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Borel on the Heap

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Abstract

In 1907 Borel published a remarkable essay on the paradox of the Heap (“Un paradoxe économique: le sophisme du tas de blé et les vérités statistiques”), in which Borel proposes what is likely the first statistical account of vagueness ever written, and where he discusses the practical implications of the sorites paradox, including in economics. Borel’s paper was integrated in his book Le Hasard, published 1914, but has gone mostly unnoticed since its publication. One of the originalities of Borel’s essay is that it puts forward a model of vagueness as imprecision, making particular use of the Gaussian law of measurement errors to model categorization. The aim of our paper is to give a presentation of the historical context of Borel’s essay, to spell out the mathematical details of his model, and to provide a critical assessment of his theory. Three aspects of Borel’s account are particularly discussed: the first concerns the comparison between Borel’s statistical account and posterior degree-theoretic accounts of vagueness. The second concerns the anti-epistemicist flavor of Borel’s approach, whereby the idea of statistical fluctuation is used to undermine the notion of sharp boundary for vague predicates. The third concerns the problematic link between Borel’s model of vagueness as imprecision and the notion of semantic indeterminacy. An English translation of Borel’s original essay is appended to this paper (Erkenntnis, this issue).

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Notes

  1. The English translation does not include the original footnotes of the 1950 French edition, incidentally, in which Borel makes cross-reference to his earlier work on the sorites.

  2. Borel exerted several academic responsabilities in relation to primary as well as higher eduction. See Guiraldenq (1999) and Gispert (2012) for more biographical details.

  3. See Graham and Kantor (2009: 62). Borel wrote to his wife in (1909a, b, c): “Not having any more strength for high mathematics, I will go safely to work in probability and statistics following your uncle Bertrand. It is not much compared to my earlier works in mathematics, but it is useful!” (from Marbo 1967). Borel in this note alludes to Joseph Bertrand, a relative of Paul Appell’s wife, and prominent probability theorist.

  4. Borel wrote at least three important contributions to the field of the foundations of statistical mechanics, namely Borel (1906, 1913), and Borel (1915).

  5. This is also the topic of the last part of Borel’s Traité du calcul des probabilités, entitled “Valeur pratique et philosophique du calcul des probabilités”. The general topic of the practical value of science was much discussed at the time, for instance by Poincaré (1902–1908).

  6. Talk of a positivist inspiration in Borel’s philosophy of science is not fortuitous. The title of chapter 5 of Le Hasard, “Les sciences sociologiques et biologiques”, for instance, is inherited from A. Comte’s division of the sciences.

  7. A good example of Borel’s rather dismissive attitude towards logic is Borel (1907a) entitled “La logique et l’intuition en mathématiques”, where Borel criticizes the views of L. Couturat. Borel writes, for instance: “A logical formula is a phenomenon like the fall of a body or like a tree and mathematics is a natural science in which logic plays no more role than in the other natural sciences” (“Une formule logique est un phénomène comme la chute d’un corps ou comme un arbre et les mathématiques sont une science naturelle dans laquelle la logique ne joue pas plus de réle que dans les autres sciences naturelles”). Despite this, Borel maintained a profound interest for the set-theoretic paradoxes, throughout his career. See in particular Borel (1946).

  8. Peirce uses the word “sorites” and “heap” in his writings, but mostly to talk of chains of arguments, as is quite common at the time. Viz. Peirce (1931–1935) CP 4.45: “A necessary inference from a single premiss is called an immediate inference, from two premisses a syllogism, from more than two a sorites.”. The entry “Sorites” of the Dictionary of Philosophy and Psychology (1902: 557) mentions both meanings to the word, namely “a chain of syllogisms” as first meaning (given by Peirce), and “applied to a Megarian sophism of the ‘Heap”’ as second meaning (given by Baldwin). The latter gives a cross-reference to the entry “Sophism”, also coauthored by Peirce and Baldwin, where the Heap appears under the second definition given there for “sophism”, namely “a false argument which, without deceiving, is difficult to refute logically” (1902: 556).

  9. Russell in particular writes: “Baldness is a vague conception; some men are certainly bald, some are certainly not bald, while between them there are men of whom it is not true to say they must either be bald or not bald”

  10. Peirce (1902) entry notes that the French word for “vague” is “vague”, possibly an indication that Peirce had found it used in French writings.

  11. “Mon argument n’a que l’apparence de l’argument célèbre du Tas, autrement dit du Chauve, lequel passe à très bon droit pour sophistique. Ce qui fait le sophisme, dans ce dernier, c’est l’idée qu’un tas comparativement à un nombre de grains de blé, par exemple, ou qu’une tête chauve, comparativement à un nombre de cheveux fixe, sont des idées vagues, qui de leur nature excluent la précision numérique. De là vient que demander combien de grains ou de cheveux en plus ou en moins font ou ne font pas le tas ou la calvitie, c’est demander quel nombre déterminé d’objets il faut pour constituer un total dont l’idée répond à un nombre indéterminé. La question est donc absurde.”

  12. Frege is another major figure and contemporary of Renouvier, Peirce and Borel, who emphasizes the connection between the sorites and indeterminateness in a brief passage. In the paragraph \(\S\) 26 of Begriffsschrift, he makes a short comment about the fact if the concept ‘heap of beans’ was hereditary in the sequence determined by the relation for b to contain one bean less than a (i.e. if the concept was closed under the relation in question) then 0 beans would make a heap. He points out that heredity fails because of “the indetermination of the concept “heap””. Similarly Peirce (1902: 748) insists that vagueness gets its origin in the fact that “the speaker’s habits of language [are] indeterminate”. Borel himself in the text talks of the “necessary indeterminacy of verbal definitions”.

  13. According to Moline (1969), most Greek scholars seemed to agree, already in Borel’s time, that the paradox should be attributed to Eubulides rather than Zeno. For example Gomperz’s book Greek Thinkers vol. 2, whose English translation from German appeared in 1905 (and whose first volume was translated in French in 1908), clearly attributes the sorites to Eubulides, also giving an explicit discussion of Zeno’s Millet Seed for comparison. Interestingly, Black (1937) mentions that the paradox is “sometimes attributed to Zeno” and cites J. Burnet Greek Philosophy (1928: 114) as a reference. As a matter of fact, Burnet does not refer to the sorites in that passage, he cites a dialogue from Simplicius about Zeno’s Millet Seed with a different purpose in mind. Also, although our research is certainly not exhaustive, the various texts written about Zeno’s paradox by French scholars around the turn of the nineteenth century, including Bergson (1907), do not contain any explicit mention of the sorites paradox but only discuss Zeno’s four classic problems.

  14. Broader interest for Zeno’s paradoxes at that period in the French philosophical community is testified by the publication, in the first issue of the Revue de Métaphysique et de Morale in 1893 of several papers on Zeno’s paradoxes, some of which in relation to Renouvier’s work. See the papers by G. Brochard, F. Evellin, G. Milhaud, G. Lechalas, G. Noël, all grouped in the same issue. Those papers, however, deal with the classic paradoxes about the impossibility of movement, not with the Heap.

  15. “Rien ne serait plus facile, d’ailleurs, que d’étendre l’argumentation de Zénon au devenir qualitatif et au devenir évolutif. On retrouverait les mêmes contradictions. Que l’enfant devienne adolescent, puis homme mûr, enfin vieillard, cela se comprend quand on considère que l’évolution vitale est ici la réalité même. Enfance, adolescence, maturité, vieillesse sont de simples vues de l’esprit, des arrêts possibles imaginés pour nous, du dehors, le long de la continuité d’un progrès.”

  16. See Borel (1907b) and also Borel (1908a), a short follow up to Bergson’s response in the Revue de Métaphysique et de Morale.

  17. “Lorsqu’on s’est habitué à ces formes de pensée, on éprouve devant les sophismes de Zénon d’Elée le même étonnement indulgent que devant un enfant de quatre ans qui demande qu’on lui décroche les étoiles: quelques années plus tard, on pourra lui expliquer pourquoi cela est impossible, mais il ne le demandera plus.”

  18. The word “tolerance” does not appear in Borel (1907), but it is used in Borel (1950) to qualify the permitted error around some standard.

  19. See in particular Hobbs (2000) for a recent perspective on this problem.

  20. Note that Borel writes: “There appears no logical way out of this dead-end; it is therefore not possible to know what is a heap of wheat” (our emphasis). Borel’s remark—which sounds like free indirect speech in this context—is nothing like an endorsement of necessary ignorance about some determinate fact of the matter, since Borel denies that there is anything like absolute boundaries to be known.

  21. See Égré (2009) and Lassiter (2011) for probabilistic versions of the tolerance principle along the lines of (2), though based on assumptions distinct from Borel’s; see Égré (2011a) for the suggestion to interpret Smith’s closeness principle in probabilistic terms.

  22. Borel (1950: 108) is even more explicit on this: “the use of a coefficient of probability allows one to introduce a discontinuity in place where there is an apparent continuity”.

  23. On the relation between probability theory and many-valued logic, see Dubois and Prade’s (2001) valuable clarification. See also Edgington (1997) and Schiffer (2003) regarding conceptual differences between probabilities viewed as reflecting subjective uncertainty, and intermediate degrees of truth taken to represent ambivalence in the case of vagueness. Schiffer’s perspective, like Edgington’s, rests upon a subjective or Bayesian conception of probability. However Borel’s (1907) use of probability in the text relies exclusively on frequentist considerations, from which Borel departed only later (see Borel 1950). See Borel (1914, chapter 1 and \(\S\) 87) for an early discussion of the distinction between subjective and objective probability, which indicates that Borel saw frequentism as a way to bridge the two perspectives.

  24. Hajek (1998: 4) in particular describes as a “frequentist temptation” the idea of interpreting “non extremal truth degrees as relative frequencies”. Borel’s discussion of linguistic judgments may be compared to the following remark by Hajek:

    “there have been attempts to explain non-extremal truth degrees as some relative frequencies. For example, take the sentence: “Sagrada Familia is beautiful” and ask n people “is Sagrada Familia beautiful?” allowing them to say only “yes” or “no”. Imagine 70 % of them say “yes”. Can you take 0.7 to be the truth degree of our sentence? This causes problems because “beautiful” in your question was two-valued (yes-no, say “beautiful”2) whereas in your sentence you deal with fuzzy “beautiful” (“beautiful” f , say).”

  25. See Lewis (1970), Kamp (1975: 134), and Kamp and Partee (1995). See also McGee and McLaughlin (1995). We are indebted to T. Williamson for pointing out this connection.

  26. Compare with Lassiter (2011) and Égré (2009, 2011b).

  27. Black’s definition is more precise, since it defines the consistency of application as the limit of this ratio when the number of observations and the number of observers tend to infinity. See Black (1937: 442) for details.

  28. Black’s account, unlike Borel’s, did have a significant influence on the development of fuzzy logic. See in particular Goguen (1969) and Zadeh (1975) for reference to Black’s work. Goguen, in particular, stresses the importance of Black’s consistency profiles toward the project of solving the sorites paradox within a deductive system. Lakoff (1973), another influential degree-theoretic account of vagueness, pursues the same tradition, but taking additional inspiration from the work of psychologist E. Rosch on typicality ratings to introduce membership degrees.

  29. Łukasiewicz’s infinite-valued logic was proposed in 1922 as a generalization of his three-valued logic. About the infinite-valued logic, Machina (1976: 191) writes: “I know of no place in which Łukasiewicz himself indicated any interest in thinking of his logic as a logic of vagueness. In fact he seems to have had a quite different interpretation in mind – an interpretation of the values as probabilities.” Possibility was in fact the driving notion for Łukasiewicz, although inspection of Łukasiewicz’s work reveals several connections between probability and many-valuedness, starting with Łukasiewicz (1913), where “the truth value of an indefinite proposition” is defined as “the ratio between the number of values of the variables for which the proposition yields true judgments and the total number of values of the variable.” About his infinite-valued system, Łukasiewicz (1930: 173) writes: “it would be most natural to suppose (as in the theory of probabilities) that there are infinitely many degrees of possibility, which leads to the infinite-valued calculus.” However, Łukasiewicz presents as an open issue the specification of the exact link of his infinite-valued calculus with probability theory, and points out in the same text (fn. 22) that his 1913’s account of probability does not rest on the logical notion of many-valuedness introduced in his later work.

  30. See Stiegler (1986) and Fisher (2010) for the mathematical and historical details of Laplace’s and Gauss’s works.

  31. See Borel and Deltheil (1923), \(\S\)8, p. 39: “On donne souvent le nom de loi de Gauss à la loi des écarts que nous venons d’exprimer” (One often gives the name of Gaussian Law to the law of deviations we just stated).

  32. The expression “déviation” or “écart” used by Borel as early as 1909a, b, c is not exactly what is currently known as the “standard deviation” (a term first introduced by Pearson, see Pearson 1896, and Stiegler 1986: 328), though it is closely related. The most complete reference on the notion of “deviation” and its use by Borel is certainly Borel, Deltheil and Huron’s (1954) revised edition of Borel and Deltheil (1923), itself a revised edition of Borel (1909a, b, c). BDH compare several notions of deviations and introduce the notion of “écart-type” (standard deviation) that was absent from earlier editions. The basic expression “écart” is used by Borel (1909a, b, c) and until BDH 1954 to denote k − np where k is the number of favorable outcomes of a Bernoulli trial with probability p. Thus, for n trials, the expression “écart” refers to the deviation of successes from the mean. They use the term “écart réduit” (reduced deviation) for \(\lambda=\frac{k-np}{\sqrt{npq}}, \) that is the ratio of “écart” to “écart-type” or standard deviation (q = 1 − p). As BDH point out on \(\S38\), Borel and Deltheil (1923) define the notion of “écart réduit” as \(\mu=\frac{k-np}{\sqrt{2npq}}\) (there they mention the name “écart étalon” for \(\sqrt{2npq}\)). The version BDH give for the “law of deviations” is that the probability for λ to be between a and b is \(\frac{1}{\sqrt{2\pi}}\int_{a}^{b} e^{-\frac{x^{2}}{2}}dx. \) BDH on \(\S\)39: 77 introduce a notion of “écart probable ou écart moyen”, which is defined as the mathematical expectation of the absolute value of the deviation. Borel and Deltheil 1923: 41 have a notion of “écart quadratique moyen”, defined as “la valeur dont le carré est égal à la moyenne du carré de l’écart” (the value whose square is equal to the mean of the square of the deviation). This corresponds to the notion of standard deviation.

  33. We refer the reader to LeCam (1986) and Fischer (2010) for a detailed history of the central limit theorem.

  34. See Œuvres complètes de P. Lévy, vol. 3. In particular: “Le Théorème fondamental de la théorie des erreurs”, pp. 71–83. Lévy presents interesting reflections on different ways to state and establish the law of large numbers in chap. 5 of the second part of his treatise Calcul des probabilités (Lévy 1925), which includes the now called central limit theorem. His remarks suggest that Borel had obtained the theorem only by analytic methods and under certain restrictions. Cf. p. 250.

  35. The simulation was run with Matlab. We used 25,000 instead of 25,0000 samplings for reasons of memory limitation.

  36. On the definition and properties of measurement scales, see Stevens (1946). Stevens distinguishes between two kinds of nominal scales, Type A and Type B scales. As a typical example of a Type A scale, he gives the numbering of football players in a team. He describes a Type B scale as one in which “each member of a class is assigned the same numeral”. What we describe as a Yes versus No scale is a type B nominal scale in Stevens’ sense.

  37. As a consequence, the decrease of retail price can be viewed as a binomial random variable taking two values, either 5 or 0, with respective probabilities \(\frac{1}{10}\) and \(\frac{9}{10}\). For 52 trials, one for each week, the expectation of this random variable is 26, which is the expected decrease on retail price for a constant weekly diminution of half a cent over one year.

  38. This kind of ‘as if’ arguments occurs at several points in Borel’s analysis. For instance, when Borel, in section 52 of Le Hasard, discusses the status or normal distributions in nature, he takes inspiration from Quételet’s example of the average man to write:

    “Generally speaking, one can say that measured sizes satisfy the same laws as measurement errors; everything happens as if the same man, whose size were equal to the mean, had been measured a large number of times by quite awkward observers or only having very imperfect measurement instruments at their disposal.”

  39. We use the terms “criterion” and “threshold” interchangeably in the current context, and irrespective of their distinct meanings in modern psychophysics (viz. McNicol 1972). The notion of “criterion” is convenient to convey the idea of a subject-relative acceptance/rejection value.

  40. Compare with the standard semantic treatment of gradable adjectives, where “tall” means “significantly taller than s”, with s some standard value, as in Fara (2000).

  41. In particular Borel’s overbrief account of lexical vagueness seems to us to fit with most of Fara’s remarks about the interest-relativity of vague expressions.

  42. As pointed out above (see fn. 23), Borel’s (1907) use of probability rests almost exclusively on frequentist considerations. Only later did Borel express an inclination toward the Bayesian definition of probability. See for instance his review of Keynes in Borel (1924), which Savage’s 1972 bibliographical supplement to Savage (1954) presents as “the earliest account of the modern concept of personal probability known to me”.

  43. “If one sticks to actual speakers of English, there is no prospect of reducing the truth conditions of vague sentences to the statistics of assent and dissent, whether or not one accepts the epistemic view of vagueness.” Williamson considers that the supervenience relation is rather one on which our use of a vague expression tracks a sharp property.

  44. See in particular Borel (1950: 100), probably the most explicit text on this element of arbitrariness; see also Borel 1914: 263–62: “If, out of 1,000 trials, 520 declare A heavier and 480 B heavier (I am supposing that dubious answers are not included), the deviation observed is among those that would happen frequently in a series of 1000 answers drawn from a toin coss; one must not conclude anything, except a strong presumption that a new series of 1000 trials will produce an outcome just as uncertain” (Si, sur 1000 expériences, 520 déclarent A plus lourd et 480 B plus lourd (je suppose qu’on laisse de côté les réponses douteuses), l’écart observé est de ceux qui se produiraient fréquemment dans une série de 1000 réponses tirées à pile ou face; on ne doit donc rien en conclure, sinon une forte présomption pour qu’une nouvelle série de 1000 expériences fournisse une résultat aussi incertain)”. On the value of opinion polls, and their application in psychophysics, see the “Appendix”.

  45. See Borel’s own emphasis in the text on talk of probabilities as “a reasonable convention”.

  46. On vagueness and coordination, see in particular Parikh (1994) and Lipman (2009).

  47. Faulkner (2010) proposes a suggestive reconstruction of the status of vague boundaries for Wittgenstein, making use of the Law of Large Numbers to model boundary fluctuations around a mean value, but based mostly on passing remarks by Wittgenstein.

  48. Decision-making in a broad sense, since utilities are not part of Borel’s picture. Note that several other remarks in Borel’s text are likely to surprise readers well acquainted with some of the recent aspects of the literature on vagueness, such as his passing emphasis on the importance of round numbers, a topic that has also received a significant amount of attention in the recent literature in natural language semantics (see Hobbs 2000; Krifka 2007; Bastiaanse 2011).

  49. “Si l’on mesure une grandeur par comparaison avec un étalon fixe, l’on obtient un certain nombre de décimales sérement exactes suivies d’une dernière décimale sur laquelle plane quelque incertitude; des expérimentateurs divers, également soigneux, obtiendront des valeurs diverses pour cette dernière décimale, de sorte que les observations se résumeront sous une forme telle que la suivante: 30 p. 100 des observateurs trouvent un 5, tandis que 50 p. 100 trouvent un 6 et 20 p. 100 trouvent un 7; une autre grandeur très voisine de la première conduira à des résultats analogues, mais non en général identiques. Il se passe là un phénomène analogue à celui qui se produit dans le sophisme du tas de blé”. (from Borel 1922, English edition, translated by A. Rappoport and J. Dougall, modified by us).

  50. “le continu physique se distingue du continu mathématique en ce que l’expérience ne permettant jamais d’atteindre qu’une approximation limitée, une certaine différence minimum est nécessaire pour l’on puisse discerner deux éléments très voisins. [Fn 2. Il ne faudrait pas croire que nous retombons, en faisant cette constatation, sous le coup de l’objection de M. Poincaré ; la différence minimum n’est pas une constante absolue, mais dépend des conditions expérimentales et souvent aussi de la valeur ou de la nature des quantités que l’on mesure. Par suite, deux grandeurs A et B que des procédés directs de mesure ne permettraient pas de distinguer, pourront être différenciées si on a l’heureuse idée— ou la chance—de les comparer toutes deux à une grandeur convenablement choisie C, que l’expérience directe ne distingue pas de B, mais distingue de A; ces nouvelles expériences diminuent pour A et B le minimum separabile. Un autre procédé pour la diminution de ce minimum est la répétition des expériences et l’application du calcul des probabilités ; c’est là une question fort intéressante sur laquelle j’aurai sans doute l’occasion de revenir un jour”.]

  51. See Fechner (1860: 84):

    “In studying the theory of probability, to which my interest in the development of our methods drove me again and again, the following considerations occurred to me: (1) according to our procedure the measure of sensitivity for differences could be represented by the value usually designated by h, which, according to Gauss, affords a measure of the precision of observations, as long as precision depends only on the sensitivity for the perception of differences under comparable modes of procedures.”

    Interestingly, Fechner got the assistance of the German mathematician Möbius to define a rigorous probabilistic solution of the problem of discriminating the difference between two weights. See Stigler (1986) for a detailed presentation of Fechner’s account and its relation to Gauss’s law.

  52. Titchener’s chapter 2 in that manual is entirely devoted to a presentation of the Law of Error and its relation to measurement in physics and psychophysics.

  53. In Titchener’s setup, this includes cases for which the subjects is unsure either way.

  54. Borel in a footnote uses the Gaussian approximation to the binomial distribution to compute the significance of this result. He calculates what Fisher will call the p-value of the difference, namely the probability of obtaining a difference at least as large in case that difference were due to chance.

  55. Several aspects of Borel’s (1908b) paper would call for further commentary, but would take us too far afield. In particular, Borel makes several interesting comments on the use of statistical methods to establish differences. One comment he makes is that “if one blindly took on board Gauss’s law, or any other precise mathematical law, to express individual errors, one would necessarily be led to the conclusion that precision increases as the square root of the number of observations and can, as a result, be made as high as one wishes).” [“si l’on acceptait d’une manière aveugle la loi de Gauss, ou toute autre loi mathématique précise, pour exprimer les erreurs individuelles, on serait forcément conduit à la conclusion que la précision croît comme la racine carré du nombre des observations et peut, par suite, être rendue aussi grande que l’on veut” (1908b: 268)]. This problem, incidentally, is the starting point of Peirce and Jastrow’s 1885 contribution to psychophysics.

  56. See Borel (1950), p. 104 sqq., the section entitled “Physical continuity according to Poincaré”.

  57. That remark, and the footnotes to Borel (1950), do not appear in the English translation of Borel (1963)’s reissue.

  58. A more recent account of the connection between intransitivity, semi-order, and the way to obtain a weak order out of a semi-order is van Rooij (2011). On indirect indiscriminability, see also Williamson (1994) and Raffman (2011). On the discussion of probabilistic treatments of phenomenal predicates and the problem of transitivity, see Hardin (1988), whose account comes out close in spirit to the first of Borel’s proposed answer to Poincaré’s predicament, and also Voorhoeve and Binmore (2006). For a discussion of Luce’s canonical weak order as resulting from a semi-order—which coincides exactly with Borel’s argument in the example at hand—see Lehrer and Wagner (1985).

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Acknowledgments

We thank three anonymous referees of Erkenntnis for helpful comments, and Marcus Rossberg for his editorial assistance. We are most grateful to Ms. Geneviève Appell for granting us the permission to publish the English translation of Borel’s 1907 essay. For discussions and input at various stages of the construction of this paper, we are particularly grateful to Rachel Briggs, Maria Cerezo, Pablo Cobreros, Benoît Collins, Elie During, Igor Douven, Laurent Fédi, Bas van Fraassen, Mathias Girel, Laurence Goldstein, Erik Gray, Dan Lassiter, Jean-Roch Lauper, Paloma Pérez-Ilzarbe, David Ripley, Katie Steele, Timothy Williamson, and to audiences in Pamplona, Konstanz (Formal Epistemology Festival 2012), Bologna (EEN Meeting 2012), Paris and Oxford. Special thanks to D. Lassiter, P. Pérez-Ilzarbe and J-R. Lauper for detailed comments, to Erik Gray for his collaboration to the translation, to Hélène Gispert and Pierre Guiraldenq for helping us to reach Borel’s estate, and to Laurence Goldstein for catching typos and English mistakes. We thank several institutions for their support: the Agence Nationale de la Recherche under grant ANR-07-JCJC-0070 (program ‘Cognitive Origins of Vagueness’); the European Research Council under the European Community’s Seventh Framework Program (FP7/2007–2013); the project ‘Borderlineness and Tolerance’ (FFI2010-16984), Ministerio de Economía y Competitividad, Government of Spain; the European Epistemology Network (EEN). This research was conducted at the Institut d’Etudes de la Cognition (ENS), with the general support of ANR-10-LABX-0087 IEC and ANR-10-IDEX-0001-02 PSL*.

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Appendix: Psychophysics and the Problem of Nontransitivity

Appendix: Psychophysics and the Problem of Nontransitivity

To a modern eye, Borel’s emphasis on the law of errors in relation to problems of categorization is reminiscent of the importance of the same law of error in psychophysics for questions of discrimination. Psychophysics was still a young science in 1907, and Borel showed interest for the problem of psychophysical measurement. Two other essays of the same period testify that Borel saw a connection between the sorites paradox and the problem of nontransitivity in discrimination. The most explicit discussion of that connection, however, only appeared much later in Borel (1950), when Borel revisited the paradox of the Heap.

In 1909, Borel published a short paper entitled “Le continu mathématique et le continu physique”, in which he responds to Poincaré’s paper “Le continu mathématique” (Poincaré 1893). In that paper, Poincaré discusses the Fechnerian problem of the comparison between different sensations of weight (making explicit reference to Fechner). The puzzle discussed by Poincaré concerns the lack of transitivity in the indiscriminability of sensations. A weight A of 10 g and a weight B of 11 g can be felt to produce identical sensations, and so will the weight B and a weight C of 12 g, however the weight A and the weight C will be felt to be distinct. Poincaré presents the relations A = BB = C and A < C as a “the formula of the physical continuum”. Based on it, Poincaré makes the hypothesis that the invention of the mathematical continuum might have been prompted by an effort to analyze away the “intolerable” contradiction resulting from that definition of the physical continuum.

A section of Borel’s 1909 paper (reproduced in Borel 1922 under Note III) is entirely devoted to the discussion of this example, and part of its interest in the present context is that Borel establishes an explicit connection between this example and his 1907 paper on the Heap. Borel disagrees with Poincaré on the idea that the definition of the identity between physical magnitudes should lead to contradiction. He contends that intransitivities only result from imperfection in measurement. About physical measurement, Borel writes the following:

“If we measure a magnitude by comparing it to a fixed standard, we obtain so many decimals certainly correct followed by a last decimal about which there is some uncertainty; different experimenters, equally careful, obtain different values for this last decimal, so that the observations can be summed up in such a form as the following: 30 % of observers find a 5, while 50 % found a 6 and 20 % found a 7; another magnitude very near to the first leads to results which are analogous, but not in general identical. The phenomenon is analogous to that which takes place in the sophism of the heap of wheat.”Footnote 49

Borel’s analogy with the heap paradox is very allusive in this passage, but what the last sentence suggests is that for him small differences in physical magnitude will necessarily imply corresponding statistical differences regarding the judgments based on them. The remark may be connected to the comment Borel makes in the last section of his 1907 paper regarding individual decisions to buy in relation to slight differences in price. Borel presents as a “psychological illusion” the idea that a slight difference in price makes no difference regarding the propensity to buy.

The rest of Borel’s 1909 note adds some indications about the way to evade intransitivities in the measurement of physical magnitudes. Borel in particular makes the following addition in the same text and in a footnote:

“the physical continuum differs from the mathematical continuum in that because experience only allows us to achieve a limited approximation, a certain minimum difference is necessary for us to be able to distinguish two very near elements. [Fn. 2. In making this observation, we should not be thought to fall prey to Mr. Poincaré’s objection; the minimum difference is not an absolute constant, but depends on the experimental conditions and often also on the value or nature of the measured quantities. Consequently, two magnitudes A and B that direct measurement procedures would not allow us to distinguish can be differentiated if one has the good idea—or good luck—to compare them both to a suitably chosen magnitude C, that direct experience does not distinguish from B, but distinguishes from A; those new experiments diminish the minimum separabile for A and B. Another procedure for the diminution of this minimum is the repetition of experiments and the application of the probability calculus; this is a very interesting issue to which I will very likely have the occasion to return some day.]”Footnote 50

In talking about the “miminum separabile” between two magnitudes, Borel points to the central question investigated by Fechner in his Elements of Psychophysics, namely the problem of measuring the difference threshold or limen required between two magnitudes to produce a difference in sensation. Fechner himself, in his Elements, first made use of the probability calculus and of Gauss’s law of errors to give a practical and theoretical answer to his problem.

From Borel’s 1909 note, it is not possible to determine whether Borel had a clear notion of Fechner’s contribution to psychophysics. However, a paper published in 1908 entitled ‘Le calcul des probabilités et la méthode des majorités’ in the psychology journal L’Année psychologique shows that Borel was well acquainted with Fechner’s work, at least through the manual of experimental psychology of Titchener (1905), which Borel discusses to some length in that paper. The content of Borel’s (1908b) paper was later included as chapter 9 of Le Hasard (\(\S\)9 to 107, with the addition of the introduction and conclusion paragraphs 98 and 108), entitled “La valeur scientifique des lois du hasard”, where Borel discusses the applications of the probability calculus to several issues in psychology.

Borel’s paper on the majority method is related to his paper on the heap paradox in two ways. First of all, Borel questions the value of opinion polls to establish facts. One example he discusses, with cross-reference to his paper on the heap, concerns the value of opinion surveys for the establishment of facts about language use. Secondly, a large section of that paper, corresponding to paragraph \(\S\) 105 in Borel (1914), is a discussion of the method Fechner defined as the method of right and wrong cases (also called the method of constant stimuli), specifically designed by Fechner to propose a probabilistic model for the measurement of the threshold. Footnote 51

Fechner’s method of constant stimuli is introduced and illustrated at paragraph 22 of Titchener’s manual on the example of lifted weights, and Borel uses Titchener’s figures (1905: 107) to make his point. Footnote 52 Borel’s use of Titchener’s data is particularly striking, for it shows that Borel thought that an adaptation of the method of constant stimuli was indeed a correct way to get a reliable estimate of the “minimum separabile” between two magnitudes. In Titchener’s experiment, the same subject is shown pairs of weights taken from a fixed set, each time with the same standard weight S present in the pair, and asked for each trial to issue a comparative judgment on the pair (which weight is felt heavier, or whether the two weights are felt equal).Footnote 53 The experiment thus provides, for each pair, the frequency of judgments of each kind (C > SC = SC < S). Based on those frequencies, the difference limen, or increase of weight d such that S + d is felt as minimally distinct from S is determined by statistical methods.

The proposal made by Borel is to transpose this method to an analysis of collective judgments. That is, instead of considering 100 trials performed by the same observer, Borel proposes to consider each trial as made by a different observer. First, Borel explains how probabilities can help to determine, relative to the series of measurement, when the two weights C and S are equal, and then the first lower value in the series for which C is reliably felt as lighter than S. Basically, in that passage Borel illustrates how the binomial distribution can help to infer whether the difference between two means is statistically significant. For instance, in Titchener’s table, a weight C of 1,071 g is felt lighter than the standard S of 1,071 g 33 % of the time, and heavier 37 % of the time, while a weight C of 1,021 g is felt lighter 53 % of the time, and heavier 10 % only. This points to a difference threshold of about 40 g, a result confirmed independently by Titchener’s calculations. Borel’s main point, however, is to argue that, relative to a set of actual data like Titchener’s, one can linearly interpolate data points between 1,021 and 1,071, so as to refine the estimate of the threshold by the same statistical method. He finds that a weight of 1,051 g would give 41 % of ‘lighter’ judgments versus 22 ‘heavier’, still a reliable correct difference, thereby reducing the differential threshold to 20 g.Footnote 54 The general conclusion Borel draws is that the majority method, applied to collective judgments, allows one to refine what he in 1909 calls the “minimum separabile” between two physical magnitudes.Footnote 55

Borel’s 1908 remarks on psychophysical measurement only cast light on the second of Borel’s proposed responses to Poincaré, however, namely on the value of repeated measurements. They say nothing about his second suggestion, that is about the introduction of a third element to improve on the direct comparison between two elements. In paragraph 49 of Borel (1950), however, Borel eventually picked up the problem where he had left it.Footnote 56 Forty years after his response to Poincaré, Borel wrote about his own paper: “I made, at that time, some objections to that definition [of the physical continuum by Poincaré] to which Poincaré did not respond, which allows one to think that he accepted them”.Footnote 57 Borel, however, is much more explicit about his solution than he was in 1908. His proposal is that, from the assumptions A ∼ BB ∼ C, and A > C one can actually logically infer that A > B > C. His brief argument is that if it were the case that B ≤ C on the continuum, then it would follow that (A − B) ≥ (A − C), contradicting the assumption that A ∼ B, and similarly if it were true that B ≥ A, then it would follow that (B − C) ≥ (A − C), contradicting B ∼ C.

Borel’s solution to the problem is particularly striking in that it antedates an abstract method for undoing intransitivities first sketched by Goodman (1951)—more briefly Russell (1940)—and only made fully precise in 1956 by Luce in the context of his theory of semi-orders. Specifically, Goodman assumes that two sensory qualities or qualia can be directly indiscriminable (can ‘match’, in Goodman’s terminology), but he accepts, like Poincaré, that direct indiscriminability (matching) is a nontransitive relation. However, Goodman considers that for two qualia to be identical, it is not sufficient for them to be directly indiscriminable: they must also be indiscriminable from exactly the same qualia. Consequently, two qualia a and b can be directly indiscriminable, but will be indirectly discriminable if there is a quale c such that c is directly indiscriminable from the one but directly discriminable from the other.

Goodman’s method only focuses on identity, however, and not so much on ordering proper. In that respect, the closest elaboration of Borel’s thoughts is definitely Luce’s theory of semi-orders, which Luce presents as a way of obtaining “a non-statistical analogue of the ‘just noticeable difference’ concept of psychophysics”. What Luce (1956) spells out is indeed a general method for extracting a weak order preference relation (inducing a transitive indifference relation) from a semi-order preference relation (in which the associated relation of indifference is typically nontransitive). Luce’s method relies on essentially the same reasoning as the one detailed independently by Borel (1950) on Poincaré’s intransitivity problem, but using a more algebraic perspective. The significance of Luce’s result is explained in greater detail in Luce (1959), where Luce discusses the correspondence between the probabilistic treatment of the notion of Just Noticeable Difference and the way in which a ratio scale can be obtained from a set of imperfect discriminations.Footnote 58

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Égré, P., Barberousse, A. Borel on the Heap. Erkenn 79 (Suppl 5), 1043–1079 (2014). https://doi.org/10.1007/s10670-013-9596-3

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