.

Returning to the issue of what is representor and what is artefact, it isclear that, in addition to large differences in real number assignment,what seems crucial on Edgingtons approach are the rules for findingthe verities of compound statements from the verities of their constitu-ent sentences. Although Edgington time and again denies that there is asingle, correct real number to assign to each sentence, she spills evenmore ink determining what the correct treatment of the logical connec-tives is. In addition, it is the particular rules that she settles on for theconnectives that allow her to prove that the Constraining Propertyholds, thus vindicatingclassical logic. The conclusion to draw is thatthe formal relationships that hold between a sentence and its subsen-tences and, more generally, the logical relations guaranteed to holdbetween sentences are representors. Exactly which numbers are chosenfor the atomic sentences is largely a matter of convenience, but, oncethese numbers are assigned, the rules produce exactly the right order-ings between the verities of, and exactly the right logical entailmentsbetween, any compound statement and its subsentences.

The fact that these rules are representative complicates our account

of when a difference in real number assignment is representative of areal difference in verity and when it is not. Notice that earlier it wasclaimed that small differences in real number assignment are not neces-sarily indicative of a real difference in verity, but the possibility of arbi-trarily small differences that are representative was left open. Theexistence of such small representative differences is guaranteed by theArbitrarily Close Verity Theorem(assuming that, for any sentence Pof


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244 Roy T. Cook

our language, there are arbitrarily many sentences Qthrough Qnthatare all independent of P and of each other.):

Theorem:Given a sentence Psuch that v(P) > and any small realnumber , there is a sentence Ssuch that:

|v(P) v(S)| < ,

v(P) > v(S),

And this inequality represents a real difference in the verities of PandS.

The reasoning goes as follows: Given a sentence Pand an arbitrarilysmall , let nbe the least positive integer such that < . Now, let Qthrough Qnbe sentences independent of Pand of each other, and con-sider the ndifferent n-ary conjunctions that contain exactly one of Qior Qifor each i, in. The disjunction of these conjunctions is atautology and thus gets the value , and the disjuncts are pairwiseincompatible, so it follows by the simplified rule for disjunction dis-cussed in above that at least one of these conjunctions gets a value. Let Rbe the disjunction of all of the conjunctions except for onewhose value is less than or equal to . It follows that >v(R).Since Ris contingent and is constructed from sentences that are allindependent of P, Ritself is independent of P. Thus, v(P! R) = v(P) v(R). Let the sentence Sbe the conjunction of Pand R. Then:

|v(P) v(S)| = v(P) [v(P) v(R)] = v(P) [v(R)]v(P)[+ ]

= v(P) < .

The difference between the reals assigned to Pand Sis less than ourarbitrarily small , yet the difference must be representative of a realdifference in verity, since this difference in real number assignment is adirect consequence of the logical relations between the sentences

involved (and the assumption regarding the availability of enoughindependent sentences). Thus, although very small differences in thereal numbers assigned to sentences often do not reflect a real differencein verity between the sentences, sometimes this is not the case.

Combining the insights we have reached thus far, we can draw twoconclusions. First, it is not the case that every aspect of the ordering ofthe real numbers assigned to sentences actually represents some realdifference in verity, since two sentences could be assigned reals whose


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difference is extremely small yet artefactual. Second, as theArbitrarilyClose Verity Theoremdemonstrates, two sentences having real numberassignments whose difference is very small is not enough to guaranteethat the ordering is artefactual.

Although we could certainly go on examining in more detail whichaspects of Edgingtons degree-theoretic account are artefacts and whichare representors, the conclusions drawn so far are enough to formulatea general description of where the distinction lies. The notion of verityin general is representative. On Edgingtons account truth (and falsity)do come in gradations, and both large differences in real number

assignments and the logical relations between complex sentences andtheir constituents are indicative of real aspects of vague natural lan-guage. On the other hand, the assignment of particular real numbers toparticular sentences, and the resulting sharp boundaries, are just con-veniences, incorporated into the semantics for the sake of simplicity,but reflecting nothing actually present in the discourse being modelled.

6. Edgingtons solution and the artefact/representor distinction

There is a final task that needs to be undertaken. Now that we knowthat parts of the degree-theoretic semantics are merely artefactual, andwe know at least roughly which parts these are, we need to revisit Edg-ingtons solution to the Sorites paradox to determine whether theaspects of the semantics she utilizes are representative or artefactual.

It should be clear that a successful solution to the Sorites paradoxshould make use of characteristics of the semantics that are representa-tive. If the solution took advantage of too much that is artefactual, thenit would be no solution at all since the artefacts do not correspond togenuine features of the phenomenon being modelled. A solution to aparadox of any sort should clarify what is actually going on in the dis-course in order to show us where we erred, not make use of tricky rea-soning afforded by irrelevant parts of the semantics originallyintroduced for the sake of simplicity. Fortunately, Edgingtons solution

makes use of representative parts of the model.Recall that Edgingtons solution to the paradox depends on what she

called the Constraining Property. Briefly, this is just the fact that the realnumber representing the unverity, or degree of falsity, of the conclusionof a classically valid argument cannot be greater that the sum of the realnumbers representing the unverities of the premisses of the same argu-ment, but what is most important in this context is that the realnumber representing the unverity of the conclusion can be as large as


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246 Roy T. Cook

the sum of the reals assigned to represent the unverities of the prem-isses.

It was stressed above that one aspect of the semantics that is repre-sentative is the logical relations between sentences. This is all that Edg-ington needs to get her solution going the Constraining Propertyfollows directly from her rules for determining real number assign-ments for complex sentences in terms of the real number assignmentsof the component sentences. Since these relations are representative,similar constraints must hold for the actual verities. Thus, Edgingtonsproposed solution to the Sorites paradox is in fact a genuine solution.

To summarize what has been accomplished: we can admit thatvagueness is characterized by the lack of the very sort of precision usu-ally found in set-theoretic semantics, and, in addition, we can grantSainsbury and Tyes point that to attribute this sort of precision tovague discourse would be a mistake. Against Sainsbury and Tye, how-ever, we have seen how we can have mathematical precision in thesemantics without attributing it to the natural language being studiedby making use of the logic as modelling picture. In particular, thevagueness that they rightly argue needs to occur somewhere in ouraccount of vagueness does not need to occur in the formal semanticsitself but instead occurs in our description of the connection betweenthe formalization and the informal discourse. The representor/artefactdistinction allows us to claim, first, that the inappropriate precisionismerely an artefact, and, second, that the distinction between what isartefact and what is representor itself relies on vague notions. In thisway, we can retain what is right from Sainsbury and Tyes observationregarding precision while resisting their conclusion that precise mathe-matics ought to be abandoned.17

Department of Logic and Metaphysics roy t. cookUniversity of St Andrews

St Andrews

Fife KY16 [email protected]

17 A version of this paper was presented to the philosophy departments at the University ofMinnesota, Michigan State University, and the Ohio State University. The paper has also benefitedfrom discussion with and cri ticism by Jack Arnold, Bob Batterman, Jon Cogburn, Diana Raffman,Agustin Rayo, George Schumm, Stewart Shapiro, and Neil Tennant, and the comments of threeanonymous referees.


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References:

Edgington, D. : Vagueness by Degrees, in Keefe and Smith ,pp. .

Evans, G. and J. McDowell : Truth and Meaning: Essays in Seman-tics. Oxford: Clarendon Press.

Fine, K. : Vagueness, Truth, and Logic, in Keefe and Smith ,pp. . Originally published in Synthese, .

Keefe, R. and P. Smith (eds) : Vagueness: A Reader. Cambridge MA:MIT Press.

Machina, K. : Truth, Belief and Vagueness, in Keefe and Smith

, pp. . Originally published inJournal of PhilosophicalLogic, .

Sainsbury, R. : Concepts without Boundaries, in Keefe and Smith, pp. . Originally published asInaugural Lecture by theKings College London Department of Philosophy.

Schirn, M. (ed) : Philosophy of Mathematics Today: Proceedings ofan InternationalCongress in Munich.Oxford: Oxford UniversityPress.

Shapiro, S. : Logical Consequence: Models and Modality, inSchirn , pp. .

Tomberlin, J. E. : Philosophical Perspectives 8: Logic and Language.Atascadero, CA: Ridgeview Publ. Co.

Tye, M. a: Vagueness: Welcome to the Quicksand. Southern Jour-nal of Philosophy, (supplement), pp. .

b: Sorites Paradoxes and the Semantics of Vagueness, in Keefeand Smith , pp. . Originally published in Tomberlin .

Williamson, T. : Vagueness, London: Routledge.Wright, C. : Language Mastery and the Sorites Paradox, in Keefe

and Smith , pp. . Originally published in Evans andMcDowell .


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