The Logical Interpretation of the Lattice Lq

  • Peter Mittelstaedt
Part of the Synthese Library book series (SYLI, volume 126)


In this chapter, the lattice of subspaces of a Hilbert space is investigated with respect to its logical interpretation. In Section 2.1, we introduce the abstract lattice L q, which has as a model the lattice of subspaces of a Hilbert space, and we mention some interesting properties of this lattice. In Section 2.2, the relation of commensurability is defined, which is of special interest from a formal point of view as well as for the physical interpretation of L q. In addition to the operations already defined in L q, we introduce, in Section 2.3, a further operation, the material quasi-implication, the existence of which is of great importance for the logical interpretation of the lattice L q. Keeping these formal results in mind we shall consider in Section 2.4 the question of what kind of requirements must be fulfilled by a lattice in order that it be interpretable as a logical calculus.


Intuitionistic Logic Modular Lattice Boolean Lattice Logical Interpretation Material Implication 
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Notes and References

  1. 1.
    As a standard reference on lattice theory, we mention here G. Birkhoff, Lattice Theory, third edn., American Mathematical Society, Providence, Rhode Island (1973).Google Scholar
  2. 5.
    K.R. Popper, Nature 219 (1968) 682.CrossRefGoogle Scholar
  3. 6.
    J.M. Jauch and C. Piron, in Quanta, P.G.O. Freund et al., (Eds). University of Chicago Press, Chicago (1970) p. 166.Google Scholar
  4. 7.
    M. Jammer, The Philosophy of Quantum Mechanics, John Wiley and Sons, New York (1974) p. 351ff.Google Scholar
  5. 8.
    E. Scheibe, Br. J. Philos. Sci. 25 (1974) 319.CrossRefGoogle Scholar
  6. 9.
    M. Nakamura, Kodai Math. Series, Rep. 9 (1957) 158.Google Scholar
  7. 10.
    F. Kamber, Math. Ann. 158 (1965) 158.CrossRefGoogle Scholar
  8. 11.
    S. Holland, in: S.C. Abbott, (Ed.), Trends in Lattice Theory, Van Nostrand Rheinhold, New York (1970) p. 41ff.Google Scholar
  9. 12.
    D. Foulis, Portugaliae Mathematica 21 (1962) 65.Google Scholar
  10. 13.
    F. Kamber (ref. 10); also Nach. Akad. Wiss. Math. Phys. Klasse 10, Göttingen, (1964) p. 103; English translation in: C.A. Hooker, (Ed.) The Logico-Algebraic Approach to Quantum Mechanics I, D. Reidel Publishing Co., Dordrecht Holland (1975) p. 221.Google Scholar
  11. 14.
    C. f. H.B. Curry, Foundations of Mathematical Logic, McGraw-Hill, New York (1963) in particular Chapter 5, “The theory of implication”.Google Scholar
  12. 15.
    P. Mittelstaedt, Z. Naturforsch. 27a (1972) 1358.Google Scholar
  13. 18.
    A.M. Gleason, J. Math. Mech. 6 (1957) 885.Google Scholar
  14. 19.
    P. Lorenzen, Formal Logic, Reidel Publishing Co., Dordrecht, Holland (1965).Google Scholar

Copyright information

© D. Reidel Publishing Company, Dordrecht, Holland 1978

Authors and Affiliations

  • Peter Mittelstaedt
    • 1
  1. 1.University of CologneGermany

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