Quantum Logic pp 27-47 | Cite as

# The Logical Interpretation of the Lattice *L*_{q}

## Abstract

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.

## Keywords

Intuitionistic Logic Modular Lattice Boolean Lattice Logical Interpretation Material Implication## Preview

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## Notes and References

- 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 - 5.
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*Quanta*, P.G.O. Freund et al., (Eds). University of Chicago Press, Chicago (1970) p. 166.Google Scholar - 7.M. Jammer,
*The Philosophy of Quantum Mechanics*, John Wiley and Sons, New York (1974) p. 351ff.Google Scholar - 8.
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*Trends in Lattice Theory*, Van Nostrand Rheinhold, New York (1970) p. 41ff.Google Scholar - 12.D. Foulis,
*Portugaliae Mathematica*21 (1962) 65.Google Scholar - 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 - 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 - 15.
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