  In a previous paper, we have proposed assigning as the value of a physical
quantity in quantum theory, a certain kind of set (a sieve) of quantities that
are functions of the given quantity. The motivation was in part physical---such
a valuation illuminates the Kochen-Specker theorem; and in part
mathematical---the valuation arises naturally in the topos theory of
presheaves.
  This paper discusses the conceptual aspects of this proposal. We also
undertake two other tasks. First, we explain how the proposed valuations could
arise much more generally than just in quantum physics; in particular, they
arise as naturally in classical physics. Second, we give another motivation for
such valuations (that applies equally to classical and quantum physics). This
arises from applying to propositions about the values of physical quantities
some general axioms governing partial truth for any kind of proposition.
