Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Graduate Text in Mathematics. Springer, New York (1981)
Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37, 823–843 (1936)
Dieks, D.: The formalism of quantum theory: an objective description of reality. Ann. Phys. 7, 174–190 (1988)
Dieks, D.: Quantum mechanics: an intelligible description of reality? Found. Phys. 35, 399–415 (2005)
Domenech, G., Freytes, H., de Ronde, C.: Scopes and limits of modality in quantum mechanics. Ann. Phys. 15, 853–860 (2006)
Domenech, G., Freytes, H., de Ronde, C.: Modal type orthomodular logic. Math. Log. Q. 55, 287–299 (2009)
Greechie, R.: On generating distributive sublattices of orthomodular lattices. Proc. Am. Math. Soc. 67, 17–22 (1977)
Halmos, P.: Algebraic logic I, monadic Boolean algebras. Compos. Math. 12, 217–249 (1955)
Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)
Jauch, J.M.: Foundations of Quantum Mechanics. Addison-Wesley, Reading (1968)
Kalman, J.A.: Lattices with involution. Trans. Am. Math. Soc. 87, 485–491 (1958)
Kalmbach, G.: Orthomodular Lattices. Academic Press, London (1983)
Maeda, F., Maeda, S.: Theory of Symmetric Lattices. Springer, Berlin (1970)
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)
van Fraassen, B.C.: Quantum Mechanics: An Empiricist View. Clarendon, Oxford (1991)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Basel
About this chapter
Cite this chapter
Freytes, H., de Ronde, C., Domenech, G. (2012). The Square of Opposition in Orthomodular Logic. In: Béziau, JY., Jacquette, D. (eds) Around and Beyond the Square of Opposition. Studies in Universal Logic. Springer, Basel. https://doi.org/10.1007/978-3-0348-0379-3_13
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0379-3_13
Publisher Name: Springer, Basel
Print ISBN: 978-3-0348-0378-6
Online ISBN: 978-3-0348-0379-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)