DOI: 10.1111/j.1467-9205.2009.01381.x Philosophical Investigations 33:2 April 2010 ISSN 0190-0536
Wittgenstein’s Philosophical Grammar : A Neglected Discussion Discuss ion of Vaguene agueness ss Nadine Faulkner, Carleton University I. Intr Introductio oduction n In Philo Wittgen tgenstei stein n pr pro ovides a sho short rt bu butt dens densee Philosophica sophicall Gra Grammar mmar (PG),1 Wit discussion of the phenomenon of vagueness. In particular, he discusses the adverb “about” and “the problem of the heap.” 2 The problem of the heap is a paradigm case of the Sorites paradox, and much has been written on it.3 Although discussions concerning “about” are less common, the topic raises the same issue: the adverb, combined with a precise word, results in a vague phrase. In both cases, there is an absence of sharp boundaries for the concepts involved. Wittgen Wit tgenstei stein’s n’s disc discuss ussion ion occu occurs rs after the Tractatus Logico-Philosophicus (TLP) but before Philosophical Investigations (PI).According toWittgenstein’s view in TLP, any vagueness in language is a surface phenomenon; analysis will wi ll sh sho ow th that at se seem emin ingl gly y vag ague ue pr prop opos osit itio ions ns ar aree in fac factt de dete termin rminat atee. 4 Later Lat er,, in PI PI,, Wi Wittg ttgens enstei tein n cha challe llenge ngess hi hiss Trac Tractaria tarian n bel belief ief in an ide ideal al of exactness underlying ordinary language. He also explores the related view that vague words words must somehow somehow be deficient. It is here that his well-known well-known discussions discussio ns of family resemblance, resemblance, language game and rule-follo rule-following wing occur. occur. 1. PG was composed composed between between 1932 and 1934 and comprises TS 213; MSS MSS 114–115 and 140. All page numbers in this article refer to the 1974 publication of PG. 2. Wit Wittgen tgenstei stein n (1969, 236) 236) writes “Der Begriff Begr iff ‘ungef ‘ungef ähr ähr,,’ ” the concept “about/roughly,” butt he als bu also o spe speaks aks of “di “diee Gra Gramma mmatik tik des Wort Wortes es ‘ungef ‘ungef ähr ähr,,’ ” the the gram grammar mar of the word “abo “a bout ut/r /rou ough ghly ly” ” (1 (196 969, 9, 23 236) 6).. Si Simi mila larl rly y, he sp spea eaks ks of “d “dem em Um Umfa fang ng de dess Be Begri griff ffss [’Sandhaufen’],” the extension of the concept (“he (“heap ap of sand”), and of “Die Unbestimmth Unbestimmtheit eit des Wortes ‘Haufen,’ ” the indeterminacy of the word “heap” (1969, 240). “Unbestimmth “Unbestimmtheit” eit” 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 describe both words, including predicates, and concepts, but I shall use “indeterminate” to describe only concepts. 3. Wi Witt ttge gens nste tein in (1 (196 969, 9, 24 240) 0) do does es me mean an th thee So Sorit rites es pa para rado dox; x; he st stat ates es “P “Prrob oble lem m de dess
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The work I look at from PG marks an interim position in Wittgenstein’s stei n’s thin thinkin king. g. Wit Wittgen tgenstei stein n is rea reasses ssessing sing ho how w lang language uage fun functio ctions ns in the absence absen ce of the metap metaph hysic ysical al back backdr drop op pr present esented ed in TLP TLP. He speak speakss of the grammar of words, as in PI, but does not yet directly employ the idea of a la lang ngua uage ge ga game me..5 Co Compa mparat rativ ively ely,, hi hiss di discu scussi ssion on in PG of offers fers a mor moree specifi spe cificc tr treat eatmen mentt of th thee pr prob oblem lem of vagu aguene eness ss th than an is pr pro ovid vided ed in PI PI,, where he does not speak of the Sorites paradox at all. Wittgenstein’s piece is not just of exegetical interest; it marks, in the history of philosophy philosophy, an important struggle to understand the workings workings of ourr lan ou langu guage age and the pr prob oblem lem of vagu aguene eness. ss. Mo More reo over er,, Witt Wittgen genste stein in’s ’s discussion is fruitful. He presents a novel way to conceive of the boundarylessness6 of vague concepts. Employing an analogy with coin tossing and co con nvergi erging ng int interv ervals als,, Wi Witt ttgen genst stein ein off offers ers a ne new w pic pictur turee of th thee re relalationships between the clear cases of the application of a predicate 7 and the unclear cases. In particular, the idea of a progression towards the penumbra 8 does not arise in his analogical case of coin tossing. Given these considerations erati ons,, one one can see that Wi Wittge ttgenste nstein in not onl only y engag engaged ed with some of the more typical problems of vagueness before coming to his views in PI but also had something significant to contribute to them. II. The Attracti Attractiv ve Pict Pictur uree When writ When writing ing PG PG,, Wi Witt ttgen genste stein in no lon longer ger bel belie iev ved th that at a pr propo oposi sitio tion n could be analysed into a combination of simple names that name simple objects. Of course, this applied to seemingly vague propositions as well; he no lon longer ger bel belie iev ved th that at ana analys lysis is wou ould ld sh sho ow tha thatt th they ey ha hav ve un under derlyi lying ng determinate determin ate truth truth-con -conditi ditions ons.. He no now w tak takes es the indet determin erminacy acy of su such ch 5. He do does es us usee an analo analogy wi with th the ga game me of chess chess to show show ho how w we ca can n us usee th thee sa same me words with different senses (Wittgenstein 1974b, 238). 6. Wit Withou houtt ascribi ascribing ng to Wittge Wittgenste nstein in any of Sain Sainsbu sbury’s ry’s vie views, ws, we we can (use (usefull fully) y) accep acceptt this much of Sain Sainsbu sbury’s ry’s (1997 (1997,, 257) definition definition of “bou “boundaryl ndaryless”: ess”: “[a] vague conc concept ept is boundaryless in that no boundary marks the things which fall under it from the things which do not, and no boundary marks the things which definitely fall under it from those which do not definitely do so; and so on. Manifestations are the unwillingness of knowing subjec sub jects ts to dra draw w an any y su such ch bo boun undarie daries, s, the co cogni gniti tiv ve imp imposs ossibi ibilit lity y of ide ident ntify ifyin ing g su such ch boundaries.” 7. I will speak both of the unclear cases cases of the applicatio application n of a predicate predicate (a word) word) and the boundarylessness of a concept (not a word). 8. Wittgens Wittgenstein tein does not use the wo word rd “penumbr “penumbra. a.” ” I use it to name the area between the
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propositions as part of their essence. Without the Tractarian view of analysis and its reliance on a metaphysics of simple objects, Wittgenstein needs a new account of how these propositions have sense. In order to understand how vague words9 function, Wittgenstein explores two analogies. The first is an analogy with space that will be familiar to those who have read TLP and PI. 10 His basic idea is that vague words occupy a different “space” or have a different grammar from non-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 the details of such a “different space.” By contrast, his second analogy with coin tossing furnishes us with a specific way to think about the functioning of vague words. I focus on the second analogy. Wittgenstein (1974b, 236) begins his discussion in PG with three statements, 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 with a precise word, resulting in a vague phrase. 11 Thus, even if one were to deny that words such as “heap” (discussed later) are vague, sentences containing these sorts of words would still need to be accounted for. The first two statements listed above involve pointing to a place; the third is a request. Referring to the request, Wittgenstein asks, “In order to say this, must I know of limits which determine the margin of tolerance of this length?”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 Wittgenstein gives.The range does not provide an answer to the question, “Is 1.25 cm acceptable or not?” For Wittgenstein, it is an “essential” part of the sense of these propositions that one is “not in a position to give ‘precise’ bounds to the margin.” Wittgenstein highlights two characteristics of the sorts of statement he has listed. First, the margin of tolerance involved – in this case, the length under 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
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the plank about 2 m long” – cannot be fixed by experiment without leading to contradictory results. We may, at one time, allow that a certain length is permissible, and at another not. This is generally accepted in contemporary 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 fall under the concept (boundarylessness); (2) the existence of some clear cases of things that do and things that do not fall under the concept; and (3) an area, or penumbra, that is itself not clearly demarcated and represents cases for which we are undecided or for which we give contradictory responses. These combined observations give rise to a particular picture of vagueness that I think Wittgenstein finds both attractive and troubling.The picture is one that fosters the very natural idea that we can approach a would-be cut-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 earlier work, Philosophical Remarks (PR), by providing an experiment in which lines parallel to an original line are drawn continuously. After each line is drawn, the subject is asked to say whether it looks smaller than the first.
Figure 1
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 and less than ten grains do not make a heap: but this has to be taken in such a way that ten and a hundred are not regarded as limits which could be essential to the concept “heap.”
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And this is the same problem as the one specifying which of the vertical 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 is acceptable but 2 is not. In both cases, the “limits” are not essential to the concepts. He states the main problem this way: And here I come up against the cardinal difficulty, since it seems as though an exact demarcation of the inexactitude is impossible. For the demarcation [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 that vagueness cannot be made exact. However, he notices that there is an urge to think of it in an exact way, and it stems from the observation that these concepts have cases that are clear: 1 is acceptable but 2 is not; 10 grains of sand is not a heap, but 100, he thinks, surely is.Thus, on the one hand, while it seems obvious to acknowledge that vagueness involves the absence of a sharp boundary, on the other hand, we may also be pulled towards thinking that vague concepts and non-vague concepts differ only in respect to the former having a penumbra. We think the two sorts of concepts are similar because both involve clear cases. In this sense, Wittgenstein’s discussion in PG and PR is not just one that applies to his Tractarian view of vagueness but also to current conceptions of it that maintain the view that we can approach the penumbra.13 This shall be explained in more detail when I discuss the coin tossing analogy later. Wittgenstein is troubled with this picture and gives yet a third way to conceive it. He considers a visual circle and its relation to a Euclidean one. He is exploring the relation between the object, here a Euclidean circle, and what is taken as a vague representation of it in experience. He provides the following figure in PR. He then writes, There seems to be something attractive [Etwas zieht zu . . .] about the following explanation [of the vagueness involved in the visual circle]:
13. This would include not just degree theorists but also someone like Sainsbury who suggests an analogy of magnetic poles to describe vague concepts. These conceptions involve a notion of “proximity” to the clear case. But just to be clear, I am not at all
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Figure 2 everything that is within a a appears as the visual circle C, everything that is outside b b does not appear as C.We would then have the case of the word “heap.” There would be an indeterminate zone left open, and the 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, signifies his belief that there is something awry in it. As we can see from the picture, 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 elucidate Figure 2: The boundaries of a and b are still only like the walls of the forecourts. They are drawn arbitrarily at a point where we can still draw something firm. – Just as if we were to border off a swamp with a wall, where the wall is not the boundary of the swamp, it only stands around it on firm ground. It is a sign which shows there is a swamp inside it, but not, that the swamp is exactly the same size as that of the surface bounded by it14 (1975, 264, §211). 14. The only reference to this section of PR that I have found is in an article by Neil Cooper (1995, 261) entitled “Paradox Lost: Understanding Vague Predicates.” Cooper puts forward a verdict theory of vagueness and uses Wittgenstein’s swamp analogy to support his explanation of what he calls “the transition problem”; that is, the problem of explaining “the nature of the transition from applying a predicate to applying its negation.” As Cooper sees it, the swamp analogy shows us a way of cordoning off the penumbra that in turn provides for the clear cases and a transition stretch.Thus, he sees Wittgenstein’s analogy of the swamp as offering a solution: We can draw three sharp areas by drawing the beginning of the
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Where the swamp starts is left indeterminate, but it is somewhere inside the wall.What is outside the wall is definitely not part of the swamp, just as a board that is 2.2 m is not “about 2 metres” and 10 grains of sand is not a heap. The wall is arbitrary in the sense that it is built where one can still build a wall; it “stands around [the swamp] on firm ground” in the same way that 10 grains of sand and 100 grains 15 of sand are numbers of grains of sand for which no issue arises as to whether they comprise a heap or not. The boundary around the indeterminate zone is not the limit of the concept, and this is what it means to say that “an exact demarcation of the inexactitude is impossible” (Wittgenstein 1975, 264, §211). For those who are familiar with the topic of vagueness, Wittgenstein’s remarks thus far seem to suggest no more than that there is higher-order vagueness. 16 But Wittgenstein, I think, has something else in mind as well: what is it about these pictures that is attractive and yet awry? As I see it, the picture of the indeterminate zone as residing between two determinate zones – in the circle example and in the line example as well as the analogy with swamps – gives rise to the alluring idea that within the clear areas, the concept in question functions just like a non-vague one. In looking at Figure 2 from PR, we may, for example, think that “by constantly reducing the interval between the figures shown [we] shall be able to reduce the indeterminate interval indefinitely, be able ‘to approach indefinitely close to a limit between what we see as C [the circle] and what [we] see as not C’ ” (Wittgenstein 1975, 265, §211). In the case of vagueness, we acknowledge that there is no sharp cut-off, and yet Wittgenstein thinks that we still may feel that we can approach the penumbra. With non-vague concepts, it makes sense to speak of “approaching” the cut-off point between, say, “greater than or equal to 2 metres” and “not greater 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 of thinking about vagueness that is both attractive and problematic. Concerning the diagram of the circles, Wittgenstein (1975, 264) states in PR that “[t]here is something attractive about the following picture . . . /Something pulls towards the following . . .” (italics and second translation (Wittgenstein 1964) mine). In Wittgenstein’s work, this usually means that there is also something misleading about it – the Augustinian picture of language, for example, is a 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 is highlighting in the swamp analogy is the very one Cooper thinks he is solving, namely the urge to think of a transition point as residing somewhere between the clear cases.
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centimetre by centimetre. Similarly, the pictures that Wittgenstein has provided give rise to the idea that we can, from the clearly determinate zone, approach the penumbra, just as in the non-vague cases we can approach the cut-off point. As I understand Wittgenstein, he thinks that this picture blocks us from seeing how vague concepts really function. That is, on this view, we conceive of a concept as functioning just like a non-vague one in that we can approach a cut-off point even though we acknowledge there is not one. On this conception, we do not yet see why, for example, the Sorites paradox arises with vague concepts. The urge to think that we can approach the penumbra is strong. After all, is not the swamp somewhere inside the wall? And when looking at Wittgenstein’s diagrams, it seems very natural to think of three areas even though the in-between area is fuzzy.What we need,Wittgenstein thinks, is an alternative way to conceive of vague concepts, one in which the idea of approaching the penumbra or indeterminate zone does not arise. An analogy with coin tossing, he thinks, provides just such an alternative.
III. A Psychological Experiment: Set-Up for the Alternative Picture How does a concept with no sharp boundaries function? Wittgenstein provides an example of coin tossing that involves converging intervals, probability and the Law of Large Numbers. His modification and use of the mathematical concept of an interval applied to vague concepts is used only in PG, as far as I know.Wittgenstein likens the natural way to conceive of vague concepts as similar to point convergence in which a sequence approaches a limit, say zero, but never reaches it. By contrast, he wants us to try to think of vague concepts as involving interval convergence. Interval convergence can be understood as convergence to an even proportion of two values – say, heads and tails – instead of a sequence approaching a single value. To explain these ideas, Wittgenstein first sets the stage with a psychological experiment similar to the one involving Figure 1 from PR. In PG, Wittgenstein (1974b, 237) sets up the following imaginary psychological experiment using Figure 3. We are given a curved line, g1, and a straight line below it, g2.A straight line, A (not shown on the diagram), is then drawn across them, and the section of line A between g1 and g2 is called Another line lin b is the
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Figure 3
suppose that the subject answers that line b is bigger.The distance between a and b is then halved, and a new line is drawn, line c .The same question is asked, and the person again answers that line c looks bigger than line a.The halving 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 is halved, and line e is drawn. This is the first point at which the person answers 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 are closing in on what one would naturally think of as the penumbra for the concept “bigger than line a” or, alternatively, the concept “same as line a.” The experiment continues, and now the section between line e and line d is halved by a new line f . The question asked is: do you see line f as bigger than e (recall that line e is seen as the same length as the original line a)? The answer is yes. Since e is seen as the same length as a, line f can be understood as belonging to the group of lines that are seen as bigger than a. Now e-f is halved by line h. Wittgenstein stops here. We are left with a group of lines that are seen to be bigger than a (lines b, c , d , f ) and a line that is seen as no bigger or the same as line a (line e ).The status of line h is left unstated. I present the three sections in succeeding discussions by 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 line a.”The purpose of Figure 4 is to show the two groups of lines that flank the undecided line h. The result is very much like the case of the circle at Figure 2.We have two clear sections of line for which the subject has said “yes” or “no” to the 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 the
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a
e
f
d
c
b
Figure 4
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-hand side as well. Since g1 slopes upward on the left, presumably the section marked out from line a to line e would then be flanked on the left-hand side with another section that would also represent “bigger than line a.” For simplicity, I shall focus on one end only, the right-hand side (as Wittgenstein does). Keeping in mind Figure 4, we can now explore Wittgenstein’s suggestion to think of the concepts “same as line a” and “bigger than line a” as similar to intervals: interval a – e and interval f – b, respectively, on the diagram. These intervals, however, are importantly different from our normal conception of them 17; in particular, they have no endpoints.Their so-called endpoints are “blurry” and in “flux” (1974b, 238). To say they have no endpoints is to say that the concept “same as line a”18 is vague in exactly the same way that the concept of a heap is: there is no sharp cut-off between a line being seen as the same as line a and being seen as bigger than line a, just as there is no sharp cut-off between something being 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 is included 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 of endpoints for these intervals. Wittgenstein (1974b, 238) suggests that a way to 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 precise limits although they may be open or closed; for example (a,b) {x: a < x < b}or [a,b]
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. . . bounded not by points, but by converging intervals which do not converge upon a point (Like the series of binary fractions that we get by throwing heads and tails.).
The interval – in our example, the interval of lengths that corresponds to the seen line a – is to be understood as itself bounded by converging intervals. I take this to mean that to understand how the interval “same as line a” is bounded (at one end), we need to use the interval “bigger than line a,” which is to the right of it. I shall explain. Consider the two (sharp) intervals marked out in Figure 4:“same as line a” = lines a – e; “bigger than line a” = lines f – b. These intervals get physically closer to each other on the diagram as we continue to draw lines and ask for responses.When we were marking the responses for the imagined experiment, the result on the page was a narrowing of the distance between the original line a and the original line b (recall that line b is clearly bigger than line a). Now, given that we seem to be able to narrow the gap, we may be pulled towards the attractive picture of these intervals progressing towards a 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 that we 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-off point. This is the attractive picture Wittgenstein thinks Figure 2 and the swamp 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 converging intervals that do not converge on a point,Wittgenstein considers the series of binary fractions that we get by throwing heads and tails (tossing a coin). Wittgenstein does not elaborate, but presumably he has in mind the sequence of 0s and 1s that one could use to record the results of coin tosses, 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 be translated 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 recorded results of particular coin tosses. Keynes (1921, 340), in his book Probability, which Wittgenstein (1974a, 116) was sent in 1924, uses “series” when discussing coin tosses but not in the technical sense.
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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, the sequence oscillates between one of two values, 0 or 1. But the results of the coin tosses do converge to an even proportion of heads and tails, and this is what Wittgenstein means by converging intervals. This convergence to an even proportion of heads and tails follows from the Law of Large Numbers, 20 which states that as the sample size increases (in this case the number of tosses) the proportion will get closer and closer to the proportion that is calculated a priori for an infinite number of tosses. 21 In lay terms, the Law states that the more one tosses the coin, the more likely the number of heads will be close to the number of tails; that is, the likelihood of obtaining a proportion of heads and tails that is close to an even one is increased. 22 In our case of coin tossing, the probability, p, of obtaining a head on each toss is 0.5, and to calculate the number of heads expected for a very large number of flips one simply multiplies the probability, p, by the total number of tosses. (So for 100 000 tosses in a trial, the expected number of heads is 50 000 (100 000 ¥ 0.5).) Graph No. 1 on page 172 is a normal curve showing the probability of obtaining particular proportions of heads given 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 a simpler 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 Great Numbers), (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 Bernouilli’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 “. . . if the series is a long one the proportion is very unlikely to differ widely from p.” Keynes states 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 of heads and tails will approximate the a priori proportion, which is 50–50. (note: Keynes uses the word “series”). 21. The proportion converges on an even one, but the chance of getting a perfectly even proportion actually decreases as the number of tosses increases (in the case of an uneven number of tosses, it is impossible; in the case of a large even number of tosses it just becomes less probable that one will obtain a perfectly even proportion). But this does not conflict with the general point expressed by the Law of Large Numbers, namely that as we continue tossing, 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 as
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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 be represented by intervals, shown on Graph No. 2. These are the intervals Wittgenstein wants us to think of as binding the “blurry ends” of an interval that represents a vague concept; these are the intervals that converge (but not to a point). We can understand their convergence by considering Graph No. 3 on page 174 first. Graph No. 3 represents a lower number of tosses in the trials than Graph No. 2. That is why the intervals are larger; we do not expect our results to cluster closely around the most probable result of an even proportion. But as the number of tosses gets larger, following the Law of Large Numbers, the probability of obtaining some proportion of heads and tails that is close to even is greater, as shown by the smaller intervals on Graph No. 2 compared to Graph No. 3. As we continue tossing, the intervals “converge” in the sense that they get smaller and smaller as we have a greater probability of obtaining results that are closer to an even proportion. There are two aspects to interval convergence that are important to providing a different conception of vagueness. First and foremost, the difference 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 flip progressively brings the proportion closer and closer to even.This is shown on Graph No. 4. I may obtain, for example, 5 heads out of 8 flips, and my ninth flip may in fact lead me to a more disproportionate number of heads and tails, 6 heads and 3 tails.Thus a particular flip may momentarily pull the proportion further away from an even one rather than closer to it. 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 two aspects of the analogy that dispel the urge to think of a progression towards 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 of
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This is a normal curve showing the probability of particular proportions of heads given a large number of flips (tosses), n, per trial. The baseline shows all the possible numbers of heads, k, out of n.These range from 0, no heads, to all heads, where k = n (0 k n).The two ends of the curve show that the probability of obtaining 0 heads out of n trials is extremely low as is the probability of obtaining all heads. In our case, p is the 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 more likely 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 of heads and tails. The narrow curve of the graph shows that when n is a very large number, we are most likely to get a proportion of heads and tails that is close to even. This is shown by the fact that the greater part of the area 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 most probable al . W st likel th to obtain be of
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Graph No. 2
As we saw in the discussion accompanying Graph No. 1, the greater part of the area under the curve is clustered around the most probable value, which in our case is k(np) where p is 0.5 (an even proportion of heads and tails). The intervals we are interested in are the portions of the baseline between 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 fall given a large number of tosses. We could plot Jevons’ results for 20 480 tosses: 10 353 heads and 10 127 tails with a “J” on the graph (Keynes 1921, 326 & see my footnote 22). He obtained a greater number of heads, but note that his result falls within the range of the most probable outcomes; it is within our intervals. Second, the intervals represent disproportion. A perfectly even proportion would be represented by the absence of any interval. Jevons’ result, while it falls within our range of probable results, has more heads than tails. An explanation of how these intervals converge is given with Graph No. 3.
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This graph has larger intervals than Graphs No. 1 and 2. Notice also that the curve is flatter. This graph represents probable outcomes of proportions of heads and tails when the number of tosses, n, in a trial is significantly fewer than in Graph No. 2. (Like Graphs No. 1 & 2, it represents 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 probable results. This means that we should expect a greater disproportion of heads than when n was larger, as in Graphs No. 1 and 2. The degree of disproportion we are to expect then, can be shown by the size of the intervals. 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 the intervals are smaller in Graph No. 2.And the more tosses we do, the more the intervals “converge”, or shrink, to our most probable value of an even proportion, represented by k(np). We can apply this analogically to the psychological experiment in which lines are drawn and a person is asked if the line drawn is seen as bigger than line a or the same as line a. A vague concept can be understood as having fuzzy ends and these ends can be thought of as bounded 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 we
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Graph No. 4
In order to further show how interval convergence is not like point convergence, and in order to quell the urge to think that we are approaching the penumbra, we can graph, or track, the results of one particular trial. First, the Law of Large Numbers says that as we increase n, we are more and more likely to obtain a result that is close to an even proportion of heads and tails. This is very unlike the sequence {1/2, 1/3, 1/4 . . .} that converges to 0; our interval convergence is to an even proportion. In this way,Wittgenstein hopes to give us a way of conceiving of the drawn lines as continuing ad infinitum, yet not progressing to a point. Second, each toss or each drawn line in our experiment does not progressively lead to the even proportion in a successive fashion (unlike the sequence that converges to 0 above). The convergence is not tied to the particular result of each toss; rather, it is cumulative. In the graph above, I have plotted our example from the text of 9 tosses. On the 8 th toss we obtained 5 heads; on the 9 th toss, we obtained 6. Note how the result of a particular flip can in fact increase the disproportion 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, the likelihood of obtaining a proportion that is close to even increases; in that case, 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 the question “Is this line the same as line a or bigger than line a?” in our imagined experiment. We are to think of the concepts “same as line a” and “bigger than line a” as being bounded by converging intervals at each end – and this just means that we are to think of the lines drawn in our experiment not as progressing towards the penumbra or some elusive cut-off but rather as contributing to either the proportion of lines that are seen as the same as line a or those that are seen as bigger than line a.This is the main thrust of the analogy. As we continue drawing the lines, just as when we continue flipping a coin, the proportion gets closer and closer to even – represented by converging intervals – but there is no convergence to a point. One warning is in order: I do not think Wittgenstein at all means that when we perform such experiments we shall in fact find an even proportion if we continue experimenting – rather, I think he means that if we look at vague concepts this way, we shall be less inclined to think that we approach a penumbra. This picture, involving converging intervals, gets us away from the idea – arising quite naturally from the diagrams – that we are closing in on the penumbra. Now, if we think of vague concepts as bounded by converging intervals, the idea of closing in on some elusive cut-off point simply does not arise. Recall that when we drew the lines in the imagined experiment at Figure 3, they got physically closer. The result was a narrowing of the gap that one saw on the page, and this gave rise to an idea that the penumbra was shrinking: we seemed to be closing in on some elusive cut-off point; that is, closing in on some demarcation between “same as line a” and “bigger than line a.” By contrast, with the coin flips, we have absolutely no urge to say, “Each flip brings me closer and closer to the penumbra residing between heads and tails.” Nor do we have the urge to ask, “But where does heads start and tails end?” The picture we now have is of the number of heads and tails or the number of “Same!” and “Bigger!” responses in the case of our experiment, getting larger and their proportions varying. The idea of a progression is entirel absent.
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VI. The Problem of the Heap Wittgenstein leaves us with the task of applying this analogy to the problem of the heap or, more generally, the Sorites paradox. The paradox starts with the uncontroversial claim that one grain of sand does not make a heap. In acknowledging that the concept of a heap is vague, 24 one further accepts that one grain of sand is not sufficient to make a difference between something being a heap and something not being a heap (for if it were, the concept would have a boundary). One is then led by successive additions of one grain of sand to the absurd result that 100 000 grains of sand is not a heap.25 I think Wittgenstein’s point is that our uneasiness with the Sorites paradox rests on our confusing vague and non-vague concepts. On the one hand, we acknowledge that the concept of a heap has no sharp cut-off point, and that is why we accept that the addition (or subtraction) of one grain of sand makes no difference to something being a heap. But on the other hand, we retain the idea, taken from the functioning of non-vague concepts, that we can approach the penumbra that replaces the sharp cut-off point.26 But, Wittgenstein wants to say, this is not how vague concepts function. On his view, there is no scale along which one progresses towards the penumbra. But to see this clearly is to see that the Sorites reasoning that involves the addition and subtraction of a grain of sand subjects our concept of a heap to a use for which it is not fitted; that is, the Sorites treats the concept of a heap as if it functioned in a way that rests on its component grains. We might now say that while heaps of sand are indeed made up of grains of sand, the grammar of the concept “heap” (of sand) is not one that ties it to individual grains of sand, the addition (or subtraction) of which 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, an epistemicist might accept Wittgenstein’s way of describing how vague concepts function as a way of describing how they function given our ignorance of their extensions.Wittgenstein likely would find the epistemic view confused, but he does not here offer any argument against it. 25. For an extensive discussion of the history of the paradox, see T. Williamson’s (1994, 8–35) Vagueness
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This is similar to the idea that each coin toss not only does not bring us closer to a point, but also does not directly bring us closer to an even proportion (see Graph No. 4).We should not be surprised now that when we submit such concepts to a form of reasoning that involves the addition or subtraction of (in this case) a grain of sand that moves us along a scale, we arrive at an absurdity. Instead of progression along a scale, 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 a heap?” (after each addition or subtraction of a grain of sand) as two converging intervals. For Wittgenstein, taking seriously the idea of boundarylessness or the lack of a cut-off point importantly means ceasing to see a progression towards one. Lastly, notice that there is no obvious analogue in the coin tossing example for the undecided cases or the contradictory responses that may arise 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 answers to “Is it the same or bigger than line a?” Presumably, one could stop the experiment when a person could not decide and answered “I don’t know.” I take it that Wittgenstein did not find this problematic in so far as his discussion of fuzzy intervals shows that it makes no more sense to try and demarcate the undecided cases than it does to think one is nearing the penumbra. As we shall see later, questions such as whether the concepts “same as line a” or “bigger than line a” overlap are undecided; thus, any division into three (or more) sections would be arbitrary. Arbitrary cut-offs could be made at the first place that a person becomes undecided, for example, or on “firm ground,” as was the case with the example of the swamp. The case of contradictory results between one subject’s own responses or the responses of different subjects is important. Wittgenstein does not discuss it. I do, however, think that the coin tossing analogy sheds light on how to think of these contradictory results in a rather simple way: the graphs allow us to plot the actual results of any number of possible outcomes. In Graph No. 2, for example, we plotted Jevons’ result of an uneven number of heads and tails.This proportion fell within the range of expected results. Similarly, we could plot a second trial that had a different proportion of heads and tails. We can think of these different results analogously as different results of our imagined psychological experiments with lines.
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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 merely adding to the number of “Same!” or “Bigger!” responses in a single trial. Alternatively, we can treat the subject’s different responses as part of two different trials.Thus, we would have two trials with a different proportion of “Same!” and “Bigger!” responses. Similarly, in the case of conflicts between subjects, we simply plot two different trials, as we plotted Jevons’s results. If they have agreed on the size of all lines up to that point, the different responses in this case would result in different proportions of “Same!” and “Bigger!” responses. Thus, “contradiction” in these cases means 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 all heads 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 still wonder what the relationship is between the concepts expressed by “same as line a” and “bigger than line a.” What sort of space does the grammar of these concepts give rise to? Wittgenstein anticipates this question and tells us in PG that . . . the special thing about two intervals that are bounded in this blurred way instead of by points is that in certain cases the answer to the question whether they overlap or are quite distinct is “undecided”; and the question whether they touch, whether they have an end-point in common, 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 question it 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 or whether the endpoints touch since they simply do not have endpoints. The absurdity of such a question is brought out when we think of the coins – no one would think it makes sense to say “Where does heads start and tails begin?” or “Do heads and tails overlap?” In Figure 5, Wittgenstein (1974b, 239) provides seven examples of the
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than line a” III, IV, and V would seem to apply; that is, it is undecided whether the intervals are separate or overlap. 27
Figure 5
Note that in II, the contact is de facto since the intervals can only have de facto ends. Such would be the case if we produced a sharp but arbitrary interval by stipulation or experiment.
VIII. Arbitrary Cut-offs and Changes in Grammar Focusing on arbitrary cut-offs and the urge to make “an exact demarcation of the inexactitude,” Wittgenstein suggests in PG that when we make a vague concept sharp, we alter its form and thus its grammar (1974b, 239). Referring to the psychological experiment with lines, Figure 3, he suggests that one could make an arbitrary cut-off by moving a straight edge from the starting edge of b.The place at which the subject first displays a particular reaction could then be taken as the cut-off point (1974b, 239). Similarly, he further suggests in PG that one could give a definition of a heap as a body that has a volume of K cubic meters, and anything less could be called a “heaplet” (1974b, 240). But on this view, “it is senseless to speak of a largest heaplet” (1974b, 240). Perhaps Wittgenstein means that it is senseless because there is no largest heaplet: if V is taken as a volume that is smaller than K, and so is a heaplet, there will always be another 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
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is that it is idle if “we mean measurement [of volumes] in the normal sense; 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 not eradicated vagueness as we wished to, since results of measurement have the form of “V v.” But Wittgenstein (1974b, 240) then suggests that it is not idle if used as a comparison: “. . . otherwise the distinction would be no more idle than the distinction between threescore apples [60 apples] and 61 apples” (1974b, 240). 28 In these cases, arbitrary stipulations can be made, 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 we make one, “is not the concept we normally use” (1974b, 240). Wittgenstein also discusses different concepts that result when arbitrary stipulations are made. He uses the example of a butcher weighing meat to the nearest ounce. One can describe this just as one describes the command “Make the plank about 2 meters long” in so far as giving the weight this way is like saying that the piece of meat does not weigh more than P1 and does not weigh less than P 2 (1974b, 236).This is the same as his 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 sense there would be an absolutely exact weighing, that is, one whose result did not have the form “!W w”. We would thus have altered our expression, and we would have to say that the weight of bodies varied according 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 quite different grammar (1974b, 238). Since it here refers to the result of a weighing on a particular occasion, it is conceivable that on another occasion the result may be slightly different; we would then have to say that the weight varied and that we were ignorant about the law governing the variation in weights. In parentheses, Wittgenstein adds: The distinction between “absolutely exact” weighing and “essentially inexact weighing” is a grammatical distinction connected with two different meanings of the expression “result of weighing” (1974b, 238).
The “result of weighing” can be seen at one time as the result of a particular weighing of an object on a particular occasion, or at another time as
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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 approximate to, as in the example “Make the plank about 2 meters long” and the case of volumes of heaplets (1974b, 238; 240) But in these cases, the grammar of the concept is different. IX. Summary Wittgenstein’s remarks in PG show that he not only thought of the venerable problem of vagueness but worked out responses to it. In trying to understand how these vague concepts function, he offers a coin tossing analogy, and several examples and diagrams in both PG and PR. He is particularly concerned with what he sees as a problematic picture that arises from the following three common observations about vagueness: (i) boundarylessness, (ii) clear cases and (iii) a penumbra. The picture that these observations give rise to is the seemingly obvious and appealing one that we can approach the penumbra or fuzzy region between the clear cases, as depicted in Figures 2 and 4.Although in the case of vagueness we readily accept that there is no cut-off point, the picture we use leaves us with a residual belief that we can approach the penumbra the way we approach the cut-off point of non-vague concepts. Wittgenstein’s analogy with coin tossing that involves converging intervals provides a fruitful way to conceive of the very different logical space that vague concepts determine. His analogy provides us with a novel conception of boundarylessness. This new conception quells the urge to think of a progression towards the penumbra: the “blurry” ends of vague concepts can be conceived of in terms of interval convergence as opposed to point convergence. On this view, it makes no sense to speak of a progression towards a penumbra or an elusive cut-off point between heaps just as it makes no sense to speak of a penumbra or an elusive cut-off point between heads and tails. In an approach similar to the one found in PI, Wittgenstein’s aim is to get clear on how language functions, and his method is to provide an alternative picture, one that does not give rise to the problems and paradoxes that confound and perplex us. 29
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References Cooper, N. (1995). “Paradox Lost: Understanding Vague Predicates.” International 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: Unwin Hyman, pp. 147–154. Sainsbury, R. M. (1997).“Concepts without Boundaries.” In R. Keefe and P. 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 University 3A Paterson Hall 1125 Colonel By Dr. Ottawa, Ont. K1S 5B6
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