Advertisement

Studia Logica

, Volume 106, Issue 1, pp 167–189 | Cite as

Correspondence Between Kripke Frames and Projective Geometries

  • Shengyang ZhongEmail author
Article

Abstract

In this paper we show that some orthogeometries, i.e. projective geometries each defined using a ternary collinearity relation and equipped with a binary orthogonality relation, which are extensively studied in mathematics and quantum theory, correspond to Kripke frames, each defined using a binary relation, satisfying a few conditions. To be precise, we will define four special kinds of Kripke frames, namely, geometric frames, irreducible geometric frames, complete geometric frames and quantum Kripke frames; and we will show that they correspond to pure orthogeometries (or, equivalently, projective geometries with pure polarities), irreducible pure orthogeometries, Hilbertian geometries and irreducible Hilbertian geometries, respectively. The discovery of these correspondences raises interesting research topics and will enrich the study of logic.

Keywords

Kripke frame Projective geometry Quantum logic Spatial logic Modal logic 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aiello, M., M. I. Pratt-Hartmann, and J. van Benthem, Handbook of Spatial Logics, Springer Netherlands, Dordrecht, 2007.Google Scholar
  2. 2.
    Balbiani, P., The modal multilogic of geometry, Journal of Applied Non-Classical Logics 8(3):259–281, 1998.CrossRefGoogle Scholar
  3. 3.
    Balbiani, P., L. F. D. Cerro, T. Tinchev, and D. Vakarelov, Modal logics for incidence geometries, Journal of Logic and Computation 7(1):59–78, 1997.CrossRefGoogle Scholar
  4. 4.
    Balbiani, P., and V. Goranko, Modal logics for parallelism, orthogonality, and affine geometries, Journal of Applied Non-Classical Logics 12(3-4):365–397, 2002.CrossRefGoogle Scholar
  5. 5.
    Baltag, A., and S. Smets, Complete axiomatizations for quantum actions, International Journal of Theoretical Physics 44(12):2267–2282, 2005.CrossRefGoogle Scholar
  6. 6.
    Berberian, S. K., Introduction to Hilbert Space, Oxford University Press, Oxford, 1961.Google Scholar
  7. 7.
    Birkhoff, G., and J. von Neumann, The logic of quantum mechanics, The Annals of Mathematics 37:823–843, 1936.CrossRefGoogle Scholar
  8. 8.
    Blackburn, P., M. de Rijke, and Y. Venema, Modal Logic, Cambridge University Press, Cambridge, 2001.CrossRefGoogle Scholar
  9. 9.
    Buekenhout, F., A theorem of parmentier characterizing projective spaces by polarities, in F. de Clerck and J. Hirschfeld (eds.), Finite Geometry and Combinatorics, Cambridge University Press, Cambridge, 1993, pp. 69–72.CrossRefGoogle Scholar
  10. 10.
    Coxeter, H. S. M., Projective Geometry, 2nd edn, Springer, New York, 1987.Google Scholar
  11. 11.
    Dishkant, H., Semantics of the minimal logic of quantum mechanics, Studia Logica 30(1):23–30, 1972.CrossRefGoogle Scholar
  12. 12.
    Faure, C.-A., and A. Frölicher, Dualities for infinite-dimensional projective geometries, Geometriae Dedicata 56(3):225–236, 1995.Google Scholar
  13. 13.
    Faure, C.-A., and A. Frölicher, Modern Projective Geometry, Springer Netherlands, Dordrecht, 2000.Google Scholar
  14. 14.
    Goldblatt, R. I., Semantic analysis of orthologic, Journal of Philosophical Logic 3:19–35, 1974.CrossRefGoogle Scholar
  15. 15.
    Hedlíková, J., and S. Pulmannová, Orthogonality spaces and atomistic orthocomplemented lattices, Czechoslovak Mathematical Journal 41:8–23, 1991.Google Scholar
  16. 16.
    Moore, D. J., Categories of representations of physical systems, Helvetica Physica Acta 68:658–678, 1995.Google Scholar
  17. 17.
    Piron, C., Foundations of Quantum Physics, W. A. Benjamin Inc., Reading, 1976.Google Scholar
  18. 18.
    Stebletsova, V., Modal Logic of Projective Geometries of Finite Dimension. Technical report, Department of Philosophy, Utrecht University, 1998.Google Scholar
  19. 19.
    Stubbe, I., and B. van Steirteghem, Propositional systems, Hilbert lattices and generalized Hilbert spaces, in K. Engesser, D. M. Gabbay, and D. Lehmann, (eds.), Handbook of Quantum Logic and Quantum Structures: Quantum Structures, Elsevier, Amsterdam, 2007, pp. 477–523.Google Scholar
  20. 20.
    Varadarajan, V. S., Geometry of Quantum Theory, 2nd edn, Springer, New York, 1985.Google Scholar
  21. 21.
    Veblen, O., and J. W. Young, Projective Geometry, Blaisdell Publishing Company, New York, 1910.Google Scholar
  22. 22.
    Venema, Y., Points, lines and diamonds: A two-sorted modal logic for projective planes, Journal of Logic and Computation 9(5):601–621, 1999.CrossRefGoogle Scholar
  23. 23.
    Zhong, S., Orthogonality and Quantum Geometry: Towards a Relational Reconstruction of Quantum Theory, PhD thesis, University of Amsterdam, 2015. http://www.illc.uva.nl/Research/Publications/Dissertations/DS-2015-03.text.pdf.

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Institute of Logic and CognitionSun Yat-sen UniversityGuangzhouPeople’s Republic of China

Personalised recommendations