A Topos Perspective on the Kochen-Specker Theorem II. Conceptual Aspects and Classical Analogues

Abstract

In a previous paper, we proposed assigning asthe value of a physical quantity in quantum theory acertain kind of set (a sieve) of quantities that arefunctions of the given quantity. The motivation was in part physical — such a valuationilluminates the Kochen–Specker theorem — andin part mathematical — the valuation arisesnaturally in the topos theory of presheaves. This paperdiscusses the conceptual aspects of this proposal. We also undertake two othertasks. First, we explain how the proposed valuationscould arise much more generally than just in quantumphysics; in particular, they arise as naturally in classical physics. Second, we give anothermotivation for such valuations (that applies equally toclassical and quantum physics). This arises fromapplying to propositions about the values of physical quantities some general axioms governingpartial truth for any kind of proposition.