Between Semantics and Pragmatics

7/30/2019 Between Semantics and Pragmatics

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BETWEEN SEMANTICS AND PRAGMATICS:

IN SEARCH OF A LOGIC FOR VAGUENESS

Bart VAN KERKHOVE*

Abstract This paper offers a survey of current logical investigation in the field of

vagueness. Its ambitions are, first, to reach every interested philosopher, remaining as

general and non-technical as possible, without however being incomplete or inaccurate,

and, second, efficiently to refer the readers keen on more to further literature. The article

begins by spelling out the problems raised by sorites-paradoxes both formally and

informally. Further paragraphes then focus on traditional, strictly semantical treatments

for these paradoxes (such as Epistemicism, Supervaluation and Fuzzy Logic), followed by

more recent attempts to expand the field of investigation to pragmatics. Finally, the heavy

metaphysical debate concerning the origin of vagueness (word orobject) is highlighted.

1. Introduction

Despite their antiquity and numerous ambitious attempts at their resolution, Sorites Paradoxes

remain of the most astonishing philosophical knots. The famous question being how to turn a

non-heap of wheat into a heap of it (i.e. the original Sorites),1

or a non-bald person into a bald

one (i.e. the so-called Falakros), when adding grains or removing hairs only one at a time,

indeed things soon appear unintelligible. For how could we reasonably expect anyone ever tobecome bald merely by losing one single hair? However, if not so, we are driven into an

equally unpleasant regressus: no hairy person can ever be turned into a bald one, provided the

hairs (even all of them!) are plucked from his scull one by one. Clearly, from the moment we

involve in slippery-slope or little-by-little arguments like these, something inevitably goes

wrong.2

Before turning to some proposals for unraveling the mystery, I will begin my

exposition, in section 2, by carefully formulating the modern version(s) of this puzzle, and

making some introductory, as objective as possible, remarks on it. As I will further focus on

modern theorizing on the topic, the reader particularly interested in historical aspects, should

be recommended to consult other material, e.g. Burnyeat [1983], or the first chapter of

Williamson [1994].

During the last twenty-five years, various claims to a logically coherent solution for similar

paradoxes have been made. Probably the most simple, yet highly contra-intuitive one is the

semantically realist Epistemic View, as heavily defended by Roy Sorensen and Timothy

Williamson. This account allows classical propositional or predicate logic (CL and PL

respectively) to be entirely retained in the face of vagueness.3

However, to date, the more

*I am highly indebted to my mentor Jean Paul Van Bendegem for his encouragement and constructive criticism

throughout my work on this topic. Also a big thanks to Brick de Bois, for his valuable comments on an earlierversion of the article.

1 As probably conceived by Eubulides, a Megarian contemporary of Aristotle.

2

Another popular version of the paradox involves a colour spectrum varying from, e.g., red to orange, in asmooth way, i.e. non-observational with the bare eye.

3 See their [1988a] and [1994] respectively.


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popular coherent remedies remain the many-valued approaches known as Supervaluation and

Fuzzy Logic. The supervaluational account, which combines classical models with the

possibility of higher-order indeterminacy, was initiated by Bas van Fraassen, and applied to

vagueness by Fine [1975]. Fuzzy logicians, who surely paved the way for substantial practical,

notably technological successes, offer a diagnosis for vagueness by generalizing two-valued

truth-functionality ad infinitum. I elaborate on these three strategies in section 3. Less debatedcoherent solutions, which I will by-pass here, involve the Intuitionistic one put forward by

Hilary Putnam in his [1983] and some three-valued paths using Kleene Tables (be it the strong

or weak variant).4

Contrasting sharply with these fairly optimistic approaches are some inconstructive ones, such

as the logicistic lets get rid of vague concepts (as Frege decidedly advises us) and the

nihilistic theres no use in trying to get a firm grasp on vague concepts (most elegantly

defended in Dummett [1975]). In section 4, I elaborate on these views and some of their

variants.

Increasing dissatisfaction with the aforementioned approaches, the constructive as well as the

non-constructive ones, has led a number of logicians to consider fundamentally other, notably

pragmatical strategies to treat sorites-infected paradoxes. In section 5, Ill be dealing with two

of them in some detail: that ofDiana Raffman and that ofRuth Manor.5

Their inspiration is, in

my view, beautifully exemplified byMark Sainsbury in his [1990]. The too strict and therefore

alledgedly contra-productive adherence to semantics the former treatments exhibit drives him

to promote a radically different view on sorites-related issues, and particularly on our

classifying ability. He says: What I suggest is that almost all concepts lackboundaries, so that

the classical picture is of very little use to us ([1990], p.7, my emphasis). Classical should

here be interpreted in a much broader sense than CL/PL-wise, as to include at least all

solutions gathered in section 3, and probably more. The most important message seems to me

that we should not at any moment forget it is elusive natural language we are dealing with afterall.

Before reaching a conclusion, a final regular section (6) will be dedicated to the metaphysical

dimension of vagueness. It questions what we should impute vagueness to: objective

characteristics of the world we live in, or rather language and the way we use it. In other

words: is the world itself a fuzzy place (the ontological view) or are we just bound to perceive

it being such (the semanto-pragmatical view)? The latter view is to date the more popular. I

will particularly highlight the logical debate triggered by Evans [1978], containing a formal

proofcontra ontological vagueness.

A final introductory remark. This article has been written in order to provide with a concise

(yet as complete as possible) overview of a vast field of inquiry, and at the same time, it isaimed at a wide audience of philosophers with a logical interest. For both these reasons, I

restricted the technicalities to an absolute minimum. As a consequence, the technically more

hungry readers shouldnt feel underestimated at any point, but are encouraged to explore

further literature, as mentioned in the course of every section.

2. Sorites and Vagueness

4 For the latter, see, e.g., Williamson [1994] ( 4.4 and 4.5).5 These and other pragmatical accounts of vagueness are addressed to more thoroughly in my [200+].


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The paradoxical argument called sorites can be formulated in several ways. I distinguish

three of them. Afirstone uses mathematical induction (MI) and universal instantiation (UI) on

an ordered domain of objects. The induction can be carried out on the conditions that a

predicate P is at least applicable to one member of the domain (call such an object x 0), and

that, if P is applicable to an xi, then this is also the case for its successor xi+1. Formally, we get:

1. Px0 basic premiss (minor)2. xi (Pxi Pxi+1) quantified premiss (major)3. xi (Pxi) 1,2; MI4. Pxn 3; UIHereby, xn stands for any x, so that the paradox is established. In the second form,

mathematical induction as an inference pattern is replaced by modus ponens (MP), viz. applied

to the first premiss and the instantiation of the second (and so on until xn is reached). In most

cases, the formal proof takes some more writing, but it also shows that MI is in fact not

needed:

1. Px0 basic premiss (minor)2. xi (Pxi Pxi+1) quantified premiss (major)3. Px0 Px1 2; UI4. Px1 1,3; MP5. Px1 Px2 2; UI6. Px2 4,5; MP7.

Pxn-1 Pxn .,.; UI

Pxn .,.; MP

A third and final presentation of the sorites is most straightforward. It opposes two

incompatible observations, e.g. that a man with zero hairs is definitely bald (Px0) and a man

with a hundred thousand hairs is clearly not (Px100.000). It is not difficult then to conclude that

there must be a number i between 0 and 100.000, so that a man with i hairs is not bald,

whereas a man with i+1 hairs is:

1. Px0 basic premiss2. Pxn final premiss3. (xi)(Pxi & Pxi+1) quantified conclusionHow beautiful! We have identified an obvious paradox within the borders of classical logic.But now we may start wondering whether this carefully formulated paradox has anything to do

with what ordinary people as well as philosophers usually call vagueness. Lets image a real-

life situation to try and find out. Youre in your favourite pub together with a bunch of friends,

and being in an experimental mood, as youve all had a few drinks, you agree to mark off the

set of bald men in the pub. Unlikely that there will be a consensus among you (problem 1),

unlikely that when repeating the task half an hour later (meanwhile some people have left,

some have entered, there might be more, or less, customers present, and you had a few more

drinks) the pattern of your demarcating behaviour will have developed similarly (problem 2).

I will come back to these remarks in a further section. For present purposes I only want to

illustrate one simple point, namely that it is very unclear (to me) whether this fictitious

experiment has got anything to do with the kind of paradoxes spelled out in the first part of this

section. Contrast the pub scene (form 2) with the following one. Youre all by yourself in a


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clean lab, observing a single, isolated test-person. This man is being mechanically robbed of

his carefully counted hairs one at a time, after each removal of which youre expected to

answer the question Is this person bald? by a simple yes or no (form 1). Ill gladly admit

there being a certain amount of indeed vagueness between the two forms,6,7

but that could

do nothing but strenghten my case, namely that it is very hard, and virtually impossible, to pin

down the phenomenon of vagueness.

In this respect, it is furthermore peculiar to observe, first, that the literature most often

concentrates on predicate expressions, as the central or exemplary cases, but, secondand most

telling, that it does so by means of examples of an extraordinary variety, making it tough to

specify what, if anything, they have in common.8

And thats exactly the reason why I didnt

introduce the topic from the ordinary point of view (form 2), but, on the contrary, chose to go

for the most rigorous, i.e. strictly logical, way first (form 1),9

be it that, when done carefully, in

a lot of cases the two forms can be (more or less) reduced to one another. The trap I wanted to

keep the reader from stepping in, is that, when confronted with real-life examples, one is

tempted to start by defining a vague concept (even implicitly) as one lacking sharp boundaries,

as clearly there is a grey zone or no mans landbetween the areas of certain application andnon-application respectively. The actual danger is that of taking a theoretical move too soon,

e.g. by immediately valuating the law of excluded middle (LEM) and/or the principle of

bivalence (PB) as invalid.10,11

On the other hand, I find this second, non-formal, expression of

vagueness far more instructive, and indeed essential, as in the end, it is real-life situations that

matter to (most of) us. To catch my point, simply think about the application of age limits in

criminal law.

There is however a deeper consideration to be made here. After all, one could say, isnt CL,

the language in which we discovered the sorites paradox, a logical theory as good (or bad) as

any of its alternatives? Well, yes, and no. There are, admittedly, strong reasons to prefer good

old two-valued CL. As one simply has to employ a logic, it might as well be kept as coherentand consistent as possible under the circumstances. So why not stick to powerful CL, as long

as it works? We start out dividing (well formalized) sentences, within particular theories, in

true and false ones. Whenever we run into trouble, theres room for argument whether to

change our logical instrument or not, and in the latter case e.g. to add extra-logical, c.q.

pragmatical, constraints to our account. It should be noted that the actual dominance of CL has

allowed paradoxes like the sorites to pop up in the first place, dragging along long discarded

fundamental questions such as Whats validity after all? It is the major, and seemingly endless

(though significantly so) task of the discipline called Philosophical Logic to sort out essential

matters like these. It therefore should and does investigate the foundations of logic(s), in their

relations to other philosophical branches, such as ontology, epistemology, or ethics. In

particular, this task is fulfilled in designing and promoting proper formal tools to handlephilosophical puzzles such as the semantic paradoxes. There obviously is far from a consensus

in doing so, as in fact this entire article may testify. But this, I think, is good news. At the very

6 The two problems I just spelled out will have fundamentally transformed, but in no way have disappeared.

7 More on a most unpleasant manifestation of the vagueness of vague we touch upon here, as causingparticular problems for the design of logical tools trying to cope with it, is to be found in 3.4.

8 Susan Haack pointed this out, in her [1974] (p.110).

9 Which is anyhow rarely considered necessary across the hundreds of pages of specialized logical literature.

10 Im not acting paranoid here. This tendency is at least reported by Sorensen [1985] (p.134-135), Horwich

[1997] (p.929-930) and Schiffer [1998] (p.199).11 LEM states that the proposition PP is a theorem, PB that for every proposition P: P is true orP is false.


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least, it secures us from an unfruitful, and in a sense even dangerous, monopoly of CL. Not any

logic should ever be taken for granted (let alone venerated), but its very basics should be

constantly open to scrutiny. As Read [1995] in a perhaps somewhat dramatic fashion

proclaims: Teaching in philosophy departments across the world exhibits this schizophrenia,

in which the dogmatic approach to logic sits uncomfortably side by side with the ceaseless

critical examination which is encouraged and demanded in philosophy (p.2).

Finally, and closing down this section, I can hardly left unmentioned the warning, echoed by

some eminent philosophers,12

not to reduce vagueness to the apparently related phenomena of

generality or ambiguity. As far as Im concerned, most (if not all) of the subsequent discussion

can be brought back to a terminological matter, so that, provided we define the different terms

in play properly, no problem whatsoever should arise. One could for instance use Max Blacks

clearcut distinction, made in his [1949]: The former is constituted by the application of a

symbol to a multiplicity of objects in the field of reference, the latter by the association of a

finite number of alternative meanings having the same phonetic form (p.29).13

Nevertheless,

various philosophers have found it necessary to relate or even reduce vagueness to one of these

phenomena. Bertrand Russell, for instance, is reputed to have unwillingly confused vaguenesswith generality.

14For some outstanding further work on this topic, see Sorensen [1989] and

[1998].

3. Coherentism

3.1. Epistemic View

No doubt the most conservative reaction to our sorites paradox is to say that, yes, there is a factof the matter whether a certain, problematic, amount of wheat is a heap or not, but that, alas,

we humans are in no position (nor could ever be so) to find out. More specifically, it is due to a

fundamental epistemic shortcoming that human access is denied to some crucial missing

knowledge about the borderline case. In other words, the Epistemic View (EV) is about

ignorance as an essential feature of borderline cases (Williamson [1994], p.201). And once

hidden lines are admitted, the argument runs, why should a line between truth and falsity not

be one of them (ditto), so that bivalence holds, be it that we cant find out which of both truth-

values applies. Short: CL shouldnt be abandoned when confronted with vagueness, and this is

used as a major trump against virtually every rival account. As might be expected (see the

previous section), I have my doubts about this policy. Nevertheless, next to this alledgedly

comparative advantage, in fact reducing to simplicity, EV-theorists have formulatedindependent reasons to prefer their solution. I will concentrate on the case for EV as presented

in Williamson [1994], canonical as it seems to have become in recent years.15

12 See, e.g., Black [1949] (p.29), Quine [1960] (27), Alston [1964] (p.85).

13 As a simple illustratation, take the word bank. Its ambiguous because, depending on its use, it denotes apiece of furniture, a financial institution or a strip of land alongside a river. Its general because it capturesdifferent kinds of furniture and institutions, which can be broken down further. And finally its vague, because

in some cases, its unclear whether to consider a particular object as a bank or not, e.g. as opposed to a chair.

14 See his [1988], dating from 1923, and, for some interesting comments, Williamson [1994] (2.4), attacking

Russells account, and Hyde [1992], defending it. Ill come back to Russell in section 4.15 An alternative account is offered in Sorensen [1988a].


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To start with, let me introduce EVs view on the nature of vague facts, which amounts to a

supervenience thesis. It consists, I think, of two crucial components. First, if two possible

situations are identical in all precisely specified respects, then they are identical in all vaguely

specified respects too. [] There could not be two situations differing vaguely but not precisely

(p.202-203). Second, if nature does not draw a line for us, then a line is drawn only if we draw it

ourselves, by our use. [] To say that use determines meaning is just to say that meaningsupervenes on use (p.206). The central part of the account then, is to demonstrate the failure of

the so-called KK-principle, which states that if I know something, then I know that I know it.

Williamson uses the notion margin for errorto invalidate KK. In the case of the sorites paradox, it

involves something like If we know that n grains make a heap, then n-1 grains make a heap

()(p.232). The idea is that if we know, i.e. are sure and aware, that (it is true that) theres a heap

ofn grains in front of us, then (it is true that) 1 grain less could never magic it away. This m argin

for error-principle gives rise to inexact knowledge, i.e. the failure of KK, in the following way. In

order to produce the sorites argument, we need a sequence of statements of the form (), one

containing the antecedent of () for an n, to start up the argument, and MP as an inference rule.

But now watch closely. In order to know that n-1 grains make a heap, we should have to knowthe premises from which we deduced n-1 grains make a heap. [] Thus we should have to know

that we know that n grains make a heap. But the previous stage of the argument showed only that

we know that n grains make a heap. With the KK-principle, a sorites paradox would indeed be

forthcoming. Without it, one iteration of knowledge is lost at each stage of the argument (p.233,

my emphasises). As a result, the sorites paradox melts away before our very eyes. The source of

this inexactness of knowledge is to be found precisely in the supervenience of meaning on use,

introduced at the beginning of this paragraph. Small changes in the use of words not ruled by

precise divisions (such as vague ones) can imperceptibly shift their meanings, so at the same time

the truth-value of statements containing them. The sharp boundary between rival vague concepts

entirely depends on the unstable, non-graspable dispositions of speakers. In other words: a man

might be (borderline) bald, but that very same man might as well have been (borderline) non-bald.

This account is often considered as not touching the very core of vagueness. According to

Horwich [1997], it is far-fetched to label the obvious(?) knowledge problem as an external failure,

viz. to realize, in common language, compatibility with the alledged objective meaning of vague

terms. He himself sticks to an internal paralysis of judgement for a diagnosis.16

EV explains our

problematic phenomenon in a most simple, i.e. economic, fashion. True, but that does not warrant

the explaining away of it, making use of supervenience.17

Both Schiffer [1997] and Burgess

[1998] scrutinize this point. Briefly, their message is one of disbelief in the essence of ignorance

as presented by Williamson. Why should the lack of an epistemic procedure to discover the

connection between semantic and underlying fact, exclude this knowledge in principle?

3.2. Supervaluation

The logical technique called Supervaluation (SV) tries to reconcile two apparently conflicting

opinions when confronted with a typical sorites series as introduced in section 2: that there

admittedly is no one transition point which is valuable for each and every observer (1), but that

at the same time, for each and every observer there surely is a transition point (2). In

16 The constructive part of his account, which somewhat disappointingly reduces to a version of Max Blacks

(see below 3.3), I will leave unmentioned.17 I refer to my warning concerning CL, somewhere near the end of the previous section.


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reconciling these positions, SV will preserve CL on the object-level (accomodating 2), while

allowing dissensus on an extra meta-level (accomodating 1). The following simple technical

presentation will prove SV to be a transparent and user-friendly way to deal with the sorites-

paradox.

We start out with a CL-model M for all well-formed formulas A of a language L, so that

vM(A)=0 or vM(A)=1, A being false or true respectively. Taking M to be a set of models like

M, a SV can be defined as follows: - SVM(A)=1 iff for all M M: vM(A)=1

- SVM(A)=0 iff for all M M: vM(A)=0

- SVM(A)=n in all other cases, n standing for neutral

The idea behind all this should be intuitively clear. When confronted with a vague expression,

different speakers judge it to be either true or false (they are restricted to the level of their

particular two-valued M). The supervaluationist gathers their models (called sharpenings or

precisifications of the vague expression) in an M. In case there appears to be a consensus

among these speakers, the expression at hand is called supertrue or superfalse on this SV-

level, depending they all have judged it to be true or false. In case of a dissensus, the

expression is awarded the status undecidedor neutral SV-wise. Note that every participant to

the natural (object) language is forced to allocate a problematic (i.e. vague) expression one of

two truth-values, notably these of CL. The ultimate truth or falsity as it were, of expressions

like these, can only be decided (or left undecided) at a meta-level unattainable for these

ordinary speakers.

What to think of all this? To be honest, from my first confrontation with SV, it has made me

feel uncomfortable even more than EV does. You see, it seems to amount to a neat trick, trying

to handle vagueness in a surveyable way, to add a level of judgement which is completely

foreign to common talk. In doing so, undoubtedly, SV clusters the advantages of a number of

accounts. Thats nice, were it not that it also shares the drawbacks with them.

Let me elaborate. In SV, one is obliged to determine a boundary between the vague concept

and its equally vague opposite. On the basic level, that is, the only area an ordinary speaker is

allowed to hang about. Clearly, this meets EVs wishes, as well as those of any others wanting

to retain CL. To know: all inference rules remain unaffected,18

no words (such as definitely)

are added to the object language, and no theorem whatsoever is banished. This is possible only

because a level on top of the usual meta-language is introduced, the SV-level, where PB no

longer holds. The logic turns three-valuedon this level: a statement is supertrue, superfalse, or

else remains undecided. This stage of the analysis is supposed to satisfy another group of

theorists, viz. those prefering logic with several thruth-values, from three up to an infinite

amount of them (see next paragraph), as well as those defending fundamental under- or

overdetermination of formal models compared to the natural language theyre trying to capture(see further sections). If youre wondering why all this is so, just note that the value undecided

is not interpreted here.

But now for the drawbacks, throwing a bit more light on my resentment with SV. Firstof all,

there is the somewhat strange role of the quantifier , as compared to the phrase there

exists. It is unclear what its truth-conditions are, if, on the one hand, (xi)(Pxi Pxi+1) holds,

but, on the other, (xi)(Pxi & Pxi+1) doesnt.19

This can be reformulated as follows: SV creates

itself an extra metalanguage, in which LEM is retained, while PB is rejected.11

A severe difficulty

arises when asked to justify this move independently, i.e. other then as an ad hoc one. Second,

18 As theyre to operate only on the basic level, in the end they all reduce to the classical ones.19 This point was made in Hyde [1997].


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viewed the fact that PB is not done away with in the different models M, it is still assumed that a

(every) vague concept is, in principle, sharpenable, i.e. eliminable. The question now amounts to

what an admittable sharpening is. Yet thats nothing but the initial problem all over again, so what

has been gained? As in the case of EV, from the perspective of daily practice, the nature of

vagueness (whatever it may be, and without doubt thats itself a vague matter) seems thoroughly

violated whenever a cut-off point is accepted, be it on an underlying level. More on all this, asconcerning the very essence of vagueness, quite naturally is to be found in the sections to follow.

In my view, the next few lines from Roy Sorensensum up this discussion, and at the same time

take it to another level: Deviant logic was recruited because it is supposed to be better suited to

genuinely vague predicates. But now we find that these purported predicates have been

expurgated by the meta-language used by the deviant logician. The very process of setting up

the apparatus needed to recognize genuine vagueness precludes it (Sorensen [1996], p.207).

The way I understand it, in the models M, vagueness is eliminated, while in the model M, it is

reintroduced as a kind ofdisagreementconcerning application of the concepts in question. But

difference of opinion can not be what vagueness is all about. It is just as much an individual

problem, over time and place, but even when confronted with a single, isolated sorites-like case.Sorensen goes on with a suggestion sometimes met (which he doesnt support): At this point, a

deviant logician [] might agree that any logic of indeterminacy must be a meta-logic of

indeterminacy (p.209). It is indeed a vague meta-language that is refered to here. Is this idea

intelligible? Or do matters tend to get trivialized, as in this inspiration, nothing less than a

natural language seems to be acceptable as an adequate model of natural language, to

paraphrase the later Wittgenstein.20

3.3. Fuzzy Logic

To make clear the idea behind Fuzzy Logic (FL), I will go back to a famous philosopher, who

is said to be one of the grandfathers of this alternative logical school.21

This philosopher was

called Max Black, and in his [1949], first published in 1937, he set out on an early attempt to

try and capture the phenomenon going under the name of vagueness. Black was looking for a

way to map the defective correlation between the ideal world of theoretical objects and

empirical reality, to indicate an appropriate symbolism for vagueness by means of which

deviations from a standard can be absorbed by a reinterpretation of the same standard (p.27).

It was Blacks ambition actually to measure degrees of terms vagueness, compiling so-called

consistency profiles of their use, on the basis of linguistic input delivered by a representative

group of observers. What this kind of empirical analysis was supposed to circumvent was the

need to locate, in a sorites series, one or more cut-off points, in the case ofa two-valued logic

and that ofa three-valued logic (demarcating a region of uncertain application; as, e.g., in SV)

respectively. These crude distinctions indeed become unnecessary, once one admits

differentiation to develop smoothly. This, in Blacks eyes, could be achieved taking into

account that the uncertainty is a matter of degree, varying quantitatively, though not

regularly, with the position of an object in the series (p.32).

The practical, viz. statistical, problems that accompany an enterprise like this might seem, even

at first sight, to be insurmountable.22

Nevertheless, during the 60s and 70s, grandfather

20 Williamson [1994] (p.84).

21 See, e.g., Goguen [1969] (p.326-327), Williamson [1994] (p.103), and Trillas [1996] (p.91).22 For a tentative overview, see, e.g., Burns [1991] (pp.34-36).


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Black, along with some others, apparently inspired a group of people, mostly (this is still so)

mathematicians and computer scientists, to pursue and elaborate his initial idea. The person

who grew up to be the fatherof FL, turned out to be Lofti Zadeh (or so is said). Now what

distinguishes the efforts from this era on, is the explicitly formal ambitions. FL aims to extend

ordinary deductive methods to situations in which the information available may be only partly

or approximately true. In FL, each statement has some numerical degree of truth or falsity(Copeland [1997], p.517, my emphasis). What I shall do in the sequel of this paragraph, is to

give a very rudimentary sketch of what FL standardly looks like, show how it can be applied to

vagueness, viz. the sorites, and formulate some objections. Doing so, Ill further rely on

secondary sources as the one just quoted. Examination of the best known basic litterature, such

as Goguen [1969], Zadeh [1975], Machina [1972] and [1976], or Sanford [1975] and [1979],

would require too much of the space available, and is kindly left to the reader. I should warn

that all of this commendable work is generally quite hermetic (perhaps less so in the case of

Machina and Sanford), or at least thats my appreciation of things, as theres very little

philosophical preoccupation to be noted.23

In FL, the multitude of truth-values between falsity and truth, i.e. in the interval [0,1], isdirected by a number of semantical rules, which, in their most simple format, look something

like this: For the four classical connectives:

FL1 (p) = 1 (p);

FL2 (pq) = max [(p),(q)];

FL3 (p&q) = min [(p),(q)];

FL4 (pq) = min [1,1(p)+(q)];

Once modal quantifiers are added, the following clauses could e.g. be added:

FL5 ((x)p) = glbx(v(p)); which stands for the greatest lower bound, viz. of

all (p)s only differing in value due to variable x, freely occurring in p

FL6 ((x)p) = lubx(v(p)); which stands for the least upper bound, viz. of all

(p)s only differing in value due to variable x, freely occurring in p

Using this framework, one can actually force any sorites paradox to melt away (Copeland

op.cit., p.520). The quantified premiss indentified in section 2 is judged not entirely true, be it true

enough in most of our daily practice, where we very rarely run through an entire sorites series.

When forced to, what FL does, is to rob the statement under consideration of a tiny fraction of its

truth, every time MP is applied. That this subtle mechanism uncovered by FL remains however

unnoticed by the ordinary speaker, explains why the latter gets embarrassed in the course of a

typical sorites argument.

Intuitively acceptable as this all may sound, one runs in serious difficulties trying to justify any

attempt in writing out this account formally. To begin with, how to interpret every time MP isapplied? In other words, how many truth-values (i.e. how big a single fixed or variable? jump

in value) should we take into consideration, and not to forget: why?24

It seems almost inescapable

that this will depend on the case at hand. But this opens the route to an endless debate, in which

23 No discussion whatsoever on the nature of philosophy is being elicited here. Roughly, what I mean is that theauthors write from a practical, as opposod to theoretical (also in the sense of reflecting), point of view. And that,

I think, is the major reason for their tending to overlook the crucial shortcomings in their account so obvious tomost philosophers (certainly the professional logicians among them), which are, first, that the possible

interpretations of the formal framework are not sufficiently screened or justified, and second, most important,that the philosophical problem raised by the sorites paradox is not addressed after all, but on the contraryaggravated. More details to follow.

24

It is very difficult to argue why certain parameters should be used. [] It is also difficult to argue what aparticular membership degree [e.g. to the set of bald men, bvk] for a certain object really means (Hellendoorn[1990], p.19).


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the most various deciding factors, which should be processed empirically, are being put forward.

Apart from the danger of loosing ourselves in it, in stead of concentrating on a suitable semantics,

at this particular point, the account drifts away from the very question at hand. More particularly,

the practical-descriptive preoccupation of most fuzzy logicians, which has been very fruitful

indeed,25

makes us lose contact with what the sorites paradox, for a philosopher, is or should be all

about: how to justify a certain threshold rather than another or a multiplicity of them, and how tojustify a threshold at all.

In founding a FL, it is looking for easy answers trying to reduce a scale of truth-values to the

alledged average of use in a certain population (as Blacks original proposal was). Writes

Sainsbury: To what in our actual use of language does this division correspond? [] We do not

know, cannot know, and do not need to know these supposed boundaries to use language

correctly ([1990], p.12). What is in fact being questioned here (see section 5 for more details), is

the point in trying to map linguistic concepts (the propositions in which they appear) and truth-

values in a rigourous way, the sense of determining boundaries after all. This is a problematic

affair, for FL as for every alternative constructive account. Indeed, the criticism can as easily be

directed towards two- or three-valued proposals, but evidently is to be radicalized in the case ofFL, as it does so itself with the notion of truth-value. In this perspective, one could even say FL is

nothing but an extreme version of CL.26

According toMichael Tye, in his [1994b], the result is

a commitment to precise dividing lines that is not only unbelievable but also thoroughly contrary

to [] vagueness [as we conceive of it in connection with everyday examples] (p.11).

3.4. Higher-Order Vagueness

Facing the accounts presented in both the preceding two points, is a problem that can hardly be

left unmentioned, cause its particularly nasty, and therefore indeed deserves a (brief)paragraph of its own: the phenomenon named Higher-Order Vagueness (HOV). In the

previous subsection, we learned that FL (as any multi-valued logic) runs into difficulties trying

to characterize truth-functionality, i.e. determining the proper number of values and the way

they should relate to one another. Now, suppose we achieve to do this. Then still, a far more

serious obstacle remains: that of the attribution of the different truth-values to statements, i.e.

that of linking the vague object-language on the one hand, and the precise meta-language on

the other. In any multi-valued logic, one has at least three options when asked to validate a

formula, for either it is true, false, or something in between these two, where the something in

between gets filled in depending of the number of intermediate values available (from one to

an infinite amount, i.e. a continuum, of them).

At this point, the objection raised against the CL-account by the sorites paradox itself, can be

generalized towards multi-valued ones. In the words of Williamson: As grain is piled on

grain, we cannot identify a precise point at which That is a heap switches from false to true.

We are equally unable to identify two precise points, one for a switch from false to neutral, the

other for a switch from neutral to true. If two values are not enough, three are not enough

(op.cit., p.111). And so on: for the very same reason, four, five, six or more truth-values wont

suffice. Fuzzy logics might indeed be able, to a large extent, to circumvent or ease out

obstacles met in real-life applications, reaching from washing machines to expert systems

25 Zadeh/Kacprzyk [1991] offers a nice view of this branch of research and development.

26 I owe this point to Jean Paul Van Bendegem, who in his turn pointed to the late Leo Apostel as hisinspiration. See also the problem of higher-order vagueness set out in 3.4.


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providing with medical diagnosis. These uncontested practical successes do not, however,

answer the philosophical question, which gets duplicated at every threshold: why should we

switch from one answer to another at this very point? On the contrary, we are dragged ever

more deeper in trouble, as long as more truth-values are being added to the analysis.

Eventually, FL leaves us philosophers with an infinity of these arbitrary thresholds to justify.

Nothing is gained. The regress into which we are driven only highlights the problem whichgave rise to the initial problem (Burgess [1990b], p.418).

27

4. Incoherentism

Solutions in the previous section recommended an alternative interpretation of the quantified

premiss (3.1), an alternative inference-tool (3.3), or something in between these two (3.2). The

current section will concentrate on those who believe the sorites argument as a whole is of an

irreducible incoherence, i.e. that it forms a genuine paradox. This means that, in their view, a

coherent logical interpretation as in section 3 is in principle impossible. Part of them, viz. anumber of rather traditional(ist) logicians, propagate the radical banishment of vague components

(i.e. the incoherence) out of natural languages realm, while another group devotes itself to the

joyful acceptance of this incoherence. The latter school can be further subdivided. Some think

that, matters being incoherent as they are, the quest for a sensible semantics should be abandoned

in the face of vagueness. Others (i.e. few) opt for paraconsistent models. Still others aspire to

develop approaches much richer and complicated than the logical ones in the strict sense, i.e.

having the field of examination contain pragmatical criteria next to the well known semantical

ones. The latter tendency will be dealt with extensively in the next section, the others right here.

Lets start out with the logicians whom I somewhat scornfully called traditional(ist). They are

being exemplified by Gottlob Frege and Bertrand Russell. Where the formers attitude towardsvague language will strike the reader as pretty straightforward, the latters is rather dubious.

Frege leaves no doubt whether vagueness should be taken seriously or not, as every symbol in

his newly developed formal language can have nothing but a stipulative meaning. In this

Begriffsschrift-logic of 1879, a complete and consistent set of definitions and recursive rules

saveguard the logically perfect language against the creeping in of semantically defective

propositions, such as the ones containing vague terms. Every formula can have but one

referent or truth-value. Clear enough that CL fails for sorites paradoxes: because vague

predicates generate borderline cases, LEM would end up under unbearable pressure. This

leaves us, according to Frege, with no choice but to eradicate vagueness. Susan Haack

captured the essence of a similar program quite nicely: The recalcitrant sentences, those the

assignment of true or false to which is thought to give rise to difficulty, do not make

statements, or, do not express propositions ([1974], p.50). And thats it. Why bother precisifying

vague concepts for instance, if in doing so, the meaning of the terms involved unavoidably gets

altered?

Frege never even considered a systematic study of the matter and only mentioned it briefly.28

Russell, on the contrary, devoted an entire article to it (the reflection of a lecture held at the

Jowett Society in 1922). Thefirstof its two major conclusions comes down to Freges: CL, i.c.

27 A lot more could be said here, but I will kindly leave it to the reader to discover. There is, e.g., an interestingdiscussion brought about by Sorensen [1985] (viz. Deas [1989], Burgess [1990b], Hyde [1994], Tye [1994a]),during the study of which I couldnt help wondering whether HOV really reduces to vagueness of vague (this

remains an open question to me). Also notice that section 5 will open an alternative route for dissolving HOV.28 For instance, see indeed his famous Begriffsschrift (e.g., in Frege [1971], p.62).


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LEM, doesnt hold for vague language. The other part of the message brought by Russell

however, is not that vague terms should be banned, for that would leave us with empty hands.

The fact is that all words are attributable without doubt over a certain area, but become

questionable within a penumbra, outside which they are again certainly not attributable (Russell

[1988], p.149, my emphasis). When saying all words, Russell means all words, including the

logical ones, such as connectives and truth-values.29

The result of this is even more drastic than inFreges case: All traditional logic habitually assumes that precise symbols are being employed. It

is therefore not applicable to this terrestrial life, but only to an imagined celestial existence (op.

cit., p.151). Thats the story of earthly life versus logical heaven in a nutshell. Williamson

comments that, as Frege, Russell never meant to seek some non-traditional system of logic better

adapted to vagueness, for no logic worth the name is reliable when applied to vague symbols

([1994], p.59). Nevertheless, both Freges and Russells solution come down to good old CL,

getting rid of vague concepts all together (see, in fact, 3.1). The argument is that logic is an

ideal realm, aimed at, yet finally unattainable for poor humans, who as a consequence have but

one option: to live their miserable lives infected with such terrible diseases as vagueness. Once

logic gets in, vagueness gets thrown out. It is, however, left extremely mysterious how we

should be able to conceive a perfectly precise CL-like language, when at the same time, all of

our words are vague.

In the (substantial) rest of this section, I shall focus attention on the theorists ready to embrace

incoherent vague language. They are, perhaps somewhat surprisingly, called nihilists, for

holding that inconsistency is a prevalent feature of natural language (Burns [1991], p.28).

How is that? Well, if the paradox, i.e. the reasoning involved, is accepted as genuine, clearly

no cut-off point can ever be admitted. Easy enough to see, that either both or none of basic

premiss and conclusion then are to be accepted. Any which of these two options we take, we

are left with contradiction. The nihilistic attitude consists, of course, in embracing the latter,

rather than withdrawing in the face of it. Formally, this can be done using, or even devising, a

paraconsistent logic. In fact, this option is all too often brushed aside without any serious

consideration. Reviewing Williamsons [1994], Dominic Hyde notices: The broad scope of

the historical survey, extending to a comprehensive bibliography, is marred only by the

omission of dialectical or paraconsistent analyses ([1995], p.925). This is particularly strange,

he adds, for our daily practice more often seems to amount to overdetermination rather than

underdetermination.30

Just imagine, or rather realize, how many times questions like Is it

raining?, Is he bald? or Are you tired? tend to get answered by a Yes andno.31

According to Sorensen [1985], the most straightforward acceptance lf the incoherence thesis

appears in the work of Peter Ungeranj Samuel Wheeler (p.135). Indeed, after inspection of their

[1979] and [1875] respectively, both his and Williamsons [1994] characterisation of these

authors, viz. as global nihilists, prove to be incontestable. The standpoint they promote is mostradical: it doesnt allow any sorites-like argumeot ever to start up, for it denies the legitimacy of

its basic premiss. Let me illustrate. Assuming ths paradox to be a genuine one (A1), we start out

with a definite heap of grains (A2).32

Now we begin removing the grains one by one. For it has

been rgreed that the argument is acceptable (A1), , three, two, one, and even zero grains will

make a heap. The latter assertion clearly amounts to absurdity, but rather than rejecting (A1), as is

29 Some criticism on this is to be found, e.g., in Sorensen [1988b] (p.268) and Williamson [1994] (p.55-56).

30 The latter of which is remedied, for instance, by the popular SV-method explained in 3.2.

31 For some introductory literature on paraconsistency, see Priest et al. [1989]. For an interesting paraconsistent

variant of paracomplete SV (3.2), called subvaluation, see Hyde [1997], and, particularly, Varzi [199+].32 Say a million of them. If youre not satisfied, add more grains until you agree the configuration forms a heap.


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done elsewhere, global nihilists reject assumption (A2), and hold that a million, or whatever

numser of, grains do notform a heap. Not unsurprisingly, the title of Ungers [1979] said there

are no ordinary things.33

Any other response, he notes, either directly forces us to accupt a

metaphysical and/or conceptual miracle, as the existence of a principal cut-off pnint in a gradual

sorites-process is called, or indirectly does so, as in the case of multi-valued logics, no rational

counter is provided with, for the miracles just mentioned remain necessary, be it on a higher level(see 3.4).

Undoubtedly, with his [1975], Michael Dummett has delivered one of the most impressive

philosophical writings on our topic. Nevertheless, it is primarily concerned with something

else, viz. strict finitism as a philosophy of mathematics, an item which is combined with the

status of observational predicates and phenomenal qualities. The articles conclusion is the

declaration of both finitisms and observationals incoherence. I will here quite naturally

concentrate on the latter. Throughout, Dummett reflects on variants of the sorites, e.g. (Hao)

Wangs Paradox (formulated in the domain of natural numbers),34

or little-by-little arguments

exploiting the non-transitivity of indistinguishability (e.g. try the minute hand of your watch).

The heart of the paradox, for Dummett, is that either we shall have to say that a contradictorystate of affairs may appear to obtain, or we shall have to say that, from It appears to be the case

that A and It appears to be the case that B, it is illicit to infer It appears to be the case that A

and B ([1975], p.321). Any which way, it should be clear, for him, that a workable logic for

vagueness is illusory, since the use of vague language is intrinsically incoherent, and the

paradoxes involved are insoluble. This is a confirmation of what Frege said, but Dummett goes on

to appreciate vagueness as an essential feature of any natural language that truly functions, instead

of wanting to eradicate this problematical quality. Vague fragments nonwithstanding, we speakers

usually understand ourselves and one another, at least to a large enough extent so as to

accomodate daily life. In other words, most of the times, language works. If we would demand,

on top of that, mutual (or indeed even individual) consistency in the meanings of the vocabulary

we share, all linguistic efficiency would desperately be lost, because awaiting the impossible,

(internal) communication would inevitably come to a hold.

A final philosopher of logic and language I cant leave unmentioned here, for having on his

curriculum some of the key texts on the topic of my article: Crispin Wright. Moreoer, treating

him at the end of this section, is not without significance, for in his writings, an important

evolution takes place. It is this movement, viz. towards a non-propositional account, that Ill

try to highlight now, stripped of all possible detail. What Wright has been increasingly

criticizing, in fact, is the principle of rule-boundedness in linguistics. Even admitted its

actually there, he further wondered how we could ever, and why we would ever want to,

penetrate to the understanding of any such rule, which speakers implicitly (have to) follow, no

matter what. His initial position towards vague concepts is that of Dummett: the linguisticitems in question lead to semantical incoherentism, not because of our intellectual laziness

(Frege), but inherently so: The utility and point of the classifications expressed by many vague

predicates would be frustrated if we supplied them with sharp boundaries. [] It is not generally a

matter simply of lacking an instruction where to draw the line; rather the instructions we already

have determine that the line is notto be drawn (Wright [1975], p.330).

Wright goes on to confirm the large actual success of the language-game, as he calls it (after later

Wittgenstein), but his eventual diagnosis, fully elaborated in his [1987], deviates from, or rather

33 Not just traditional sorites-prone concepts as scalps or heaps are subject to this law. As there are no ordinary

things whatsoever, every object we name, e.g. a stone, lacks an existential status (i.e. genuine nominalism).34 In short, the paradox states that every number n is small, since 0 is small, and if n is small, then so is n+1.


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goes beyond, Dummetts. In fact, according to him, the latter does not offer one after all, as he is

unable or unwilling to identify a fallacy in our inconsistent, yet functional fabric called natural

language. For Wright on the other hand, it is the very idea of incoherently codified practice that is

in itself incoherent: If the rules for the use of red really do sanction the paradox, why do we

have absolutely no sense of disturbance, no sense that a real case has been made for the

inferential ingredient at all? Are we so abjectly irrational that we cannot recognize ourconfusion even when it is completely explicit? A different account is called for.

35His spinned

out [1987] is an attempt to get a grip on such an account, which has to be non-proposional by

nature. This is not to say that the notion of meaning becomes useless. The account will concern

non-cognitive, yet meaningful knowledge, like the one constituting practical skills, which we

just seem to be able to perform, but couldnt ever (want to) describe in full detail. We learn

them by doing, in the course of an ever-lasting ostensive training, so that their meaning is

always under construction. Note that, together with Wright, we have already stepped with one

foot in the next section.36

5. The Pragmatical Turn

So finally now for the radical departure from all traditional semantico-logical theories of

language brought together in the previous two sections, theories which consider the linguistic

vehicles of a concept as having a meaning which fixes its extension, the set of things of which it is

true. Following Mark Sainsbury however, no sharp cut-off to the shadow of vagueness is

marked in our linguistic practice, so to attribute it to the predicate is to misdescribe it ([1990],

p.11). In the course of section 2, we have been confronted with one particularly tough problem

over and over again: that of HOV (2.4). No wonder, says Sainsbury, for you do not improve a

bad idea by iterating it (ibidem), and thats exactly what multi-valued semantics try to do. Hence,

the argument in favour of pragmatical accounts, concerned with language-users rather than the

world they talk about, will be the absolute uselessness of rigid linguistic boundaries, inextricably

bound up with the apparent unwillingness and/or inability of speakers to draw them. Doing

pragmatics, contextual constraints, whether internal, as in the case of Diana Raffman, or external,

as in the case of Ruth Manor, will prove to be extremely important.37

According to Raffman, an adequate treatment of vague predicates and their sorites puzzles must

appeal to the character of our judgments about the items in the series ([1994], p.44). Lets regain

the red-orange continuum of colour patches to illustrate. It is easily agreed upon that any subject

has to (and will) report a shift in colour sometime, be it sooner or later, when running through the

series, in order to avoid the paradoxical conclusion. But does this not inevitably mean then, that

two particular adjacent patches have to be reported as having a different colour? Raffman thinks

not, as neighbouring patches can never be in different categories when judged pairwise, only

when judged singly. To be able to explain this, the traditional (cartesian) picture of the integrated

mind has to be challenged, for at least two different mentalpersonae are said to be at the basis of

any observational decision: one categorizing, i.e. vertically comparing with a (possibly

imaginary) model, another discriminating, i.e. horizontally comparing two objects. So, the latter

35 Cited from the abridged reprint of Wright [1987] in Keefe/Smith [1996] (p.213).

36 In fact, Wright also seems to have inspired others to look for a more behaviouristic semantics, as Raffmanconfides in her [1994] (p.43). See the sequel.

37

Some further pragmatically inspired accounts can be found in the writings ofHans Kamp (viz. his [1981],preceded by [1975]), Linda Burns (viz. her [1991] and [1994]), Graham Priest (viz. his [1998], prepared in[1991]), and Terry Horgan (viz. his [1994] and [1998]). In my [200+], I additionaly elaborated on the latter two.


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only comes into play from the moment the patches are (to be) considered pairwise. Finding a pairto be marginally different, he will constrain his colleague to categorize them identically: Cate-gorize them as you like, hell say, but categorize them together (op.cit., p.47).

A suitable diagnosis for sorites is straightforward: if judgments alongside the series are madepairwise, a shift in colour cannot happen but pairwise. Hence, it becomes possible that twoadjacent patches x5 and x6 are said to be red, while just after (or before) that, the patches x6 and x7jointly receive(d) the label orange. And theres more to it. A phenomenon called judgmentalinertia prevents categorical shifts from being local. As long as pairs of patches are judged as red,the polar attraction of the original red reference will remain fairly strong. Thats the reason why,once a shift to orange has occured, it will often be observed that the brand new, orange referencespreads its force way back in the series, so that red pairs now have become orange ones. Thisdoes not however, at any moment, imply that couples of patches are seen as red and orange at thesame time. Perception is episodic, and it is precisely because of this kind of shifts that nodiscontinuities whatsoever are perceived. In this respect, also, and finally, notice that thepresentation of the colour-spectrum can turn out to be extremely important. Just think about the

force of the initial red reference when all of the patches (also the clear orange ones at the extremeof the continuum we are moving towards) are disposed during the entire experiment, as comparedto its force when only a couple of patches are uncovered at a time. The account certainly woulddeserve to be dealt with more thoroughly, for its foundations are far more elaborated than evensuggested here, but that would really take me too far, so I have to leave it at that. 38

In the interpretation of Ruth Manor, who in her turn wishes to challenge the sharp distinctionbetween semantics and pragmatics, the relevant contexts are not internal or psychological, asRaffmans, but rather external or physical. More precisely, it is the domain of objects to whichproperties are (not) predicated, that, in her view, plays the crucial role in the varying meaningsvague sentences exhibit. As Manor, in her [1997], also proposes a formal framework that does

justice to this particular contextual constraint, I will set it out, meanwhile allowing the underlyingidea gradually to become clear.

One of the basic ideas is that vague predicates may denote so-called foggy objects, whose partscannot unequivocally be determined. In order to formalize them, objects dare represented by setsof (alternative) sets of atoms belonging to the domain A: d

. (A). As a set of subsets of A, anyobject dis characterized as a set of possible delineations. It will then be called a distinctobject dd,if it has only one such a delineation (dd = {s}, with s A), or afoggy object df, if it has at leasttwo (df= {s1,s2,}, with s1,s2, A). The opposite #dof an object drelative to A is defined as#d={A\s with s d} (A\s reads: the complement of s relative to A). An object d1 is then aboundary case of an object d2, iff every member of the former is a subset of some member(s) ofthe latter and a subset of some member(s) of its complement #d2. So the boundary case d1 for anobject d2 is partly enclosed in that object d2 (more exactly: some of the delineations of d1 areenclosed in delineations ofd2), and partly enclosed in its complement #d2. I have to skip someother notions here, such as parthood, n-ary objects, universal vs. resticted domains, as well as theformal model Manor builds on top of all this. Nevertheless, we are already in a position to get anidea of how a similar model can be used in dissolving sorites-paradoxes.

Consider the domain of people all over the world. In it, for every n, there is at least one person

with n hairs on his scull. Lets therefore call Hn the non-empty extension of has n hairs. Now

consider an objectp = {H0, H0H1, , H0H1H2... Hn, }, the elements of which are

the sets of people with less than n hairs. Obviously, one can define its complement #p. In this

38 See, however, the [1994] article refered to and cited here, as well as Raffman [1996].


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particular domain,p and #p then denote the bald and non-bald people respectively. One can verify

that theyre both vague (or foggy) in nature, for counting different subsets of the domain asmembers (as the delineation of the n-haired clearly cant be unequivocal). But more importantly,

every set of people with a particular number of hairs, e.g. {Hn}, neither belongs top nor to #p.

This means that the corresponding object {{Hn}} can never belong to the extension of is bald or

is non-bald, or, put differently, that it is always a boundary case of these two. The contextualaccount proves to be fruitful, for it depends on how the domain gets divided, i.e. by determining n

(from occasion to occasion), whether the group of n-haired will count as bald (togetherwith the

rest of the bald) or non-bald (togetherwith the rest of the non-bald), while on their own they cant

be but borderline.

6. On the Origin of Vagueness

Gareth Evans, in a mere one page article, provided with a formal proof that met an enormousamount of fierce criticism in the twenty years that have passed since its appearance. However,I am disappointed (yet amazed), not about the quantity, but about the arguments that havegiven those who reacted and contra-reacted to this [1978]. As I see it, the article couldnt havedone other than arouse suspicion, for one is claiming to decide on an objective characteristic ofthe world in a strictly logical way. Since logic clearly belongs to humans view upon thisworld (more particularly, the part of it expressed or expressible in language), Evans shouldhave explained why his proof nevertheless is able to go beyond logic. He never anticipatedthis, nor did anyone ever after. But more and most surprisingly: nobody ever bothered askingor wondering. I will set this case aside for the time being, to present the famous proof andsome of its major criticisms, but only after having quoted some inspiring words to be found in

Lejewski [1976]: The ontologist is anxious to justify, with the aid ofinformal arguments, hisexistential answers to the fundamental question: what kinds of entities are there? The logicianis not concerned with these arguments. The theories he develops do not imply, in the logicalsense of imply, the existence of anything or should not imply the existence of anything, if hecares for logical purity (p.28, my emphasis). Ill come back to it at the end of this section.

Now for Evans proof, with first some preparations: a and b are names for objects, a=b is anidentity statement with undecided truth value, and the operator expresses this uncertainty.The actual proof then, is as follows.

(1) (a=b)

(2) x [(x=a)]b

(3) ~(a=a)(4) ~x [(x=a)]a

(5) ~(a=b)

---------------

(5) ~(a=b)

Explanation. Line (1) expresses the uncertainty of the statement a=b (the object with thename a is the same as that with the name b).39 What happens in line (2) is called intensional or-abstraction: object b has the property of being thus that it is uncertain whether it is identical toobject a. This is a major step, and one who doesnt accept it, can halt the proof here, before it has

actually started, as does Lowe [1994], saying there can be no such thing as the property an39 Henceforth, I will abbreviate the object with the name a/b to object a/b or even a/b simpliciter.


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object has just in case there is no objective fact of the matter as to whether or not it is identicalwith the object a (p.112).

Line (3) seems to be as evidently clear as true: it is not uncertain whether object a is identical toobject a. But he who defends objective vagueness would simply be giving in accepting it, for thisparticular statement resides at the core of the ongoing discussion: are there vague, throughout thisproof interpreted as non self-identical, objects? Somewhere near the end of this section, I willrefer to Quantum Mechanics (QM) as a positive line on this ontic vagueness or non-identity. Ihave qualms about reducing vagueness and identity to one another, but explaining this in detailwould lead us too far here.40 Anyway, one can wonder if Evans is not begging the question byinserting (3).

Lets move on to line (4) then, which reformulates (3) in terms of-abstraction. In the quotedarticle, Lowe goes on scrutinizing this logical technique, claiming that the symmetrical propertiesx [(x=a)] and x [(x=b)] are in fact one and the same, as they only differ by permutation of aand b. As a consequence, and even if (2) were true, (4) could never be so: the property expressesuncertain self-identity, and object a is equally submitted to it as b is.

Line (5) concludes the proof, by applying CPOS41 of Leibniz Law (LL)42: as (2) attributes aproperty to b which (4) denies to a, both objects are non-identical (so not of uncertain mutualidentity, ergo non-vague compared to one another).

In (5), Evans strenghtens the former conclusion: a and b definitely are distinct objects, providedlines (1) to (4) can be prefixed by a definitely-operator . This whole operation presupposes and its freshly introduced dual to be strong enough to generate a modal logic as strong asS5, Evans confines us. But as Johnson [1989] points out, this is far too strong a claim:Evanss sentential operator , representing objective indeterminacy is in fact not the dual ofthat symbol which Evans defines as its dual, , or else the modal logic Evans proposes togenerate obviously is not nearly as strong as S5, contrary to his indications (p.104). We turnto Michael Dummett, who explains: If it is possible to give a coherent account of this matter,then the result will be in effect a modal logic weaker than S4, in which each reiteration of themodal operator Definitely yields a strenghtened statement ([1975], p.311). Indeed, thecharacteristic S4-axiom pp would be disastrous for any vagueness-logic. Besides, whatabout the following S5-axioms: pp and pp, the first of which trivializes vagueness(as every statement is definitely indefinite), while the latter neglects any possibility of HOV(see 3.4).43

Thus far, I have only considered the technical (non)merits of the proof. Lets now focus on itsphilosophical interpretation, as have done numerous commentators. The most important line ofcriticism is the one questioning reference in Evans proof. Ill try to summarize the main

points of it. We have seen that statements of the form p can be made true or false bysharpening or precisifying them (in a SV; see 3.2). When doing so for object a in (3) and (4),notice that both expressions are equivalent only in case of exact reference. But then we have tosuppose what were trying to prove, which means that inference (3)-(4) is invalid, bothThomason [1982] and Noonan [1982] conclude. The latter specifies that an a with multiplesharpenings leaves us with an expression (4) uncertain in truth-value, and an a without any

40 See note 46 for some references.

41 From a b classically to infer ba.

42 (a=b) (Ea Eb) for all predicates or properties E. Remarkably, and despite its crucial role in this very proof,

Evans himself never bothered formalizing it (Johnsen [1989], p.104).43 On Evans careless use of operators, see also Over [1989].


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possible sharpening even with a false expression (4).44 In his [1990], Noonan puts theseremarks in some philosophical context, reminding us of the quest for vague objects thisdiscussion is all about, and Evans failure to demonstrate any necessary incoherence in the ideaof their very existence. However, he adds that if whenever there is denotation of vagueobjects there is also semantic indeterminacy i.e. in denotation then [] the idea that the

objects indeterminately denoted are themselves ontologically indeterminate ceases to do anywork in explaining the indeterminacy in truth-value (p.159-160).45 The whole questionconcerning relative identity is said to be closely linked to all this, but that would really take meto far in view of the purposes of this paper.46

Another, less essential, line of criticism focuses on the use of LL, or more exactly, of CPOS ofLL, by Evans. This precision is important, as both Broome [1984] and Garrett [1991] note.They stand in defence of a three-valued semantics for vagueness in which the inference ruleMT (which amounts to CPOS, followed by MP) is not valid, blocking the derivation of (5)from (2) and (4).47

To close off this final section, Ill come back to its opening lines, with some critical questions

about the nature of logical investigation, as applied here to the case of ontological vagueness.It is a well known principle that no logic, exotic as it may be, can be allowed, in the course of aformal proof, to addinformation not contained by its premisses. This means that if there werevagueness in the world, the evidence about the latter would already have to be enclosed in thepremisses feeding any formal proof of it, as an argumentation can never be valid if itsconclusion speaks about an object with property G, while the premisses dont (VanBendegem [1997], p.107, my translation).48 Our logical tools are able to nothing more thandisclose this concealed or entangled knowledge, using some sophisticated technique. As far asIm concerned, a suitable motto for all articles refered to in this section would be the one of

Richard Heck Jr.: I shall not, however, attempt to decide the question whether there are vague

objects. The conclusion of the present paper is just that logic alone does not preclude the existenceof such objects ([1998], p.274). Nor can it provide evidence of the opposite in its own right.

Theres another form of reduction, making use of which one can easily trivialize this verydiscussion (as well as hundreds of others), and Ill suffice with raising it. The argument is thatevery piece of knowledge about the world, objective as it may be, is mediated through ourrepresentations. As a consequence, all potential vagueness should be imputed to the latter, so thatnothing but an epistemical question can ever arise, as the so-called ontological problem, whetherthere actually is one or not, is de jure inaccesible to us. French and Krause raise this point at thebeginning of their commendable [1996], which offers a survey of their joint work in the field of a

44 Mutatis mutandis, the same line of argument applies to object b in steps (1)-(2) of the proof. For aninteresting, i.c. restricted interpretation of both inferences (and the proof as a whole), see Garrett [1991] (p.345).

45 Compare Everett [1996]: The claim that we cannot rigidly designate vague or indeterminate objects seemsdreadfully ad hoc. [] It seems that if we can dub or causally interact with a determinate object in such a wayas to provide it a with rigidly designating name, surely we should be able to dub or causally interact with avague object in a similar way. [] The friend of vague objects really owes us some independent philosophicalmotivation [] and an account of how rigid designation is possible for determinate objects but why it fails forindeterminate ones (p.217).

46 See however Burgess [1989]-[1990a], Noonan [1990], Tye [1990], and Zemach [1991], to mention but a few.

47 This step can be written out in some more detail as follows. From (2) Eb, (4) Ea, to infer (4) (EaEb), ascan be easily shown, and from (4) and (4) (a=b)(EaEb), i.e. LL, by MT to infer (5) (a=b). See Broomesarticle for some truth-tables (p.9), and Garretts for an instructive philosophical comment.

48

It should be added here that an element of relevance has to be in play. Clearly, one is warranted, e.g., to movefrom (x)(Px) to (x)(PxGx). As clearly as that, nothing can ever be gained by this move. Except for beingallowed to speak about property G in connection with object x, that is.


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quantum logical account, defending (a sensible approach to) ontic vagueness. Apart from them,also Rolf [1980] and Sainsbury [1989] have recognized the danger of cheap reduction at the heartof this discussion, and have tried to anticipate it, proposing more classically looking formalframeworks. In them, the distinction epistemic-ontic vagueness is (or better: can be) maintained,at least in the first stages of analysis.

7. Conclusion

In conclusion of my paper, I shall not recapitulate the sections.49 Instead, here are some tips forthose who now might plan on devoting some of their available library space (and money) tovagueness. In listing the following books, viz. two extensive surveys, two specials ofphilosophical journals and a reader, I do not in the least pretend completeness, but neverthelessfirmly believe that they offer an economical basis for further exploration.

One cant help noticing that I made frequent use of Williamson [1994], arguably the bestreference to date, irrespective of its (quite natural?) pronounced theoretical position (3.1). Inits turn, Burns [1991] sets out for an ambitious survey, with explicit sympathy for pragmaticalaspects (section 5), but the result, however commendable, is both less complete and systematicthan Williamsons. Taking a big leap backwards in time, allow me to draw your attention to1975s special issue ofSynthese, a milestone in the study of vagueness, for it collects some ofthe key articles in various schools of thought.50 Recently, The Monistalso devoted a specialissue to vagueness (the 2nd number of 1998s 81st volume, to be precise), with contributionsranging from both Hecks and Hydes on ontology (section 6), and Chambers on theintuitionistic view (refered to in section 1), to exercises in renewing strategies (leaning towardswhat was sketched in section 5), e.g. indeterminism (Burgess) and transvaluationism (Horgan),

the latter of which is explicitly based on what Sainsbury [1990] told. The few gaps of primaryliterature still remaining after consultation of these journals, can be filled by Keefe/Smith[1996]. True, this reader will add some doubles to your archive (viz. Fines, Dummetts andWrights [1975]), but next to them youll now be in possession of some further essentialpapers: Black [1949], Machina [1976] (both 3.3), Russell [1988], an abridged version ofWright [1987] (both section 4), Sainsbury [1990] (section 5), Evans [1978] (section 6), andsome more. Finally, a copy of The Southern Journal of Philosophys 33th supplementaryvolume (1994), reporting on the Spindell Conference 1993 on vagueness, could prove useful,especially as an introduction to some pragmatically inspired accounts (section 5).

49 Please refer to the introductory one for a brief overview of the article.50 Namely the [1975]s of Fine (3.2), Zadeh (3.3), Wheeler, Dummett, and Wright (all section 4).


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REFERENCES

ALSTON William P., Philosophy of Language, Prentice Hall, Englewood Cliffs N.J., 1964.

BLACK Max, Vagueness: An Exercise in Logical Analysis, in his Language and Philosophy.Studies in Method, Cornell University Press, Ithaca NY, 1949 (pp.25-58).

BROOME John, Indefiniteness in Identity, Analysis Vol.44 1984 (pp.6-12).

BURGESS J.A., Vague Identity: Evans Misrepresented, Analysis Vol.49 1989 (pp.112-119).

BURGESS J.A., Vague Objects and Indefinite Identity, Philosophical Studies Vol.59 1990a(pp.263-287).

BURGESS J.A., The Sorites Paradox and Higher-Order Vagueness, Synthese Vol.85 1990b(pp.417-474).

BURGESS J.A., In Defence of an Indeterminist Theory of Vagueness, The Monist Vol.811998 (pp.233-252).

BURNS Linda, Vagueness. An Investigation into Natural Languages and the Sorites Paradox,Kluwer Academic Publishers, Dordrecht/Boston/New York, 1991.

BURNYEAT M.F., Gods and Heaps, in: SCHOFIELD Malcolm et al. (eds.), Language andLogos. Studies in Ancient Greek Philosophy, Cambridge University Press, London/NewYork, 1982.

CHAMBERS Timothy, On Vagueness, Sorites, and Putnams Intuitionistic Strategy, TheMonist Vol.81 1998 (pp.343-348).

COPELAND B. Jack, Vague Identity and Fuzzy Logic, The Journal of Philosophy Vol.941997 (pp.514-534).

DEAS Robert, Sorensens Sorites, Analysis Vol.49 1989 (pp.26-31).

DUMMETT Michael, Wangs Paradox, Synthese Vol.30 1975 (pp.301-324).

EVANS Gareth, Can There Be Vague Objects?, Analysis Vol.38 1978 (p.208).

EVERETT Anthony, Qualia and Vagueness, Synthese Vol.106 1996 (pp.205-226).FINE Kit, Vagueness, Truth and Logic, Synthese Vol.30 1975 (pp.265-300).

FREGE Gottlob, Begriffsschrift. A Formula Language, Modeled upon That of Arithmetic, forPure Thought, 1879, in: VAN HEIJENOORT Jean (ed.), From Frege to Gdel. ASource Book in Mathematical Logic, 1879-1931, Harvard University Press, CambridgeMass., 1967, 19712 (pp.1-82; translated from German by Stefan Bauer-Mengelberg).

FRENCH Steven, KRAUSE Dcio, Quantum Objects are Vague Objects, Sorites (ElectronicQuarterly of Analytical Philosophy, edited by Lorenzo PEA, Madrid) Vol.6 1996(pp.21-33).

GARRETT Brian J., Vague Identity and Vague Objects, Nos Vol.25 1991 (pp.341-353).

GOGUEN J.A., The Logic of Inexact Concepts, Synthese Vol.19 1968-69 (pp.325-373).

HAACK Susan, Deviant Logic. Some Philosophical Issues, Cambridge University Press,London/New York, 1974.

HECK Richard G. (Jr.), That There Might Be Vague Objects (So Far as Concerns Logic), TheMonist Vol.81 1998 (pp.274-296).

HELLENDOORN Hans, Reasoning with Fuzzy Logic, Ph.D. Technical University Delft,published on his own, 1990.

HORGAN Terry, Transvaluationism: A Dionysian Approach to Vagueness, The SouthernJournal of Philosophy Vol.33 Supplement 1994 (pp.97-126).

HORGAN Terry, The Transvaluationist Conception of Vagueness, The Monist Vol.81 1998(pp.313-330).

HORWICH Paul, The Nature of Vagueness, Philosophy and Phenomenological ResearchVol.57 1997 (pp.929-935).

HYDE Dominic, Rehabilitating Russell, Logique et Analyse Vol.35 1992 (pp.139-173).


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21

HYDE Dominic, Why Higher-Order Vagueness Is a Pseudo Problem, Mind Vol.103 1994(pp.35-41).

HYDE Dominic, Vagueness (Review), Mind Vol.104 1995 (pp.919-925).

HYDE Dominic, From Heaps and Gaps to Heaps of Gluts, Mind Vol.106 1997 (pp.641-660).

HYDE Dominic, Vagueness, Ontology and Supervenience, The Monist Vol.81 1998 (pp.297-

312).JOHNSON Bruce, Is Vague Identity Incoherent?, Analysis Vol.49 1989 (pp.103-112).

KAMP Hans, Two Theories about Adjectives, in: KEENAN Edward L. (ed.), FormalSemantics of Natural Language, Cambridge University Press, Cambridge Mass., 1975(pp.123-155).

KAMP Hans, The Paradox of the Heap, in: MNNICH Uwe (ed.), Aspects of PhilosophicalLogic. Some Logical Forays into Central Notions of Linguistics and Philosophy, Reidel,Dordrecht/Boston/ London, 1981 (pp.225-277).

KEEFE Rosanna, SMITH Peter (eds.), Vagueness. A Reader, MIT Press, Cambridge Mass.,1996.

LEJEWSKI Czeslaw, Ontology and Logic, in: KRNER Stephan (ed.), Philosophy of Logic,Basil Blackwell, Oxford, 1976 (pp.1-28).

LOWE E.J., Vague Identity and Quantum Indeterminacy, Analysis Vol.54 1994 (pp.110-114).

MACHINA Kenton F., Vague Predicates, American Philosophical Quarterly Vol.9 1972(pp.225-233).

MACHINA Kenton F., Truth, Belief, and Vagueness, Journal of Philosophical Logic Vol.51976 (pp.47-78).

MANOR Ruth, Solving the Heap Paradox without Sacrificing Vagueness, draft presented atthe Centre for Logic and Philosophy of Science, University of Ghent, Belgium, 1997.

NOONAN Harold W., Vague Objects, Analysis Vol.42 1982 (pp.3-6).

NOONAN Harold W., Vague Identity yet again, Analysis Vol.50 1990 (pp.157-162).

OVER D.E., Vague Objects and Identity, Analysis Vol.49 1989 (pp.97-99).

PRIEST Graham, ROUTLEY Richard, NORMAN Jean (eds.), Paraconsistent Logic. Essays onthe Inconsistent, Philosophia Verlag, Mnchen, 1989.

PRIEST Graham, Sorites and Identity, Logique et Analyse Vol.135-136 1991 (pp.293-296).

PRIEST Graham, Fuzzy Identity and Local Validity, The Monist Vol.81 1998 (pp.331-342).

PUTNAM Hilary, Vagueness and Alternative Logic, Erkenntnis Vol.19 1983 (pp.297-314).

QUINE W.V.O., Word and Object, MIT Press, Cambridge Mass., 1960.

RAFFMAN Diana, Vagueness without Paradox, The Philosophical Review Vol.103 1994(pp.41-74).

RAFFMAN Diana, Vagueness and Context-Relativity, Philosophical Studies Vol.81 1996

(pp.175-192).

ROLF Bertil, A Theory of Vagueness, Journal of Philosophical Logic Vol.9 1980 (pp.315-325).

RUSSELL Bertrand, Vagueness, The Australasian Journal of Psychology and PhilosophyVol.1 1923 (pp.84-92), reprinted in: The Collected Papers Volume 9: Essays onLanguage, Mind and Matter 1919-1923, Edited by SLATER John G., with theassistance of FROHMANN Bernd, Unwin Hyman, London, 1988 (pp.145-154).

SAINSBURY Mark, What is a Vague Object?, Analysis Vol.49 1989 (pp.99-103).

SAINSBURY Mark, Concepts without Boundaries, Kings College, London, 1990.

SANFORD David H., Borderline Logic, American Philosophical Quarterly Vol.12 1975(pp.29-39).

SANFORD David H., Competing Semantics of Vagueness: Many Values versus Super-Truth,Synthese Vol.33 1976 (pp.195-210).


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SCHIFFER Stephen, Two Issues of Vagueness, The Monist Vol.81 1998 (pp.193-214).

SORENSEN Roy, An Argument for the Vagueness of Vague, Analysis Vol.45 1985(pp.134-137).

SORENSEN Roy, Blindspots, Clarendon Press, Oxford, 1988a.

SORENSEN Roy A., Precisification by Means of Vague Predicates, Notre Dame Journal of

Formal Logic Vol.29 1988b (pp.267-275).SORENSEN Roy, The Ambiguity of Vagueness and Precision, Pacific Philosophical

Quarterly Vol.70 1989 (pp.174-183).

SORENSEN Roy, The Metaphysics of Words, Philosophical Studies Vol.81 1996 (pp.193-214).

SORENSEN Roy, Ambiguity, Discretion, and the Sorites, The Monist Vol.81 1998 (pp.215-232).

THOMASON Richmond, Identity and Vagueness, Philosophical Studies Vol.42 1982 (pp.329-332).

TRILLAS Enric, Mengers Trace in Fuzzy Logic, Theoria Vol.11 1996 (pp.89-96).

TYE Michael, Vague Objects, Mind Vol.99 1990 (pp.535-557).TYE Michael, Why the Vague Need Not be Higher-Order Vague, Mind Vol.103 1994a

(pp.43-45).

TYE Michael, Vagueness: Welcome to the Quicksand, The Southern Journal of PhilosophyVol.33 Supplement 1994b (pp.1-23).

VAN BENDEGEM Jean Paul, Tot in der Eindigheid. Over Wetenschap, New Age en Religie,Hadewijch, Antwerpen/Baarn, 1997.

VAN KERKHOVE Bart, Vagueness Unlimited: A Look at Some Pragmatic Accounts of theSorites Paradox, to appear, 200+.

VARZI Achille C., Supervaluationism and Paraconsistency, to appear in the proceedings ofthe First World Congress on Paraconsistency (Summer 1997, University of Ghent,

Belgium), 199+.WILLIAMSON Timothy, Vagueness, Routledge, London, 1994.

WRIGHT Crispin, On the Coherence of Vague Predicates, Synthese Vol. 30 1975 (pp.325-365).

WRIGHT Crispin, Further Reflections on the Sorites Paradox, Philosophical Topics Vol.151987 (pp.227-290).

ZADEH Lofti, Fuzzy Logic and Approximate Reasoning, Synthese Vol.30 1975 (pp.407-428).

ZADEH Lofti, KACPRZYK Janusz (eds.), Fuzzy Logic for the Management of Uncertainty,John Wiley & Sons Inc., New York, 1992.

ZEMACH Eddy M., Vague Objects, Nos Vol.25 1991 (pp.323-340).

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