The Square of Opposition in Orthomodular Logic

Part of the Studies in Universal Logic book series (SUL)


In Aristotelian logic, categorical propositions are divided in Universal Affirmative, Universal Negative, Particular Affirmative and Particular Negative. Possible relations between two of the mentioned type of propositions are encoded in the square of opposition. The square expresses the essential properties of monadic first order quantification which, in an algebraic approach, may be represented taking into account monadic Boolean algebras. More precisely, quantifiers are considered as modal operators acting on a Boolean algebra and the square of opposition is represented by relations between certain terms of the language in which the algebraic structure is formulated. This representation is sometimes called the modal square of opposition. Several generalizations of the monadic first order logic can be obtained by changing the underlying Boolean structure by another one giving rise to new possible interpretations of the square.


Square of opposition Modal orthomodular logic Classical consequences 

Mathematics Subject Classification

03G12 06C15 03B45 


  1. 1.
    Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Graduate Text in Mathematics. Springer, New York (1981) MATHCrossRefGoogle Scholar
  2. 2.
    Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37, 823–843 (1936) CrossRefGoogle Scholar
  3. 3.
    Dieks, D.: The formalism of quantum theory: an objective description of reality. Ann. Phys. 7, 174–190 (1988) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dieks, D.: Quantum mechanics: an intelligible description of reality? Found. Phys. 35, 399–415 (2005) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Domenech, G., Freytes, H., de Ronde, C.: Scopes and limits of modality in quantum mechanics. Ann. Phys. 15, 853–860 (2006) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Domenech, G., Freytes, H., de Ronde, C.: Modal type orthomodular logic. Math. Log. Q. 55, 287–299 (2009) CrossRefGoogle Scholar
  7. 7.
    Greechie, R.: On generating distributive sublattices of orthomodular lattices. Proc. Am. Math. Soc. 67, 17–22 (1977) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Halmos, P.: Algebraic logic I, monadic Boolean algebras. Compos. Math. 12, 217–249 (1955) MathSciNetGoogle Scholar
  9. 9.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998) MATHCrossRefGoogle Scholar
  10. 10.
    Jauch, J.M.: Foundations of Quantum Mechanics. Addison-Wesley, Reading (1968) MATHGoogle Scholar
  11. 11.
    Kalman, J.A.: Lattices with involution. Trans. Am. Math. Soc. 87, 485–491 (1958) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Kalmbach, G.: Orthomodular Lattices. Academic Press, London (1983) MATHGoogle Scholar
  13. 13.
    Maeda, F., Maeda, S.: Theory of Symmetric Lattices. Springer, Berlin (1970) MATHGoogle Scholar
  14. 14.
    van Fraassen, B.C.: A modal interpretation of quantum mechanics. In: Beltrametti, E.G., van Fraassen, B.C. (eds.) Current Issues in Quantum Logic, pp. 229–258. Plenum, New York (1981) CrossRefGoogle Scholar
  15. 15.
    van Fraassen, B.C.: Quantum Mechanics: An Empiricist View. Clarendon, Oxford (1991) Google Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Universita degli Studi di CagliariCagliariItaly
  2. 2.Instituto Argentino de MatemáticaBuenos AiresArgentina
  3. 3.Departamento de Filosofía “Dr. A Korn”Universidad de Buenos Aires-CONICETBuenos AiresArgentina
  4. 4.Center Leo Apostel and Foundations of the Exact SciencesVrije Universiteit BrusselBrusselsBelgium
  5. 5.Instituto de Astronomía y Física del EspacioBuenos AiresArgentina

Personalised recommendations