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Classic Works of the Dempster-Shafer Theory of Belief Functions pp 117–182Cite as

Non-Additive Probabilities in the Work of Bernoulli and Lambert

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Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 219))

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References

  1. Anonymous (1699), A calculation of the credibility of human testimony. Phil. Trans. Roy. Soc. London, 21:359–365. The author was George Hooper; see S5.2 above.

    Google Scholar 

  2. Antoine Arnauld and Pierre Nicole (1662), la Logique ou l’art de penser. Paris.

    Google Scholar 

  3. Thomas Bayes (1764), An essay toward solving a problem in the doctrine of chances. Phil. Trans. Roy. Soc, London for 1763, 53:370–418. Reprinted in Biometrika, 45, 293–315 (1958), and in [63].

    Google Scholar 

  4. Rudolph Beran (1970), Upper and lower risks and minimax procedures. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1, 1–16.

    Google Scholar 

  5. Daniel Bernoulli (1778), Dijudicatio maxime probabilis plurium observationum discrepantium atque verisimillima inductio inde formanda. Acta Acad. Petrop. for 1777, pars prior: 3–23. An English translation by G. D. Allen was published in Biometrika, 48:1–18 (1961), and reprinted in Pearson and Kendall [63].

    Google Scholar 

  6. Jakob Bernoulli (1713), Ars Conjectandi. Basel. Reprinted in 1968 by Culture et Civilisation, Brussels, and in 1975 in [7], Volume 3. An English translation of Part IV, by Bing Sung, was issued as Research Report No. 2 of the Department of Statistics, Harvard University, Feb. 12, 1966. An English translation of the entire book, by Edith D. Sylla, was published by Johns Hopkins University Press in 2006. Sylla lists other translations, in various languages, on pp.406–408.

    Google Scholar 

  7. Jacob Bernoulli (1969–1999), Die Werke von Jakob Bernoulli. Basel. Five volumes have appeared: Volume 1 in 1969, Volume 2 in 1989, Volume 3 in 1975, Volume 4 in 1993, and Volume 5 in 1999. Volume 3, which is cited repeatedly in this article, was edited by B. L. van der Waerden.

    Google Scholar 

  8. Nicolaus Bernoulli (1709), Dissertationis de Usu Artis Conjectandi in Jure. Basel. Excerpts were published on pp. 159–170 of Volume VI of Acta Eruditorum, Supplementa, 1711. The entire dissertation was reprinted in [7], Volume 3, 287–326, with commentary by Karl Kohli on pp. 541–556.

    Google Scholar 

  9. Kurt-Reinhard Biermann and Margot Faak (1957), G. W. Leibniz’ “De incerti aestimatione”. Forschungen und Fortschritte, 31:45–50.

    Google Scholar 

  10. George Boole (1854), An Investigation into the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities. Macmillan, London. Reprinted by Dover, 1958.

    Google Scholar 

  11. J. van Brakel (1976), Some remarks on the prehistory of the concept of statistical probability. Archive for History of Exact Sciences, 16:119–136.

    Article  MATH  MathSciNet  Google Scholar 

  12. Edmund F. Byrne (1968), Probability and opinion: A study in the medieval presuppositions of post-medieval theories of probability. Martinus Nijhoff, The Hague.

    Google Scholar 

  13. Marie Jean Condorcet (1785), Essai sur l’application de l’analyse ‘a la probabilit’e des d’ecisions rendues ‘a la pluralit’e des voix. Paris. Reprinted by Chelsea in 1973.

    Google Scholar 

  14. Antoine-Augustin Cournot (1843), Exposition de la th’eorie des chances et des probabilit’es. Paris. Reprinted as Volume I of Cournot’s Oeuvres compl‘etes in 1984 by Vrin, Paris, with introduction, notes, and index by Bernard Bru.

    Google Scholar 

  15. Louis Couturat (1901), La Logique de Leibniz. Paris. Reprinted by Georg Olms, Hildesheim, 1961.

    MATH  Google Scholar 

  16. John Craig (1699), Theologiae Christianae Principia Mathematica. London. The part dealing with probability was reprinted by Mouton as Craig’s Rules of Historical Evidence, Beiheft 4 of History and Theory: Studies in the Philosophy of History, 1964.

    Google Scholar 

  17. F. N. David (1962), Games, Gods and Gambling. Griffin, London.

    MATH  Google Scholar 

  18. Abraham De Moivre (1711), De mensura sortis, seu, de probabilitate eventum in ludis a casu fortuito pendentibus. Phil. Trans. Roy. Soc. London, 27: 213–264.

    Google Scholar 

  19. Abraham De Moivre (1718), The Doctrine of Chances, or a Method of Calculating the Probability of Events in Play. London.

    Google Scholar 

  20. A. P. Dempster (1966), New methods for reasoning towards posterior distributions based on sample data. Annals of Mathematical Statistics, 37:355–374.

    Google Scholar 

  21. A. P. Dempster (1967), Upper and lower probabilities induced by a multivalued mapping. Annals of Mathematical Statistics, 38:325–339.

    Article  MathSciNet  Google Scholar 

  22. A. P. Dempster (1967), Upper and lower probability inferences based on a sample from a finite univariate population. Biometrika, 54:515–528.

    MathSciNet  Google Scholar 

  23. A. P. Dempster (1968), Upper and lower probabilities generated by random closed interval. Annals of Mathematical Statistics, 39:957–966.

    Article  MathSciNet  Google Scholar 

  24. A. P. Dempster (1968), A generalization of Bayesian inference (with discussion). Journal of the Royal Statistical Society Series B, 30:205–247.

    MathSciNet  Google Scholar 

  25. A. P. Dempster (1968), Probability (Chapter 2 of The Theory of Statistical Inference: A Critical Analysis). Department of Statistics, Harvard University, Research Report S-3, September 27.

    Google Scholar 

  26. A. P. Dempster (1969), Upper and lower probability inferences for families of hypotheses with monotone density ratios. Annals of Mathematical Statistics, 40:953–969.

    Article  MathSciNet  Google Scholar 

  27. A. P. Dempster (1974), Discussion of a paper by A. W. F. Edwards. In Blaesild, Barndorff-Nielsen, and Schou, editors, Memoirs No. 1 of the Dept. of Theoretical Statistics, 57–58. Aarhus.

    Google Scholar 

  28. Denis Diderot et al. (1765), Encyclop’edie, ou Dictionnaire raisonn’e des sciences, des arts et des m’etiers. Paris.

    Google Scholar 

  29. M. E. Eisenring (1942), Johann Heinrich Lambert und die wissenschaftliche Philosophie der Gegenwart. Zurich.

    Google Scholar 

  30. Pierre de Fermat (1891–1922), Oeuvres. Paris. Edited by P. Tannery and C. Henry, 4 volumes.

    Google Scholar 

  31. Terry Fine (1973), Theories of Probability. Academic Press.

    Google Scholar 

  32. Bernard Fontenelle (1706), Eloge de Jacques Bernoulli. Histoire de l’Acad’emie royale des sciences for 1705:148–149.

    Google Scholar 

  33. Robin Giles (1976), A logic for subjective belief. In citech05:harper/hooker:1976, Volume I, 41–72.

    Google Scholar 

  34. I.J. Good (1962), The measure of a non-measurable set. In Ernest Nagal, Patrick Suppes, and Alfred Tarski, editors, Logic, Methodology and Philosophy of Science, 319–329. Stanford University Press, Stanford.

    Google Scholar 

  35. Thomas Granger. Divine Logike. London, 1620.

    Google Scholar 

  36. Ian Hacking (1975), The Emergence of Probability. Cambridge University Press, New York.

    MATH  Google Scholar 

  37. Theodore Hailperin (1976), Boole’s Logic and Probability. North-Holland, Amsterdam.

    MATH  Google Scholar 

  38. William L. Harper and C. A. Hooker, editors (1976), Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science. Reidel, Dordrecht, 3 volumes.

    Google Scholar 

  39. Julian Henny (1975), Niklaus und Johann Bernoullis Forschungen auf dem Gebieter ahrscheinlichkeitsrechnung in ihrem Briefwechsel mit Pierre R’emond de Montmort. In citech05:bernoulli1969, Volume 3, 457–507.

    Google Scholar 

  40. Joseph E. Hofmann (1974), Leibniz in Paris 1672–1676; His Growth to Mathematical Maturity. Cambridge University Press.

    Google Scholar 

  41. Peter Huber (1973), The use of Choquet capacities in statistics. Bulletin of the International Statistical Institute, 45, book 4:181–188.

    Google Scholar 

  42. Christiaan Huygens (1657), De ratiociniis in aleae ludo. In F. Van Schooten, editor, Exercitionum Mathematicorum. Amsterdam. Incorporated into Part I of Bernoulli’s Ars Conjectandi. The original Dutch version, published in 1660, is in Volume XIV of [43].

    Google Scholar 

  43. Christiaan Huygens, (1888–1950), Oeuvres compl‘etes. The Hague, 22 volumes.

    Google Scholar 

  44. M. G. Kendall (1956), The beginnings of a probability calculus. Biometrika, 43:1–14, 1956. Reprinted in [63].

    MATH  MathSciNet  Google Scholar 

  45. Karl Kohli and B. L. van der Waerden (1975), Bewertung von Leibrenten. In [7], Volume 3, 515–539.

    Google Scholar 

  46. Karl Kohli (1975), Zur Publikationsgeschichte der Ars Conjectandi. In [7], Volume 3, 391–401.

    Google Scholar 

  47. Karl Kohli (1975), Spieldauer: Von Jakob Bernoullis L"osung der f"unften Aufgabe von Huygens bis zu den Arbeiten von de Moivre. In [7], Volume 3, pp. 403–455.

    Google Scholar 

  48. Karl Kohli (1975), Aus dem Briefwechsel zwischen Leibniz und Jakob Bernoulli. In [7], Volume 3, pp.509–513.

    Google Scholar 

  49. Bernard O. Koopman (1940), The bases of probability. Bulletin of the American Mathematical Society, 46:763–774.

    Google Scholar 

  50. Henry E. Kyburg (1961), Probability and the logic of rational belief. Wesleyan University Press, Middletown.

    Google Scholar 

  51. Johann Heinrich Lambert (1764), Neues Organon, oder Gedanken uber die Erforschung und Bezeichnung des Wahren und dessen Unterscheidung Von Irrtum und Schein. Leipzig. Reprinted in 1965 by Olms of Hildesheim as the first two volumes of Lambert’s textitPhilosophische Schriften.

    Google Scholar 

  52. Pierre Simon Laplace (1774), M’emoire sur la probabilit’e des causes par les ’ev’enements. M’emoires de l’Acad’emie royale des Sciences de Paris (Savants ’etrangers), 6:621–656. Reprinted in [54], Volume 8, pp. 27–65.

    Google Scholar 

  53. Pierre Simon Laplace (1814), Th‘eorie analytique des probabilit’es. Paris, 2nd edition.

    Google Scholar 

  54. Pierre Simon Laplace (1878–1912), Oeuvres compl‘etes. Paris, 14 volumes.

    Google Scholar 

  55. Gottfried Wilhelm Leibniz (1669), De conditionibus. Leipzig. Reprinted in [56], Volume IV, part III, p. 148ff.

    Google Scholar 

  56. Gottfried Wilhelm Leibniz (1768), Opera Omnia. Geneva. Edited by L. Dutens, 6 volumes.

    Google Scholar 

  57. Gottfried Wilhelm Leibniz (1849–1863), Mathematische Schriften. Halle. Edited by C. I. Gerhardt, 7 volumes.

    Google Scholar 

  58. Genevi‘eve Lewis (1952), Lettres de Leibniz ‘a Arnauld. Paris.

    Google Scholar 

  59. John Locke (1690), An essay concerning human understanding. London.

    Google Scholar 

  60. John Locke (1936), An early draft of Locke’s essay. Oxford. Edited by R. I. Aaron and J. Gibb.

    Google Scholar 

  61. Pierre R’emond de Montmort (1708), Essay d’analyse sur les jeux de hazard. Paris.

    Google Scholar 

  62. Oystein Ore (1953), Cardano, the gambling scholar. Princeton University Press. Includes Gould’s translation of Cardano’s de Ludo Aleae.

    Google Scholar 

  63. E. S. Pearson and M. G. Kendall, editors (1970), Studies in the History of Statistics and Probability. Hafner, Darien.

    MATH  Google Scholar 

  64. Sim’eon-Denis Poisson (1837), Recherches sur la probabilit’e des jugements en mati‘ere criminelle et en mati‘ere civile, pr’ec’ed’ees des r‘egles g’en’erales du calcul des probabilit’es. Paris.

    Google Scholar 

  65. Pierre Prevost and Simon Lhuilier (1800), M’emoire sur l‘application du Calcul des probabilit’es ‘a la valeur du t’emoignage. M’emoires de l’Acad’emie Royal des Sciences et Belles Lettres ‘a Berlin for 1797:120–152.

    Google Scholar 

  66. Emile Ravier (1937), Bibliographie des Oeuvres de Leibniz. Paris.

    Google Scholar 

  67. Alfred Renyi (1973), Letters on probability. Wayne State University Press.

    Google Scholar 

  68. Joseph Saurin (1706), Eloge de Jacques Bernoulli. Journal des Sçavans, 34, Part I:126–139.

    Google Scholar 

  69. Ivo Schneider (1968), Der Mathematiker Abraham de Moivre (1667–1754). Archive for History of Exact Sciences, 5:258–300.

    Google Scholar 

  70. Glenn Shafer (1976), A Mathematical Theory of Evidence. Princeton University Press, Princeton, NJ.

    MATH  Google Scholar 

  71. Glenn Shafer (1976), Review of Ian Hacking’s The Emergence of Probability. Journal of the American Statistical Association, 71:519–521.

    Article  Google Scholar 

  72. Glenn Shafer (1976), A theory of statistical evidence. In cite[38], Volume II, pp. 365–436.

    Google Scholar 

  73. O. B. Sheynin (1971), J. H. Lambert’s work on probability. Archive for History of Exact Sciences, 7:244–256.

    Google Scholar 

  74. C. A. B. Smith (1961), Consistency in statistical inference and decision (with discussion). Journal of the Royal Statistical Society Series B, 23:1–25.

    MATH  Google Scholar 

  75. C. A. B. Smith (1965), Personal probability and statistical analysis (with discussion). Journal of the Royal Statistical Society, Series A, 128:469–499.

    Article  Google Scholar 

  76. Stephen M. Stigler (1975), Napoleonic statistics: The work of Laplace. Biometrika, 62:503–517.

    MATH  MathSciNet  Google Scholar 

  77. Patrick Suppes (1974), The measurement of belief. Journal of Royal Statistical Society Series B, 36:160–175.

    MATH  Google Scholar 

  78. Isaac Todhunter (1865), A History of the Mathematical Theory of Probability from the time of Pascal to that of Laplace. Cambridge University Press. Reprinted by Chelsea 1949, 1965.

    Google Scholar 

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  1. Glenn Shafer
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  1. Machine Intelligence Institute, Iona College, 715 N. Avenue, New Rochelle, NY 10801-1890, USA

    Roland R. Yager (Dr.) (Dr.)

  2. Department of Management and Information Systems, University of Akron College of Business Administration, 351 Akron, OH 4325-4801, USA

    Liping Liu

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Shafer, G. (2008). Non-Additive Probabilities in the Work of Bernoulli and Lambert. In: Yager, R.R., Liu, L. (eds) Classic Works of the Dempster-Shafer Theory of Belief Functions. Studies in Fuzziness and Soft Computing, vol 219. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44792-4_6

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