Is Modern Logic Non-Aristotelian?

Part of the Synthese Library book series (SYLI, volume 387)


In this paper we examine up to which point Modern logic can be qualified as non-Aristotelian. After clarifying the difference between logic as reasoning and logic as a theory of reasoning, we compare syllogistic with propositional and first-order logic. We touch the question of formal validity, variable and mathematization and we point out that Gentzen’s cut-elimination theorem can be seen as the rejection of the central mechanism of syllogistic – the cut-rule having been first conceived as a modus Barbara by Hertz. We then examine the non-Aristotelian aspect of some non-classical logics, in particular paraconsistent logic. We argue that a paraconsistent negation can be seen as neo-Aristotelian since it corresponds to the notion of subcontrary in Boethius’ square of opposition. We end by examining if the comparison promoted by Vasiliev between non-Aristotelian logic and non-Euclidian geometry makes sense.



I would like to thanks my colleagues of Moscow State University and Russian Academy of Science in Moscow, in particular Dmitry Zaitsev, Vladimir Markin and Vladimir Vasyukov as well as Valentin Bazhanov, Ioannis Vandoulakis, Jean Paul van Bendegem, Jean-Lous Hudry and José Veríssimo for help in providing me useful information.

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  1. Ables, F., & Fuller, M. E. (Eds.). (2016). Modern logic 1850–1950, east and west. Basel: Birkhäuser.Google Scholar
  2. Alvarez-Fontecilla, E. (2016). Canonical syllogistic moods in traditional Aristotelian logic. Logica Universalis, 10(2016), 517–531.CrossRefGoogle Scholar
  3. Alvarez-Fontecilla, E., & Correia, M. (2016). “Conversion and opposition: Traditional and theoretical formulations” in. Beziau&Gerogiorgakis, 2016.Google Scholar
  4. Anderson, A. R., & Belnap, N. N. (1975). Entailment, the logic of relevance and necessity (Vol. I). Princeton: Princeton University Press.Google Scholar
  5. Arruda, A. I. (Ed.). (1990). N.A.Vasiliev e a lógica paraconsistente. Campinas: CLE.Google Scholar
  6. Asenjo, F. (1966). A calculus of antinomies. Notre Dame Journal of Formal Logic, 7(1), 103–115.CrossRefGoogle Scholar
  7. Bazhanov, V. A. (1990). The fate of one forgotten idea: N.A. Vasiliev and his imaginary logic. Studies in Soviet Thought, 39(3–4), 333–344.CrossRefGoogle Scholar
  8. Bazhanov, V. A. (1993). C.S. Peirce’s influence on logical ideas of N.A. Vasiliev. Modern Logic, 3(1), 45–51.Google Scholar
  9. Bazhanov, V. A. (1994). The imaginary geometry of N.I. Lobachevsky and the imaginary logic of N.A. Vasiliev. Modern Logic, 4(2), 148–156.Google Scholar
  10. Bazhanov, V. A. (1998). Toward the reconstruction of the early history of Paraconsistent logic: The prerequisites of N. A. Vasiliev’s imaginary logic. LogiqueetAnalyse, 161(63), 17–20.Google Scholar
  11. Bazhanov, V. (2001). The origins and becoming of non-classical logic in Russia (XIX – the turn of XX century). In Zwischen traditioneller und moderner Logik. Nichtklassiche Ansatze (pp. 205–217). Paderborn: Mentis-Verlag.Google Scholar
  12. Bazhanov, V. A. (2008). Non-classical stems from classical: N. A. Vasiliev’s approach to logic and his reassessment of the square of proposition. LogicaUniversalis, 2(1), 71–76.Google Scholar
  13. Bazhanov, V. A. (2011). The dawn of paraconsistency: Russia’s logical thought in the turn of XXth century. Manuscrito, 34, 89–98.CrossRefGoogle Scholar
  14. Bazhanov, V. A. (2016a). Russian origins of non-classical logics. In F. Abeles & M. Fuller (Eds.), Modern logic 1850–1950, east and west (pp. 197–203). Cham: Springer.Google Scholar
  15. Bazhanov, V. A. (2016b). Nicolai Alexandrovich Vasiliev. Internet Encyclopedia of Philosophy. Google Scholar
  16. Becker Arenhart, J. R. (2016). Liberating paraconsistency from contradiction. LogicaUniversalis, 10, 523–544.Google Scholar
  17. Bernays, P. (1926). Axiomatische Untersuchungen des Aussagen Kalküls der Principia Mathematica. Mathematische Zeitschrift, 25, 305-320, translated as “Axiomatic investigations of the propositional calculus of the Principia Mathematica” in (Beziau 2012a), p. 43–57.Google Scholar
  18. Beziau, J.-Y. (1994). Théorie législative de la négation pure. Logique et Analyse, 147–148, 209–225.Google Scholar
  19. Beziau, J.-Y. (1997). What is many-valued logic? In Proceedings of the 27th International Symposium on Multiple-Valued Logic (pp. 117–121). Los Alamitos: IEEE Computer Society.Google Scholar
  20. Beziau, J.-Y. (2003a). Bivalence, excluded middle and non contradiction. In L. Behounek (Ed.), The Logic a yearbook 2003 (pp. 73–84). Prague: Academy of Sciences.Google Scholar
  21. Beziau, J.-Y. (2003b). New light on the square of oppositions and its nameless corner. Logical Investigations, 10(2003), 218–232.Google Scholar
  22. Beziau, J.-Y. (2008). What is “formal logic?”. In Myung-Hyun-Lee (Ed.), Proceedings of the XXII world congress of philosophy (Vol. 13, pp. 9–22). Seoul: Korean Philosophical Association.Google Scholar
  23. Beziau, J.-Y. (2010a). Logic is not logic. Abstracta, 6, 73–102.Google Scholar
  24. Beziau, J.-Y. (2010b). Truth as a mathematical object. Principia, 14, 31–46.Google Scholar
  25. Beziau, J.-Y. (Ed.). (2012a). Universal logic: An anthology. Basel: Birkhäuser.Google Scholar
  26. Beziau, J.-Y. (2012b). History of truth-values. In D. M. Gabbay & J. Woods (Eds.), Handbook of the history of logic, Vol. 11 – Logic: A history of its central concepts (pp. 233–305). Amsterdam: Elsevier.Google Scholar
  27. Beziau, J.-Y. (2012c). Paralogics and the theory of valuation. in (Beziau 2012a), pp. 361–372.Google Scholar
  28. Beziau, J.-Y. (2013). The metalogical hexagon. Argumentos, 10(2013), 111–122.Google Scholar
  29. Beziau, J.-Y. (2015a). The relativity and universality of logic. Synthese, 192(2015), 1939–1954.CrossRefGoogle Scholar
  30. Beziau, J.-Y. (2015b). Round squares are no contradictions. In New directions in paraconsistent logic (pp. 131–145). New Dehli: Springer.CrossRefGoogle Scholar
  31. Beziau, J.-Y. (2016a). Disentangling contradiction from contrariety via incompatibility. LogicaUniversalis, 10, 157–170.Google Scholar
  32. Beziau, J.-Y. (2016b). Two genuine 3-valued Paraconsistent logics. In S. Akama (Ed.), Towards Paraconsistent engineering (pp. 35–47). Heidelberg: Springer.CrossRefGoogle Scholar
  33. Beziau, J.-Y., & Basti, G. (Eds.). (2016). The square of opposition: A cornerstone of thought. Basel: Birkhäuser.Google Scholar
  34. Beziau, J.-Y., & Franceschetto, A. (2015). Strong three-valued paraconsistent logics. In New directions in paraconsistent logic (pp. 131–145). New Dehli: Springer.CrossRefGoogle Scholar
  35. Beziau, J.-Y., & Gerogiorgakis, S. (Eds.). (2016). New dimensions of the square of opposition. Munich: PhilosophiaVerlag.Google Scholar
  36. Beziau, J.-Y., & Jacquette, D. (Eds.). (2012). Around and beyond the square of opposition. Basel: Birkhäuser.Google Scholar
  37. Beziau, J.-Y., Payette, G. (Eds.). (2008). The square of opposition, Special Issue of Logica Universalis, 1(2).Google Scholar
  38. Beziau, J.-Y., & Payette, G. (Eds.). (2012). The square of opposition – A general framework for cognition. Bern: Peter Lang.Google Scholar
  39. Beziau, J.-Y., & Read, S. (Eds.). (2014). The square of opposition in historical perspective, special issue of History and Philosophy of Logic, 35(3).Google Scholar
  40. Bocheński, J.-M. (1927). On logical relativism, translated an reprinted in Axiomathes, 2, 1993, pp. 193–209.Google Scholar
  41. Bocheński, J.-M. (1951). Ancient formal logic. Amsterdam: North Holland.Google Scholar
  42. Boole, G. (1854). An investigation of the laws of thought, on which are founded the mathematical theories of logic and probabilities. London/Cambridge: Macmillan & Co..CrossRefGoogle Scholar
  43. Carnielli, W. A. (2012). Paul Bernays and the Eve of Non-standard Models in Logic. in (Beziau 2012), pp. 33–42.Google Scholar
  44. Corcoran, J. (1973). A mathematical model of Aristotle’s syllogistic. Archivfür Geschichte der Philosophie, 55, 191–219.Google Scholar
  45. Corcoran, J. (1974a). Ancient logic and its modern interpretations, Proceedings of the buffalo symposium on modernist interpretations of ancient logic. Dordrecht: Reidel.CrossRefGoogle Scholar
  46. Corcoran, J. (1974b) Aristotle’s Natural Deduction System, in (Corcoran 1974), pp. 85–132.Google Scholar
  47. Corcoran, J. (1974c). Aristotelian syllogisms: Valid arguments or true generalized conditionals? Mind, 83, 278–281.CrossRefGoogle Scholar
  48. Corcoran, J. (2003). Aristotle’s prior analytics and Boole’s Laws of thought. History and Philosophy of Logic, 24, 261–288.CrossRefGoogle Scholar
  49. Correia, M. (2012). Boethius on the square of opposition, in (Beziau and Jacquettei 2012), pp. 41–52.Google Scholar
  50. Correia, M. (2016) The proto-exposition of Aristotelian categorical logic, in (Beziau and Basti).Google Scholar
  51. D’Ottaviano, I. M. L., & da Costa, N. C. A. (1970). Sur un problème de Jaśkowski. Comptes Rendus de l’Académie des Sciences de Paris, 270, 1349–1353.Google Scholar
  52. Da Costa, N. C. A. (1980). Ensaio sobre os fundamentos da lógica. São Paulo: HUCITEC.Google Scholar
  53. da Costa, N. C. A., & Beziau, J.-Y. (1994). Théorie de la valuation. Logique et Analyse, 146, 95–117.Google Scholar
  54. Da Costa, N. C. A., Beziau, J.-Y., & Bueno, O. A. S. (1995). Paraconsistentlogic in a historical perspective. Logique et Analyse, 150–152, 111–125.Google Scholar
  55. Dieudonné, J. (1987). Pour l’honneur de l’esprit humain, Hachette, Paris. (English translation: Mathematics – The music of reason, Springer, Berlin, 1992).Google Scholar
  56. Geach, P. (1950). Subject and predicate. Mind, 59, 461–482.CrossRefGoogle Scholar
  57. Gomes, E. L., & D’Ottaviano, I. M. L. (2010). Aristotle’s theory of deduction and paraconsistency. Principia, 14, 71–98.Google Scholar
  58. Gourinat, J.-B. (2009). Le traité de De l’interprétation entre logique classique et logique non-classique. In S. Husson (Ed.), Interpréter le De interpretatione (pp. 164–192). Paris: Vrin.Google Scholar
  59. Gourinat, J.-B. (2010). La dialectique des stoïciens. Vrin: Paris.Google Scholar
  60. Granger, G.-G. (1976). La théorie aristotélicienne de la science. Paris: Aubier.Google Scholar
  61. Hertz, P. (1931). Von Wesen des Logischen, insbesondere der Bedeutung des modus Barbara. Erkenntnis, 2, 369–392.CrossRefGoogle Scholar
  62. Hilbert, D. (1899) Grundlagen der Geometrie, Leipzig: Teubner (English translation: Foundations of Geometry).Google Scholar
  63. Hintikka, J. (2012) Which mathematical logic is the logic of mathematics? LogicaUniversalis, 6, 459–475.Google Scholar
  64. Horn, L. (2015). On the contrary: Disjunctive syllogism and pragmatic strengthening. In A. Koslow & A. Uchsbaum (Eds.), The road to universal logic – festschrift for 50th birthday of Jean-Yves Béziau volume I (pp. 241–265). Basel: Birkhäuser.Google Scholar
  65. Jones, R. E. (2010). Truth and contradiction in Aristotle’s De Interpretatione 6-9. Phronesis, 55, 26–67.CrossRefGoogle Scholar
  66. Kneale, W., & Kneale, M. (1962). The development of logic. Oxford: Clarendon.Google Scholar
  67. Korzybski, A. (1933). Science and sanity – An introduction to non-Aristotelian systems and general semantics. New York: Institute of general semantics.Google Scholar
  68. Lachance, G. (2016). Platonic contrariety (enantia): Ancestor of the Aristotelian notion of contradiction (antiphasis)? Logica Universalis, 10, 143–156.CrossRefGoogle Scholar
  69. Largeault, J. (1993). La Logique. Paris: Presses universitaires de France.Google Scholar
  70. Lear, J. (1982). Aristotle’s philosophy of mathematics. Philosophical Review, 91, 169–192.CrossRefGoogle Scholar
  71. Legris, J. (2012). Paul Hertz and the origins of structural reasoning, in (Beziau 2012a).Google Scholar
  72. Lewis, C. I. (1932). Alternative systems of logic. The Monist, 42, 481–507.CrossRefGoogle Scholar
  73. Łukasiewicz, J. (1910a). Ozasadziesprzecznosci u Arystotelesa. Kraków: Studiumkrytyczne.Google Scholar
  74. Łukasiewicz, J. (1910b) Über den Satz von Widerspruchbei Aristoteles. Bulletin International de l’Académie des Sciences de Cracovíe: Classe de Philosophie, pp. 15–38.Google Scholar
  75. Łukasiewicz, J. (1920). O logicetrójwartosciowej. RuchFilozoficny, 5, 170–171.Google Scholar
  76. Łukasiewicz, J. (1927). O logicestoików. PrzegladFilozoficny, 30, 278–299.Google Scholar
  77. Łukasiewicz, J. (1951). Aristotle’s syllogistic: From the standpoint of modern formal logic. Oxford: Oxford University Press.Google Scholar
  78. Łukasiewicz, J. (1953). A system of modal logic. Journal of Computing Systems, 1, 111–149.Google Scholar
  79. Markin, V. I. (2013). What trends in non-classical logic were anticipated by Nikolai Vasiliev?. Logical Investigations, 19, 122–135.Google Scholar
  80. Moisil, G. (1972). Essais sur les logiques non-Chrysipiennes. Bucarest: Académie de la République Socialiste de Roumanie.Google Scholar
  81. Murinová, P., & Novák, V. (2016). Syllogisms and 5-square of opposition with Intermediate Quantifiers in fuzzy natural logic. LogicaUniversalis, 10, 339–357.Google Scholar
  82. Patterson, R. (1995). Aristotle’s modal logic. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  83. Priest, G. (1979). The logic of paradox. Journal of Philosophical Logic, 8, 219–241.CrossRefGoogle Scholar
  84. Raspa, V. (1999). Łukasiewicz on the principle of contradiction. Journal of Philosophical Research, 24, 57–112.CrossRefGoogle Scholar
  85. Reiser, O. (1935). Non-Aristotelianlogics. The Monist, 45, 110–117.CrossRefGoogle Scholar
  86. Resnik, M. (1973/4). The Frege-Hilbert controversy. Philosophy and Phenomenological Research, 34, 386–403.Google Scholar
  87. Rini, A. (2011). Aristotle’s modal proofs. Dordrecht: Springer.CrossRefGoogle Scholar
  88. Scholz, H. (1931). Abriss der Geschichte der Logik. Freiburg: Karl Alber.Google Scholar
  89. Seddon, F. (1996). Aristotle & Łukasiewicz on the principle of contradiction. Ames: Modern Logic Publishing.Google Scholar
  90. Slater, B. H. (1995). Paraconsistentlogics? Journal of PhilosophicalLogic, 24, 451–154.Google Scholar
  91. Smirnov, V. A. (1989). The logical ideas of N. A. Vasiliev and modern logic. In Logic, methodology and philosophy of science (Studies in Logic and Foundations of Mathematics, Vol. 126, pp. 625–640). Amsterdam.Google Scholar
  92. Smith, H. B. (1918). Non-Aristotelian logic. The Journal of Philosophy, Psychology and Scientific Methods, 15, 453–458.CrossRefGoogle Scholar
  93. Smith, R. (1984). Aristotle as proof theorist. Philosophia Naturalis, 21, 590–597.Google Scholar
  94. Suchon, W. (1999). Vasiliev: What did he exactly do? Logic and Logical Philosophy, 7, 131–141.CrossRefGoogle Scholar
  95. Szabó, A. (1969) Anfänge der griechischen Mathematik, Akademiai Kiádo, Budapest (English translation: The Beginnings of Greek mathematics, Kluwer, Dordrecht).Google Scholar
  96. Tarski, A. (1936) 0 logice matematyczne jimetodziededukcyjnej, Ksiaznica-Atlas, Lwów and Warsaw (4th English edition by Jan Tarski: Introduction to Logic and to the Methodology of the Deductive Sciences, OUP, Oxford, 1994).Google Scholar
  97. Tarski, A. (1937). Sur la méthode déductive. In Travaux du IXe Congrès International de Philosophie (Vol. VI, pp. 95–103). Paris: Hermann.Google Scholar
  98. Van Heijenoort, J. (1974). Subject and predicate in western logic. Philosophy East and West, 24, 253–268.CrossRefGoogle Scholar
  99. van Rooij, R. (2012). The propositional and relational syllogistic. Logique et Analyse. Google Scholar
  100. Van Vogt, A. E. (1945). The world of null a. In Astounding stories. New York: Tom Doherty.Google Scholar
  101. Vandoulakis, M. (1998). Was Euclid’s approach ta arithmetic axiomatic ? OriensOccidens, 2, 141–181.Google Scholar
  102. Vasiliev, N. A. (1910) On partial judgments, triangle of opposition, law of excluded forth, Kazan.Google Scholar
  103. Vasiliev, N. A. (1912) Imaginary (non-Aristotelian) logic. The Journal of the Ministry of Education, 40, 207–246, English translation in Logique et Analyse, 182 (2003),pp. 127–163.Google Scholar
  104. Vasiliev, N. A. (1913). Logic and metalogic, Logos 1/2, pp. 53–81, English translation in Axiomathes, 3 (1993), pp. 329–351.Google Scholar
  105. Westerståhl, D. (1989). Aristotelian syllogisms and generalized quantifiers. StudiaLogica, 48, 577–585.Google Scholar

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Authors and Affiliations

  1. 1.University of BrazilRio de JaneiroBrazil
  2. 2.Brazilian Research CouncilBrasíliaBrazil
  3. 3.Brazilian Academy of PhilosophyRio de JaneiroBrazil

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