La gramática filosófica de Wittgenstein

Wittgenstein’s Philosophical Grammar: A NeglectedDiscussion of Vagueness

Nadine Faulkner, Carleton University

I. Introduction

In Philosophical Grammar (PG),1 Wittgenstein provides a short but densediscussion of the phenomenon of vagueness. In particular, he discusses theadverb “about” and “the problem of the heap.”2 The problem of the heapis a paradigm case of the Sorites paradox, and much has been written onit.3 Although discussions concerning “about” are less common, the topicraises the same issue: the adverb, combined with a precise word, results ina vague phrase. In both cases, there is an absence of sharp boundaries forthe concepts involved.

Wittgenstein’s discussion occurs after the Tractatus Logico-Philosophicus(TLP) but before Philosophical Investigations (PI).According toWittgenstein’sview in TLP, any vagueness in language is a surface phenomenon; analysiswill show that seemingly vague propositions are in fact determinate.4

Later, in PI, Wittgenstein challenges his Tractarian belief in an ideal ofexactness underlying ordinary language. He also explores the related viewthat vague words must somehow be deficient. It is here that his well-knowndiscussions of family resemblance, language game and rule-following occur.

1. PG was composed between 1932 and 1934 and comprises TS 213; MSS 114–115 and140. All page numbers in this article refer to the 1974 publication of PG.2. Wittgenstein (1969, 236) writes “Der Begriff ‘ungef ähr,’ ” the concept “about/roughly,”

but he also speaks of “die Grammatik des Wortes ‘ungef ähr,’ ” the grammar of the word“about/roughly” (1969, 236). Similarly, he speaks of “dem Umfang des Begriffs[’Sandhaufen’],” the extension of the concept (“heap of sand”), and of “Die Unbestimmtheitdes Wortes ‘Haufen,’ ” the indeterminacy of the word “heap” (1969, 240). “Unbestimmtheit”can also be translated as “indefiniteness” or “vagueness.”Wittgenstein does not use the word“vage/Vagheit [vague/vagueness]” in these passages. I shall myself use “vague” to describeboth words, including predicates, and concepts, but I shall use “indeterminate” to describeonly concepts.3. Wittgenstein (1969, 240) does mean the Sorites paradox; he states “Problem des

‘Sandhaufens [pile of sand].’ ”When he just wants to refer to the concept or word “heap,”he uses “Haufen.”4. Notebooks (Wittgenstein 1961, 68);TLP (Wittgenstein 1922, 3.23, 4.002, 4.463, 5.156).

DOI: 10.1111/j.1467-9205.2009.01381.xPhilosophical Investigations 33:2 April 2010ISSN 0190-0536

© 2009 Blackwell Publishing Ltd., 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148,USA.

The work I look at from PG marks an interim position in Wittgen-stein’s thinking.Wittgenstein is reassessing how language functions in theabsence of the metaphysical backdrop presented in TLP. He speaks of thegrammar of words, as in PI, but does not yet directly employ the idea ofa language game.5 Comparatively, his discussion in PG offers a morespecific treatment of the problem of vagueness than is provided in PI,where he does not speak of the Sorites paradox at all.

Wittgenstein’s piece is not just of exegetical interest; it marks, in thehistory of philosophy, an important struggle to understand the workings ofour language and the problem of vagueness. Moreover, Wittgenstein’sdiscussion is fruitful. He presents a novel way to conceive of the bound-arylessness6 of vague concepts. Employing an analogy with coin tossingand converging intervals, Wittgenstein offers a new picture of the rela-tionships between the clear cases of the application of a predicate7 and theunclear cases. In particular, the idea of a progression towards the penumbra8

does not arise in his analogical case of coin tossing. Given these consid-erations, one can see that Wittgenstein not only engaged with some of themore typical problems of vagueness before coming to his views in PI butalso had something significant to contribute to them.

II. The Attractive Picture

When writing PG, Wittgenstein no longer believed that a propositioncould be analysed into a combination of simple names that name simpleobjects. Of course, this applied to seemingly vague propositions as well; heno longer believed that analysis would show that they have underlyingdeterminate truth-conditions. He now takes the indeterminacy of such

5. He does use an analogy with the game of chess to show how we can use the samewords with different senses (Wittgenstein 1974b, 238).6. Without ascribing to Wittgenstein any of Sainsbury’s views, we can (usefully) accept

this much of Sainsbury’s (1997, 257) definition of “boundaryless”: “[a] vague concept isboundaryless in that no boundary marks the things which fall under it from the thingswhich do not, and no boundary marks the things which definitely fall under it from thosewhich do not definitely do so; and so on. Manifestations are the unwillingness of knowingsubjects to draw any such boundaries, the cognitive impossibility of identifying suchboundaries.”7. I will speak both of the unclear cases of the application of a predicate (a word) and the

boundarylessness of a concept (not a word).8. Wittgenstein does not use the word “penumbra.” I use it to name the area between the

cases for which predicates clearly do and do not apply. The penumbra itself is also notclearly defined, implying higher-order vagueness. For an early discussion of this issue, seeRussell’s (1983) “Vagueness” article in Papers 9: 152.

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© 2009 Blackwell Publishing Ltd.

propositions as part of their essence. Without the Tractarian view ofanalysis and its reliance on a metaphysics of simple objects, Wittgensteinneeds a new account of how these propositions have sense.

In order to understand how vague words9 function, Wittgensteinexplores two analogies. The first is an analogy with space that willbe familiar to those who have read TLP and PI.10 His basic idea is thatvague words occupy a different “space” or have a different grammar fromnon-vague ones, so certain questions such as “What is the smallest heap?”make no sense. This is an important analogy but does not provide us thedetails of such a “different space.” By contrast, his second analogy withcoin tossing furnishes us with a specific way to think about the function-ing of vague words. I focus on the second analogy.

Wittgenstein (1974b, 236) begins his discussion in PG with threestatements, each containing the word “about”:

He came from about there→.About there is the brightest point of the horizon.Make the plank about 2 m long.

In all three, the word “about” is used as an adverb that is combined witha precise word, resulting in a vague phrase.11 Thus, even if one were todeny that words such as “heap” (discussed later) are vague, sentencescontaining these sorts of words would still need to be accounted for.

The first two statements listed above involve pointing to a place; the thirdis a request. Referring to the request, Wittgenstein asks, “In order to saythis, must I know of limits which determine the margin of tolerance of thislength?”Wittgenstein replies in the negative and adds, “Isn’t it enough e.g.to say ‘A margin of �1 cm is perfectly permissible; 2 would be too much?’ ”(1974b, 236). Importantly, there is a gap in the range that Wittgensteingives.The range does not provide an answer to the question, “Is 1.25 cmacceptable or not?” For Wittgenstein, it is an “essential” part of the sense ofthese propositions that one is “not in a position to give ‘precise’ bounds tothe margin.”

Wittgenstein highlights two characteristics of the sorts of statement hehas listed. First, the margin of tolerance involved – in this case, the lengthunder or over 2 m that would still count as fulfilling the command “Make

9. See footnote 2.10. In TLP Wittgenstein speaks of “logical space,” and later in PI he speaks of the“grammar” of a word.11. Wittgenstein (1974b, 236) points out that his comments about this word depend on thecontext of its use.

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the plank about 2 m long” – cannot be fixed by experiment withoutleading to contradictory results.We may, at one time, allow that a certainlength is permissible, and at another not. This is generally accepted incontemporary discussions as one of the characteristics of vagueness.12

Second – and this, too, is part of the contemporary discussion of vagueness– when asked, we simply do not seem to know of any margin of tolerance(Wittgenstein 1974b, 237; 239).

Wittgenstein’s example, coupled with the two preceding characteristics,provides the following three observations about vague concepts:

(1) the absence of a cut-off point between things that do and do not fallunder the concept (boundarylessness);

(2) the existence of some clear cases of things that do and things that donot fall under the concept; and

(3) an area, or penumbra, that is itself not clearly demarcated and repre-sents cases for which we are undecided or for which we give con-tradictory responses.

These combined observations give rise to a particular picture of vaguenessthat I think Wittgenstein finds both attractive and troubling.The picture isone that fosters the very natural idea that we can approach a would-becut-off point or boundary even if we acknowledge the absence of one.This is the picture, I shall argue, that Wittgenstein wants to replace.

Wittgenstein (1975,263, §211) sets out this attractive picture in an earlierwork, Philosophical Remarks (PR), by providing an experiment in whichlines parallel to an original line are drawn continuously. After each line isdrawn, the subject is asked to say whether it looks smaller than the first.

Comparing this situation with the concept of a heap, he writes in PR:

You might say: any group with more than a hundred grains is a heap andless than ten grains do not make a heap: but this has to be taken in sucha way that ten and a hundred are not regarded as limits which could beessential to the concept “heap.”

12. Contemporary discussions also typically include the fact that the margins of toleranceprovided by different people may be contradictory.

Figure 1



And this is the same problem as the one specifying which of thevertical strokes we first notice to have a different length from the first(1975, 263, §211).

“One hundred grains” and “ten grains” function just like the command“Make me a plank about 2 meters long” coupled with the claim that �1 isacceptable but �2 is not. In both cases, the “limits” are not essential to theconcepts. He states the main problem this way:

And here I come up against the cardinal difficulty, since it seems asthough an exact demarcation of the inexactitude is impossible. For thedemarcation [e.g., ten grains and one hundred grains] is arbitrary . . .(1975, 264, §211).

As I see it, one of Wittgenstein’s points, in contrast to TLP, is thatvagueness cannot be made exact. However, he notices that there is anurge to think of it in an exact way, and it stems from the observation thatthese concepts have cases that are clear: �1 is acceptable but �2 is not; 10grains of sand is not a heap, but 100, he thinks, surely is.Thus, on the onehand, while it seems obvious to acknowledge that vagueness involves theabsence of a sharp boundary, on the other hand, we may also be pulledtowards thinking that vague concepts and non-vague concepts differ onlyin respect to the former having a penumbra. We think the two sorts ofconcepts are similar because both involve clear cases. In this sense, Wit-tgenstein’s discussion in PG and PR is not just one that applies to hisTractarian view of vagueness but also to current conceptions of it thatmaintain the view that we can approach the penumbra.13 This shall beexplained in more detail when I discuss the coin tossing analogy later.

Wittgenstein is troubled with this picture and gives yet a third wayto conceive it. He considers a visual circle and its relation to a Euclideanone. He is exploring the relation between the object, here a Euclideancircle, and what is taken as a vague representation of it in experience. Heprovides the following figure in PR.

He then writes,

There seems to be something attractive [Etwas zieht zu . . .] about thefollowing explanation [of the vagueness involved in the visual circle]:

13. This would include not just degree theorists but also someone like Sainsbury whosuggests an analogy of magnetic poles to describe vague concepts. These conceptionsinvolve a notion of “proximity” to the clear case. But just to be clear, I am not at allsuggesting that Wittgenstein presents an argument that could be used against such views;rather, I am suggesting that his conception differs because it does not carry with it anynotion of proximity to clear cases or the penumbra.

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everything that is within a a appears as the visual circle C, everythingthat is outside b b does not appear as C.We would then have the case ofthe word “heap.” There would be an indeterminate zone left open, andthe boundaries a and b are not essential to the concept defined (1975,264, §211).

That Wittgenstein points out the attraction of this view, I think, signifieshis belief that there is something awry in it. As we can see from thepicture, there is a sense of exactness: we seem to have three areas.However, as Wittgenstein notes, any limit is arbitrary.

Wittgenstein then offers a fourth example in the same text to elucidateFigure 2:

The boundaries of a and b are still only like the walls ofthe forecourts. They are drawn arbitrarily at a point where we can stilldraw something firm. – Just as if we were to border off a swamp witha wall, where the wall is not the boundary of the swamp, it only standsaround it on firm ground. It is a sign which shows there is a swampinside it, but not, that the swamp is exactly the same size as that of thesurface bounded by it14 (1975, 264, §211).

14. The only reference to this section of PR that I have found is in an article by NeilCooper (1995, 261) entitled “Paradox Lost: Understanding Vague Predicates.” Cooper putsforward a verdict theory of vagueness and uses Wittgenstein’s swamp analogy to support hisexplanation of what he calls “the transition problem”; that is, the problem of explaining “thenature of the transition from applying a predicate to applying its negation.” As Cooper seesit, the swamp analogy shows us a way of cordoning off the penumbra that in turn providesfor the clear cases and a transition stretch.Thus, he sees Wittgenstein’s analogy of the swampas offering a solution: We can draw three sharp areas by drawing the beginning of thepenumbra “on firm ground.” On Cooper’s (1995, 263) view, since the boundaries are drawnon firm ground in this picture, “it avoids the problems of higher-order vagueness withoutmaking vague words precise.”

Figure 2



Where the swamp starts is left indeterminate, but it is somewhere inside thewall.What is outside the wall is definitely not part of the swamp, just as aboard that is 2.2 m is not “about 2 metres” and 10 grains of sand is not aheap. The wall is arbitrary in the sense that it is built where one can stillbuild a wall; it “stands around [the swamp] on firm ground” in the same waythat 10 grains of sand and 100 grains15 of sand are numbers of grains of sandfor which no issue arises as to whether they comprise a heap or not.

The boundary around the indeterminate zone is not the limit of theconcept, and this is what it means to say that “an exact demarcation of theinexactitude is impossible” (Wittgenstein 1975, 264, §211). For those whoare familiar with the topic of vagueness, Wittgenstein’s remarks thus farseem to suggest no more than that there is higher-order vagueness.16 ButWittgenstein, I think, has something else in mind as well: what is it aboutthese pictures that is attractive and yet awry?

As I see it, the picture of the indeterminate zone as residing betweentwo determinate zones – in the circle example and in the line example aswell as the analogy with swamps – gives rise to the alluring idea that withinthe clear areas, the concept in question functions just like a non-vagueone. In looking at Figure 2 from PR, we may, for example, think that “byconstantly reducing the interval between the figures shown [we] shall beable to reduce the indeterminate interval indefinitely, be able ‘to approachindefinitely close to a limit between what we see as C [the circle] andwhat [we] see as not C’ ” (Wittgenstein 1975, 265, §211). In the case ofvagueness, we acknowledge that there is no sharp cut-off, and yet Witt-genstein thinks that we still may feel that we can approach the penumbra.With non-vague concepts, it makes sense to speak of “approaching” thecut-off point between, say, “greater than or equal to 2 metres” and “notgreater than or equal to 2 metres.” We can progress towards 2 metres,

By contrast, I think Wittgenstein provides the swamp analogy to show us a way ofthinking about vagueness that is both attractive and problematic. Concerning the diagramof the circles, Wittgenstein (1975, 264) states in PR that “[t]here is something attractive aboutthe following picture . . . /Something pulls towards the following . . .” (italics and secondtranslation (Wittgenstein 1964) mine). In Wittgenstein’s work, this usually means that thereis also something misleading about it – the Augustinian picture of language, for example, isa natural one, and in PI he states that we are “seduced” by certain pictures and “dazzled”by certain ideals (1953, §63 and §100). As I see it, the problem that Wittgenstein ishighlighting in the swamp analogy is the very one Cooper thinks he is solving, namely theurge to think of a transition point as residing somewhere between the clear cases.15. Wittgenstein uses 100 grains, but if that is contentious for anyone, 100 000 could besubstituted.16. See footnote 14 for an alternative view.

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centimetre by centimetre. Similarly, the pictures that Wittgenstein hasprovided give rise to the idea that we can, from the clearly determinatezone, approach the penumbra, just as in the non-vague cases we canapproach the cut-off point.

As I understand Wittgenstein, he thinks that this picture blocks us fromseeing how vague concepts really function. That is, on this view, weconceive of a concept as functioning just like a non-vague one in that wecan approach a cut-off point even though we acknowledge there is notone. On this conception, we do not yet see why, for example, the Soritesparadox arises with vague concepts. The urge to think that we canapproach the penumbra is strong. After all, is not the swamp somewhereinside the wall? And when looking at Wittgenstein’s diagrams, it seemsvery natural to think of three areas even though the in-between area isfuzzy.What we need,Wittgenstein thinks, is an alternative way to conceiveof vague concepts, one in which the idea of approaching the penumbra orindeterminate zone does not arise. An analogy with coin tossing, hethinks, provides just such an alternative.

III. A Psychological Experiment: Set-Up for the Alternative Picture

How does a concept with no sharp boundaries function? Wittgensteinprovides an example of coin tossing that involves converging intervals,probability and the Law of Large Numbers. His modification and use of themathematical concept of an interval applied to vague concepts is used onlyin PG, as far as I know.Wittgenstein likens the natural way to conceive ofvague concepts as similar to point convergence in which a sequenceapproaches a limit, say zero, but never reaches it. By contrast, he wants us totry to think of vague concepts as involving interval convergence. Intervalconvergence can be understood as convergence to an even proportion oftwo values – say, heads and tails – instead of a sequence approaching a singlevalue. To explain these ideas, Wittgenstein first sets the stage with a psy-chological experiment similar to the one involving Figure 1 from PR.

In PG, Wittgenstein (1974b, 237) sets up the following imaginarypsychological experiment using Figure 3.

We are given a curved line, g1, and a straight line below it, g2.A straightline, A (not shown on the diagram), is then drawn across them, and thesection of line A between g1 and g2 is called a.Another line, line b, is thendrawn parallel to a.The subject is then asked to state whether line b is biggerthan line a or whether he or she cannot distinguish the two. We are to



suppose that the subject answers that line b is bigger.The distance betweena and b is then halved, and a new line is drawn, line c.The same question isasked, and the person again answers that line c looks bigger than line a.Thehalving is then repeated, and a new line is drawn between a and c, line d.Line d is still seen as bigger than line a.The process is repeated, and a-d ishalved, and line e is drawn. This is the first point at which the personanswers that the line drawn, line e, does not look bigger than line a.

Thus far, we have a series of lines: b, c and d that look bigger than a.And we have one line, line e, that does not look bigger. Notice that we areclosing in on what one would naturally think of as the penumbra for theconcept “bigger than line a” or, alternatively, the concept “same as line a.”

The experiment continues, and now the section between line e andline d is halved by a new line f. The question asked is: do you see line fas bigger than e (recall that line e is seen as the same length as the originalline a)? The answer is yes. Since e is seen as the same length as a, line f canbe understood as belonging to the group of lines that are seen as biggerthan a. Now e-f is halved by line h. Wittgenstein stops here. We are leftwith a group of lines that are seen to be bigger than a (lines b, c, d, f ) anda line that is seen as no bigger or the same as line a (line e).The status ofline h is left unstated. I present the three sections in succeeding discussionsby using (i) bold letters for the section that constitutes the clear cases for“same as line a”; (ii) regular font for line h, which is undecided; and (iii)italics for lines b, c, d and f, which are the clear cases of “bigger than linea.”The purpose of Figure 4 is to show the two groups of lines that flankthe undecided line h.

The result is very much like the case of the circle at Figure 2.We havetwo clear sections of line for which the subject has said “yes” or “no” tothe question of whether they are the same length as a or bigger than a.I think Wittgenstein means us to think that any line drawn within thesection a–e is the same length as a, while any line drawn within thesection f–b is bigger than line a. But the section between e and f is

Figure 3

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undetermined. Understood in this way, our arbitrary limits (lines e and f ),however, are drawn where we are still on “firm ground.”

Wittgenstein suggests that line a can be approached from the left-handside as well. Since g1 slopes upward on the left, presumably the sectionmarked out from line a to line e would then be flanked on the left-handside with another section that would also represent “bigger than line a.”For simplicity, I shall focus on one end only, the right-hand side (asWittgenstein does).

Keeping in mind Figure 4, we can now explore Wittgenstein’s sugges-tion to think of the concepts “same as line a” and “bigger than line a” assimilar to intervals: interval a–e and interval f–b, respectively, on thediagram. These intervals, however, are importantly different from ournormal conception of them17; in particular, they have no endpoints.Theirso-called endpoints are “blurry” and in “flux” (1974b, 238). To say theyhave no endpoints is to say that the concept “same as line a”18 is vague inexactly the same way that the concept of a heap is: there is no sharpcut-off between a line being seen as the same as line a and being seen asbigger than line a, just as there is no sharp cut-off between somethingbeing a heap and something not being a heap.

Looking at Figure 4, we can see that the interval for “same as line a”has no precise limits. Looking on the right side, we know that line e isincluded and that line f is not, but line e is not an endpoint for the interval“same as line a” since whether line h is included is left undecided.

IV. The Alternative Picture: Coin Tossing and Interval Convergence

The difficulty now is to show how to conceive of the absence ofendpoints for these intervals. Wittgenstein (1974b, 238) suggests that a wayto conceive of the limits as non-precise is to think of the intervals as

17. Normally, we think of an interval as possessing definite endpoints or preciselimits, although they may be open or closed; for example, (a,b) {x: a < x < b}or [a,b]{x: a � x � b}.18. I should qualify this, as Wittgenstein does, by saying that the concept is vague withrespect to visual space ( the “seen length”).

h

a e f d c b

Figure 4



. . . bounded not by points, but by converging intervals which do notconverge upon a point (Like the series of binary fractions that we get bythrowing heads and tails.).

The interval – in our example, the interval of lengths that corresponds tothe seen line a – is to be understood as itself bounded by convergingintervals. I take this to mean that to understand how the interval “same asline a” is bounded (at one end), we need to use the interval “bigger thanline a,” which is to the right of it. I shall explain.

Consider the two (sharp) intervals marked out in Figure 4:“same as linea” = lines a–e; “bigger than line a” = lines f–b. These intervals get physi-cally closer to each other on the diagram as we continue to draw lines andask for responses.When we were marking the responses for the imaginedexperiment, the result on the page was a narrowing of the distancebetween the original line a and the original line b (recall that line b isclearly bigger than line a).

Now, given that we seem to be able to narrow the gap, we may bepulled towards the attractive picture of these intervals progressing towardsa point that is never reached, the way the sequence (1/2, 1/3, 1/4,1/5 . . .) converges to but never reaches the limit 0. When we think thatwe are decreasing the gap between the clear cases of “same as line a” and“bigger than line a,” we are thinking in terms of point convergence; that is,we are thinking that we are approaching but never reaching a cut-offpoint. This is the attractive picture Wittgenstein thinks Figure 2 and theswamp analogy give rise to. To replace this picture of point convergence,he is suggesting interval convergence.

What is interval convergence? To elucidate his notion of convergingintervals that do not converge on a point,Wittgenstein considers the seriesof binary fractions that we get by throwing heads and tails (tossing a coin).Wittgenstein does not elaborate, but presumably he has in mind thesequence of 0s and 1s that one could use to record the results of cointosses, say heads = 1, tails = 0.19 One sequence for 10 tosses might be 1, 0,

19. Translation note: First, “Reihe” has been translated as “series,” but it can also betranslated as “sequence.”There is a mathematical difference between a sequence and a series;I think in this context, “sequence” makes more sense in so far as it is simply the recordedresults of particular coin tosses. Keynes (1921, 340), in his book Probability, which Wittgen-stein (1974a, 116) was sent in 1924, uses “series” when discussing coin tosses but not in thetechnical sense.

Second, Wittgenstein calls the results of coin tosses a sequence/series of “Dualbrüche.”This can be translated as “binary fractions” or “dual fractions.” Perhaps “dual” is better,simply reflecting two possibilities. Wittgenstein may have been thinking of the results as

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0, 1, 0, 0, 1, 0, 1, 1. Unlike the sequence of fractions (1/2, 1/3, 1/4,1/5 . . .), this sequence does not convergeon any value; rather, thesequence oscillates between one of two values, 0 or 1.

But the results of the coin tosses do converge to an even proportion ofheads and tails, and this is what Wittgenstein means by converging inter-vals. This convergence to an even proportion of heads and tails followsfrom the Law of Large Numbers,20 which states that as the sample sizeincreases (in this case the number of tosses) the proportion will get closerand closer to the proportion that is calculated a priori for an infinitenumber of tosses.21 In lay terms, the Law states that the more one tossesthe coin, the more likely the number of heads will be close to the numberof tails; that is, the likelihood of obtaining a proportion of heads and tailsthat is close to an even one is increased.22

In our case of coin tossing, the probability, p, of obtaining a head oneach toss is 0.5, and to calculate the number of heads expected for a verylarge number of flips one simply multiplies the probability, p, by the totalnumber of tosses. (So for 100 000 tosses in a trial, the expected number ofheads is 50 000 (100 000 ¥ 0.5).) Graph No. 1 on page 172 is a normalcurve showing the probability of obtaining particular proportions of headsgiven a trial comprising a large number of flips.That the curve is narrow

being represented by dual fractions as follows: H/HT, H/HT, T/HT, etc. I am using asimpler expression of the results of tosses with 0 and 1.20. Keynes’s book Probability, which was sent to Wittgenstein (1974a, 116) in 1924,contains (i) a detailed discussion of the Law of Large Numbers (also called the Law of GreatNumbers), (ii) results from actual coin tossing experiments (see Graph No. 2) and (iii)Bernoulli’s particular formulation of the law. Keynes (1921, 333, 341, cf 338) gives Ber-nouilli’s Theorem as follows: “. . . if the a priori probability is known throughout, then . . .in the long run a certain determinate frequency of occurrence is to be expected” and “. . . ifthe series is a long one the proportion is very unlikely to differ widely from p.” Keynesstates the Law of Large Numbers in approximate terms, using the phrase “in the long run.”This means that after a sufficiently large number of coin tosses, the actual proportion ofheads and tails will approximate the a priori proportion, which is 50–50. (note: Keynes usesthe word “series”).21. The proportion converges on an even one, but the chance of getting a perfectly evenproportion actually decreases as the number of tosses increases (in the case of an unevennumber of tosses, it is impossible; in the case of a large even number of tosses it just becomesless probable that one will obtain a perfectly even proportion). But this does not conflictwith the general point expressed by the Law of Large Numbers, namely that as we continuetossing, we are more likely to obtain a result that is close to even.22. Keynes (1921, 326) gives the results of three actual coin tossings that were used ascomparisons with the a priori probabilities.The results cited were as follows: (i) Buffon: 1992tails to 2048 heads; (ii) Mr. H.: 2044 tails to 2048 heads; and (iii) Jevons: 10 127 tails to10 353 heads. Jevons’s result is plotted on Graph No. 2.



shows that our expected results cluster around an even proportion; that is,the majority of our results are expected to be close to an even proportion.This is just what the Law of Large Number says.23

The range within which the result of our trial is likely to fall can berepresented by intervals, shown on Graph No. 2. These are the intervalsWittgenstein wants us to think of as binding the “blurry ends” of aninterval that represents a vague concept; these are the intervals thatconverge (but not to a point). We can understand their convergence byconsidering Graph No. 3 on page 174 first. Graph No. 3 represents alower number of tosses in the trials than Graph No. 2. That is why theintervals are larger; we do not expect our results to cluster closely aroundthe most probable result of an even proportion. But as the number oftosses gets larger, following the Law of Large Numbers, the probability ofobtaining some proportion of heads and tails that is close to even isgreater, as shown by the smaller intervals on Graph No. 2 compared toGraph No. 3.As we continue tossing, the intervals “converge” in the sensethat they get smaller and smaller as we have a greater probability ofobtaining results that are closer to an even proportion.

There are two aspects to interval convergence that are important toproviding a different conception of vagueness. First and foremost, thedifference is that the convergence involved is to a proportion, not a point.We shall see below how this applies to our psychological experiment.Second, the Law of Large Numbers does not state that each particular flipprogressively brings the proportion closer and closer to even.This is shownon Graph No. 4. I may obtain, for example, 5 heads out of 8 flips, andmy ninth flip may in fact lead me to a more disproportionate number ofheads and tails, 6 heads and 3 tails.Thus a particular flip may momentarilypull the proportion further away from an even one rather than closer toit. So there is no successive progression the way there is in the sequence{1/2, 1/3, 1/4, 1/5 . . .} that converges to the limit zero. It is these twoaspects of the analogy that dispel the urge to think of a progressiontowards the penumbra.

23. All the graphs assume a very large number of trials and for that reason are symmetrical;but our interest is in the intervals and the relationship between their size as the number offlips (per trial) increase. For a low number of trials (comprising infinite flips) we would havea narrow but skewed graph. Graph No. 3 shows a large number of trials comprising fewerflips, so the intervals are larger, but the graph is symmetrical.

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Graph No. 1

This is a normal curve showing the probability of particular propor-tions of heads given a large number of flips (tosses), n, per trial. Thebaseline shows all the possible numbers of heads, k, out of n.These rangefrom 0, no heads, to all heads, where k = n (0 � k � n).The two ends ofthe curve show that the probability of obtaining 0 heads out of n trials isextremely low as is the probability of obtaining all heads. In our case, p isthe probability of obtaining a head on any given trial; it is 0.5.

The Law of Large Numbers states that as n gets larger, we are morelikely to obtain a number of heads that is close to the most probable value.We can calculate that probable value by multiplying the number of trials,n, by the probability, p, of obtaining a head as an outcome, which is 0.5.So k(np) = 0.5(n). In our case, k(np) means the same as an even number ofheads and tails.

The narrow curve of the graph shows that when n is a very largenumber, we are most likely to get a proportion of heads and tails thatis close to even. This is shown by the fact that the greater part of thearea under the curve is located close to the most probable value k(np),and it predicts that our result will deviate only slightly from the mostprobable value. We are most likely, then, to obtain some number ofheads that differs from k(np) by only a small portion of n flips (thedisproportion between heads and tails will be small).



Graph No. 2

As we saw in the discussion accompanying Graph No. 1, the greaterpart of the area under the curve is clustered around the most probablevalue, which in our case is k(np) where p is 0.5 (an even proportion ofheads and tails).

The intervals we are interested in are the portions of the baselinebetween a and k(np) and between b and k(np) respectively.

For our purposes, there are two important aspects to these intervals.First, the intervals show the range within which our results might fallgiven a large number of tosses. We could plot Jevons’ results for 20 480tosses: 10 353 heads and 10 127 tails with a “J” on the graph (Keynes1921, 326 & see my footnote 22). He obtained a greater number of heads,but note that his result falls within the range of the most probableoutcomes; it is within our intervals.

Second, the intervals represent disproportion. A perfectly even propor-tion would be represented by the absence of any interval. Jevons’ result,while it falls within our range of probable results, has more heads thantails.

An explanation of how these intervals converge is given with GraphNo. 3.

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Graph No. 3

This graph has larger intervals than Graphs No. 1 and 2. Notice alsothat the curve is flatter. This graph represents probable outcomes ofproportions of heads and tails when the number of tosses, n, in a trial issignificantly fewer than in Graph No. 2. (Like Graphs No. 1 & 2, itrepresents an infinite number of trials and for that reason it is symmetrical;non-symmetrical graphs need not concern us here).

The larger intervals at the baseline show more of a range of probableresults. This means that we should expect a greater disproportion ofheads than when n was larger, as in Graphs No. 1 and 2. The degree ofdisproportion we are to expect then, can be shown by the size of theintervals. No intervals at all would represent a perfectly even proportion.

If we compare this graph with Graph No. 2, it is easy to see that theintervals are smaller in Graph No. 2.And the more tosses we do, the morethe intervals “converge”, or shrink, to our most probable value of an evenproportion, represented by k(np).

We can apply this analogically to the psychological experiment inwhich lines are drawn and a person is asked if the line drawn is seenas bigger than line a or the same as line a. A vague concept can beunderstood as having fuzzy ends and these ends can be thought of asbounded by two such converging intervals.

Each time a line is drawn and the subject answers “Same!” or “Bigger!”to “Do you see the drawn line as the same as line a or bigger than line a?”we are to imagine it as equivalent to obtaining a head or a tail. As weincrease the drawn lines, our intervals converge, but to an even propor-tion, not to a point.



Graph No. 4

In order to further show how interval convergence is not like pointconvergence, and in order to quell the urge to think that we are approach-ing the penumbra, we can graph, or track, the results of one particulartrial.

First, the Law of Large Numbers says that as we increase n, we are moreand more likely to obtain a result that is close to an even proportion ofheads and tails. This is very unlike the sequence {1/2, 1/3, 1/4 . . .} thatconverges to 0; our interval convergence is to an even proportion. In thisway,Wittgenstein hopes to give us a way of conceiving of the drawn linesas continuing ad infinitum, yet not progressing to a point.

Second, each toss or each drawn line in our experiment does notprogressively lead to the even proportion in a successive fashion (unlikethe sequence that converges to 0 above). The convergence is not tied tothe particular result of each toss; rather, it is cumulative.

In the graph above, I have plotted our example from the text of 9tosses. On the 8th toss we obtained 5 heads; on the 9th toss, we obtained6. Note how the result of a particular flip can in fact increase thedisproportion between heads and tails.

The dashed lines represent possible results had we continued the trial.The Law of Large Numbers just says that as n gets larger and larger, thelikelihood of obtaining a proportion that is close to even increases; in thatcase, our dashed line would zigzag closer to thehorizontal line at 0.5.

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V. Applying the New Picture to the Psychological Experiment

The proportion of heads and tails can be likened to the proportion of“Same!” responses and “Bigger!” responses that we get in answer to thequestion “Is this line the same as line a or bigger than line a?” in ourimagined experiment.

We are to think of the concepts “same as line a” and “bigger than linea” as being bounded by converging intervals at each end – and this justmeans that we are to think of the lines drawn in our experiment not asprogressing towards the penumbra or some elusive cut-off but rather ascontributing to either the proportion of lines that are seen as the same asline a or those that are seen as bigger than line a.This is the main thrustof the analogy. As we continue drawing the lines, just as when wecontinue flipping a coin, the proportion gets closer and closer to even –represented by converging intervals – but there is no convergence to apoint.

One warning is in order: I do not think Wittgenstein at all meansthat when we perform such experiments we shall in fact find an evenproportion if we continue experimenting – rather, I think he meansthat if we look at vague concepts this way, we shall be less inclined tothink that we approach a penumbra. This picture, involving convergingintervals, gets us away from the idea – arising quite naturally from thediagrams – that we are closing in on the penumbra.

Now, if we think of vague concepts as bounded by convergingintervals, the idea of closing in on some elusive cut-off point simply doesnot arise. Recall that when we drew the lines in the imagined experi-ment at Figure 3, they got physically closer. The result was a narrowingof the gap that one saw on the page, and this gave rise to an idea thatthe penumbra was shrinking: we seemed to be closing in on someelusive cut-off point; that is, closing in on some demarcation between“same as line a” and “bigger than line a.” By contrast, with the coinflips, we have absolutely no urge to say, “Each flip brings me closer andcloser to the penumbra residing between heads and tails.” Nor do wehave the urge to ask, “But where does heads start and tails end?” Thepicture we now have is of the number of heads and tails or the numberof “Same!” and “Bigger!” responses in the case of our experiment,getting larger and their proportions varying. The idea of a progression isentirely absent.



VI. The Problem of the Heap

Wittgenstein leaves us with the task of applying this analogy to theproblem of the heap or, more generally, the Sorites paradox.The paradoxstarts with the uncontroversial claim that one grain of sand does not makea heap. In acknowledging that the concept of a heap is vague,24 onefurther accepts that one grain of sand is not sufficient to make a differencebetween something being a heap and something not being a heap (for ifit were, the concept would have a boundary). One is then led by succes-sive additions of one grain of sand to the absurd result that 100 000 grainsof sand is not a heap.25

I think Wittgenstein’s point is that our uneasiness with the Soritesparadox rests on our confusing vague and non-vague concepts. On theone hand, we acknowledge that the concept of a heap has no sharp cut-offpoint, and that is why we accept that the addition (or subtraction) of onegrain of sand makes no difference to something being a heap. But on theother hand, we retain the idea, taken from the functioning of non-vagueconcepts, that we can approach the penumbra that replaces the sharpcut-off point.26 But, Wittgenstein wants to say, this is not how vagueconcepts function. On his view, there is no scale along which oneprogresses towards the penumbra. But to see this clearly is to see that theSorites reasoning that involves the addition and subtraction of a grain ofsand subjects our concept of a heap to a use for which it is not fitted; thatis, the Sorites treats the concept of a heap as if it functioned in a way thatrests on its component grains.

We might now say that while heaps of sand are indeed made up ofgrains of sand, the grammar of the concept “heap” (of sand) is not onethat ties it to individual grains of sand, the addition (or subtraction) ofwhich moves us along the scale towards being a heap (or not being a heap).

24. On the epistemic view, concepts do have sharp cut-offs, but we are ignorant of them.But the epistemicist does not deny the phenomenon of vagueness. Perhaps, then, anepistemicist might accept Wittgenstein’s way of describing how vague concepts function asa way of describing how they function given our ignorance of their extensions.Wittgensteinlikely would find the epistemic view confused, but he does not here offer any argumentagainst it.25. For an extensive discussion of the history of the paradox, see T. Williamson’s (1994,8–35) Vagueness.26. This discussion applies to “degree vagueness” as opposed to what is sometimes called“combinatory vagueness”. In the latter case, vagueness arises because it is indefinite howmany characteristics one should count from a group of defining characteristics.

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This is similar to the idea that each coin toss not only does not bring uscloser to a point, but also does not directly bring us closer to an evenproportion (see Graph No. 4).We should not be surprised now that whenwe submit such concepts to a form of reasoning that involves theaddition or subtraction of (in this case) a grain of sand that moves usalong a scale, we arrive at an absurdity. Instead of progression along ascale, the more correct conception of the workings of the concept “heap”would be to think of our assents and dissents to the question “Is it aheap?” (after each addition or subtraction of a grain of sand) as twoconverging intervals. For Wittgenstein, taking seriously the idea ofboundarylessness or the lack of a cut-off point importantly means ceasingto see a progression towards one.

Lastly, notice that there is no obvious analogue in the coin tossingexample for the undecided cases or the contradictory responses that mayarise for one subject or between subjects.The coin tosses have two values,and with the lines we only looked at cases for which we had clear answersto “Is it the same or bigger than line a?” Presumably, one could stop theexperiment when a person could not decide and answered “I don’tknow.” I take it that Wittgenstein did not find this problematic in so faras his discussion of fuzzy intervals shows that it makes no more sense totry and demarcate the undecided cases than it does to think one is nearingthe penumbra. As we shall see later, questions such as whether theconcepts “same as line a” or “bigger than line a” overlap are undecided;thus, any division into three (or more) sections would be arbitrary. Arbi-trary cut-offs could be made at the first place that a person becomesundecided, for example, or on “firm ground,” as was the case with theexample of the swamp.

The case of contradictory results between one subject’s own responsesor the responses of different subjects is important. Wittgenstein does notdiscuss it. I do, however, think that the coin tossing analogy sheds light onhow to think of these contradictory results in a rather simple way: thegraphs allow us to plot the actual results of any number of possibleoutcomes. In Graph No. 2, for example, we plotted Jevons’ result of anuneven number of heads and tails.This proportion fell within the range ofexpected results. Similarly, we could plot a second trial that had a differentproportion of heads and tails. We can think of these different resultsanalogously as different results of our imagined psychological experimentswith lines.

With this in mind, a contradictory result is simply a case for which onesubject responds “Same!” (a head) to the question “Is this line bigger than



or the same as line a?” at one time, while at another time she responds“Bigger!” for the same line. We can think of these responses as merelyadding to the number of “Same!” or “Bigger!” responses in a single trial.Alternatively, we can treat the subject’s different responses as part of twodifferent trials.Thus, we would have two trials with a different proportionof “Same!” and “Bigger!” responses. Similarly, in the case of conflictsbetween subjects, we simply plot two different trials, as we plotted Jevons’sresults. If they have agreed on the size of all lines up to that point, thedifferent responses in this case would result in different proportions of“Same!” and “Bigger!” responses. Thus, “contradiction” in these casesmeans two different results for one trial or a result for two different trials.Extreme cases may also occur, just as we may actually obtain a result of allheads in a trial, however unlikely it may be.

VII. The Grammar of Blurry Concepts

Given this different conception of how vague concepts work, we may stillwonder what the relationship is between the concepts expressed by “sameas line a” and “bigger than line a.”What sort of space does the grammarof these concepts give rise to? Wittgenstein anticipates this question andtells us in PG that

. . . the special thing about two intervals that are bounded in this blurredway instead of by points is that in certain cases the answer to thequestion whether they overlap or are quite distinct is “undecided”; andthe question whether they touch, whether they have an end-point incommon, is always a senseless one since they don’t have end-points at all(1974b, 238).

Once we think of the endpoints in this blurred way, the sorts of questionit makes sense to ask change. It is senseless to ask whether the concepts“same as line a” and “bigger than line a” have an endpoint in common orwhether the endpoints touch since they simply do not have endpoints.The absurdity of such a question is brought out when we think of thecoins – no one would think it makes sense to say “Where does heads startand tails begin?” or “Do heads and tails overlap?”

In Figure 5, Wittgenstein (1974b, 239) provides seven examples of thepossible relationships between two intervals or concepts that are boundedin a blurry way. In our case of the intervals “same as line a” and “bigger

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than line a” III, IV, and V would seem to apply; that is, it is undecidedwhether the intervals are separate or overlap.27

Note that in II, the contact is de facto since the intervals can only havede facto ends. Such would be the case if we produced a sharp but arbitraryinterval by stipulation or experiment.

VIII. Arbitrary Cut-offs and Changes in Grammar

Focusing on arbitrary cut-offs and the urge to make “an exact demarcationof the inexactitude,” Wittgenstein suggests in PG that when we make avague concept sharp, we alter its form and thus its grammar (1974b, 239).Referring to the psychological experiment with lines, Figure 3, he sug-gests that one could make an arbitrary cut-off by moving a straight edgefrom the starting edge of b.The place at which the subject first displays aparticular reaction could then be taken as the cut-off point (1974b, 239).

Similarly, he further suggests in PG that one could give a definition ofa heap as a body that has a volume of K cubic meters, and anything lesscould be called a “heaplet” (1974b, 240). But on this view, “it is senselessto speak of a largest heaplet” (1974b, 240). Perhaps Wittgenstein meansthat it is senseless because there is no largest heaplet: if V is taken as avolume that is smaller than K, and so is a heaplet, there will always beanother heaplet with a volume that answers the formula V + (K - V)/2.Wittgenstein then asks, “But isn’t this distinction an idle one?” His reply

27. Those who view vagueness in terms of truth-value gaps perhaps can be understood assaying the concepts are separate; those who view vagueness in terms of truth-value glutsperhaps can be understood as saying the concepts overlap.

Figure 5



is that it is idle if “we mean measurement [of volumes] in the normalsense; for such a result has the form of ‘V � v’ ” (1974b, 240).

His point seems to be that in giving a precise definition, we have noteradicated vagueness as we wished to, since results of measurement havethe form of “V � v.” But Wittgenstein (1974b, 240) then suggests that itis not idle if used as a comparison: “. . . otherwise the distinction would beno more idle than the distinction between threescore apples [60 apples]and 61 apples” (1974b, 240).28 In these cases, arbitrary stipulations can bemade, but whether or not they are idle will depend on how they are used.Moreover, as he suggests in the next paragraph, such a stipulation, if wemake one, “is not the concept we normally use” (1974b, 240).

Wittgenstein also discusses different concepts that result when arbitrarystipulations are made. He uses the example of a butcher weighing meatto the nearest ounce. One can describe this just as one describes thecommand “Make the plank about 2 meters long” in so far as giving theweight this way is like saying that the piece of meat does not weigh morethan P1 and does not weigh less than P2 (1974b, 236).This is the same ashis point earlier that the results of measurement have the form of“V � v.”We could make the expression exact, he tells us, by choosing to

. . . call the result of a weighing “the weight of a body” and in that sensethere would be an absolutely exact weighing, that is, one whose resultdid not have the form “!W � w”. We would thus have altered ourexpression, and we would have to say that the weight of bodies variedaccording to a law that was unknown to us (1974b, 239).

The point is that we are now using “the weight of a body” with a quitedifferent grammar (1974b, 238). Since it here refers to the result of aweighing on a particular occasion, it is conceivable that on anotheroccasion the result may be slightly different; we would then have to saythat the weight varied and that we were ignorant about the law governingthe variation in weights.

In parentheses, Wittgenstein adds:

The distinction between “absolutely exact” weighing and “essentiallyinexact weighing” is a grammatical distinction connected with twodifferent meanings of the expression “result of weighing” (1974b, 238).

The “result of weighing” can be seen at one time as the result of a particularweighing of an object on a particular occasion, or at another time as

28. This point is repeated in PR (Wittgenstein 1975, 266).

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something that weights approximate to. Similarly, I can use the word“length” to mean the measure of a board today at 5:00 – which is exact –or I can use the word to mean something that measurements approximateto, as in the example “Make the plank about 2 meters long” and the case ofvolumes of heaplets (1974b,238;240) But in these cases, the grammar of theconcept is different.

IX. Summary

Wittgenstein’s remarks in PG show that he not only thought of thevenerable problem of vagueness but worked out responses to it. In tryingto understand how these vague concepts function, he offers a coin tossinganalogy, and several examples and diagrams in both PG and PR. He isparticularly concerned with what he sees as a problematic picture thatarises from the following three common observations about vagueness: (i)boundarylessness, (ii) clear cases and (iii) a penumbra. The picture thatthese observations give rise to is the seemingly obvious and appealing onethat we can approach the penumbra or fuzzy region between the clearcases, as depicted in Figures 2 and 4.Although in the case of vagueness wereadily accept that there is no cut-off point, the picture we use leaves uswith a residual belief that we can approach the penumbra the way weapproach the cut-off point of non-vague concepts.

Wittgenstein’s analogy with coin tossing that involves converging inter-vals provides a fruitful way to conceive of the very different logical spacethat vague concepts determine. His analogy provides us with a novelconception of boundarylessness. This new conception quells the urge tothink of a progression towards the penumbra: the “blurry” ends of vagueconcepts can be conceived of in terms of interval convergence as opposedto point convergence. On this view, it makes no sense to speak of aprogression towards a penumbra or an elusive cut-off point between heapsjust as it makes no sense to speak of a penumbra or an elusive cut-offpoint between heads and tails. In an approach similar to the one found inPI, Wittgenstein’s aim is to get clear on how language functions, and hismethod is to provide an alternative picture, one that does not give rise tothe problems and paradoxes that confound and perplex us.29

29. I should like to thank Keith Arnold, whose discussions and constant encouragementhelped me tremendously. I am also grateful to Stephen Talmage, P. M. S. Hacker, PeterMason and I. P. Knight for the time they gave me and their comments.



References

Cooper, N. (1995). “Paradox Lost: Understanding Vague Predicates.” Inter-national Journal of Philosophical Studies 3 (2): 244–269.

Keynes, J. M. (1921). A Treatise on Probability [P]. London: Macmillan.Russell, B. (1983). “Vagueness.” In J. G. Slater (ed.), The Collected Papers:

Essays on Language, Mind and Matter 1919–1926, vol. 9. London: UnwinHyman, pp. 147–154.

Sainsbury, R. M. (1997).“Concepts without Boundaries.” In R. Keefe andP. Smith (eds.), Vagueness: A Reader. Cambridge, MA: The MIT Press,pp. 251–264.

Williamson, T. (1994). Vagueness. London: Routledge.Wittgenstein, L. (1922). Tractatus Logico-Philosophicus [TLP], C. K. Ogden,

trans., London: Routledge.——— (1953). Philosophical Investigations [PI], G. E. M. Anscombe, trans.,

Oxford: Blackwell.——— (1961). Notebooks, 1914–1916 [NB]. Oxford: Blackwell.——— (1964). Philosophische Bemerkungen [PR2]. Oxford: Blackwell.——— (1969). Philosophische Grammatik [PG2]. Oxford: Blackwell.——— (1974a). Letters to Russell, Keynes, and Moore. Oxford: Blackwell.——— (1974b). Philosophical Grammar [PG], A. Kenny, trans., R. Rhees

(ed.), Oxford: Blackwell.——— (1975). Philosophical Remarks [PR], R. Hargreaves and R. White,

trans., Oxford: Blackwell.

Department of Philosophy, Carleton University3A Paterson Hall1125 Colonel By Dr.Ottawa, Ont. K1S 5B6

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