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Leibniz and the Calculus Ratiocinator

Part of the Philosophy of Engineering and Technology book series (POET,volume 30)


This paper deals with the interconnections between mathematics, metaphysics, and logic in the work of Leibniz. On the one hand, it touches upon some practical aspects such as Leibniz’s construction of a Four-species calculating machine, a mechanical digital calculating machine, and even a cipher machine. On the other hand, it examines how far Leibniz’s metaphysical dreams concerning the “calculus ratiocinator” and its underlying “characteristica universalis” have in fact been realized by the great philosopher. In particular it will be shown that Leibniz not only developed an “intensional” algebra of concepts which is provably equivalent to Boole’s “extensional” algebra of sets, but that he also discovered some basic laws of quantifier logic which allowed him to define individual concepts as maximally-consistent concepts. Moreover, Leibniz had the ingenious idea of transforming the basic principles of arithmetical addition and subtraction into a theory of “real” addition and subtraction thus obtaining some important building blocks of elementary set-theory.


  • Leibniz
  • Calculating machine
  • Metaphysic
  • Logic
  • Set-theory

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  1. 1.; online access January 3, 2017. The concluding quote comes from Rogers (1963), p. 943.

  2. 2.

    Cf. Mackensen (1990), p. 56–57 (my translation).

  3. 3.

    In “Machina arithmetica in qua non additio tantum et subtractio sed et multiplicatio nullo, divisio vero paene nullo animi labore peragantur” Leibniz wrote: “Indignum est excellentium virorum horas servili calculandi perire quia Machina adhibita velissimo cuique secure transcribi possit.” The translation is taken from “Leibniz on his calculating machine” in Smith (1929), 173–181.

  4. 4.

    Cf. Mackensen (1990), p. 58 (my translation). A facsimile of Leibniz’s manuscript “De progressione dyadica” may be found in Stein and Heinekamp (1990), p. 109.

  5. 5.

    Cf. Mackensen (1990), p. 59 (my translation).

  6. 6.

    Cf. A I 2, p. 125; the translation is from Rescher (2012), p. 35–36.

  7. 7.

    Cf. A IV 4, p. 68; translation from Rescher (2012), p. 37.

  8. 8.

    This remark, which Louis Couturat chose as motto for his ground-breaking book (1901), was written by Leibniz on the margin of the “Dialogus” of August 1677; cf. GP 7, p. 191. As far as I know, Leibniz nowhere seriously discussed the problem of the proper creation of the world, i.e. the transition from the mere idea to its physical actualization.

  9. 9.

    Cf. Lenzen (1990), especially Chap. 6.

  10. 10.

    Cf. GP 7, p. 200; the translation has been adopted from

  11. 11.

    Cf. A VI, 4, p. 443: “Itaque profertur hic calculus quidam novus et mirificus, qui in omnibus nostris ratiocinationibus locum habet, et qui non minus accurate procedit, quam Arithmetica aut Algebra”.

  12. 12.

    Parkinson (1966), p.10.

  13. 13.

    Cf. the fragment “De Numeris Characteristicis ad Linguam universalem constituendam” in GP 7, p. 184–9. The translation has been adopted with some modifications from Ariew & Garber (1989), p. 6–8.

  14. 14.

    Cf. GP 7, p. 185 and p. 187.

  15. 15.

    Cf. GP 7, p. 189, and Ariew and Garber (1989), p. 9–10.

  16. 16.

    Cf. “Elementa Calculi” in Couturat (1903), p. 49–57; the translation has been adopted from Parkinson (1966), p. 17–24.

  17. 17.

    According to Leibniz’s condition, the valid mood Darii would become invalid. The assignment of numbers B = 3, C = 6, D = 2 satisfies the premise ‘All C are D’, because 6 can be divided by 2; furthermore ‘Some B are C’ becomes true because the number of the predicate, C = 6, is divisible by the number of the subject, B = 3. But the conclusion ‘Some B are D’ would result as false since neither B = 3 can be divided by D = 2, nor conversely D by B. Thus also Leibniz soon noticed that for the truth of a particular affirmative proposition “it is not necessary that the subject can be divided by the predicate or the predicate divided by the subject”; cf. C., p. 57.

  18. 18.

    Cf. “Regulae ex quibus de bonitate consequentiarum […] judicari potest, per numeros”, in C. p. 77–84; an English version may be found in Parkinson (1966), p. 25–32.

  19. 19.

    Cf. C., p. 25–28; condition (vi) was put forward only in another fragment. Cf. C., p. 245–7: “Ex hoc calculo omnes modi et figurae derivari possunt per solas regulas Numerorum. Si nosse volumus an aliqua figura procedat vi formae, videmus an contradictorium conclusionis sit compatibile cum praemissis, id est an numeri reperiri possint satisfacientes simul praemissis et contradictoriae conclusionis; quodsi nulli reperiri possunt, concludet argumentum vi formae.”

  20. 20.

    Cf. GP 7, p. 189, or Ariew and Garber (1989), p. 10.

  21. 21.

    Cf. Arnauld and Nicole (1683).

  22. 22.

    Cf. C., p. 80. In Arnauld & Nicole (1683) the principle of subalternation is put forward informally as follows: “Les propositions particulières sont enfermés dans les générales de même nature, et non les générales dans les particulières, I dans A, et O dans E, et non A dans I, ni E dans O”.

  23. 23.

    Cf. C., p. 410–411.

  24. 24.

    Cf. C., p. 411.

  25. 25.

    As will turn out below, this weak condition of existential import is tantamount to the assumption that concept B is self-consistent!

  26. 26.

    Cf. A VI, 4, p. 274: “Subjectum a in exemplo praecedenti, Omnis homo. Semper enim signum universale subjecto praefixum intelligatur”.

  27. 27.

    Cf. GP 7, p. 218 or the translation in Parkinson (1966), p. 33. For the sake of uniformity, Leibniz’s small letters ‘a’, ‘b’ have been replaced by capitals ‘A’, ‘B’.

  28. 28.

    Cf. C., p. 367 or the translation in Parkinson (1966), p. 57.

  29. 29.

    Cf. C., p. 53 or the translation in Parkinson (1966), p. 20–21. A similar distinction may also be found in Arnauld & Nicole (1683), p. 51–2.

  30. 30.

    Cf. GP 5, p. 469.

  31. 31.

    Cf. C., p. 235.

  32. 32.

    Cf. Quine (1953), p. 21.

  33. 33.

    Cf. A VI, 3, p. 506.

  34. 34.

    Cf. A VI, 4, p. 154.

  35. 35.

    Leibniz stated these laws especially in the “Generales Inquisitiones”. Cf. A VI, 4, p. 751 “Propositio per se vera est A coincidit ipsi A”; p. 750: “(6) Si A coincidit ipsi B, B coincidit ipsi A […] (8) Si A coincidit ipsi B, et B coincidit ipsi C, etiam A coincidit ipsi B”.

  36. 36.

    Cf. A VI, 4, p. 148: “AB est A pendet a significatione huiusmodi compositionis literarum. Hoc ipsum enim vult AB, nempe id quod est A, itemque B”.

  37. 37.

    Cf. GP 7, p. 221–2, or the translation in Parkinson (1966), p. 40.

  38. 38.

    Cf. Parkinson (1966), p. 40.

  39. 39.

    Cf. Parkinson (1966), p. 58, fn. 4.

  40. 40.

    Cf. C., p. 378, or the translation in Parkinson (1966), p. 67.

  41. 41.

    The first quotation is from April 1679, the second from around 1686; cf. A VI, 4, p. 248 and p. 804.

  42. 42.

    Cf. § 96 of the “General Inquiries”, e.g., A VI, 4, p. 767.

  43. 43.

    Cf. § 77 of the “General Inquiries”, e.g. A VI, 4, p 764, or the translation in Parkinson (1966), p. 67.

  44. 44.

    In the “General Inquiries”, the above principles had been formulated as follows: “A proposition false in itself is ‘A coincides with not-A’” (§ 11); “If A = B, then A ≠ not-B” (§ 171, Seventh); “It is false that B contains not-B, that is, B doesn’t contain not-B” (§ 43); and “A is B, therefore A isn’t not-B” (§ 91). Cf. A VI, 4, p. 751, p. 783, p. 755, and p. 766, or the translation in Parkinson (1966), p. 56, p. 83, p. 59, and p. 68.

  45. 45.

    Cf. Grua, p. 536.

  46. 46.

    Cf. A VI, 4, p. 766: “Non valet consequentia: Si A non est non-B, tunc A est B, seu Omne animal esse non hominem falsum est, quidem; sed tamen hinc non sequitur Omne animal esse hominem.”

  47. 47.

    Cf., e.g., § 21 of “Specimina calculi rationalis” in A VI, 4, p. 813: “A non est B idem est quod A est non B.”

  48. 48.

    Cf. A VI, 4, p. 218; the quoted example of Apostle Peter only appears in the critical apparatus of variants; Leibniz later replaced it by the less fortunate example ‘this piece of gold is a metal’ vs. ‘this piece of gold is a non-metal’.

  49. 49.

    Cf. A VI, 4, p. 218; critical apparatus, variant (d): “Imo hic patet me errasse, neque enim procedit regula.”

  50. 50.

    Cf. A VI, 4, p. 218; in order to avoid confusions, I have interchanged Leibniz’s symbolic letters ‘B’ and ‘A’.

  51. 51.

    Cf. A VI, 4, p. 749, fn 8: “A non-A contradictorium est. Possibile est quod non continet contradictorium seu A non-A. Possibile est quod non est Y, non-Y”.

  52. 52.

    Cf. GP 7, p. 211–217, or the translation in Parkinson (1966), p. 115–121.

  53. 53.

    Cf. § 55 of the “General Inquiries”, e.g. A VI, 4, p. 757, or the translation in Parkinson (1966), p. 60.

  54. 54.

    More exactly, this holds only for the implication ¬P(AB) → AB, while the converse AB → ¬P(AB) is easily proven: If AB, then (by Cont 3) A = AB, hence (by Iden 6) AB = ABB, and thus (AB)∈(BB), i.e. ¬P(AB). Cf. A VI, 4, p. 863: “Vera propositio categorica affirmativa universalis est: A est B, si A et AB coincidat et A sit possibile, et B sit possibile. Hinc sequitur, si A est B, vera propositio est, A non-B implicare contradictionem, nam pro A substituendo aequivalens AB fit AB non-B quod manifeste est contradictorium”.

  55. 55.

    Cf. Parkinson (1966), p. 81.

  56. 56.

    Consider the concept A(∼A(∼B)) which contains A(∼A). Since AA is contradictory, it follows by Poss 2 that A(∼A(∼B)) is also impossible; but from ¬P(A(∼A(∼B))) it immediately follows by Poss 3 that A(∼A)∈B!.

  57. 57.

    The inference from a contradictory pair of premises, α, ¬α to an arbitrary conclusion β was well known in Medieval logic, but Leibniz wasn’t convinced of its validity. In his excerpts from Caramuel’s Leptotatos (A VI 4, p. 1334–1343) he considered the “argumentatio curiose” by means of which, e.g., the conclusion ‘Circulus habet 4 angulos’ is derived from the premises ‘Petrus currit’ and ‘Petrus non currit’. Although the deduction is based on two impeccable formal principles, Leibniz annotated: “Videtur esse sophisma”.

  58. 58.

    Leibniz knew quite well that the corresponding propositional connective (α∨β) can similarly be defined as ¬(¬α ∧¬β). For a closer discussion cf. Lenzen (1983), p. 132–133.

  59. 59.

    Cf. A VI, 4, p. 751 or the translation in Parkinson (1966), p. 56

  60. 60.

    Cf. A VI, 4, p. 751, fn. 13, or Parkinson (1966), p. 56, fn. 1: “It is noteworthy that for ‘A = BY’ one can also say ‘A = AB’ so that there is no need to introduce a new letter”.

  61. 61.

    This proof was given by Leibniz himself in the important paper “Primaria Calculi Logic Fundamenta” of August 1690; cf. C., 235.

  62. 62.

    Cf. C., 259–261, or the text-critical edition in A VI, 4, p. 807–814.

  63. 63.

    Cf. C., p. 261.

  64. 64.

    Cf. C., p. 260.

  65. 65.

    Cf. A VI, 4, p. 753: “(32) Propositio Negativa. A non continet B, seu A esse (continere) B falsum est., seu A non coincidit BY.”

  66. 66.

    Cf. A VI, 4, p. 762 or the translation in Parkinson (1966), p. 65, § 72 including fn. 1; for a closer interpretation of Leibniz’s logical criteria for individual concepts cf. Lenzen (2004).

  67. 67.

    Cf. A VI, 4, p. 217–218.

  68. 68.

    Cf. A VI, 4, p. 218, lines 3–6, variant (d). The long story of Leibniz’s cardinal mistake of mixing up ‘A isn’t B’ and ‘A is not-B’ is analyzed in detail in Lenzen (1986).

  69. 69.

    Cf. the text-critical edition in A VI, 4, p. 830–845 and 845–855; English translations may be found in Parkinson (1966), p. 122–130, 131–144.

  70. 70.

    Cf., e.g., Parkinson (1966), p. 131–144 (Definition 3, Propositions 13, 14 and 17).

  71. 71.

    Cf. Parkinson (1966), p. 132 (Axiom 1).

  72. 72.

    Cf. Parkinson (1966), p. 132.

  73. 73.

    Cf. C., p. 267.

  74. 74.

    Cf. Parkinson (1966), p. 124 (Axiom 2).

  75. 75.

    Cf. C., p. 267, # 29: “Itaque si A + B = C, erit A = C-B […] Sed opus est A et B nihil habere commune”.

  76. 76.

    Cf. Parkinson (1966), p. 123, who misleadingly inserts the word ‘term’ before the entities M, A, B, while Leibniz himself spoke more neutrally of “aliquid M”!

  77. 77.

    Cf. Parkinson (1966), p. 130.

  78. 78.

    Cf. Parkinson (1966), p. 127, Case 2 of Theorem IX: “Let us assume meanwhile that E is everything which A and G have in common – if they have something in common, so that if they have nothing in common, E = Nothing”.

  79. 79.

    Cf. Parkinson (1966), p. 128, Theorem X.


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Lenzen, W. (2018). Leibniz and the Calculus Ratiocinator. In: Hansson, S. (eds) Technology and Mathematics. Philosophy of Engineering and Technology, vol 30. Springer, Cham.

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