From Aristotle’s Square of Opposition to the “Tri-unity’s Concordance”: Cusanus’ Non-classical Reasoning

Chapter
Part of the Studies in Universal Logic book series (SUL)

Abstract

It is well-known that Cusanus (1401–1464) introduced the surprising notion of the coincidence of opposites, which in fact shrinks Aristotle’s square of opposition into a segment. Almost a century ago Cassirer suggested that Cusanus had looked for a new kind of logic. Indeed, an accurate inspection of Cusanus’ texts shows that in order to discover new names of God by means of coincidences of opposites, Cusanus invented several names belonging to different kinds of non-classical logic—i.e. positive, paraconsistent, modal and intuitionist—, which were formalised in the last century. When, in his more important book, he invented an intuitionist name he implicitly reasoned about it according to the intuitionist square of opposition so precisely that he was able to organize his theories in a new way; it was based not on axioms- principles, but on the search for a new method for solving a general problem. Moreover, he wanted to refer his reasoning not to the square of opposition but to a new logical scheme, a “tri-unity of concordance”, for which he suggested an original definition. Here this tri-unity is represented by means of a geometrical figure.

Keywords

Cusanus Square of opposition Non-classical logics 

Mathematical Subject Classification

Primary 03A05 Secondary 03B53 03B22 03B45 03B20 

Notes

Acknowledgement

I am grateful to Prof. David Braithwaite for having revised my poor English and to an anonymous referee for an important suggestion.

Bibliography

  1. 1.
    D. Albertson, Mathematical Theologies. Nicholas of Cusa and the Legacy of Thierry of Chartres (Oxford U.P, Oxford, 2014)CrossRefGoogle Scholar
  2. 2.
    Aristotle, in Categories Google Scholar
  3. 3.
    Aristotle, in Metaphysics Γ□ Google Scholar
  4. 4.
    Aristotle, in De Interpretation and Topics. Second Analytics Google Scholar
  5. 5.
    V. Bazhanov, Non-classical stems from classical: N. A. Vasiliev’s approach to logic and his reassessment of the square of opposition. Log. Univers. 2(1), 71–76 (2008)CrossRefGoogle Scholar
  6. 6.
    E.W. Beth, Foundations of Matheatics (North-Holland, Amsterdam, 1959)Google Scholar
  7. 7.
    P. Blackburn, M. de Rijke, Y. Venema, Modal Logic (Cambridge U.P., Cambridge, 2001)CrossRefGoogle Scholar
  8. 8.
    J.M. Bochenski, Formale Logik (Alber, Muenchen, 1956)Google Scholar
  9. 9.
    A. Bonetti, La ricerca metafisica nel pensiero di Niccolò Cusano (Brescia, Paideia, 1993)Google Scholar
  10. 10.
    M. Capozzi, G. Roncaglia, Logic and Philosophy from Humanism to Kant, in The Development of Modern Logic, ed. by L. Haaparanta (Oxford U.P, Oxford, 2009), pp. 78–158CrossRefGoogle Scholar
  11. 11.
    S. Caramella, Il problema di una logica trascendente nell’ultima fase del pensiero di Nicola Cusano, in Nicola Cusano agli inizi del mondo moderno, ed. by G. Santinello (Firenze, Sansoni, 1970), pp. 365–373, p. 370Google Scholar
  12. 12.
    L. Carnot, Essai sur les machines en général (Dijon, Defay, 1783)Google Scholar
  13. 13.
    E. Cassirer, in Das Erkenntinis Problem in der Philosophie und Wissenschaft in Nueerer Zeit, vol. I, ch. I (Bruno Cassirer, Berlin, 1911)Google Scholar
  14. 14.
    E. Cassirer, in Individuum und Kosmos in der Philosophie der Renaissance, ch. I (1927)Google Scholar
  15. 15.
    B.F. Chellas, Modal Logic (Cambridge U.P, Cambridge, 1980)CrossRefGoogle Scholar
  16. 16.
    F.E. Cranz, The De Aequalitate and the De Principio of Nichola of Cusa, in Nicholas of Cusa and Christ and the Church, ed. by G. Christianson, T.M.I. Izbicki (Brill, New York, 1996), pp. 271–280Google Scholar
  17. 17.
    C.M. Cubillos Munoz, Los múltiples nombres del Dios innombrable (Eunsa U. de Navarra, Baranain, 2013)Google Scholar
  18. 18.
    N. Cusano, in Scritti Filosofici, ed. by G. Federici Vescovini (UTET, Torino, 1972)Google Scholar
  19. 19.
    N. Cusanus, in Sermo XX (h xvi/3 n. 5) (1439-40)Google Scholar
  20. 20.
    N. Cusanus, in De Docta Ignorantia (1440)Google Scholar
  21. 21.
    N. Cusanus, in De Coniecturis (1442)Google Scholar
  22. 22.
    N. Cusanus, in De Deo Abscondito (1445a)Google Scholar
  23. 23.
    N. Cusanus, in De Quaerendo Deum (1445b)Google Scholar
  24. 24.
    N. Cusanus, in Apologia Doctae Ignorantiae (1449)Google Scholar
  25. 25.
    N. Cusanus, in De Mente (1450)Google Scholar
  26. 26.
    N. Cusanus, in De Visione Dei (1453)Google Scholar
  27. 27.
    N. Cusunus, in De Aequalitate (1459a)Google Scholar
  28. 28.
    N. Cusunus, in De Principio (1459b)Google Scholar
  29. 29.
    N. Cusanus, in Kribatio Alkorani (1460–1461)Google Scholar
  30. 30.
    N. Cusanus, in De Possest (1460)Google Scholar
  31. 31.
    N. Cusanus, in De Li Non Aliud (De Aequalitate) (1462)Google Scholar
  32. 32.
    N. Cusanus, in De Venatione Sapientiae (1463)Google Scholar
  33. 33.
    N. Cusanus, in De Apice Theoriae (1464)Google Scholar
  34. 34.
    J. D’Alembert, Élémens des sciences, in Éncyclopédie Française, ed. by Jean Le Ronde d'Alembert, D. Diderot (Briasson, Paris, 1751-1772, t. 5), pp. 491–497Google Scholar
  35. 35.
    A. Drago, The beginnings of a pluralist history of mathematics: L. Carnot and Lobachevsky, In Mem. N. I. Lobachevskii, 3, pt. 2, 134–144 (1995)Google Scholar
  36. 36.
    A. Drago, Il ruolo del principio di ragione sufficiente nella scienza secondo Federico Enriques, in Federico Enriques Filosofia e storia del pensiero scientifico, ed. by O. Pompeo Faracovi, F. Speranza (Belforte Ed., Livorno, 1998), pp. 223–265Google Scholar
  37. 37.
    A. Drago, Vasiliev’s paraconsistent logic interpreted by means of the dual role played by the double negation law. J. Appl. Non-Classical Log. 11, 281–294 (2001)CrossRefGoogle Scholar
  38. 38.
    A. Drago, Nicholas of Cusa’s logical way of reasoning interpreted and re-constructed according to modern logic. Metalogicon 22, 51–86 (2009)Google Scholar
  39. 39.
    A. Drago, Dialectics in Cusanus (1401-1464), Lanza del Vasto (1901-1981) and beyond. Epistemologia 33, 305–328 (2010)Google Scholar
  40. 40.
    A. Drago, Pluralism in logic: the square of opposition, Leibniz’ principle of sufficient reason and Markov’s principle, in Around and Beyond the Square of Opposition, ed. by J.-Y. Béziau, D. Jacquette (Birkhaueser, Basel, 2012), pp. 175–189CrossRefGoogle Scholar
  41. 41.
    A. Drago, Il ruolo centrale di Nicola Cusano nella nascita della scienza moderna, in Intorno a Galileo: La storia della fisica e il punto di svolta Galileiano, ed. by M. Toscano, G. Giannini, E. Giannetto (Guaraldi, Rimini, 2012), pp. 17–25Google Scholar
  42. 42.
    P. Duhem, Le Système du Monde (Hermann, Paris, 1913), pp. 272–286Google Scholar
  43. 43.
    M. Dummett, Elements of Intuitionism (Oxford U. P, Oxford, 1977)Google Scholar
  44. 44.
    F. Enriques, Il principio di ragion sufficiente nella costruzione scientifica. Scientia 5, 1–20 (1909)Google Scholar
  45. 45.
    K. Flasch, in Nikolaus von Kues. Geschichte einer Entwicklung. Vorlesungen fuer Einfuehrung in seine Philosophie (Frankfurt am Mein, Klosterman, 20012) (tr. it.: Torino, Aragno, 2010)Google Scholar
  46. 46.
    L. Gabriel, Il pensiero dialettico in Cusano e in Hegel. Filosofia 21, 537–547 (1970)Google Scholar
  47. 47.
    M. de Gandillac, Explicatio-complicatio chez Nicolas de Cues (Padova, Editrice Antenore, 1993)Google Scholar
  48. 48.
    J.-L. Gardiès, Le raisonnement par l’absurde (PUF, Paris, 1991)Google Scholar
  49. 49.
    J. Garson, Modal logic, in Stanford Encyclopedia of Philosophy, ed. by M. Zalta (2014, Online)Google Scholar
  50. 50.
    J.B. Grize, Logique, in Logique et la connaissance scientifique, dans Encyclopédie de la Pléyade, ed. by J. Piaget (Gallimard, Paris, 1970), 135–288, pp. 206–210Google Scholar
  51. 51.
    S. Haack, Deviant Logic (Cambridge U.P, Cambridge, 1974)Google Scholar
  52. 52.
    J. Hopkins, in Introduction à Nicholas of Cusa on God as Not-Other. A Translation and Appraisal of De Li Non Aliud (Banning, Minneapolis, 19873)Google Scholar
  53. 53.
    L.R. Horn, The logic of logical double negation, in Proceedings of the Sophia Symposium on Negation, Tokyo, U. of Sophia, 2001, pp. 79–112Google Scholar
  54. 54.
    L.R. Horn, Multiple negations in English and other languages, in The Expression of Negation (de Gruyter, Mouton, 2010), pp. 111–148Google Scholar
  55. 55.
    Lanza del Vasto, La Montée des Ames Vivantes (Denoel, Paris, 1968). p. 78Google Scholar
  56. 56.
    N.I. Lobachevsky, The Theory of Parallels, An Appendix to R. Bonola, Non-Eclidean Geometry (Dover, New York, 1955)Google Scholar
  57. 57.
    A.A. Markov, On constructive mathematics. Trudy Math. Inst. Steklov 67, 8–14 (1962); Engl. tr. Am. Math. Soc. Translations 98(2), 1–9 (1971)Google Scholar
  58. 58.
    C.L. Miller, in Reading Cusanus. Metaphor and Dialectics in a Conceptual Universe (Cath. Univ. America P., Washington, DC, 2003)Google Scholar
  59. 59.
    D. Monaco, Deus trinitas. Dio come non altro nel pensiero di Nicola Cusano (Città Nuova, Roma, 2010)Google Scholar
  60. 60.
    J.-M. Nicolle, Les écrits mathématiques. Nicolas de Cues (Champion, Paris, 2007)Google Scholar
  61. 61.
    A.D. Prawitz, Natural Deduction. A Proof Theoretical Study (Almquist and Wiksel, Stokholm, 1965)Google Scholar
  62. 62.
    A.D. Prawitz, P.-E. Malmnaes, A survey of some connections between classical, intuitionistic and minimal logic, in Contributions to Mathematical Logic, ed. by A. Schmidt, K. Schuette, H.J. Thiele (North-Holland, Amsterdam, 1968), pp. 215–229Google Scholar
  63. 63.
    G. Priest, K. Tanaka, Z. Weber, Paraconsistent logic, in Stanford Encyclopedia of Philosophy, ed. by N.E. Zalta (2013)~http://plato.stanford.edu/entries/logic-paraconsistent/
  64. 64.
    B. Russell, The Principles of Mathematics (Cambridge U.P, Cambridge, 1903)Google Scholar
  65. 65.
    G. Santinello, Per una interpretazione del pensiero del Cusano in generale, in N. Cusano: Scritti Filosfici (Zanichelli, Bologna, 1965), pp. 3–33Google Scholar
  66. 66.
    G. Santinello, Introduzione a Niccolò Cuano (Laterza, Bari, 1987)Google Scholar
  67. 67.
    H. Schwaetzer, ”Non Autre” comme la Trinité, in La Trinité chez Eckhart et Nicolas de Cues, ed. by M.A. Vannier (Cerf, Paris, 2009), pp. 145–153Google Scholar
  68. 68.
    A. Troelstra, D. Van Dalen, Constructivism in Mathematics (North-Holland, Amsterdam, 1988)Google Scholar
  69. 69.
    M. Ursic, Paraconsistency as Coincidentia Oppositorum. Paraconsistency and dialectics as coincidentia oppositorum in the philosophy of Nicholas of Cusa (1998), http://tonymarmo.tripod.com/linguistix/index.blog?start=1087301823
  70. 70.
    E. Vanstenbeerghe, Le Cardinal Nicolas de Cues (Frankfurt, Minerva, 1920), p. 301Google Scholar
  71. 71.
    J. van Heijenoort, Goedel’s theorem, in Macmillan Encyclopaedia of Philosophy (Macmillan, London, 1967)Google Scholar
  72. 72.
    N.A. Vasiliev, Logic and metalogic, in Logos, 1912-13, bd. 1-2, 53-b1 (Engl. Trans. by R. Poli, in Axiomathes, n. 3 dec.) (1993), pp. 329–351Google Scholar
  73. 73.
    G. Wenk, in De Ignota Literatura (1442-1443)Google Scholar
  74. 74.
    E.A. Wyller, Idantitaet und Kontradiktion. Ein Weg zu Cusanus’ Unendlichkeitsidee, MFCG 15, 104–120, p. 120 (1982)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.University of PisaPisaItaly

Personalised recommendations