Around and Beyond the Square of Opposition pp 175-189 | Cite as

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

## 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## Notes

### Acknowledgements

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

## References

- 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.Beth, E.W.: Foundations of Mathematics, ch. I, 2. Harper, New York (1959) MATHGoogle Scholar
- 3.Carnot, L.: Essai sur le machines en général, pp. 101–103. Defay, Dijon (1783) Google Scholar
- 4.Carnot, L.: Principes de l’équilibre et du mouvement, pp. xii–xv. Deterville, Paris (1803) Google Scholar
- 5.Carnot, S.: Réflexions sur la puissance motrice du feu, p. 23. Blanchard, Paris (1824) Google Scholar
- 6.Le Ronde d’Alembert, J.: Elémens. In: Le Ronde d’Alembert, J., Diderot, D. (eds.) Éncyclopédie Française, p. 17 (1754) Google Scholar
- 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.Drago, A.: Mathematics and alternative theoretical physics: the method for linking them together. Epistemologia
**19**, 33–50 (1996) Google Scholar - 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.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.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.Drago, A.: The square of opposition and the four fundamental choices. Logica Univers.
**2**, 127–141 (2008) MathSciNetMATHCrossRefGoogle Scholar - 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.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.Drago, A.: The coincidentia oppositorum in Cusanus (1401–1464), Lanza del Vasto (1901–1981) and beyond. Epistemologia
**33**, 305–328 (2010) Google Scholar - 16.Drago, A., Oliva, R.: Atomism and the reasoning by non-classical logic. Hyle
**5**, 43–55 (1999) Google Scholar - 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.Dummett, M.: Principles of Intuitionism. Clarendon, Oxford (1977) Google Scholar
- 19.Einstein, A.: Zur Elektrodynamik bewegter Koerper. Ann. Phys.
**17**, 891–921 (1905) MATHCrossRefGoogle Scholar - 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.Goedel, K.: Collected Works, papers 1932 and 1933f. Oxford University Press, Oxford (1986) MATHGoogle Scholar
- 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.Haak, S.: Deviant Logic. Oxford University Press, Oxford (1978) Google Scholar
- 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.Heyting, A.: Intuitionism. An Introduction, Sect. 2. North-Holland, Amsterdam (1971) MATHGoogle Scholar
- 26.Kleene, C.: Introduction to Metamathematics, pp. 318–319. Van Nostrand, Princeton (1952) MATHGoogle Scholar
- 27.Klein, M.J.: Thermodynamics in Einstein’s thought. Science
**57**, 505–516 (1967) Google Scholar - 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.Leibniz, G.W.: Letter to Arnauld, 14-7-1686, Gerh. II, Q., Opusc. 402, 513 Google Scholar
- 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.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.Miller, A.I.: Albert Einstein’s Special Theory of Relativity, pp. 123–142. Addison-Wesley, Reading (1981) Google Scholar
- 33.Parsons, T.: Things that are right wit the traditional square of opposition. Logica Univers.
**2**, 3–11 (2008) MathSciNetMATHCrossRefGoogle Scholar - 34.Poincaré, H.: La Science et l’Hypothèse, Ch. “Optique et Electricité”. Hermann, Paris (1903) Google Scholar
- 35.Poincaré, H.: La Valeur de la Science, Ch. VII. Flammarion, Paris (1905) Google Scholar
- 36.Poincaré, H.: Science et Méthode, p. 187. Flammarion, Paris (1912) Google Scholar
- 37.Prawitz, D.: Meaning and proof. The conflict between classical and intuitionist logic. Theoria
**43**, 6–39 (1976) MathSciNetGoogle Scholar - 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.Tennant, N.: Natural Logic, p. 57. Edinburgh University Press, Edinburgh Google Scholar
- 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.Troelstra, A., van Dalen, D.: Constructivism in Mathematics, vol. 1. North-Holland, Amsterdam (1988) Google Scholar
- 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.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