Advertisement

Synthese

pp 1–26 | Cite as

Intuitionistic mereology

  • Paolo Maffezioli
  • Achille C. VarziEmail author
S.I.: Mereology and Identity
  • 23 Downloads

Abstract

Two mereological theories are presented based on a primitive apartness relation along with binary relations of mereological excess and weak excess, respectively. It is shown that both theories are acceptable from the standpoint of constructive reasoning while remaining faithful to the spirit of classical mereology. The two theories are then compared and assessed with regard to their extensional import.

Keywords

Mereology Intuitionism Apartness Excess Extensionality 

Notes

References

  1. Baroni, M. A. (2005). Constructive suprema. Journal of Universal Computer Science, 11, 1865–1877.Google Scholar
  2. Bridges, D. (1999). Constructive mathematics: A foundation for computable analysis. Theoretical Computer Science, 219, 95–109.CrossRefGoogle Scholar
  3. Brouwer, L. E. J. (1919). Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten. Zweiter Teil: Theorie der Punktmengen. Koninklijke Akademie van Wetenschappen te Amsterdam. Verhandelingen (Eerste Sectie), 7, 1–33.Google Scholar
  4. Brouwer, L. E. J. (1923). Intuïitionistische splitsing van mathematische grondbegrippen. Koninklijke Akademie van Wetenschappen te Amsterdam, Verslagen, 32, 877–880.Google Scholar
  5. Brouwer, L. E. J. (1925). Intuitionistische Zerlegung mathematischer Grundbegriffe. Jahresbericht der Deutschen Mathematiker-Vereinigung, 33, 251–256.Google Scholar
  6. Casati, R., & Varzi, A . C. (1999). Parts and places: The structures of spatial representation. Cambridge (MA): MIT Press.Google Scholar
  7. Ciraulo, F. (2013). Intuitionistic overlap structures. Logic and Logical Philosophy, 22, 201–212.CrossRefGoogle Scholar
  8. Ciraulo, F., Maietti, M. E., & Toto, P. (2013). Constructive version of Boolean algebra. Logic Journal of the IJPL, 21, 44–62.CrossRefGoogle Scholar
  9. Cotnoir, A. J. (2010). Anti-symmetry and non-extensional mereology. The Philosophical Quarterly, 60, 396–405.CrossRefGoogle Scholar
  10. Cotnoir, A. J., & Bacon, A. (2012). Non-wellfounded mereology. The Review of Symbolic Logic, 5, 187–204.CrossRefGoogle Scholar
  11. Eberle, R. A. (1970). Nominalistic systems. Dordrecht: Reidel.CrossRefGoogle Scholar
  12. Forrest, P. (2002). Nonclassical mereology and its application to sets. Notre Dame Journal of Formal Logic, 43, 79–94.CrossRefGoogle Scholar
  13. Gentzen, G. (1933). Über das Verhältnis zwischen intuitionistischer und klassischer Logik. Published posthumously in Archiv für mathematische Logik und Grundlagenforschung, 16, 119–132, 1974.Google Scholar
  14. Gentzen, G. (1935). Untersuchungen über das logische Schliessen. Mathematische Zeitschrift, 39, 176–210 and 405–431.Google Scholar
  15. Gödel, K. (1933). Zur intuitionistischen Arithmetik und Zahlentheorie. Ergebnisse eines mathematischen Kolloquiums, 4, 34–38.Google Scholar
  16. Goodman, N. (1951). The structure of appearance. Cambridge (MA): Harvard University Press. Third edition: Dordrecht, Reidel, 1977.Google Scholar
  17. Greenleaf, N. (1978). Linear order in lattices: A constructive study. In G.-C. Rota (Ed.), Studies in foundations and combinatorics (pp. 11–30). New York: Academic Press.Google Scholar
  18. Heyting, A. (1928). Zur intuitionistischen Axiomatik der projektiven Geometrie. Mathematische Annalen, 98, 491–538.CrossRefGoogle Scholar
  19. Heyting, A. (1956). Intuitionism. An introduction. Amsterdam: North-Holland.Google Scholar
  20. Hovda, P. (2009). What is classical mereology? Journal of Philosophical Logic, 38, 55–82.CrossRefGoogle Scholar
  21. Hovda, P. (2016). Parthood-like relations: Closure principles and connections to some axioms of classical mereology. Philosophical Perspectives, 30, 183–197.CrossRefGoogle Scholar
  22. Johnstone, P. T. (1982). Stone spaces. Cambridge: Cambridge University Press.Google Scholar
  23. Kripke, S. A. (1965). Semantical analysis of intuitionistic logic I. In J. N. Crossley & M. A. E. Dummett (Eds.), Formal systems and recursive functions. Proceedings of the 8th logic colloquium (pp. 92–130). Amsterdam: North-Holland.CrossRefGoogle Scholar
  24. Lando, G. (2017). Mereology. A philosophical introduction. London: Bloomsbury.Google Scholar
  25. Leonard, H. S., & Goodman, N. (1940). The calculus of individuals and its uses. The Journal of Symbolic Logic, 5, 45–55.CrossRefGoogle Scholar
  26. Leśniewski, S. (1916). Podstawy ogólnej teoryi mnogości. I. Moskow, Prace Polskiego Koła Naukowego w Moskwie (Sekcya matematyczno-przyrodnicza).Google Scholar
  27. Leśniewski, S. (1927–1931). O podstawach matematyki. Przeglad Filozoficzny, 30–34, 164–206, 261–291, 60–101, 77–105, 142–170.Google Scholar
  28. Lewis, D. K. (1991). Parts of classes. Oxford: Blackwell.Google Scholar
  29. Mormann, T. (2013). Heyting mereology as a framework for spatial reasoning. Axiomathes, 23, 137–164.CrossRefGoogle Scholar
  30. Negri, S. (1999). Sequent calculus proof theory of intuitionistic apartness and order relations. Archive for Mathematical Logic, 38, 521–547.CrossRefGoogle Scholar
  31. Niebergall, K.-G. (2011). Mereology. In L. Horsten & R. Pettigrew (Eds.), The Continuum companion to philosophical logic (pp. 271–298). New York: Continuum.Google Scholar
  32. Pietruszczak, A. (2014). A general concept of being a part of a whole. Notre Dame Journal of Formal Logic, 55, 359–381.CrossRefGoogle Scholar
  33. Pietruszczak, A. (2015). Classical mereology is not elementarily axiomatizable. Logic and Logical Philosophy, 24, 485–498.Google Scholar
  34. von Plato, J. (1999). Order in open intervals of computable reals. Mathematical Structures in Computer Science, 9, 103–108.CrossRefGoogle Scholar
  35. von Plato, J. (2001). Positive lattices. In P. Schuster, U. Berger, & H. Osswald (Eds.), Reuniting the antipodes. Constructive and nonstandard views of the continuum (pp. 185–197). Dordrecht: Kluwer.CrossRefGoogle Scholar
  36. Polkowski, L. T. (2001). Approximate reasoning by parts. An introduction to rough mereology. Berlin: Springer.Google Scholar
  37. Russell, J. S. (2016). Indefinite divisibility. Inquiry, 59, 239–263.CrossRefGoogle Scholar
  38. Sambin, G. (Forthcoming). Positive topology and the basic picture. New structures emerging from constructive mathematics. Oxford: Clarendon Press.Google Scholar
  39. Scott, D. (1968). Extending the topological interpretation to intuitionistic analysis. Compositio Mathematica, 20, 194–210.Google Scholar
  40. Simons, P. M. (1987). Parts. A study in ontology. Oxford: Clarendon Press.Google Scholar
  41. Simons, P. M. (1991). Free part-whole theory. In K. Lambert (Ed.), Philosophical applications of free logic (pp. 285–306). New York: Oxford University Press.Google Scholar
  42. Smith, N. J. J. (2005). A plea for things that are not quite all there: Or, is there a problem about vague composition and vague existence? The Journal of Philosophy, 102, 381–421.CrossRefGoogle Scholar
  43. Smorynski, C. A. (1973). Applications of Kripke models. In A. S. Troelstra (Ed.), Metatmathematical investigation of intuitionistic arithmetic and analysis (pp. 324–391). Berlin: Springer.CrossRefGoogle Scholar
  44. Tennant, N. (2013). Parts, classes and Parts of classes An anti-realist reading of Lewisian mereology. Synthese, 190, 709–742.CrossRefGoogle Scholar
  45. Troelstra, A. S., & van Dalen, D. (1988). Constructivism in mathematics. An introduction. Volume I. Amsterdam: North-Holland.Google Scholar
  46. Varzi, A. C. (2008). The extensionality of parthood and composition. The Philosophical Quarterly, 58, 108–133.CrossRefGoogle Scholar
  47. Varzi, A. C. (2016). Mereology. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Winter edition. https://plato.stanford.edu/archives/win2016/entries/mereology.
  48. Weber, Z., & Cotnoir, A. J. (2015). Inconsistent boundaries. Synthese, 192, 1267–1294.CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of BarcelonaBarcelonaSpain
  2. 2.Department of PhilosophyColumbia UniversityNew YorkUSA

Personalised recommendations