Pluralism in Logic: The Square of Opposition, Leibniz’ Principle of Sufficient Reason and Markov’s Principle

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


According to the present pluralism in mathematical logic, I translate from classical logic to non-classical logic the predicates of the classical square of opposition. A similar unique structure is obtained. In order to support this new logical structure, I investigate on the rich legacy of the non-classical arguments presented by ingenuity by several authors of scientific theories. A comparative analysis of their ways of arguing shows that each of these theories is severed in two parts; the former one proves a universal predicate by an ad absurdum proof. This conclusion of every theory results to be formalised by the A thesis of the new logical structure. Afterwards, this conclusion is changed in the corresponding affirmative predicate, which in the latter part plays the role of a new hypothesis for a deductive development. This kind of change is the same suggested by Leibniz’ principle of sufficient reason. Instead, Markov’s principle results to be a weaker logical change, from the intuitionist thesis I in the affirmative thesis I. The relevance of all the four theses of the new logical structure is obtained by studying all the conversion implications of intuitionist predicates. In the Appendix, I analyse as an example of the above theories, Markov’s presentation of his theory of real numbers.


Square of opposition Non-classical logic Doubly negated statements Logical principles 

Mathematics Subject Classification

01A20 03A05 03B20 



I acknowledge two anonymous referees for an improvement of my previous logical framework, and David Braithwhaite for the correction on my English text.


  1. 1.
    Avogadro, A.: Essay d’une manière de determiner les masses relatives des molecules élémentaires des corps…J. Phys. Chim. Hist. Nat. 73, 58–76 (1811) Google Scholar
  2. 2.
    Beth, E.W.: Foundations of Mathematics, ch. I, 2. Harper, New York (1959) MATHGoogle Scholar
  3. 3.
    Carnot, L.: Essai sur le machines en général, pp. 101–103. Defay, Dijon (1783) Google Scholar
  4. 4.
    Carnot, L.: Principes de l’équilibre et du mouvement, pp. xii–xv. Deterville, Paris (1803) Google Scholar
  5. 5.
    Carnot, S.: Réflexions sur la puissance motrice du feu, p. 23. Blanchard, Paris (1824) Google Scholar
  6. 6.
    Le Ronde d’Alembert, J.: Elémens. In: Le Ronde d’Alembert, J., Diderot, D. (eds.) Éncyclopédie Française, p. 17 (1754) Google Scholar
  7. 7.
    Drago, A.: Poincaré vs. Peano and Hilbert about the mathematical principle of induction. In: Greffe, J.-L., Heinzmann, G., Lorenz, K. (eds.) Henri Poincaré. Science et Philosophie, pp. 513–527. Blanchard/Springer, Paris/Berlin (1996) Google Scholar
  8. 8.
    Drago, A.: Mathematics and alternative theoretical physics: the method for linking them together. Epistemologia 19, 33–50 (1996) Google Scholar
  9. 9.
    Drago, A.: The birth of an alternative mechanics: Leibniz’ principle of sufficient reason. In: Poser, H., et al. (eds.) Leibniz-Kongress. Nihil Sine Ratione, vol. I, pp. 322–330. Berlin (2001) Google Scholar
  10. 10.
    Drago, A.: A.N. Kolmogoroff and the relevance of the double negation law in science. In: Sica, G. (ed.) Essays on the Foundations of Mathematics and Logic, pp. 57–81. Polimetrica, Milano (2005) Google Scholar
  11. 11.
    Drago, A.: Four ways of logical reasoning in theoretical physics and their relationships with both the kinds of mathematics and computer science. In: Cooper, S.B., et al. (eds.) Computation and Logic in Real World, pp. 132–142. Univ. Siena Dept. of Math. Sci., Siena (2007) Google Scholar
  12. 12.
    Drago, A.: The square of opposition and the four fundamental choices. Logica Univers. 2, 127–141 (2008) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Drago, A.: Lo scritto di Einstein sulla relatività. Esame secondo il modello di organizzazione basata su un problema. Atti Fond. G. Ronchi 64, 113–131 (2009) Google Scholar
  14. 14.
    Drago, A.: A logical analysis of Lobachevsky’s geometrical theory. Atti Fond. G. Ronchi 64(4), 453–481 (2010). Italian version in: Period. Mat., Ser. VIII 4, 41–52 (2004) Google Scholar
  15. 15.
    Drago, A.: The coincidentia oppositorum in Cusanus (1401–1464), Lanza del Vasto (1901–1981) and beyond. Epistemologia 33, 305–328 (2010) Google Scholar
  16. 16.
    Drago, A., Oliva, R.: Atomism and the reasoning by non-classical logic. Hyle 5, 43–55 (1999) Google Scholar
  17. 17.
    Drago, A., Pisano, R.: Interpretation and reconstruction of Sadi Carnot’s Réflexions through non-classical logic. Atti Fond. G. Ronchi 59, 615–644 (2004). Italian version in: G. Fis. 41, 195–215 (2000) Google Scholar
  18. 18.
    Dummett, M.: Principles of Intuitionism. Clarendon, Oxford (1977) Google Scholar
  19. 19.
    Einstein, A.: Zur Elektrodynamik bewegter Koerper. Ann. Phys. 17, 891–921 (1905) MATHCrossRefGoogle Scholar
  20. 20.
    Gabbey, D., Olivetti, N.: Goal-oriented deductions. In: Gabbey, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 9, pp. 134–199. Reidel, Boston (2002) CrossRefGoogle Scholar
  21. 21.
    Goedel, K.: Collected Works, papers 1932 and 1933f. Oxford University Press, Oxford (1986) MATHGoogle Scholar
  22. 22.
    Grize, J.B.: Logique. In: Piaget, J. (ed.) Logique et connaissance scientifique. Éncyclopédie de la Pléiade, pp. 135–288 (206–210). Gallimard, Paris (1970) Google Scholar
  23. 23.
    Haak, S.: Deviant Logic. Oxford University Press, Oxford (1978) Google Scholar
  24. 24.
    Hand, M.: Antirealism and Falsity. In: Gabbey, D.M., Wansing, H. (eds.) What is Negation?, pp. 185–198 (188). Kluwer, Dordrecht (1999) Google Scholar
  25. 25.
    Heyting, A.: Intuitionism. An Introduction, Sect. 2. North-Holland, Amsterdam (1971) MATHGoogle Scholar
  26. 26.
    Kleene, C.: Introduction to Metamathematics, pp. 318–319. Van Nostrand, Princeton (1952) MATHGoogle Scholar
  27. 27.
    Klein, M.J.: Thermodynamics in Einstein’s thought. Science 57, 505–516 (1967) Google Scholar
  28. 28.
    Kolmogorov, A.N.: On the principle ‘tertium non datur’. Mat. Sb. 32, 646–667 (1924/1925). Engl. translation in: van Heijenoorth, J. (ed.) From Frege to Goedel, pp. 416–437. Harvard University Press, Cambridge (1967) Google Scholar
  29. 29.
    Leibniz, G.W.: Letter to Arnauld, 14-7-1686, Gerh. II, Q., Opusc. 402, 513 Google Scholar
  30. 30.
    Lobachevsky, N.I.: Geometrische Untersuchungen zur der Theorien der Parallellineen. Finkl, Berlin (1840). Engl. translation as an Appendix to: Bonola, R.: Non-Euclidean Geometry. Dover, New York (1955) Google Scholar
  31. 31.
    Markov, A.A.: On constructive mathematics. Tr. Mat. Inst. Steklova 67, 8–14 (1962). Also in: Am. Math. Soc. Transl. 98(2), 1–9 (1971) MATHGoogle Scholar
  32. 32.
    Miller, A.I.: Albert Einstein’s Special Theory of Relativity, pp. 123–142. Addison-Wesley, Reading (1981) Google Scholar
  33. 33.
    Parsons, T.: Things that are right wit the traditional square of opposition. Logica Univers. 2, 3–11 (2008) MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Poincaré, H.: La Science et l’Hypothèse, Ch. “Optique et Electricité”. Hermann, Paris (1903) Google Scholar
  35. 35.
    Poincaré, H.: La Valeur de la Science, Ch. VII. Flammarion, Paris (1905) Google Scholar
  36. 36.
    Poincaré, H.: Science et Méthode, p. 187. Flammarion, Paris (1912) Google Scholar
  37. 37.
    Prawitz, D.: Meaning and proof. The conflict between classical and intuitionist logic. Theoria 43, 6–39 (1976) MathSciNetGoogle Scholar
  38. 38.
    Prawitz, D., Melmnaas, P.-E.: A survey of some connections between classical intuitionistic and minimal logic. In: Schmidt, A., Schuette, H. (eds.) Contributions to Mathematical Logic, pp. 215–229. North-Holland, Amsterdam (1968) Google Scholar
  39. 39.
    Tennant, N.: Natural Logic, p. 57. Edinburgh University Press, Edinburgh Google Scholar
  40. 40.
    Troesltra, A.: Remarks on intuitionism and the philosophy of mathematics. In: Costantini, D., Galavotti, M.C. (eds.) Nuovi Problemi della Logica e della Filosofia della Scienza, pp. 211–228 (223). CLUEB, Bologna (1991) Google Scholar
  41. 41.
    Troelstra, A., van Dalen, D.: Constructivism in Mathematics, vol. 1. North-Holland, Amsterdam (1988) Google Scholar
  42. 42.
    van Dalen, D.: Intuitionist logic. In: Gabbey, D.M., Guenthner, F. (eds.) Handbook of Mathematical Logic, vol. II, pp. 225–449 (226 and 230). Reidel, Dordrecht (2002) Google Scholar
  43. 43.
    Woods, J., Irvine, A.: Aristotle’s early logic. In: Gabbey, D.M., Woods, J. (eds) Handbook of History of Logic, vol. 1, pp. 27–99 (66). Elsevier, New York (2004) Google Scholar

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© Springer Basel 2012

Authors and Affiliations

  1. 1.University of PisaPisaItaly

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