Abstract
We extend the topos-theoretic treatment given in previous papers (Butterfield, J. and Isham, C. J. (1999). International Journal of Theoretical Physics 38, 827–859; Hamilton, J., Butterfield, J., and Isham, C. J. (2000). International Journal of Theoretical Physics 39, 1413–1436; Isham, C. J. and Butterfield, J. (1998). International Journal of Theoretical Physics 37, 2669–2733) of assigning values to quantities in quantum theory. In those papers, the main idea was to assign a sieve as a partial and contextual truth value to a proposition that the value of a quantity lies in a certain set \(\Delta \subseteq \mathbb{R}\). Here we relate such sieve-valued valuations to valuations that assign to quantities subsets, rather than single elements, of their spectra (we call these “interval” valuations). There are two main results. First, there is a natural correspondence between these two kinds of valuation, which uses the notion of a state's support for a quantity (Section 3). Second, if one starts with a more general notion of interval valuation, one sees that our interval valuations based on the notion of support (and correspondingly, our sieve-valued valuations) are a simple way to secure certain natural properties of valuations, such as monotonicity (Section 4).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Butterfield, J. (2001). Topos theory as aframework for partial truth. In Proceedings of the 11th International Congress of Logic Methodology and Philosophy of Science, P. Gardenfors, K. Kijania-Placek, and J. Wolenski, eds., Kluwer Academic, Norwell, MA. Available at: http://philsciarchive. pitt.edu, ID code D PITT-PHIL-SCI00000192.
Butterfield, J. and Isham, C. J. (1999). Atoposperspective on theKochen-Specker theorem: II. Conceptual aspects, and classical analogues. International Journal of Theoretical Physics 38, 827–859.
Hamilton, J., Butterfield, J., and Isham, C. J.(2000). A topos perspective on the Kochen-Specker theorem: III. Von Neumann algebras as the base category. International Journal of Theoretica Physics 39, 1413–1436.
Isham, C. J. and Butterfield, J. (1998). Atopos perspective on theKochen-Specker theorem: I. Quantum states as generalised valuations. International Journal of Theoretical Physics 37, 2669–2733.
Kadison, R. V. and Ringrose, J. R. (1983). Fundamentals of theTheory of Operator Algebras Vol. 1: Elementary Theory, Academic Press, New York.
Kochen, S. and Specker, E. P. (1967). The problem of hiddenvariables in quantum mechanics. Journal of Mathematics and Mechanics 17, 59–87.
Vermaas, P. (2000). A Philosopher's Understanding of QuantumMechanics, Cambridge University Press, Cambridge, UK.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Butterfield, J., Isham, C.J. Topos Perspective on the Kochen–Specker Theorem: IV. Interval Valuations. International Journal of Theoretical Physics 41, 613–639 (2002). https://doi.org/10.1023/A:1015276209768
Issue Date:
DOI: https://doi.org/10.1023/A:1015276209768