Abstract
An orthoalgebra, which is a natural generalization of an orthomodular lattice or poset, may be viewed as a “logic” or “proposition system” and, under a welldefined set of circumstances, its elements may be classified according to the Aristotelian modalities: necessary, impossible, possible, and contingent. The necessary propositions band together to form a local filter, that is, a set that intersects every Boolean subalgebra in a filter. In this paper, we give a coherent account of the basic theory of Orthoalgebras, define and study filters, local filters, and associated structures, and prove a version of the compactness theorem in classical algebraic logic.
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References
Alfsen, E., and Shultz, F. (1976). Non-commutative spectral theory for affine function spaces on convex sets,Memoirs of the American Mathematical Society,172.
Beran, L. (1984).Orthomodular Lattices, Reidel/Academia, Dordrecht, Holland.
Birkhoff, G., and von Neumann, J. (1936). The logic of quantum mechanics,Annals of Mathematics,37, 823–843.
Cook, T. (1985). Banach spaces of weights on quasimanuals,International Journal of Theoretical Physics,24(11), 1113–1131.
D'Andrea, A., and De Lucia, P. (1991). The Brooks-Jewett theorem on an Orthomodular lattice,Journal of Mathematical Analysis and Applications,154, 507–522.
Foulis, D. (1962). A note on orthomodular lattices,Portugaliae Mathematica,21(1), 65–72.
Foulis, D. (1989). Coupled physical systems,Foundations of Physics,7, 905–922.
Foulis, D., and Randall, C. (1981). Operational statistics and tensor products, inInterpretations and Foundations of Quantum Theory, H. Neumann, ed., Vol. 5, Wissenschaftsverlag, Bibliographisches Institut, Mannheim, pp. 21–28.
Golfin, A. (1987). Representations and products of lattices, Ph.D. thesis, University of Massachusetts, Amherst, Massachusetts.
Greechie, R., and Gudder, S. (1971). Is a quantum logic a logic?,Helvetica Physica Acta,44, 238–240.
Greechie, R., and Gudder, S. (1973). Quantum logics, inContemporary Research in the Foundations and Philosophy of Quantum Theory, C. Hooker, ed., Reidel, Boston.
Gudder, S. (1988).Quantum Probability, Academic Press, Boston.
Hardegree, G., and Frazer, P. (1981). Charting the labyrinth of quantum logics, inCurrent Issues in Quantum Logic, E. Beltrametti and B. van Fraassen, eds., Ettore Majorana International Science Series, 8, Plenum Press, New York.
Janowitz, M. (1963). Quantifiers on quasi-orthomodular lattices, Ph.D. thesis, Wayne State University.
Jauch, J. (1968).Foundations of Quantum Mechanics, Addison-Wesley, Reading, Massachusetts.
Kalmbach, G. (1983).Orthomodular Lattices, Academic Press, New York.
Kläy, M., Randall, C., and Foulis, D. (1987). Tensor products and probability weights,International Journal of Theoretical Physics,26(3), 199–219.
Lock, P., and Hardegree, G. (1984a). Connections among quantum logics, Part 1, Quantum prepositional logics,International Journal of Theoretical Physics,24(1), 43–53.
Lock, P., and Hardegree, G. (1984b). Connections among quantum logics, Part 2, Quantum event logics,International Journal of Theoretical Physics,24(1), 55–61.
Piron, C. (1964). Axiomatique quantique,Helvetica Physica Acta,37, 439–468.
Piron, C. (1976).Foundations of Quantum Physics, A. Wightman, ed., Benjamin, Reading, Massachusetts.
Ramsay, A. (1966). A theorem on two commuting observables,Journal of Mathematics and Mechancis,15(2), 227–234.
Randall, C., and Foulis, D. (1979). New definitions and theorems, University of Massachusetts Mimeographed Notes, Amherst, Massachusetts, Autumn 1979.
Randall, C., and Foulis, D. (1981a). Empirical logic and tensor products, inInterpretations and Foundations of Quantum Theory, H. Neumann, ed., Vol. 5, Wissenschaftsverlag, Bibliographisches Institut, Mannheim, pp. 9–20.
Randall, C., and Foulis, D. (1981b). What are quantum logics and what ought they to be?, inCurrent Issues in Quantum Logic, E. Beltrametti and B. van Fraassen, eds., Ettore Majorana International Science Series 8, Plenum Press, New York, pp. 35–52.
Rüttimann, G. (1979). Non-commutative measure theory, Habilitationsschrift, University of Bern, Switzerland.
Rüttimann, G. (1989). The approximate Jordan-Hahn decomposition,Canadian Journal of Mathematics,41(6), 1124–1146.
Sasaki, U. (1954). On orthocomplemented lattices satisfying the exchange axiom,Hiroshima Japan University Journal of Science Series A,17(3), 293–302.
Schindler, C. (1986). Decompositions of measures on orthologics, Ph.D. thesis, University of Berne, Switzerland.
Svetlichny, G. (1986). Quantum supports and modal logic,Foundations of Physics,16(12), 1285–1295.
Svetlichny, G. (1990). On the inverse FPR problem: Quantum is classical,Foundations of Physics,20(6), 635–650.
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Foulis, D.J., Greechie, R.J. & Rüttimann, G.T. Filters and supports in orthoalgebras. Int J Theor Phys 31, 789–807 (1992). https://doi.org/10.1007/BF00678545
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DOI: https://doi.org/10.1007/BF00678545