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Refinement Conditions on Operations in Sample Spaces

Published online by Cambridge University Press:  20 November 2018

Robert J. Weaver
Robert J. Weaver*
Affiliation:
Mount Holyoke College, South Hadley, Massachusetts
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The recent study of operational statistics (see [2; 4; 5; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16]) describes a generalized sample space which represents the set of all possible outcomes of a collection of coherently related operations (experiments). This approach to probability generalizes the classical notion of a sample space due to A. N. Kolmogorov [8], and it gives the concept somewhat wider applicability. For instance in [4] and [14], D. J. Foulis and C. H. Randall set out the start of a program wherein a generalized sample space (hereafter called a GSS) and its affiliated partially ordered set of generalized propositions could be a framework within which a genuinely operational interpretation of the so called “logic” of quantum mechanical systems may be found.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 5 , 01 October 1975 , pp. 991 - 999
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Birkhoff, G., Lattice theory, Amer. Math. Soc. Colloquim Publ. 3rd éd., 1967.Google Scholar
2. Dacey, J. C., Orthomodular spaces and additive measurement, Carribbean J. of Sci. and Math. 1 (1969), 51–66.Google Scholar
3. Foulis, D. J., A note on orthomodular lattices, Portugal. Math. 27 (1962), 65–72.Google Scholar
4. Foulis, D. J. and Randall, C. H., Operational statistics, I. Basic concepts, J. Mathematical Phys. 13 (1972), 1667–1675.Google Scholar
5. Foulis, D. J. and Randall, C. H., The stability of pure weights under conditioning (to appear in Glasgow Math. J.).Google Scholar
6. Holland, S. S. Jr., The current interest in orthomodular lattices, Trends in lattice theory, ed. J. C. Abbott (Van Nostrand, Princeton, N.J., 1969).Google Scholar
7. Jauch, J., Foundations of quantum mechanics (Addison-Wesley, Reading, Mass., 1968).Google Scholar
8. Kolmogorov, A. N., Foundations of the theory of probability, 2nd ed. (Chelsea, New York, N.Y., 1956 (German ed., 1933)).Google Scholar
9. Mackey, G. W., Mathematical foundations of quantum mechanics (Benjamin, New York, N.Y., 1963).Google Scholar
10. Randall, C. H., A mathematical foundation for empirical science with special reference to quantum theory. Part I: A calculus of experimental propositions, Tech. Rep. KAPL-3147, Knolls Atomic Power Laboratory, 1966.Google Scholar
11. Randall, C. H. and Foulis, D. J., An approach to empirical logic, Amer. Math. Monthly 77 (1970), 363–374.Google Scholar
12. Randall, C. H. and Foulis, D. J., Lexicographic orthogonality, J. Combinatorial Theory Ser. A 11 (1971), 157–162.Google Scholar
13. Randall, C. H. and Foulis, D. J., States and the free orthogonality monoid, Math. Systems Theory 6 (1972), 262–276.Google Scholar
14. Randall, C. H. and Foulis, D. J., Operational statistics, II. Manuals of operations and their logics, J. Mathematical Phys. J4(1973), 1472–1480.Google Scholar
15. Weaver, R. J., Admissible operations in sample spaces over the free orthogonality monoid, Colloq. Math. 25 (1972), 153–158.Google Scholar
16. Weaver, R. J., Closed sets in the free orthogonality monoid, Amer. J. Math. 96 (1974), 593–601.Google Scholar