Intuitionistic mereology

  1. Maffezioli, Paolo
  2. Varzi, Achille C.
Revista:
Synthese

ISSN: 0039-7857,  1573-0964

Año de publicación: 2019

Volumen: 198

Número: S18

Páginas: 4277-4302

Tipo: Artículo

DOI: 10.1007/S11229-018-02035-2 GOOGLE SCHOLAR lock_openAcceso abierto editor

Otras publicaciones en: Synthese

Resumen

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.

Referencias bibliográficas

  • Baroni, M. A. (2005). Constructive suprema. Journal of Universal Computer Science, 11, 1865–1877.
  • Bridges, D. (1999). Constructive mathematics: A foundation for computable analysis. Theoretical Computer Science, 219, 95–109.
  • 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.
  • Brouwer, L. E. J. (1923). Intuïitionistische splitsing van mathematische grondbegrippen. Koninklijke Akademie van Wetenschappen te Amsterdam, Verslagen, 32, 877–880.
  • Brouwer, L. E. J. (1925). Intuitionistische Zerlegung mathematischer Grundbegriffe. Jahresbericht der Deutschen Mathematiker-Vereinigung, 33, 251–256.
  • Casati, R., & Varzi, A . C. (1999). Parts and places: The structures of spatial representation. Cambridge (MA): MIT Press.
  • Ciraulo, F. (2013). Intuitionistic overlap structures. Logic and Logical Philosophy, 22, 201–212.
  • Ciraulo, F., Maietti, M. E., & Toto, P. (2013). Constructive version of Boolean algebra. Logic Journal of the IJPL, 21, 44–62.
  • Cotnoir, A. J. (2010). Anti-symmetry and non-extensional mereology. The Philosophical Quarterly, 60, 396–405.
  • Cotnoir, A. J., & Bacon, A. (2012). Non-wellfounded mereology. The Review of Symbolic Logic, 5, 187–204.
  • Eberle, R. A. (1970). Nominalistic systems. Dordrecht: Reidel.
  • Forrest, P. (2002). Nonclassical mereology and its application to sets. Notre Dame Journal of Formal Logic, 43, 79–94.
  • Gentzen, G. (1933). Über das Verhältnis zwischen intuitionistischer und klassischer Logik. Published posthumously in Archiv für mathematische Logik und Grundlagenforschung, 16, 119–132, 1974.
  • Gentzen, G. (1935). Untersuchungen über das logische Schliessen. Mathematische Zeitschrift, 39, 176–210 and 405–431.
  • Gödel, K. (1933). Zur intuitionistischen Arithmetik und Zahlentheorie. Ergebnisse eines mathematischen Kolloquiums, 4, 34–38.
  • Goodman, N. (1951). The structure of appearance. Cambridge (MA): Harvard University Press. Third edition: Dordrecht, Reidel, 1977.
  • 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.
  • Heyting, A. (1928). Zur intuitionistischen Axiomatik der projektiven Geometrie. Mathematische Annalen, 98, 491–538.
  • Heyting, A. (1956). Intuitionism. An introduction. Amsterdam: North-Holland.
  • Hovda, P. (2009). What is classical mereology? Journal of Philosophical Logic, 38, 55–82.
  • Hovda, P. (2016). Parthood-like relations: Closure principles and connections to some axioms of classical mereology. Philosophical Perspectives, 30, 183–197.
  • Johnstone, P. T. (1982). Stone spaces. Cambridge: Cambridge University Press.
  • 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.
  • Lando, G. (2017). Mereology. A philosophical introduction. London: Bloomsbury.
  • Leonard, H. S., & Goodman, N. (1940). The calculus of individuals and its uses. The Journal of Symbolic Logic, 5, 45–55.
  • Leśniewski, S. (1916). Podstawy ogólnej teoryi mnogości. I. Moskow, Prace Polskiego Koła Naukowego w Moskwie (Sekcya matematyczno-przyrodnicza).
  • Leśniewski, S. (1927–1931). O podstawach matematyki. Przeglad Filozoficzny, 30–34, 164–206, 261–291, 60–101, 77–105, 142–170.
  • Lewis, D. K. (1991). Parts of classes. Oxford: Blackwell.
  • Mormann, T. (2013). Heyting mereology as a framework for spatial reasoning. Axiomathes, 23, 137–164.
  • Negri, S. (1999). Sequent calculus proof theory of intuitionistic apartness and order relations. Archive for Mathematical Logic, 38, 521–547.
  • Niebergall, K.-G. (2011). Mereology. In L. Horsten & R. Pettigrew (Eds.), The Continuum companion to philosophical logic (pp. 271–298). New York: Continuum.
  • Pietruszczak, A. (2014). A general concept of being a part of a whole. Notre Dame Journal of Formal Logic, 55, 359–381.
  • Pietruszczak, A. (2015). Classical mereology is not elementarily axiomatizable. Logic and Logical Philosophy, 24, 485–498.
  • von Plato, J. (1999). Order in open intervals of computable reals. Mathematical Structures in Computer Science, 9, 103–108.
  • 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.
  • Polkowski, L. T. (2001). Approximate reasoning by parts. An introduction to rough mereology. Berlin: Springer.
  • Russell, J. S. (2016). Indefinite divisibility. Inquiry, 59, 239–263.
  • Sambin, G. (Forthcoming). Positive topology and the basic picture. New structures emerging from constructive mathematics. Oxford: Clarendon Press.
  • Scott, D. (1968). Extending the topological interpretation to intuitionistic analysis. Compositio Mathematica, 20, 194–210.
  • Simons, P. M. (1987). Parts. A study in ontology. Oxford: Clarendon Press.
  • 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.
  • 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.
  • 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.
  • Tennant, N. (2013). Parts, classes and Parts of classes An anti-realist reading of Lewisian mereology. Synthese, 190, 709–742.
  • Troelstra, A. S., & van Dalen, D. (1988). Constructivism in mathematics. An introduction. Volume I. Amsterdam: North-Holland.
  • Varzi, A. C. (2008). The extensionality of parthood and composition. The Philosophical Quarterly, 58, 108–133.
  • 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.
  • Weber, Z., & Cotnoir, A. J. (2015). Inconsistent boundaries. Synthese, 192, 1267–1294.