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)

Abstract

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.

Keywords

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

Mathematics Subject Classification

01A20 03A05 03B20 

References

  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

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.University of PisaPisaItaly

Personalised recommendations